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Xu +Uri Alon +Graham Neubig +Language Technologies Institute +Carnegie Mellon University +{fangzhex,ualon,gneubig}@cs.cmu.edu +Abstract +Language models (LMs) compute the probability of a text by sequentially computing +a representation of an already-seen context and using this representation to predict the +next word. Currently, most LMs calculate these representations through a neural network +consuming the immediate previous context. However recently, retrieval-augmented LMs +have shown to improve over standard neural LMs, by accessing information retrieved from a +large datastore, in addition to their standard, parametric, next-word prediction. In this paper, +we set out to understand why retrieval-augmented language models, and specifically why +k-nearest neighbor language models (kNN-LMs) perform better than standard parametric +LMs, even when the k-nearest neighbor component retrieves examples from the same +training set that the LM was originally trained on. To this end, we perform a careful +analysis of the various dimensions over which kNN-LM diverges from standard LMs, and +investigate these dimensions one by one. Empirically, we identify three main reasons +why kNN-LM performs better than standard LMs: using a different input representation +for predicting the next tokens, approximate kNN search, and the importance of softmax +temperature for the kNN distribution. Further, we incorporate these insights into the +model architecture or the training procedure of the standard parametric LM, improving +its results without the need for an explicit retrieval component. The code is available at +https://github.com/frankxu2004/knnlm-why. +1 +Introduction +Language modeling is the task of predicting the probability of a text (often conditioned on context), with +broad-spanning applications across natural language processing (Bengio et al., 2003; Merity et al., 2018; +Baevski and Auli, 2018; Brown et al., 2020). This modeling is usually done by sequentially encoding a context +ct using a trained neural network function f, and computing the probability of the next word wt according to +f (ct) and a vector representation of wt. +Recently, retrieval-augmented LMs have shown a series of impressive results (Grave et al., 2017; Guu et al., +2018; He et al., 2020; Khandelwal et al., 2020b; Borgeaud et al., 2022; Alon et al., 2022). Retrieval-augmented +LMs compute next token distributions based not only on the immediately preceding context ct and the model +parameters, but also on an external datastore, from which examples are retrieved and incorporated into the +base LM’s prediction. +One retrieval-augmented model that is notable for both its simplicity and efficacy is the k-nearest neighbor +language model (kNN-LM; Khandelwal et al., 2020b). It extends a trained base LM by linearly interpolating +the output word distribution with a kNN model. The nearest neighbors are retrieved according to the distances +between the current context embedding of the base LM and all the context embeddings in the datastore. The +datastore is created by encoding all contexts from any text collection, including the original LM training data. +One of the most surprising results from Khandelwal et al. (2020b) is that kNN-LM reduces the perplexity of +the base LM even when the kNN component is retrieving examples from the same training set that the LM +was originally trained on, indicating that the kNN-LM improves the ability to model the training data and is +Preprint. Under review. +arXiv:2301.02828v1 [cs.CL] 7 Jan 2023 + +Multi Headed +Attention +Feed Forward +Network +Layer Norm +ℎ𝑠𝑚 +𝑊𝑠𝑚 +𝐷 +𝑉 +ℎ𝑑𝑠 +𝑊𝑑𝑠 +𝐷 +𝑁𝑑𝑠 ++ +𝑃𝐿𝑀 parametric component +𝑃𝑘𝑁𝑁 non-parametric component +In 𝑘NN-LM: +𝑁𝑑𝑠: up to 5000𝑉 +𝐷 +𝐷 +mask-to-k() +In 𝑘NN-LM: +top-𝑘() +FFN +ATT +softmax() +softmax() +Figure 1: An illustration of the generalized formulation of kNN-LM in Equation 5. +not simply benefiting from access to more data. Intrigued by this, we ask questions like, could kNN-LM be +improving because of capacity issues in the parametric base LM? In this paper, we set out to understand why +kNN-LMs work even in this setting. +In the following sections, we first elucidate connections between the added kNN component and the standard +LM component. Specifically, we note that word distributions from the two components are both calculated +using a softmax function, based on the similarity of the current context embedding with a set of embeddings +that corresponds to different next words. With this intuition, we formalize and generalize the non-parametric +distribution calculation with the softmax layer and word embedding layer used in parametric LMs. We then +show that this generalized form exposes a variety of design choices, e.g., the number of context embeddings +in the datastore, the input representation used in softmax layer, different similarity functions, as well as the +approximation and sparsification implementations in the kNN search. This provides a general framework for +analyzing kNN-LM and similar models and allows us to perform ablation studies that test the importance of +various design decisions. +We proceed to propose multiple hypotheses for why kNN-LM works, which are testable by adjusting the +various parameters exposed by our generalized formulation. Based on these hypotheses, we perform ablation +experiments and analyze the nuances between different implementations of the generalized version of PkNN. +As the answer to our question, “why kNN-LMs work”, we eventually show that the most probable reasons are +threefold: +1. Ensembling the output of softmax using two representations from different layers of the transformer +is important; in our experiments, this accounts for 55% of the performance gain of kNN-LM, or 6.5% +relative perplexity improvement compared to the base LM. +2. kNN-LM uses approximate nearest neighbor search to handle the large number of candidates, and +the lack of this preciseness in this algorithm actually helps kNN-LM to generalize better than using +exact nearest neighbor search and distance calculation, possibly due to a regularization effect. The +relative perplexity improvement from this factor is about 2.6%. +3. Depending on the design decisions that are chosen for modeling, adding a temperature term to +the kNN non-parametric component can become crucial to the success of modeling (although +coincidentally, in the original settings of Khandelwal et al. (2020b), a temperature of 1.0 is close to +optimal, which hid the importance of this term). In some settings, the relative perplexity gap between +the default and optimal temperature can be as high as 3.7%. +Finally, one significant drawback to the current kNN-LM is the inefficiency of kNN search performed at each +step (He et al., 2021; Borgeaud et al., 2022; Alon et al., 2022; Wang et al., 2022). Because of the similarity +between kNN-LM and the parametric LM’s last layers and the many design choices, we also demonstrate that +we are able to make kNN-LM more efficient by substituting the kNN search with another matrix operation +that can fit in accelerator memory while maintaining more than half the perplexity improvement, or more than +6.5% relative improvement compared to the base LM. +2 + +2 +Formalizing and Generalizing kNN-LM +kNN-LM (Khandelwal et al., 2020b) is a linear interpolation between a base LM and a kNN model. Given a +set of contexts ci and their corresponding next token wi as a pair (ci, wi) ∈ D, kNN-LMs create a datastore +(K, V) = {(ki, vi)}, as a set of key-value pairs: +(K, V) = {(f (ci) , wi) | (ci, wi) ∈ D} +(1) +During inference, the parametric component of the LM generates the output distribution pLM(wt|ct; θ) over +the next tokens and produces the corresponding context representation f(ct), given the test input context ct. +Then the non-parametric component of the LM queries the datastore with the f(ct) representation to retrieve +its k-nearest neighbors N according to a distance function d(·, ·). Next, the kNN-LM computes a probability +distribution over these neighbors using the softmax of their negative distances, and aggregates the probability +mass for each vocabulary item across all of its occurrences in the retrieved targets: +pkNN(wt|ct) ∝ +� +(ki,vi)∈N +1wt=vi exp(−d(ki, f(ct))) +(2) +Finally, this distribution is interpolated with the parametric LM distribution pLM to produce the final kNN-LM +distribution: +p(wt|ct; θ) = (1 − λ)pLM(wt|ct; θ) + λpkNN(wt|ct) +(3) +where λ is a scalar that controls the weights of the interpolation between two components, with higher λ +putting more weight on the non-parametric component. +Looking closely at Equation 2, we can notice a similarity between the calculation of PkNN and the standard +PLM. The kNN distribution is based on the distances between the current context and the nearest neighbors +from the datastore, normalized by a softmax function. Recall that in (standard) parametric language models, +the distribution over the vocabulary is also based on a measure of distance, the inner product between the +current context embedding and the word embeddings of every token in the vocabulary. Because each context +embedding in the datastore (K, V) corresponds to a target token, we can also view this datastore as a large +word embedding matrix with multiple word embeddings for each of the vocabulary words. Theoretically, +given unlimited computation, we could calculate the distribution based on the distances to every embedding in +the datastore, and aggregate by vocabulary items, making it more closely resemble PLM. In this case, k = |D|, +the size of the entire datastore, and Equation 2 becomes the following, based on the distances to every context +in the datastore D instead of a subset of nearest neighbors N. +pD(wt|ct) ∝ +� +(ki,vi)∈D +1wt=vi exp(−d(ki, f(ct))) +(4) +In practice, we use kNN search as a way of approximation, by limiting the calculation to only k nearest +neighbors to avoid the computational cost of calculating the distribution over the entire datastore. +If we re-write and generalize Equation 2, both the kNN-LM of Khandelwal et al. (2020b) and a large number +of related models can be expressed through the following equation: +Pinterp = (1 − λ) softmax(Wsm · hsm) +� +�� +� +PLM parametric component ++λ Msoftmax(mask-to-k(Wds ⊗ hds)/τ) +� +�� +� +PkNN non-parametric component +. +(5) +Figure 1 provides an illustration of Equation 5. The first term of the equation is the standard parametric +language model, whereas the second represents a generalized version of utilizing an external datastore. The +first component, the output layer of a common parametric language model, is relatively straightforward. Wsm +of size V × D is the embedding matrix of the output token, and hsm is the context vector used to calculate the +distribution of the output token, usually the output of the final feedforward layer in the transformer. +In the second component, Wds represents the datastore, of size Nds × D. Nds is the number of entries in +the datastore, and D is the size of each context vector. hds represents the context vector used to query the +datastore. As shown in Figure 1, these vectors can come from different layers of the transformer architecture. +⊗ represents the operation type used to calculate the similarity between context vectors and the query vector, +which also has several alternatives that we discuss below. +mask-to-k(·) represents a function to sparsify similarity scores across the datastore, setting all but k similarity +scores to −∞, which results in probabilities of zero for all masked similarity scores after the softmax. +3 + +Practically, this is necessary for kNN-LMs because the size of the datastore Nds makes it infeasible to +calculate all outputs at the same time. With masked logits, we apply a more generalized version of softmax +with temperature τ. Intuitively adding the temperature can adjust the peakiness or confidence of the softmax +probability distribution output. After the softmax, the matrix M of dimension V × Nds sums the probability of +the Nds datastore entries corresponding to each of the V vocabulary entries. Each column in this matrix consists +of a one-hot vector with a value of 1 and the index corresponding to the vocabulary item wi corresponding to +the datastore entry for ci. +Within this formulation, it becomes obvious that there are many design choices for kNN-LM-like models. One +important thing to note is that the right side of Equation 5 is actually very similar to the left side representing +the standard parametric language model, with a few additional components: M, mask-to-k, and ⊗. More +specifically, some of the design decisions that go into the kNN-LM, and parallels with standard parametric +models are: +1. Size of Wds: In the standard parametric model, the size of Wsm is V embedding vectors, each with +D dimensions. In the kNN-LM it is very large: Nds, the size of the datastore, usually the number of +tokens in the entire training corpus. +2. Input representation: In the parametric model, hsm is the output from the feedforward layer in the +last transformer block, which we abbreviate “ffn”. In contrast, Khandelwal et al. (2020b) rather use +as hds the output from the multi-headed attention layer of the last transformer block (before running +the representations through the feed-forward network, and after the LayerNorm (Ba et al., 2016)), +which we abbreviate as “att”. +3. Similarity & Temperature: In the parametric model, the functional form of ⊗ is the inner product +(abbreviated IP), whereas Khandelwal et al. (2020b) use negative squared L2 distance (abbreviated +L2) as a similarity function between Wds and hds. As the similarity scores are turned into probability +distributions with the softmax function, the choice of softmax temperature (τ) can control the scaling +of the similarity scores and thus the peakiness of the non-parametric component distribution. +4. Approximation & Sparsification: In the parametric model, k = V , and no values are masked, +but in the kNN-LM, k ≪ V , and most of the datastore entries are pruned out. The definition of +the mask-to-k(·) function, i.e. how to select the important datastore embeddings to include in the +similarity calculation (in kNN-LM’s case the k nearest neighbors), is a crucial open design choice. +In the following sections, we set out to better understand how each of these design decisions contributes to the +improvement in accuracy due to the use of kNN-LMs. +3 +Baseline kNN-LM Results +First, we evaluate the kNN-LM baseline on the Wikitext-103 dataset (Merity et al., 2016), and examine the +importance of two design choices: the input representation hds and the similarity function ⊗. +In models examined in this paper, the parametric model is a transformer language model with mostly the +same architecture as in Khandelwal et al. (2020b). However, We do make modifications to the original base +LM (Baevski and Auli, 2018) to accommodate our experimentation need. We using BPE tokenization (Sennrich +et al., 2015) to train a smaller vocabulary (33K) than the original (260K) on the training corpus of Wikitext-103, +as subword tokenization is ubiquitous in many state-of-the-art language models (Radford et al., 2019; Devlin +et al., 2018; Liu et al., 2019; Brown et al., 2020). Using subword tokenization also eliminates the need for +adaptive softmax (Joulin et al., 2017). It makes the output layer more generalized, sharing more resemblance +to the kNN component as described in Section 2, and facilitates the ablation studies in this paper.1 This base +LM has 268M parameters. To get a perspective on how large the datastore is, it is built on the training data +that contains nearly 150M BPE tokens, each paired with a context vector of size 1024. This datastore has a +total memory consumption of about 300GB. At every retrieval step, we take the top 1024 nearest neighbors, +i.e., k = 1024, following Khandelwal et al. (2020b). The interpolated perplexity is computed with optimal +interpolation parameter λ tuned according to the perplexity on the development set. λ is fixed during the +inference for all predictions, the same as the standard kNN-LM. +1By training our own version of the base LM from scratch with BPE tokenization and a standard output softmax layer, +our LM’s perplexity is worse than that used in the original kNN-LM paper. However, we observe similar relative gains +from the additional kNN component. We argue that the base LM’s performance is orthogonal to the study of the factors +behind kNN-LM’s improvements. +4 + +hds +⊗ ++#params +PPL +Interp. PPL +Oracle +Base LM +- +- +0 +21.750 +- +- +kNN-LM-L2 +att +L2 +Nds × D +∞ +19.174 +14.230 +kNN-LM-IP +att +IP +Nds × D +∞ +19.095 +14.077 +kNN-LM-L2 +ffn +L2 +Nds × D +∞ +20.734 +15.594 +kNN-LM-IP +ffn +IP +Nds × D +∞ +21.101 +16.254 +Table 1: Performance of the parametric language model and several kNN-LM variants. +Results comparing multiple kNN-LM variants are shown in Table 1. The first row represents the base +parametric language model’s perplexity. The second is a formulation analogous to that of Khandelwal et al. +(2020b), and in the remaining rows, we vary the input representation hds and distance function ⊗ from +Equation 5. All of them use a large datastore with size Nds, approximately 5000 times the size of the +vocabulary V , as also reflected in “+#params”, the number of additional parameters other than the base LM. +We report several important quantities with respect to each model. +• “PPL” shows the perplexity of only the kNN component of the model pkNN(). This is ∞ for all kNN- +LM models in all cases, as when the kNN search does not retrieve any datastore entries corresponding +to the true target word wt the probability of the target word will be zero. +• “Oracle” shows the lower bound of the interpolation performance by choosing the best λ for each +token in the evaluation dataset, which will either be λ = 0 or λ = 1 depending on whether +PLM(wt|ct) > Pknn(wt|ct) or not, respectively. +From the table, we can see that: +1. Using the output of the multi-headed attention layer (“att”) as hds (instead of the standard “ffn” layer) +is crucial for better performance of kNN-LM. +2. In general, using negative squared L2 distance or inner product as a similarity function does not result +in a large and consistent difference, although in our setting, IP provides slightly better performance +when using the “att” inputs, and slightly worse when using “ffn” inputs. +3. Interestingly, when using “ffn” and “IP”, the same input and distance metric used in the parametric +model, the results are the worst, indicating that the kNN-LM is particularly benefiting when the +kNN-LM achieves a different view of the data from the parametric model. +We found in preliminary experiments that kNN-LM is generalizable to other base language models as well, +ranging from small models with 82M parameters to larger models with 774M parameters. The gain from +kNN-LM is more significant when used with a smaller, less capable base language model, as expected. The +details are shown in Appendix A. In this paper, we are mainly focused on the factors contributing to the +relative improvements from kNN-LM, instead of the absolute performance, so we use the 268M model for the +remainder of the paper. +In the next sections, we perform further experiments with ablations on the general formulation Equation 5 to +elucidate the key elements contributing to the performance improvements in kNN-LM. +4 +Effect of Different Wds Formulations +4.1 +Replacing the Datastore with Trainable Embeddings +From the observation in Section 3, we see that the choice of hds has a large impact on the performance of +kNN-LM. This intrigues us to explore if one key to the improvements afforded by kNN-LM lies in the use +of different input representations together, namely the attention output (hds = att) and feedforward output +(hds = ffn). However, from only the experiments above, it is not possible to disentangle the effect of the +choice of hds and that of other design choices and factors in Equation 5. +To test the effect of hds in a more controlled setting, we remove the non-parametric datastore entirely, and +initialize Wds in Equation 5 with a randomly initialized word embedding matrix with the same size (Nds = V ) +5 + +as the LM’s output embedding Wsm, and train Wds with all other parameters fixed.2 The loss function for +training is the cross-entropy loss of softmax(Wds · hds) with respect to the ground-truth tokens, identically +to how the base LM is trained. We compare how using hds = att or hds = ffn affects the interpolated +performance. The results are shown in Table 2, and we also show results from kNN-LMs using these two +varieties of input representation for reference. +From these experiments we can find several interesting conclusions: +Effectiveness of re-training Wds: In the case of “Learned Wds w/ FFN”, we are essentially re-learning the +weights feeding into the softmax function separately from the underlying LM encoder. Despite this fact, we +can see the model achieves a PPL of 20.920, which is 0.83 points better than the base model. This suggests +that there is some benefit in learning the parameters of Wds after freezing the body of the transformer encoder. +Effectiveness of ensembling two predictors: In both cases for Wds, the interpolated perplexity is significantly +better than that of using a single predictor. This is particularly the case when using the “att” representation for +hds, suggesting that the utility of ensembling predictions from two views of the data is not only useful when +using kNN-LM, but also in standard parametric models as well. +Parametric ensembles as an alternative to kNN-LM?: Overall, by using a separate word embedding matrix +with size V × D as an alternative to kNN, we can recover about 55% of the performance gain achieved by +kNN-LM, with only a limited number of parameters and without the necessity for slow kNN retrieval every +time a token is predicted. This suggests that the majority of the gain afforded by kNN-LM could be achieved +by other more efficient means as well. +hds +Nds +⊗ ++#params +PPL +Interp. +Oracle +Base LM +- +- +- +0 +21.750 +- +- +kNN-LM w/ ATT +att +Big +IP +Nds × D +∞ +19.095 +14.077 +Learned Wds w/ ATT +att +1x +IP +V × D +22.584 +20.353 +16.954 +kNN-LM w/ FFN +ffn +Big +IP +Nds × D +∞ +21.101 +16.254 +Learned Wds w/ FFN +ffn +1x +IP +V × D +20.920 +20.694 +18.772 +Table 2: Performance comparison how the choice of hds, input representation, affects kNN baselines and +models with learnable embeddings as datastore alternative. hds is the attention layer output. +4.2 +Increasing the Softmax Capacity +One premise behind kNN-LM is that the large datastore is the key reason for the model working well: the +larger the softmax capacity, the better the performance. Naturally, as a first step, we need to check whether +such a big datastore is warranted and whether the high rank of Wds leads to better performance. We test +the effect of the datastore size for kNN retrieval on kNN-LM interpolated perplexity. If a bigger datastore +(a high rank Wds) is better in kNN-LM than a smaller datastore, then the hypothesis of softmax capacity is +more probable. We randomly subsample the full datastore in varying percentages and the results are shown +in Figure 2. The full datastore contains more than 150M entries and storing them takes 293GB when using +half-precision floating points (fp16). We can see that whether or not approximate kNN is used, the final +perplexity decreases almost linearly with more percentage of the original datastore. Even with just 5% of +the datastore size (15G), kNN-LM still provides a benefit over just using the base LM. However, even when +the subsampling percentage reaches 90%, having more entries in the datastore still provides benefits without +having significant diminishing returns, suggesting that a large datastore is beneficial. +One possible reason why a larger datastore is helpful is that words can be difficult to predict. There are several +reasons: (1) They are rare, or (2) they are frequent, but they have multiple meanings and appear in different +contexts. The softmax bottleneck (Yang et al., 2017) suggests that the final dot product of language model +Wsm · hsm limits the expressivity of the output probability distributions given the context; that is, a single +output vector of a fixed (1024) size cannot express all the possible mappings between 100M training examples +and 33K vocabulary outputs. We hypothesize that kNN-LM improves performance by alleviating the problem, +since Wds ⊗ hds has a higher rank and is more expressive than just Wsm · hsm. In other words, kNN is a +sparse approximation of the full softmax over all the embeddings in the datastore Wds. To test this hypothesis, +2Because we previously found little difference between IP and L2 as similarity functions, we use IP in the experiments. +For simplicity, we set temperature τ = 1. +6 + +we disentangle the effect of the high rank in Wds from the actual saved context embeddings in Wds, by training +an embedding matrix of the same desired size to test from scratch. +Ratio to Full Datastore Size +Interpolated Perplexity +19.000 +20.000 +21.000 +22.000 +0.00 +0.25 +0.50 +0.75 +1.00 +Figure 2: The effect of the size of the datastore used for kNN retrieval on final interpolated perplexity. +We explore several potential solutions for increasing the capacity of softmax, and examine if they can achieve +a similar effect of kNN-LM. The first and easiest solution is to increase the embedding matrix size by adding +more embedding vectors for each word type in the vocabulary. To test this, we replace Wsm with a much +smaller matrix of size nV × D, where we allocate n embedding vectors for each word type. When calculating +the probability from this component, we compute the softmax over nV items and sum the probabilities for +each vocabulary entry to calculate the final probability. mask-to-k(·) is no longer needed, as this formulation +is small enough to fit the entire matrix in the GPU. We then finetune the new Wds on the training data until +convergence. +Figure 3 compares the base LM and the original kNN-LM versus using either attention layer output (“att”) +or feedforward layer output (“ffn”) as hds. We plot the number of embeddings for each word type (nV total +embeddings in Wds) versus the interpolated perplexity, with full details found in Appendix B. In both cases, +comparing with the top horizontal line which represents the perplexity of the base LM, replacing the datastore +with a much smaller weight matrix (from Nds to nVds) by assigning only a few more embeddings for each +word helps, although only about half as effective as kNN-LM. To give a perspective, the original datastore +size is about 5000V . Surprisingly, we find that increasing n does not always bring better performance, even +though a larger datastore is better than using a small datastore in kNN-LM. We can see that when hds = ffn, +over-parameterization provides very limited improvements, while for hds = att it does not bring consistent +improvements at all. Comparing the trend of increasing the embeddings in Wds, with the bottom horizontal line +in the plot, which represents the perplexity of the standard kNN-LM using the full datastore (Wds with approx. +5000V embeddings), we can see no clear trend that more trainable embeddings result in better perplexity, and +that the gap between using trained embeddings and using full datastore is still significant. This suggests that +simply over-parameterizing Wds is not an effective method of achieving accuracy gains similar to kNN-LM. +We hypothesize that this is because by just adding more embeddings, while still using the same training +procedure as the original LM, the multiple embeddings for each word type after learning could still be very +close to each other, and thus do not increase the softmax capacity much. This suggests that some regularization +terms may be needed during training to make the multiple embeddings not converge to the same vector, +rendering over-parameterization useless. +Besides simply increasing the number of embedding vectors equally for each word type, we also propose +other alternatives to increase softmax capacity. First, we hypothesize that different word types have different +difficulties for the language model to predict. For those words that appear very frequently, they may appear +in many different contexts. As a result, instead of adding an equal number of additional embeddings to +each word type, we propose to adaptively increase the number of embeddings for word types based on word +frequency, or total training loss for the word. Second, we try to break the softmax bottleneck with a Mixture +of Softmax. Yang et al. (2017) proposes a solution to the problem using a Mixture of Softmax (MoS) to +produce more linearly independent probability distributions of words given different contexts. Last, opposite +to training the word embeddings of increased size, we also consider ways to compress the datastore down to a +similar-sized embedding matrix for softmax computation by clustering the whole datastore and allowing for +further finetuning of the embedding matrix consisting of cluster centroids. However, none of these alternative +methods provided additional benefits over the simple multi-embedding approach. More details on these +attempts can be found in Appendix C. +7 + +Number of Trained Embeddings (nV) +Interpolated Perplexity +19 +20 +21 +22 +2 +4 +6 +8 +att + +ffn + +Figure 3: The number of embeddings per word type (nV total embeddings in Wds) versus interpolated +perplexity. The horizontal line at the top represents the perplexity of the base LM. The horizontal line at the +bottom represents the interpolated perplexity using a full datastore with kNN-LM. +5 +Approximate kNN Search & Softmax Temperature +5.1 +Comparing Approximate kNN Search +To calculate PkNN of the non-parametric component in Equation 5, it is usually prohibitive to use exhaustive +kNN search, and thus Khandelwal et al. (2020a) use approximate kNN search using the FAISS library (Johnson +et al., 2019). The use of FAISS (similarly to other approximate search libraries) results in two varieties of +approximation. +• Approximate Neighbors: Because the search for nearest neighbors is not exact, the set of nearest +neighbors might not be equivalent to the actual nearest neighbors. Recall the function mask-to-k(·) in +Equation 5, it is the function where we select the kNN entries from the datastore Wds. We denote +“real mask” as the accurate nearest neighbors for mask-to-k(·) selection, and “FAISS mask” as the +approximate nearest neighbors returned by the FAISS library.3 +• Approximate Scores: In addition, FAISS makes some approximations in calculating the distances +between the query and the retrieved neighbors for efficiency purposes. We denote “real score” as the +scores calculated from ground truth distances between the embeddings, and “FAISS score” as the +distances returned by FAISS approximate search. +The comparison of the different approximation settings is shown in Table 3. Quite surprisingly, we actually +find that the interpolated perplexity with approximate search is better than that with exact search, both with +respect to the mask and the score calculation. Intrigued by this counter-intuitive result, we explore the effect of +kNN search approximation. +hds +⊗ ++#params +PPL +λ +Interp. PPL +Oracle +Base LM +- +- +0 +21.750 +- +- +- +kNN-LM w/ FAISS mask, FAISS score +att +L2 +Nds × D +∞ +0.271 +19.174 +14.230 +kNN-LM w/ FAISS mask, real score +att +L2 +Nds × D +∞ +0.176 +19.672 +14.393 +kNN-LM w/ real mask, real score +att +L2 +Nds × D +∞ +0.172 +19.735 +14.480 +Table 3: Performance of the parametric language model and comparison of kNN-LMs using the approximate +versus ground truth kNN. +First, we plot the subsampled size of the datastore with the interpolated perplexity Figure 4, a similar plot +to Figure 2, but showcasing the comparison between approximate and real masks, between approximate and +real scores in both the full datastore as well as a small subsampled datastore setting. We find that using an +approximate FAISS mask to find nearest neighbors is better than using the ground truth nearest neighbors and +that using the approximate score returned by FAISS is better than recomputing the ground truth distances +3To calculate the real mask over a large datastore, we shard the datastore into several smaller datastores, calculate the +nearest neighbors for each of the smaller datastores, and combine them back together to get the final result. +8 + +between embeddings for the kNN distribution at different levels of datastore size, both at 5% or 100%. +Interestingly, the gap between using an approximate score or real score given the same approximate nearest +neighbors (“FAISS mask, FAISS score” vs. “FAISS mask, real score”) is larger than that between using +approximate or real nearest neighbors given the same ground truth method of calculating the distance (“real +mask, real score” vs. “FAISS mask, real score”), for reasons we will elucidate in the next section. +Ratio to Full Datastore Size +Interpolated Perplexity +19.000 +20.000 +21.000 +22.000 +0.00 +0.25 +0.50 +0.75 +1.00 +FAISS mask, FAISS score +FAISS mask, real score +real mask, real score +Figure 4: The differences between using approximate and accurate kNN search on varying size of the datastore. +5.2 +Adding Softmax Temperature to kNN Distribution +Because the number of retrieved nearest neighbors, k is usually much smaller than the vocabulary size V , +intuitively, the kNN distribution PkNN used for interpolation tends to be more peaky than the standard LM +output distribution. When k = 1024 and V = 33000, as in our experiments, PkNN will only have a few +vocabulary items with a non-zero probability. Furthermore, many of the retrieved neighbors share the same +target token and thus make the kNN distribution even peakier. One way to control the entropy, or peakiness of +the distribution is to add temperature to the logits that go into the softmax function (Holtzman et al., 2019). +We calculate the probability of non-parametric component PkNN with the following equation where t is the +softmax temperature: +PkNN = Msoftmax(mask-to-k(Wds ⊗ hds)/t) +(6) +In general, the higher the temperature, the less “peaky” the distribution would become. We experiment with +both the 5% as well as the full datastore using different temperatures ranging from 0 to 3 at 0.1 intervals. The +results are shown in Figure 5a and Figure 5b respectively. +(a) On 5% subsampled datastore. +(b) On full datastore. +Figure 5: The interpolated perplexity varies with different softmax temperature values. +We can see that the default temperature t = 1 does not always result in the best-interpolated perplexity and +tuning softmax temperature is desirable for all sizes of datastore. The lesson learned here is that tuning the +9 + +real mask, real score +21.70 +FAISS mask, FAISS score +FAlSS mask, real score +21.65 +21.60 +21.55 +21.50 +21.45 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0real mask, real score +20.6 +FAISS mask, FAISS score +FAiss mask, real score +20.4 +20.2 +20.0 +19.8 +19.6 +19.4 +19.2 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0softmax temperature for the kNN distribution is crucial for getting optimal results from each setting. Only +coincidentally, a temperature of 1.0 was close to optimal in the original settings of Khandelwal et al. (2020b), +which hid the importance of this hyperparameter. +In both the 5% subsampled datastore and the full datastore scenarios, temperature t = 1 is close to optimal +when using “FAISS mask, FAISS score”. When using either “real mask” or “real score”, this is not true +anymore. Even at the optimal temperature for each setting, “real mask, real score” somewhat underperforms +“FAISS mask, real score”. It is consistent with the counter-intuitive phenomenon discussed in Section 5.1. +There are also differences between the two scenarios of different datastore sizes. With the full datastore, using +“real score” outperforms “FAISS score” given the same “FAISS mask”. However, the opposite is true when +using the 5% datastore. This suggests that as the datastore size grows, using accurate distance values are better +than the approximate ones. The relatively small gap between using “real score” and “FAISS score” in both +datastore settings shows that the main contributor to the improvements is using approximate nearest neighbors +(“FAISS mask”) rather than using approximate distance values (“FAISS score”). +We hypothesize that this is related to regularization for preventing overfitting, and approximate search provides +fuzziness that functions as a regularizer. We can think of the non-parametric part in kNN-LM, the kNN +component as a model, where the datastore size is its model capacity, and the datastore is its training data. +Considering that the kNN component uses the exact same training data as the base parametric LM, having +ground truth, accurate kNN search may cause the kNN component to overfit the training data. Comparing the +small datastore with only 5% with the original datastore, we see that a small datastore means a small training +set for the kNN “model” and it thus it benefits more from this regularization, both both through using the +FAISS mask and FAISS score (at optimal temperature settings). From these experiments, we can see that, +surprisingly, one of the important ingredients in kNN-LM seems to be approximate kNN search, which likely +prevents overfitting to the datastore created from the same training set. We further analyze this unexpected +result in Appendix D, where we find that longer words and words that appear in many different contexts have +slightly better results with approximate nearest neighbors. +Notably, similar effects, where an approximation component lead to better generalization, have been reported in +other NLP tasks as well, and are sometimes referred to as “beneficial search bias”, when modeling errors cause +the highest-scoring solution to not be the correct one: Meister et al. (2020b) suggest that “quite surprisingly, +beam search often returns better results than exact inference due to beneficial search bias for NLP tasks.” +Stahlberg and Byrne (2019) also conclude that “vanilla NMT in its current form requires just the right amount +of beam search errors, which, from a modeling perspective, is a highly unsatisfactory conclusion indeed, as +the model often prefers an empty translation”. +6 +Probably Wrong Hypotheses for Why kNN-LMs Work +The results in the previous sections are the result of extensive analysis and experimentation, in which we also +tested a number of hypotheses that did not turn out to have a significant effect. Additional details of these +hypotheses are detailed in Appendix E, and we hope that they may provide ideas for future improvements of +retrieval-based LMs. +Ensemble of Distance Metrics +We hypothesized that the ensemble of two distance metrics: the standard +inner product distance (which the LM uses) and the L2 distance (which the kNN component uses), is the key +to the improvement. However, we found that similar gains can be achieved using the inner-product metric for +the retrieved kNN. More details can be found in Appendix E.1. +Ensembling of Two Models +We hypothesized that the kNN component merely provides another model +for ensembling. The improvement from kNN-LM is purely due to the ensembling effect of different models. +However, we found that kNN-LM’s improvement is orthogonal to ensembling with a different base LM. More +details can be found in Appendix E.5. +Sparsification +The mask-to-k(·) used by kNN retrieval induces sparsity in the distribution over the vocab- +ulary, due to a small k (typically 1024) compared to the size of the vocabulary V (33K in our experiments +and 260K in the original settings of Khandelwal et al. (2020b)). We hypothesized that kNN-LM increases +the probability of the top-k entries while taking “probability mass” from the long tail of unlikely word types. +However, we could not gain any benefits solely from sparsifying the output probability of a standard LM and +interpolating it with the original LM. More details can be found in Appendix E.2. +10 + +Stolen Probabilities +The stolen probabilities effect (Demeter et al., 2020) refers to the situation where the +output embeddings of an LM are learned such that some words are geometrically placed inside the convex +hull that is formed by other word embeddings and can thus never be “selected” as the argmax word. We +hypothesized that kNN-LM solves the stolen probabilities problem by allowing to assign the highest probability +to any word, given a test context that is close enough to that word’s datastore key. However, we found that +none of the vectors in our embedding matrix and in the original embedding matrix of Khandelwal et al. (2020b) +is located in the convex hull of the others, which is consistent with the findings of Grivas et al. (2022). More +details can be found in Appendix E.4. +Memorization +We hypothesized that the kNN component simply provides memorization of the training set. +However, we could not improve a standard LM by interpolating its probability with another standard LM that +was further trained to overfit the training set. More details can be found in Appendix E.6.1. +Soft Labels +We hypothesized that kNN-LM’s improvement lies in reducing the “over-correction” error +when training with 1-hot labels, as hypothesized by Yang et al. (2022), and that retrieving neighbors is not +important. If only “soft labels” are the key, we could hypothetically improve the performance of another +fresh LM with the same model architecture but trained with the soft labels from the base LM, instead of from +kNN-LM. This separates the effect of “soft labeling” from the additional guidance provided by kNN. However, +this does not help with the interpolated perplexity at all. More details can be found in Appendix E.6.2. +Optimizing Interpolated Loss +We hypothesized that the standard LM cross-entropy training loss does +not emphasize the examples where base LM performs badly which could benefit from kNN, and directly +optimizing the interpolated loss of standard LM and a separate trainable softmax layer could be a better +alternative. However, we could not gain any benefits by training an additional softmax layer together with a +base LM using the interpolated loss. More details can be found in Appendix E.6.3. +7 +Conclusion +In this paper, we investigate why kNN-LM improves perplexity, even when retrieving examples from the same +training data that the base LM was trained on. By proposing and testing various hypotheses and performing +extensive ablation studies, we find that the key to kNN-LM’s success is threefold: +1. Ensembling different input representations – the feedforward layer output and the attention layer +output – can recover 55% of the performance, even without retrieval. +2. One of the most unexpected discoveries in the paper is that using approximate nearest neighbor +search allows kNN-LMs to generalize better than exact nearest neighbor search, possibly due to a +regularization effect. +3. 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Association for Computational Linguistics. doi: 10.18653/v1/2022.naacl-main.406. URL +https://aclanthology.org/2022.naacl-main.406. +13 + +A +kNN-LM Generalization to Other LMs +#params +Base LM PPL +kNN-LM PPL +Absolute PPL Gain +Ours +268M +21.75 +19.17 +2.58 +Distilled-GPT2 +82M +18.25 +14.84 +3.41 +GPT2-small +117M +14.84 +12.55 +2.29 +GPT2-medium +345M +11.55 +10.37 +1.18 +GPT2-large +774M +10.56 +9.76 +0.80 +Table 4: Performance of kNN-LM applied to other pretrained language models of different sizes. +To test the generalizability of kNN-LM, we follow the same experimental setup as used in Section 3. We +select several pretrained models from the GPT2 family (Radford et al., 2019) of various parameter counts, +plus a distilled version of GPT2, DistillGPT2. (Sanh et al., 2019) We take the pretrained model checkpoint, +build the datastore and evaluate on the Wikitext-103 dataset splits. The results are shown in Table 4. We can +see that kNN-LMs has good generalizability on other models. It improves the perplexity of all the base LMs +tested. However, the larger the model is, and usually the better the base LM’s perplexity is, the less gain can +be achieved from adding kNN. Note that our model is trained from scratch on Wikitext-103 dataset and thus +even with a relatively large model size, the perplexity and perplexity gain from adding kNN is still less than +models with pretraining. Without loss of generalizability, we will use our own trained-from-scratch model as +the base LM in the following sections for ablation study. +B +Detailed Results for Increasing the Softmax Capacity +hds +Nds +⊗ ++#params +PPL +Interp. +Oracle +- +- +- +0 +21.750 +- +- +att +Big +IP +Nds × D +∞ +19.095 +14.077 +att +1x +IP +V × D +22.584 +20.353 +16.954 +att +2x +IP +2V × D +21.903 +20.529 +17.432 +att +3x +IP +3V × D +22.434 +20.395 +17.132 +att +4x +IP +4V × D +21.936 +20.521 +17.423 +att +5x +IP +5V × D +22.025 +20.643 +17.560 +att +6x +IP +6V × D +21.972 +20.519 +17.422 +att +9x +IP +9V × D +22.084 +20.696 +17.631 +ffn +Big +IP +Nds × D +∞ +21.101 +16.254 +ffn +1x +IP +V × D +20.920 +20.694 +18.772 +ffn +2x +IP +2V × D +20.889 +20.646 +18.701 +ffn +3x +IP +3V × D +20.829 +20.603 +18.717 +ffn +4x +IP +4V × D +20.769 +20.629 +18.876 +ffn +5x +IP +5V × D +20.720 +20.594 +18.878 +ffn +6x +IP +6V × D +20.726 +20.599 +18.902 +ffn +9x +IP +9V × D +20.687 +20.567 +18.887 +Table 5: Performance comparison of kNN baselines and models with learnable embeddings as datastore +alternative. hds is either attention layer output (att) or feedforward layer output (ffn). +C +Alternative Methods for Increasing Softmax Capacity +C.1 +Adaptive Increasing Embedding Size +We hypothesize that different word types have different difficulties for the language model to predict. For +those words that appear very frequently, they may appear in many different contexts. As a result, instead +of adding equal number of additional embeddings to each word type, we propose to adaptively increase the +number of embeddings for word types based on word frequency, or total training loss for the word. Based on +the intuition of Zipf’s law (Clauset et al., 2009), we assign 1 + logb fv for each word type v ∈ V , based on +14 + +either the frequency or the total training loss of the word, fv. The b is a hyperparameter that could be tuned. +To ensure fair comparison, we tune b so that for each experiment the total number of embeddings matches: +� +v∈V 1 + logb fv = nV . The results are shown in Table 6. We can see that although nice in paper, given the +same number of total embeddings, adaptively increasing the number of embeddings assigned for each word +type does not make a significant difference in the final perplexity, when compared with the models that use +equal number of embeddings for each word type. +hds +Nds +⊗ ++#params +PPL +λ +Interp. PPL +Oracle +Base LM +- +- +- +0 +21.750 +- +- +- +KNN +att +Big +L2 +Nds × D +∞ +0.271 +19.174 +14.230 +KNN +att +Big +IP +Nds × D +∞ +0.266 +19.095 +14.077 +Equal Per Word +att +3x +IP +3V × D +22.434 +0.417 +20.395 +17.132 +Loss Weighted +att +3x +IP +3V × D +21.948 +0.437 +20.440 +17.303 +Freq. Weighted +att +3x +IP +3V × D +22.507 +0.412 +20.387 +17.105 +KNN +ffn +Big +L2 +Nds × D +∞ +0.065 +20.734 +15.594 +KNN +ffn +Big +IP +Nds × D +∞ +0.050 +21.101 +16.254 +Equal Per Word +ffn +3x +IP +3V × D +20.829 +0.622 +20.603 +18.717 +Loss Weighted +ffn +3x +IP +3V × D +20.764 +0.713 +20.659 +18.978 +Freq. Weighted +ffn +3x +IP +3V × D +20.757 +0.658 +20.572 +18.782 +Table 6: Performance comparison of kNN baselines and several configurations that adaptively increase the +embedding size with training loss or word frequency. +C.2 +Mixture of Softmaxes +Yang et al. (2017) proposes a solution to the problem using a Mixture of Softmax (MoS) to produce more +linearly independent probability distributions of words given different contexts. Suppose that there are a +total of R mixture components. MoS first uses R linear layers with weight wr to transform the current query +context vector hds into wrhds. With a shared word embedding matrix Wsm, we can calculate each softmax +component’s probability distribution with softmax(Wsm · wrhds). The mixture distribution is then given by: +PMoS = +R +� +r +πr,hdssoftmax(Wsm · wrhds) +(7) +The prior weights are calculated using another linear layer with weight wπ, as πr,hds = softmax(wπhds). +The softmax ensures that �R +r πr,hds += 1. +Comparing the MoS with the first term in Equation 5, +Msoftmax(mask-to-k(Wds ⊗ hds)), we can see that there are some connections between the two. MoS +eliminates the mask-to-k(·) operation, and replaces the single softmax across a very large vector (size of +datastore), into multiple smaller softmaxes, each across only a vector of the size of vocabulary. As a result, +the huge Wds is replaced by several linear layers to project the word embedding matrix. Now the first term +becomes: +M(⊕R +r softmax(Wsm · wrhds)) +(8) +Mir = πr,hds, ∀i ≤ V +(9) +where ⊕ represents the vector concatenation operation, and the aggregation matrix M now contains the mixture +weights for each softmax being concatenated. We perform experiments with a varying number of mixtures (R), +different definitions hds, and whether to fine-tune the output word embeddings Wsm. We allow fine-tuning the +word embedding when we use attention layer output as context vector, since the word embedding matrix is +trained with feedforward layer output originally. The results for this formulation are shown in Table 7. MoS +models on its own increase the performance of the language model marginally. When compared with Table 5, +we find that these models are worse than those that simply increases the number of embeddings. This is +expected because MoS has fewer added parameters compared to those, as it only requires several additional +linear projection layers for the embeddings. +C.3 +Clustering Datastore +Opposite to training the word embeddings of an increased size, we also consider ways to compress the datastore +down to a similar-sized embedding matrix for softmax computation. The intuition is that the datastore contains +15 + +hds +R +⊗ ++#params +PPL +λ +Interp. PPL +Oracle +Base LM +- +- +- +0 +21.750 +- +- +- +KNN +att +- +L2 +Nds × D +∞ +0.271 +19.174 +14.230 +KNN +att +- +IP +Nds × D +∞ +0.266 +19.095 +14.077 +KNN +ffn +- +L2 +Nds × D +∞ +0.065 +20.734 +15.594 +KNN +ffn +- +IP +Nds × D +∞ +0.050 +21.101 +16.254 +Ft. MoS+embed +att +2 +IP +V D + 2D2 + 2D +21.986 +0.437 +20.720 +17.573 +Ft. MoS+embed +att +3 +IP +V D + 3D2 + 3D +22.106 +0.422 +20.779 +17.609 +Ft. MoS Only +att +2 +IP +2D2 + 2D +22.552 +0.371 +21.011 +17.796 +Ft. MoS Only +att +3 +IP +3D2 + 3D +22.573 +0.371 +21.024 +17.812 +Ft. MoS Only +ffn +2 +IP +2D2 + 2D +21.351 +0.843 +21.338 +20.258 +Ft. MoS Only +ffn +3 +IP +3D2 + 3D +21.495 +0.733 +21.460 +20.322 +Ft. MoS Only +ffn +4 +IP +4D2 + 4D +21.321 +0.994 +21.321 +20.396 +Ft. MoS Only +ffn +5 +IP +5D2 + 5D +21.371 +0.909 +21.367 +20.406 +Table 7: Performance comparison of kNN baselines and several MoS configurations. R is the number of +mixtures. +redundant context vectors, and thus compression could make the datastore smaller without sacrificing too +much performance gain. He et al. (2021) shows that we can safely compress the datastore by clustering to 50% +of the original size without losing performance. We test this idea further by clustering the entire datastore +into a size that could fit in GPU memory (e.g. 2V , 3V ) and thus could be easily fine-tuned further and use the +resulting centroids to replace Wds. Within each cluster, there will be a distribution of different words with +contexts, and we use the frequency of words within each cluster to calculate the aggregation matrix M in +Equation 5. This would have the added benefit of “multi-sense” embedding, which allows similar meanings to +be clustered to form a new “meta word” while the same word with different meanings would form different +“meta words”. A notable example is bank, shore, and financial institution. However, this does not work, mostly +because of the high compression loss after clustering and the imbalanced distribution of word types among +each cluster. +D +Which Words Benefit from Approximation? +To further understand the unexpected results when using the different kNN approximate retrieval settings +in Section 5.1 and Section 5.2, we analyze on a token level, based on how many times each ground truth +token’s probability in the evaluation set are helped by each kNN setting. It means that for each ground truth +token in the evaluation, we count the times when the kNN distribution is higher than the base LM distribution +PLM, i.e., PkNN > PLM. +Since we found previously that approximate kNN provides an additional performance boost compared to +ground truth kNN, we thus compare “real mask, real score” versus “FAISS mask, real score” in this analysis. +To prevent outliers, we filter out words with less than 10 occurrences in the evaluation set. For each setting, we +calculate the percentage of occurrences in the evaluation set where each token in the vocabulary where the +kNN module achieves a better probability than base LM. We then plot the absolute difference between the +percentages of the two settings, with respect to various possible attributes of the token that achieves better +probability using each setting. +Figure 6 shows that the longer the token is, which usually suggests proper nouns and harder and less common +words in English, are better with approximate neighbors than ground truth ones, and vice versa. We hypothesize +that this is due to longer words are more prone to overfitting in kNN-LM and thus using approximate kNN +provides an effect similar to smoothing and regularization. +We also compare words that could appear in more diverse contexts with words that co-occur with few distinct +contexts. To measure how diverse the contexts of each word in the vocabulary is, we calculate both the forward +and backward bigram entropy for each word in the evaluation set that has more than 10 occurrences. The +bigram entropy is a simple yet good indicator of context diversity for a given word, as used in Kneser–Ney +smoothing (Ney et al., 1994). We calculate both the forward and backward bigram entropy for each word w as +16 + +Figure 6: The effect of the token character length on how much accurate nearest neighbors are better than +approximate FAISS neighbors. Negative values mean worse. The trend line of the scatter points is shown. +follows, where wafter and wbefore represent the word after and before the given word w. +Hforward(w) = − +� +wafter +p(wafter|w) log p(wafter|w) +(10) +Hbackward(w) = − +� +wbefore +p(wbefore|w) log p(wbefore|w) +(11) +Forward and backward entropy represents how diverse the context after and before the given word is. Intuitively, +bigram entropy is supposed to indicate words that can appear in lots of different contexts. The higher the +entropy of a word, the more diverse its context is, and vice versa. For example, words like “Francisco” would +have a low entropy because it mostly comes after “San”. +Figure 7: The effect of the forward and backward entropy of words on how accurate nearest neighbors are +better than approximate FAISS neighbors. Negative values mean worse. The trend line of the scatter points are +shown. +The comparison is shown in Figure 7. We can see that the higher the entropy in both forward and backward +cases, the better using approximate nearest neighbor search becomes. This suggests that words that appear +in many different contexts are better off with an approximate kNN, and “easy-to-predict” examples such +as “Jersey” and “Fransisco” is better with accurate kNN, possibly because these examples are less prone to +overfitting errors and thus requires less regularization from approximation. +17 + +E +Failed Hypotheses +E.1 +Distance Metric +We hypothesize that the key to kNN-LM’s performance gain is the ensemble of two distance metrics: the +standard dot product distance (which the LM uses) with the L2 distance (which the kNN component uses as +⊗). We tried to replace the kNN component with a component that just takes the tokens retrieved by the kNN +search and returns their L2 distance to the LM output word embeddings: Wsm ⊗ hds instead of Wds ⊗ hds, +where ⊗ represents the negative L2 distance. We tried this with both variants of hds, attention layer output, +and feedforward layer output. None of these helped. +E.2 +Sparsification +In Equation 5, mask-to-k(·) used by kNN retrieval induces sparsity in the distribution over the vocabulary, +due to a small k compared to the number of vocabulary V . We hypothesize that the in kNN-LM, the kNN +distribution is sparse, practically increasing the probability of the top-k entries. The kNN distribution has +up to 1024 entries that are non-zero, concentrating more probability mass over the most likely tokens. This +effect is similar to the redistribution of probability mass for text generation in Holtzman et al. (2019). We +test this hypothesis only by taking top 32, 64, 128, 512, or 1024 tokens in the parametric LM probability and +zeroing out the probabilities of the rest of the tokens. To compensate, we experiment with different softmax +temperatures and then interpolate with the parametric LM probability. This isolates the effect of the datastore +and retrieval at all, and this does not help at all, suggesting that sparsification of the output probability alone is +not enough. +Another attempt is to hypothesize that the key in kNN-LM is that it selects “which tokens to include” in the +kNN distribution, and not their distances. The intuition behind is that maybe the selection of the top tokens +according to the kNN search is better than that from the dot-product distance between the language model’s +output vector and all the vocabulary embeddings. We perform experiments similar to the previous attempt, +sparsifying the output probability with the tokens retrieved by the kNN search (but ignoring the distances +provided by the kNN search) rather than the top k tokens of the LM, with and without removing duplicates. In +the best case, they manage to reduce the perplexity by 0.5 (whereas kNN-LM reduces by nearly 2). +E.3 +Location within Context Window +Supposedly, words in the beginning of the “context window” of the transformer at test time have less contextual +information than words toward the end of context window. +We hypothesized that maybe the base LM performs worse in one of these (beginning vs. end of the context +window), and maybe kNN-LM provides a higher improvement in one of these. We measured the per-token +test perplexity with respect to the location of each token in the context window. However, we did not find any +significant correlation between the performance of the base LM and the location, and no significant correlation +between the difference between kNN-LM and the base LM and the location. +We also hypothesized that maybe the beginning of every Wikipedia article is more “predictable”, and the text +becomes more difficult to predict as the article goes into details. However, we also did not find any correlation +with the location of the word within the document it appears in. +E.4 +Stolen Probabilities +The stolen probabilities effect (Demeter et al., 2020) refers to the situation where the output embeddings of +an LM are learned such that some words are geometrically placed inside the convex hull that is formed by +other word embeddings. Since language models generate a score for every output word by computing the +dot product of a hidden state with all word embeddings, Demeter et al. (2020) prove that in such a case, it is +impossible for words inside the convex hull to be predicted as the LM’s most probable word (the “argmax”). +We hypothesized that kNN-LM solves the stolen probabilities problem by allowing to assign the highest +probability to any word, given a test hidden state that is close enough to that word’s datastore key. Nevertheless, +as shown by Grivas et al. (2022), although this problem might happen in small RNN-based language models, +in modern transformers it rarely happens in practice. Using the code of Grivas et al. (2022), we checked the +embeddings matrix of our model and of the checkpoint provided by Khandelwal et al. (2020b). Indeed, we +found that in both models – no word is un-argmaxable. +18 + +E.5 +Are kNN-LM Just Ensembling? +Our hypothesis is that kNN component only provides another model for ensembling. The interpolation +process is basically an ensemble model. Technically it is unsurprising that kNN-LM will have the benefit +from ensembling, but we perform experiments to see how it compares to other ensembling. We trained +another language model with the same architecture as the base LM we used throughout the experiments, +with some variants having more than one embedding vector for each word (similar to Section 4.2). We +interpolate the models with the original base LM, and the results are shown in Table 8. We can see that even +just ensembling the base LM with another identical model, but trained with a different random seed, provides +a huge performance boost, both on interpreted perplexity and on oracle perplexity. +Prev. Layers +hds +Nds +⊗ ++#params +PPL +Interp. +Oracle +same +- +- +- +0 +21.750 +- +- +same +att +Big +L2 +Nds × D +∞ +19.174 +14.230 +same +att +Big +IP +Nds × D +∞ +19.095 +14.077 +same +ffn +Big +L2 +Nds × D +∞ +20.734 +15.594 +same +ffn +Big +IP +Nds × D +∞ +21.101 +16.254 +diff +ffn +1x +IP +F + V × D +21.569 +18.941 +14.980 +diff +ffn +2x +IP +F + 2V × D +21.914 +18.948 +14.885 +diff +ffn +3x +IP +F + 3V × D +22.206 +18.981 +14.853 +Table 8: Performance comparison of kNN baselines and models with different size output embeddings +re-trained from scratch. +However, just because ensembling two LMs of the same architecture provides better performance than +interpolating the base LM with kNN does not necessarily suggest that kNN’s performance improvement can +be fully replaced by model ensembling. In other words, we are interested in whether the kNN performance +improvements are orthogonal to that of model ensembling. To test this, we compare the performance of the +ensemble of K multiple LMs versus the ensemble of K − 1 multiple LMs plus the kNN component. The +comparison is fair because we have the same number of models in the ensemble, and the only difference is +whether the kNN component is included. The results are shown in Figure 8. For the “LM” series, each point +is K LMs ensemble, and for the “kNN” series, each point is K − 1 LMs plus kNN. We can see that even at +4-ensemble, the ensemble that contain kNN as a component still have a considerable edge over the 4-ensemble +that contain just LMs. +Ensemble Components +16 +18 +20 +22 +1 +2 +3 +4 +LM +KNN +LM and KNN +Figure 8: Ensembling effect comparison, between multiple base LMs and multiple base LMs plus kNN +component. +E.6 +Are kNN-LM Just Alternative Training Methods? +E.6.1 +Overfitting +Since kNN-LM improves perplexity even with the same training dataset as datastore, we are curious if +kNN-LM works by only “memorizing” the training data. The hypothesis is that the datastore and the kNN +19 + +Prev. Layers +hds +Nds +⊗ ++#params +PPL +Interp. +Oracle +Base LM +same +- +- +- +0 +21.750 +- +- +KNN +same +att +Big +L2 +Nds × D +∞ +19.174 +14.230 +KNN +same +att +Big +IP +Nds × D +∞ +19.095 +14.077 +KNN +same +ffn +Big +L2 +Nds × D +∞ +20.734 +15.594 +KNN +same +ffn +Big +IP +Nds × D +∞ +21.101 +16.254 +Overfit@92 +diff +ffn +V +IP +F + V × D +1702.806 +21.732 +17.764 +Overfit@129 +diff +ffn +V +IP +F + V × D +8966.508 +21.733 +17.814 +Table 9: Performance comparison of several baselines with two overfitted models, at 92 and 129 additional +epochs. +search are trying to memorize the training data. In other words, the parametric LM is under-fitting some +tokens. The intuition behind this is that the kNN component retrieves examples directly from the training set. +What if we could retrieve the same examples using an overfitted LM? We took the trained LM, removed the +dropout, and continued training until almost perfect fit (very small training loss). We then interpolated the +overfitted transformer with the original LM. The results are shown in Table 9. F represents the number of +parameters in the base LM, minus the output embedding matrix. We can see that overfitting can provide very +little help after interpolation. Looking at the oracle performance, we think that the overfitted model memorizes +some rare contexts and tokens in the training set where it could be useful during evaluation. However, the +overfitting hurts the performance on other tokens too much so that even interpolation is not able to balance the +performance. +E.6.2 +Soft-Label Training +Yang et al. (2022) claims that using “soft labels” during training is the key to kNN’s success, that interpolates +the ground truth labels with kNN-LM model outputs, effectively “distilling” kNN-LM. It is based on the +hypothesis that the room for kNN-LM’s improvement over base LM lies in the “over-correction” when training +with a 1-hot labels. This is related to the effect from label smoothing methods (Szegedy et al., 2016; Pereyra +et al., 2017; Meister et al., 2020a). However, we believe that this explanation is not satisfactory. If the key is +training with soft-labels, why do these soft labels must be provided specifically by a kNN search? If soft labels +were the key, then soft-label training where the labels come from the base LM itself should have worked as +well. To separate the effect of soft labeling from the kNN’s additional guidance, we train another LM with the +same model architecture as the base LM, with the soft labels from the base LM. This teacher-student training +is to distill the knowledge from the base LM (Hinton et al., 2015). We find that by just training with “soft +labels“ from the base LM to alleviate the alleged “over-correction” problem is not the key, as this does not help +with the interpolated perplexity at all. This suggests that even with the same training data, kNN still provides +valuable additional guidance. +E.6.3 +Training to Optimize Interpolated Loss +In Section 4.2, we discover that using over-parameterization with standard LM training loss does not further +close the gap towards kNN-LM. This suggests that some regularization term may be needed during training to +make the multiple embeddings not converge to the same vector, rendering over-parameterization useless. +From Table 2, we see that a better interpolated perplexity may not require a very low perplexity when measured +only with the extra input representation. However, we still use a standard LM loss to only train the additional +embedding matrix, that directly minimizes the perplexity using only the extra input representation. This +discrepancy between training and the evaluation with interpolation suggests that training with an alternative +loss function that interpolates the base LM’s output with the output using the extra input representation may +be beneficial. +To test the hypothesis that standard LM training loss do not emphasize the examples where base LM performs +badly, we train the extra model’s parameter Wds, with interpolated loss L: +L = CrossEntropy(λsoftmax(Wds · hds) + (1 − λ)softmax(Wsm · hsm), y) +(12) +y represents the ground truth label for each context. We only learn the parameter Wds while freezing all +other parameters, similar to all other experiments. We choose λ = 0.25 as it is the best hyper-parameter for +kNN-LM experiments and our goal for this training is to mimic the loss of kNN-LM after interpolation. This +training loss effectively assigns a higher value to the training examples where the base LM’s loss is high, +20 + +suggesting the need for the extra Wds to help with these hard cases. However, for either “att” for “ffn” for hds, +either V or 3V for the number of embeddings in Wds, we are unable to achieve a better perplexity than just +the base LM. This suggests that, while nice on paper, the interpolated loss optimization process is not trivial. +21 + diff --git a/09E0T4oBgHgl3EQf_gKx/content/tmp_files/load_file.txt b/09E0T4oBgHgl3EQf_gKx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b6c26fc9bd8f146dcebdcc9b3f53699694680d2 --- /dev/null +++ b/09E0T4oBgHgl3EQf_gKx/content/tmp_files/load_file.txt @@ -0,0 +1,972 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf,len=971 +page_content='Why do Nearest Neighbor Language Models Work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Frank F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Xu Uri Alon Graham Neubig Language Technologies Institute Carnegie Mellon University {fangzhex,ualon,gneubig}@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='edu Abstract Language models (LMs) compute the probability of a text by sequentially computing a representation of an already-seen context and using this representation to predict the next word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Currently, most LMs calculate these representations through a neural network consuming the immediate previous context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However recently, retrieval-augmented LMs have shown to improve over standard neural LMs, by accessing information retrieved from a large datastore, in addition to their standard, parametric, next-word prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In this paper, we set out to understand why retrieval-augmented language models, and specifically why k-nearest neighbor language models (kNN-LMs) perform better than standard parametric LMs, even when the k-nearest neighbor component retrieves examples from the same training set that the LM was originally trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To this end, we perform a careful analysis of the various dimensions over which kNN-LM diverges from standard LMs, and investigate these dimensions one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Empirically, we identify three main reasons why kNN-LM performs better than standard LMs: using a different input representation for predicting the next tokens, approximate kNN search, and the importance of softmax temperature for the kNN distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Further, we incorporate these insights into the model architecture or the training procedure of the standard parametric LM, improving its results without the need for an explicit retrieval component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='com/frankxu2004/knnlm-why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 1 Introduction Language modeling is the task of predicting the probability of a text (often conditioned on context), with broad-spanning applications across natural language processing (Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Merity et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Baevski and Auli, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This modeling is usually done by sequentially encoding a context ct using a trained neural network function f, and computing the probability of the next word wt according to f (ct) and a vector representation of wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Recently, retrieval-augmented LMs have shown a series of impressive results (Grave et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Guu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Borgeaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Retrieval-augmented LMs compute next token distributions based not only on the immediately preceding context ct and the model parameters, but also on an external datastore, from which examples are retrieved and incorporated into the base LM’s prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' One retrieval-augmented model that is notable for both its simplicity and efficacy is the k-nearest neighbor language model (kNN-LM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' It extends a trained base LM by linearly interpolating the output word distribution with a kNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The nearest neighbors are retrieved according to the distances between the current context embedding of the base LM and all the context embeddings in the datastore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The datastore is created by encoding all contexts from any text collection, including the original LM training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' One of the most surprising results from Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b) is that kNN-LM reduces the perplexity of the base LM even when the kNN component is retrieving examples from the same training set that the LM was originally trained on, indicating that the kNN-LM improves the ability to model the training data and is Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='02828v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='CL] 7 Jan 2023 Multi Headed Attention Feed Forward Network Layer Norm ℎ𝑠𝑚 𝑊𝑠𝑚 𝐷 𝑉 ℎ𝑑𝑠 𝑊𝑑𝑠 𝐷 𝑁𝑑𝑠 + 𝑃𝐿𝑀 parametric component 𝑃𝑘𝑁𝑁 non-parametric component In 𝑘NN-LM: 𝑁𝑑𝑠: up to 5000𝑉 𝐷 𝐷 mask-to-k() In 𝑘NN-LM: top-𝑘() FFN ATT softmax() softmax() Figure 1: An illustration of the generalized formulation of kNN-LM in Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' not simply benefiting from access to more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Intrigued by this, we ask questions like, could kNN-LM be improving because of capacity issues in the parametric base LM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In this paper, we set out to understand why kNN-LMs work even in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In the following sections, we first elucidate connections between the added kNN component and the standard LM component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Specifically, we note that word distributions from the two components are both calculated using a softmax function, based on the similarity of the current context embedding with a set of embeddings that corresponds to different next words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' With this intuition, we formalize and generalize the non-parametric distribution calculation with the softmax layer and word embedding layer used in parametric LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We then show that this generalized form exposes a variety of design choices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', the number of context embeddings in the datastore, the input representation used in softmax layer, different similarity functions, as well as the approximation and sparsification implementations in the kNN search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This provides a general framework for analyzing kNN-LM and similar models and allows us to perform ablation studies that test the importance of various design decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We proceed to propose multiple hypotheses for why kNN-LM works, which are testable by adjusting the various parameters exposed by our generalized formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Based on these hypotheses, we perform ablation experiments and analyze the nuances between different implementations of the generalized version of PkNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' As the answer to our question, “why kNN-LMs work”, we eventually show that the most probable reasons are threefold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Ensembling the output of softmax using two representations from different layers of the transformer is important;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' in our experiments, this accounts for 55% of the performance gain of kNN-LM, or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='5% relative perplexity improvement compared to the base LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' kNN-LM uses approximate nearest neighbor search to handle the large number of candidates, and the lack of this preciseness in this algorithm actually helps kNN-LM to generalize better than using exact nearest neighbor search and distance calculation, possibly due to a regularization effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The relative perplexity improvement from this factor is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Depending on the design decisions that are chosen for modeling, adding a temperature term to the kNN non-parametric component can become crucial to the success of modeling (although coincidentally, in the original settings of Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b), a temperature of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='0 is close to optimal, which hid the importance of this term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In some settings, the relative perplexity gap between the default and optimal temperature can be as high as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Finally, one significant drawback to the current kNN-LM is the inefficiency of kNN search performed at each step (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Borgeaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Because of the similarity between kNN-LM and the parametric LM’s last layers and the many design choices, we also demonstrate that we are able to make kNN-LM more efficient by substituting the kNN search with another matrix operation that can fit in accelerator memory while maintaining more than half the perplexity improvement, or more than 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='5% relative improvement compared to the base LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 2 2 Formalizing and Generalizing kNN-LM kNN-LM (Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2020b) is a linear interpolation between a base LM and a kNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Given a set of contexts ci and their corresponding next token wi as a pair (ci, wi) ∈ D, kNN-LMs create a datastore (K, V) = {(ki, vi)}, as a set of key-value pairs: (K, V) = {(f (ci) , wi) | (ci, wi) ∈ D} (1) During inference, the parametric component of the LM generates the output distribution pLM(wt|ct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' θ) over the next tokens and produces the corresponding context representation f(ct), given the test input context ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Then the non-parametric component of the LM queries the datastore with the f(ct) representation to retrieve its k-nearest neighbors N according to a distance function d(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Next, the kNN-LM computes a probability distribution over these neighbors using the softmax of their negative distances, and aggregates the probability mass for each vocabulary item across all of its occurrences in the retrieved targets: pkNN(wt|ct) ∝ � (ki,vi)∈N 1wt=vi exp(−d(ki, f(ct))) (2) Finally, this distribution is interpolated with the parametric LM distribution pLM to produce the final kNN-LM distribution: p(wt|ct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' θ) = (1 − λ)pLM(wt|ct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' θ) + λpkNN(wt|ct) (3) where λ is a scalar that controls the weights of the interpolation between two components, with higher λ putting more weight on the non-parametric component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Looking closely at Equation 2, we can notice a similarity between the calculation of PkNN and the standard PLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The kNN distribution is based on the distances between the current context and the nearest neighbors from the datastore, normalized by a softmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Recall that in (standard) parametric language models, the distribution over the vocabulary is also based on a measure of distance, the inner product between the current context embedding and the word embeddings of every token in the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Because each context embedding in the datastore (K, V) corresponds to a target token, we can also view this datastore as a large word embedding matrix with multiple word embeddings for each of the vocabulary words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Theoretically, given unlimited computation, we could calculate the distribution based on the distances to every embedding in the datastore, and aggregate by vocabulary items, making it more closely resemble PLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In this case, k = |D|, the size of the entire datastore, and Equation 2 becomes the following, based on the distances to every context in the datastore D instead of a subset of nearest neighbors N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' pD(wt|ct) ∝ � (ki,vi)∈D 1wt=vi exp(−d(ki, f(ct))) (4) In practice, we use kNN search as a way of approximation, by limiting the calculation to only k nearest neighbors to avoid the computational cost of calculating the distribution over the entire datastore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' If we re-write and generalize Equation 2, both the kNN-LM of Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b) and a large number of related models can be expressed through the following equation: Pinterp = (1 − λ) softmax(Wsm · hsm) � �� � PLM parametric component +λ Msoftmax(mask-to-k(Wds ⊗ hds)/τ) � �� � PkNN non-parametric component .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (5) Figure 1 provides an illustration of Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The first term of the equation is the standard parametric language model, whereas the second represents a generalized version of utilizing an external datastore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The first component, the output layer of a common parametric language model, is relatively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Wsm of size V × D is the embedding matrix of the output token, and hsm is the context vector used to calculate the distribution of the output token, usually the output of the final feedforward layer in the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In the second component, Wds represents the datastore, of size Nds × D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Nds is the number of entries in the datastore, and D is the size of each context vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' hds represents the context vector used to query the datastore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' As shown in Figure 1, these vectors can come from different layers of the transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' ⊗ represents the operation type used to calculate the similarity between context vectors and the query vector, which also has several alternatives that we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' mask-to-k(·) represents a function to sparsify similarity scores across the datastore, setting all but k similarity scores to −∞, which results in probabilities of zero for all masked similarity scores after the softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 3 Practically, this is necessary for kNN-LMs because the size of the datastore Nds makes it infeasible to calculate all outputs at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' With masked logits, we apply a more generalized version of softmax with temperature τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Intuitively adding the temperature can adjust the peakiness or confidence of the softmax probability distribution output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' After the softmax, the matrix M of dimension V × Nds sums the probability of the Nds datastore entries corresponding to each of the V vocabulary entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Each column in this matrix consists of a one-hot vector with a value of 1 and the index corresponding to the vocabulary item wi corresponding to the datastore entry for ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Within this formulation, it becomes obvious that there are many design choices for kNN-LM-like models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' One important thing to note is that the right side of Equation 5 is actually very similar to the left side representing the standard parametric language model, with a few additional components: M, mask-to-k, and ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' More specifically, some of the design decisions that go into the kNN-LM, and parallels with standard parametric models are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Size of Wds: In the standard parametric model, the size of Wsm is V embedding vectors, each with D dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In the kNN-LM it is very large: Nds, the size of the datastore, usually the number of tokens in the entire training corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Input representation: In the parametric model, hsm is the output from the feedforward layer in the last transformer block, which we abbreviate “ffn”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In contrast, Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b) rather use as hds the output from the multi-headed attention layer of the last transformer block (before running the representations through the feed-forward network, and after the LayerNorm (Ba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2016)), which we abbreviate as “att”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Similarity & Temperature: In the parametric model, the functional form of ⊗ is the inner product (abbreviated IP), whereas Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b) use negative squared L2 distance (abbreviated L2) as a similarity function between Wds and hds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' As the similarity scores are turned into probability distributions with the softmax function, the choice of softmax temperature (τ) can control the scaling of the similarity scores and thus the peakiness of the non-parametric component distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Approximation & Sparsification: In the parametric model, k = V , and no values are masked, but in the kNN-LM, k ≪ V , and most of the datastore entries are pruned out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The definition of the mask-to-k(·) function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' how to select the important datastore embeddings to include in the similarity calculation (in kNN-LM’s case the k nearest neighbors), is a crucial open design choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In the following sections, we set out to better understand how each of these design decisions contributes to the improvement in accuracy due to the use of kNN-LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 3 Baseline kNN-LM Results First, we evaluate the kNN-LM baseline on the Wikitext-103 dataset (Merity et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2016), and examine the importance of two design choices: the input representation hds and the similarity function ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In models examined in this paper, the parametric model is a transformer language model with mostly the same architecture as in Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, We do make modifications to the original base LM (Baevski and Auli, 2018) to accommodate our experimentation need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We using BPE tokenization (Sennrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2015) to train a smaller vocabulary (33K) than the original (260K) on the training corpus of Wikitext-103, as subword tokenization is ubiquitous in many state-of-the-art language models (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Using subword tokenization also eliminates the need for adaptive softmax (Joulin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' It makes the output layer more generalized, sharing more resemblance to the kNN component as described in Section 2, and facilitates the ablation studies in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='1 This base LM has 268M parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To get a perspective on how large the datastore is, it is built on the training data that contains nearly 150M BPE tokens, each paired with a context vector of size 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This datastore has a total memory consumption of about 300GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' At every retrieval step, we take the top 1024 nearest neighbors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', k = 1024, following Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The interpolated perplexity is computed with optimal interpolation parameter λ tuned according to the perplexity on the development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' λ is fixed during the inference for all predictions, the same as the standard kNN-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 1By training our own version of the base LM from scratch with BPE tokenization and a standard output softmax layer, our LM’s perplexity is worse than that used in the original kNN-LM paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, we observe similar relative gains from the additional kNN component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We argue that the base LM’s performance is orthogonal to the study of the factors behind kNN-LM’s improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 4 hds ⊗ +#params PPL Interp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' PPL Oracle Base LM 0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='750 kNN-LM-L2 att L2 Nds × D ∞ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='174 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='230 kNN-LM-IP att IP Nds × D ∞ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='095 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='077 kNN-LM-L2 ffn L2 Nds × D ∞ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='734 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='594 kNN-LM-IP ffn IP Nds × D ∞ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='101 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='254 Table 1: Performance of the parametric language model and several kNN-LM variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Results comparing multiple kNN-LM variants are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The first row represents the base parametric language model’s perplexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The second is a formulation analogous to that of Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b), and in the remaining rows, we vary the input representation hds and distance function ⊗ from Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' All of them use a large datastore with size Nds, approximately 5000 times the size of the vocabulary V , as also reflected in “+#params”, the number of additional parameters other than the base LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We report several important quantities with respect to each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' “PPL” shows the perplexity of only the kNN component of the model pkNN().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This is ∞ for all kNN- LM models in all cases, as when the kNN search does not retrieve any datastore entries corresponding to the true target word wt the probability of the target word will be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' “Oracle” shows the lower bound of the interpolation performance by choosing the best λ for each token in the evaluation dataset, which will either be λ = 0 or λ = 1 depending on whether PLM(wt|ct) > Pknn(wt|ct) or not, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' From the table, we can see that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Using the output of the multi-headed attention layer (“att”) as hds (instead of the standard “ffn” layer) is crucial for better performance of kNN-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In general, using negative squared L2 distance or inner product as a similarity function does not result in a large and consistent difference, although in our setting, IP provides slightly better performance when using the “att” inputs, and slightly worse when using “ffn” inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Interestingly, when using “ffn” and “IP”, the same input and distance metric used in the parametric model, the results are the worst, indicating that the kNN-LM is particularly benefiting when the kNN-LM achieves a different view of the data from the parametric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We found in preliminary experiments that kNN-LM is generalizable to other base language models as well, ranging from small models with 82M parameters to larger models with 774M parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The gain from kNN-LM is more significant when used with a smaller, less capable base language model, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The details are shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In this paper, we are mainly focused on the factors contributing to the relative improvements from kNN-LM, instead of the absolute performance, so we use the 268M model for the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In the next sections, we perform further experiments with ablations on the general formulation Equation 5 to elucidate the key elements contributing to the performance improvements in kNN-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 4 Effect of Different Wds Formulations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='1 Replacing the Datastore with Trainable Embeddings From the observation in Section 3, we see that the choice of hds has a large impact on the performance of kNN-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This intrigues us to explore if one key to the improvements afforded by kNN-LM lies in the use of different input representations together, namely the attention output (hds = att) and feedforward output (hds = ffn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, from only the experiments above, it is not possible to disentangle the effect of the choice of hds and that of other design choices and factors in Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To test the effect of hds in a more controlled setting, we remove the non-parametric datastore entirely, and initialize Wds in Equation 5 with a randomly initialized word embedding matrix with the same size (Nds = V ) 5 as the LM’s output embedding Wsm, and train Wds with all other parameters fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2 The loss function for training is the cross-entropy loss of softmax(Wds · hds) with respect to the ground-truth tokens, identically to how the base LM is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We compare how using hds = att or hds = ffn affects the interpolated performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The results are shown in Table 2, and we also show results from kNN-LMs using these two varieties of input representation for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' From these experiments we can find several interesting conclusions: Effectiveness of re-training Wds: In the case of “Learned Wds w/ FFN”, we are essentially re-learning the weights feeding into the softmax function separately from the underlying LM encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Despite this fact, we can see the model achieves a PPL of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='920, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='83 points better than the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This suggests that there is some benefit in learning the parameters of Wds after freezing the body of the transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Effectiveness of ensembling two predictors: In both cases for Wds, the interpolated perplexity is significantly better than that of using a single predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This is particularly the case when using the “att” representation for hds, suggesting that the utility of ensembling predictions from two views of the data is not only useful when using kNN-LM, but also in standard parametric models as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Parametric ensembles as an alternative to kNN-LM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' : Overall, by using a separate word embedding matrix with size V × D as an alternative to kNN, we can recover about 55% of the performance gain achieved by kNN-LM, with only a limited number of parameters and without the necessity for slow kNN retrieval every time a token is predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This suggests that the majority of the gain afforded by kNN-LM could be achieved by other more efficient means as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' hds Nds ⊗ +#params PPL Interp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Oracle Base LM 0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='750 kNN-LM w/ ATT att Big IP Nds × D ∞ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='095 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='077 Learned Wds w/ ATT att 1x IP V × D 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='584 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='353 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='954 kNN-LM w/ FFN ffn Big IP Nds × D ∞ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='101 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='254 Learned Wds w/ FFN ffn 1x IP V × D 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='920 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='694 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='772 Table 2: Performance comparison how the choice of hds, input representation, affects kNN baselines and models with learnable embeddings as datastore alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' hds is the attention layer output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2 Increasing the Softmax Capacity One premise behind kNN-LM is that the large datastore is the key reason for the model working well: the larger the softmax capacity, the better the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Naturally, as a first step, we need to check whether such a big datastore is warranted and whether the high rank of Wds leads to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We test the effect of the datastore size for kNN retrieval on kNN-LM interpolated perplexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' If a bigger datastore (a high rank Wds) is better in kNN-LM than a smaller datastore, then the hypothesis of softmax capacity is more probable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We randomly subsample the full datastore in varying percentages and the results are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The full datastore contains more than 150M entries and storing them takes 293GB when using half-precision floating points (fp16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We can see that whether or not approximate kNN is used, the final perplexity decreases almost linearly with more percentage of the original datastore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Even with just 5% of the datastore size (15G), kNN-LM still provides a benefit over just using the base LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, even when the subsampling percentage reaches 90%, having more entries in the datastore still provides benefits without having significant diminishing returns, suggesting that a large datastore is beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' One possible reason why a larger datastore is helpful is that words can be difficult to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' There are several reasons: (1) They are rare, or (2) they are frequent, but they have multiple meanings and appear in different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The softmax bottleneck (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2017) suggests that the final dot product of language model Wsm · hsm limits the expressivity of the output probability distributions given the context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' that is, a single output vector of a fixed (1024) size cannot express all the possible mappings between 100M training examples and 33K vocabulary outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We hypothesize that kNN-LM improves performance by alleviating the problem, since Wds ⊗ hds has a higher rank and is more expressive than just Wsm · hsm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In other words, kNN is a sparse approximation of the full softmax over all the embeddings in the datastore Wds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To test this hypothesis, 2Because we previously found little difference between IP and L2 as similarity functions, we use IP in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' For simplicity, we set temperature τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 6 we disentangle the effect of the high rank in Wds from the actual saved context embeddings in Wds, by training an embedding matrix of the same desired size to test from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Ratio to Full Datastore Size Interpolated Perplexity 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='000 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='000 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='000 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='00 Figure 2: The effect of the size of the datastore used for kNN retrieval on final interpolated perplexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We explore several potential solutions for increasing the capacity of softmax, and examine if they can achieve a similar effect of kNN-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The first and easiest solution is to increase the embedding matrix size by adding more embedding vectors for each word type in the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To test this, we replace Wsm with a much smaller matrix of size nV × D, where we allocate n embedding vectors for each word type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' When calculating the probability from this component, we compute the softmax over nV items and sum the probabilities for each vocabulary entry to calculate the final probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' mask-to-k(·) is no longer needed, as this formulation is small enough to fit the entire matrix in the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We then finetune the new Wds on the training data until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Figure 3 compares the base LM and the original kNN-LM versus using either attention layer output (“att”) or feedforward layer output (“ffn”) as hds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We plot the number of embeddings for each word type (nV total embeddings in Wds) versus the interpolated perplexity, with full details found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In both cases, comparing with the top horizontal line which represents the perplexity of the base LM, replacing the datastore with a much smaller weight matrix (from Nds to nVds) by assigning only a few more embeddings for each word helps, although only about half as effective as kNN-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To give a perspective, the original datastore size is about 5000V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Surprisingly, we find that increasing n does not always bring better performance, even though a larger datastore is better than using a small datastore in kNN-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We can see that when hds = ffn, over-parameterization provides very limited improvements, while for hds = att it does not bring consistent improvements at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Comparing the trend of increasing the embeddings in Wds, with the bottom horizontal line in the plot, which represents the perplexity of the standard kNN-LM using the full datastore (Wds with approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 5000V embeddings), we can see no clear trend that more trainable embeddings result in better perplexity, and that the gap between using trained embeddings and using full datastore is still significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This suggests that simply over-parameterizing Wds is not an effective method of achieving accuracy gains similar to kNN-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We hypothesize that this is because by just adding more embeddings, while still using the same training procedure as the original LM, the multiple embeddings for each word type after learning could still be very close to each other, and thus do not increase the softmax capacity much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This suggests that some regularization terms may be needed during training to make the multiple embeddings not converge to the same vector, rendering over-parameterization useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Besides simply increasing the number of embedding vectors equally for each word type, we also propose other alternatives to increase softmax capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' First, we hypothesize that different word types have different difficulties for the language model to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' For those words that appear very frequently, they may appear in many different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' As a result, instead of adding an equal number of additional embeddings to each word type, we propose to adaptively increase the number of embeddings for word types based on word frequency, or total training loss for the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Second, we try to break the softmax bottleneck with a Mixture of Softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2017) proposes a solution to the problem using a Mixture of Softmax (MoS) to produce more linearly independent probability distributions of words given different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Last, opposite to training the word embeddings of increased size, we also consider ways to compress the datastore down to a similar-sized embedding matrix for softmax computation by clustering the whole datastore and allowing for further finetuning of the embedding matrix consisting of cluster centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, none of these alternative methods provided additional benefits over the simple multi-embedding approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' More details on these attempts can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 7 Number of Trained Embeddings (nV) Interpolated Perplexity 19 20 21 22 2 4 6 8 att ffn Figure 3: The number of embeddings per word type (nV total embeddings in Wds) versus interpolated perplexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The horizontal line at the top represents the perplexity of the base LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The horizontal line at the bottom represents the interpolated perplexity using a full datastore with kNN-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 5 Approximate kNN Search & Softmax Temperature 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='1 Comparing Approximate kNN Search To calculate PkNN of the non-parametric component in Equation 5, it is usually prohibitive to use exhaustive kNN search, and thus Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020a) use approximate kNN search using the FAISS library (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The use of FAISS (similarly to other approximate search libraries) results in two varieties of approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Approximate Neighbors: Because the search for nearest neighbors is not exact, the set of nearest neighbors might not be equivalent to the actual nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Recall the function mask-to-k(·) in Equation 5, it is the function where we select the kNN entries from the datastore Wds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We denote “real mask” as the accurate nearest neighbors for mask-to-k(·) selection, and “FAISS mask” as the approximate nearest neighbors returned by the FAISS library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='3 Approximate Scores: In addition, FAISS makes some approximations in calculating the distances between the query and the retrieved neighbors for efficiency purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We denote “real score” as the scores calculated from ground truth distances between the embeddings, and “FAISS score” as the distances returned by FAISS approximate search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The comparison of the different approximation settings is shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Quite surprisingly, we actually find that the interpolated perplexity with approximate search is better than that with exact search, both with respect to the mask and the score calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Intrigued by this counter-intuitive result, we explore the effect of kNN search approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' hds ⊗ +#params PPL λ Interp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' PPL Oracle Base LM 0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='750 kNN-LM w/ FAISS mask, FAISS score att L2 Nds × D ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='271 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='174 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='230 kNN-LM w/ FAISS mask, real score att L2 Nds × D ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='176 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='672 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='393 kNN-LM w/ real mask, real score att L2 Nds × D ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='172 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='735 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='480 Table 3: Performance of the parametric language model and comparison of kNN-LMs using the approximate versus ground truth kNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' First, we plot the subsampled size of the datastore with the interpolated perplexity Figure 4, a similar plot to Figure 2, but showcasing the comparison between approximate and real masks, between approximate and real scores in both the full datastore as well as a small subsampled datastore setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We find that using an approximate FAISS mask to find nearest neighbors is better than using the ground truth nearest neighbors and that using the approximate score returned by FAISS is better than recomputing the ground truth distances 3To calculate the real mask over a large datastore, we shard the datastore into several smaller datastores, calculate the nearest neighbors for each of the smaller datastores, and combine them back together to get the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 8 between embeddings for the kNN distribution at different levels of datastore size, both at 5% or 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Interestingly, the gap between using an approximate score or real score given the same approximate nearest neighbors (“FAISS mask, FAISS score” vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' “FAISS mask, real score”) is larger than that between using approximate or real nearest neighbors given the same ground truth method of calculating the distance (“real mask, real score” vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' “FAISS mask, real score”), for reasons we will elucidate in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Ratio to Full Datastore Size Interpolated Perplexity 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='000 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='000 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='000 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='00 FAISS mask, FAISS score FAISS mask, real score real mask, real score Figure 4: The differences between using approximate and accurate kNN search on varying size of the datastore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2 Adding Softmax Temperature to kNN Distribution Because the number of retrieved nearest neighbors, k is usually much smaller than the vocabulary size V , intuitively, the kNN distribution PkNN used for interpolation tends to be more peaky than the standard LM output distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' When k = 1024 and V = 33000, as in our experiments, PkNN will only have a few vocabulary items with a non-zero probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Furthermore, many of the retrieved neighbors share the same target token and thus make the kNN distribution even peakier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' One way to control the entropy, or peakiness of the distribution is to add temperature to the logits that go into the softmax function (Holtzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We calculate the probability of non-parametric component PkNN with the following equation where t is the softmax temperature: PkNN = Msoftmax(mask-to-k(Wds ⊗ hds)/t) (6) In general, the higher the temperature, the less “peaky” the distribution would become.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We experiment with both the 5% as well as the full datastore using different temperatures ranging from 0 to 3 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='1 intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The results are shown in Figure 5a and Figure 5b respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (a) On 5% subsampled datastore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (b) On full datastore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Figure 5: The interpolated perplexity varies with different softmax temperature values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We can see that the default temperature t = 1 does not always result in the best-interpolated perplexity and tuning softmax temperature is desirable for all sizes of datastore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The lesson learned here is that tuning the 9 real mask, real score 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='70 FAISS mask, FAISS score FAlSS mask, real score 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='65 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='60 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='55 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='50 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='0real mask, real score 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='6 FAISS mask, FAISS score FAiss mask, real score 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='0softmax temperature for the kNN distribution is crucial for getting optimal results from each setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Only coincidentally, a temperature of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='0 was close to optimal in the original settings of Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b), which hid the importance of this hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In both the 5% subsampled datastore and the full datastore scenarios, temperature t = 1 is close to optimal when using “FAISS mask, FAISS score”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' When using either “real mask” or “real score”, this is not true anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Even at the optimal temperature for each setting, “real mask, real score” somewhat underperforms “FAISS mask, real score”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' It is consistent with the counter-intuitive phenomenon discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' There are also differences between the two scenarios of different datastore sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' With the full datastore, using “real score” outperforms “FAISS score” given the same “FAISS mask”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, the opposite is true when using the 5% datastore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This suggests that as the datastore size grows, using accurate distance values are better than the approximate ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The relatively small gap between using “real score” and “FAISS score” in both datastore settings shows that the main contributor to the improvements is using approximate nearest neighbors (“FAISS mask”) rather than using approximate distance values (“FAISS score”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We hypothesize that this is related to regularization for preventing overfitting, and approximate search provides fuzziness that functions as a regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We can think of the non-parametric part in kNN-LM, the kNN component as a model, where the datastore size is its model capacity, and the datastore is its training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Considering that the kNN component uses the exact same training data as the base parametric LM, having ground truth, accurate kNN search may cause the kNN component to overfit the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Comparing the small datastore with only 5% with the original datastore, we see that a small datastore means a small training set for the kNN “model” and it thus it benefits more from this regularization, both both through using the FAISS mask and FAISS score (at optimal temperature settings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' From these experiments, we can see that, surprisingly, one of the important ingredients in kNN-LM seems to be approximate kNN search, which likely prevents overfitting to the datastore created from the same training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We further analyze this unexpected result in Appendix D, where we find that longer words and words that appear in many different contexts have slightly better results with approximate nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Notably, similar effects, where an approximation component lead to better generalization, have been reported in other NLP tasks as well, and are sometimes referred to as “beneficial search bias”, when modeling errors cause the highest-scoring solution to not be the correct one: Meister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b) suggest that “quite surprisingly, beam search often returns better results than exact inference due to beneficial search bias for NLP tasks.” Stahlberg and Byrne (2019) also conclude that “vanilla NMT in its current form requires just the right amount of beam search errors, which, from a modeling perspective, is a highly unsatisfactory conclusion indeed, as the model often prefers an empty translation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 6 Probably Wrong Hypotheses for Why kNN-LMs Work The results in the previous sections are the result of extensive analysis and experimentation, in which we also tested a number of hypotheses that did not turn out to have a significant effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Additional details of these hypotheses are detailed in Appendix E, and we hope that they may provide ideas for future improvements of retrieval-based LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Ensemble of Distance Metrics We hypothesized that the ensemble of two distance metrics: the standard inner product distance (which the LM uses) and the L2 distance (which the kNN component uses), is the key to the improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, we found that similar gains can be achieved using the inner-product metric for the retrieved kNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' More details can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Ensembling of Two Models We hypothesized that the kNN component merely provides another model for ensembling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The improvement from kNN-LM is purely due to the ensembling effect of different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, we found that kNN-LM’s improvement is orthogonal to ensembling with a different base LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' More details can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Sparsification The mask-to-k(·) used by kNN retrieval induces sparsity in the distribution over the vocab- ulary, due to a small k (typically 1024) compared to the size of the vocabulary V (33K in our experiments and 260K in the original settings of Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We hypothesized that kNN-LM increases the probability of the top-k entries while taking “probability mass” from the long tail of unlikely word types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, we could not gain any benefits solely from sparsifying the output probability of a standard LM and interpolating it with the original LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' More details can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 10 Stolen Probabilities The stolen probabilities effect (Demeter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2020) refers to the situation where the output embeddings of an LM are learned such that some words are geometrically placed inside the convex hull that is formed by other word embeddings and can thus never be “selected” as the argmax word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We hypothesized that kNN-LM solves the stolen probabilities problem by allowing to assign the highest probability to any word, given a test context that is close enough to that word’s datastore key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, we found that none of the vectors in our embedding matrix and in the original embedding matrix of Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b) is located in the convex hull of the others, which is consistent with the findings of Grivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' More details can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Memorization We hypothesized that the kNN component simply provides memorization of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, we could not improve a standard LM by interpolating its probability with another standard LM that was further trained to overfit the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' More details can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Soft Labels We hypothesized that kNN-LM’s improvement lies in reducing the “over-correction” error when training with 1-hot labels, as hypothesized by Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2022), and that retrieving neighbors is not important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' If only “soft labels” are the key, we could hypothetically improve the performance of another fresh LM with the same model architecture but trained with the soft labels from the base LM, instead of from kNN-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This separates the effect of “soft labeling” from the additional guidance provided by kNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, this does not help with the interpolated perplexity at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' More details can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Optimizing Interpolated Loss We hypothesized that the standard LM cross-entropy training loss does not emphasize the examples where base LM performs badly which could benefit from kNN, and directly optimizing the interpolated loss of standard LM and a separate trainable softmax layer could be a better alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, we could not gain any benefits by training an additional softmax layer together with a base LM using the interpolated loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' More details can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 7 Conclusion In this paper, we investigate why kNN-LM improves perplexity, even when retrieving examples from the same training data that the base LM was trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' By proposing and testing various hypotheses and performing extensive ablation studies, we find that the key to kNN-LM’s success is threefold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Ensembling different input representations – the feedforward layer output and the attention layer output – can recover 55% of the performance, even without retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' One of the most unexpected discoveries in the paper is that using approximate nearest neighbor search allows kNN-LMs to generalize better than exact nearest neighbor search, possibly due to a regularization effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Tuning the softmax temperature for the kNN distribution is crucial to adjust the standard LM output distribution with the distribution created by the retrieved neighbors’ distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We performed extensive experiments which ruled out other hypotheses as to why kNN-LMs work, such as over-parameterization, datastore clustering, sparsification, overfitting, ensembling of distance metrics, and alternative training methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We believe that this work unlocks a variety of exciting research directions for efficient kNN alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' For example, exploring methods that replace the kNN component with trainable parameters and achieve comparable results without the latency burden of kNN-LM.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' OpenAI blog, 1(8):9, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Victor Sanh, Lysandre Debut, Julien Chaumond, and Thomas Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Distilbert, a distilled version of bert: smaller, faster, cheaper and lighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' arXiv preprint arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='01108, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Rico 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' arXiv preprint arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='10090, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Rethinking the inception architecture for computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2818–2826, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Dexin Wang, Kai Fan, Boxing Chen, and Deyi Xiong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Efficient cluster-based k-nearest-neighbor machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' ArXiv, abs/2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='06175, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Zhilin Yang, Zihang Dai, Ruslan Salakhutdinov, and William W Cohen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Breaking the softmax bottleneck: A high-rank rnn language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' arXiv preprint arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='03953, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Zhixian Yang, Renliang Sun, and Xiaojun Wan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Nearest neighbor knowledge distillation for neural machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 5546–5556, Seattle, United States, July 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='18653/v1/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='naacl-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' URL https://aclanthology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='org/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='naacl-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 13 A kNN-LM Generalization to Other LMs #params Base LM PPL kNN-LM PPL Absolute PPL Gain Ours 268M 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='75 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='58 Distilled-GPT2 82M 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='25 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='41 GPT2-small 117M 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='84 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='29 GPT2-medium 345M 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='55 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='18 GPT2-large 774M 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='56 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='80 Table 4: Performance of kNN-LM applied to other pretrained language models of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To test the generalizability of kNN-LM, we follow the same experimental setup as used in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We select several pretrained models from the GPT2 family (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2019) of various parameter counts, plus a distilled version of GPT2, DistillGPT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2019) We take the pretrained model checkpoint, build the datastore and evaluate on the Wikitext-103 dataset splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The results are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We can see that kNN-LMs has good generalizability on other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' It improves the perplexity of all the base LMs tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, the larger the model is, and usually the better the base LM’s perplexity is, the less gain can be achieved from adding kNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Note that our model is trained from scratch on Wikitext-103 dataset and thus even with a relatively large model size, the perplexity and perplexity gain from adding kNN is still less than models with pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Without loss of generalizability, we will use our own trained-from-scratch model as the base LM in the following sections for ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' B Detailed Results for Increasing the Softmax Capacity hds Nds ⊗ +#params PPL Interp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Oracle 0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='750 att Big IP 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='694 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='772 ffn 2x IP 2V × D 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='889 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='646 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='701 ffn 3x IP 3V × D 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='829 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='603 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='717 ffn 4x IP 4V × D 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='769 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='629 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='876 ffn 5x IP 5V × D 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='720 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='594 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='878 ffn 6x IP 6V × D 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='726 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='599 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='902 ffn 9x IP 9V × D 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='687 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='567 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='887 Table 5: Performance comparison of kNN baselines and models with learnable embeddings as datastore alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' hds is either attention layer output (att) or feedforward layer output (ffn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' C Alternative Methods for Increasing Softmax Capacity C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='1 Adaptive Increasing Embedding Size We hypothesize that different word types have different difficulties for the language model to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' For those words that appear very frequently, they may appear in many different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' As a result, instead of adding equal number of additional embeddings to each word type, we propose to adaptively increase the number of embeddings for word types based on word frequency, or total training loss for the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Based on the intuition of Zipf’s law (Clauset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2009), we assign 1 + logb fv for each word type v ∈ V , based on 14 either the frequency or the total training loss of the word, fv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The b is a hyperparameter that could be tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To ensure fair comparison, we tune b so that for each experiment the total number of embeddings matches: � v∈V 1 + logb fv = nV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The results are shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We can see that although nice in paper, given the same number of total embeddings, adaptively increasing the number of embeddings assigned for each word type does not make a significant difference in the final perplexity, when compared with the models that use equal number of embeddings for each word type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' hds Nds ⊗ +#params PPL λ Interp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' PPL Oracle Base LM 0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='750 KNN att Big L2 Nds × D ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='271 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='174 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='230 KNN att Big IP Nds × D ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='266 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='095 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='077 Equal Per Word att 3x IP 3V × D 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='434 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='417 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='395 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='132 Loss Weighted att 3x IP 3V × D 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='948 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='437 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='440 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='303 Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Weighted att 3x IP 3V × D 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='507 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='412 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='387 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='105 KNN ffn Big L2 Nds × D ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='065 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='734 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='594 KNN ffn Big IP Nds × D ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='050 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='101 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='254 Equal Per Word ffn 3x IP 3V × D 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='622 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='603 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='717 Loss Weighted ffn 3x IP 3V × D 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='713 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='659 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='978 Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Weighted ffn 3x IP 3V × D 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='658 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='572 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='782 Table 6: Performance comparison of kNN baselines and several configurations that adaptively increase the embedding size with training loss or word frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2 Mixture of Softmaxes Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2017) proposes a solution to the problem using a Mixture of Softmax (MoS) to produce more linearly independent probability distributions of words given different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Suppose that there are a total of R mixture components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' MoS first uses R linear layers with weight wr to transform the current query context vector hds into wrhds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' With a shared word embedding matrix Wsm, we can calculate each softmax component’s probability distribution with softmax(Wsm · wrhds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The mixture distribution is then given by: PMoS = R � r πr,hdssoftmax(Wsm · wrhds) (7) The prior weights are calculated using another linear layer with weight wπ, as πr,hds = softmax(wπhds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The softmax ensures that �R r πr,hds = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Comparing the MoS with the first term in Equation 5, Msoftmax(mask-to-k(Wds ⊗ hds)), we can see that there are some connections between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' MoS eliminates the mask-to-k(·) operation, and replaces the single softmax across a very large vector (size of datastore), into multiple smaller softmaxes, each across only a vector of the size of vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' As a result, the huge Wds is replaced by several linear layers to project the word embedding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Now the first term becomes: M(⊕R r softmax(Wsm · wrhds)) (8) Mir = πr,hds, ∀i ≤ V (9) where ⊕ represents the vector concatenation operation, and the aggregation matrix M now contains the mixture weights for each softmax being concatenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We perform experiments with a varying number of mixtures (R), different definitions hds, and whether to fine-tune the output word embeddings Wsm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We allow fine-tuning the word embedding when we use attention layer output as context vector, since the word embedding matrix is trained with feedforward layer output originally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The results for this formulation are shown in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' MoS models on its own increase the performance of the language model marginally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' When compared with Table 5, we find that these models are worse than those that simply increases the number of embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This is expected because MoS has fewer added parameters compared to those, as it only requires several additional linear projection layers for the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='3 Clustering Datastore Opposite to training the word embeddings of an increased size, we also consider ways to compress the datastore down to a similar-sized embedding matrix for softmax computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The intuition is that the datastore contains 15 hds R ⊗ +#params PPL λ Interp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' PPL Oracle Base LM 0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='750 KNN att L2 Nds × D ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='271 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='174 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='230 KNN att IP Nds × D ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='266 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='095 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='077 KNN ffn L2 Nds × D ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='065 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='734 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='594 KNN ffn IP Nds × D ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='050 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='101 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='254 Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' MoS+embed att 2 IP V D + 2D2 + 2D 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='437 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='720 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='573 Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' MoS+embed att 3 IP V D + 3D2 + 3D 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='422 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='779 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='609 Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' MoS Only att 2 IP 2D2 + 2D 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='371 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='011 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='796 Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' MoS Only att 3 IP 3D2 + 3D 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='573 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='371 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='024 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='812 Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' MoS Only ffn 2 IP 2D2 + 2D 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='843 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='338 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='258 Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' MoS Only ffn 3 IP 3D2 + 3D 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='733 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='460 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='322 Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' MoS Only ffn 4 IP 4D2 + 4D 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='994 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='321 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='396 Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' MoS Only ffn 5 IP 5D2 + 5D 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='371 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='909 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='367 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='406 Table 7: Performance comparison of kNN baselines and several MoS configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' R is the number of mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' redundant context vectors, and thus compression could make the datastore smaller without sacrificing too much performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2021) shows that we can safely compress the datastore by clustering to 50% of the original size without losing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We test this idea further by clustering the entire datastore into a size that could fit in GPU memory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 2V , 3V ) and thus could be easily fine-tuned further and use the resulting centroids to replace Wds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Within each cluster, there will be a distribution of different words with contexts, and we use the frequency of words within each cluster to calculate the aggregation matrix M in Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This would have the added benefit of “multi-sense” embedding, which allows similar meanings to be clustered to form a new “meta word” while the same word with different meanings would form different “meta words”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' A notable example is bank, shore, and financial institution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, this does not work, mostly because of the high compression loss after clustering and the imbalanced distribution of word types among each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' D Which Words Benefit from Approximation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To further understand the unexpected results when using the different kNN approximate retrieval settings in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='1 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2, we analyze on a token level, based on how many times each ground truth token’s probability in the evaluation set are helped by each kNN setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' It means that for each ground truth token in the evaluation, we count the times when the kNN distribution is higher than the base LM distribution PLM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', PkNN > PLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Since we found previously that approximate kNN provides an additional performance boost compared to ground truth kNN, we thus compare “real mask, real score” versus “FAISS mask, real score” in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To prevent outliers, we filter out words with less than 10 occurrences in the evaluation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' For each setting, we calculate the percentage of occurrences in the evaluation set where each token in the vocabulary where the kNN module achieves a better probability than base LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We then plot the absolute difference between the percentages of the two settings, with respect to various possible attributes of the token that achieves better probability using each setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Figure 6 shows that the longer the token is, which usually suggests proper nouns and harder and less common words in English, are better with approximate neighbors than ground truth ones, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We hypothesize that this is due to longer words are more prone to overfitting in kNN-LM and thus using approximate kNN provides an effect similar to smoothing and regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We also compare words that could appear in more diverse contexts with words that co-occur with few distinct contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To measure how diverse the contexts of each word in the vocabulary is, we calculate both the forward and backward bigram entropy for each word in the evaluation set that has more than 10 occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The bigram entropy is a simple yet good indicator of context diversity for a given word, as used in Kneser–Ney smoothing (Ney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We calculate both the forward and backward bigram entropy for each word w as 16 Figure 6: The effect of the token character length on how much accurate nearest neighbors are better than approximate FAISS neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Negative values mean worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The trend line of the scatter points is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' follows, where wafter and wbefore represent the word after and before the given word w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Hforward(w) = − � wafter p(wafter|w) log p(wafter|w) (10) Hbackward(w) = − � wbefore p(wbefore|w) log p(wbefore|w) (11) Forward and backward entropy represents how diverse the context after and before the given word is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Intuitively, bigram entropy is supposed to indicate words that can appear in lots of different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The higher the entropy of a word, the more diverse its context is, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' For example, words like “Francisco” would have a low entropy because it mostly comes after “San”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Figure 7: The effect of the forward and backward entropy of words on how accurate nearest neighbors are better than approximate FAISS neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Negative values mean worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The trend line of the scatter points are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The comparison is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We can see that the higher the entropy in both forward and backward cases, the better using approximate nearest neighbor search becomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This suggests that words that appear in many different contexts are better off with an approximate kNN, and “easy-to-predict” examples such as “Jersey” and “Fransisco” is better with accurate kNN, possibly because these examples are less prone to overfitting errors and thus requires less regularization from approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 17 E Failed Hypotheses E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='1 Distance Metric We hypothesize that the key to kNN-LM’s performance gain is the ensemble of two distance metrics: the standard dot product distance (which the LM uses) with the L2 distance (which the kNN component uses as ⊗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We tried to replace the kNN component with a component that just takes the tokens retrieved by the kNN search and returns their L2 distance to the LM output word embeddings: Wsm ⊗ hds instead of Wds ⊗ hds, where ⊗ represents the negative L2 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We tried this with both variants of hds, attention layer output, and feedforward layer output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' None of these helped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2 Sparsification In Equation 5, mask-to-k(·) used by kNN retrieval induces sparsity in the distribution over the vocabulary, due to a small k compared to the number of vocabulary V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We hypothesize that the in kNN-LM, the kNN distribution is sparse, practically increasing the probability of the top-k entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The kNN distribution has up to 1024 entries that are non-zero, concentrating more probability mass over the most likely tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This effect is similar to the redistribution of probability mass for text generation in Holtzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We test this hypothesis only by taking top 32, 64, 128, 512, or 1024 tokens in the parametric LM probability and zeroing out the probabilities of the rest of the tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To compensate, we experiment with different softmax temperatures and then interpolate with the parametric LM probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This isolates the effect of the datastore and retrieval at all, and this does not help at all, suggesting that sparsification of the output probability alone is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Another attempt is to hypothesize that the key in kNN-LM is that it selects “which tokens to include” in the kNN distribution, and not their distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The intuition behind is that maybe the selection of the top tokens according to the kNN search is better than that from the dot-product distance between the language model’s output vector and all the vocabulary embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We perform experiments similar to the previous attempt, sparsifying the output probability with the tokens retrieved by the kNN search (but ignoring the distances provided by the kNN search) rather than the top k tokens of the LM, with and without removing duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In the best case, they manage to reduce the perplexity by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='5 (whereas kNN-LM reduces by nearly 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='3 Location within Context Window Supposedly, words in the beginning of the “context window” of the transformer at test time have less contextual information than words toward the end of context window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We hypothesized that maybe the base LM performs worse in one of these (beginning vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' end of the context window), and maybe kNN-LM provides a higher improvement in one of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We measured the per-token test perplexity with respect to the location of each token in the context window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, we did not find any significant correlation between the performance of the base LM and the location, and no significant correlation between the difference between kNN-LM and the base LM and the location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We also hypothesized that maybe the beginning of every Wikipedia article is more “predictable”, and the text becomes more difficult to predict as the article goes into details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, we also did not find any correlation with the location of the word within the document it appears in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='4 Stolen Probabilities The stolen probabilities effect (Demeter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2020) refers to the situation where the output embeddings of an LM are learned such that some words are geometrically placed inside the convex hull that is formed by other word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Since language models generate a score for every output word by computing the dot product of a hidden state with all word embeddings, Demeter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020) prove that in such a case, it is impossible for words inside the convex hull to be predicted as the LM’s most probable word (the “argmax”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We hypothesized that kNN-LM solves the stolen probabilities problem by allowing to assign the highest probability to any word, given a test hidden state that is close enough to that word’s datastore key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Nevertheless, as shown by Grivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2022), although this problem might happen in small RNN-based language models, in modern transformers it rarely happens in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Using the code of Grivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2022), we checked the embeddings matrix of our model and of the checkpoint provided by Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Indeed, we found that in both models – no word is un-argmaxable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 18 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='5 Are kNN-LM Just Ensembling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Our hypothesis is that kNN component only provides another model for ensembling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The interpolation process is basically an ensemble model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Technically it is unsurprising that kNN-LM will have the benefit from ensembling, but we perform experiments to see how it compares to other ensembling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We trained another language model with the same architecture as the base LM we used throughout the experiments, with some variants having more than one embedding vector for each word (similar to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We interpolate the models with the original base LM, and the results are shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We can see that even just ensembling the base LM with another identical model, but trained with a different random seed, provides a huge performance boost, both on interpreted perplexity and on oracle perplexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Layers hds Nds ⊗ +#params PPL Interp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Oracle same 0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='750 same att Big L2 Nds × D ∞ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='174 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='230 same att Big IP Nds × D ∞ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='095 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='077 same ffn Big L2 Nds × D ∞ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='734 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='594 same ffn Big IP Nds × D ∞ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='101 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='254 diff ffn 1x IP F + V × D 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='569 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='941 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='980 diff ffn 2x IP F + 2V × D 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='914 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='948 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='885 diff ffn 3x IP F + 3V × D 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='206 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='981 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='853 Table 8: Performance comparison of kNN baselines and models with different size output embeddings re-trained from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, just because ensembling two LMs of the same architecture provides better performance than interpolating the base LM with kNN does not necessarily suggest that kNN’s performance improvement can be fully replaced by model ensembling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In other words, we are interested in whether the kNN performance improvements are orthogonal to that of model ensembling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To test this, we compare the performance of the ensemble of K multiple LMs versus the ensemble of K − 1 multiple LMs plus the kNN component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The comparison is fair because we have the same number of models in the ensemble, and the only difference is whether the kNN component is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The results are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' For the “LM” series, each point is K LMs ensemble, and for the “kNN” series, each point is K − 1 LMs plus kNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We can see that even at 4-ensemble, the ensemble that contain kNN as a component still have a considerable edge over the 4-ensemble that contain just LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Ensemble Components 16 18 20 22 1 2 3 4 LM KNN LM and KNN Figure 8: Ensembling effect comparison, between multiple base LMs and multiple base LMs plus kNN component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='6 Are kNN-LM Just Alternative Training Methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='1 Overfitting Since kNN-LM improves perplexity even with the same training dataset as datastore, we are curious if kNN-LM works by only “memorizing” the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The hypothesis is that the datastore and the kNN 19 Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Layers hds Nds ⊗ +#params PPL Interp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Oracle Base LM same 0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='750 KNN same att Big L2 Nds × D ∞ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='174 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='230 KNN same att Big IP Nds × D ∞ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='095 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='077 KNN same ffn Big L2 Nds × D ∞ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='734 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='594 KNN same ffn Big IP Nds × D ∞ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='101 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='254 Overfit@92 diff ffn V IP F + V × D 1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='806 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='732 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='764 Overfit@129 diff ffn V IP F + V × D 8966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='508 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='733 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='814 Table 9: Performance comparison of several baselines with two overfitted models, at 92 and 129 additional epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' search are trying to memorize the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' In other words, the parametric LM is under-fitting some tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The intuition behind this is that the kNN component retrieves examples directly from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' What if we could retrieve the same examples using an overfitted LM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We took the trained LM, removed the dropout, and continued training until almost perfect fit (very small training loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We then interpolated the overfitted transformer with the original LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' The results are shown in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' F represents the number of parameters in the base LM, minus the output embedding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We can see that overfitting can provide very little help after interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Looking at the oracle performance, we think that the overfitted model memorizes some rare contexts and tokens in the training set where it could be useful during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, the overfitting hurts the performance on other tokens too much so that even interpolation is not able to balance the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2 Soft-Label Training Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' (2022) claims that using “soft labels” during training is the key to kNN’s success, that interpolates the ground truth labels with kNN-LM model outputs, effectively “distilling” kNN-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' It is based on the hypothesis that the room for kNN-LM’s improvement over base LM lies in the “over-correction” when training with a 1-hot labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This is related to the effect from label smoothing methods (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Pereyra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' Meister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, we believe that this explanation is not satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' If the key is training with soft-labels, why do these soft labels must be provided specifically by a kNN search?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' If soft labels were the key, then soft-label training where the labels come from the base LM itself should have worked as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To separate the effect of soft labeling from the kNN’s additional guidance, we train another LM with the same model architecture as the base LM, with the soft labels from the base LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This teacher-student training is to distill the knowledge from the base LM (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We find that by just training with “soft labels“ from the base LM to alleviate the alleged “over-correction” problem is not the key, as this does not help with the interpolated perplexity at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This suggests that even with the same training data, kNN still provides valuable additional guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='3 Training to Optimize Interpolated Loss In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='2, we discover that using over-parameterization with standard LM training loss does not further close the gap towards kNN-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This suggests that some regularization term may be needed during training to make the multiple embeddings not converge to the same vector, rendering over-parameterization useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' From Table 2, we see that a better interpolated perplexity may not require a very low perplexity when measured only with the extra input representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, we still use a standard LM loss to only train the additional embedding matrix, that directly minimizes the perplexity using only the extra input representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This discrepancy between training and the evaluation with interpolation suggests that training with an alternative loss function that interpolates the base LM’s output with the output using the extra input representation may be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' To test the hypothesis that standard LM training loss do not emphasize the examples where base LM performs badly, we train the extra model’s parameter Wds, with interpolated loss L: L = CrossEntropy(λsoftmax(Wds · hds) + (1 − λ)softmax(Wsm · hsm), y) (12) y represents the ground truth label for each context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We only learn the parameter Wds while freezing all other parameters, similar to all other experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' We choose λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content='25 as it is the best hyper-parameter for kNN-LM experiments and our goal for this training is to mimic the loss of kNN-LM after interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This training loss effectively assigns a higher value to the training examples where the base LM’s loss is high, 20 suggesting the need for the extra Wds to help with these hard cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' However, for either “att” for “ffn” for hds, either V or 3V for the number of embeddings in Wds, we are unable to achieve a better perplexity than just the base LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' This suggests that, while nice on paper, the interpolated loss optimization process is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQf_gKx/content/2301.02828v1.pdf'} diff --git a/2tAzT4oBgHgl3EQfuP3I/vector_store/index.faiss b/2tAzT4oBgHgl3EQfuP3I/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..297efe6f2be366c6c41df2f62491cabb95fda213 --- /dev/null +++ b/2tAzT4oBgHgl3EQfuP3I/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0ccb7692db76875ab91c60805d7901d1e416fec3a569290f6f7e5c5eff26d65a +size 5505069 diff --git a/39FKT4oBgHgl3EQf8y5R/content/tmp_files/2301.11951v1.pdf.txt b/39FKT4oBgHgl3EQf8y5R/content/tmp_files/2301.11951v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..586dcaab261938999b2bead060a7c4042744bdea --- /dev/null +++ b/39FKT4oBgHgl3EQf8y5R/content/tmp_files/2301.11951v1.pdf.txt @@ -0,0 +1,3146 @@ +arXiv:2301.11951v1 [cond-mat.str-el] 27 Jan 2023 +Journal of Superconductivity and Novel Magnetism manuscript No. +(will be inserted by the editor) +Topological structures in unconventional scenario for 2D +cuprates +A.S. Moskvin · Yu.D. Panov +Received: date / Accepted: date +Abstract Numerous +experimental +data +point +to +cuprates as d-d +charge transfer unstable systems +whose description implies the inclusion of the three +many-electron valence states CuO7−,6−,5− +4 +(nominally +Cu1+,2+,3+) on an equal footing as a well-defined charge +triplet. We introduce a minimal model to describe the +charge degree of freedom in cuprates with the on-site +Hilbert space reduced to only the three states and make +use of the S=1 pseudospin formalism. The formalism +constitutes a powerful method to study complex phe- +nomena in interacting quantum systems characterized +by the coexistence and competition of various ordered +states. Overall, such a framework provides a simple and +systematic methodology to predict and discover new +kinds of orders. In particular, the pseudospin formal- +ism provides the most effective way to describe different +topological structures, in particular, due to a possibil- +ity of a geometrical two-vector description of the on-site +states. We introduce and analyze effective pseudospin +Hamiltonian with on-site and inter-site charge corre- +lations, two types of a correlated one-particle trans- +fer and two-particle, or the composite boson transfer. +The latter is of a principal importance for the HTSC +perspectives. The 2D S=1 pseudospin system is prone +to a creation of different topological structures, which +form topologically protected inhomogeneous distribu- +tions of the eight local S=1 pseudospin order parame- +ters. We present a short overview of localized topologi- +cal structures, typical for S=1 (pseudo)spin systems, fo- +cusing on unexpected antiphase domain walls in parent +cuprates and so-called quadrupole skyrmion, which are +believed to be candidates for a topological charge ex- +A.S. Moskvin · Yu.D. Panov +Ural Federal University, Ekaterinburg, 620083, Russia +E-mail: alexander.moskvin@urfu.ru +citation in parent or underdoped cuprates. Puzzlingly, +these unconventional structures can be characterized by +an uniform distribution of the mean on-site charge, that +makes these invisible for X-rays. Quasiclassical approx- +imation and computer simulation are applied to ana- +lyze localized topological defects and evolution of the +domain structures in ”negative-U” model under charge +order-superfluid phase transition. +Keywords high-Tc cuprates · charge degree of +freedom · S=1 pseudospin formalism · topological +structures · unconventional skyrmions +1 Introduction +The origin of high-Tc superconductivity [1] is presently +still a matter of great controversy. Both copper and +novel non-copper based layered high-Tc materials re- +veal normal and superconducting state properties very +different from that of standard electron-phonon coupled +”conventional” +superconductors. +Copper oxides start out life as insulators in con- +trast with BCS superconductors being conventional +metals. Unconventional behavior of these materials un- +der charge doping, in particular, a remarkable interplay +of charge, lattice, orbital, and spin degrees of freedom, +strongly differs from that of ordinary metals and merely +resembles that of a doped Mott insulator. In addition to +the occurrence of unconventional d-wave superconduc- +tivity the phase diagram of the high-Tc cuprates does +reveal a flurry of various anomalous electronic prop- +erties. In normal state, these materials exhibit non- +Fermi liquid properties and enter a mysterious pseu- +dogap (PG) regime, characterized by the observation +of multiple crossover PG temperatures T∗’s. + +2 +A.S. Moskvin, Yu.D. Panov +The exotic superconductors differ from ordinary +Bardeen-Cooper-Schrieffer (BCS) superconductors in +many other points. Thus, muon spin relaxation (µSR) +measurements of the magnetic field penetration depth +revealed nearly linear relationship between Tc and the +superfluid density in high-Tc cuprates and many other +exotic superconductors that cannot be expected in BCS +theory, but is typical for Bose-Einstein condensation +(BEC) of preformed pairs[2]. Bosonic scenario for high- +Tc cuprates [3] has been elaborated by many authors, in +particular, by Alexandrov (see, e.g., Ref. [4]) who con- +sidered real-space bipolarons. It is worth noting that +many reasonable predictions of the bipolaronic theory +are valid for any local bosons irrespective of its micro- +scopic mechanism. Numerous observations point to the +possibility that high-Tc cuprate superconductors may +not be conventional BCS or BEC superconductors, but +rather manifest a boson-fermion competition in a strug- +gle for the electronic ground state, in particular, a com- +petition between the two- and one-particle transport +with resistivity ∝T and ∝T2, respectively. +The most part of the current scenarios, includ- +ing the Hubbard and t − J-models, spin fluctuations, +Alexandrov-Mott bipolarons [4] consider cuprates to be +homogeneous systems and ignore numerous signatures +of the electron and crystalline inhomogeneity [5]. +Both the normal and high-transition-temperature +(high-Tc) superconducting (SC) state in cuprates is be- +lieved to be electronically inhomogeneous, in particu- +lar, due to a quenched disorder, arising from dopants +and/or nonisovalent substitution. However, the dopant- +induced impurity potential, seemingly being a natural +source of electron inhomogeneity, varies widely among +the cuprates that cannot explain observation of an uni- +versal, scaling behavior evidencing for an intrinsic elec- +tronic tendency toward inhomogeneity in CuO2 planes. +This intrinsic propensity can be stimulated, firstly, by +a local out-of-plane nonisovalent substitution, toward +formation of in-plane universal inhomogeneity centers. +Another stimulating factor of the intrinsic electronic in- +homogeneity is related with a two-dimensionality and +a competition and intertwinnig of charge, spin and or- +bital degrees of freedom in CuO2 planes. Concept of +phase separation and percolation phenomena [6,7,8,9, +10], stripes [11,12,13], large polarons [14,15], nucleation +of the mixed valence PJT-(pseudo-Jahn-Teller) phase +[16] has appeared to be very fruitful for explanation +of many puzzling properties of cuprates. Furthermore, +some authors [17,18,19] associate the anomalous prop- +erties of cuprates with quasi-2D structure of the active +CuO2 layers and different topological defects, or vortex- +like solitons to be specific collective excitation modes of +the 2D vector fields. +The topological order inherent in the doped cuprate +endows it with tremendous amount of robustness to var- +ious unavoidable ”real-life” material complications [20], +such as impurities and other coexisting broken sym- +metries. In general, such a complex, multiscale phase +separation does challenge theories of high-temperature +superconductivity that include complexity [21]. +Recently [22] we argued that an unique prop- +erty of high-Tc cuprates is related with a dual na- +ture of the Mott insulating +state of the parent +compounds that manifests itself in two distinct en- +ergy scales for the charge transfer (CT) reaction: +Cu2+ + Cu2+ → Cu1+ + Cu3+. Indeed, the d - d CT gap +as derived from the optical measurements in parent +cuprates such as La2CuO4 is 1.5-2.0 eV while the true +(thermal) d - d CT gap, or effective correlation param- +eter Ud, appears to be as small as 0.4-0.5 eV. It means +cuprates should be addressed to be d-d CT unstable +systems whose description implies accounting of the +three many-electron valence states CuO7−,6−,5− +4 +(nom- +inally Cu1+,2+,3+) on an equal footing as a well-defined +charge triplet. This allows us to introduce a minimal +model for cuprates with the on-site Hilbert space re- +duced to only three states, three effective valence cen- +ters CuO7−,6−,5− +4 +(Cu1+,2+,3+) where the electronic +and lattice degrees of freedom get strongly locked to- +gether, and make use of the S=1 pseudospin formal- +ism [22,23,24,25,26,27]. Such a formalism constitutes +a powerful method to study complex phenomena in in- +teracting quantum systems characterized by the coex- +istence and competition of various ordered states [28]. +Overall, such a framework provides a simple and sys- +tematic methodology to predict and discover new kinds +of orders. In particular, the pseudospin formalism pro- +vides the most effective way to describe different topo- +logical structures. +The paper is organized as follows. In Sec. 1 we in- +troduce a working model for the CuO4 centers based +on assumption that the three many-electron valence +states CuO7−,6−,5− +4 +(nominally Cu1+,2+,3+) form the +“on-site” Hilbert space of the CuO4 plaquettes. We +restricted ourselves only by the consideration of the +charge degree of freedom and have suggested simple +geometrical vector representation for the on-site charge +states. In Sec. 2 we have addressed an effectuve pseu- +dospin Hamiltonian for the model cuprate. In Sec. 3 we +have considered several simplified versions of the gen- +eral Hamiltonian. Sec. 4 is devoted to description of un- +conventional localized topological structures typical for +2D S=1 (pseudo)spin systems. In Sec. 5 we considered +localized topological structures in a limiting case of the +model, or so-called ”negative-U” model. A brief sum- +mary is given in Sec. 6. + +Topological structures in unconventional scenario for 2D cuprates +3 +2 Working model of the CuO4 centers +Hereafter we consider the CuO4 plaquette to be a main +element of crystal and electron structure of high-Tc +cuprates and introduce a simplified toy model with the +“on-site” Hilbert space of the CuO4 plaquettes reduced +to states formed by only three effective valence centers +[CuO4]7−,6−,5− (nominally Cu1+,2+,3+, respectively). +The centers are characterized by different conventional +spin: s=1/2 for Cu2+ center and s=0 for Cu1+,3+ cen- +ters, and different orbital symmetry:B1g for the ground +states of the Cu2+ center, A1g for the Cu1+ centers, +and the Zhang-Rice (ZR) A1g or more complicated low- +lying non-Zhang-Rice states for the Cu3+ center. Elec- +trons of such configurations cannot be treated through +a mean-field independent particle approach; therefore, +their behavior is studied in terms of auxiliary neither +Fermi nor Bose quasiparticles, representing combina- +tions of atomic-like many-electron configurations[29]. +The key problem that arises from the strong corre- +lations in the normal state of the copper-oxide super- +conductors is identifying the weakly interacting entities +that make a particle interpretation of the current possi- +ble. All formulations of superconductivity are reduced +to a pairing instability of such well-defined quasiparti- +cles. However, there is good reason to believe that the +construction of such entities may not be possible [30]. +|Ψ⟩ = c−1|Cu1+⟩ + c0|Cu2+⟩ + c1|Cu3+⟩, +(1) +Such an approach immediately implies introduction +of the unconventional on-site quantum superpositions +that points to many novel effects related with local +CuO4 centers. Validity of such a model implies well iso- +lated ground states of the three centers. This surely +holds for the 1A1g singlet ground state of the Cu1+ +centers with nominally filled 3d shell whose excita- +tion energy does usually exceed 2 eV (see, e.g., Ref. [31] +and references therein). The b1g ∝ dx2−y2 character of +the ground hole state in CuO6− +4 +cluster (Cu2+ center) +seems to be one of a few indisputable points in cuprate +physics. A set of low-lying excited states with the en- +ergy ≥ 1.5 eV includes bonding molecular orbitals with +a1g ∝ dz2, b2g ∝ dxy, and eg ∝ dyz, dxz symmetry, as +well as purely oxygen nonbonding orbitals with a2g(π) +and eu(π) symmetry (see, e.g., Refs. [32,33]). +In 1988 Zhang and Rice [34] have proposed that the +doped hole in a parent cuprate forms a Cu3+ center +with a well isolated local spin and orbital 1A1g singlet +ground state which involves a phase coherent combi- +nation of the 2pσ orbitals of the four nearest neighbor +oxygens with the same b1g symmetry as for a bare Cu +3dx2−y2 hole. The Zhang-Rice (ZR) singlet is a leading +paradigm in modern theories of high-temperature su- +perconductivity. However, both numerous experimental +data and the cluster model calculations suggest the in- +volvement of some other physics which introduces low- +lying states into the excitation of the doped-hole state, +or competition of conventional ZR singlet with another +electron removal state(s), in particular, formed by the +hole occupation of the oxygen nonbonding a2g(π) and +eu(π) orbitals[32,33,35,36,37], the a2g(π) orbital to be +the lowest in energy. +Unified selfconsistent description of the charge, spin, +and orbital degrees of freedom for CuO4 centers with +mixed valence is a hardly solvable task so we are forced +to address simplified model approaches focusing on the +quantum description of the charge degree of freedom +that is responsible for superconductivity in cuprates. +2.1 The charge triplet model: S=1 pseudospin +formalism +To describe the diagonal and off-diagonal, or quantum +local charge order we start with a simplified charge +triplet model that implies a full neglect of spin and or- +bital degrees of freedom [26]. Three charge states of the +CuO4 plaquette: a bare center M 0=CuO6− +4 , a hole cen- +ter M +=CuO5− +4 , and an electron center M −=CuO7− +4 +are assigned to three components of the S=1 pseu- +dospin (isospin) triplet with the pseudospin projections +MS = 0, +1, −1, respectively. Obviously, the model +resembles that of so-called semi-hard-core bosons [38], +which are described by extended Bose-Hubbard model +that assumes a truncation of the on-site Hilbert space +to the three lowest occupation states n = 0, 1, 2 with +further mapping to an anisotropic spin-1 model (see, +e.g., Refs.[39,40,41]). For 2D cuprates these states cor- +respond to a ”electron” CuO7− +4 +(Cu1+), ”bare” CuO6− +4 +(Cu2+), and ”hole” CuO5− +4 +(Cu3+) centers, respec- +tively. +The S=1 (pseudo)spin algebra includes eight inde- +pendent nontrivial pseudospin operators, three dipole +and five quadrupole operators: +ˆSz; +ˆS± = ∓ 1 +√ +2 +(Sx ± iSy); +(2) +ˆS2 +z; +ˆT± = {Sz, S±}; +ˆS2 +± = 1 +2( ˆS2 +x − ˆS2 +y ± i{ ˆSx, ˆSy}). +One should note a principal difference between the +s=1/2 and S=1 quantum systems. The only on-site or- +der parameter in the former case is an average spin +moment ⟨Sx,y,z⟩, whereas in the latter one has five ad- +ditional ”spin-quadrupole”, or spin-nematic order pa- +rameters described by traceless symmetric tensors +Qij = ⟨1 +2{Si, Sj} − 2 +3δij⟩. +(3) + +4 +A.S. Moskvin, Yu.D. Panov +Interestingly, that in a sense, the S = 1 +2 quantum spin +system is closer to a classic one (S → ∞) with all the +order parameters defined by a simple on-site vectorial +order parameter ⟨S⟩ than the S=1 quantum spin system +with its eight independent on-site order parameters. +It is worth noting that the three spin-linear (dipole) +operators ˆSx,y,z and five independent spin-quadrupole +operators Qij = 1 +2{ ˆSi, ˆSj} − 1 +3 ˆS2δij at S=1 form eight +Gell-Mann operators being the generators of the SU(3) +group [42]. +To describe different types of pseudospin ordering in +a mixed-valence system we have to introduce eight local +(on-site) order parameters: two classical (diagonal) or- +der parameters: ⟨Sz⟩ being a ”valence”, or charge den- +sity with an electro-neutrality constraint, and ⟨S2 +z⟩ be- +ing the density of polar centers M ±, or ”ionicity”, and +six off-diagonal order parameters. The off-diagonal or- +der parameters describe different types of the valence +mixing. +It +should +be +emphasized +that +for +the +S=1 +(pseudo)spin algebra there are two operators: S± and +T± = {Sz, S±} that change the pseudo-spin projection +by ±1, with slightly different properties +⟨0| ˆS±| ∓ 1⟩ = ⟨±1| ˆS±|0⟩ = ∓1, +(4) +but +⟨0| ˆT±| ∓ 1⟩ = −⟨±1|( ˆT±|0⟩ = +1. +(5) +It is worth noting the similar behavior of the both op- +erators under the hermitian conjugation: ˆS† +± = − ˆS∓; +ˆT † +± = − ˆT∓. +The ˆS2 +± operator changes the pseudospin projection +by ±2 with the local order parameter +⟨S2 +±⟩ = 1 +2(⟨S2 +x − S2 +y⟩ ± i⟨{Sx, Sy}⟩) = +(6) += c∗ ++c− = c2 +x − c2 +y ± 2icxcy. +Obviously, this on-site off-diagonal order parameter is +nonzero only when both c+ and c− are nonzero, or for +the on-site ”electron-hole” M −(Cu1+)-M +(Cu3+) su- +perpositions. It is worth noting that the ˆS2 ++ ( ˆS2 +−) opera- +tor creates an on-site hole (electron) pair, or composite +boson, with a kinematic constraint ( ˆS2 +±)2 = 0, that un- +derlines its ”hard-core” nature. +Both ˆS+( ˆS−) and ˆT+( ˆT−) can be associated with +the single particle creation (annihilation) operators, +however, these are not standard fermionic ones, as well +as ˆS2 ++( ˆS2 +−) operators are not standard bosonic ones. +Nevertheless, namely ⟨S2 +±⟩ can be addressed as a local +superconducting order parameter +The two operators, S± and T± are related with the +two different types of a correlated single-particle trans- +port, these change the pseudospin projection by ±1. In +lieu of these operators one may use two novel operators: +ˆP± = 1 +2( ˆS± + ˆT±); ˆN± = 1 +2( ˆS± − ˆT±) , which do real- +ize transformations Cu2+↔Cu3+ and Cu1+↔Cu2+, re- +spectively. In other words, for parent cuprates these are +the hole and electron creation operators, respectively. +The boson-like pseudospin raising/lowering operators +ˆS2 +± do change the pseudo-spin projection by ±2 and +define a local nematic order parameter +⟨S2 +±⟩ = 1 +2(⟨S2 +x − S2 +y⟩ ± i⟨{Sx, Sy}⟩). +(7) +This on-site off-diagonal order parameter with the +d-type symmetry is nonzero only for the on-site +M −(Cu1+)-M +(Cu3+) superpositions. It is worth not- +ing that the +ˆS2 ++ ( ˆS2 +−) operator creates an on-site +hole (electron) pair, or composite boson, with a kine- +matic constraint ( ˆS2 +±)2 = 0, that underlines its ”hard- +core” nature. Obviously, the pseudospin nematic aver- +age ⟨S2 +±⟩ can be addressed to be a local complex super- +conducting order parameter: +⟨S2 +±⟩ = |⟨S2 +±⟩|e±iϕ. +(8) +Both ˆS+( ˆS−) and ˆT+( ˆT−) can be anyhow related with +conventional single particle creation (annihilation) op- +erators, however, these are not standard fermionic ones, +as well as ˆS2 ++( ˆS2 +−) operators are not standard bosonic +ones. +It should be noted again that the pseudospin op- +erators are not to be confused with real physical spin +operators; they act in a pseudo-space. +2.2 Simple ”geometrical” representation of the on-site +charge states +Making use of a simple classical representation of on- +site spin states using arrows is a popular and useful +method for describing spin structures through vector +fields. However, such an approach works only for classi- +cal spins and, under certain limitations, for spin s=1/2. +Indeed, for the classical spin all the on-site spin order +parameters are derived through ⟨S⟩, while for s=1/2 +⟨S⟩ is the only local spin order parameter. At variance +with s=1/2 systems for S=1 systems we have addi- +tional spin-quadrupole order parameters whose descrip- +tion cannot be realized within framework of a classical +”single-arrow” representation. Nevertheless, hereafter +we propose a novel ”geometrical” representation that +allows us to selfconsistently describe all the on-site S=1 +states and make use of the 2D vector fields to describe +uniform and nonuniform configurations for model 2D +cuprate. In particular, the vector field patterns are of a +great importance for physically clear representation of +the complex topological structures. + +Topological structures in unconventional scenario for 2D cuprates +5 +Instead of the three |1M⟩ states one may use the +Cartesian basis set Ψ, or |x, y, z⟩: +|10⟩ = |z⟩ , |1 ±1⟩ = ∓ 1 +√ +2(|x⟩ ± i|y⟩), +(9) +so that the on-site wave function can be written in the +matrix form as follows [42]: +ψ = + + +c1 +c2 +c3 + + = + + +R1 exp(iΦ1) +R2 exp(iΦ2) +R3 exp(iΦ3) + + ; +|R|2 = 1 , +(10) +with R = {sin Θ cos η, sin Θ sin η, cos Θ}. Obviously, +the minimal number of dynamic variables describing +an isolated on-site S=1 (pseudo)spin center equals to +four, however, for a more general situation, when the +(pseudo)spin system represents only the part of the big- +ger system, and we are forced to consider the coupling +with the additional degrees of freedom, one should con- +sider all the five non-trivial parameters. +The pseudospin matrix has a very simple form +within the |x, y, z⟩ basis set: +⟨i| ˆSk|j⟩ = iǫikj. +(11) +We start by introducing the following set of S=1 +coherent states characterized by vectors a and b satis- +fying the normalization constraint[42] +|c⟩ = |a, b⟩ = c · Ψ = (a + ib) · Ψ, +(12) +where a and b are real vectors that are arbitrarily ori- +ented with respect to some fixed coordinate system in +the pseudospin space with orthonormal basis e1,2,3. +The two vectors are related by the normalization +condition, so the minimal number of dynamic variables +describing the S=1 (pseudo)spin system appears to be +equal to four. Hereafter, we would like to emphasize +the director nature of the c vector field: |c⟩ and | − c⟩ +describe the physically identical states. +It should be noted that in a real space the |c⟩ state +corresponds to a quantum on-site superposition +|c⟩ = c−1|Cu1+⟩ + c0|Cu2+⟩ + c+1|Cu3+⟩ . +(13) +Existence of such unconventional on-site superpositions +is a princial point of our model. +Below instead of a and b we will make use of a pair +of unit vectors m and n, defined as follows [43]: +a = cos ϕ m, b = sin ϕ n. +(14) +For the averages of the principal pseudospin opera- +tors we obtain +⟨S⟩ = sin 2ϕ [m × n], +(15) +⟨{Si, Sj}⟩ = 2(δij − cos2 ϕ mimj − sin2 ϕ ninj). +(16) +Figure 1 shows orientations of the m and n vectors +which provide extremal values of different on-site pseu- +dospin order parameters given ϕ = π/4. The monova- +lent Cu2+, or M 0 center, is described by a pair of m and +n vectors directed along Z-axis with |mz| = |nz| = 1. +We arrive at the Cu2+-Cu3+ (M 0-M +) or Cu2+-Cu1+ +(M 0-M −) mixtures if turn c−1 or c+1, respectively, into +zero. The mixtures are described by a pair of m and +n vectors whose projections on the XY-plane, m⊥ and +n⊥, are of the same length and orthogonal to each other: +m⊥·n⊥ = 0, m⊥ = n⊥ with [m⊥ ×n⊥] = ⟨Sz⟩ = ± sin2 θ +for M 0-M ± mixtures, respectively (see Fig. 1). +It is worth noting that for ”conical” configurations +in Figs. 1b-1d: +⟨Sz⟩ = 0, ⟨S2 +z⟩ = sin2 θ, ⟨S2 +±⟩ = −1 +2 sin2 θ e±2iϕ, +⟨S±⟩ = − i +√ +2 +sin 2θ e±iϕ, ⟨T±⟩ = 0, +(17) +(Fig. 1b) +⟨Sz⟩ = 0, ⟨S2 +z⟩ = sin2 θ, ⟨S2 +±⟩ = −1 +2 sin2 θ e±2iϕ, +⟨S±⟩ = 0, ⟨T±⟩ = ∓ 1 +√ +2 sin 2θ e±iϕ, +(18) +(Fig. 1c) +⟨Sz⟩ = −⟨S2 +z⟩ = − sin2 θ, ⟨S2 +±⟩ = 0, +⟨S±⟩ = ⟨T±⟩ = ±1 +2e∓i π +4 sin 2θ e±iϕ, +(19) +(Fig. 1d). Figures 1e,f do show the orientation of m and +n vectors for the local binary mixture Cu1+-Cu3+, and +Fig.1g does for monovalent Cu3+ center. It is worth +noting that for binary mixtures Cu1+-Cu2+ and Cu3+- +Cu2+ we arrive at the same algebra of the ˆS± and ˆT± +operators with ⟨S±⟩ = ⟨T±⟩, while for ternary mixtures +Cu1+-Cu2+-Cu3+ these operators describe different ex- +citations. Interestingly that in all the cases the local +Cu2+ fraction can be written as follows: +ρ(Cu2+) = 1 − ⟨S2 +z⟩ = cos2 θ. +(20) +3 Effective S=1 pseudospin Hamiltonian +Effective S=1 pseudospin Hamiltonian which does com- +mute with the z-component of the total pseudospin +� +i Siz thus conserving the total charge of the system +can be written to be a sum of potential and kinetic +energies: +ˆH = ˆHpot + ˆHkin , +(21) +where +ˆHpot = +� +i +(∆iS2 +iz − µSiz) + +� +i ∆1 we arrive +at insulating monovalent quantum paramagnetic M 0 +(Cu2+)-phase, a typical one for Mott-Hubbard insula- +tors. In parent cuprates, such as La2CuO4, the Cu2+ +ions form an antiferromagnetically (AF) coupled square +lattice of s = 1/2 spins, which could possibly realize +the resonant valence bond (RVB) liquid of singlet spin +pairs. In the RVB state the large energy gain of the +singlet pair state, resonating between the many spa- +tial pairing configurations, drives strong quantum fluc- +tuations strong enough to suppress long range AF or- +der. However, by lowering the ∆ below ∆1 the undoped +cuprate can be turned first into metallic and supercon- +ducting XY 123 phase, and given ∆ < ∆2 into a fully +disproportionated MV-2 system of electron M − and +hole M + centers (M ±-phase) with ⟨S2 +iz⟩ = 1 (Fig. 1), +or electron-hole Bose liquid (EHBL) [22,23,24,25,16]. + +8 +A.S. Moskvin, Yu.D. Panov +There is no single particle transport: ⟨S±⟩ = ⟨T±⟩ = 0, +while the bosonic one may exist, and, in common, +⟨S2 +±⟩ ̸= 0. +Strictly speaking the S=1 pseudospin Hamiltonian +(21) describes an extended bosonic Hubbard model +(EBHM) with truncation of the on-site Hilbert space +to the three lowest occupation states n = 0, 1, 2, or the +model of semi-hard-core bosons [38]. The EBHM Hamil- +tonian is a paradigmatic model for the highly topical +field of ultracold gases in optical lattices, however, this +is one of the working models to describe the insulator- +metal transition and high-temperature superconductiv- +ity. +The pseudospin Hamiltonian (21) can be general- +ized for cuprates to include spin and orbital degrees +of freedom [27], in particular, to take into account an +intra-plaquette charge nematicity. Indeed, different or- +bital symmetry, B1g and A1g of the ground states for +Cu2+ and Cu1+,3+, respectively, unequivocally should +result in a spontaneous orbital symmetry breaking ac- +companying the formation of the on-site superpositions +with emergence of the on-site orbital order parame- +ter of the B1g = B1g × A1g (∝ dx2−y2) symmetry. In +frames of the CuO4 cluster model the rhombic B1g-type +symmetry breaking may be realized both by the b1g- +a1g (dx2−y2-dz2) mixing for central Cu ion or through +the oxygen subsystem either by emergence of different +charge densities on the oxygens placed symmetrically +relative to the central Cu ion and/or by the B1g-type +distortion of the CuO4 plaquette resulting in different +Cu-O separations for these oxygens. The latter effect +seems to be natural for Cu1+ admixtures. Indeed, at +variance with Cu2+ and Cu3+ ions the Cu1+ ion due +to a large intra-atomic s - d - hybridization does prefer +a dumbbell O-Cu-O linear configuration thus making +large rhombic distortions of the CuO4 cluster. Tak- +ing into account two energetically equivalent B1g-type +charge imbalance/distortions of the isolated CuO4 pla- +quette in both cases we can introduce a dichotomic ne- +matic variable that can be build in into effective pseu- +dospin Hamiltonian. The STM [50] and 17O NQR [51] +measurements of a static nematic order in cuprates sup- +port a charge imbalance between the density of holes at +the oxygen sites oriented along a- and b-axes, however, +there are clear signatures of the B1g-type distortion +(half-breathing mode) instabilities even in hole-doped +superconducting cuprates which can be addressed to +be a true ”smoking gun” for electronic Cu1+ centers. +The two dynamically coexisting sets of CuO4 clusters +with different in-plane Cu-O interatomic distances have +been really found by polarized Cu K-edge EXAFS in +La1.85Sr0.15CuO4 [12]. Giant phonon softening and line +broadening of electronic origin of the longitudinal Cu- +O bond stretching phonons near half-way to the zone +boundary was observed in hole-doped cuprates (see, +e.g., Ref. [52] and references therein). Their amplitude +follows the superconducting dome that supports our +message about a specific role of electron-hole Cu1+- +Cu3+ pairs in high-Tc superconductivity. +Conventional spin s=1/2 degree of freedom can be +build in our effective Hamiltonian, if we transform con- +ventional Heisenberg spin exchange Cu2+-Cu2+ cou- +pling as follows +ˆHex = +� +i 0 since, +in general, this is the case of more interest. However, +the Hamiltonian (29) is invariant under the transfor- +mation J, λ → −J, −λ and a shift of the Brillouin zone +k → k + (π, π) for 2D square lattice. The system de- +scribed by the Hamiltonian (29) can be characterized +by local (on-site) spin-linear order parameters ⟨S⟩ and +spin-quadratic (quadrupole spin-nematic) order param- +eters Q2 +0 = Qzz = ⟨S2 +z − 2 +3⟩ and Q2 +±2 = ⟨S2 +±1⟩. +The model has been studied rather extensively +in recent years by several methods, e.g., molecular +field approximation, spin-wave theories, exact numer- +ical diagonalizations, nonlinear sigma model, quantum +Monte Carlo, series expansions, variational methods, +coupled cluster approach, self-consistent harmonic ap- +proximation, and generalized SU(3) Schwinger boson +representation (see, e.g., Refs. [54,55,56] and references +therein). +The spectrum of the spin Hamiltonian (29) in the +absence of external magnetic field changes drastically +as ∆ varies from very small to very large positive or +negative values. A strong ”easy-plane” anisotropy for +large positive ∆ > 0 favors a singlet phase where all the +spins are in the Sz = 0 ground state. This quadrupole +(Qzz = - 2 +3) phase has no magnetic order, and is aptly +referred to as a quantum paramagnetic phase (QPM), +which is separated from the ”ordered” state by a quan- +tum critical point (QCP) at some ∆ = ∆1. A strong +”easy-axis” anisotropy for large negative ∆ ≤ ∆2, fa- +vors a spin ordering along Z, the ”easy axis”, with the +on-site Sz = ±1 (Z-phase). The order parameter will +be ”Ising-like” and long-range (staggered) diagonal or- +der will persist at finite temperature, up to a critical +line Tc(∆). For intermediate values ∆1 > ∆ > ∆2 the +Hamiltonian will have O(2) symmetry and the system +is in a gapless XY phase. At T = 0 the O(2) symme- +try will be spontaneously broken and the system will +exhibit spin order in some direction. Although there +will be no ordered phase at finite temperature one ex- +pects a finite temperature Kosterlitz-Thouless transi- +tion. At finite effective field hz but λ = 1 the XY phase +transforms into a canted antiferromagnetic XY -ZF M +phase, the spins acquire a uniform longitudinal com- +ponent which increases with field and saturates at the +fully polarized (FP) state (all Sz = 1, ZF M phase) above +the saturation field hs. However, at D > 0 and λ > 1 +the phase diagram contains an extended spin supersolid +or conical phase XY -ZF IM with ferrimagnetic z-order +that does exist over a range of magnetic fields [54,55, +56]. +4.2 ”Negative”-U model and its relevance for 2D +cuprates +At large negative values of the on-site correlation pa- +rameter ∆ = U/2 we arrive at the ground state of our +model cuprate to be a system of electron CuO7− +4 +and +hole CuO5− +4 +centers coupled by inter-site correlations +and two-particle transport, while single-particle trans- +port described by ˆH(1) +kin is suppressed due to large value +of the transfer energy. This electron-hole liquid is equiv- +alent to the lattice hard-core (hc) Bose system with +an inter-site repulsion and can be termed as electron- +hole Bose liquid (EHBL). Indeed, one may address the +electron M − center to be a system of a local com- +posite boson (e2) localized on the hole M + center: +M − = M + + e2. For such a system, the pseudo-spin +Hamiltonian (21) can be mapped onto the Hamiltonian +of hc Bose gas on a lattice (see Refs. [3,57,58] and ref- +erences therein) +Hhc = − +� +⟨ij⟩ +tij ˆP(ˆb† +iˆbj + ˆb† +jˆbi) ˆP + ++ +� +⟨ij⟩ +Vijninj − µ +� +i +ni, +(30) +where ˆP is the projection operator which removes dou- +ble occupancy of any site, ˆb†(ˆb) are the Pauli creation +(annihilation) operators which are Bose-like commuting +for different sites [ˆbi,ˆb† +j] = 0, if i ̸= j, [ˆbi,ˆb† +i] = 1 − 2ni, +ni = ˆb† +iˆbi; N is a full number of sites, µ the chemical po- +tential determined from the condition of fixed full num- +ber of bosons � +i⟨ni⟩ or concentration n = 1 +N +� +i⟨ni⟩ ∈ +[0, 1]. The tij denotes an effective transfer integral, Vij is +an intersite interaction between the bosons. Hereafter, +we’ll consider only a nearest neighbor boson-boson re- +pulsion, Vij = Vnn = V > 0, and tij = tnn = t > 0. +It is worth noting that near half-filling (n ≈ 1/2) one +might introduce the renormalization ni → (ni−1/2), or +neutralizing background, that immediately provides the + +10 +A.S. Moskvin, Yu.D. Panov +particle-hole symmetry. The model of hard-core bosons +with an intersite repulsion is equivalent to a system +of s = 1/2 spins exposed to an external magnetic field +in the z-direction[59]. For the system with neutraliz- +ing background we arrive at an effective pseudo-spin +Hamiltonian +Hhc = +� +⟨ij⟩ +Jxy +ij (ˆs+ +i ˆs− +j +ˆs+ +j ˆs− +i )+ +� +⟨ij⟩ +Jz +ijˆsz +i ˆsz +j−µ +� +i +ˆsz +i ,(31) +where Jxy +ij = 2tij, Jz +ij = Vij, ˆs− +i = +1 +√ +2ˆbi, ˆs+ +i = − 1 +√ +2ˆb† +i, +ˆsz +i = − 1 +2 + ˆb† +iˆbi, ˆs± +i = ∓ 1 +√ +2(ˆsx +i ± iˆsy +i ). +Local on-site order is characterized by the three or- +der parameters: ⟨ˆsz⟩ =n - 1 +2; ⟨ˆs±⟩ = |⟨ˆs±⟩|e±iϕ, related +with the charge and superfluid degree of freedom, re- +spectively. +The +EHBL +model +exhibits +many +fascinating +quantum phases and phase transitions. Early in- +vestigations[3] +point +to +the +T += +0 +charge or- +der (CO=Z13 +AF M), Bose superfluid (BS=XY -Z13 +F M) +and mixed (BS+CO=XY -Z13 +F IM) supersolid uniform +phases with an Ising-type melting transition (CO- +NO=Z13 +AF M-Z13 +F M) and Kosterlitz-Thouless-type (BS- +NO=XY -Z13 +F M-Z13 +F M) +phase +transitions to +a +non- +ordered normal fluid (NO=Z13 +F M) in 2D systems. At +half-filling (nB = 0.5, ∆n = 0) given tb > Vnn, Vnnn = 0 +the EHBL system obviously prefers a superconducting +BS=XY 13 phase while at tb < Vnn, Vnnn = 0 it prefers +an insulating checkerboard charge order CO=Z13 +AF M. +The mean-field phase diagram for the hard-core +bosons is well-known (see, e.g., Ref. [3]). First of all +the MFA points to emergence of an uniform supersolid +CO+SF phase with deviation away from half-filling. At +a critical concentration: +∆nc = 1 +2 +�V − 2t +V + 2t +� 1 +2 +(32) +the supersolid phase does transform into the SF phase +at T = 0. +In Fig. 2 we present the phase diagram of the square +lattice hc-boson model with the nearest neighbour (nn) +transfer integral tb +nn = t (the Josephson coupling) and +repulsion Vnn = 3t, derived from the quantum Monte- +Carlo (QMC) calculations by Schmid et al. [60]. Differ- +ent filling points to CO phase, BS phase, and phase sep- +arated supersolid BS+CO phase. The AB line TKT (x) +points to 2D Kosterlitz-Thouless phase transition; the +C-B-D-C′ line points to the first order phase transi- +tion; the D-E line TCO(∆n) which can be termed as +the pseudogap onset temperature T ∗(∆n) points to the +second order Ising kind melting phase transition CO- +NO=Z13 +AF M-Z13 +F M into a nonordered, or normal fluid +phase. It is worth noting that the QMC calculations [60] +show that under doping away from half filling, the +Fig. 2 Fig. 2. (Color online) The QMC T-n phase diagram +of the 2D hc-boson system (reconstruction of Fig. 2 from +Ref.[60]. Blank painting marks the HTSC phase for LSCO. +checkerboard solid undergoes phase separation: the su- +perfluid (BS) and solid (CO) phases coexist but not as +a single thermodynamic BS+CO phase. +Simple uniform EHBL model truly reproduces many +important aspects of the cuprate physics, in particular, +the pseudogap phenomenon as a result of the charge +ordering and high values of the critical temperatures +for superconducting transition. Indeed, the phase dia- +gram in Fig. 2 points to critical temperatures for 2D +superconductivity as large as 0.5 t that is on the or- +der of 500 K, if we take into account reasonable esti- +mations for the transfer integral in cuprates[22]: t ∼ +I ∼ 1000 K. Thus, the model is very promising for find- +ing paths to room-temperature superconductivity [61]. +At the same time the model cannot explain a number +of well-known properties, in particular, manifestation +of the Cu2+ valence states in doped cuprates over wide +doping range [62] and suppression of the superconduc- +tivity for overdoped cuprates. Such a behaviour cannot +be derived from the EHBL scenario and points to real- +ization of the more complicated ”boson-fermion” dual +XY-Z123 +F IM phase with coexisting spin and pseudospin +(charge) orders in a wide doping range from parent to +overdoped compounds including all the superconduct- +ing phase. The suppression of the superconductivity for +the hole overdoped cuprates can be explained as a tran- +sition from the trivalent superconducting (M123) phase +to a bivalent nonsuperconducting M23 phase. Indeed, +the M−=Cu1+ centers could be energetically gainless +under hole doping particularly for overscreened EH cou- +pling. Some properties of nonsuperconducting phases +M23 and M12, or XY -Z23 +F M and XY -Z12 +F M, can be un- +derstood if we address limiting insulating phases M + or + +Topological structures in unconventional scenario for 2D cuprates +11 +M − (Z1 +F M or Z3 +F M) with precisely M + or M −-centers +on each of the lattice sites. In frames of the pseudospin +formalism these phases correspond to fully polarized +ferromagnetic states with Stot +z += ± N, where N is the +number of Cu sites. Interestingly, in frames of the pseu- +dospin formalism the ”heavily overdoped” XY -Z23 +F M +and XY -Z12 +F M phases with x ≈ 1 can be represented +as ferromagnets where the charge constraint is real- +ized through the occurrence of (1−x)N non-interacting +pseudospin magnons (∆Sz = ± 1), that is Cu2+ cen- +ters, obeying Fermi statistics due to s=1/2 conventional +spin. These heavily overdoped cuprates could be ad- +dressed to be conventional Fermi liquids. Indeed, the +standard Fermi liquid theory hinges on the key assump- +tion that although the electrons (holes) interact, the +low-energy excitation spectrum stands in a one-to-one +correspondence with that of a non-interacting system. +In other words, the Fermi liquid behavior can be typ- +ical for overdoped XY -Z23 +F M and XY -Z12 +F M phases in +a rather wide range of ”overdoping”. The phases are +expected to manifest all the signatures of an electron +(hole) Fermi liquid with a large Fermi surface which +contains (1 − x) electrons in the hole-doped cuprates +such as La2−xSrxCuO4 or (1 − x) holes in the electron- +doped cuprates such as Nd2−xCexCuO4. However, this +conventional Fermi liquid behavior can fail for hole or +electron overdoped cuprates with a rather large content +of Cu2+ centers and the electron pairing due to dispro- +portionation reaction Cu2++Cu2+→Cu3++Cu1+ with +formation of EH dimers and local bosons. In the frame- +work of the pseudospin formalism we arrive at a binding +of two pseudomagnons with ∆Sz = -1 to a single bi- +magnon ∆Sz = -2. In other words, we may think of the +local boson as a long-lived virtual two-pseudomagnon +bound state, or bimagnon, where the pseudomagnons +are bound on the same site. Electron pairing due to +formation of the EH dimers, or CT exciton concept of- +fers a fruitful insight into challenging issues of the cop- +per oxide superconductors [63,64,65,5]. The EH bind- +ing/unbinding energy ∆EH, that is the energy of the +local boson binding in the EH dimer, can be identified +with the pseudogap observed in the extended part of +the cuprate phase diagram from heavily underdoped +to overdoped systems. On the other hand, this en- +ergy defines an effective gap for the thermal activation +of hole carriers which has been found from the high- +temperature Hall data in La2−xSrxCuO4 [63,64,66,67]. +Effective number of hole carriers derived from RH(T ) +were excelently fitted by a simple two-component for- +mula where the first component is related with the tem- +perature independent itinerant carriers, while the sec- +ond one bears the activation character. Interestingly, +the activation energy was shown to coincide with the +ARPES measured excitation energies needed to trans- +fer electron from the antinodal points (0,π), (π, 0) in +the Brillouin zone to the chemical potential [63,64]. The +equal energy for creation of an electron (ARPES) and +a hole (thermal activation) can be understood only +in terms of bound states for electron-hole pairs[63, +64], EH-dimers, or condensation of the CT excitons +whose coupling/decoupling energy is revealed in both +types of experiments. These states, seen near antin- +odal points, according to ARPES, form quasi-periodic +structures close to the double periodicity along the Cu- +Cu bonds directions. The number of itinerant carriers +rapidly increases with x [63,64,66,67] that results in +an effective screening of the ∆EH(x) parameter with a +sharp fall of the ”ionization” energy of the EH dimers +from ∆EH(x) ≈ 0.5 eV given x = 0.01 up to values close +to zero given x > 0.2. Given zero values of the ∆EH +parameter the XY -Z123 +F IM phase becomes energetically +gainless as compared with XY -Z23 +F M phase, hence we +arrive at a QCP separating superconducting XY -Z123 +F IM +phase and nonsuperconducting XY -Z23 +F M phase with a +large Fermi surface (FS) and other attributes of a Fermi +liquid. The onset of the pseudogap below QCP natu- +rally explains the FS reconstruction with a number of +unusual properties of the doped cuprates, such as the +Fermi arc and/or pocket formation [65]. +It should be noted that the ”negative-U” model is +a limiting case of more complicated model with sup- +pressed single-particle transport but with finite, posi- +tive or negative, values of the on-site correlation param- +eter U. Such a model was analyzed recently both within +the mean-field approximation and quantum Monte- +Carlo calculation [68]. +5 Topological defects in 2D S=1 pseudospin +systems +5.1 Short overview +In the framework of our charge triplet model the +cuprates prove to be in the universality class of the +(pseudo)spin 2D systems whose description incorpo- +rates static or dynamic topological defects to be natural +element both of micro- and macroscopic physics. Like +domain walls, the vortices and skyrmions are stable for +topological reasons. Depending on the structure of ef- +fective pseudo-spin Hamiltonian in 2D-systems the lat- +ter could correspond to either in-plane and out-of-plane +vortices or skyrmions[69]. Under certain conditions ei- +ther topological defects could determine the structure of +the ground state. In particular, this could be a generic +feature of electric multipolar systems with long-range + +12 +A.S. Moskvin, Yu.D. Panov +multipolar interactions. Indeed, a Monte-Carlo simula- +tion of a ferromagnetic Heisenberg model with dipolar +interaction on a 2D square lattice L × L shows that, +as L is increased, the spin structure changes from a +ferromagnetic one to a novel one with a vortex-like ar- +rangement of spins even for rather small magnitude of +dipolar anisotropy[70]. +Quasi-classical continuous description of the 2D +magnetic +systems +reveals +their +striking +features, +namely, the collective localized inhomogeneous states +with nontrivial topology and finite excitation en- +ergy. These include topological solitons [69,71], magnon +drops [72], +in- +and +out-of-plane +vortex-antivortex +pairs [73], and various spiral solutions [74,75,76]. Ba- +sically these solutions have been obtained for the +isotropic and anisotropic ferromagnet. +A vector representation is useful, if not a single in- +strument of a visual qualitative description of complex +spin and pseudospin structures. A striking example of +a single-vector representations are the N´eel and Bloch +domain walls in classic ferromagnets. Situation in quan- +tum S=1 systems is more intricate, however, our ”two- +vector” (m, n)-description of the on-site S=1 states can +be successfully applied to predict and analyse differ- +ent uniform and nonuniform, in particular, topological +structures. One of the most surprising is the prediction +of the existence of unusual antiphase 180◦-domain walls +for parent cuprates. Indeed, the bare on-site Cu2+ state +in parent cuprate is described by the collinear pair of +vectors m and n, or −m and −n directed along Z-axis. +It means that two ”parent” domains can be separated +by a domain wall in which a collinear pair of vectors +m and n can rotate by 180 degrees (see Fig. 1). De- +viation from Z-axis corresponds to emergence of the +on-site electron-hole Cu1+-Cu3+ component due to the +gradual suppression of the bare parent Cu2+ compo- +nent until its complete disappearance in the domain +wall center with the maximum value of the electron-hole +Cu1+-Cu3+ component. which ensures the maximum +value of the electron-hole component and, accordingly, +the maximum value of the modulus of the supercon- +ducting order parameter ⟨ ˆS2 +±⟩. In other words, such a +domain wall can be considered as a potential source of +filamentary superconductivity. +Interestingly, the domain wall structure is charac- +terized by an uniform distribution of the mean on-site +charge as ⟨ ˆSz⟩ = 0 for collinear (m, n)-pair. In other +words, the domain wall structure and bare parent Cu2+ +monovalent (insulating) phase have absolutely the same +distribution of the mean on-site charges. From the one +hand, this point underlines an unconventional quantum +nature of the on-site states indomain wall, while from +the other hand it makes the domain wall textures to be +invisible, in particular, for X-rays. +It is obvious that the formation of such domain walls +in the parent cuprates is energetically unfavorable, how- +ever, the situation changes radically with electron/hole +doping due to the fact that the doping into a domain +wall stabilizes the domain configuration[57,58]. The an- +tiphase domain wall in the parent cuprate appears to +be a very efficient potential well for the localization of +extra electron/hole pairs thus forming a novel type of a +neutral or charged topological defect. We believe that +the stripe structures in underdoped cuprates [11,12,13] +can somehow be associated with these antiphase do- +main walls. +Topological defects are stable non-uniform spin +structures with broken translational symmetry and +non-zero topological charge (chirality, vorticity, wind- +ing number). Vortices are stable states of anisotropic +2D Heisenberg Hamiltonian +ˆH = +� +i λ > +λc stable solution corresponds to the out-of-plane OP- +vortex (Sz ̸= 0), at which center the spin vector appears +to be oriented along z-axis, and at infinity it arranges +within xy-plane. The in-plane vortex is described by the +formulas Φ = qϕ, cos θ = 0. The θ(r) dependence for +the out-of-plane vortex cannot be found analytically. +Both kinds of vortices have the energy logarithmically +dependent on the size of the system. +The cylindrical domains, or bubble like solitons +with spins oriented along the z-axis both at infin- +ity and in the center (naturally, in opposite direc- +tions), exist for the ”easy-axis” anisotropy λ > 1. +Their energy has a finite value. Skyrmions are gen- +eral static solutions of classical continuous limit of +the isotropic (λ = 1) 2D Heisenberg ferromagnet, ob- +tained by Belavin and Polyakov [69] from classical non- +linear sigma model. Belavin-Polyakov skyrmion and +out-of-plane vortex represent the simplest toy model +(pseudo)spin textures [69,77]. +The simplest skyrmion spin texture looks like a bub- +ble domain in ferromagnet and consists of a vortex-like +arrangement of the in-plane components of spin with +the z-component reversed in the centre of the skyrmion +and gradually increasing to match the homogeneous +background at infinity. The spin distribution within + +Topological structures in unconventional scenario for 2D cuprates +13 +such a classical skyrmion with a topological charge q +is given as follows [69] +Φ = qϕ + ϕ0; +cos Θ = r2q − λ2q +r2q + λ2q , +(34) +where r, ϕ are polar coordinates on plane, q += +±1, ±2, ... the chirality. For q = 1, ϕ0 = 0 we arrive +at +nx = +2rλ +r2 + λ2 cos ϕ; ny = +2rλ +r2 + λ2 sin ϕ; nz = r2 − λ2 +r2 + λ2 . +(35) +In terms of the stereographic variables the skyrmion +with radius λ and phase ϕ0 centered at a point z0 is +identified with spin distribution w(z) = +Λ +z−z0 , where +z = x + iy = reiϕ is a point in the complex plane, +Λ = λeiα. For a multicenter skyrmion we have [69] +w(z) = cot Θ +2 eiΦ = +� +i +�z − zj +Λ +�mj � +j +� +Λ +z − zj +�nj +, +(36) +where � mi > � nj, q = � mj. Skyrmions are char- +acterized by the magnitude and sign of its topological +charge, by its size (radius), and by the global orien- +tation of the spin. The scale invariance of skyrmionic +solution reflects in that its energy Esk = 4π|q|IS2 is +proportional to topological charge and does not depend +on radius and global phase [69]. Like domain walls, vor- +tices and skyrmions are stable for topological reasons. +Skyrmions cannot decay into other configurations be- +cause of this topological stability no matter how close +they are in energy to any other configuration. +In a continuous field model, such as, e.g., the nonlin- +ear σ-model, the ground-state energy of the skyrmion +does not depend on its size [69], however, for the +skyrmion on a lattice, the energy depends on its size. +This must lead to the collapse of the skyrmion, making +it unstable. Strong anisotropic interactions, in particu- +lar, long range dipole-dipole interactions may, in prin- +ciple, dynamically stabilize the skyrmions in 2D lat- +tices [78,79,80]. +Wave function of the spin system, which corre- +sponds to a classical skyrmion, is a product of spin +coherent states [81]. In case of spin S = 1 +2 +Ψsk(0) = +� +i +[cos θi +2 ei ϕi +2 |↑⟩ + sin θi +2 e−i ϕi +2 |↓⟩], +(37) +where θi = arccos r2 +i −λ2 +r2 +i +λ2 . Coherent state provides a +maximal equivalence to classical state with minimal +uncertainty of spin components. The motion of such +skyrmions has to be of highly quantum mechanical na- +ture. However, this may involve a semi-classical perco- +lation in the case of heavy non-localized skyrmions or +variable range hopping in the case of highly localized +skyrmions in a random potential. Effective overlap and +transfer integrals for quantum skyrmions are calculated +analytically by Istomin and Moskvin [82]. The skyrmion +motion has a cyclotronic character and resembles that +of electron in a magnetic field. +The interest in skyrmions in ordered spin sys- +tems received much attention soon after the discovery +of high-temperature superconductivity in copper ox- +ides [83,84,85,86,87,88,89,90,91,92,93]. Initially, there +was some hope that interaction of electrons and holes +with spin skyrmions could play some role in supercon- +ductivity, but this was never successfully demonstrated. +Some indirect evidence of skyrmions in the magnetore- +sistance of the litium doped lanthanum copper oxide +has been recently reported [94] but direct observation of +skyrmions in 2D antiferromagnetic lattices is still lack- +ing. In recent years the skyrmions and exotic skyrmion +crystal (SkX) phases have been discussed in connection +with a wide range of condensed matter systems includ- +ing quantum Hall effect, spinor Bose condensates and +especially chiral magnets [95,96,97]. It is worth noting +that the skyrmion-like structures for hard-core 2D bo- +son system were considered by Moskvin et al. [57,58] in +frames of the s=1/2 pseudospin formalism. +5.2 Unconventional skyrmions in S=1 (pseudo)spin +systems +Different +skyrmion-like topological defects +for +2D +(pseudo)spin S=1 systems as solutions of isotropic spin +Hamiltonians were addressed in Ref. [43] and in more +detail in Ref. [42]. In general, isotropic non-Heisenberg +spin-Hamiltonian for the S=1 quantum (pseudo)spin +systems should include both bilinear Heisenberg ex- +change term and biquadratic non-Heisenberg exchange +term: +ˆH = − ˜J1 +� +i,η +ˆSiˆSi+η − ˜J2 +� +i,η +(ˆSiˆSi+η)2 = +(38) += −J1 +� +i,η +ˆSiˆSi+η − J2 +� +i,η +3 +� +k≥j +({ ˆSk ˆSj}i{ ˆSk ˆSj}i+η), +where Ji are the appropriate exchange integrals, J1 = +˜J1 − ˜J2/2, J2 = ˜J2/2, i and η denote the summation +over lattice sites and nearest neighbours, respectively. +Having substituted our trial wave function (12) to +⟨ ˆH⟩ provided ⟨ˆS(1)ˆS(2)⟩ = ⟨ˆS(1)⟩⟨ˆS(2)⟩ we arrive at +the Hamiltonian of the isotropic classical spin-1 model + +14 +A.S. Moskvin, Yu.D. Panov +in the continual approximation as follows: +H = J1 +� +d2r +� 3 +� +i=1 +(∇⟨Si⟩)2 +� ++ ++ J2 +� +d2r + + +3 +� +i≤j=1 +(∇aiaj + ∇bibj)2 + + + ++ 4(J2 − J1) +c2 +� +|⟨ˆS⟩|2d2r, +(39) +where ⟨ˆS⟩ = 2[a × b]. It should be noted that the third +”gradientless” term in the Hamiltonian breaks the scal- +ing invariance of the model. +5.2.1 Dipole (pseudo)spin skyrmions +Dipole, or magnetic skyrmions as the solutions of bilin- +ear Heisenberg (pseudo)spin Hamiltonian when J2 = 0 +were obtained in Ref. [43] given the restriction a ⊥ b +and the lengths of these vectors were fixed. +The model reduces to the nonlinear O(3)-model +with the solutions for a and b described by the fol- +lowing formulas (in polar coordinates): +√ +2a = (ez sin θ − er cos θ) sin ϕ + eϕ cos ϕ, +√ +2b = (ez sin θ − er cos θ) cos ϕ − eϕ sin ϕ. +(40) +For dipole ”magneto-electric” skyrmions the m, n +vectors are assumed to be perpendicular to each other +(m ⊥ n) and the (pseudo)spin structure is determined +by the skyrmionic distribution (34) of the l = [m × n] +vector [43]. In other words, the fixed-length spin vec- +tor ⟨S⟩ = 2[a × b] is distributed in the same way as +in the usual skyrmions (34). However, unlike the usual +classic skyrmions, the dipole skyrmions in the S=1 the- +ory have additional topological structure due to the +existence of two vectors m and n. Going around the +center of the skyrmion the vectors can make N turns +around the l vector. Thus, we can introduce two topo- +logical quantum numbers: N and q [43]. In addition, it +should be noted that q number may be half-integer. The +dipole-quadrupole skyrmion is characterized by nonzero +both pseudospin dipole order parameter ⟨S⟩ with usual +skyrmion texture (34) and quadrupole order parame- +ters +⟨{ ˆSi ˆSj}⟩ = 2⟨ ˆSi⟩⟨ ˆSj⟩ = lilj. +(41) +5.2.2 Quadrupole (pseudo)spin skyrmions +Hereafter we address another situation with purely bi- +quadratic (pseudo)spin Hamiltonian (J1=0) and treat +the non-magnetic (“electric”) degrees of freedom. The +topological classification of the purely electric solutions +is simple because it is also based on the usage of sub- +group instead of the full group. We address the solutions +given a ∥ b and the fixed lengths of the vectors, so we +use for the classification the same subgroup as above. +After simple algebra the biquadratic part of the +Hamiltonian can be reduced to the expression familiar +for nonlinear O(3)-model: +Hbq = J2 +� +d2r + + +3 +� +i,j=1 +(∇ninj)2 + + = += 2J2|n|2 +� +d2r +� 3 +� +i=1 +(∇ni)2 +� +, +(42) +where a = αn, b = βn, and α + iβ = exp(iκ), κ ∈ R, +|n|2 =const. Its solutions are skyrmions, but instead of +the spin distribution in magnetic skyrmion we have so- +lutions with zero spin, but the non-zero distribution of +five spin-quadrupole moments Qij, or ⟨{SiSj}⟩ which +in turn are determined by the ”skyrmionic” distribu- +tion of the n vector (34) with classical skyrmion energy: +Eel = 16πqJ2. The distribution of the spin-quadrupole +moments ⟨{SiSj}⟩ can be easily obtained: +⟨S2 +z⟩ = +4r2qλ2q +(r2q + λ2q)2 , ⟨ ˆS2 +±⟩ = +2r2qλ2q +(r2q + λ2q)2 e±2iqϕ, +⟨ ˆT±⟩ = −i +√ +2(λ2q − r2q)rqλq +(r2q + λ2q)2 +e∓iqϕ. +(43) +One should be emphasized that the distribution of +five independent quadrupole order parameters for the +quadrupole skyrmion are straightforwardly determined +by a single vector field m(r) (n(r)) while ⟨ˆS⟩ = 0. +The quadrupole skyrmion supposedly can be a typ- +ical topological charge excitation for parent or under- +doped cuprates. Fig.3 demonstrates the radial distribu- +tion of different order parameters for the quadrupole +skyrmion, the modulus of the SC order parameter, the +Cu2+ spin density, and the T -type order parameter. +We see a circular layered structure with clearly visi- +ble anticorrelation effects due to a pseudospin kinemat- +ics. Interestingly, at the center (r = 0) and far from +the center (r → ∞) for such a skyrmion we deal with +a parent Cu2+ monovalent (insulating) state while for +the domain wall center (r = λ) we arrive at a fully +disproportionated ”superconducting” +Cu1+-Cu3+ su- +perposition whose weight diminishes with moving away +from the center. In other words, the ring shaped do- +main wall is an area with a circular distribution of the +superconducting order parameter, or circular ”bosonic” +supercurrent. Nonzero T -type order parameter distri- +bution points to a circular ”fermionic” current with a +puzzlingly opposite sign of the ⟨ ˆT±⟩ parameter for ”in- +ternal” (0 < r < λ) and ”external” (r > λ) parts of +the skyrmion. Given the simplest winding number q = 1 + +Topological structures in unconventional scenario for 2D cuprates +15 +-0.4 +-0.2 +0.2 +0.4 +0.6 +0.8 +1.0 +1 +2 +3 +4 +5 +r/� +=0 +z +1- +z +2 +Cu +2+ +Cu +2+ +“Fermi” rings +“Bose” ring +, Re +z +2 +2 +� +Re +� +a +b +c +d +Fig. 3 (Color online) a) Radial distribution of the diagonal +and off-diagonal nematic order parameters (the Cu2+ spin +density, the modulus of the SC order parameter, and the T- +type order parameter) for a quadrupole pseudospin skyrmion +(q=1): b) the ring shaped distribution of the Cu2+ spin den- +sity and off-diagonal order parameters describing the Fermi- +and Bose-like transport, arrows point to opposite sign of the +T-type order parameter for the ”internal” and ”external” +rings, the ± signs point to the signs of the Re⟨ ˆS2 +±⟩: c) and +d) the spatial distribution of Re⟨ ˆS2 +±⟩ and ⟨ ˆS2 +z⟩, respectively. +It should be noted the dx2−y2 symmetry of the SC order +parameter. +we arrive at the d-wave (dx2−y2/dxy symmetry of the +superconducting order parameter. +First of all we should note that such a skyrmionic +structure is characterized by an uniform distribution of +the mean on-site charge as ⟨ ˆSz⟩ = 0, that is why it can +be termed as a neutral skyrmion. Indeed, all over the +skyrmion the m and n vectors form a collinear config- +uration, thus ⟨S⟩ turns into zero. In other words, the +quadrupole skyrmionic structure and bare parent Cu2+ +monovalent (insulating) phase have absolutely the same +distribution of the mean on-site charges. From the one +hand, this point underlines an unconventional quantum +nature of the quadrupole skyrmion under considera- +tion, while from the other hand it makes the quadrupole +skyrmion texture to be invisible, in particular, for X- +rays. At the same time, the skyrmion has a well de- +veloped layered ring-shaped distribution of the ⟨ ˆT±⟩ +and ⟨ ˆS2 +±⟩ order parameters that points to its instability +with regard to circular fermionic- and bosonic-like cir- +cular currents with maximal current density around the +skyrmion radius. In this connection it is worth noting +a scenario of the circulating charge currents for under- +doped cuprates [98,37]. +An interesting example of a topological inhomogene- +ity is provided by a multi-center skyrmion [69] which +energy does not depend on the position of the centers. +The latter are believed to be addressed as an additional +degree of freedom, or positional order parameter. Fig. 4 +shows an example of the in-plane distribution of the +modulus of the SC order parameter ⟨ ˆS2 +±⟩ for a multi- +center quadrupole pseudospin skyrmion with a random +distribution of the centers. An individual skyrmion in +this multi-center entity can be characterized by its po- +sition (i.e., the center of a skyrmionic texture), its size +(i.e., the radius of domain wall), and the orientation of +the in-plane components of pseudospin (U(1) degree of +freedom). +The domain wall center of the quadrupole skyrmion +provides maximal values of the pseudospin susceptibil- +ity χzz, or charge susceptibility [57,58]. It means the do- +main wall appears to form a very efficient ring-shaped +potential well for the charge carrier localization thus +giving rise to a novel type of a charged topological +defect. In the framework of the pseudospin formalism +the skyrmion charging corresponds to a single-magnon +∆Sz = ± 1 (single particle) or a two-magnon ∆Sz = ± 2 +(two-particle) excitations. It is worth noting that for +large negative ∆ the single-magnon (single-particle) ex- +citations may not be the lowest energy excitations of +the strongly anisotropic pseudospin system. Their en- +ergy may surpass the energy of a two-magnon bound +state (bimagnon), or two-particle, local boson-like, ex- +citation, created at a particular site. Thus we arrive at +a competition of the two types of charged quadrupole +skyrmions. Such a charged topological defect can be ad- +dressed to be an extended skyrmion-like mobile quasi- +particle. However, at the same time it should’nt be for- +gotten that skyrmion corresponds to a collective state +(excitation) of the whole system. +Skyrmionic scenario allows us to make several im- +portant predictions for cuprates. First, the parent insu- +lating antiferromagnetic monovalent Cu2+ phase may +be unstable with regard to nucleation of a topologi- +cal defect in the unconventional form of a single- or +multi-center skyrmion-like object with ring-shaped su- +perfluid regions. The parent Cu2+ phase may gradu- +ally lose its stability under non-isovalent substitution +(electron/hole doping), while a novel topological self- +organized texture is believed to become stable. The +most probable possibility is that every domain wall ac- +cumulates single boson, or boson hole. Then, the num- +ber of centers in a multi-center skyrmion nucleated with +doping has to be equal to the number of bosons/holes. +In such a case, we anticipate the near-linear dependence +of the total SC volume fraction on the doping. Gener- +ally speaking, one may assume scenario when the nucle- + +30:0 +O'J +30'5 +0'0 +0'316 +A.S. Moskvin, Yu.D. Panov +Fig. 4 +Contour plot of the in-plane distribution of the mod- +ulus of the off-diagonal SC order parameter ⟨ ˆS2 +±⟩ for a mul- +ticenter quadrupole pseudospin skyrmion with a random dis- +tribution of the centers. +ation of a multi-center skyrmion occurs spontaneously +with no doping. In such a case we should anticipate the +existence of neutral multi-center skyrmion-like object +with equal number of positively and negatively charged +single skyrmions. However, in practice, namely the bo- +son/hole doping is likely to be a physically clear driv- +ing force for a nucleation of a multi-center skyrmion- +like self-organized collective mode which may be (not +strictly correctly) referred to as multi-skyrmion system +akin in a quantum Hall ferromagnetic state of a two- +dimensional electron gas [99]. It seems likely that for +a light doping any doped particle results in a nucle- +ation of a new single-skyrmion state, hence its density +changes gradually with particle doping. Therefore, as +long as the separation between skyrmionic centers is +sufficiently large so that the inter-skyrmion coupling is +negligible, the energy of the system per particle remains +almost constant. This means that the chemical poten- +tial remains unchanged with doping. Nucleation of the +skyrmionic textures eventually leads to the destruction +of the antiferromagnetic Ne´el ordering which is known +to exist even at very low doping. Furthermore, the +skyrmion structure with insulating spin s=1/2 core iso- +lated by spinless nonmagnetic Cu1+-Cu3+ ring-shaped +domain wall from surrounding Cu2+ entity provides a +physically clear mechanism of the nucleation of a spin +glass phase typical for underdoped cuprates. Further- +more, the nucleation of the unconventional quadrupole +skyrmions does provide a physically clear mechanism +for the unconventional vortex Nernst signal and local +diamagnetism universally observed in many hole doped +cuprates at the temperatures above Tc [100]. +Meanwhile we discuss the quadrupole skyrmion +to be a classical solution of the continual isotropic +model, however, this idealized object is believed to +preserve their main features for strongly anisotropic +(pseudo)spin lattice quantum systems. Both quantum +effects and the discreteness of skyrmion texture can re- +sult in substantial deviations from the predictions of a +classical model. The continuous model is relevant for +discrete lattices only if we deal with long-wave length +inhomogeneities when their size is much bigger than the +lattice spacing. In the discrete lattice the very notion of +topological excitation seems to be inconsistent. At the +same time, the discreteness of the lattice itself does not +prohibit from considering the nanoscale (pseudo)spin +textures whose topology and spin arrangement is that +of a skyrmion[88]. It is worth to note that skyrmions +cannot decay into other configurations because of the +topological stability no matter how close they are in +energy to any other configuration. +The boson addition or removal in the half-filled +(n = 1) boson system can be a driving force for a nu- +cleation of a multi-center “charged” skyrmions. Such +topological structures, rather than uniform phases pre- +dicted by the mean-field approximation, are believed +to describe the evolution of the EBHM systems away +from half-filling. It is worth noting that the multi-center +skyrmions one considers as systems of skyrmion-like +quasiparticles forming skyrmion liquids and skyrmion +lattices, or crystals (see, e.g., Refs. [101,99]). +5.2.3 Dipole-quadrupole (pseudo)spin skyrmions +In the continual limit for J1 = J2 = J the Hamilto- +nian (39) can be transformed into the classical Hamilto- +nian of the fully SU(3)-symmetric scale-invariant model +which can be rewritten as follows [42]: +Hisotr = 2J +� +d2r{(∇Θ)2 + sin2 Θ(∇η)2 + ++ sin2 Θ cos2 Θ +� +cos2 η(∇Ψ1)2 + sin2 η(∇Ψ2)2� ++ ++ sin4 Θ cos2 η sin2 η(∇Ψ1 − ∇Ψ2)2}, +(44) +where we have used the representation (10) and intro- +duced Ψ1 = Φ1−Φ3, Ψ2 = Φ3−Φ2. The topological solu- +tions for the Hamiltonian (44) can be classified at least +by three topological quantum numbers (winding num- +bers): phases η, Ψ1,2 can change by 2π after the passing +around the center of the defect. The appropriate modes +may have very complicated topological structure due + +Topological structures in unconventional scenario for 2D cuprates +17 +to the possibility for one defect to have several differ- +ent centers (while one of the phases η, Ψ1,2,3 changes +by 2π given one turnover around one center (r1, ϕ1), +other phases may pass around other centers (ri, ϕi)). +It should be noted that for such a center the winding +numbers may take half-integer values. Thus we arrive at +a large variety of topological structures to be solutions +of the model. Below we will briefly address two sim- +plest classes of such solutions. One type of skyrmions +can be obtained given the trivial phases Ψ1,2. If these +are constant, the R vector distribution (see (10)) rep- +resents the skyrmion described by the usual formula +(34). All but one topological quantum numbers are zero +for this class of solutions. It includes both dipole and +quadrupole solutions: depending on selected constant +phases one can obtain both ”electric” and different +”magnetic” skyrmions. The substitution Φ1 = Φ2 = Φ3 +leads to the electric skyrmion which was obtained above +as a solution of more general SU(3)-anisotropic model. +Another example can be Φ1 = Φ2 = 0, Φ3 = π/2. +This substitution implies b∥Oz, a∥Oxy, S∥Oxy, and +S = sin Θ cos Θ{sin η, − cosη, 0}. Nominally, this is the +in-plane spin vortex with a varying length of the spin +vector +|S| = 2rλ|r2 − λ2| +(r2 + λ2)2 , +(45) +which is zero at the circle r = λ, at the center r = 0 +and at the infinity r → ∞, and has maxima at r = +λ( +√ +2 ± 1). In addition to the non-zero in-plane compo- +nents of spin-dipole moment ⟨Sx,y⟩ this vortex is char- +acterized by a non-zero distribution of (pseudo)spin- +quadrupole moments. Here we would like to emphasize +the difference between spin-1/2 systems in which there +are such the solutions as in-plane vortices with the en- +ergy having a well-known logarithmic dependence on +the size of the system and fixed spin length, and spin- +1 systems in which the in-plane vortices also can exist +but they may have a finite energy and a varying spin +length. The distribution of quadrupole components as- +sociated with in-plane spin-1 vortex is non-trivial. Such +solutions can be termed as ”in-plane dipole-quadrupole +skyrmions”. +Other types of the simplest solutions with the +phases Ψ1 = Q1ϕ, Ψ2 = Q2ϕ governed by two inte- +ger winding numbers Q1,2 and η = η(r), Θ = Θ(r) are +considered in Ref. [42]. +6 Topological excitations in ”negative-U” +model +6.1 Quasi-classical approximation +Let start with the Hamiltonian of the ”negative-U” +model in terms of the pseudospin σ, σz = ±1: +H = −t +� +⟨ij⟩ +� +σi+σj− + σi−σj+ +� ++ V +� +⟨ij⟩ +σizσjz. +(46) +Here σα, α = x, y, z are Pauli matrices, σ± = (σx ± +iσy)/2. The z-component of the pseudospin describes +the local density of composite bosons, so that antifer- +romagnetic z-z exchange corresponds to the repulsive +density-density interaction, while isotropic ferromag- +netic planar exchange corresponds to the kinetic energy +of the bosons. The constant total number of the bosons +leads to the constraint on the total z-component of the +pseudospin. We define the n, as the density of the total +doped charge counted from the state with a zero total z- +component, or parent Cu2+ state. Then n is the sum of +z-components of the pseudospin: � +i σiz = nN, where +N is the total number of sites. If ρ is the density of +hc bosons, then n is the deviation from the half-filling: +ρ = (1 + n)/2. +The energy functional E = ⟨Ψ |H| Ψ⟩ in a quasi- +classical approximation with +|Ψ⟩ = +� +i +� +cos θi +2 e−i φi +2 |+1⟩ + sin θi +2 ei φi +2 |−1⟩ +� +, +(47) +where |±1⟩ are the eigenfunctions of the σz on i-th site, +takes the form +ε = − +� +⟨ij⟩ +sin θi sin θj cos(φi − φj) + +(48) ++ λ +� +⟨ij⟩ +cos θi cos θj − ξ +� � +i +cos θi − nN +� +. +Here ⟨i, j⟩ denotes summation over nearest neighbors +in a square lattice. The θi and φi are the polar and +azimuthal angles of the quasiclassical pseudospin vector +at an i-th site. We define ε = 2E/t, λ = 2V/t, ξ = 2µ/t, +where the chemical potential µ takes into account the +bosons density constraint. +Hereafter, we introduce two sublattices A and B +with the checkerboard ordering. The first sum in the +Exp.(48) has its lowest value if cos(φi − φj) = 1. This +allows us to make a simplifying assumption, that +φA(r) = φB(r) ≡ φ(r). It is worth to note that this +assumption is confirmed by the results of our numer- +ical simulations. We define functions u(r) ≡ θA(r), +v(r) ≡ θB(r), and their combinations f = cos u cosv, +F = −f +λfuv, where subscripts u and v denote deriva- +tives with respect to these quantities. Then the Euler + +18 +A.S. Moskvin, Yu.D. Panov +equations in the continuous approximation take a com- +pact form + + + + + + + +fuvφkk − 2fvukφk = 0, +fuvφkk − 2fuvkφk = 0, +Fukk + Fuukuk − fuφkφk − 4Fu + ξ sin v = 0, +Fvkk + Fvvkvk − fvφkφk − 4Fv + ξ sin u = 0. +(49) +Here we assume summation over pair indices. These +equations need to add the boson density constraint. +With the relevant exchange constants, the equations +(49) lead to the equations of Ref. [102]. +6.2 The asymptotic behavior of localized solutions +The system (49) along with the boson density con- +straint has uniform solutions, φ = φ0, u = u0, v = v0, +that determine the well-known ground-state phase di- +agram[3] of the hc boson system in the mean-field ap- +proximation. +Given λ < 1 or n2 > (λ−1)/(λ+1) the ground state +of the system is a superfluid (SF) with cos u0 = cos v0 = +n, ε0 = −2+2 (λ + 1) n2. Given n2 < (λ−1)/(λ+1) the +ground state is a supersolid (SS) with cos u0 = n + z, +cos v0 = n− z, where z2 = 1 + n2 − 2|n|λ/ +√ +λ2 − 1, and +ε0 = −2λ+4|n| +√ +λ2 − 1. In all phases, the value of ξ sat- +isfies the regular expression ξ0 = ∂ε0/∂n. When λ > 1 +and n = 0 the SS phase transforms into a conventional +charge ordered (CO) phase with the checkerboard or- +dering. +The equations (49) in the case of λ < 0 have lo- +calized solutions with nonzero topological charge and +finite energy [69,71,72,73,74,75,76]. However, in our +case λ > 0, the numerical calculations with the con- +jugate gradient method for minimizing of the energy +functional (47) on the lattice 256×256 indicate the exis- +tence of similar solutions, at least, as metastable states. +The results are shown in Fig.5. The actual stability of +these solutions in our calculation was different. The SF- +phase solutions, similar to that of in Fig.5, cases a and +b, quickly evolved to an uniform one. The SS- and CO- +phase solutions, similar to that of in Fig.4, cases c and +d, retained its form for more than 106 iterations. +We investigated the asymptotic behavior of local- +ized solutions, suggesting that at r → ∞ they have the +form φ(r) = φ0+ ˜φ(r), u(r) = u0+˜u(r), v(r) = v0+˜v(r), +where ˜φ, ˜u, ˜v → 0 at r → ∞. Hereinafter, the index 0 +means the corresponding values for constant solutions. +The linearized system (49) for the functions ˜φ, ˜u, ˜v takes +the form + + + + + + + + + + + +fuv0 ˜φkk = 0, +F0 ˜ukk + 4F0˜u + 4 +� +−Fuv0 + ξ0 +4 cos v0 +� +˜v = 0, +F0 ˜vkk + 4F0˜v + 4 +� +−Fuv0 + ξ0 +4 cos u0 +� +˜u = 0. +(50) +In the case of the SF and SS phases, the solutions +for the first equation can be written as +˜φ(r) = +∞ +� +m=1 +cm +rm cos m(ϕ − ϕm), +(51) +with cm and ϕm determined by the boundary condi- +tions. In the case of the CO phase, the first equation +reduces to an identity since fuv0 = 0. +In the case of the SF phase, the second and the third +equations become independent Helmholtz equations for +the ferro- and antiferro-type combinations U = ˜u + ˜v, +V = ˜u − ˜v: +Ukk + A1 U = 0, +A1 = 4(λ + 1)(1 − n2) +λ − (λ + 1)n2 ; +(52) +Vkk + A2 V = 0, +A2 = 4λ − 1 − (λ + 1)n2 +λ − (λ + 1)n2 +. +(53) +The corresponding solutions have the form +Φ(r, ri) = +∞ +� +l=0 +alKl(r/ri) cos l(ϕ − αl), +(54) +Ψ(r, ri) = +∞ +� +l=0 +� +b1lJl(r/ri) cos l(ϕ − β1l) + ++ b2lYl(r/ri) cos l(ϕ − β2l) +� +, +(55) +where Kl are the Macdonald functions, Jl and Yl are +the Bessel functions of the first and second kind, and +al, αl, bkl, βkl, k = 1, 2 are some constants. An analysis +of the asymptotic behavior of solutions (54,55) and the +requirement that the omitted nonlinear terms in equa- +tions (50) are small as compared with the remaining +linear terms point to Ψ = 0. The account in the low- +est order of the mixing with the function ˜φ does not +change V (r) and gives additional term in U(r) having +asymptotic behavior: +U1(r) ≈ − +nc2 +m +2(λ + 1) +√ +1 − n2 +m2 +r2m+2 , +(56) +where m is the number that specifies first nonzero term +in (51). +Line n2 = λ/(λ + 1) is the boundary of areas of +the SF-phase with a different behavior of the U and V +functions +n2 > +λ +λ + 1 : U(r) = Φ(r, r1) + U1(r), V (r) = 0; +(57) +n2 < +λ +λ + 1 : U(r) = U1(r), V (r) = Φ(r, r2). +(58) +Here we define characteristic lengths r−2 +i += |Ai|. +In the case of the SS phase, we need to define the +ferri-type combinations: ˜U = A˜u + ˜v and ˜V = A˜u − ˜v, +where A = −Fuv0+ ξ0 +4 cos u0. The equations (50) lead to + +Topological structures in unconventional scenario for 2D cuprates +19 +Fig. 5 (color online) Nonuniform states in 2D system of +charged hard-core bosons. The left panels show local charge +density ni = σzi = cos θi. The difference of sublattice states +is clearly evident in the cases b, c and d. The right panels show +phase flow of planar components of the pseudospin, σxi and +σyi. The phase flow reveals the vortex-antivortex pair struc- +ture in the core of inhomogeneity of the local charge density +in the cases a, b and c. The parameters of the model are: a) +n = 0.2, λ = 0.5 (SF phase); b) n = 0.1, λ = 0.9 (SF phase); +c) n = 0.2, λ = 1.5 (SS phase); d) n = 0.0, λ = 1.5 (CO +phase). These sets of parameters are denoted with letters a-d +on the ground-state phase diagram in Fig.6. +Helmholtz equation for the ˜U function having solution +Ψ(r, r3), r−2 +3 += 8. As in previous case we have to put +Ψ = 0. The ˜V function obeys the Laplace equation. +Taking into account the mixing in the lowest order with +the function ˜φ we come to the expressions as follows +˜U(r) = c2 +m +8 (Afu0 + fv0) +m2 +r2m+2 , +(59) +˜V (r) = +∞ +� +l=1 +Cl +rl cos l(ϕ − γl) + ++ c2 +m +4 (Afu0 − fv0) +1 +r2m , +(60) +where m is the number that specifies first non-zero term +in (51), and the expressions fu0 = − sin u0 cos v0, fv0 = +Fig. 6 (color online) The ground state phase diagram of the +hc bosons in the mean-field approximation. The solid line +corresponds to SF-SS phase boundary, n2 = (λ − 1)/(λ + 1). +The thick line at λ > 1, n = 0 shows CO phase. The dotted +line, n2 = λ/(λ+1), separates the two types of the asymptotic +behavior in accordance with the expressions (57) and (58). In +the shaded areas inside the SF phase region the characteristic +lengths satisfy to inequalities ri > 1. The letters a-d in the +circles correspond to the parameter sets in Fig.5. +− cosu0 sin v0 are determined by the expressions for the +uniform solutions. +Similarly the case of the CO phase, we obtain +˜φ(r) = 0, +U(r) = 0, +V (r) = Φ(r, r4), +(61) +where r−2 +4 += 4(λ − 1). +The analysis of the asymptotic behavior of the lo- +calized states reveals qualitative differences of the finite +energy excitations in the SF, SS, and CO phases. +In the SF phase an asymptotic of the polar angle of +the pseudospin vector is determined by the expressions +(57, 58). When comparing these results with numerical +calculations it is worth to note that the characteris- +tic lengths obey to inequality ri < 1 in the most part +of the phase diagram in the SF phase except for the +areas indicated shadowed in Fig.6, so the function Φ +goes to zero value very fast with increasing of r. On +the contrary, the asymptotic behavior of the azimuthal +angle of the pseudospin (51) has no characteristic scale. +This means that in the SF phase the main excitations +are almost in-plane vortex-antivortex pairs. They have +well localized out-of-plane core of the ferro-type, with +σzA = σzB, as shown in Fig.5a, and become the pure +in-plane ones at n = 0 in accordance with expression +(56). The same type of localized solutions was found by +the authors of Ref. [73]. +For the hc bosons, the polar angle is related with the +density of bosons, while the azimuthal angle is respon- +sible for the superfluid density, hence these states cor- +respond to the excitation of the superfluid component + +2.0 +? +O'S20 +A.S. Moskvin, Yu.D. Panov +with highly localized heterogeneity of bosons density in +the foci of the vortex-antivortex pairs. In the shaded +region in the SF phase near the border of the SF-SS +phases in Fig.6, the antiferro type vortices (see Fig.5b), +with σzA ̸= σzB, begin to dominate, their inflation is +preceded by a change of the homogeneous ground state +from SF to SS phase. +The CO phase has no linear excitation of ˜φ. The +characteristic lengths of the azimuthal excitations (61) +are small except the region near λ = 1. This results +in a high stability of the homogeneous CO phase. A +typical picture of the nonuniform state shown in Fig.5d +is represented by linear domains of the CO phase. The +non-zero values of the SF order parameter are realized +within the domain walls, thus giving rise to appearance +of a filamentary superfluidity for hc bosons. +In the SS phase the asymptotic behavior of the po- +lar and azimuthal excitations is qualitatively the same +without characteristic scales. Hence in this case there +are skyrmion-like excitations as shown in Fig.5c. For +the hc bosons these coherent states include both the +excitations of the superfluid component and the boson +density. In a center of skyrmion the difference σzA−σzB +has maximal magnitude, that corresponds to CO phase, +and near there is a region where σzA − σzB = 0, that +corresponds to SF phase. So, the skyrmion-like excita- +tions in the SS phase generate the topological phase +separation. Note, that another type of instability in SS +phase were also found by Quantum Monte Carlo calcu- +lations [103]. +6.3 Computer simulation of the domain structures +Making use of a special algorithm for CUDA architec- +ture for NVIDIA graphics cards that implies a nonlinear +conjugate-gradient method to minimize energy func- +tional and Monte-Carlo technique we have been able +to directly observe formation of the ground state con- +figuration for the 2D hard-core bosons with lowering +the temperature and its transformation with increase +the temperature and boson concentration, allowing us +to examine earlier implications and uncover novel fea- +tures of the phase transitions, in particular, look upon +the nucleation of the odd domain structure, the local- +ization of the bosons doped away from half-filling, and +the phase separation regime. The accuracy of numeri- +cal calculations has been limited making it possible to +reproduce the effect of minor inhomogeneities common +to any real crystal. +We started with the hard-core boson Hamiltonian +(30) on a 256×256 square lattice at half-filling (n = 1/2) +given the value of the inter-site repulsion Vnn = V = +3tnn, which is the typical one for many papers with +QMC calculations for 2D hard-core bosons [103,104, +60]. It should be noted that hereafter we follow the +notations in these papers. +First, we addressed the formation of the phase state +under the annealing (thermalization) procedure. The +algorithm starts initially with T set to a high value +T ∼ 2Tcr. The annealing is accompanied by formation +a fragile unstable CO domain structure with antiphase +180◦ domain walls whose center is characterized by a +large nonzero SF order parameter which is suppressed +as one runs deep into the CO domain thus suggest- +ing the presence of a fragile filamentary superfluidity +(FLSF) nucleated at the antiphase domain walls. The +term ”filamentary superfluidity” is a full analogue of +a more familiar term ”filamentary superconductivity”. +Here, in the paper, ”filamentary superfluidity” is re- +lated with an antiphase domain wall (the wall between +two CO domains) which is characterized by a nonzero +superfluid order parameter. +Typically for small and moderate anisotropy the an- +nealing is finished by formation a system of domains +with closed-loop domain walls which quickly collapse +thus setting an uniform single-domain CO ground state +with a hardly noticeable remnant inhomogeneity. At the +lowest temperatures we can form an almost ideal charge +ordered checkerboard structure at half-filling that does +not modify with increasing the temperature up to TCO. +Weak deviation away from half-filling in such a case +gives rise to nucleation of two types of topological de- +fects. Unconventional small nanoscopic defects with an +effective radius of several lattice separations accommo- +date a single boson and are characterized by a strong +distortion of the CO order extended on the three-four +coordination spheres with emergence of the local super- +fluid order. Large topological defects (droplets, ”blobs”) +having mainly a circular shape can accommodate many +bosons. These are comprised of a superfluid core and a +ring shaped supersolid boundary. Computer simulation +reveals occurrence of a critical radius for stability of the +”large” cylindrical defects. Given the increasing doping +we arrive at the growing volume fraction of the large +defects, the change of their shape and their confluence +up to a full phase transformation. At the same time, it +is worth noting the persistence of the decreasing vol- +ume fraction of the checkerboard charge order up to +very large doping. +However, systematic studies have indicated that +in some cases there occurs a low-temperature CO +domain structure with stable stripe-like disconnected +(within our lattice size) domain walls oriented along +main lattice axes. Along with a simple uniform (”ferro- +magnetic”) SF phase parameter distribution these 1D +walls can have unconventional multidomain topological + +Topological structures in unconventional scenario for 2D cuprates +21 +0.03 +0.01 +0.10 +0.15 +0.20 +0.22 +0.05 +SS +CO +SF +0.0 +1.0 +1.0 +0.03 +0.05 +0.01 +0.00 +0.10 +0.20 +0.15 +0.25 +0.20 +0.30 +0.22 +0.35 +0.05 +0.10 +a) +b) +c) ++ +- +- +V/t=9 +V/t=3 +V/t=3 +CO +SF +CO +SF +Fig. 7 (Color online) Evolution of the hc-boson ground state configuration under doping away from half-filling. a) At ∆n = 0.01 +one observes a sudden nucleation of a rather large ”blob” composed of a SF core and a ring shaped SS boundary that does +accommodate all the ”injected” bosons. Under further deviation from half-filling the blob grows up to a full CO-SF phase +transformation close to a critical value of ∆n = 0.225; b) Doped bosons do localize in narrow FLSF DWs of the striped CO +phase leading to their broadening. In a well developed phase separation regime we arrive at a system of nearly parallel CO +and SF domains separated by the SS DWs. The ”plus” and ”minus” signs point to different CO domains. Orientation of the +phase angle ϕ within domain walls is demonstrated schematically for ∆n = 0.1.; c) At ∆n < 0.01 the doped bosons do localize +in the center of the narrow DWs breaking the FLSF without any visible transformation of the domains. The regular DW shape +breaks under further doping, these are nonuniformly ”swelled” with the emergence and rise of widenings. Step by step these +widenings and ”blobs” nucleated inside domains spread until these cover all lattice. The CO domain topology survives up to +very high doping. Different color in a), b), c) does highlight the value of the order parameters. +structure of the SF phase order parameter with a high +density of 2π domain walls separating the 1D phase +domains. +Evolution of the ”uniform” and ”striped” hc-boson +ground state configurations under doping away from +half-filling is shown in Fig. 7a and Fig. 7b, respectively, +for a ”moderate” anisotropy V = 3t. +Minor deviation away from half-filling at ∆n ≤ 0.01 +practically does not lead to visible effects by slightly +perturbing the remnant inhomogeneity of the initial +”uniform” state. However, at ∆n ≈ 0.01 we observe a +sudden nucleation of rather large topological defect(s) +(droplets, ”blobs”) having mainly a circular shape that +can accommodate all the ”injected” bosons thus mak- +ing the surrounding CO phase more uniform. These +droplets are comprised of a superfluid core and a ring +shaped supersolid boundary. Given the increasing dop- +ing we arrive at a well developed phase separation with +the growing volume fraction of the large defects, the +change of their shape and their confluence up to a full +CO-SF phase transformation close to a critical value of +∆ncr ≈ 0.22. +Evolution of the ”striped” CO phase with a filamen- +tary superfluidity under deviation away from half-filling +goes in different scenario because of the doped bosons +do localize in the center of the narrow domain walls +leading to their uniform broadening up to formation +of SF domains. Interestingly, the structure of the final +SF phase in this case depends on the initial topologi- +cal structure of the SF phase parameter (ϕ) within 1D +domain walls. In Fig. 7b we started with the two 1D do- +main walls with an uniform distribution of the SF phase +order parameter for lower wall and with 2π domain wall +separating the 1D phase domains for the upper wall. +Orientation of the phase angle ϕ within domain walls +is demonstrated schematically in Fig. 7b for ∆n = 0.1. +In a well developed phase separation regime we arrive +at a system of nearly parallel CO and SF domains sep- +arated by supersolid domain walls. However, the regu- +lar domain structure becomes more and more unstable +the nearer we get to the CO-SF phase transition point. +Near ∆n ≈∆ncr the CO ”remnants” do collapse to a 0D +skyrmion-like topological defect obviously related with +2π domain wall separating the 1D phase domains. This +point defect survives up to a maximal doping. Interest- +ingly, such evolution cannot be replicated under SF-CO +transition with lowering the doping. The initial stripe +structure does not restore, instead, we arrive at creation +of an unconventional skyrmion-like cylindrical defect in +the CO matrix that does collapse with approaching to +the half-filling. +We have also performed computer modeling of the +CO-SF phase transition in 2D hc-boson system with +a strong Ising anisotropy V = 9t. At variance with + +22 +A.S. Moskvin, Yu.D. Panov +the previous situation of moderate anisotropy the ther- +malization procedure for strong anisotropy results in +emergence of a well developed rigid domain structure +with 180◦ domain walls whose center is characterized +by a large nonzero superfluid order parameter which +is suppressed as one runs deep into the CO domain +thus suggesting the presence of a filamentary superflu- +idity nucleated at the antiphase domain walls, that is a +gossamer superfluidity, or superfluidity existing in the +presence of the ”insulating” charge order. It is worth +noting that the wall width increases under rise of the +boson transfer integral. +Weak deviation away from half-filling (∆n ≤ 0.01) +does not give rise to visible modification of the domain +structure (see Fig. 7c) because the doped bosons do lo- +calize in the center of the narrow domain walls breaking +the filamentary superfluidity without any visible trans- +formation of the domains. However, further rise of the +domain wall loading results in their puzzling transfor- +mation in a very small doping range. Tiny amounts +of excess bosons suffice to destroy regular narrow do- +main walls. At the very beginning, the overloading is +accompanied by formation of different unstable in-wall +structures, in particular, ladder-like patterns formed by +bosonic dimers. However, then the ”completely filled” +domain wall becomes regularly broadened because ex- +tra bosons prefer to occupy delocalized states beyond +the wall center thus forming a rather extended shell +with an inhomogeneous distribution of the SF and +SS order parameters. However, the regular wall shape +breaks under further doping, these are nonuniformly +”swelled” with the emergence and rise of widenings. +Step by step these widenings and ”blobs” nucleated +inside domains spread until these cover all the lattice +(see Fig. 7c). The Fig. 7c does well illustrate a peculiar +”memory” effect: the CO domain topology survives up +to very high doping though the domain wall structure +changes significantly. +7 Conclusion +We addressed a minimal toy model to describe the +charge degree of freedom in the CuO2 planes with +the on-site Hilbert space reduced to only a charge +triplet of the three effective valence centers (nominally +Cu1+;2+;3+), and made use of the S=1 pseudospin for- +malism. It does introduce the on-site mixed valence +quantum superpositions, that, at variance with classical +spins or quantum s=1/2 spins, should be described by +two classical vectors. The formalism provides an uni- +fied standpoint for classification of the ”myriad” of +electronic charge phases in cuprates and their evolu- +tion under a nonisovalent doping. Despite its simplic- +ity the S=1 formalism is shown to constitute a power- +ful method to describe and study complex phenomena +in parent and doped cuprates, in particular , a com- +prehensive description of the correlated one- and two- +particle transport, coexistence of p- and n-type car- +riers, electron-hole asymmetry, anticorrelation of con- +ventional spin and superconducting order parameters. +Concept of the electron and hole centers, differing by +a composite (two electrons/holes) local boson, is be- +lieved to explain central points of the cuprate puzzles, +in particular, the HTSC itself as a condensation of com- +posite bosons and the pseudogap phenomena to be re- +sult of the charge order. Our scenario points namely +to a charge bosonic degree of freedom engaged by a +strong electron-lattice polarization to be responsible +for the high-Tc effect in cuprates while the spin de- +gree of freedom does compete with it to reduce high- +Tc’s. The 2D S=1 pseudospin system is prone to a +creation of different topological structures which form +topologically protected inhomogeneous distributions of +the eight local S=1 pseudospin order parameters in- +cluding charge density and superfluid order parame- +ters. Pseudospin formalism with the two-vector geo- +metrical representation of the on-site states is shown +to be the most powerful technique to describe topolog- +ical structures. We presented a short overview of differ- +ent localized topological structures which are typical for +the S=1 (pseudo)spin systems, in particular, localized +topological excitations in “negative-U” model which is +equivalent to s=1/2 pseudospin system. We argue that +even the parent insulating cuprates may be unstable +with regard to nucleation of topological defects such +as “strange“ antiphase domain walls or unconventional +localized single- or multi-center skyrmion-like objects +with filamentary or ring-shaped superfluid regions. Puz- +zlingly these topological structures with complex distri- +bution of the off-diagonal order parameters can be in- +visible for X-rays due to the same uniform distribution +of the mean on-site charge density as in the bare par- +ent monovalent (insulating) phase. Making use of the +computer simulation we have demonstrated evolution +of different starting charge ordered phases in a model +cuprate with large “negative-U” under deviation from +half-filling and a step-by-step transformation into su- +perfluid phase. +In summary, the S=1 pseudospin formalism is be- +lieved to provide a conceptual framework for an in- +depth understanding and a novel starting point for ana- +lytical and computational studies of high-Tc supercon- +ductivity and other puzzles in cuprates. +Acknowledgements One of the authors (ASM) would like +to thank A. Bianconi, R. Micnas, A. 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DOI 10.1103/PhysRevB.65.014513 + diff --git a/39FKT4oBgHgl3EQf8y5R/content/tmp_files/load_file.txt b/39FKT4oBgHgl3EQf8y5R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e8036a4c0a46b0607a63a0340e714573f8dea2a --- /dev/null +++ b/39FKT4oBgHgl3EQf8y5R/content/tmp_files/load_file.txt @@ -0,0 +1,1769 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf,len=1768 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='11951v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='str-el] 27 Jan 2023 Journal of Superconductivity and Novel Magnetism manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' (will be inserted by the editor) Topological structures in unconventional scenario for 2D cuprates A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Moskvin · Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Panov Received: date / Accepted: date Abstract Numerous experimental data point to cuprates as d-d charge transfer unstable systems whose description implies the inclusion of the three many-electron valence states CuO7−,6−,5− 4 (nominally Cu1+,2+,3+) on an equal footing as a well-defined charge triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' We introduce a minimal model to describe the charge degree of freedom in cuprates with the on-site Hilbert space reduced to only the three states and make use of the S=1 pseudospin formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The formalism constitutes a powerful method to study complex phe- nomena in interacting quantum systems characterized by the coexistence and competition of various ordered states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Overall, such a framework provides a simple and systematic methodology to predict and discover new kinds of orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In particular, the pseudospin formal- ism provides the most effective way to describe different topological structures, in particular, due to a possibil- ity of a geometrical two-vector description of the on-site states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' We introduce and analyze effective pseudospin Hamiltonian with on-site and inter-site charge corre- lations, two types of a correlated one-particle trans- fer and two-particle, or the composite boson transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The latter is of a principal importance for the HTSC perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The 2D S=1 pseudospin system is prone to a creation of different topological structures, which form topologically protected inhomogeneous distribu- tions of the eight local S=1 pseudospin order parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' We present a short overview of localized topologi- cal structures, typical for S=1 (pseudo)spin systems, fo- cusing on unexpected antiphase domain walls in parent cuprates and so-called quadrupole skyrmion, which are believed to be candidates for a topological charge ex- A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Moskvin · Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Panov Ural Federal University, Ekaterinburg, 620083, Russia E-mail: alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='moskvin@urfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='ru citation in parent or underdoped cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Puzzlingly, these unconventional structures can be characterized by an uniform distribution of the mean on-site charge, that makes these invisible for X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Quasiclassical approx- imation and computer simulation are applied to ana- lyze localized topological defects and evolution of the domain structures in ”negative-U” model under charge order-superfluid phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Keywords high-Tc cuprates · charge degree of freedom · S=1 pseudospin formalism · topological structures · unconventional skyrmions 1 Introduction The origin of high-Tc superconductivity [1] is presently still a matter of great controversy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Both copper and novel non-copper based layered high-Tc materials re- veal normal and superconducting state properties very different from that of standard electron-phonon coupled ”conventional” superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Copper oxides start out life as insulators in con- trast with BCS superconductors being conventional metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Unconventional behavior of these materials un- der charge doping, in particular, a remarkable interplay of charge, lattice, orbital, and spin degrees of freedom, strongly differs from that of ordinary metals and merely resembles that of a doped Mott insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In addition to the occurrence of unconventional d-wave superconduc- tivity the phase diagram of the high-Tc cuprates does reveal a flurry of various anomalous electronic prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In normal state, these materials exhibit non- Fermi liquid properties and enter a mysterious pseu- dogap (PG) regime, characterized by the observation of multiple crossover PG temperatures T∗’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Moskvin, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Panov The exotic superconductors differ from ordinary Bardeen-Cooper-Schrieffer (BCS) superconductors in many other points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Thus, muon spin relaxation (µSR) measurements of the magnetic field penetration depth revealed nearly linear relationship between Tc and the superfluid density in high-Tc cuprates and many other exotic superconductors that cannot be expected in BCS theory, but is typical for Bose-Einstein condensation (BEC) of preformed pairs[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Bosonic scenario for high- Tc cuprates [3] has been elaborated by many authors, in particular, by Alexandrov (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' [4]) who con- sidered real-space bipolarons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' It is worth noting that many reasonable predictions of the bipolaronic theory are valid for any local bosons irrespective of its micro- scopic mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Numerous observations point to the possibility that high-Tc cuprate superconductors may not be conventional BCS or BEC superconductors, but rather manifest a boson-fermion competition in a strug- gle for the electronic ground state, in particular, a com- petition between the two- and one-particle transport with resistivity ∝T and ∝T2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The most part of the current scenarios, includ- ing the Hubbard and t − J-models, spin fluctuations, Alexandrov-Mott bipolarons [4] consider cuprates to be homogeneous systems and ignore numerous signatures of the electron and crystalline inhomogeneity [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Both the normal and high-transition-temperature (high-Tc) superconducting (SC) state in cuprates is be- lieved to be electronically inhomogeneous, in particu- lar, due to a quenched disorder, arising from dopants and/or nonisovalent substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' However, the dopant- induced impurity potential, seemingly being a natural source of electron inhomogeneity, varies widely among the cuprates that cannot explain observation of an uni- versal, scaling behavior evidencing for an intrinsic elec- tronic tendency toward inhomogeneity in CuO2 planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' This intrinsic propensity can be stimulated, firstly, by a local out-of-plane nonisovalent substitution, toward formation of in-plane universal inhomogeneity centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Another stimulating factor of the intrinsic electronic in- homogeneity is related with a two-dimensionality and a competition and intertwinnig of charge, spin and or- bital degrees of freedom in CuO2 planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Concept of phase separation and percolation phenomena [6,7,8,9, 10], stripes [11,12,13], large polarons [14,15], nucleation of the mixed valence PJT-(pseudo-Jahn-Teller) phase [16] has appeared to be very fruitful for explanation of many puzzling properties of cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Furthermore, some authors [17,18,19] associate the anomalous prop- erties of cuprates with quasi-2D structure of the active CuO2 layers and different topological defects, or vortex- like solitons to be specific collective excitation modes of the 2D vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The topological order inherent in the doped cuprate endows it with tremendous amount of robustness to var- ious unavoidable ”real-life” material complications [20], such as impurities and other coexisting broken sym- metries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In general, such a complex, multiscale phase separation does challenge theories of high-temperature superconductivity that include complexity [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Recently [22] we argued that an unique prop- erty of high-Tc cuprates is related with a dual na- ture of the Mott insulating state of the parent compounds that manifests itself in two distinct en- ergy scales for the charge transfer (CT) reaction: Cu2+ + Cu2+ → Cu1+ + Cu3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Indeed, the d - d CT gap as derived from the optical measurements in parent cuprates such as La2CuO4 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='0 eV while the true (thermal) d - d CT gap, or effective correlation param- eter Ud, appears to be as small as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' It means cuprates should be addressed to be d-d CT unstable systems whose description implies accounting of the three many-electron valence states CuO7−,6−,5− 4 (nom- inally Cu1+,2+,3+) on an equal footing as a well-defined charge triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' This allows us to introduce a minimal model for cuprates with the on-site Hilbert space re- duced to only three states, three effective valence cen- ters CuO7−,6−,5− 4 (Cu1+,2+,3+) where the electronic and lattice degrees of freedom get strongly locked to- gether, and make use of the S=1 pseudospin formal- ism [22,23,24,25,26,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Such a formalism constitutes a powerful method to study complex phenomena in in- teracting quantum systems characterized by the coex- istence and competition of various ordered states [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Overall, such a framework provides a simple and sys- tematic methodology to predict and discover new kinds of orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In particular, the pseudospin formalism pro- vides the most effective way to describe different topo- logical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 1 we in- troduce a working model for the CuO4 centers based on assumption that the three many-electron valence states CuO7−,6−,5− 4 (nominally Cu1+,2+,3+) form the “on-site” Hilbert space of the CuO4 plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' We restricted ourselves only by the consideration of the charge degree of freedom and have suggested simple geometrical vector representation for the on-site charge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 2 we have addressed an effectuve pseu- dospin Hamiltonian for the model cuprate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 3 we have considered several simplified versions of the gen- eral Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 4 is devoted to description of un- conventional localized topological structures typical for 2D S=1 (pseudo)spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 5 we considered localized topological structures in a limiting case of the model, or so-called ”negative-U” model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' A brief sum- mary is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Topological structures in unconventional scenario for 2D cuprates 3 2 Working model of the CuO4 centers Hereafter we consider the CuO4 plaquette to be a main element of crystal and electron structure of high-Tc cuprates and introduce a simplified toy model with the “on-site” Hilbert space of the CuO4 plaquettes reduced to states formed by only three effective valence centers [CuO4]7−,6−,5− (nominally Cu1+,2+,3+, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The centers are characterized by different conventional spin: s=1/2 for Cu2+ center and s=0 for Cu1+,3+ cen- ters, and different orbital symmetry:B1g for the ground states of the Cu2+ center, A1g for the Cu1+ centers, and the Zhang-Rice (ZR) A1g or more complicated low- lying non-Zhang-Rice states for the Cu3+ center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Elec- trons of such configurations cannot be treated through a mean-field independent particle approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' therefore, their behavior is studied in terms of auxiliary neither Fermi nor Bose quasiparticles, representing combina- tions of atomic-like many-electron configurations[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The key problem that arises from the strong corre- lations in the normal state of the copper-oxide super- conductors is identifying the weakly interacting entities that make a particle interpretation of the current possi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' All formulations of superconductivity are reduced to a pairing instability of such well-defined quasiparti- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' However, there is good reason to believe that the construction of such entities may not be possible [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' |Ψ⟩ = c−1|Cu1+⟩ + c0|Cu2+⟩ + c1|Cu3+⟩, (1) Such an approach immediately implies introduction of the unconventional on-site quantum superpositions that points to many novel effects related with local CuO4 centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Validity of such a model implies well iso- lated ground states of the three centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' This surely holds for the 1A1g singlet ground state of the Cu1+ centers with nominally filled 3d shell whose excita- tion energy does usually exceed 2 eV (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' [31] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The b1g ∝ dx2−y2 character of the ground hole state in CuO6− 4 cluster (Cu2+ center) seems to be one of a few indisputable points in cuprate physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' A set of low-lying excited states with the en- ergy ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='5 eV includes bonding molecular orbitals with a1g ∝ dz2, b2g ∝ dxy, and eg ∝ dyz, dxz symmetry, as well as purely oxygen nonbonding orbitals with a2g(π) and eu(π) symmetry (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' [32,33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In 1988 Zhang and Rice [34] have proposed that the doped hole in a parent cuprate forms a Cu3+ center with a well isolated local spin and orbital 1A1g singlet ground state which involves a phase coherent combi- nation of the 2pσ orbitals of the four nearest neighbor oxygens with the same b1g symmetry as for a bare Cu 3dx2−y2 hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The Zhang-Rice (ZR) singlet is a leading paradigm in modern theories of high-temperature su- perconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' However, both numerous experimental data and the cluster model calculations suggest the in- volvement of some other physics which introduces low- lying states into the excitation of the doped-hole state, or competition of conventional ZR singlet with another electron removal state(s), in particular, formed by the hole occupation of the oxygen nonbonding a2g(π) and eu(π) orbitals[32,33,35,36,37], the a2g(π) orbital to be the lowest in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Unified selfconsistent description of the charge, spin, and orbital degrees of freedom for CuO4 centers with mixed valence is a hardly solvable task so we are forced to address simplified model approaches focusing on the quantum description of the charge degree of freedom that is responsible for superconductivity in cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='1 The charge triplet model: S=1 pseudospin formalism To describe the diagonal and off-diagonal, or quantum local charge order we start with a simplified charge triplet model that implies a full neglect of spin and or- bital degrees of freedom [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Three charge states of the CuO4 plaquette: a bare center M 0=CuO6− 4 , a hole cen- ter M +=CuO5− 4 , and an electron center M −=CuO7− 4 are assigned to three components of the S=1 pseu- dospin (isospin) triplet with the pseudospin projections MS = 0, +1, −1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Obviously, the model resembles that of so-called semi-hard-core bosons [38], which are described by extended Bose-Hubbard model that assumes a truncation of the on-site Hilbert space to the three lowest occupation states n = 0, 1, 2 with further mapping to an anisotropic spin-1 model (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='[39,40,41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' For 2D cuprates these states cor- respond to a ”electron” CuO7− 4 (Cu1+), ”bare” CuO6− 4 (Cu2+), and ”hole” CuO5− 4 (Cu3+) centers, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The S=1 (pseudo)spin algebra includes eight inde- pendent nontrivial pseudospin operators, three dipole and five quadrupole operators: ˆSz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' ˆS± = ∓ 1 √ 2 (Sx ± iSy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' (2) ˆS2 z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' ˆT± = {Sz, S±};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' ˆS2 ± = 1 2( ˆS2 x − ˆS2 y ± i{ ˆSx, ˆSy}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' One should note a principal difference between the s=1/2 and S=1 quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The only on-site or- der parameter in the former case is an average spin moment ⟨Sx,y,z⟩, whereas in the latter one has five ad- ditional ”spin-quadrupole”, or spin-nematic order pa- rameters described by traceless symmetric tensors Qij = ⟨1 2{Si, Sj} − 2 3δij⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' (3) 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Moskvin, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Panov Interestingly, that in a sense, the S = 1 2 quantum spin system is closer to a classic one (S → ∞) with all the order parameters defined by a simple on-site vectorial order parameter ⟨S⟩ than the S=1 quantum spin system with its eight independent on-site order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' It is worth noting that the three spin-linear (dipole) operators ˆSx,y,z and five independent spin-quadrupole operators Qij = 1 2{ ˆSi, ˆSj} − 1 3 ˆS2δij at S=1 form eight Gell-Mann operators being the generators of the SU(3) group [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' To describe different types of pseudospin ordering in a mixed-valence system we have to introduce eight local (on-site) order parameters: two classical (diagonal) or- der parameters: ⟨Sz⟩ being a ”valence”, or charge den- sity with an electro-neutrality constraint, and ⟨S2 z⟩ be- ing the density of polar centers M ±, or ”ionicity”, and six off-diagonal order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The off-diagonal or- der parameters describe different types of the valence mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' It should be emphasized that for the S=1 (pseudo)spin algebra there are two operators: S± and T± = {Sz, S±} that change the pseudo-spin projection by ±1, with slightly different properties ⟨0| ˆS±| ∓ 1⟩ = ⟨±1| ˆS±|0⟩ = ∓1, (4) but ⟨0| ˆT±| ∓ 1⟩ = −⟨±1|( ˆT±|0⟩ = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' (5) It is worth noting the similar behavior of the both op- erators under the hermitian conjugation: ˆS† ± = − ˆS∓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' ˆT † ± = − ˆT∓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The ˆS2 ± operator changes the pseudospin projection by ±2 with the local order parameter ⟨S2 ±⟩ = 1 2(⟨S2 x − S2 y⟩ ± i⟨{Sx, Sy}⟩) = (6) = c∗ +c− = c2 x − c2 y ± 2icxcy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Obviously, this on-site off-diagonal order parameter is nonzero only when both c+ and c− are nonzero, or for the on-site ”electron-hole” M −(Cu1+)-M +(Cu3+) su- perpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' It is worth noting that the ˆS2 + ( ˆS2 −) opera- tor creates an on-site hole (electron) pair, or composite boson, with a kinematic constraint ( ˆS2 ±)2 = 0, that un- derlines its ”hard-core” nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Both ˆS+( ˆS−) and ˆT+( ˆT−) can be associated with the single particle creation (annihilation) operators, however, these are not standard fermionic ones, as well as ˆS2 +( ˆS2 −) operators are not standard bosonic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Nevertheless, namely ⟨S2 ±⟩ can be addressed as a local superconducting order parameter The two operators, S± and T± are related with the two different types of a correlated single-particle trans- port, these change the pseudospin projection by ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In lieu of these operators one may use two novel operators: ˆP± = 1 2( ˆS± + ˆT±);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' ˆN± = 1 2( ˆS± − ˆT±) , which do real- ize transformations Cu2+↔Cu3+ and Cu1+↔Cu2+, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In other words, for parent cuprates these are the hole and electron creation operators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The boson-like pseudospin raising/lowering operators ˆS2 ± do change the pseudo-spin projection by ±2 and define a local nematic order parameter ⟨S2 ±⟩ = 1 2(⟨S2 x − S2 y⟩ ± i⟨{Sx, Sy}⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' (7) This on-site off-diagonal order parameter with the d-type symmetry is nonzero only for the on-site M −(Cu1+)-M +(Cu3+) superpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' It is worth not- ing that the ˆS2 + ( ˆS2 −) operator creates an on-site hole (electron) pair, or composite boson, with a kine- matic constraint ( ˆS2 ±)2 = 0, that underlines its ”hard- core” nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Obviously, the pseudospin nematic aver- age ⟨S2 ±⟩ can be addressed to be a local complex super- conducting order parameter: ⟨S2 ±⟩ = |⟨S2 ±⟩|e±iϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' (8) Both ˆS+( ˆS−) and ˆT+( ˆT−) can be anyhow related with conventional single particle creation (annihilation) op- erators, however, these are not standard fermionic ones, as well as ˆS2 +( ˆS2 −) operators are not standard bosonic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' It should be noted again that the pseudospin op- erators are not to be confused with real physical spin operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' they act in a pseudo-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='2 Simple ”geometrical” representation of the on-site charge states Making use of a simple classical representation of on- site spin states using arrows is a popular and useful method for describing spin structures through vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' However, such an approach works only for classi- cal spins and, under certain limitations, for spin s=1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Indeed, for the classical spin all the on-site spin order parameters are derived through ⟨S⟩, while for s=1/2 ⟨S⟩ is the only local spin order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' At variance with s=1/2 systems for S=1 systems we have addi- tional spin-quadrupole order parameters whose descrip- tion cannot be realized within framework of a classical ”single-arrow” representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Nevertheless, hereafter we propose a novel ”geometrical” representation that allows us to selfconsistently describe all the on-site S=1 states and make use of the 2D vector fields to describe uniform and nonuniform configurations for model 2D cuprate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' In particular, the vector field patterns are of a great importance for physically clear representation of the complex topological structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Topological structures in unconventional scenario for 2D cuprates 5 Instead of the three |1M⟩ states one may use the Cartesian basis set Ψ, or |x, y, z⟩: |10⟩ = |z⟩ , |1 ±1⟩ = ∓ 1 √ 2(|x⟩ ± i|y⟩), (9) so that the on-site wave function can be written in the matrix form as follows [42]: ψ = \uf8eb \uf8ed c1 c2 c3 \uf8f6 \uf8f8 = \uf8eb \uf8ed R1 exp(iΦ1) R2 exp(iΦ2) R3 exp(iΦ3) \uf8f6 \uf8f8 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' |R|2 = 1 , (10) with R = {sin Θ cos η, sin Θ sin η, cos Θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Obviously, the minimal number of dynamic variables describing an isolated on-site S=1 (pseudo)spin center equals to four, however, for a more general situation, when the (pseudo)spin system represents only the part of the big- ger system, and we are forced to consider the coupling with the additional degrees of freedom, one should con- sider all the five non-trivial parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The pseudospin matrix has a very simple form within the |x, y, z⟩ basis set: ⟨i| ˆSk|j⟩ = iǫikj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' (11) We start by introducing the following set of S=1 coherent states characterized by vectors a and b satis- fying the normalization constraint[42] |c⟩ = |a, b⟩ = c · Ψ = (a + ib) · Ψ, (12) where a and b are real vectors that are arbitrarily ori- ented with respect to some fixed coordinate system in the pseudospin space with orthonormal basis e1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The two vectors are related by the normalization condition, so the minimal number of dynamic variables describing the S=1 (pseudo)spin system appears to be equal to four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Hereafter, we would like to emphasize the director nature of the c vector field: |c⟩ and | − c⟩ describe the physically identical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' It should be noted that in a real space the |c⟩ state corresponds to a quantum on-site superposition |c⟩ = c−1|Cu1+⟩ + c0|Cu2+⟩ + c+1|Cu3+⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' (13) Existence of such unconventional on-site superpositions is a princial point of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Below instead of a and b we will make use of a pair of unit vectors m and n, defined as follows [43]: a = cos ϕ m, b = sin ϕ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' (14) For the averages of the principal pseudospin opera- tors we obtain ⟨S⟩ = sin 2ϕ [m × n], (15) ⟨{Si, Sj}⟩ = 2(δij − cos2 ϕ mimj − sin2 ϕ ninj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' (16) Figure 1 shows orientations of the m and n vectors which provide extremal values of different on-site pseu- dospin order parameters given ϕ = π/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The monova- lent Cu2+, or M 0 center, is described by a pair of m and n vectors directed along Z-axis with |mz| = |nz| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' We arrive at the Cu2+-Cu3+ (M 0-M +) or Cu2+-Cu1+ (M 0-M −) mixtures if turn c−1 or c+1, respectively, into zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' The mixtures are described by a pair of m and n vectors whose projections on the XY-plane, m⊥ and n⊥, are of the same length and orthogonal to each other: m⊥·n⊥ = 0, m⊥ = n⊥ with [m⊥ ×n⊥] = ⟨Sz⟩ = ± sin2 θ for M 0-M ± mixtures, respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' It is worth noting that for ”conical” configurations in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 1b-1d: ⟨Sz⟩ = 0, ⟨S2 z⟩ = sin2 θ, ⟨S2 ±⟩ = −1 2 sin2 θ e±2iϕ, ⟨S±⟩ = − i √ 2 sin 2θ e±iϕ, ⟨T±⟩ = 0, (17) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 1b) ⟨Sz⟩ = 0, ⟨S2 z⟩ = sin2 θ, ⟨S2 ±⟩ = −1 2 sin2 θ e±2iϕ, ⟨S±⟩ = 0, ⟨T±⟩ = ∓ 1 √ 2 sin 2θ e±iϕ, (18) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 1c) ⟨Sz⟩ = −⟨S2 z⟩ = − sin2 θ, ⟨S2 ±⟩ = 0, ⟨S±⟩ = ⟨T±⟩ = ±1 2e∓i π 4 sin 2θ e±iϕ, (19) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Figures 1e,f do show the orientation of m and n vectors for the local binary mixture Cu1+-Cu3+, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content='1g does for monovalent Cu3+ center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' It is worth noting that for binary mixtures Cu1+-Cu2+ and Cu3+- Cu2+ we arrive at the same algebra of the ˆS± and ˆT± operators with ⟨S±⟩ = ⟨T±⟩, while for ternary mixtures Cu1+-Cu2+-Cu3+ these operators describe different ex- citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' Interestingly that in all the cases the local Cu2+ fraction can be written as follows: ρ(Cu2+) = 1 − ⟨S2 z⟩ = cos2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FKT4oBgHgl3EQf8y5R/content/2301.11951v1.pdf'} +page_content=' (20) 3 Effective S=1 pseudospin Hamiltonian Effective S=1 pseudospin Hamiltonian which does com- mute with the z-component of the total pseudospin � i Siz thus conserving the total charge of the system can be written to be a sum of potential and kinetic energies: ˆH = ˆHpot + ˆHkin , (21) where ˆHpot = � i (∆iS2 iz − µSiz) + � i 0. All the waveguide’s +optical properties relies on the parameters ∆n and d. +the fabrication process by developing the flash-TPP con- +cept, which combines TPP-DLW with ultraviolet (UV) +blanked illumination to efficiently polymerize an IC’s +non-light guiding volume in a single step9. We achieve +very symmetric splitting ratios in optical couplers, and +(for a first proof of concept) low propagation losses of +∼ 1.3 dB/mm and insertion losses of ∼ 0.26 dB. Fi- +nally, we printed optical waveguides on semiconductor +substrates hosting micro-lasers, demonstrating that our +concept is CMOS compatible. +II. +BASICS OF ADDITIVE FABRICATION +In the past 15 years, DLW and TPP have become +a versatile fabrication tool of polymer structures with +sub-micron dimensions10–12. +In contrast to 2D pla- +nar methods such as electron-beam lithography or +mask based lithography, +DLW allows for fabricat- +ing three-dimensional structures13. +DLW has played +a crucial role for many proof-of-concept designs in +optics7, acoustics14,15, elasticity13,16–18, robotics19 and +even electric transport20. +Major challenges such as +inclusion of conductive resins21, quantum-dots doped +resins22, liquid-crystals doped resins23 are still in the +development phase. +Recently, great progress towards +parallel direct-laser writing has been made, +which +enables a substantially accelerated fabrication process24. +Finally, different polymerization concepts are constantly +being developed, some of which use novel approaches +to high-resolution 3D printing based on polymer resins25. + +3 +III. +PHOTONIC INTEGRATION VIA +PHOTO-INDUCED POLYMERIZATION +Standard photonic waveguides covered in this review +rely the guiding element called the core having a higher +refractive index ncore than the refractive index of the +confining part called the cladding ncladding, i.e. ∆n = +ncore − ncladding > 0. +As schematically illustrated in +Fig. 1 (c), in such a configuration optical rays imping- +ing on the core-cladding interface with an angle smaller +than the critical angle θc = arcsin(1−(∆n/ncore)) exhibit +total internal reflection. As a consequence, they are con- +fined to the waveguide’s core and propagate along this +structure, allowing to direct optical propagation along +pre-designed paths via an integrated and solid core. +Refractive index contrast ∆n combined with the core +diameter d are a waveguide’s determining characteris- +tics, which determine a waveguide’s numerical aperture +NA = +� +n2core − n2 +cladding. The same holds for the num- +ber of spatial modes allowed to propagated through the +waveguide M ≈ V 2/2 = (4πd/λ)NA for large M, where λ +is the optical wavelength.. Here, V is the normalized fre- +quency a central indirect property of optical waveguides; +for V ≤ 2.405 a waveguide is single-mode, otherwise it +allows for higher modes to propagate. Finally, ∆n also +determines the minimal bending radius for which light +can be directed without exceedingly high losses. This in +turn is the limiting factor for integration density inside a +photonic IC. +In work covered in this review, we used the commer- +cial 3D direct-laser writing Nanoscribe GmbH (Photon- +ics Professional GT) system, which is equipped with a +femtosecond (fs) laser operating at 780 nm, and galvo- +mirrors for rapid beam movement in the lateral direc- +tions. The fs-laser is usually tightly focused into the resin +through an objective lens of high numerical aperture. Af- +ter finishing the TPP-DLW step, the unpolymerized resin +was removed in a two-step development process, immers- +ing the structure first in propylene-glycol-methyl-ether- +acetate (PGMEA) acting as a developer for 20 minutes, +followed by rinsing in isopropyl alcohol (2-propanol) for +3-5 minutes. For OPP, we deposited samples in the com- +mercial UV-chamber Rolence Enterprise Inc., LQ-Box +model, 405 nm wavelength, 150 mW/cm2 average light +intensity. +A. +Two-photon polymerization +Two-photon polymerization is a maskless direct-laser +writing technique26. A highly focused pulsed laser beam +in the femtosecond regime is used to induce the absorp- +tion of two-photons in the exposed volume inside the +photo-resist (which is a monomer in its liquid phase), +c.f Fig. 2 (a). This two-photon activated polymerization +creates long-chained polymer molecules that in turn form +a solid volume due to molecule interlinkage. Forming al- +most arbitrary 3D structures can then be achieved by +translating the laser through the overall volume of the +photo-resist along all three spatial dimensions. +Grav- +ity can impose limitations on attainable shapes, yet this +aspect usually does not have a too strong impact: the +polymer and the original monomer resin have very simi- +lar mass densities, and thus the Archimedes forces keep a +polymerized voxel locked in its position due to the resin’s +viscosity. +FIG. 2. Principle of direct-laser writing (DLW). (a) The fs- +writing laser is scanned through the photo-resist through the +monomer resin using high-speed galvo-mirrors for the dis- +placement in the (x, y)-plane, while a piezo controls the z- +position. (b) The resin is two-photon polymerized only inside +a small voxel volume, and voxels are placed on a grid deter- +mined by hatching distance h in the (x, y)-plane, and slicing +distance s in the z-direction. The laser power (LP) as well as +s, h determine the overlap of neighboring voxels and through +this the minimum feature size and the smoothness of printed +surfaces. (c) In our work we use the ’dip-in’ technique, where +a drop of resin is located between the microscope objective +and the substrate. The printing direction is downwards, and +the maximum size of 3D-printed structures is around 6 mm +in height. +Originally, +the writing laser spot was translated +through the resin using piezo stages. +This approach +is highly accurate as the stages readily have nanomet- +ric precision. However, it does not allow for large dis- +placement, is very slow and hence cannot be used for +large printing areas/volumes. +A major breakthrough +resulted from using galvo-mirrors for moving the writ- +ing laser’s focal spot through the resin (see Fig. 2 (a)). +As a consequence, printing speed increased by orders of +magnitude27, and fabricating large-scale 2.5 metasurfaces +or 3D volumes became possible. +Crucial for the quality of 3D structures and for inte- +gration in general is the feature size of a single, polymer- +ized voxel relative to the the scanning speed of the print- +ing laser. The photoinitiation of the chemical reaction +which essentially is instantaneous relative to the the writ- +ing speed, and hence the writing-volume directly follows +laser’s scanning. However, polymerization is a chemical +reaction with an associated time scale, like any diffusion +phenomenon. Typically, this timescale is orders of mag- +nitude slower than the galvo-controlled laser scanning28. +This aspect is crucial, since as a consequence polymer- +ization is taking place for several neighboring voxels at +overlapping times. It makes the polymerization process + +(a) +(q) +h +LP1 < LP2 +LP1 +Sample holder and substrate +3D Scanning piezo +LP2 +Positioning stage +S +High-resolution objective +Zm +(c) +DiLL (Dip-in) +substrate +resin +Galvo high-speed 2D scanning +Femtosecond laser +Xg4 +more homogeneous, and the obtained structures do not +suffer from (unintended) variations of material properties +resulting from stitching countless small voxels together +to form a large structure. As schematically illustrated +in Fig. 2 (b), the writing laser power (LP), the hatch- +ing h and slicing s distances as well as the scan speed +modify the overlap between neighboring polymer voxels. +Through this, the smoothness of surfaces and the ho- +mogeinity of the polymer-medium can be controlled to a +good degree. For much faster polymerization, the peri- +odic voxels would results in a photonic crystal like struc- +ture, thus introduce scattering and all related phenomena +inside the produced polymer. Thanks to diffusion, this +aspect is almost not observable, yet it potentially is a +source of optical losses in long waveguides. +A powerful technique, called ’dip-in’ mode, c.f. Fig. 2 +(c), where the liquid resin is held between the substrate +and the microscope objective, was introduced in 2013. +This avoids having to print through the substrate (con- +trary to immersion-oil techniques), which reduces aber- +rations and removes the thickness of the substrate as a +limitation of the maximal height of printed structures. +Importantly for CMOS compatibility, it enables printing +on materials that are not transparent at fs-laser’s wave- +length. Piezo actuators and/or the writing field (deter- +mined by the microscope objective of the printer) are +usually quite limited in area, usually below mm-scales. +For printing larger structures stitching various writing +fields together is required, and in that it is not dissimilar +to the stepper-process used in 2D semiconductor lithog- +raphy. One can select a lower NA microscope objective to +increase the writing field, however, this can only be em- +ployed on the cost of a reduced printing low-resolution29. +Generally, 3D printing via direct-laser writing creates +structures of high quality, and their optical and ellastical +properties have been characterized with high accuracy +using Brillouin light scattering30. In this paper, the au- +thors demonstrate an excellent quality check of the poly- +mer in the GHz regime for elastic waves. For example, +the 3D-printed samples can have an elastic quality fac- +tor only ten times smaller than fused silica at hypersonic +frequencies. +Importantly for printing photonic waveguides, the de- +gree of polymerization and through the Clausius rela- +tionship also the refractive index n, is mainly determined +by the type of photo-resist and the dose parameters D +of the fs-laser, i.e. scanning speed and LP. Within the +window between the TPP-threshold and the breakdown +point above which the polymerized voxel contains defects, +the so-called dynamic power range of the photo-resist26, +the size of the TPP-voxel can be further modified by +adapting D and fabrication parameters distances h and +s. +Figure 3 (a-b) depicts the experimental optimization +of the dynamic power range of the liquid negative- +tone IP-S photo-resist, with n ≈ 1.51 when fully TPP- +polymerized31,32 and using a 25X magnification NA = 0.8 +microscope objective for writing. We printed, on a fused +h = 0.3 μm 5 μm +5 μm +LP = 11 mW +LP = 7mW +5 μm +LP = 15 mW5 μm +LP = 17 mW5 μm +LP = 19 mW5 μm +h = 0.4 μm 5 μm +h = 0.5 μm 5 μm +h = 0.6 μm 5 μm +5 μm +h = 0.7 μm +(a) +(b) +FIG. 3. Dynamic power range characterization of waveguide +cores printed via TPP using the IP-S photo-resist. +Image +taken with permission from9. (a) SEM micrograph of pilars, +printed to reassemble the cores of waveguides, with 20 µm +height and d = 5 µm, with laser power LP ∈ {7, . . . , 19} mW, +using hatching h = 0.4 µm and slicing distance s = 1 µm. +(b) Impact of hatching distance h ∈ {0.3 : 0.1 : 0.7} µm, with +fixed LP = 15 mW and s = 1 µm. +silica substrate, a set of five free-standing pillars to em- +ulate waveguide cores with 20 µm height and diame- +ter d += +5 µm using a range of TPP laser power LP +∈ {7, . . . , 19} mW and hatching distances h ∈ {0.3 : 0.1 : +0.7} µm. As globally fixed parameters in all our fabrica- +tions we use a scanning speed of 10 mm/s and a slicing +distance of s = 1 µm. The scanning electron microscopy +(SEM) micrograph in Fig. 3 (a) shows the effect of grad- +ually modifying the LP with a hatching distance constant +h = 0.4 µm. Structures printed with LP = 7 mW and +LP = 11 mW have ondulated surfaces, whereas when in- +creasing the laser power to LP = 15 mW results in larger +TPP voxels and therefore smoother surfaces. Exceeding +LP = 15 mW leads to overpolymerization of the IP-S +photo-resist (see two last micrographs of Fig 3 (a)). We +therefore select LP = 15 mW and proceed to optimize +the second fabrication parameter by scanning the hatch- +ing distance from h ∈ {0.3 : 0.1 : 0.7} µm, and Fig. 3 (b) +shows the results. We found that for h = 0.3 µm results +are not always reproducible since smaller hatching dis- +tance increases local exposure dose D and hence moves +the process above the available power range. +B. +One-photon polymerization +One-photon polymerization is widely used to process +thin material layers in the current 2D photo-lithography +technology used for electronic semiconductor ICs. The +process is based on the exposure of a photosensitive resin, +usually at the UV range, through a photo-mask including +specific design patterns. +Repeating this process layer- +by-layer is possible to process and stack different thin +material layers and fabricate 3D structures33. For highly +structured patterns like SD memory cards, this has led to + +HV +curr +usecase +det +mag +贝 +WD +tilt +20μm +5.00kv +25pA +Standard +ETD +2500x +10.0mm +40.0 +FEMTO-STHV +curr +use case +det +mag +贝 +WD +tilt +20μm +5.00kv +25pA +Standard +ETD +2500x +10.0mm +40.0 +FEMTO-STHV +curr +use case +det +mag +贝 +WD +tilt +20μm +5.00kv +25pA +Standard +ETD +2500x +10.0mm +40.0 +FEMTO-STHV +curr +use case +det +mag +WD +tilt +20μm +5.00kv +25pA +Standard +ETD +2500x +10.0mm +40.0 +FEMTO-STHV +curr +use case +det +mag只WD +tilt +50um +5.00 kV +0.25 nA + Standard +ETD +800x +10.0mm40.0 +FEMTO-STHV +curr +det +mag 只 +WD +tilt +50μm +use case +5.00kv +0.25nA +Standard +ETD +800x +10.0mm +40.0° +FEMTO-STHV +curr +use case +det +mag贝 +WD +tilt +20 μm +5.00kV +25pA +Standard +ETD +2500x +10.0mm +40.0 +FEMTO-STHV +curr +usecase +det +mag +贝 +WD +tilt +20μm +5.00kv +25pA +Standard +ETD +2500x +10.0mm +40.0 +FEMTO-STHV +curr +use case +det +mag +贝 +WD +tilt +20μm +5.00kv +25pA +Standard +ETD +2500x +10.0mm +40.0 +FEMTO-STHV +curr +use case +det +mag +贝 +WD +tilt +20μm +5.00kv +25pA +Standard +ETD +2500x +10.0mm +40.0 +FEMTO-ST5 +ICs with up 100 or more circuit layers2. However, such +stacking of layers created via a generically 2D fabrication +concept has several severe drawbacks. For one, it requires +to precisely align the photo-mask multiple times in each +photo-lithographic step, which is challenging and time- +consuming. Secondly, one of the strongest features of 2D +lithography is its economic appeal. Between each layer, +each of the process step have to be repeated in a loop- +like manner. A process where the entire IC’s volume is +created during few of such process steps will potentially +have the upper hand economically speaking. Still, such +stacked 2D lithography has also been used of complex 3D +photonic integration, c.f. Fig. 4. +(a) (b) + (c) +FIG. 4. Multilayer 3D waveguide fabrication using OPP. Im- +age taken with permission from34. (a) Schematic diagram of +the fabrication sequence for the stacking waveguide using spin +coating and simple direct UV photolithography curing (s1); +UV irradiation of the waveguides using a mask (s2); develop- +ment (s3); UV irradiation of the cladding (s4). (b) Layout +of the 3D interconnect polymer structure with an array of +4x8 waveguides. (c) Cross-section microscope optical image +of 4x8 stack waveguides. +Just as with TPP, the refractive index of the poly- +merized resin is a function of the optical exposure does +D31,35–37. However, in OPP the refractive index of the +resin is modified for substantially larger volumes, and in +particular volumes outside the intended plane of exposure +do strongly accumulate unintended irradiation doses. It +is therefore a formidable challenge to precisely control a +3D refractive index distribution, i.e. a volume hologram, +with high spatial resolution. +OPP is therefore better +suited for simultaneous polymerization of, either, large +areas like in classical 2D lithography, or for large uni- +form volumes. +C. +Flash-TPP: combining one- and two-photon +polymerization for photonic integration +One can combine one- and two-photon polymeriza- +tion as an hybrid configuration to accelerate the fabri- +cation of 3D photonic chips. +Several approaches com- +50 μm +Waveguide core +TPP +OPP +(a) +Mechanical supports +(a) +(b) +(c) +FIG. 5. Flash-TPP printing concept for 3D integrated pho- +tonics. +Image taken with permission from9. +(a) Classical +’dip-in’ process for the DLW-TPP fabrication of 3D photonic +waveguides. (b) UV chamber that polymerizes the unexposed +regions of the 3D structure via OPP. (c) SEM micrograph +of a 3D-printed cuboid cross-section embedding 16 photonic +waveguides. The waveguide cores (mechanical supports) are +printed with large (small) hatching distances, which defines +the resolution of each component of the 3D photonic circuit. +Red colour represents regions polymerized via TPP, while +blue colour regions via OPP. +bining UV lithography with DLW-TPP have been pre- +viously demonstrated in38 and39 for the fabrication of +high resolution 3D optical microcomponents. However, +those methodologies require the polymerization of multi- +ple photo-resists in two separated fabrication steps and +become time-consuming if used for 3D fabrication due to +the layer-by-layer approach. +We demonstrated a novel lithographic strategy that +combines OPP and TPP, flash-TPP9, where we combine +high resolution and quality TPP with unstructured and +uniform OPP in order to accelerate the fabrication pro- +cess by one order of magnitude when compared to us- +ing TPP-only. Importantly, the concept only requires a +single resin and adding the OPP step does not add ad- +ditional development and washing steps. In flash-TPP, +TPP and OPP are used for the fabrication of the dif- +ferent sections of a photonic circuit, Fig. 5 illustrates +the working principle, here for the liquid negative-tone +IP-S photo-resist. +Waveguide cores accommodate the +large majority of an optical signal’s electromagnetic field, +hence cores are printed via TPP with a precisely opti- +mized laser power and fine resolution in the (x, y)-plane, +i.e. small hatching distance. This ensures smooth core- +cladding interfaces and hence low propagation losses. +Mechanical supports, i.e. surfaces that define the outer +limits of the volumetric circuit, are printed with larger +hatching distance and high LP. +Figure 5 (a) depicts the typically ’dip-in’ DLW-TPP +printing procedure. After development, the photonic cir- +cuit is transferred to a UV chamber, c.f. Fig. 5 (b), and +the OPP dosage D of the 3D circuit’s volume is con- + +a +s1) +buffer layer +silicon +substrate +s2) +mask +spin- +coated +waveguide +layer +s3) +s4) +506 +trolled via the duration of the UV exposure, through +which we tailor the refractive index of the waveguides’ +cladding ncladding and hence ∆n. The SEM micrograph +from Fig. 5 (c) shows the cross-section of an exemplary +3D photonic chip fabricated via flash-TPP consisting of +a cuboid integrating 16 waveguides. The cores and me- +chanical supports, printed via TPP, are highlighted in +red region, while the cladding volume, polymerized via +OPP, is highlighted in blue. +Via flash-TPP, we fabricated photonic waveguides +with a refractive index contrast between core and +cladding in the order of ∆n ≈ 5·10−39. +Figure 6 (a) +shows the evolution of the the average numerical aperture + and refractive index of the cladding < ncladding > +polymerized via OPP versus D. We used UV exposure +doses D of 0, 750, 3000 and 9000 mJ/cm2, respectively. +Assuming a constant ncore ≈ 1.51, we can precisely con- +trol, both, and < ncladding >. Waveguides are +single-mode for d ≤ 4.9 µm, which are feasible to fab- +ricate via standard DLW-TPP processes. We obtained +1.3 dB/mm (0.26 dB) propagation (injection) losses for +the fundamental LP01 mode of waveguides printed via +flash-TPP. Crucially, our 3D circuits did not degrade +over time, and we evaluated the NA of waveguides under +continuous operating condition across several months9. +Overall, this demonstrates the reliability of the flash- +TPP lithography methodology for an ultra-fast, single- +step and high performance fabrication of 3D photonic +components. +Printing via flash-TPP consist in polymerizing only +the sections vital for communication and mechanical in- +tegrity. Importantly, the majority of a circuit’s area or +volume is not involved in either, and they can hence be +rapidly fabricated via UV blanket exposure. The print- +ing times in flash-TPP is therefore drastically reduced, +and in particular cases also scales different with the cir- +cuit’s size9. This agrees with our experience; flash-TPP +reduces the printing time to only 10% compared to only- +TPP. As an example, printing a large structure that +integrates waveguides with heights ranging from 0.1 to +6 mm9, shown in Fig. 6 (b), requires ∼24 hours only +using TPP but only ∼3 hours using flash-TPP. +IV. +AIR-CLADDED WAVEGUIDES +Polymer waveguides with an air cladding have a rel- +atively large ∆n ≈ 0.5 with ncore = 1.51. On the one +hand, this leads to very strong confinement and a large +NA = 1.13, which enables very small bending radii of +25 µm (14 µm) at λ = 1550 nm (λ = 650 nm), and +in turn dense photonic integration40–42. The large ∆n +makes fabricating single-mode waveguide circuits chal- +lenging. To be single-mode, air-cladded waveguides have +to have a core diameter d ≤ 1 µm (d ≤ 0.43 µm) at +λ = 1550 nm (λ = 650 nm). Printing waveguides with +d ≤ 1 µm is possible7, and strongly confined photonic +IC at λ = 1550 nm are within reach. For photonic 3D +(a) (b) +FIG. 6. Optical performance of waveguides printed via flash- +TPP. Image taken with permission from9. (a) Average numer- +ical aperture and cladding’s refractive index < n2 > +over OPP dose D of photonic waveguides printed via flash- +TPP. The (< n2 >) decreases (increases) over D, +meaning that we can control the degree of polymerization of +the cladding via the dosage of UV light. +(b) Macroscopic +structure scaled to a match that integrates waveguides with +heights ranging from 0.1 to 6 mm. +ICs close to the visible wavelength of light this remains +a challenge. +Recently, 3D optical splitter/combiners based on air- +cladded waveguides with a 1 to 4, 1 to 9 and 1 to 16 +configuration were printed using TPP43,44. Figure 7 (a) +shows an SEM image of the 1 to 4 fractal splitter/coupler, +with its optical characterization at λ = 632 nm shown in +Fig. 7 (b). +There, the distance between output ports +was scanned within the range D0 ∈ [10, 12, ..., 20] µm +while keeping their height constant at 52 µm. Losses do +not substantially increase for smaller distance between +the output ports, which validates the estimated mini- +mal bending radii given before. Furthermore, this per- +formance was evaluated for two different LP settings. No +clear difference can be seen between the two data-sets, +and hence the printing power for air-cladded 3D polymer +waveguides is not a critical parameter, as long one stays +within the dynamic power range. +For large-scale network interconnect, Moughames et al. +demonstrated 3D parallel interconnects with high con- +nectivity, shown in Figure 7 (c), by cascading two layers +of 1 to 9 splitters and spatially multiplexing an arrays of +such 1 to 81 splitters to allows for an array of 15x15 input +waveguides. The entire circuits only occupies a volume +of 460x460x300 µm3, in which an interconnect for 225 +inputs and 529 outputs is realized7. Figure 7 (d) shows +a higher magnification of this interconnect. Individual +wavegudies have a low surface roughness, and the incor- +porated chirality of the fractal splitters/couplers avoids +intersections of individual waveguides. +V. +STEP AND GRADED INDEX WAVEGUIDES +Based on the previous discussed concepts and fabrica- +tion technologies, we addressed step- (STIN) and graded- +index (GRIN) waveguides. In STIN waveguides, the re- +fractive index of the waveguide’s core is constant, while +for GRIN waveguides it is a function of the radial distance +to the core’s center. Usually, GRIN waveguides follow + +7 +FIG. 7. Air-cladded waveguides and couplers fabricated via +DLW-TPP. Image taken with permission from7,43. +(a) 2x2 +optical splitter/coupler with 1 input and 4 outputs with dis- +tance D0 = 16 µm between waveguides, and 1.2 µm waveguide +diameter43. (b) Optical losses of 2x2 splitters/couplers as a +function of the distance D0 between waveguides, for hatching +distances h = 0.1 µm (in blue) and h = 0.2 µm (in red). Data +on top correspond to splitters/couplers written with laser +power LP = 10.4 mW, and data at the bottom correspond to +splitters/couplers written with laser power LP = 11.2 mW. +(c) SEM micrographs of 3D-printed waveguides realizing par- +allel interconnects with high connectivity7. (d) Zoom-in of +(c). +a parabolic refractive index distribution. For the STIN +waveguides, all bound rays propagate at angles within +the total internal reflection condition θc at any position +in the core cross-section, while for GRIN waveguides, the +range of angles varies with position45. +We proposed a single-step additive fabrication tech- +nique, (3+1)D printing8, by which we spatially modify +the refractive index of a single resin over the TPP expo- +sure dose during fabrication. Using the (3+1)D-printing +concept, we constructed volume holograms and photonic +waveguides with, both, STIN and GRIN profiles in a +single-step, single-material fabrication with a commer- +cially available process. This demonstrates the versatility +of the 3D photonic integration approach based on DLW; +optical manipulation based on integrated and monolithic +3D structures can either rely on discrete components, i.e. +waveguides, or leverage continuous manipulations of free +optical propagation, i.e. holograms8. Both schemes can +be exploited on the same photonic IC and be realized +using the same fabrication concept and during the same +fabrication step. We used the negative tone IP-Dip resin +(n ≈ 1.547)36 and a 63X magnification NA = 1.4 micro- +scope objective, c.f. Fig. 5 (a). +The SEM micrograph of Fig. 8 (a) shows an exem- +plary cuboid embedding 20 STIN waveguides fabricated +via (3+1)D-printing. Contrary to flash-TPP, in (3+1)D- +printing all the 3D photonic chip volume is fabricated +via TPP-only. The refractive index contrast ∆n between +core-cladding waveguides is achieved from the control +over the TPP dosage D for individual writing voxels. For +(a) +100 µm +(b) +(c) +FIG. 8. Step- (STIN) and graded-index (GRIN) waveguides +fabricated via (3+1)D-printing. +Image taken with permis- +sion from8. (a) SEM micrograph of an exemplary 3D-printed +cuboid integrating 20 STIN waveguides of 300 µm heigh. +Waveguide cores (cladding) are printed via TPP with high +(low) laser power, which ensures a refractive index contrast +∆n ≈ 2.4·10−3. Panels (b) and (c) depict the output intensi- +ties (triangles) and fundamental LP01 mode fits (dashed lines) +of a 3 µm radius STIN and GRIN waveguide, respectively. +a higher (lower) refractive index as needed for the waveg- +uide cores (claddings), one requires an accordingly higher +(lower) LP, i.e. D. STIN waveguides result from a con- +stant LP all across their core, while for GRIN waveguides +the writing power changes from high to low following a +parabolic profile. +To evaluate the optical performance, we fitted the ex- +perimental output intensities for diameters d below the +cut-off condition of the second propagation mode. The +output intensity of the LP01 mode of a STIN waveguides +is described by J2 +0(u r +R) for | r | < R and K2 +0(v r +R) for | +r | > R, while for GRIN waveguides is given by an in- +finite parabolic refractive index profile as exp − 1 +2V r2 +R2 45. +Figure 8 (b-c) depicts the fit of fundamental LP01 mode +to the normalized output of STIN and GRIN waveguides +with radius R += +3 µm, respectively. Considering the +refractive index of the core constant (ncore ≈ 1.547), +we obtained an averaged numerical aperture = +0.08 ± 0.01 (i.e. ncore = ncladding + 2.4 · 10−3) for STIN +and of = 0.18 ± 0.02 for GRIN waveguides. As +expected, the core-confinement of GRIN waveguides is +significantly higher than for STIN waveguides due to the +inner core refractive index modification, which offers a +crucial advantage for photonic integration schemes7. +As seen, STIN waveguides with a polymer cladding +have a refractive index contrast in the order of ∆n ≈ +2.4·10−3, with low NA ≈ 0.12. Contrary than for air- +cladded waveguides, this leads to large bending radii of +15 mm (7 mm) at λ = 1550 nm (λ = 650 nm), and in +turn dense photonic integration is much more challeng- +ing for STIN waveguides. However, the low ∆n allows to +have single-mode propagation for waveguide diameters +d ≤ 9.8 µm (d ≤ 4.2 µm) at λ = 1550 nm (λ = 650 nm), +which is standard with the current DLW-TPP fabrica- +tion technology. Future efforts include combining poly- +mer and air-cladded waveguides, taking the strengths of +each configuration in a single platform, i.e. air cladding +waveguides providing highly-densed photonic integration + +P = 10.4 mW +-5 +-7 +6- +-11 +p = 11.2 mW +-5 +-7 +-9 +30 μm +.11 +200 μm +50gmexp(cs291.2μmcs291.2μmcs291.2μmcs291.2μmcs291.2μmcs291.2μm8 +with their small bending radii, while STIN waveguides +serving as tools for single-mode propagation with large +waveguides diameters over wide distances. +VI. +FLASH-TPP PRINTED WAVEGUIDES +Recently, we demonstrated the fabrication of large +scale 3D integrated photonic components via flash-TPP. +Several features of flash-TPP make it an enabling tech- +nology for integration of larger circuits. +Of primary +importance is the substantial accelerated fabrication; +without, fabrication of larger integrated circuits would +quickly approach timescales beyond 24h9. Based on this +approach, we demonstrated long (6 mm) single-mode +waveguides, and we achieved exceptionally low injection +(≈ 0.26 dB) and propagation (≈ 1.3 dB/mm) losses9. +Next as the demonstration of optical splitters and com- +biners based on this concept. These are the backbone of +any photonic IC, and 3D integration enables interesting +alternatives for creating 1 to M optical couplers without +using sensitive optical interference units46. In 3D, 1 to M +optical couplers can simply be realized by arranging nu- +merous output waveguides around the input waveguide, +something impossible to realize in a purely 2D integra- +tion setting. We demonstrated broadband 1 to M split- +ters leveraging adiabatic coupling6,47. +Adiabatic cou- +pling achieves low-loss single-mode optical transfer from +1 to M waveguides through evanescent waves, where the +optical mode adiabatically leaks from a tapered core of +an input waveguide towards the cladding into inversely- +tapered cores of the output waveguides48,49. All the pre- +vious studies consider the 2D case of only one to one +adiabatic coupling between optical components50. +In our work, we showed efficient single-mode adiabatic +transfer with 1 input and up to 4 outputs via a single +component. Figure 9 (a) illustrates the design for the +exemplary case of a 1 to 2 adiabatic couplers. The waveg- +uide’s circular core cross-section continuously changes as +a function of propagation direction z. The originally cir- +cular core is reduced in size exclusively along the direc- +tions where an output waveguide is located; the core is +essentially cut along plane surfaces. +These cut-planes +move towards the input core’s center during the taper- +length lt at equal rate d/lt along the (x, y)-plane in order +to match their relative effective modal indices45. Output +waveguides follow exactly the same concept, yet in an in- +verted direction. We separated in and output waveguides +via gap g and studied the evanescence coupling efficiency +between coupled waveguides6. The same tapering strat- +egy was applied to 1 to 3 and 1 to 4 as depicted in the +output intensity profiles from Fig. 9 (b). +We obtained record optical coupling losses of 0.06 dB +for the optimal case of 1 to 2 adiabatic couplers, with +a difference between the two outputs intensities of only +∼ 3.4 %. We furthermore demonstrated broadband func- +tionality from 520 nm to 980 nm during which losses re- +main below 2 dB6. Importantly, these adiabatic couplers +can be cascaded in order to exponentially increase the +number of M outputs, c.f. Fig. 7 (c). We arranged a +double-layer of 1 to 4 adiabatic couplers and the result- +ing 1 to 16 single-mode output intensities can be seen in +the last diagram of Fig. 9 (b). Importantly, the global +losses of the entire device is only 1 dB , and the entire +circuit was realized within (0.08 × 0.08 × 1.5) mm36. +x +y +z +Norm. Intensity +0 1 +(a) + + (b) +FIG. 9. Adiabatic 1 to M broadband-scalable couplers fabri- +cated via flash-TPP. Image taken with permission from6. (a) +Design of the 1 to 2 adiabatic couplers printed via flash-TPP. +The same tapering strategy can be applied to higher-order +couplers, i.e. 1 to 3 and 1 to 4 couplers. (b) Output intensity +profiles of the 1 to 2, 3 and 4 adiabatic couplers. The last +output intensity corresponds to a cascaded 1 to 16 adiabatic +coupler. +VII. +TOWARDS A SCALABLE AND CMOS +COMPATIBLE INTEGRATION OF PHOTONIC +NETWORKS +High-density photonic integration requires the inter- +connection of several photonic platforms. +Most of the +current photonic devices are based on silicon-on-insulator +(SOI) and CMOS technology. Combining the strength of +multiple photonic and electronic systems in one hybrid +and multi-chip platform can result in the diversification +of specific computing tasks while increasing the overall +performance. +A versatile fabrication technology with low-losses is of +vital importance for the scalability of free-form as well as +integrated optical interconnects in three-dimensions. The +polymer-based 3D printing technology based on DLW- +TPP is excellently suited to address these challanges, and +several proof-of-concept studies have been realized50–52. +Figure 10 (a) shows photonic wire-bonding, realising a +3D photonic waveguide forming a point to point com- +munication for a chip-to-chip connection between SOI +chips hosting individual waveguides. The photonic wire- +bond was fabricated via DLW-TPP using the negative- +tone MicroChem SU-8 2075 photo-resist (n ≈ 1.51 at +1550 nm)53, and it connected two SOI waveguides sep- +arated a distance of 100 µm on different CMOS chips. +This demonstrated for the fist time the basic viability of +TPP-based 3D printing as a tool for CMOS compatible, +wafer-scale as well as chip-to-chip connections. +A major challenge of the polymer-based 3D fabrication + +g +C15.8 +Intensity +11.9 +20 +Size (μm) +0 +7.9 +4.0 +0.0 +4.0 +7.9 +11.9 +Size (μm)9 +and the CMOS technology is the interaction of the CMOS +substrate with the photo-resist during the TPP printing +process. In a standard fabrication setting, the interac- +tion between the fs-pulsed laser and the glass substrate +is negligible since the substrate material, i.e. fused silica, +is transparent at the wavelength of the fs-laser (780 nm), +and low specular reflection. However, the CMOS tech- +nology is based on 2D stacking of multiple thin layers +of semiconductor materials such as GaAs, InP or Sili- +con. These often have a bandgap energy below that of +the writing laser, and in that case printing through the +semiconductor substrate is impossible; only the ’dip-in’ +concept is therefore a viable general approach for fabri- +cating 3D photonic integrated circuits directly on top of +a CMOS substrate based on DLW-TPP. Another chal- +lenge is the higher specular reflection, as these semicon- +ductor materials have a higher refractive index. The re- +sulting optical reflection of the fs-laser laser at the semi- +conductor substrate leads to a overpolymerization of the +photo-resist if not compensated for. The LP therefore +needs to be continuously adjusted at the vicinity of the +CMOS/photonic circuit interface in order to achieve the +intended degree of polymerization of the photo-resist. A +further requirement is the precise alignment of the 3D +photonic chip with the semiconductor device patterned +on the CMOS substrate. +(a) +(b) +25 μm +IP-S +GaAs +IP-S +FIG. 10. Polymer-based 3D printing and CMOS technology +compatibility. (a) Chip-to-chip photonic wire bonding con- +cept. +A 3D polymer waveguide fabricated via DLW-TPP +connects two SOI waveguides sitting on distant CMOS chips. +SEM image taken with permission from 53. (b) SEM micro- +graph of and exemplary 3D-cuboid integrating a cascaded 1 +to 16 adiabatic couplers printed via flash-TPP on top of a +quantum dot micropillar laser array. +Figure 10 (b) depicts an exemplary 3D-printed cuboid +integrating a cascaded 1 to 16 adiabatic coupler (cf. +Fig. 9 (b)) printed via flash-TPP on top of a semiconduc- +tor substrate integrating quantum dot micropillar laser +arrays. Each of the micropillar lasers consists of a cylin- +drical microcavity (a vertical arrangement of highly re- +flective distributed Bragg reflectors (DBR) alternating +Al(Ga)As and GaAs mirror pairs) sandwiching a cen- +tral gain section based on InGaAs self-assembled quan- +tum dots (QDs). Further details about the fabrication +and optical properties of the quantum dot micropillars +laser arrays from Fig. 10 (b) can be found in54–56. We +used IP-S photo-resist for the fabrication, with a lower +laser power LP = 6.5 mW (compared to the previously +LP = 15 mW) in order to avoid microexplosions of the +photo-resist at the semiconductor-polymer interface dur- +ing TPP printing. After development, the 3D photonic +chip is then polymerized via OPP with a exposure dose +D += 3000 mJ/cm2. The SEM micrograph shows the +perfectly aligned 3D photonic structure with the angle +of the periodic GaAs micropillar array. We checked the +adherence of the polymer over time, and after a continu- +ously observation over more than 4 months no deteriora- +tion has been found. This confirms the reliability of in- +tegrating our 3D printing technology with CMOS-based +micro-laser arrays. +VIII. +CONCLUSION +Here, we present a review over our recent work address- +ing additive manufacturing towards future 3D photonic +integration of optical components that is CMOS com- +patible. Based on one- and two-photon polymerization +processes combined with direct-laser writing systems, we +demonstrated the fabrication of high performance indi- +vidual photonic waveguides as well as scalabale optical +splitters. All such 3D structures have been fabricated in +our local FEMTO-ST RENATECH infrastructure. +We demonstrated that using the commercial DLW- +TPP Nanoscribe GmbH (Photonics Professional GT) +system and the ’dip-in’ DLW strategy, we are able to +the construct, both, air- and polymer-claddded photonic +waveguides. For air-cladded waveguides, we used a TPP- +only, a single-step and single resin (IP-Dip resist). A 3D +IC comprising a network of fractal optical splitter with +225 input and 529 output waveguides only occupies a +volume of 460x460x300 µm3. Such air-cladded waveg- +uide ICs are prime candidates for highly-dense photonic +packaging thanks to their low bending-radii on 10s of µm +scale. For polymer-cladded waveguides, we presented two +different strategies in which we 3D-printed the waveguide +cores via TPP while achieving a precise control over the +refractive index contrast ∆n via, (i), the adjustment of +the fs-laser dose D on an single-voxel level, i.e. (3+1)D- +printing, and (ii), the duration of UV blanket exposure +that determines the OPP dosage D to fix the index of the +cladding material for the entire photonic IC in a single +shot, i.e. flash-TPP. Noteworthy, both fabrication con- +cepts require a single procedure writing step and a single +resin (IP-S resist). Importantly, with flash-TPP fabri- +cation times are reduced by up to ≈ 90 % compared to +(3+1)D-printing thanks to the additional OPP process. +Via flash-TPP, we achieved polymer-cladded waveguides +with refractive index contrast ∆n ≈ 5·10−3, with low +1.3 dB/mm (0.26 dB) propagation (injection) losses while +printing waveguides up to 6 mm heigh. This allows to +have single-mode propagation over large distances. We +demonstrated the fabrication, via flash-TPP, of scalable- +boadband couplers leveraging adiabatic transfer from 1 +input up to 4 outputs. Using a tapered/inversely-tapered +waveguide sequence, we achieved record 0.06 dB optical +coupling losses with very symmetric splitting ratios. We + +HV +curr +use case +det +mag +只 +WD +tilt +50 μm +5.00 kV +0.20 nA +Standard +LVD +1 000 x +10.0 mm +45.0° +FEMTO-ST(a) +(b) +Photonic wire +Photonicwire +bond +bond +SOlwaveguide +SOI +25μm +waveguides +Chip1 +20 μm +10 μm +Chip2 +(c) +Input fiber +Qutput fiber +Photonic wire bonds +Chip1 +Chip.2 +Grating couplers10 +arranged a double-layer of 1 to 4 adiabatic couplers, re- +sulting in a device with 16 single-mode outputs with only +1 dB global losses. +Importantly, we demonstrated the compatibility of +our fabrication methodology based on DLW-TPP with +CMOS substrates. +As a proof-of-concept, we success- +fully 3D-printed our cascaded 1 to 16 adiabatic couplers +on top of a CMOS substrate integrating GaAs quantum +dot micropillar laser arrays. +Preliminary characteriza- +tion of these structures shows encouraging performance +in terms of losses and stability. +Overall, in this review we have covered our novel 3D- +printing technology, which represents a breakthrough +with the potential to become a high-impact tool for the +hybrid, highly-dense and hence compact packaging of, +both, electronic and photonic devices. +The concepts +opens several potential avenues for future exploration. +The combination of air- and polymer-cladded waveguides +could enable dense integration with simultaneous precise +control over optical signal properties such as mode num- +ber, polarization and phase. +As the concept leverages +photo-polymerization, in principle the large-scale and +exceptionally performing production facilities of CMOS +electronic integration could be amended with 3D pho- +tonic integration capability. Due to the excellent compat- +ibility of standard photo-resins, the approach is largely +agnostic to the underlying substrate. In this it is more +flexible than integrated silicon photonics, and fabricat- +ing additively on a already processed CMOS substrate +removes many of the challenges compared to fabricating +photonic ICs based on different process - such as DLW di- +rectly into bulk dielectrics followed by bonding to CMOS. +IX. +ACKNOWLEDGMENT +The authors would like to thank Stephan Reitzen- +stein for his contribution through fabricating the semi- +conductor laser sample used for producing the circuit +shown in Fig. +10 (b) and Erik Jung for the valuable +help on the design of 3D waveguides. +This work was +partly supported by the french RENATECH network and +its FEMTO-ST technological facility. +The authors ac- +knowledge the support of the Region Bourgogne Franche- +Comt´e. +This work was supported by the EUR EIPHI +program (Contract No. ANR-17-EURE- 0002), by the +Volkswagen Foundation (NeuroQNet II), by the French +Investissements d’Avenir program, project ISITE-BFC +(contract ANR-15-IDEX-03), by the European Union’s +Horizon 2020 research and innovation programme un- +der the Marie Sk�lodowska-Curie grant agreements No. +713694 (MULTIPLY). +1N. 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Forchel, “Quantum dot micropillars,” +Journal of Physics D: Applied Physics, vol. 43, no. 3, p. 033001, +2010. + diff --git a/3NAzT4oBgHgl3EQfDvrh/content/tmp_files/load_file.txt b/3NAzT4oBgHgl3EQfDvrh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..30ce20181b2d5d8236cdf469f8fea6fffc09404f --- /dev/null +++ b/3NAzT4oBgHgl3EQfDvrh/content/tmp_files/load_file.txt @@ -0,0 +1,1172 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf,len=1171 +page_content='Additive 3D photonic integration that is CMOS compatible Adria Grabulosa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='1 Johnny Moughames,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='1 Xavier Porte,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 2 Muamer Kadic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='1 and Daniel Brunner1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' a) 1)FEMTO-ST Institute/Optics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' CNRS & University Franche-Comt´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 15B avenue des Montboucons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Besan¸con Cedex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 25030,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' France 2)Now with: Institute of Photonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' University of Strathclyde,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Glasgow G1 1RD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' UK (Dated: 4 January 2023) Today,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' continued miniaturization in electronic integrated circuits (ICs) appears to have reached its funda- mental limit at ∼2 nm feature-sizes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' from originally ∼1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' At the same time, energy consumption due by communication becomes the dominant limitation in high performance electronic ICs for computing, and modern computing concepts such a neural networks further amplify the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Communication based on co-integrated photonic circuits is a promising strategy to address the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' As feature size has leveled out, adding a third dimension to the predominantly two dimensional integrated circuits appears the most promising future strategy for further IC architecture improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Crucial for efficient electronic-photonic co-integration is CMOS compatibility of the associated photonic integration fabrication process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Here, we review our latest results obtained in the FEMTO-ST RENATECH facilities on using additive photo-induced polymerization of a standard photo-resin for truly 3D photonic integration according to these principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Based on one- and two-photon polymerization and combined with direct-laser writing, we 3D-printed air- and polymer-cladded photonic waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' An important application of such circuits are the interconnects of optical neural networks, where 3D integration enables scalability in terms of network size versus its geometric dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In particular via flash-TPP, a fabrication process combining blanket one- and high-resolution two-photon polymerization, we demonstrated polymer-cladded step-index waveguides with up to 6 mm length, low insertion (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='26 dB) and propagation (∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='3 dB/mm) losses, realized broadband and low loss (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='06 dB splitting losses) adiabatic 1 to M couplers as well as tightly confining air-cladded waveguides for denser in- tegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' By stably printing such integrated photonic circuits on standard semiconductor samples, we show the concept’s CMOS compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' With this, we lay out a promising, future avenue for scalable integration of hybrid photonic and electronic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' INTRODUCTION The backbone behind most of today’s cutting-edge technology is dense integration of two dimensional (2D) electronic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' However, by now these do experience several challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Further pushing the performance of 2D computing chips becomes increasingly difficult, while new applications, in particular neural networks (NNs), challenge the hegemony of such 2D circuits - and this on a fundamental level1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' New integration concepts and fab- rication technologies are needed if we are to continue the astonishing technological progress of the past decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Crucially, these integration concepts need to take the es- sential features behind the success of 2D electronic inte- grated circuits (ICs) into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Elevating a new integration technology even close to the level of 2D electronic ICs is a daunting and certainly a long-term challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Since the first demonstration of a planar, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 2D, monolithic IC at Fairchild, this clas- sical integration has continuously been advanced for 60 years plus in an almost world-wide effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The concept’s success is a testimony to what can be achieved when previously individual components are integrated inside a single, monolithic circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' It typically led to substan- tial miniaturization and increased reliability as well as a)Electronic mail: daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='brunnerfemto-st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='fr robustness, all while fabrication costs plummeted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Com- bined, these factors enabled decades of exponential scal- ing for electronic ICs: around every two years the num- bers of transistors per chips doubled (Moore’s law) while the power consumption per component halves (Dennard scaling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Monolithic ICs comprising different compo- nents and functionalities are therefore also indispensable for 3D photonic integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' While still far from the levels of today’s electronic IC, photonic integration also has considerably advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In order to maximize compatibility and synergy with electronics, photonic integration based on silicon sub- strates emerged in the 1980s with the demonstration of the silicon waveguide3,4, the photonic equivalent to a metallic or polysilicon wire in integrated electronics ICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Electronic ICs are almost exclusively based on comple- mentary metal–oxide–semiconductor (CMOS) technol- ogy that uses mostly silicon as semiconductor host lever- aging boron, gallium, indium, phosphorus, arsenic and bismuth as dopants, and CMOS compatibility is consid- ered fundamentally important for photonic ICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' By a vast majority, both, electronic and photonic inte- gration leverages fabrication concepts developed for pla- nar, 2D substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The layout of a circuit’s single layer is etched into a thin surface of either mostly metal or semi- conductor materials, which is the process of 2D lithogra- phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Typically, coating said surface with a photo-resist protects certain surface-areas from etching, which is de- termined by photo-resist illumination that is structured arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='00983v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='optics] 3 Jan 2023 2 by a photo-mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The appeal of such 2D lithography is that each of the involved process steps, photo-resist ap- plication, exposure by photo-mask, etching and several washing sequences, can be carried out in a single pro- cedure for a large area or even an entire wafer, which strongly reduces fabrication costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' A new challenge to classical electronics computers based on 2D substrates arose with the breakthrough of NN computing around a decade ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Conceptually, NNs link a large number of neurons through the network’s connections, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In an physical hardware implementation that mirrors this topology, these con- nections correspond to electronic or photonic signaling ’wires’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Currently, these connections are emulated, which creates substantial energy and speed overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Future NN circuits that abolish this overhead require ICs with a far higher degree of connectivity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' much more wires to communicate signals across the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This causes sev- eral problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Energetically speaking, electronic com- munication is the factor limiting performance even for classical computing concepts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' communicating a floating point number costs around 80-times more energy than creating a new floating point number5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' NN computation dramatically escalates this problem, as the number of a NN’s connections by far out-scale the number of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Photonic and 3D integration provide promising solutions, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Optical communication is (i) energetically superior for ever shorter distances and (ii) mitigates heat dissipation challenges that arise for volumetric circuits, while (iii) 3D integration shortens the length of commu- nication links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Most importantly, in many NN topologies the number of connections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' wires, increases quadratic or faster with the number of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Consequently, in- tegrating a NN’s interconnect in 2D results in a quadratic scaling (or worse) of chip-area with the size of a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Recently, the number of neurons in a NN has turned into the parameter of fundamental relevance, and alternative strategies for integrating NNs are of funda- mental importance for the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In this review for the RENATECH special issue, we describe our recent work addressing such photonic ICs based on standard techniques and fabrication infrastruc- ture available in our local RENATECH cleanroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In those efforts, we have demonstrated additive, 3D pho- tonic integration, which importantly is using concepts and materials that make the entire fabrication and result- ing photonic IC CMOS compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Based on additive two-photon polymerization (TPP) in a direct-laser writ- ing (DLW) system, combined with rapid and large area one-photon polymerization (OPP), we integrated large 3D photonic waveguide circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We demonstrate indi- vidual waveguides as well as optical splitters and net- works of splitter6 based on (i) air-cladded waveguides comprising polymer cores7, and (ii) step-index waveg- uides where we induce the refractive index difference between core and cladding required for guiding by dy- namically controlling the optical power used for printing our 3D structures8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Finally, we substantially accelerate FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 3D photonic integration and optical waveguide basics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (a) Schematics of a typical neural network where a large num- ber of neurons are highly interconnected through a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (b) Integrating a large number connections in 2D leads to an exponential growth of the number of channels over the chip’s area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' whereas leveraging integration in 3D results in a efficient and linear scalability of optical interconnects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (c) In photonic waveguides, the light is confined within the core of diameter d due to total internal reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For this, the refractive index of the core ncore must be larger than the cladding’s ncladding, and hence ∆n = ncore − ncladding > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' All the waveguide’s optical properties relies on the parameters ∆n and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' the fabrication process by developing the flash-TPP con- cept, which combines TPP-DLW with ultraviolet (UV) blanked illumination to efficiently polymerize an IC’s non-light guiding volume in a single step9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We achieve very symmetric splitting ratios in optical couplers, and (for a first proof of concept) low propagation losses of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='3 dB/mm and insertion losses of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='26 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Fi- nally, we printed optical waveguides on semiconductor substrates hosting micro-lasers, demonstrating that our concept is CMOS compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' BASICS OF ADDITIVE FABRICATION In the past 15 years, DLW and TPP have become a versatile fabrication tool of polymer structures with sub-micron dimensions10–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In contrast to 2D pla- nar methods such as electron-beam lithography or mask based lithography, DLW allows for fabricat- ing three-dimensional structures13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' DLW has played a crucial role for many proof-of-concept designs in optics7, acoustics14,15, elasticity13,16–18, robotics19 and even electric transport20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Major challenges such as inclusion of conductive resins21, quantum-dots doped resins22, liquid-crystals doped resins23 are still in the development phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Recently, great progress towards parallel direct-laser writing has been made, which enables a substantially accelerated fabrication process24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Finally, different polymerization concepts are constantly being developed, some of which use novel approaches to high-resolution 3D printing based on polymer resins25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 3 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' PHOTONIC INTEGRATION VIA PHOTO-INDUCED POLYMERIZATION Standard photonic waveguides covered in this review rely the guiding element called the core having a higher refractive index ncore than the refractive index of the confining part called the cladding ncladding, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' ∆n = ncore − ncladding > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' As schematically illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 1 (c), in such a configuration optical rays imping- ing on the core-cladding interface with an angle smaller than the critical angle θc = arcsin(1−(∆n/ncore)) exhibit total internal reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' As a consequence, they are con- fined to the waveguide’s core and propagate along this structure, allowing to direct optical propagation along pre-designed paths via an integrated and solid core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Refractive index contrast ∆n combined with the core diameter d are a waveguide’s determining characteris- tics, which determine a waveguide’s numerical aperture NA = � n2core − n2 cladding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The same holds for the num- ber of spatial modes allowed to propagated through the waveguide M ≈ V 2/2 = (4πd/λ)NA for large M, where λ is the optical wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='. Here, V is the normalized fre- quency a central indirect property of optical waveguides;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' for V ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='405 a waveguide is single-mode, otherwise it allows for higher modes to propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Finally, ∆n also determines the minimal bending radius for which light can be directed without exceedingly high losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This in turn is the limiting factor for integration density inside a photonic IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In work covered in this review, we used the commer- cial 3D direct-laser writing Nanoscribe GmbH (Photon- ics Professional GT) system, which is equipped with a femtosecond (fs) laser operating at 780 nm, and galvo- mirrors for rapid beam movement in the lateral direc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The fs-laser is usually tightly focused into the resin through an objective lens of high numerical aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Af- ter finishing the TPP-DLW step, the unpolymerized resin was removed in a two-step development process, immers- ing the structure first in propylene-glycol-methyl-ether- acetate (PGMEA) acting as a developer for 20 minutes, followed by rinsing in isopropyl alcohol (2-propanol) for 3-5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For OPP, we deposited samples in the com- mercial UV-chamber Rolence Enterprise Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=', LQ-Box model, 405 nm wavelength, 150 mW/cm2 average light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Two-photon polymerization Two-photon polymerization is a maskless direct-laser writing technique26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' A highly focused pulsed laser beam in the femtosecond regime is used to induce the absorp- tion of two-photons in the exposed volume inside the photo-resist (which is a monomer in its liquid phase), c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='f Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This two-photon activated polymerization creates long-chained polymer molecules that in turn form a solid volume due to molecule interlinkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Forming al- most arbitrary 3D structures can then be achieved by translating the laser through the overall volume of the photo-resist along all three spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Grav- ity can impose limitations on attainable shapes, yet this aspect usually does not have a too strong impact: the polymer and the original monomer resin have very simi- lar mass densities, and thus the Archimedes forces keep a polymerized voxel locked in its position due to the resin’s viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Principle of direct-laser writing (DLW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (a) The fs- writing laser is scanned through the photo-resist through the monomer resin using high-speed galvo-mirrors for the dis- placement in the (x, y)-plane, while a piezo controls the z- position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (b) The resin is two-photon polymerized only inside a small voxel volume, and voxels are placed on a grid deter- mined by hatching distance h in the (x, y)-plane, and slicing distance s in the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The laser power (LP) as well as s, h determine the overlap of neighboring voxels and through this the minimum feature size and the smoothness of printed surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (c) In our work we use the ’dip-in’ technique, where a drop of resin is located between the microscope objective and the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The printing direction is downwards, and the maximum size of 3D-printed structures is around 6 mm in height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Originally, the writing laser spot was translated through the resin using piezo stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This approach is highly accurate as the stages readily have nanomet- ric precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' However, it does not allow for large dis- placement, is very slow and hence cannot be used for large printing areas/volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' A major breakthrough resulted from using galvo-mirrors for moving the writ- ing laser’s focal spot through the resin (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 2 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' As a consequence, printing speed increased by orders of magnitude27, and fabricating large-scale 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='5 metasurfaces or 3D volumes became possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Crucial for the quality of 3D structures and for inte- gration in general is the feature size of a single, polymer- ized voxel relative to the the scanning speed of the print- ing laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The photoinitiation of the chemical reaction which essentially is instantaneous relative to the the writ- ing speed, and hence the writing-volume directly follows laser’s scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' However, polymerization is a chemical reaction with an associated time scale, like any diffusion phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Typically, this timescale is orders of mag- nitude slower than the galvo-controlled laser scanning28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This aspect is crucial, since as a consequence polymer- ization is taking place for several neighboring voxels at overlapping times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' It makes the polymerization process (a) (q) h LP1 < LP2 LP1 Sample holder and substrate 3D Scanning piezo LP2 Positioning stage S High-resolution objective Zm (c) DiLL (Dip-in) substrate resin Galvo high-speed 2D scanning Femtosecond laser Xg4 more homogeneous, and the obtained structures do not suffer from (unintended) variations of material properties resulting from stitching countless small voxels together to form a large structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' As schematically illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 2 (b), the writing laser power (LP), the hatch- ing h and slicing s distances as well as the scan speed modify the overlap between neighboring polymer voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Through this, the smoothness of surfaces and the ho- mogeinity of the polymer-medium can be controlled to a good degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For much faster polymerization, the peri- odic voxels would results in a photonic crystal like struc- ture, thus introduce scattering and all related phenomena inside the produced polymer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Thanks to diffusion, this aspect is almost not observable, yet it potentially is a source of optical losses in long waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' A powerful technique, called ’dip-in’ mode, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 2 (c), where the liquid resin is held between the substrate and the microscope objective, was introduced in 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This avoids having to print through the substrate (con- trary to immersion-oil techniques), which reduces aber- rations and removes the thickness of the substrate as a limitation of the maximal height of printed structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Importantly for CMOS compatibility, it enables printing on materials that are not transparent at fs-laser’s wave- length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Piezo actuators and/or the writing field (deter- mined by the microscope objective of the printer) are usually quite limited in area, usually below mm-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For printing larger structures stitching various writing fields together is required, and in that it is not dissimilar to the stepper-process used in 2D semiconductor lithog- raphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' One can select a lower NA microscope objective to increase the writing field, however, this can only be em- ployed on the cost of a reduced printing low-resolution29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Generally, 3D printing via direct-laser writing creates structures of high quality, and their optical and ellastical properties have been characterized with high accuracy using Brillouin light scattering30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In this paper, the au- thors demonstrate an excellent quality check of the poly- mer in the GHz regime for elastic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For example, the 3D-printed samples can have an elastic quality fac- tor only ten times smaller than fused silica at hypersonic frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Importantly for printing photonic waveguides, the de- gree of polymerization and through the Clausius rela- tionship also the refractive index n, is mainly determined by the type of photo-resist and the dose parameters D of the fs-laser, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' scanning speed and LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Within the window between the TPP-threshold and the breakdown point above which the polymerized voxel contains defects, the so-called dynamic power range of the photo-resist26, the size of the TPP-voxel can be further modified by adapting D and fabrication parameters distances h and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Figure 3 (a-b) depicts the experimental optimization of the dynamic power range of the liquid negative- tone IP-S photo-resist, with n ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='51 when fully TPP- polymerized31,32 and using a 25X magnification NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='8 microscope objective for writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We printed, on a fused h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='3 μm 5 μm 5 μm LP = 11 mW LP = 7mW 5 μm LP = 15 mW5 μm LP = 17 mW5 μm LP = 19 mW5 μm h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='4 μm 5 μm h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='5 μm 5 μm h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='6 μm 5 μm 5 μm h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='7 μm (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Dynamic power range characterization of waveguide cores printed via TPP using the IP-S photo-resist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Image taken with permission from9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (a) SEM micrograph of pilars, printed to reassemble the cores of waveguides, with 20 µm height and d = 5 µm, with laser power LP ∈ {7, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' , 19} mW, using hatching h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='4 µm and slicing distance s = 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (b) Impact of hatching distance h ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='3 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='7} µm, with fixed LP = 15 mW and s = 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' silica substrate, a set of five free-standing pillars to em- ulate waveguide cores with 20 µm height and diame- ter d = 5 µm using a range of TPP laser power LP ∈ {7, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' , 19} mW and hatching distances h ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='3 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='7} µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' As globally fixed parameters in all our fabrica- tions we use a scanning speed of 10 mm/s and a slicing distance of s = 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The scanning electron microscopy (SEM) micrograph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 3 (a) shows the effect of grad- ually modifying the LP with a hatching distance constant h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Structures printed with LP = 7 mW and LP = 11 mW have ondulated surfaces, whereas when in- creasing the laser power to LP = 15 mW results in larger TPP voxels and therefore smoother surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Exceeding LP = 15 mW leads to overpolymerization of the IP-S photo-resist (see two last micrographs of Fig 3 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We therefore select LP = 15 mW and proceed to optimize the second fabrication parameter by scanning the hatch- ing distance from h ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='3 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='7} µm, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 3 (b) shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We found that for h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='3 µm results are not always reproducible since smaller hatching dis- tance increases local exposure dose D and hence moves the process above the available power range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' One-photon polymerization One-photon polymerization is widely used to process thin material layers in the current 2D photo-lithography technology used for electronic semiconductor ICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The process is based on the exposure of a photosensitive resin, usually at the UV range, through a photo-mask including specific design patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Repeating this process layer- by-layer is possible to process and stack different thin material layers and fabricate 3D structures33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For highly structured patterns like SD memory cards, this has led to HV curr usecase det mag 贝 WD tilt 20μm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='00kv 25pA Standard ETD 2500x 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0mm 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0 FEMTO-STHV curr use case det mag 贝 WD tilt 20μm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='00kv 25pA Standard ETD 2500x 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0mm 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0 FEMTO-STHV curr use case det mag 贝 WD tilt 20μm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='00kv 25pA Standard ETD 2500x 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='25 nA Standard ETD 800x 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0mm40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0 FEMTO-STHV curr det mag 只 WD tilt 50μm use case 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='00kv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='25nA Standard ETD 800x 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0mm 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0° FEMTO-STHV curr use case det mag贝 WD tilt 20 μm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='00kV 25pA Standard ETD 2500x 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0mm 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0 FEMTO-STHV curr usecase det mag 贝 WD tilt 20μm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='00kv 25pA Standard ETD 2500x 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0mm 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0 FEMTO-STHV curr use case det mag 贝 WD tilt 20μm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='00kv 25pA Standard ETD 2500x 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0mm 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0 FEMTO-STHV curr use case det mag 贝 WD tilt 20μm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='00kv 25pA Standard ETD 2500x 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0mm 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0 FEMTO-ST5 ICs with up 100 or more circuit layers2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' However, such stacking of layers created via a generically 2D fabrication concept has several severe drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For one, it requires to precisely align the photo-mask multiple times in each photo-lithographic step, which is challenging and time- consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Secondly, one of the strongest features of 2D lithography is its economic appeal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Between each layer, each of the process step have to be repeated in a loop- like manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' A process where the entire IC’s volume is created during few of such process steps will potentially have the upper hand economically speaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Still, such stacked 2D lithography has also been used of complex 3D photonic integration, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Multilayer 3D waveguide fabrication using OPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Im- age taken with permission from34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (a) Schematic diagram of the fabrication sequence for the stacking waveguide using spin coating and simple direct UV photolithography curing (s1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' UV irradiation of the waveguides using a mask (s2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' develop- ment (s3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' UV irradiation of the cladding (s4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (b) Layout of the 3D interconnect polymer structure with an array of 4x8 waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (c) Cross-section microscope optical image of 4x8 stack waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Just as with TPP, the refractive index of the poly- merized resin is a function of the optical exposure does D31,35–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' However, in OPP the refractive index of the resin is modified for substantially larger volumes, and in particular volumes outside the intended plane of exposure do strongly accumulate unintended irradiation doses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' It is therefore a formidable challenge to precisely control a 3D refractive index distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' a volume hologram, with high spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' OPP is therefore better suited for simultaneous polymerization of, either, large areas like in classical 2D lithography, or for large uni- form volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Flash-TPP: combining one- and two-photon polymerization for photonic integration One can combine one- and two-photon polymeriza- tion as an hybrid configuration to accelerate the fabri- cation of 3D photonic chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Several approaches com- 50 μm Waveguide core TPP OPP (a) Mechanical supports (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Flash-TPP printing concept for 3D integrated pho- tonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Image taken with permission from9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (a) Classical ’dip-in’ process for the DLW-TPP fabrication of 3D photonic waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (b) UV chamber that polymerizes the unexposed regions of the 3D structure via OPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (c) SEM micrograph of a 3D-printed cuboid cross-section embedding 16 photonic waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The waveguide cores (mechanical supports) are printed with large (small) hatching distances, which defines the resolution of each component of the 3D photonic circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Red colour represents regions polymerized via TPP, while blue colour regions via OPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' bining UV lithography with DLW-TPP have been pre- viously demonstrated in38 and39 for the fabrication of high resolution 3D optical microcomponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' However, those methodologies require the polymerization of multi- ple photo-resists in two separated fabrication steps and become time-consuming if used for 3D fabrication due to the layer-by-layer approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We demonstrated a novel lithographic strategy that combines OPP and TPP, flash-TPP9, where we combine high resolution and quality TPP with unstructured and uniform OPP in order to accelerate the fabrication pro- cess by one order of magnitude when compared to us- ing TPP-only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Importantly, the concept only requires a single resin and adding the OPP step does not add ad- ditional development and washing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In flash-TPP, TPP and OPP are used for the fabrication of the dif- ferent sections of a photonic circuit, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 5 illustrates the working principle, here for the liquid negative-tone IP-S photo-resist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Waveguide cores accommodate the large majority of an optical signal’s electromagnetic field, hence cores are printed via TPP with a precisely opti- mized laser power and fine resolution in the (x, y)-plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' small hatching distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This ensures smooth core- cladding interfaces and hence low propagation losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Mechanical supports, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' surfaces that define the outer limits of the volumetric circuit, are printed with larger hatching distance and high LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Figure 5 (a) depicts the typically ’dip-in’ DLW-TPP printing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' After development, the photonic cir- cuit is transferred to a UV chamber, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 5 (b), and the OPP dosage D of the 3D circuit’s volume is con- a s1) buffer layer silicon substrate s2) mask spin- coated waveguide layer s3) s4) 506 trolled via the duration of the UV exposure, through which we tailor the refractive index of the waveguides’ cladding ncladding and hence ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The SEM micrograph from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 5 (c) shows the cross-section of an exemplary 3D photonic chip fabricated via flash-TPP consisting of a cuboid integrating 16 waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The cores and me- chanical supports, printed via TPP, are highlighted in red region, while the cladding volume, polymerized via OPP, is highlighted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Via flash-TPP, we fabricated photonic waveguides with a refractive index contrast between core and cladding in the order of ∆n ≈ 5·10−39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Figure 6 (a) shows the evolution of the the average numerical aperture and refractive index of the cladding < ncladding > polymerized via OPP versus D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We used UV exposure doses D of 0, 750, 3000 and 9000 mJ/cm2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Assuming a constant ncore ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='51, we can precisely con- trol, both, and < ncladding >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Waveguides are single-mode for d ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='9 µm, which are feasible to fab- ricate via standard DLW-TPP processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We obtained 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='3 dB/mm (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='26 dB) propagation (injection) losses for the fundamental LP01 mode of waveguides printed via flash-TPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Crucially, our 3D circuits did not degrade over time, and we evaluated the NA of waveguides under continuous operating condition across several months9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Overall, this demonstrates the reliability of the flash- TPP lithography methodology for an ultra-fast, single- step and high performance fabrication of 3D photonic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Printing via flash-TPP consist in polymerizing only the sections vital for communication and mechanical in- tegrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Importantly, the majority of a circuit’s area or volume is not involved in either, and they can hence be rapidly fabricated via UV blanket exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The print- ing times in flash-TPP is therefore drastically reduced, and in particular cases also scales different with the cir- cuit’s size9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This agrees with our experience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' flash-TPP reduces the printing time to only 10% compared to only- TPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' As an example, printing a large structure that integrates waveguides with heights ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='1 to 6 mm9, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 6 (b), requires ∼24 hours only using TPP but only ∼3 hours using flash-TPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' AIR-CLADDED WAVEGUIDES Polymer waveguides with an air cladding have a rel- atively large ∆n ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='5 with ncore = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' On the one hand, this leads to very strong confinement and a large NA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='13, which enables very small bending radii of 25 µm (14 µm) at λ = 1550 nm (λ = 650 nm), and in turn dense photonic integration40–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The large ∆n makes fabricating single-mode waveguide circuits chal- lenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' To be single-mode, air-cladded waveguides have to have a core diameter d ≤ 1 µm (d ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='43 µm) at λ = 1550 nm (λ = 650 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Printing waveguides with d ≤ 1 µm is possible7, and strongly confined photonic IC at λ = 1550 nm are within reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For photonic 3D (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Optical performance of waveguides printed via flash- TPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Image taken with permission from9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (a) Average numer- ical aperture and cladding’s refractive index < n2 > over OPP dose D of photonic waveguides printed via flash- TPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The (< n2 >) decreases (increases) over D, meaning that we can control the degree of polymerization of the cladding via the dosage of UV light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (b) Macroscopic structure scaled to a match that integrates waveguides with heights ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='1 to 6 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' ICs close to the visible wavelength of light this remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Recently, 3D optical splitter/combiners based on air- cladded waveguides with a 1 to 4, 1 to 9 and 1 to 16 configuration were printed using TPP43,44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Figure 7 (a) shows an SEM image of the 1 to 4 fractal splitter/coupler, with its optical characterization at λ = 632 nm shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 7 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' There, the distance between output ports was scanned within the range D0 ∈ [10, 12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=', 20] µm while keeping their height constant at 52 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Losses do not substantially increase for smaller distance between the output ports, which validates the estimated mini- mal bending radii given before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Furthermore, this per- formance was evaluated for two different LP settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' No clear difference can be seen between the two data-sets, and hence the printing power for air-cladded 3D polymer waveguides is not a critical parameter, as long one stays within the dynamic power range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For large-scale network interconnect, Moughames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' demonstrated 3D parallel interconnects with high con- nectivity, shown in Figure 7 (c), by cascading two layers of 1 to 9 splitters and spatially multiplexing an arrays of such 1 to 81 splitters to allows for an array of 15x15 input waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The entire circuits only occupies a volume of 460x460x300 µm3, in which an interconnect for 225 inputs and 529 outputs is realized7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Figure 7 (d) shows a higher magnification of this interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Individual wavegudies have a low surface roughness, and the incor- porated chirality of the fractal splitters/couplers avoids intersections of individual waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' STEP AND GRADED INDEX WAVEGUIDES Based on the previous discussed concepts and fabrica- tion technologies, we addressed step- (STIN) and graded- index (GRIN) waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In STIN waveguides, the re- fractive index of the waveguide’s core is constant, while for GRIN waveguides it is a function of the radial distance to the core’s center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Usually, GRIN waveguides follow 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Air-cladded waveguides and couplers fabricated via DLW-TPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Image taken with permission from7,43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (a) 2x2 optical splitter/coupler with 1 input and 4 outputs with dis- tance D0 = 16 µm between waveguides, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='2 µm waveguide diameter43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (b) Optical losses of 2x2 splitters/couplers as a function of the distance D0 between waveguides, for hatching distances h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='1 µm (in blue) and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='2 µm (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Data on top correspond to splitters/couplers written with laser power LP = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='4 mW, and data at the bottom correspond to splitters/couplers written with laser power LP = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='2 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (c) SEM micrographs of 3D-printed waveguides realizing par- allel interconnects with high connectivity7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (d) Zoom-in of (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' a parabolic refractive index distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For the STIN waveguides, all bound rays propagate at angles within the total internal reflection condition θc at any position in the core cross-section, while for GRIN waveguides, the range of angles varies with position45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We proposed a single-step additive fabrication tech- nique, (3+1)D printing8, by which we spatially modify the refractive index of a single resin over the TPP expo- sure dose during fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Using the (3+1)D-printing concept, we constructed volume holograms and photonic waveguides with, both, STIN and GRIN profiles in a single-step, single-material fabrication with a commer- cially available process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This demonstrates the versatility of the 3D photonic integration approach based on DLW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' optical manipulation based on integrated and monolithic 3D structures can either rely on discrete components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' waveguides, or leverage continuous manipulations of free optical propagation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' holograms8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Both schemes can be exploited on the same photonic IC and be realized using the same fabrication concept and during the same fabrication step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We used the negative tone IP-Dip resin (n ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='547)36 and a 63X magnification NA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='4 micro- scope objective, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 5 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The SEM micrograph of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 8 (a) shows an exem- plary cuboid embedding 20 STIN waveguides fabricated via (3+1)D-printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Contrary to flash-TPP, in (3+1)D- printing all the 3D photonic chip volume is fabricated via TPP-only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The refractive index contrast ∆n between core-cladding waveguides is achieved from the control over the TPP dosage D for individual writing voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For (a) 100 µm (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Step- (STIN) and graded-index (GRIN) waveguides fabricated via (3+1)D-printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Image taken with permis- sion from8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (a) SEM micrograph of an exemplary 3D-printed cuboid integrating 20 STIN waveguides of 300 µm heigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Waveguide cores (cladding) are printed via TPP with high (low) laser power, which ensures a refractive index contrast ∆n ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='4·10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Panels (b) and (c) depict the output intensi- ties (triangles) and fundamental LP01 mode fits (dashed lines) of a 3 µm radius STIN and GRIN waveguide, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' a higher (lower) refractive index as needed for the waveg- uide cores (claddings), one requires an accordingly higher (lower) LP, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' STIN waveguides result from a con- stant LP all across their core, while for GRIN waveguides the writing power changes from high to low following a parabolic profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' To evaluate the optical performance, we fitted the ex- perimental output intensities for diameters d below the cut-off condition of the second propagation mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The output intensity of the LP01 mode of a STIN waveguides is described by J2 0(u r R) for | r | < R and K2 0(v r R) for | r | > R, while for GRIN waveguides is given by an in- finite parabolic refractive index profile as exp − 1 2V r2 R2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Figure 8 (b-c) depicts the fit of fundamental LP01 mode to the normalized output of STIN and GRIN waveguides with radius R = 3 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Considering the refractive index of the core constant (ncore ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='547), we obtained an averaged numerical aperture = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='01 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' ncore = ncladding + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='4 · 10−3) for STIN and of = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='02 for GRIN waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' As expected, the core-confinement of GRIN waveguides is significantly higher than for STIN waveguides due to the inner core refractive index modification, which offers a crucial advantage for photonic integration schemes7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' As seen, STIN waveguides with a polymer cladding have a refractive index contrast in the order of ∆n ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='4·10−3, with low NA ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Contrary than for air- cladded waveguides, this leads to large bending radii of 15 mm (7 mm) at λ = 1550 nm (λ = 650 nm), and in turn dense photonic integration is much more challeng- ing for STIN waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' However, the low ∆n allows to have single-mode propagation for waveguide diameters d ≤ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='8 µm (d ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='2 µm) at λ = 1550 nm (λ = 650 nm), which is standard with the current DLW-TPP fabrica- tion technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Future efforts include combining poly- mer and air-cladded waveguides, taking the strengths of each configuration in a single platform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' air cladding waveguides providing highly-densed photonic integration P = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='4 mW 5 7 6- 11 p = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='2 mW 5 7 9 30 μm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='11 200 μm 50gmexp(cs291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='2μmcs291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='2μmcs291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='2μmcs291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='2μmcs291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='2μmcs291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='2μm8 with their small bending radii, while STIN waveguides serving as tools for single-mode propagation with large waveguides diameters over wide distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' FLASH-TPP PRINTED WAVEGUIDES Recently, we demonstrated the fabrication of large scale 3D integrated photonic components via flash-TPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Several features of flash-TPP make it an enabling tech- nology for integration of larger circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Of primary importance is the substantial accelerated fabrication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' without, fabrication of larger integrated circuits would quickly approach timescales beyond 24h9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Based on this approach, we demonstrated long (6 mm) single-mode waveguides, and we achieved exceptionally low injection (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='26 dB) and propagation (≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='3 dB/mm) losses9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Next as the demonstration of optical splitters and com- biners based on this concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' These are the backbone of any photonic IC, and 3D integration enables interesting alternatives for creating 1 to M optical couplers without using sensitive optical interference units46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In 3D, 1 to M optical couplers can simply be realized by arranging nu- merous output waveguides around the input waveguide, something impossible to realize in a purely 2D integra- tion setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We demonstrated broadband 1 to M split- ters leveraging adiabatic coupling6,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Adiabatic cou- pling achieves low-loss single-mode optical transfer from 1 to M waveguides through evanescent waves, where the optical mode adiabatically leaks from a tapered core of an input waveguide towards the cladding into inversely- tapered cores of the output waveguides48,49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' All the pre- vious studies consider the 2D case of only one to one adiabatic coupling between optical components50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In our work, we showed efficient single-mode adiabatic transfer with 1 input and up to 4 outputs via a single component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Figure 9 (a) illustrates the design for the exemplary case of a 1 to 2 adiabatic couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The waveg- uide’s circular core cross-section continuously changes as a function of propagation direction z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The originally cir- cular core is reduced in size exclusively along the direc- tions where an output waveguide is located;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' the core is essentially cut along plane surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' These cut-planes move towards the input core’s center during the taper- length lt at equal rate d/lt along the (x, y)-plane in order to match their relative effective modal indices45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Output waveguides follow exactly the same concept, yet in an in- verted direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We separated in and output waveguides via gap g and studied the evanescence coupling efficiency between coupled waveguides6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The same tapering strat- egy was applied to 1 to 3 and 1 to 4 as depicted in the output intensity profiles from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 9 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We obtained record optical coupling losses of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='06 dB for the optimal case of 1 to 2 adiabatic couplers, with a difference between the two outputs intensities of only ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='4 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We furthermore demonstrated broadband func- tionality from 520 nm to 980 nm during which losses re- main below 2 dB6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Importantly, these adiabatic couplers can be cascaded in order to exponentially increase the number of M outputs, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 7 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We arranged a double-layer of 1 to 4 adiabatic couplers and the result- ing 1 to 16 single-mode output intensities can be seen in the last diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 9 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Importantly, the global losses of the entire device is only 1 dB , and the entire circuit was realized within (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='08 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='08 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='5) mm36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' x y z Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Intensity 0 1 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Adiabatic 1 to M broadband-scalable couplers fabri- cated via flash-TPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Image taken with permission from6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (a) Design of the 1 to 2 adiabatic couplers printed via flash-TPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The same tapering strategy can be applied to higher-order couplers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 1 to 3 and 1 to 4 couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (b) Output intensity profiles of the 1 to 2, 3 and 4 adiabatic couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The last output intensity corresponds to a cascaded 1 to 16 adiabatic coupler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' TOWARDS A SCALABLE AND CMOS COMPATIBLE INTEGRATION OF PHOTONIC NETWORKS High-density photonic integration requires the inter- connection of several photonic platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Most of the current photonic devices are based on silicon-on-insulator (SOI) and CMOS technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Combining the strength of multiple photonic and electronic systems in one hybrid and multi-chip platform can result in the diversification of specific computing tasks while increasing the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' A versatile fabrication technology with low-losses is of vital importance for the scalability of free-form as well as integrated optical interconnects in three-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The polymer-based 3D printing technology based on DLW- TPP is excellently suited to address these challanges, and several proof-of-concept studies have been realized50–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Figure 10 (a) shows photonic wire-bonding, realising a 3D photonic waveguide forming a point to point com- munication for a chip-to-chip connection between SOI chips hosting individual waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The photonic wire- bond was fabricated via DLW-TPP using the negative- tone MicroChem SU-8 2075 photo-resist (n ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='51 at 1550 nm)53, and it connected two SOI waveguides sep- arated a distance of 100 µm on different CMOS chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This demonstrated for the fist time the basic viability of TPP-based 3D printing as a tool for CMOS compatible, wafer-scale as well as chip-to-chip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' A major challenge of the polymer-based 3D fabrication g C15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='8 Intensity 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='9 20 Size (μm) 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='9 Size (μm)9 and the CMOS technology is the interaction of the CMOS substrate with the photo-resist during the TPP printing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In a standard fabrication setting, the interac- tion between the fs-pulsed laser and the glass substrate is negligible since the substrate material, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' fused silica, is transparent at the wavelength of the fs-laser (780 nm), and low specular reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' However, the CMOS tech- nology is based on 2D stacking of multiple thin layers of semiconductor materials such as GaAs, InP or Sili- con.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' These often have a bandgap energy below that of the writing laser, and in that case printing through the semiconductor substrate is impossible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' only the ’dip-in’ concept is therefore a viable general approach for fabri- cating 3D photonic integrated circuits directly on top of a CMOS substrate based on DLW-TPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Another chal- lenge is the higher specular reflection, as these semicon- ductor materials have a higher refractive index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The re- sulting optical reflection of the fs-laser laser at the semi- conductor substrate leads to a overpolymerization of the photo-resist if not compensated for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The LP therefore needs to be continuously adjusted at the vicinity of the CMOS/photonic circuit interface in order to achieve the intended degree of polymerization of the photo-resist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' A further requirement is the precise alignment of the 3D photonic chip with the semiconductor device patterned on the CMOS substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (a) (b) 25 μm IP-S GaAs IP-S FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Polymer-based 3D printing and CMOS technology compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (a) Chip-to-chip photonic wire bonding con- cept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' A 3D polymer waveguide fabricated via DLW-TPP connects two SOI waveguides sitting on distant CMOS chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' SEM image taken with permission from 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (b) SEM micro- graph of and exemplary 3D-cuboid integrating a cascaded 1 to 16 adiabatic couplers printed via flash-TPP on top of a quantum dot micropillar laser array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Figure 10 (b) depicts an exemplary 3D-printed cuboid integrating a cascaded 1 to 16 adiabatic coupler (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 9 (b)) printed via flash-TPP on top of a semiconduc- tor substrate integrating quantum dot micropillar laser arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Each of the micropillar lasers consists of a cylin- drical microcavity (a vertical arrangement of highly re- flective distributed Bragg reflectors (DBR) alternating Al(Ga)As and GaAs mirror pairs) sandwiching a cen- tral gain section based on InGaAs self-assembled quan- tum dots (QDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Further details about the fabrication and optical properties of the quantum dot micropillars laser arrays from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 10 (b) can be found in54–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We used IP-S photo-resist for the fabrication, with a lower laser power LP = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='5 mW (compared to the previously LP = 15 mW) in order to avoid microexplosions of the photo-resist at the semiconductor-polymer interface dur- ing TPP printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' After development, the 3D photonic chip is then polymerized via OPP with a exposure dose D = 3000 mJ/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The SEM micrograph shows the perfectly aligned 3D photonic structure with the angle of the periodic GaAs micropillar array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We checked the adherence of the polymer over time, and after a continu- ously observation over more than 4 months no deteriora- tion has been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This confirms the reliability of in- tegrating our 3D printing technology with CMOS-based micro-laser arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' CONCLUSION Here, we present a review over our recent work address- ing additive manufacturing towards future 3D photonic integration of optical components that is CMOS com- patible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Based on one- and two-photon polymerization processes combined with direct-laser writing systems, we demonstrated the fabrication of high performance indi- vidual photonic waveguides as well as scalabale optical splitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' All such 3D structures have been fabricated in our local FEMTO-ST RENATECH infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We demonstrated that using the commercial DLW- TPP Nanoscribe GmbH (Photonics Professional GT) system and the ’dip-in’ DLW strategy, we are able to the construct, both, air- and polymer-claddded photonic waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For air-cladded waveguides, we used a TPP- only, a single-step and single resin (IP-Dip resist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' A 3D IC comprising a network of fractal optical splitter with 225 input and 529 output waveguides only occupies a volume of 460x460x300 µm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Such air-cladded waveg- uide ICs are prime candidates for highly-dense photonic packaging thanks to their low bending-radii on 10s of µm scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' For polymer-cladded waveguides, we presented two different strategies in which we 3D-printed the waveguide cores via TPP while achieving a precise control over the refractive index contrast ∆n via, (i), the adjustment of the fs-laser dose D on an single-voxel level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' (3+1)D- printing, and (ii), the duration of UV blanket exposure that determines the OPP dosage D to fix the index of the cladding material for the entire photonic IC in a single shot, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' flash-TPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Noteworthy, both fabrication con- cepts require a single procedure writing step and a single resin (IP-S resist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Importantly, with flash-TPP fabri- cation times are reduced by up to ≈ 90 % compared to (3+1)D-printing thanks to the additional OPP process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Via flash-TPP, we achieved polymer-cladded waveguides with refractive index contrast ∆n ≈ 5·10−3, with low 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='3 dB/mm (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='26 dB) propagation (injection) losses while printing waveguides up to 6 mm heigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This allows to have single-mode propagation over large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We demonstrated the fabrication, via flash-TPP, of scalable- boadband couplers leveraging adiabatic transfer from 1 input up to 4 outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Using a tapered/inversely-tapered waveguide sequence, we achieved record 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='06 dB optical coupling losses with very symmetric splitting ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' We HV curr use case det mag 只 WD tilt 50 μm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='00 kV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='20 nA Standard LVD 1 000 x 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0 mm 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='0° FEMTO-ST(a) (b) Photonic wire Photonicwire bond bond SOlwaveguide SOI 25μm waveguides Chip1 20 μm 10 μm Chip2 (c) Input fiber Qutput fiber Photonic wire bonds Chip1 Chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content='2 Grating couplers10 arranged a double-layer of 1 to 4 adiabatic couplers, re- sulting in a device with 16 single-mode outputs with only 1 dB global losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Importantly, we demonstrated the compatibility of our fabrication methodology based on DLW-TPP with CMOS substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' As a proof-of-concept, we success- fully 3D-printed our cascaded 1 to 16 adiabatic couplers on top of a CMOS substrate integrating GaAs quantum dot micropillar laser arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Preliminary characteriza- tion of these structures shows encouraging performance in terms of losses and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Overall, in this review we have covered our novel 3D- printing technology, which represents a breakthrough with the potential to become a high-impact tool for the hybrid, highly-dense and hence compact packaging of, both, electronic and photonic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The concepts opens several potential avenues for future exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The combination of air- and polymer-cladded waveguides could enable dense integration with simultaneous precise control over optical signal properties such as mode num- ber, polarization and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' As the concept leverages photo-polymerization, in principle the large-scale and exceptionally performing production facilities of CMOS electronic integration could be amended with 3D pho- tonic integration capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' Due to the excellent compat- ibility of standard photo-resins, the approach is largely agnostic to the underlying substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' In this it is more flexible than integrated silicon photonics, and fabricat- ing additively on a already processed CMOS substrate removes many of the challenges compared to fabricating photonic ICs based on different process - such as DLW di- rectly into bulk dielectrics followed by bonding to CMOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' ACKNOWLEDGMENT The authors would like to thank Stephan Reitzen- stein for his contribution through fabricating the semi- conductor laser sample used for producing the circuit shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' 10 (b) and Erik Jung for the valuable help on the design of 3D waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' This work was partly supported by the french RENATECH network and its FEMTO-ST technological facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQfDvrh/content/2301.00983v1.pdf'} +page_content=' The authors ac- knowledge the support of the Region Bourgogne Franche- Comt´e.' metadata={'source': 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Different Student Clusters in Functional +Programming Assignments: From Quick Learners to Struggling +Students +Chuqin Geng +McGill University +Montreal, QC, Canada +chuqin.geng@mail.mcgill.ca +Wenwen Xu +McGill University +Montreal, QC, Canada +wenwen.xu2@mail.mcgill.ca +Yingjie Xu +McGill University +Montreal, QC, Canada +yj.xu@mail.mcgill.ca +Brigitte Pientka +McGill University +Montreal, QC, Canada +bpientka@cs.mcgill.ca +Xujie Si +McGill University +Montreal, QC, Canada +xsi@cs.mcgill.ca +ABSTRACT +Instructors and students alike are often focused on the grade in +programming assignments as a key measure of how well a student +is mastering the material and whether a student is struggling. This +can be, however, misleading. Especially when students have access +to auto-graders, their grades may be heavily skewed. +In this paper, we analyze student assignment submission data +collected from a functional programming course taught at McGill +university incorporating a wide range of features. In addition to the +grade, we consider activity time data, time spent, and the number +of static errors. This allows us to identify four clusters of students: +"Quick-learning", "Hardworking", "Satisficing", and "Struggling" +through cluster algorithms. We then analyze how work habits, +working duration, the range of errors, and the ability to fix errors +impact different clusters of students. This structured analysis pro- +vides valuable insights for instructors to actively help different +types of students and emphasize different aspects of their overall +course design. It also provides insights for students themselves to +understand which aspects they still struggle with and allows them +to seek clarification and adjust their work habits. +CCS CONCEPTS +• Social and professional topics → Student assessment. +KEYWORDS +online programming platform; computer science education; cluster +analysis +1 +INTRODUCTION +Online programming environments, such as RoboProf [8] for C++, +DrScheme [13, 14] for Scheme or, more recently, Mumuki [4] , offer +immense potential to enhance the students’ educational experience +in large-scale programming-oriented courses. They not only lower +the entry barrier for beginners but often feature auto-grading facili- +ties that allow students to run and get feedback on their code while +they are developing their programs, giving them the opportunity +to fix bugs and address errors in their understanding right away. +While having access to immediate feedback on their code has been +recognized to significantly improve student learning outcomes and +engagement (see, e.g., [15, 26, 30]), instructors and students alike +are often too focused on the grade as a key measure of competency. +Especially when students have access to auto-graders, the students’ +grades may be heavily skewed and misleading. +This paper develops a data-driven approach to better understand +students’ behavior when solving programming assignments in a +functional programming course. In addition to the grade, we pro- +pose to consider additional factors such as the number of static +errors and total time spent on solving programming assignments to +identify student clusters. Using this methodology, we analyze the +assignment submission data collected in a functional programming +course taught at McGill university which uses the Learn-OCaml +online programming platform [5, 6, 17]. This allows us to identify +four student clusters: "Quick-learning", "Hardworking", "Satisficing", +and "Struggling". While the first two clusters can be characterized +as maximizers, i.e. students strive to achieve the highest possible +grades and continue to improve their work, they still differ in the +amount of time and effort spent on completing a given homework. +In contrast, satisficing1 students accept a possibly non-optimal out- +come as ”good enough” allowing them to adequately achieve their +goals by saving time and effort. We further analyze these clusters +with respect to work habits and the number and kinds of errors +that are prevalent. This leads to four key insights: +• Leveraging the notion of chronotype - a circadian typology in +humans and animals, we confirm that a work pattern where +students tend to work in the morning is related to academic +success. In particular, quick learners tend to work more in +the morning, while other clusters of students rely more on +afternoons and evenings. +• In general, starting on the homework early is related to +higher grades. However, we also noticed that satisficing stu- +dents start relatively late but finish the earliest. This further +emphasizes that satisficing students aim for satisfactory re- +sults rather than the optimal one. At the same time, satisfic- +ing students have one of the lowest numbers of programming +errors suggesting that they struggle significantly less with +static errors than for example hardworking students. +• Our analysis of static errors shows that syntax and type +errors are prevalent among all students. Further, students +1The term “satisficing” was introduced by H. Simon [27] to describe a decision-making +process in which an individual makes a choice that is satisfactory rather than optimal. +arXiv:2301.02611v1 [cs.CY] 6 Jan 2023 + +continue to struggle with these errors throughout the se- +mester. In addition, our analysis points to other common +mistakes such as non-exhaustive case analysis and the use +of unbound variables. +• Taking into account students’ ability to fix static errors, i.e. +how many tries a student needs to fix a particular error, we +notice that the failure/success ratio is particularly high for +hardworking students. This highlights both their desire and +drive to get the best possible grade, but also that their path +is full of small stumbling blocks. +We believe our proposed set of features and data-driven analysis +can provide instructors with a clearer and more detailed picture of +students’ behaviours and performance. This in turn may be used +to adjust how some concepts, such as how to avoid static errors, +are taught. It may also be used to design different strategies for +different students to enhance the students’ learning experience. +Furthermore, this data may be interesting to students themselves +to better understand how well they do in a class and identify areas +where they can actively make changes and seek help early. +2 +RELATED WORK +Analyzing student data in programming courses is a central topic +in learning analytics, and it is gaining increasing attention with +the recent advances in storing and processing data. One of the core +aims of analyzing student data is to understand student behaviours, +and in turn, improve student learning experience [21]. +Over the past decade, there have been several studies that focus +on identifying groups of students using cluster analysis. For exam- +ple, Emerson et al. [12] use cluster algorithms to identify student +misconceptions in a block-based programming environment for +non-CS major students based on program structures. Wiggins et +al. [29] finds five major clusters of hint requests in a block-based +programming system equipped with an intelligent tutor. Hossein +et al. [20] leverages clustering analysis to further investigate the +correlation between students’ programming speed and program- +ming behaviours by collecting programming snapshots whenever +an action occurs. They then divide students into two clusters by +comparing a student’s programming speed to the median speed. +Lahtinen et al. [23] uses different levels of Bloom’s Taxonomy as fea- +tures to identify six distinct student groups that instructor should +be aware of when teaching introductory programming courses. +In contrast to these existing works, our work considers multi- +categorical features involving the grade, total time spent on the +assignment, and the number of static errors encountered to identify +clusters of students. +Based on the identified clusters, we follow existing work in under- +standing the work/rest patterns of students. In particular, Claes et al. +[7] study programmers’ working patterns using clustering analysis +on time stamps of committed activities of 86 large open-source +software projects. Zavgorodniaia et al. [31] study the chronotypes +of students through cluster algorithms using keystroke data. In our +study, we use activity data (such as whether a student compiled or +graded their homework) to study the work/rest patterns of students. +It is the first study in the context of typed functional programming. +We further analyze static errors in typed functional programming +assignments and their impact on different student clusters. This +is the first such study in this setting. Previous studies focus on +compilation events in object-oriented programs written in Java. +For example, Ahmadzadeh et al. [1] investigates compiler error +frequencies of student programs and debugging activity patterns +in Java. They suggest debugging skills should be emphasized in the +teaching of programming. Altadmri et al. [2] collect a large dataset +comprising compilation events of 250,000 students, which provides +a robust analysis of error patterns and time for fixing different +errors. Denny et al. [9] also study various syntax error frequencies +and how long students spend fixing common syntax errors. They +also found that certain types of errors remain challenging even for +higher-ability students. Edwards et al. [11] analyze 10 million static +analysis errors found in over 500 thousand program submissions +made by students over a five-semester period. The experimental +results suggest error frequencies made by CS major and non-major +students are consistent. +Our analysis is one of the first that investigates in more depth +the frequency of various static errors in the typed functional pro- +gramming assignments. Here, static errors go beyond syntax and +simple type errors and include for example detection of missing +branches in a program. +3 +STUDY DESIGN +This research aims to gain a deeper understanding of how students +develop typed functional programs (TFP). We assume that the grade +alone is not a good indicator of how well a student masters basic +concepts and achieves competency. Instead, we propose that taking +into account the time spent as well as the number of errors a student +encounters can provide a more nuanced picture. Hence, the main +question that we tackle in this paper is how can we best identify +different clusters of students taking into account grades, time spent, +and the number of errors. We then analyze our clusters with respect +to five hypotheses: +H1: Even students with a high grade in programming assign- +ments may significantly struggle with a range of static errors. +H2: Despite a lower grade, students who spend less time and +have a low number of static errors do in fact well overall. +H3: Work/rest patterns of students as well as the time a student +spends on homework play a role in students achieving a high +grade. It highlights how driven a student is. +H4: Static errors in TFP range from syntax and type errors +to detecting unbound variables and missing branches in +programs. This wide range of static errors provides a fine- +grained picture of concepts students find challenging. +H5: Error fix ratio, i.e. how many tries a student needs to fix +a static error, indicates how well students understand basic +ideas in TFP and this is correlated to their understanding +and performance. +3.1 +Course Context +Our study concerns students in a second-year undergraduate com- +puter science course at McGill university. The course introduces +concepts about functional programming and programming paradigms. +It is offered every semester with more than 300 registered under- +graduate students. In this study, all data is collected in the Fall 2021 + +programming topics +#tasks +HW1 +basic expressions, recursion +7 +HW2 +data types and pattern matching +6 +HW3 +higher-order functions +11 +HW4 +references, state, memorization +5 +HW5 +exception, continuations +5 +HW6 +lazy programming, toy language +5 +Table 1: Overview of six programming assignments. +Figure 1: Data collection pipeline. Grade and Compile and Eval +events are handled by different servers, all submission data are +stored in a MongoDB database. The components highlighted in light +green are original components in the Learn-OCaml platform, while +the components highlighted in light blue are newly introduced by +us. +semester when students could attend online Zoom or in-person +sessions. +The course had six bi-weekly programming assignments each +worth 5% of the final grade. Each homework consists of several pro- +gramming tasks to implement functions and test cases. Homework +information is summarized in Table 1. All homework assignments +were hosted on Learn-OCaml [6], an online programming platform +for OCaml which allows students to edit, compile, test, and debug +code all in one place. We used a modified version of Learn-OCaml +by Hameer and Pientka [18] with additional features such as style +checking and evaluation of test cases written by students. +3.2 +Data Collection +Our data collection pipeline is built on top of the Learn-OCaml +platform and it can automatically log students’ actions. Specifically, +we send local programming events like compile and evaluation (for +testing and debugging) with asynchronous logging requests to our +backend server. Figure 1 illustrates the process of collecting the +data from the online environment Learn-OCaml. +Around 52.81% (i.e., 169 out of 320) students gave us consent +to access their data. We collect more than 270,000 programming +events, and each event stores a snapshot of the code as well as +feedback information (e.g., time-stamp, static errors, grades, etc.). +3.3 +Feature +For each homework, we collect a sequence of programming activity +events. The activity events include grade, compile, and evaluation +events. This allows us to create an activity density vector for each +student. It is a four-element vector that represents the percentage +of the student’s activity events that occurs in different ranges of +hours [0-6, 6-12, 12-18, 18-0], which is the same choice of ranges +suggested in [31]. +In addition, we design the following features based on the activity +event sequence: +• Start time. The day when a student starts actively working +on an assignment based on the activity events collected. +• End time. The day when a student finishes an assignment, +which is the last Grade event. +• Working session. Defined as the time window where ac- +tivity events occur. If there is no activity event within 30min, +then the working session is assumed to have ended. +• Total time spent. Sum over the length of all working ses- +sions. +• Number of errors. The number of static errors that a stu- +dent made while completing an assignment. +• Grade. The final grade a student receives for an assignment. +3.4 +Feature Engineering +There are two challenges to applying clustering algorithms and sta- +tistical tests to our study. The first one is skewed data . For instance, +the grade is highly skewed as students can always improve their +grades through interacting with the auto-grader. The second one +is the difference between feature scales, which renders the clus- +tering results incoherent. We use two approaches to address these +challenges. First, we use non-parametric tests including Spearman +correlations and Kruskal-Wallis H-Tests. Second, we apply the rank +transformation on features to facilitate clustering algorithms. +4 +IDENTIFYING STUDENT CLUSTERS +To identify student clusters, we run the K-means[19] clustering al- +gorithm on the aggregation (mean) of three most important features +(i.e., grade, number of errors and time spent) over six homework. We +use the elbow method to determine the optimal k (the number of +clusters) to be 4. After determining the optimal k, we re-run the +K-means algorithm and report the results in Table 2. We give the +time in hours and note that all clusters have a similar size in terms +of number of students (#𝑆𝑡𝑑). +To determine whether the resulting four clusters are different, we +run a Kruskal-Wallis H-Test, which is a nonparametric equivalent +of an ANOVA, on the three features (time spent, #errors, and grade) +of each cluster. The results are statistically significant with the +statistics of 113.26, 100.87, and 123.02 respectively, and all p-values +< 0.0001. This suggests the four clusters are statistically different. +Students in cluster A have the highest average grade (95.24) +while spending less than the expected 6h on solving the homework. +This suggests that they achieve their goal with relative ease. In fact, +Clusters +#Std +Time (Hours) +# Error +Grade +A - Quick learning +46 +5.30 (± 0.94) +66.11 (± 26.95) +95.24 (± 3.25) +B - Hardworking +46 +8.24 (± 1.52) +148.67 (± 63.26) +94.25 (± 3.90) +C - Satisficing +31 +4.47 (± 1.01) +52.26 (± 21.89) +74.43 (± 11.31) +D - Struggling +46 +6.49 (± 0.94) +118.14 (± 35.32) +72.81 (± 11.03) +Table 2: Student clusters + +Feedback from +Autograder +programming +Git repo +autograder +history +Webserver +Grade Event +Grade data entry +LearnOCaml +[id, timestamp, code, grade] +Client +Compile and +Eval Event +Compilation +Results +MongoDB +MongoDB +Webserver +Compile and Eval +data entry +[id, timstamp, code]students in this cluster outperform students in other clusters by a +large margin. We characterize this cluster as quick learning. +Students in cluster B have the second-highest average grade +(94.25). However, they also have the highest average number of +errors (148.67) and with 8.24h spend significantly more time on +homework than any other group. In particular, they spend signifi- +cantly more time than expected. This suggests that they face many +difficulties which they manage to overcome by spending a signif- +icant amount of time. These students are driven to improve their +work and to achieve the highest possible grade. Hence, we charac- +terize them as hardworking. This data supports our hypothesis +H1. +Cluster C has the lowest average number of errors (52.26) and +spent the least amount of time (4.47h) on the homework. With an +average grade of 74.43, they still achieve a “good enough” result. +These students achieve their goals by saving time and effort. At the +same time, these students reach a satisfying level of competency as +evidenced by their low number of average errors. We describe these +students as satisficing students. This supports our hypothesis H2. +Students in Cluster D are in fact closely related to students in +cluster B, which shows a similarly high average number of errors +(118.14) and a significant amount of time (6.49h). However, com- +pared to students in cluster B, they fail to overcome the difficulties +along their path. These students are struggling. +5 +UNDERSTANDING STUDENT CLUSTERS +5.1 +How do work habits vary for different +student clusters? +To investigate our hypothesis H3, we consider when students are ac- +tive based on our activity data. Prior research suggests that chrono- +type, a person’s preference in carrying out activity at certain periods +in a day, is governed by the circadian cycle which is controlled by +clock genes [10, 25]. In this section, we are interested in investigat- +ing the chronotypes, or in other words, the work habits of students. +In particular, it has been observed that “morningness” is positively +correlated with academic achievement [24, 31]. +To identify potential chronotypes, we run the K-means cluster- +ing algorithm on the feature space spanned by activity density +vectors. The elbow method yields 𝑘 = 3, suggesting three possible +chronotypes, which is different from four chronotypes reported in +[31]. We report centroids of each chronotype cluster in Table 3. +Chrono clusters +0 - 6 +6 - 12 +12 - 18 +18 - 0 +Chronotype +Cluster 1 +8% +14% +26% +52% +Evening (Eve) +Cluster 2 +4% +26% +20% +50% +Morning (Mor) +Cluster 3 +2% +19% +37% +42% +Afternoon (Aft) +Table 3: Centroids of each chronotype. +As we can see, most activities occur from 18:00 - 00:00 for all +three clusters. This is not surprising as most students may have +classes during the day. Based on this observation, we aim to define +chronotypes by considering secondary activity peaks as well. We +notice that Cluster 2 has its secondary activity peak (26%) in 6:00 - +12:00 whereas Cluster 3 has the secondary activity peak (37%) in +12:00 - 18:00. Thus, we define Cluster 2 and 3 as the morning (Mor) +and afternoon (Aft) type. Cluster 1 has only one activity peak in +18:00 - 00:00, thus we define it as evening (Eve) type. +Figure 2: Chronotype distribution in each student cluster. +As Figure 2 suggests, quick-learning students usually tend to +work in the morning and afternoon whereas satisficing students +worked on their homework in the evening. This suggests quick- +learning students were driven, motivated, and had possibly better +time management skills. In general, satisficing students were the +only group to have a strong incline to work in the evening. This +could point to other commitments that students have or a high +course load. The afternoon type occurs most frequently in strug- +gling and hardworking clusters. This may be because they were +seeking help during office hours that were offered during the day +or they simply required more time in general. Overall, our results +confirm previous findings that certain chronotypes are related to +academic achievement[24, 31]. +Figure 3: Clustering result of different types of students The +start of a time interval stands for the average start time whereas +the end represents the average end time. +5.2 +How long do different clusters of students +work on their homework? +To further investigate hypothesis H3, we investigate when students +in a given cluster start and finish their homework. We report the +average start time and end time for each cluster in Figure 3. In addi- +tion, the Kruskal-Wallis H-Test suggests start date was statistically +significantly different (stat = 22.59, p-value < 0.0001) whereas the +end date was not (stat = 3.12, p-value = 0.37). Despite that, we can +still observe some interesting patterns. + +25 +Mor +Aft +Eve +20 +T of Students +15 +Number: +10 +5 +0 +Quick learming +Hardworking +Satisficing +StrugglingTime intervals of completing homework for each student cluster +Quick learning +6.38 +10.84 +Hardworking +6.06 +11.08 +Satisficing +7.51 +10.78 +Struggling +7.22 +11.41 +6 +F7 +5 +8 +9 +10 +11 +12 +DaysafterhomeworkreleaseError Groups +Error Categories +HW1 +HW2 +HW2 +HW4 +HW5 +HW6 +A. General Static Errors +1. Type Error +38.12% +30.94% +40.93% +32.65% +36.90% +34.83% +2. Syntax Error +42.33% +21.54% +21.79% +32.68% +17.80% +25.66% +3. Unbound value +10.42% +7.19% +9.06% +13.42% +7.02% +7.27% +B. Imperative Thinking Errors +4. Missing else branch +1.92% +0.75% +0.43% +0.08% +1.03% +1.07% +5. Unused variable +0.74% +0.65% +0.63% +6.37% +21.34% +7.23% +C. Pattern Matching Errors +6. Pattern matching type error +0.84% +5.24% +2.13% +0.62% +1.37% +1.40% +7. Non-exhaustive pattern matching +1.02% +16.78% +15.74 % +2.47% +4.62% +11.92% +D. Function Applications Errors +8. Wrong number of arguments +1.67% +2.19% +3.38% +1.17% +2.09% +1.89% +9. Misuse of non-function values +2.50% +2.10% +2.07% +1.72% +1.50% +1.77% +10. Others +0.88% +12.6% +5.89% +8.83% +6.33% +6.96% +Total number of errors +7,850 +27,519 +14,331 +19,859 +22,467 +26,681 +Table 4: Error Groups and error categories together with their distribution of HWs +We note that both satisficing and struggling students start rela- +tively late on their homework, at 7.51 and 7.22 average days respec- +tively. However, satisficing students finish the earliest (10.78). This +underscores the fact that they accept a “good enough” result rather +than striving for better outcomes. Further, satisficing students had +the shortest working duration. This substantiates our claim that +these students achieve their goals by saving time and effort. +Struggling students experienced many difficulties as evidenced +by a high number of static errors that they encounter. These stu- +dents finish indeed last (finish time (11.41)). This indicates that +these students are struggling, although they do try their best until +the very end. However, they lack the skills or support to overcome +their difficulties. +Hardworking students have the longest time interval. While they +start the earliest (6.06), they finish the second latest (11.08). This +shows the commitment and dedication they bring to their work. +Quick-learning students tend to start quite earlier (6.38), al- +though not as early as hardworking students. This suggests that +these students have confidence in their abilities to finish the home- +work smoothly. +We ran Spearman correlations to examine the correlation be- +tween start time and homework grade, the statistically significant +result (correlation = -0.42, p-value < 0.0001) suggests procrastination +affects negatively on student learning outcomes, which has been +widely reported [3, 16, 22]. +5.3 +How do static errors affect students in +different clusters? +Compilers for typed functional programming languages such as +OCaml provide a wealth of errors and feedback to programmers. It +not only reports syntax and type errors but also reports, for example, +unused variables, and missing branches in case-statements and if- +expressions. This provides a basis for a better understanding of +what basic concepts students struggle with the most. +5.3.1 +Overview of static errors. To investigate our hypothesis H4, +we analyze the types of errors of each failed compile event and +group errors into four main categories: general static errors (eg. +group A), errors due to imperative thinking (Group B), and errors +related to pattern matching and function (eg. groups C and D). We +also include how often particular errors occurred in assignment +submissions (see Table 4). +The first homework shows a significant spike (42.33%) in syntax +errors encountered. This is unsurprising, as it is the first time that +students attempt to write programs in a new language. However, +it may be surprising that 20% to 30% of the errors encountered +are related to syntax and type errors (Group A) throughout the +semester. In fact, these errors constitute around 60% of errors for +every homework assignment in Table 4. This may point to the fact +that type errors in TFP catch conceptual errors in the programmer’s +thinking early rather than later during testing. This may also sug- +gest instructors dedicating more time to demystifying type error +analysis. +For some key concepts from typed functional programming such +as pattern matching, our error analysis indicates that students do +improve and gain a better understanding of it. When pattern match- +ing is first introduced in HW2, pattern matching errors and non- +exhaustive pattern matching errors (Group C) consist 22% of total +static errors. After practicing HW2 and HW3, the proportion of +Error Group C drops greatly, which suggests that students gain a +deeper understanding with more programming practice. +One of the prerequisites of this course is taking an introduc- +tory CS course, which is taught in Java or Python at our univer- +sity. This implies that all of the participants had experience in +programming before and had to deal with conceptual transfer from +imperative/object-oriented programming (Python or Java) to func- +tional programming (OCaml). Students usually report transition- +ing smoothly between procedural language and object-oriented +language for concepts such as if-conditionals and functions and +scope[28]. From our observations, students struggle more when tran- +sitioning to functional programming. In particular, they struggle +with the concept of bound or unbound variables, missing branches +in if-expressions, and function application errors. Although these +errors occur less frequently than syntax and type errors, we believe +it highlights that students struggle with thinking recursively and +considering all cases in such a recursive program (Error No.4,7). +Therefore, if-else expression without an else branch also often leads +to type errors in a language like OCaml. +Moreover, imperative programming supports variables declared +in the local or global state, while in functional languages, such +as OCaml, we distinguish between stateful variables that can be +updated and bound variables. While the concept of free variables +and bound variables and the difference between stateful variables + +are discussed frequently in this course, students continue to en- +counter errors related to variables. In particular, the unbound value +error occurs throughout the semester. This seems to be a sign that +the concept of stateful variable declarations as used in imperative +programming is persisting in how students think about a given prob- +lem. The most essential concept of functional programming is that +functions are first-class citizens. Therefore, higher-order functions, +which take a function as an argument, or return a function, are +used frequently, especially in HW3 and subsequent assignments. +If functions are not used correctly, it would most frequently be +flagged as a type error. However, OCaml also provides other error +reporting. In particular, it may report on the incorrect number of +arguments (Error NO.8) and use a function value instead of apply- +ing arguments on a non-function value (Error NO.9). These errors +form a non-negligible class indicating where students stumble. +5.3.2 +How efficiently do students in each cluster fix errors? Lastly, +we investigate hypothesis H5 and aim to understand how students +in different clusters vary in their ability to fix errors quickly. Table 5 +shows the average number of successful compile events and fail- +ure ones experienced by different student clusters throughout the +semester. The Failure/Success ratio x can be roughly interpreted +as debugging efficiency or error fix rate that it on average costs a +student x failure compile events to get a successful one. +Quick-learning +Hardworking +Satisficing +Struggling +Success +37.9 +60.4 +28.1 +40.7 +Failure +85.7 +162.3 +66.9 +118 +F/S +2.26 +2.67 +2.38 +2.90 +Table 5: Average success, failure and failure/success ratio +(F/S) of compile events in each student cluster +Struggling students have the most difficulty in fixing static errors, +requiring 2.9 failure compilations to fix the error on average. By +contrast, quick-learning students have the best ability to debug with +only a 2.26 failure compilation to get a successful one. Furthermore, +the gap between their debugging efficiency is more significant, if we +look at their average failure and success. While the average success +for struggling students (40.7) and quick learners (37.9 ) are close, +their average failures have a substantial gap: a struggling student +has around 30 more failure compilations than quick learners. +Figure 4: Distribution of static errors in each student cluster. +The row of Failure in Table 5 can be further represented by the +average number of each group of static errors for four student clus- +ters in Figure 4. Type and syntax errors (Group A) dominate for all +clusters but there are noteworthy differences. Quick learners have +fewer errors in all groups, not only general static errors but also +errors specific to functional programming. Satisficing students have +the fewest errors in Group B, C, and D which may indicate that +they in fact achieve competency. Lastly, hardworking and strug- +gling students have significantly more errors in all error groups. In +particular, they struggle more with basic concepts such as bound or +unused variables, missing branches, and the proper use of functions. +6 +CONCLUSION +In this study, we aim to understand how students develop func- +tional programming assignments based on data collected through +the Learn-OCaml programming platform. Our analysis considers +grade, total time spent, and the total number of static errors to +identify four student clusters: "Quick-learning", "Hardworking", "Sat- +isficing", and "Struggling". Using statistical tests we validate our +clustering results along with other analysis results. This provides +a nuanced picture of students’ behaviours and also exposes differ- +ent paths towards achieving academic success in the course. Our +analysis of chronotypes confirms that students who work in the +morning reach the highest grade most quickly and smoothly. The +total amount of time students spend on the homework also high- +lights the difference and similarities between the different student +clusters. Although this part of the analysis was done in the context +of a functional programming course, we expect our methodology +to be applicable to other programming courses and help identify +clusters of students who would benefit from additional support. +Our detailed analysis of static errors in typed functional pro- +gramming also highlights areas where instructors can adjust their +course content and possibly revisit topics. We believe our analysis +also provides insights for students themselves, in particular the +hardworking students, to understand which aspects they still strug- +gle with and to seek clarifications. 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Morning or evening? an examination of circadian rhythms +of CS1 students. +2021 IEEE/ACM 43rd International Conference on Software +Engineering: Software Engineering Education and Training (ICSE-SEET) (2021). +https://doi.org/10.1109/icse-seet52601.2021.00036 + diff --git a/3dE0T4oBgHgl3EQfvAER/content/tmp_files/load_file.txt b/3dE0T4oBgHgl3EQfvAER/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b0d9b889b341fc9a0fbb9338daebebe318c6a22 --- /dev/null +++ b/3dE0T4oBgHgl3EQfvAER/content/tmp_files/load_file.txt @@ -0,0 +1,696 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf,len=695 +page_content='Identifying Different Student Clusters in Functional Programming Assignments: From Quick Learners to Struggling Students Chuqin Geng McGill University Montreal, QC, Canada chuqin.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='xu@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='ca Brigitte Pientka McGill University Montreal, QC, Canada bpientka@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='ca Xujie Si McGill University Montreal, QC, Canada xsi@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='ca ABSTRACT Instructors and students alike are often focused on the grade in programming assignments as a key measure of how well a student is mastering the material and whether a student is struggling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This can be, however, misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Especially when students have access to auto-graders, their grades may be heavily skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In this paper, we analyze student assignment submission data collected from a functional programming course taught at McGill university incorporating a wide range of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In addition to the grade, we consider activity time data, time spent, and the number of static errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This allows us to identify four clusters of students: "Quick-learning", "Hardworking", "Satisficing", and "Struggling" through cluster algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We then analyze how work habits, working duration, the range of errors, and the ability to fix errors impact different clusters of students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This structured analysis pro- vides valuable insights for instructors to actively help different types of students and emphasize different aspects of their overall course design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' It also provides insights for students themselves to understand which aspects they still struggle with and allows them to seek clarification and adjust their work habits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' CCS CONCEPTS Social and professional topics → Student assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' KEYWORDS online programming platform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' computer science education;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' cluster analysis 1 INTRODUCTION Online programming environments, such as RoboProf [8] for C++, DrScheme [13, 14] for Scheme or, more recently, Mumuki [4] , offer immense potential to enhance the students’ educational experience in large-scale programming-oriented courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' They not only lower the entry barrier for beginners but often feature auto-grading facili- ties that allow students to run and get feedback on their code while they are developing their programs, giving them the opportunity to fix bugs and address errors in their understanding right away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' While having access to immediate feedback on their code has been recognized to significantly improve student learning outcomes and engagement (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=', [15, 26, 30]), instructors and students alike are often too focused on the grade as a key measure of competency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Especially when students have access to auto-graders, the students’ grades may be heavily skewed and misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This paper develops a data-driven approach to better understand students’ behavior when solving programming assignments in a functional programming course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In addition to the grade, we pro- pose to consider additional factors such as the number of static errors and total time spent on solving programming assignments to identify student clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Using this methodology, we analyze the assignment submission data collected in a functional programming course taught at McGill university which uses the Learn-OCaml online programming platform [5, 6, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This allows us to identify four student clusters: "Quick-learning", "Hardworking", "Satisficing", and "Struggling".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' While the first two clusters can be characterized as maximizers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' students strive to achieve the highest possible grades and continue to improve their work, they still differ in the amount of time and effort spent on completing a given homework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In contrast, satisficing1 students accept a possibly non-optimal out- come as ”good enough” allowing them to adequately achieve their goals by saving time and effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We further analyze these clusters with respect to work habits and the number and kinds of errors that are prevalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This leads to four key insights: Leveraging the notion of chronotype - a circadian typology in humans and animals, we confirm that a work pattern where students tend to work in the morning is related to academic success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In particular, quick learners tend to work more in the morning, while other clusters of students rely more on afternoons and evenings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In general, starting on the homework early is related to higher grades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' However, we also noticed that satisficing stu- dents start relatively late but finish the earliest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This further emphasizes that satisficing students aim for satisfactory re- sults rather than the optimal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' At the same time, satisfic- ing students have one of the lowest numbers of programming errors suggesting that they struggle significantly less with static errors than for example hardworking students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Our analysis of static errors shows that syntax and type errors are prevalent among all students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Further, students 1The term “satisficing” was introduced by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Simon [27] to describe a decision-making process in which an individual makes a choice that is satisfactory rather than optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='02611v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='CY] 6 Jan 2023 continue to struggle with these errors throughout the se- mester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In addition, our analysis points to other common mistakes such as non-exhaustive case analysis and the use of unbound variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Taking into account students’ ability to fix static errors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' how many tries a student needs to fix a particular error, we notice that the failure/success ratio is particularly high for hardworking students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This highlights both their desire and drive to get the best possible grade, but also that their path is full of small stumbling blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We believe our proposed set of features and data-driven analysis can provide instructors with a clearer and more detailed picture of students’ behaviours and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This in turn may be used to adjust how some concepts, such as how to avoid static errors, are taught.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' It may also be used to design different strategies for different students to enhance the students’ learning experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Furthermore, this data may be interesting to students themselves to better understand how well they do in a class and identify areas where they can actively make changes and seek help early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 2 RELATED WORK Analyzing student data in programming courses is a central topic in learning analytics, and it is gaining increasing attention with the recent advances in storing and processing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' One of the core aims of analyzing student data is to understand student behaviours, and in turn, improve student learning experience [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Over the past decade, there have been several studies that focus on identifying groups of students using cluster analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' For exam- ple, Emerson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' [12] use cluster algorithms to identify student misconceptions in a block-based programming environment for non-CS major students based on program structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Wiggins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' [29] finds five major clusters of hint requests in a block-based programming system equipped with an intelligent tutor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Hossein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' [20] leverages clustering analysis to further investigate the correlation between students’ programming speed and program- ming behaviours by collecting programming snapshots whenever an action occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' They then divide students into two clusters by comparing a student’s programming speed to the median speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Lahtinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' [23] uses different levels of Bloom’s Taxonomy as fea- tures to identify six distinct student groups that instructor should be aware of when teaching introductory programming courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In contrast to these existing works, our work considers multi- categorical features involving the grade, total time spent on the assignment, and the number of static errors encountered to identify clusters of students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Based on the identified clusters, we follow existing work in under- standing the work/rest patterns of students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In particular, Claes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' [7] study programmers’ working patterns using clustering analysis on time stamps of committed activities of 86 large open-source software projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Zavgorodniaia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' [31] study the chronotypes of students through cluster algorithms using keystroke data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In our study, we use activity data (such as whether a student compiled or graded their homework) to study the work/rest patterns of students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' It is the first study in the context of typed functional programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We further analyze static errors in typed functional programming assignments and their impact on different student clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This is the first such study in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Previous studies focus on compilation events in object-oriented programs written in Java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' For example, Ahmadzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' [1] investigates compiler error frequencies of student programs and debugging activity patterns in Java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' They suggest debugging skills should be emphasized in the teaching of programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Altadmri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' [2] collect a large dataset comprising compilation events of 250,000 students, which provides a robust analysis of error patterns and time for fixing different errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Denny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' [9] also study various syntax error frequencies and how long students spend fixing common syntax errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' They also found that certain types of errors remain challenging even for higher-ability students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Edwards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' [11] analyze 10 million static analysis errors found in over 500 thousand program submissions made by students over a five-semester period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The experimental results suggest error frequencies made by CS major and non-major students are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Our analysis is one of the first that investigates in more depth the frequency of various static errors in the typed functional pro- gramming assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Here, static errors go beyond syntax and simple type errors and include for example detection of missing branches in a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 3 STUDY DESIGN This research aims to gain a deeper understanding of how students develop typed functional programs (TFP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We assume that the grade alone is not a good indicator of how well a student masters basic concepts and achieves competency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Instead, we propose that taking into account the time spent as well as the number of errors a student encounters can provide a more nuanced picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Hence, the main question that we tackle in this paper is how can we best identify different clusters of students taking into account grades, time spent, and the number of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We then analyze our clusters with respect to five hypotheses: H1: Even students with a high grade in programming assign- ments may significantly struggle with a range of static errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' H2: Despite a lower grade, students who spend less time and have a low number of static errors do in fact well overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' H3: Work/rest patterns of students as well as the time a student spends on homework play a role in students achieving a high grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' It highlights how driven a student is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' H4: Static errors in TFP range from syntax and type errors to detecting unbound variables and missing branches in programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This wide range of static errors provides a fine- grained picture of concepts students find challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' H5: Error fix ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' how many tries a student needs to fix a static error, indicates how well students understand basic ideas in TFP and this is correlated to their understanding and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='1 Course Context Our study concerns students in a second-year undergraduate com- puter science course at McGill university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The course introduces concepts about functional programming and programming paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' It is offered every semester with more than 300 registered under- graduate students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In this study, all data is collected in the Fall 2021 programming topics #tasks HW1 basic expressions, recursion 7 HW2 data types and pattern matching 6 HW3 higher-order functions 11 HW4 references, state, memorization 5 HW5 exception, continuations 5 HW6 lazy programming, toy language 5 Table 1: Overview of six programming assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Figure 1: Data collection pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Grade and Compile and Eval events are handled by different servers, all submission data are stored in a MongoDB database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The components highlighted in light green are original components in the Learn-OCaml platform, while the components highlighted in light blue are newly introduced by us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' semester when students could attend online Zoom or in-person sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The course had six bi-weekly programming assignments each worth 5% of the final grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Each homework consists of several pro- gramming tasks to implement functions and test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Homework information is summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' All homework assignments were hosted on Learn-OCaml [6], an online programming platform for OCaml which allows students to edit, compile, test, and debug code all in one place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We used a modified version of Learn-OCaml by Hameer and Pientka [18] with additional features such as style checking and evaluation of test cases written by students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='2 Data Collection Our data collection pipeline is built on top of the Learn-OCaml platform and it can automatically log students’ actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Specifically, we send local programming events like compile and evaluation (for testing and debugging) with asynchronous logging requests to our backend server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Figure 1 illustrates the process of collecting the data from the online environment Learn-OCaml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Around 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='81% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=', 169 out of 320) students gave us consent to access their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We collect more than 270,000 programming events, and each event stores a snapshot of the code as well as feedback information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=', time-stamp, static errors, grades, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='3 Feature For each homework, we collect a sequence of programming activity events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The activity events include grade, compile, and evaluation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This allows us to create an activity density vector for each student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' It is a four-element vector that represents the percentage of the student’s activity events that occurs in different ranges of hours [0-6, 6-12, 12-18, 18-0], which is the same choice of ranges suggested in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In addition, we design the following features based on the activity event sequence: Start time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The day when a student starts actively working on an assignment based on the activity events collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' End time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The day when a student finishes an assignment, which is the last Grade event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Working session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Defined as the time window where ac- tivity events occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' If there is no activity event within 30min, then the working session is assumed to have ended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Total time spent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Sum over the length of all working ses- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Number of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The number of static errors that a stu- dent made while completing an assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The final grade a student receives for an assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='4 Feature Engineering There are two challenges to applying clustering algorithms and sta- tistical tests to our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The first one is skewed data .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' For instance, the grade is highly skewed as students can always improve their grades through interacting with the auto-grader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The second one is the difference between feature scales, which renders the clus- tering results incoherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We use two approaches to address these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' First, we use non-parametric tests including Spearman correlations and Kruskal-Wallis H-Tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Second, we apply the rank transformation on features to facilitate clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 4 IDENTIFYING STUDENT CLUSTERS To identify student clusters, we run the K-means[19] clustering al- gorithm on the aggregation (mean) of three most important features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=', grade, number of errors and time spent) over six homework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We use the elbow method to determine the optimal k (the number of clusters) to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' After determining the optimal k, we re-run the K-means algorithm and report the results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We give the time in hours and note that all clusters have a similar size in terms of number of students (#𝑆𝑡𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' To determine whether the resulting four clusters are different, we run a Kruskal-Wallis H-Test, which is a nonparametric equivalent of an ANOVA, on the three features (time spent, #errors, and grade) of each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The results are statistically significant with the statistics of 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='26, 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='87, and 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='02 respectively, and all p-values < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This suggests the four clusters are statistically different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Students in cluster A have the highest average grade (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='24) while spending less than the expected 6h on solving the homework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This suggests that they achieve their goal with relative ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In fact, Clusters #Std Time (Hours) # Error Grade A - Quick learning 46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='30 (± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='94) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='11 (± 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='95) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='24 (± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='25) B - Hardworking 46 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='24 (± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='52) 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='67 (± 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='26) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='25 (± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='90) C - Satisficing 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='47 (± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='01) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='26 (± 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='89) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='43 (± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='31) D - Struggling 46 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='49 (± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='94) 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='14 (± 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='32) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='81 (± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='03) Table 2: Student clusters Feedback from Autograder programming Git repo autograder history Webserver Grade Event Grade data entry LearnOCaml [id, timestamp, code, grade] Client Compile and Eval Event Compilation Results MongoDB MongoDB Webserver Compile and Eval data entry [id, timstamp, code]students in this cluster outperform students in other clusters by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We characterize this cluster as quick learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Students in cluster B have the second-highest average grade (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' However, they also have the highest average number of errors (148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='67) and with 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='24h spend significantly more time on homework than any other group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In particular, they spend signifi- cantly more time than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This suggests that they face many difficulties which they manage to overcome by spending a signif- icant amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' These students are driven to improve their work and to achieve the highest possible grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Hence, we charac- terize them as hardworking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This data supports our hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Cluster C has the lowest average number of errors (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='26) and spent the least amount of time (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='47h) on the homework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' With an average grade of 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='43, they still achieve a “good enough” result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' These students achieve their goals by saving time and effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' At the same time, these students reach a satisfying level of competency as evidenced by their low number of average errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We describe these students as satisficing students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This supports our hypothesis H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Students in Cluster D are in fact closely related to students in cluster B, which shows a similarly high average number of errors (118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='14) and a significant amount of time (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='49h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' However, com- pared to students in cluster B, they fail to overcome the difficulties along their path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' These students are struggling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 5 UNDERSTANDING STUDENT CLUSTERS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='1 How do work habits vary for different student clusters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' To investigate our hypothesis H3, we consider when students are ac- tive based on our activity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Prior research suggests that chrono- type, a person’s preference in carrying out activity at certain periods in a day, is governed by the circadian cycle which is controlled by clock genes [10, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In this section, we are interested in investigat- ing the chronotypes, or in other words, the work habits of students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In particular, it has been observed that “morningness” is positively correlated with academic achievement [24, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' To identify potential chronotypes, we run the K-means cluster- ing algorithm on the feature space spanned by activity density vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The elbow method yields 𝑘 = 3, suggesting three possible chronotypes, which is different from four chronotypes reported in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We report centroids of each chronotype cluster in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Chrono clusters 0 - 6 6 - 12 12 - 18 18 - 0 Chronotype Cluster 1 8% 14% 26% 52% Evening (Eve) Cluster 2 4% 26% 20% 50% Morning (Mor) Cluster 3 2% 19% 37% 42% Afternoon (Aft) Table 3: Centroids of each chronotype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' As we can see, most activities occur from 18:00 - 00:00 for all three clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This is not surprising as most students may have classes during the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Based on this observation, we aim to define chronotypes by considering secondary activity peaks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We notice that Cluster 2 has its secondary activity peak (26%) in 6:00 - 12:00 whereas Cluster 3 has the secondary activity peak (37%) in 12:00 - 18:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Thus, we define Cluster 2 and 3 as the morning (Mor) and afternoon (Aft) type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Cluster 1 has only one activity peak in 18:00 - 00:00, thus we define it as evening (Eve) type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Figure 2: Chronotype distribution in each student cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' As Figure 2 suggests, quick-learning students usually tend to work in the morning and afternoon whereas satisficing students worked on their homework in the evening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This suggests quick- learning students were driven, motivated, and had possibly better time management skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In general, satisficing students were the only group to have a strong incline to work in the evening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This could point to other commitments that students have or a high course load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The afternoon type occurs most frequently in strug- gling and hardworking clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This may be because they were seeking help during office hours that were offered during the day or they simply required more time in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Overall, our results confirm previous findings that certain chronotypes are related to academic achievement[24, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Figure 3: Clustering result of different types of students The start of a time interval stands for the average start time whereas the end represents the average end time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='2 How long do different clusters of students work on their homework?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' To further investigate hypothesis H3, we investigate when students in a given cluster start and finish their homework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We report the average start time and end time for each cluster in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In addi- tion, the Kruskal-Wallis H-Test suggests start date was statistically significantly different (stat = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='59, p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='0001) whereas the end date was not (stat = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='12, p-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Despite that, we can still observe some interesting patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 25 Mor Aft Eve 20 T of Students 15 Number: 10 5 0 Quick learming Hardworking Satisficing StrugglingTime intervals of completing homework for each student cluster Quick learning 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='38 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='84 Hardworking 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='06 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='08 Satisficing 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='51 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='78 Struggling 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='22 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='41 6 F7 5 8 9 10 11 12 DaysafterhomeworkreleaseError Groups Error Categories HW1 HW2 HW2 HW4 HW5 HW6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' General Static Errors 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Type Error 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='12% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='94% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='93% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='65% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='90% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='83% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Syntax Error 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='33% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='54% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='79% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='68% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='80% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='66% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Unbound value 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='42% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='19% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='06% 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='42% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='02% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='27% B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Imperative Thinking Errors 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Missing else branch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='92% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='75% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='43% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='08% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='03% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='07% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Unused variable 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='74% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='65% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='63% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='37% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='34% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='23% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Pattern Matching Errors 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Pattern matching type error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='84% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='24% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='13% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='62% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='37% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='40% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Non-exhaustive pattern matching 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='02% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='78% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='74 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='47% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='62% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='92% D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Function Applications Errors 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Wrong number of arguments 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='67% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='19% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='38% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='17% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='09% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='89% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Misuse of non-function values 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='50% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='10% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='07% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='72% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='50% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='77% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Others 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='88% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='6% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='89% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='83% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='33% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='96% Total number of errors 7,850 27,519 14,331 19,859 22,467 26,681 Table 4: Error Groups and error categories together with their distribution of HWs We note that both satisficing and struggling students start rela- tively late on their homework, at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='51 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='22 average days respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' However, satisficing students finish the earliest (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This underscores the fact that they accept a “good enough” result rather than striving for better outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Further, satisficing students had the shortest working duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This substantiates our claim that these students achieve their goals by saving time and effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Struggling students experienced many difficulties as evidenced by a high number of static errors that they encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' These stu- dents finish indeed last (finish time (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='41)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This indicates that these students are struggling, although they do try their best until the very end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' However, they lack the skills or support to overcome their difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Hardworking students have the longest time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' While they start the earliest (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='06), they finish the second latest (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This shows the commitment and dedication they bring to their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Quick-learning students tend to start quite earlier (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='38), al- though not as early as hardworking students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This suggests that these students have confidence in their abilities to finish the home- work smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We ran Spearman correlations to examine the correlation be- tween start time and homework grade, the statistically significant result (correlation = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='42, p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='0001) suggests procrastination affects negatively on student learning outcomes, which has been widely reported [3, 16, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='3 How do static errors affect students in different clusters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Compilers for typed functional programming languages such as OCaml provide a wealth of errors and feedback to programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' It not only reports syntax and type errors but also reports, for example, unused variables, and missing branches in case-statements and if- expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This provides a basis for a better understanding of what basic concepts students struggle with the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='1 Overview of static errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' To investigate our hypothesis H4, we analyze the types of errors of each failed compile event and group errors into four main categories: general static errors (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' group A), errors due to imperative thinking (Group B), and errors related to pattern matching and function (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' groups C and D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We also include how often particular errors occurred in assignment submissions (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The first homework shows a significant spike (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='33%) in syntax errors encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This is unsurprising, as it is the first time that students attempt to write programs in a new language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' However, it may be surprising that 20% to 30% of the errors encountered are related to syntax and type errors (Group A) throughout the semester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In fact, these errors constitute around 60% of errors for every homework assignment in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This may point to the fact that type errors in TFP catch conceptual errors in the programmer’s thinking early rather than later during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This may also sug- gest instructors dedicating more time to demystifying type error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' For some key concepts from typed functional programming such as pattern matching, our error analysis indicates that students do improve and gain a better understanding of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' When pattern match- ing is first introduced in HW2, pattern matching errors and non- exhaustive pattern matching errors (Group C) consist 22% of total static errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' After practicing HW2 and HW3, the proportion of Error Group C drops greatly, which suggests that students gain a deeper understanding with more programming practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' One of the prerequisites of this course is taking an introduc- tory CS course, which is taught in Java or Python at our univer- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This implies that all of the participants had experience in programming before and had to deal with conceptual transfer from imperative/object-oriented programming (Python or Java) to func- tional programming (OCaml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Students usually report transition- ing smoothly between procedural language and object-oriented language for concepts such as if-conditionals and functions and scope[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' From our observations, students struggle more when tran- sitioning to functional programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In particular, they struggle with the concept of bound or unbound variables, missing branches in if-expressions, and function application errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Although these errors occur less frequently than syntax and type errors, we believe it highlights that students struggle with thinking recursively and considering all cases in such a recursive program (Error No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='4,7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Therefore, if-else expression without an else branch also often leads to type errors in a language like OCaml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Moreover, imperative programming supports variables declared in the local or global state, while in functional languages, such as OCaml, we distinguish between stateful variables that can be updated and bound variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' While the concept of free variables and bound variables and the difference between stateful variables are discussed frequently in this course, students continue to en- counter errors related to variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In particular, the unbound value error occurs throughout the semester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This seems to be a sign that the concept of stateful variable declarations as used in imperative programming is persisting in how students think about a given prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The most essential concept of functional programming is that functions are first-class citizens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Therefore, higher-order functions, which take a function as an argument, or return a function, are used frequently, especially in HW3 and subsequent assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' If functions are not used correctly, it would most frequently be flagged as a type error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' However, OCaml also provides other error reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In particular, it may report on the incorrect number of arguments (Error NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='8) and use a function value instead of apply- ing arguments on a non-function value (Error NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' These errors form a non-negligible class indicating where students stumble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='2 How efficiently do students in each cluster fix errors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Lastly, we investigate hypothesis H5 and aim to understand how students in different clusters vary in their ability to fix errors quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Table 5 shows the average number of successful compile events and fail- ure ones experienced by different student clusters throughout the semester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The Failure/Success ratio x can be roughly interpreted as debugging efficiency or error fix rate that it on average costs a student x failure compile events to get a successful one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Quick-learning Hardworking Satisficing Struggling Success 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='9 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='4 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='7 Failure 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='7 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='3 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='9 118 F/S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='67 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='90 Table 5: Average success, failure and failure/success ratio (F/S) of compile events in each student cluster Struggling students have the most difficulty in fixing static errors, requiring 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='9 failure compilations to fix the error on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' By contrast, quick-learning students have the best ability to debug with only a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='26 failure compilation to get a successful one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Furthermore, the gap between their debugging efficiency is more significant, if we look at their average failure and success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' While the average success for struggling students (40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='7) and quick learners (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='9 ) are close, their average failures have a substantial gap: a struggling student has around 30 more failure compilations than quick learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Figure 4: Distribution of static errors in each student cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The row of Failure in Table 5 can be further represented by the average number of each group of static errors for four student clus- ters in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Type and syntax errors (Group A) dominate for all clusters but there are noteworthy differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Quick learners have fewer errors in all groups, not only general static errors but also errors specific to functional programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Satisficing students have the fewest errors in Group B, C, and D which may indicate that they in fact achieve competency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Lastly, hardworking and strug- gling students have significantly more errors in all error groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In particular, they struggle more with basic concepts such as bound or unused variables, missing branches, and the proper use of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 6 CONCLUSION In this study, we aim to understand how students develop func- tional programming assignments based on data collected through the Learn-OCaml programming platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Our analysis considers grade, total time spent, and the total number of static errors to identify four student clusters: "Quick-learning", "Hardworking", "Sat- isficing", and "Struggling".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Using statistical tests we validate our clustering results along with other analysis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' This provides a nuanced picture of students’ behaviours and also exposes differ- ent paths towards achieving academic success in the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Our analysis of chronotypes confirms that students who work in the morning reach the highest grade most quickly and smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The total amount of time students spend on the homework also high- lights the difference and similarities between the different student clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Although this part of the analysis was done in the context of a functional programming course, we expect our methodology to be applicable to other programming courses and help identify clusters of students who would benefit from additional support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Our detailed analysis of static errors in typed functional pro- gramming also highlights areas where instructors can adjust their course content and possibly revisit topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' We believe our analysis also provides insights for students themselves, in particular the hardworking students, to 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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='1145/3408877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='3432538 [30] Chris Wilcox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' The role of automation in undergraduate computer science education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' In Proceedings of the 46th ACM Technical Symposium on Computer Science Education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' ACM, 90–95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' [31] Albina Zavgorodniaia, Raj Shrestha, Juho Leinonen, Arto Hellas, and John Edwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' Morning or evening?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' an examination of circadian rhythms of CS1 students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' 2021 IEEE/ACM 43rd International Conference on Software Engineering: Software Engineering Education and Training (ICSE-SEET) (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='1109/icse-seet52601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} +page_content='00036' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfvAER/content/2301.02611v1.pdf'} diff --git a/3tE3T4oBgHgl3EQfogoT/content/tmp_files/2301.04633v1.pdf.txt b/3tE3T4oBgHgl3EQfogoT/content/tmp_files/2301.04633v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..963d093022116239e6e409f6a05f68fcb6dcb4f6 --- /dev/null +++ b/3tE3T4oBgHgl3EQfogoT/content/tmp_files/2301.04633v1.pdf.txt @@ -0,0 +1,1159 @@ +Springer Nature 2021 LATEX template +Accelerating Machine Learning Inference with GPUs in +ProtoDUNE Data Processing +Tejin Cai1, Kenneth Herner2*, Tingjun Yang2, Michael Wang2, Maria Acosta +Flechas2, Philip Harris3, Burt Holzman2, Kevin Pedro2 and Nhan Tran2 +1Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, +M3J 1P3, ON, Canada. +2Fermi National Accelerator Laboratory, Kirk Road and Pine Streets, Batavia, 60510, IL, +USA. +3Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, +Cambridge, 02139, MA, USA. +*Corresponding author(s). E-mail(s): kherner@fnal.gov; +Abstract +We study the performance of a cloud-based GPU-accelerated inference server to speed up event +reconstruction in neutrino data batch jobs. Using detector data from the ProtoDUNE experiment +and employing the standard DUNE grid job submission tools, we attempt to reprocess the data by +running several thousand concurrent grid jobs, a rate we expect to be typical of current and future +neutrino physics experiments. We process most of the dataset with the GPU version of our processing +algorithm and the remainder with the CPU version for timing comparisons. We find that a 100-GPU +cloud-based server is able to easily meet the processing demand, and that using the GPU version of the +event processing algorithm is two times faster than processing these data with the CPU version when +comparing to the newest CPUs in our sample. The amount of data transferred to the inference server +during the GPU runs can overwhelm even the highest-bandwidth network switches, however, unless +care is taken to observe network facility limits or otherwise distribute the jobs to multiple sites. We +discuss the lessons learned from this processing campaign and several avenues for future improvements. +Keywords: machine learning, heterogeneous (CPU+GPU) computing, GPU (graphics processing unit), +particle physics, cloud computing (SaaS), neutrino physics, distributed computing +1 Introduction +Machine learning (ML)-based algorithms have +been widely used in the field of neutrino physics, +for applications ranging from data acquisition to +data reconstruction and analysis [1–4]. A detec- +tor technology ideally suited for computer vision +applications in neutrino physics is that of liquid +argon time projection chambers (LArTPCs), which +are employed by the Deep Underground Neutrino +Experiment (DUNE) [5] and Short-Baseline Neu- +trino [6] experiments. ML applications are now +deeply integrated into the event reconstruction and +data analyses for the LArTPC experiments [7–9]. +1 +arXiv:2301.04633v1 [hep-ex] 11 Jan 2023 + +Springer Nature 2021 LATEX template +2 +GPUaaS in ProtoDUNE data +Event record sizes for the current generation of +LArTPC experiments are typically ≤1 GB and are +expected to increase in the next few years. With +increased event size, the event reconstruction, espe- +cially the inference of ML algorithms, will become +a challenge. Additionally, neutrino detectors are +sensitive to neutrinos from a core-collapse super- +nova in or near the Milky Way. One of DUNE’s +physics goals is to rapidly reconstruct detector +trigger records from such a supernova to provide +rapid localization information to optical telescopes, +placing a premium on short event reconstruction +times. We have demonstrated GPU-accelerated ML +inference as a service, which significantly reduced +the reconstruction time for simulated neutrino +events in the ProtoDUNE experiment [10]. Later, +we tested the same GPU-as-a-Service (GPUaaS) +approach to process the entire ProtoDUNE Run +I dataset to demonstrate the scalability of this +method. This paper reports the results of those +tests. +2 Infrastructure setup and +methods +2.1 ProtoDUNE background +The +ProtoDUNE +single +phase +detector +(ProtoDUNE-SP) [11, 12] is a liquid argon time +projection chamber (LArTPC) that serves as a +prototype for the first far detector module of +DUNE [5]. The ProtoDUNE-SP is installed at +the CERN Neutrino Platform [13]. It has an +active volume of 7.2 × 6.1 × 7.0 m3. The TPC +wires are read out by 15,360 electric channels at +a rate of 2 MHz. A typical event record consists +of 6000 time samples, corresponding to a 3 ms +time window. Between October 10 and November +11, 2018, ProtoDUNE-SP was exposed to a beam +that delivers charged pions, kaons, protons, muons +and electrons with momenta in the range 0.3 +GeV/c to 7 GeV/c. After the beam runs ended, +ProtoDUNE-SP continued to collect cosmic ray +and calibration data until July 20, 2020, after +which +the +detector +decommissioning +started. +The total number of trigger records (also called +“events”) during the beam period, which consist of +both beam interactions and non-beam interactions +such as cosmic rays, is approximately 7.2 million. +A ProtoDUNE-SP TPC waveform recorded by +a single electric channel consists of both signals +and noise. There are typically three sources of sig- +nals. During the beam runs, the beam particles +can interact with the liquid argon inside the TPC +and produce both ionization electrons and scintilla- +tion light. Since ProtoDUNE-SP is located on the +Earth’s surface, it is subject to a large flux of cos- +mic ray muons, which induce signals over the entire +detector. There are also radioactive backgrounds +such as 39Ar that generate low energy signals on +the scale of a few hundred keV to a few MeV. +Figure 1 shows the event display of a 6 GeV/c pion +interaction in the ProtoDUNE-SP detector. +The first step in the reconstruction of events +in the TPC is the signal processing. The goal of +this stage is to produce distributions of charge +arrival times and positions given the input TPC +waveforms. The effects of induced currents due +to drifting and collecting charge, as well as the +response of the front-end electronics, are removed +through de-convolution. The charge arrival distri- +butions are used in subsequent reconstruction steps, +starting with hit finding. The hit finding algorithm +fits peaks in the wire waveforms, where a hit repre- +sents a charge deposition on a single wire at a given +time. Each hit corresponds to a fitted peak. The +hits are input to pattern recognition algorithms +such as Pandora [14–16]. This stage finds the high- +level objects associated with particles, like tracks, +showers, and vertices, and assembles them into a +hierarchy of parent-daughter nodes that ultimately +point back to the candidate neutrino interaction. + +Springer Nature 2021 LATEX template +GPUaaS in ProtoDUNE data +3 +0 +100 +200 +300 +400 +Wire Number +3500 +3750 +4000 +4250 +4500 +4750 +5000 +Tick +50 cm +DUNE:ProtoDUNE-SP Run 5772 Event 15132 +2 +0 +2 +4 +6 +8 +10 +Charge/tick/channel (ke) +Fig. 1: A 6 GeV/c beam π+ interaction in the ProtoDUNE-SP detector [11]. The x axis shows the +wire number. The y axis shows the time tick in the unit of 0.5 µs. The color scale represents the charge +deposition. +More details on the reconstruction workflow are +described in Ref. [11]. +In ProtoDUNE-SP, a novel algorithm is devel- +oped based on a convolutional neural network +(CNN) to perform the classification of each recon- +structed hit as track-like or arising from electromag- +netic cascades [9]. These hit-level classifications +can be used alongside pattern recognition based +reconstruction algorithms such as Pandora to refine +the track or shower classification of reconstructed +particles. The CNN model was trained using Ten- +sorFlow [17]. Hereafter, we call this algorithm +EmTrkMichelId. +In order to improve the efficiency and speed +of the inference of ML algorithms in a large- +scale data processing, GPU acceleration specifically +for the ProtoDUNE reconstruction chain has +been integrated without disrupting the native +computing workflow using the services for opti- +mized network inference on coprocessors (SONIC) +approach [10, 18]. With the integrated framework, +the most time-consuming task, track and particle +shower hit identification, runs faster by a factor of +17. This results in a factor of 2.7 reduction in the +total processing time when compared with CPU- +only production. This initial test using a small +number of simulated ProtoDUNE events showed +a viable, cost-effective way to solve the comput- +ing challenge facing the neutrino experiments. In +this work, we report the results of reprocessing +the entire 7 million ProtoDUNE events taken dur- +ing the test beam runs with the SONIC-enabled +framework. +2.2 Inference server setup +The Nvidia Triton™ Inference Server is an open- +source inference serving software that helps stan- +dardize model deployment and execution; its goal +is to deliver fast and scalable AI in production [19]. + +Springer Nature 2021 LATEX template +4 +GPUaaS in ProtoDUNE data +NVIDIA provides multiple ways to deploy the +inference server on different cloud providers and +infrastructure types, including both bare metal +and containerized workloads. +This study uses a cloud-based deployment of +Nvidia Triton™ Inference Server within a Google +Cloud Kubernetes Engine [20] cluster on virtual +infrastructure provided by Google Cloud Platform. +The use of this technology enables us to deploy +a flexible GPUaaS model where a public end- +point takes remote inference requests from various +geographically distributed sources as depicted in +Figure 2. The Triton™ server running on the Google +cloud supports different backends. We use the Ten- +sorFlow (version 1.15.5) backend for the inference +of the EmTrkMichelId algorithm. +In a similar way as Ref. [10], this study uses sev- +eral Triton™ servers split into separate Kubernetes +deployments with common services for network- +ing and external load balancing in the form of +ingress objects [21]. One significant improvement +for the current study is the deployment of metrics +and monitoring which provided us with observ- +ability within the system in different states. In IT +and cloud computing, observability is the ability +to measure a system’s current state based on the +data it generates, such as logs, metrics, and traces. +It relies on telemetry derived from instrumenta- +tion that comes from the endpoints and services in +computing environments. Triton™ provides a built- +in metrics endpoint [22] that publishes plain-text +data in Prometheus format. Prometheus collects +and stores data to be displayed by Grafana as seen +in Figure 3. +2.3 Methods +The DUNE collaboration undertook a production +campaign in 2021 to process ProtoDUNE-SP data +using the LArSoft toolkit [23] version v09 30 00. +Each production run during the beam period com- +prises several data files, each containing between +100 and 150 data records. In contrast to the previ- +ous work, in which DUNE simulation events were +processed by submitting jobs locally to a dedicated +queue, we submit jobs to process each file via the +current standard DUNE workflow management +and job submission systems [24, 25], thus requir- +ing no special treatment. Jobs may run either at +Fermilab or one of several remote sites that we +reach with opportunistic access enabled by the +OSG Consortium [26]. +We begin from the existing reconstructed +outputs and apply the updated EmTrkMichelId +algorithm to produce new outputs. Of the 7.2 mil- +lion ProtoDUNE events during the 2018 beam +period, we process 6.4 million through the SONIC +infrastructure, and 800k with the CPU-only ver- +sion of the same algorithm for comparison. The +OSG sites included in the SONIC runs were cho- +sen to be geographically proximate to the location +of the Google Cloud GPU servers (which were in +Iowa, USA at the time) in order to minimize the +latency in data transmissions. +The difference in the time spent in the infer- +ence step is the primary metric with which we +assess the advantage of GPUaaS over traditional +CPU processing. Each job produces a log file that +statistically summarizes the time spent on each +stage of the event reconstruction for the job as +a whole. The log has no record of per-stage pro- +cessing time at the individual event level, but we +can closely approximate it by taking the difference +between the start times of consecutive events. We +estimate the per-event EmTrkMichelId duration +by subtracting the median non-EmTrkMichelId +duration from the total event duration, as the +non-EmTrkMichelId stages display very little time +variation across events. The CNN-based hit classi- +fication occurs in the EmTrkMichelId stage and is + +Springer Nature 2021 LATEX template +GPUaaS in ProtoDUNE data +5 +Internet +(gRPC) +t +Google Kubernetes Engine - protoDUNE TritonRT +Local Compute +FermiGrid farm +Offsite Compute +University of Notre Dame +Offsite Compute +Wayne State University +Offsite Compute +University of Wisconsin-Madison +Google Kubernetes Engine - protoDUNE monitoring +Grafana +Prometheus Server +TCP Network Load +Balancer +TritonRT Server +Pod +TritonRT Server +Pod +TritonRT Server +Pod +External Service +(https) +TCP Network Load +Balancer +Service +:8000 (http) +:8001 (gRPC) +:8002/metrics +Internet +(HTTPS) +User +Real-time monitoring +dashboard +Offsite Compute +MWT2 - (U.Chicago, IU, U.of FL) +Fig. 2: ProtoDUNE GPUaaS component diagram depicting local and remote batch inference runs +submitted from Fermilab and OSG Grid sites. +Fig. 3: A real-time monitoring view of a 100-GPU cluster run for ProtoDUNE (2021). +the most time-consuming step in the event recon- +struction, typically accounting for more than 90% +of the processing time. +3 Results +3.1 CPU-only runs +We process a set of 13 runs using CPU-based +Tensorflow both at Fermilab and several off-site +locations. The off-site locations are the University +of Notre Dame, the University of Victoria, and +the high performance computing center at Wayne + + General / Nvidia GPU ★ ++ + 2021-09-30 09:34:31 to 2021-09-30 11:19:58 +Host +All +Average Utilization +(3. 0% +(320% +(15.0% +(18.0% +(2.0% +(22.0% +(2.0% +(9. 0% +( 25.0% +15.0% +( 25.0% +67.0% +(1.0% +(2.0 +2.0% +(34.0 +(1.0% +25.0 +32.0 +20.0 +5.000% +26.760% +38.000% +10.000% +22.6049 +32.0009 +60.00% +2.756% +3.000% +8.6599 +18.000% +.000% +16.000% +0% +12.9179 +29.000% +20.009 +75-ee634b3e8507 +0% +1.676% +0% +13.012% +25.000% +0%09:35 +0% +1.654% +09:45 +09:50 +09:55 +GPU load - Power +120.00% +100.00% +80.00% +60.009 +fastp.d15@ 10.56.83:002Fermila.Springer Nature 2021 LATEX template +6 +GPUaaS in ProtoDUNE data +0 +100 +200 +300 +400 +500 +EmTrkMichelId Time (s) +0 +2000 +4000 +6000 +8000 +# of Events / 2 sec +CPU Series +AMD 6376 +AMD EPYC 7502 +Intel E5-2650 v2 +Intel E5-2650 v3 +Intel E5-2670 v3 +Intel E5-2680 v4 +Intel Gold 6140 +non-FNAL +Fig. 4: Timing distributions for CPU-only runs, +broken down by CPU type. +State University. The TensorFlow version used in +the CPU-only runs is 2.3.1. Table 1 summarizes +the number of events processed at each site and +the median processing times. We did not request +any specific CPU type when submitting these jobs +since typical DUNE practice is to use any and all +available CPU types. +Table 1: List of CPU-only run sites and median +processing time +OSG Site +N samples +Median processing time (s) +FermiGrid +746603 +79 +Notre Dame +36082 +68 +Victoria +10944 +52 +Wayne State +4242 +45 +There is a clear dependence on processor type +in the EmTrkMichelId processing time distribution. +In general, more recent CPUs process events faster. +Figure 4 shows the CPU-based EmTrkMichelId +timing for each of the CPU types currently avail- +able on the Fermilab general purpose batch farm. +We do not have access to CPU type information +outside of Fermilab and thus group them together. +3.2 GPU runs +Our main processing effort uses the GPUaaS infras- +tructure as described. Figure 5 shows the average +EmTrkMichelId processing time when using the +GPUaaS infrastructure for our entire running +period. The first peak at approximately 20 s repre- +sents a factor of two improvement with respect to +the fastest CPU-only runs, and a factor of roughly +11 over the slowest CPU runs. It is important to +note that the EmTrkMichelId times we report here +are wall times measured within the job, and thus +include contributions from network latency to and +from the server. There is another peak in the dis- +tribution with a median of over 100 s, to which we +now turn. +0 +25 +50 +75 +100 +125 +150 +175 +200 +Avg. EmTrkMchelID time (s) +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +Njobs +All runs 9/30 - 10/20 +Fig. 5: Average EmTrkMichelId times for GPU +runs during the period September 30, 2021 to Octo- +ber 20, 2021. The double peak structure arises from +periods during which the outbound network con- +nection from the Fermilab grid processing center +was saturated. + +Springer Nature 2021 LATEX template +GPUaaS in ProtoDUNE data +7 +3.2.1 Outbound network saturation +During the first period of GPU running we +averaged between 200 and 2000 concurrent jobs. +Figure 6 shows the overlay of network traffic and +event processing start rate during the period of +September 30, 2021 to October 6, 2021. As the +event start rate increases because of the rise in the +number of concurrent jobs, we see that the 100 +Gb/s outbound network connection used by the +Fermilab data center where the jobs run becomes +saturated. While our jobs were not solely responsi- +ble for the saturation (the connection serves the +entire cluster), the saturation did result in a sig- +nificant increase in the average EmTrkMichelId +processing time as shown in Figure 7. The highest +job concurrency levels were on October 5, when +unusually low demand for computing resources +from other Fermilab experiments resulted in a large +number of opportunistic job slots being available +at Fermilab. We were, without any direct interven- +tion, thus able to scale up to approximately 6,000 +concurrent jobs. The monitoring does show switch +saturation as early as October 1, however. After +learning of the network saturation we implemented +a concurrency limit on jobs of approximately 600; +thereafter the jobs ran without incident and the +EmTrkMichelId times returned to pre-saturation +levels (see Figure 8). +4 Discussion +In order to understand the impact of ProtoDUNE +jobs on the Fermilab network traffic, we plot the +distribution of event processing start rate versus +network traffic in Figure 9. Even though the net- +work traffic has contributions from all grid jobs at +Fermilab, there is a clear correlation between the +number of ProtoDUNE concurrent jobs and the +increase of network traffic. We fit a straight line +to the data points below the network traffic of 80 +09/30 10/1 +10/2 +10/3 +10/4 +10/5 +10/6 +10/7 +Date +0 +5 +10 +15 +20 +25 +Event Starting Rate/s +0 +20 +40 +60 +80 +100 +Traffic (Gb/s) +Event Rate +Outbound Traffic +Google Traffic +Fig. 6: Overlay of network traffic and event pro- +cessing start rate at FermiGrid as a function of +time, which is a proxy for the number of concurrent +jobs. The origin day is September 30, 2021. The +solid line is the event start rate, the blue dot-dash +line is the outbound network traffic rate through +the 100 Gb/s switch at Fermilab used by the batch +processing cluster, and the black dashed line is the +ingress rate to the Google cloud server. We are +unable to disambiguate traffic sources through the +switch, so the blue dot-dash line represents the +total traffic as opposed to only traffic generated +by our processing campaign. We see that the net- +work switch was effectively saturated in multiple +instances, though Google ingress was not. +Gb/s. The slope of the best fit line is 4.2 ± 0.2 Gb, +which is the average outbound data transmission +per event. The intercept is 44 ± 2 Gb/s, which is +the average traffic from non-ProtoDUNE grid jobs. +Based on the discussion of transmission time in +Ref. [10], for 55,000 inferences per event, with each +input a 48 × 48 image at 32 bits, the total amount +of data transmitted is about 4.1 Gigabits per event. +This is consistent with the slope of the best fit +straight line. The spread in data with respect to +the straight line could be caused by the variation +in the number of non-ProtoDUNE grid jobs during +this period. +Figure 8 indicates that the average process- +ing time is roughly 25 s/event for the GPU +jobs. Assuming the entire 100 Gb/s bandwidth + +Springer Nature 2021 LATEX template +8 +GPUaaS in ProtoDUNE data +20 +30 +40 +50 +60 +70 +80 +90 +100 +FermiGrid Outbound Traffic (Gb/s) +0 +50 +100 +150 +200 +250 +300 +EmTrkMichelId Time (s) +EmTrkMichelId Time +0 +2000 +4000 +6000 +8000 +10000 +12000 +No. of Events +20 +30 +40 +50 +60 +70 +80 +90 +100 +FermiGrid Outbound Traffic (Gb/s) +0 +50 +100 +150 +200 +250 +300 +EmTrkMichelId Time (s) +EmTrkMichelId Time, normalized to max entry per column +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Events/Column Max +Fig. 7: The average EmTrkMichelId duration +before Oct. 7 as a function of the total network +traffic through the 100 Gb/s network switch at Fer- +milab used by the batch processing cluster. The +top plot shows the real event rate. The bottom +plot is the same as the left one, with each column +scaled separately so the maximum amplitude is 1 +for each column. +is available to the ProtoDUNE jobs, the max- +imum number of concurrent ProtoDUNE jobs +we can run without saturating the network is +(100 Gb/s)/(4.1 Gb/event) · (25 s/event) ≃ 600. +This is consistent with the concurrency limit of +600 jobs that we implemented after October 7. +Based on the above discussions, we conclude +that, while overall computational time clearly +decreases using GPUaaS, one does have to take +particular care to understand what the expected +data movement requirements will be for jobs using +this architecture, and to set job concurrency limits +0 +25 +50 +75 +100 +125 +150 +175 +200 +Avg. EmTrkMchelID time (s) +0 +1000 +2000 +3000 +4000 +5000 +6000 +Njobs +All runs after Oct 8 +Fig. 8: The average time spent in the EmTrk- +MichelId task for all GPU jobs after October 8, +when the network saturation had subsided. +appropriate to the capabilities of each local comput- +ing site and input data source. HTCondor [27, 28] +in particular has the ability to define an arbitrary +kind of resource that each job requires; one could +define a “bandwidth” resource for these jobs, for +example. HTCondor additionally allows configur- +ing the job submissions to prevent more jobs to +start at a given site once the sum of consumed +resources by running jobs at that site reaches a +certain threshold. Therefore, if one knows the total +network capacity of each site hosting jobs, one can +configure per-site job limits and prevent network +saturation in an automated way. +4.1 Future improvements +A number of improvements to overall scalability +and ease of use are possible. In addition to auto- +matic job concurrency limits to prevent network +saturation as previously described, we are explor- +ing the possibility of compressing the data sent to +the GPU server to reduce the overall bandwidth +requirements. While a reduced payload would obvi- +ously increase job concurrency limits, that must +be balanced against the additional run time that + +Springer Nature 2021 LATEX template +GPUaaS in ProtoDUNE data +9 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +Average Started Events (s +1) +40 +50 +60 +70 +80 +90 +100 +Network Traffic (Gb/s) +y = mx + b +slope: 4.2 ± 0.2 (Gb) +intercept: 44 ± 2 (Gb/s) +Outbound Traffic vs #Events Started Per Second +Oct. 5 +Oct. 6 +Fit w/ Uncertainty +0 +50 +100 +150 +200 +250 +EmTrkMichelId Time(s) +Fig. 9: The outbound network traffic vs. the average event start rate per second in 2-minute sliding +windows, on October 5 and October 6. Data from each day is denoted with a different marker type. The +color coding corresponds to the median EmTrkMichelId time for events in each sliding window. The linear +fit to the traffic below 80 Gb/s indicates that each event sends 4.2 ± 0.2 Gb of outbound traffic, on top of +44 ± 2 Gb/s of baseline traffic from non-ProtoDUNE sources. +would be introduced in compressing and decom- +pressing the data on the worker node and server, +respectively. Another desirable area of improve- +ment is in overall ease of use and human effort +requirements. In the current setup we make use +of the standard DUNE Production job submission +infrastructure, which allows for a high degree of +automated job submission, but due to the current +nature of the cloud server it requires an authorized +individual to manually instantiate the GPU infer- +ence server before we submit jobs. Establishing a +method of automatically instantiating the server +at job submission time and automatically ramp- +ing it down when the associated jobs are complete +would avoid a clear possible failure point should +no authorized individuals be available when the +infrastructure is needed. +A second option to study is to use several geo- +graphically distributed inference servers instead of +a single server, while also spreading the job work- +load over a much broader range of sites. Expanding +the site pool has the advantage of making it much +less likely that any single site would get enough +work assigned to saturate its external connectivity, +and using several inference servers spread around +the world would help to mitigate the potential +problem of network latency becoming comparable +to the inference time. The cost changes in this sce- +nario (for example, the relative cost of three cloud +servers versus a single server three times the size) + +Springer Nature 2021 LATEX template +10 +GPUaaS in ProtoDUNE data +must be assessed and taken into account. Another +consideration is how the overall event processing +times would change if the worker nodes were much +more geographically diffuse than they were for this +study. Since we stream the input data over the +network, longer network paths between the worker +nodes and input data sources may lead to the non- +EmTrkMichelId portions of the event processing +taking longer, which in turn affects the total event +processing time. DUNE is able to distribute data +to various storage elements around the world via +the Rucio framework [29], and pre-placing the data +of interest at storage elements close to the sites to +be used for processing may mitigate such concerns, +though it is not required. +Another potential avenue is to use the GPU +server infrastructure, but to use sites with GPUs +available on the worker nodes, and run an inde- +pendent server on each worker node. Several +high-performance computing sites have built or are +building clusters with readily available GPUs, and +in some cases with multiple GPUs on each worker +node, that would naturally lend themselves to such +a setup. If the jobs run on worker nodes with local +GPUs, external network connectivity limitations +become unimportant for carrying out the infer- +ence calculations. In fact, Triton™ allows the use +of shared memory for direct data transfer between +CPU and GPU when the GPU is local. While it +may not be necessary to retain the server infras- +tructure in these cases, the advantage of doing so +is that the experiment software does not have to +be modified to directly access the GPU, making +it maximally portable and easier to maintain. We +plan to conduct a similar study using this type of +setup in the future. +5 Summary +We have reprocessed approximately seven million +data events from the ProtoDUNE detector installed +at CERN. We use an Nvidia Triton™ inference +server hosted on the Google Cloud Platform to +run the most computationally expensive step of +the workflow on a GPU, speeding up the required +processing time by more than a factor of two, even +comparing to the fastest CPU runs. Running at +a scale similar to that expected during regular +ProtoDUNE-II and DUNE operations, we see the +expected performance improvement until the net- +work switch through which the majority of our jobs +communicate becomes saturated. Despite that, the +cloud infrastructure easily kept up with demand +and demonstrates the viability of the GPUaaS +model at a level sufficient for current and future +high-energy physics experiments, as long as the +job concurrency levels at each site respect the +site’s network resource limits. With several promis- +ing avenues of improvement to explore, we expect +that this computing model will become even more +capable and easier to use in the future. +Author Contributions +All authors contributed to the study conception +and design. Material preparation, data collection +and analysis were performed by Tejin Cai, Ken- +neth Herner, and Tingjun Yang. The first draft of +the manuscript was prepared by Tejin Cai, Maria +Acosta Flechas, Kenneth Herner, Kevin Pedro, +Nhan Tran, and Tingjun Yang. All authors read +and approved the final manuscript. +Acknowledgments +We acknowledge the Fast Machine Learning collec- +tive as an open community of multi-domain experts +and collaborators. This community was important +for the development of this project. We acknowl- +edge the DUNE collaboration for providing the +ProtoDUNE-SP code base and data samples. The + +Springer Nature 2021 LATEX template +GPUaaS in ProtoDUNE data +11 +analysis is enabled in part by the Digital Research +Alliance of Canada. +Declarations +Competing Interests +The authors have no competing interests to declare +that are relevant to the content of this article. +Data Availability +The datasets generated during and/or analysed +during the current study are available from the +corresponding author on reasonable request. +Funding +MF, KH, BH, KP, NT, MW, and TY are sup- +ported by Fermi Research Alliance, LLC under +Contract No. DE-AC02-07CH11359 with the U.S. +Department of Energy, Office of Science, Office of +High Energy Physics. NT is partially supported +by the U.S. Department of Energy Early Career +Award. KP is partially supported by the High +Velocity Artificial Intelligence grant as part of the +U.S. Department of Energy High Energy Physics +Computational HEP program. PH is supported +by NSF grants #1934700, #193146. Cloud credits +for this study were provided by Internet2 man- +aged Exploring Cloud to accelerate Science (NSF +grant PHY-190444). TC is supported by NSERC +Canada. +References +[1] Psihas, F.: The Convolutional Visual Network +for Identification and Reconstruction of NOvA +Events. J. Phys. Conf. Ser. 898(7), 072053 +(2017). https://doi.org/10.1088/1742-6596/ +898/7/072053 +[2] Perdue, +G.N., +et +al.: +Reducing +model +bias in a deep learning classifier using +domain +adversarial +neural +networks +in +the MINERvA experiment. 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Computing and Software for Big Sci- +ence 3(1), 11 (2019). https://doi.org/10.1007/ +s41781-019-0026-3 + diff --git a/3tE3T4oBgHgl3EQfogoT/content/tmp_files/load_file.txt b/3tE3T4oBgHgl3EQfogoT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..436ad3f223317c24cdbf9c3f008b3c5ea14005d5 --- /dev/null +++ b/3tE3T4oBgHgl3EQfogoT/content/tmp_files/load_file.txt @@ -0,0 +1,678 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf,len=677 +page_content='Springer Nature 2021 LATEX template Accelerating Machine Learning Inference with GPUs in ProtoDUNE Data Processing Tejin Cai1, Kenneth Herner2*, Tingjun Yang2, Michael Wang2, Maria Acosta Flechas2, Philip Harris3, Burt Holzman2, Kevin Pedro2 and Nhan Tran2 1Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, M3J 1P3, ON, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 2Fermi National Accelerator Laboratory, Kirk Road and Pine Streets, Batavia, 60510, IL, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 3Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, 02139, MA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' E-mail(s): kherner@fnal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='gov;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Abstract We study the performance of a cloud-based GPU-accelerated inference server to speed up event reconstruction in neutrino data batch jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Using detector data from the ProtoDUNE experiment and employing the standard DUNE grid job submission tools, we attempt to reprocess the data by running several thousand concurrent grid jobs, a rate we expect to be typical of current and future neutrino physics experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We process most of the dataset with the GPU version of our processing algorithm and the remainder with the CPU version for timing comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We find that a 100-GPU cloud-based server is able to easily meet the processing demand, and that using the GPU version of the event processing algorithm is two times faster than processing these data with the CPU version when comparing to the newest CPUs in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The amount of data transferred to the inference server during the GPU runs can overwhelm even the highest-bandwidth network switches, however, unless care is taken to observe network facility limits or otherwise distribute the jobs to multiple sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We discuss the lessons learned from this processing campaign and several avenues for future improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Keywords: machine learning, heterogeneous (CPU+GPU) computing, GPU (graphics processing unit), particle physics, cloud computing (SaaS), neutrino physics, distributed computing 1 Introduction Machine learning (ML)-based algorithms have been widely used in the field of neutrino physics, for applications ranging from data acquisition to data reconstruction and analysis [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' A detec- tor technology ideally suited for computer vision applications in neutrino physics is that of liquid argon time projection chambers (LArTPCs), which are employed by the Deep Underground Neutrino Experiment (DUNE) [5] and Short-Baseline Neu- trino [6] experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' ML applications are now deeply integrated into the event reconstruction and data analyses for the LArTPC experiments [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='04633v1 [hep-ex] 11 Jan 2023 Springer Nature 2021 LATEX template 2 GPUaaS in ProtoDUNE data Event record sizes for the current generation of LArTPC experiments are typically ≤1 GB and are expected to increase in the next few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' With increased event size, the event reconstruction, espe- cially the inference of ML algorithms, will become a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Additionally, neutrino detectors are sensitive to neutrinos from a core-collapse super- nova in or near the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' One of DUNE’s physics goals is to rapidly reconstruct detector trigger records from such a supernova to provide rapid localization information to optical telescopes, placing a premium on short event reconstruction times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We have demonstrated GPU-accelerated ML inference as a service, which significantly reduced the reconstruction time for simulated neutrino events in the ProtoDUNE experiment [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Later, we tested the same GPU-as-a-Service (GPUaaS) approach to process the entire ProtoDUNE Run I dataset to demonstrate the scalability of this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' This paper reports the results of those tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 2 Infrastructure setup and methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='1 ProtoDUNE background The ProtoDUNE single phase detector (ProtoDUNE-SP) [11, 12] is a liquid argon time projection chamber (LArTPC) that serves as a prototype for the first far detector module of DUNE [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The ProtoDUNE-SP is installed at the CERN Neutrino Platform [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' It has an active volume of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='1 × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='0 m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The TPC wires are read out by 15,360 electric channels at a rate of 2 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' A typical event record consists of 6000 time samples, corresponding to a 3 ms time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Between October 10 and November 11, 2018, ProtoDUNE-SP was exposed to a beam that delivers charged pions, kaons, protons, muons and electrons with momenta in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='3 GeV/c to 7 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' After the beam runs ended, ProtoDUNE-SP continued to collect cosmic ray and calibration data until July 20, 2020, after which the detector decommissioning started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The total number of trigger records (also called “events”) during the beam period, which consist of both beam interactions and non-beam interactions such as cosmic rays, is approximately 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2 million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' A ProtoDUNE-SP TPC waveform recorded by a single electric channel consists of both signals and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' There are typically three sources of sig- nals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' During the beam runs, the beam particles can interact with the liquid argon inside the TPC and produce both ionization electrons and scintilla- tion light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Since ProtoDUNE-SP is located on the Earth’s surface, it is subject to a large flux of cos- mic ray muons, which induce signals over the entire detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' There are also radioactive backgrounds such as 39Ar that generate low energy signals on the scale of a few hundred keV to a few MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Figure 1 shows the event display of a 6 GeV/c pion interaction in the ProtoDUNE-SP detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The first step in the reconstruction of events in the TPC is the signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The goal of this stage is to produce distributions of charge arrival times and positions given the input TPC waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The effects of induced currents due to drifting and collecting charge, as well as the response of the front-end electronics, are removed through de-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The charge arrival distri- butions are used in subsequent reconstruction steps, starting with hit finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The hit finding algorithm fits peaks in the wire waveforms, where a hit repre- sents a charge deposition on a single wire at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Each hit corresponds to a fitted peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The hits are input to pattern recognition algorithms such as Pandora [14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' This stage finds the high- level objects associated with particles, like tracks, showers, and vertices, and assembles them into a hierarchy of parent-daughter nodes that ultimately point back to the candidate neutrino interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Springer Nature 2021 LATEX template GPUaaS in ProtoDUNE data 3 0 100 200 300 400 Wire Number 3500 3750 4000 4250 4500 4750 5000 Tick 50 cm DUNE:ProtoDUNE-SP Run 5772 Event 15132 2 0 2 4 6 8 10 Charge/tick/channel (ke) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 1: A 6 GeV/c beam π+ interaction in the ProtoDUNE-SP detector [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The x axis shows the wire number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The y axis shows the time tick in the unit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='5 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The color scale represents the charge deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' More details on the reconstruction workflow are described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' In ProtoDUNE-SP, a novel algorithm is devel- oped based on a convolutional neural network (CNN) to perform the classification of each recon- structed hit as track-like or arising from electromag- netic cascades [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' These hit-level classifications can be used alongside pattern recognition based reconstruction algorithms such as Pandora to refine the track or shower classification of reconstructed particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The CNN model was trained using Ten- sorFlow [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Hereafter, we call this algorithm EmTrkMichelId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' In order to improve the efficiency and speed of the inference of ML algorithms in a large- scale data processing, GPU acceleration specifically for the ProtoDUNE reconstruction chain has been integrated without disrupting the native computing workflow using the services for opti- mized network inference on coprocessors (SONIC) approach [10, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' With the integrated framework, the most time-consuming task, track and particle shower hit identification, runs faster by a factor of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' This results in a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='7 reduction in the total processing time when compared with CPU- only production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' This initial test using a small number of simulated ProtoDUNE events showed a viable, cost-effective way to solve the comput- ing challenge facing the neutrino experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' In this work, we report the results of reprocessing the entire 7 million ProtoDUNE events taken dur- ing the test beam runs with the SONIC-enabled framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2 Inference server setup The Nvidia Triton™ Inference Server is an open- source inference serving software that helps stan- dardize model deployment and execution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' its goal is to deliver fast and scalable AI in production [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 4 GPUaaS in ProtoDUNE data NVIDIA provides multiple ways to deploy the inference server on different cloud providers and infrastructure types, including both bare metal and containerized workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' This study uses a cloud-based deployment of Nvidia Triton™ Inference Server within a Google Cloud Kubernetes Engine [20] cluster on virtual infrastructure provided by Google Cloud Platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The use of this technology enables us to deploy a flexible GPUaaS model where a public end- point takes remote inference requests from various geographically distributed sources as depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The Triton™ server running on the Google cloud supports different backends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We use the Ten- sorFlow (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='5) backend for the inference of the EmTrkMichelId algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' In a similar way as Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' [10], this study uses sev- eral Triton™ servers split into separate Kubernetes deployments with common services for network- ing and external load balancing in the form of ingress objects [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' One significant improvement for the current study is the deployment of metrics and monitoring which provided us with observ- ability within the system in different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' In IT and cloud computing, observability is the ability to measure a system’s current state based on the data it generates, such as logs, metrics, and traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' It relies on telemetry derived from instrumenta- tion that comes from the endpoints and services in computing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Triton™ provides a built- in metrics endpoint [22] that publishes plain-text data in Prometheus format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Prometheus collects and stores data to be displayed by Grafana as seen in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='3 Methods The DUNE collaboration undertook a production campaign in 2021 to process ProtoDUNE-SP data using the LArSoft toolkit [23] version v09 30 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Each production run during the beam period com- prises several data files, each containing between 100 and 150 data records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' In contrast to the previ- ous work, in which DUNE simulation events were processed by submitting jobs locally to a dedicated queue, we submit jobs to process each file via the current standard DUNE workflow management and job submission systems [24, 25], thus requir- ing no special treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Jobs may run either at Fermilab or one of several remote sites that we reach with opportunistic access enabled by the OSG Consortium [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We begin from the existing reconstructed outputs and apply the updated EmTrkMichelId algorithm to produce new outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Of the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2 mil- lion ProtoDUNE events during the 2018 beam period, we process 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='4 million through the SONIC infrastructure, and 800k with the CPU-only ver- sion of the same algorithm for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The OSG sites included in the SONIC runs were cho- sen to be geographically proximate to the location of the Google Cloud GPU servers (which were in Iowa, USA at the time) in order to minimize the latency in data transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The difference in the time spent in the infer- ence step is the primary metric with which we assess the advantage of GPUaaS over traditional CPU processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Each job produces a log file that statistically summarizes the time spent on each stage of the event reconstruction for the job as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The log has no record of per-stage pro- cessing time at the individual event level, but we can closely approximate it by taking the difference between the start times of consecutive events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We estimate the per-event EmTrkMichelId duration by subtracting the median non-EmTrkMichelId duration from the total event duration, as the non-EmTrkMichelId stages display very little time variation across events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The CNN-based hit classi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='fication occurs in the EmTrkMichelId stage and is ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 3: A real-time monitoring view of a 100-GPU cluster run for ProtoDUNE (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' the most time-consuming step in the event recon- struction, typically accounting for more than 90% of the processing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 3 Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='1 CPU-only runs We process a set of 13 runs using CPU-based Tensorflow both at Fermilab and several off-site locations.' 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09:45 09:50 09:55 GPU load - Power 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='00% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='00% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='00% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='009 fastp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='d15@ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='83:002Fermila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='Springer Nature 2021 LATEX template 6 GPUaaS in ProtoDUNE data 0 100 200 300 400 500 EmTrkMichelId Time (s) 0 2000 4000 6000 8000 # of Events / 2 sec CPU Series AMD 6376 AMD EPYC 7502 Intel E5-2650 v2 Intel E5-2650 v3 Intel E5-2670 v3 Intel E5-2680 v4 Intel Gold 6140 non-FNAL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 4: Timing distributions for CPU-only runs, broken down by CPU type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The TensorFlow version used in the CPU-only runs is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Table 1 summarizes the number of events processed at each site and the median processing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We did not request any specific CPU type when submitting these jobs since typical DUNE practice is to use any and all available CPU types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Table 1: List of CPU-only run sites and median processing time OSG Site N samples Median processing time (s) FermiGrid 746603 79 Notre Dame 36082 68 Victoria 10944 52 Wayne State 4242 45 There is a clear dependence on processor type in the EmTrkMichelId processing time distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' In general, more recent CPUs process events faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Figure 4 shows the CPU-based EmTrkMichelId timing for each of the CPU types currently avail- able on the Fermilab general purpose batch farm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We do not have access to CPU type information outside of Fermilab and thus group them together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2 GPU runs Our main processing effort uses the GPUaaS infras- tructure as described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Figure 5 shows the average EmTrkMichelId processing time when using the GPUaaS infrastructure for our entire running period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The first peak at approximately 20 s repre- sents a factor of two improvement with respect to the fastest CPU-only runs, and a factor of roughly 11 over the slowest CPU runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' It is important to note that the EmTrkMichelId times we report here are wall times measured within the job, and thus include contributions from network latency to and from the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' There is another peak in the dis- tribution with a median of over 100 s, to which we now turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 0 25 50 75 100 125 150 175 200 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' EmTrkMchelID time (s) 0 1000 2000 3000 4000 5000 6000 7000 8000 Njobs All runs 9/30 - 10/20 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 5: Average EmTrkMichelId times for GPU runs during the period September 30, 2021 to Octo- ber 20, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The double peak structure arises from periods during which the outbound network con- nection from the Fermilab grid processing center was saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Springer Nature 2021 LATEX template GPUaaS in ProtoDUNE data 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='1 Outbound network saturation During the first period of GPU running we averaged between 200 and 2000 concurrent jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Figure 6 shows the overlay of network traffic and event processing start rate during the period of September 30, 2021 to October 6, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' As the event start rate increases because of the rise in the number of concurrent jobs, we see that the 100 Gb/s outbound network connection used by the Fermilab data center where the jobs run becomes saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' While our jobs were not solely responsi- ble for the saturation (the connection serves the entire cluster), the saturation did result in a sig- nificant increase in the average EmTrkMichelId processing time as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The highest job concurrency levels were on October 5, when unusually low demand for computing resources from other Fermilab experiments resulted in a large number of opportunistic job slots being available at Fermilab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We were, without any direct interven- tion, thus able to scale up to approximately 6,000 concurrent jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The monitoring does show switch saturation as early as October 1, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' After learning of the network saturation we implemented a concurrency limit on jobs of approximately 600;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' thereafter the jobs ran without incident and the EmTrkMichelId times returned to pre-saturation levels (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 4 Discussion In order to understand the impact of ProtoDUNE jobs on the Fermilab network traffic, we plot the distribution of event processing start rate versus network traffic in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Even though the net- work traffic has contributions from all grid jobs at Fermilab, there is a clear correlation between the number of ProtoDUNE concurrent jobs and the increase of network traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We fit a straight line to the data points below the network traffic of 80 09/30 10/1 10/2 10/3 10/4 10/5 10/6 10/7 Date 0 5 10 15 20 25 Event Starting Rate/s 0 20 40 60 80 100 Traffic (Gb/s) Event Rate Outbound Traffic Google Traffic Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 6: Overlay of network traffic and event pro- cessing start rate at FermiGrid as a function of time, which is a proxy for the number of concurrent jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The origin day is September 30, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The solid line is the event start rate, the blue dot-dash line is the outbound network traffic rate through the 100 Gb/s switch at Fermilab used by the batch processing cluster, and the black dashed line is the ingress rate to the Google cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We are unable to disambiguate traffic sources through the switch, so the blue dot-dash line represents the total traffic as opposed to only traffic generated by our processing campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We see that the net- work switch was effectively saturated in multiple instances, though Google ingress was not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Gb/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The slope of the best fit line is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2 Gb, which is the average outbound data transmission per event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The intercept is 44 ± 2 Gb/s, which is the average traffic from non-ProtoDUNE grid jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Based on the discussion of transmission time in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' [10], for 55,000 inferences per event, with each input a 48 × 48 image at 32 bits, the total amount of data transmitted is about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='1 Gigabits per event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' This is consistent with the slope of the best fit straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The spread in data with respect to the straight line could be caused by the variation in the number of non-ProtoDUNE grid jobs during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Figure 8 indicates that the average process- ing time is roughly 25 s/event for the GPU jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Assuming the entire 100 Gb/s bandwidth Springer Nature 2021 LATEX template 8 GPUaaS in ProtoDUNE data 20 30 40 50 60 70 80 90 100 FermiGrid Outbound Traffic (Gb/s) 0 50 100 150 200 250 300 EmTrkMichelId Time (s) EmTrkMichelId Time 0 2000 4000 6000 8000 10000 12000 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' of Events 20 30 40 50 60 70 80 90 100 FermiGrid Outbound Traffic (Gb/s) 0 50 100 150 200 250 300 EmTrkMichelId Time (s) EmTrkMichelId Time, normalized to max entry per column 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='0 Events/Column Max Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 7: The average EmTrkMichelId duration before Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 7 as a function of the total network traffic through the 100 Gb/s network switch at Fer- milab used by the batch processing cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The top plot shows the real event rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The bottom plot is the same as the left one, with each column scaled separately so the maximum amplitude is 1 for each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' is available to the ProtoDUNE jobs, the max- imum number of concurrent ProtoDUNE jobs we can run without saturating the network is (100 Gb/s)/(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='1 Gb/event) · (25 s/event) ≃ 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' This is consistent with the concurrency limit of 600 jobs that we implemented after October 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Based on the above discussions, we conclude that, while overall computational time clearly decreases using GPUaaS, one does have to take particular care to understand what the expected data movement requirements will be for jobs using this architecture, and to set job concurrency limits 0 25 50 75 100 125 150 175 200 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' EmTrkMchelID time (s) 0 1000 2000 3000 4000 5000 6000 Njobs All runs after Oct 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 8: The average time spent in the EmTrk- MichelId task for all GPU jobs after October 8, when the network saturation had subsided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' appropriate to the capabilities of each local comput- ing site and input data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' HTCondor [27, 28] in particular has the ability to define an arbitrary kind of resource that each job requires;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' one could define a “bandwidth” resource for these jobs, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' HTCondor additionally allows configur- ing the job submissions to prevent more jobs to start at a given site once the sum of consumed resources by running jobs at that site reaches a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Therefore, if one knows the total network capacity of each site hosting jobs, one can configure per-site job limits and prevent network saturation in an automated way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='1 Future improvements A number of improvements to overall scalability and ease of use are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' In addition to auto- matic job concurrency limits to prevent network saturation as previously described, we are explor- ing the possibility of compressing the data sent to the GPU server to reduce the overall bandwidth requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' While a reduced payload would obvi- ously increase job concurrency limits, that must be balanced against the additional run time that Springer Nature 2021 LATEX template GPUaaS in ProtoDUNE data 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='0 Average Started Events (s 1) 40 50 60 70 80 90 100 Network Traffic (Gb/s) y = mx + b slope: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2 (Gb) intercept: 44 ± 2 (Gb/s) Outbound Traffic vs #Events Started Per Second Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 5 Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 6 Fit w/ Uncertainty 0 50 100 150 200 250 EmTrkMichelId Time(s) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 9: The outbound network traffic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' the average event start rate per second in 2-minute sliding windows, on October 5 and October 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Data from each day is denoted with a different marker type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The color coding corresponds to the median EmTrkMichelId time for events in each sliding window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The linear fit to the traffic below 80 Gb/s indicates that each event sends 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='2 Gb of outbound traffic, on top of 44 ± 2 Gb/s of baseline traffic from non-ProtoDUNE sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' would be introduced in compressing and decom- pressing the data on the worker node and server, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Another desirable area of improve- ment is in overall ease of use and human effort requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' In the current setup we make use of the standard DUNE Production job submission infrastructure, which allows for a high degree of automated job submission, but due to the current nature of the cloud server it requires an authorized individual to manually instantiate the GPU infer- ence server before we submit jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Establishing a method of automatically instantiating the server at job submission time and automatically ramp- ing it down when the associated jobs are complete would avoid a clear possible failure point should no authorized individuals be available when the infrastructure is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' A second option to study is to use several geo- graphically distributed inference servers instead of a single server, while also spreading the job work- load over a much broader range of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Expanding the site pool has the advantage of making it much less likely that any single site would get enough work assigned to saturate its external connectivity, and using several inference servers spread around the world would help to mitigate the potential problem of network latency becoming comparable to the inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The cost changes in this sce- nario (for example, the relative cost of three cloud servers versus a single server three times the size) Springer Nature 2021 LATEX template 10 GPUaaS in ProtoDUNE data must be assessed and taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Another consideration is how the overall event processing times would change if the worker nodes were much more geographically diffuse than they were for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Since we stream the input data over the network, longer network paths between the worker nodes and input data sources may lead to the non- EmTrkMichelId portions of the event processing taking longer, which in turn affects the total event processing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' DUNE is able to distribute data to various storage elements around the world via the Rucio framework [29], and pre-placing the data of interest at storage elements close to the sites to be used for processing may mitigate such concerns, though it is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Another potential avenue is to use the GPU server infrastructure, but to use sites with GPUs available on the worker nodes, and run an inde- pendent server on each worker node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Several high-performance computing sites have built or are building clusters with readily available GPUs, and in some cases with multiple GPUs on each worker node, that would naturally lend themselves to such a setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' If the jobs run on worker nodes with local GPUs, external network connectivity limitations become unimportant for carrying out the infer- ence calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' In fact, Triton™ allows the use of shared memory for direct data transfer between CPU and GPU when the GPU is local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' While it may not be necessary to retain the server infras- tructure in these cases, the advantage of doing so is that the experiment software does not have to be modified to directly access the GPU, making it maximally portable and easier to maintain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We plan to conduct a similar study using this type of setup in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' 5 Summary We have reprocessed approximately seven million data events from the ProtoDUNE detector installed at CERN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We use an Nvidia Triton™ inference server hosted on the Google Cloud Platform to run the most computationally expensive step of the workflow on a GPU, speeding up the required processing time by more than a factor of two, even comparing to the fastest CPU runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Running at a scale similar to that expected during regular ProtoDUNE-II and DUNE operations, we see the expected performance improvement until the net- work switch through which the majority of our jobs communicate becomes saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Despite that, the cloud infrastructure easily kept up with demand and demonstrates the viability of the GPUaaS model at a level sufficient for current and future high-energy physics experiments, as long as the job concurrency levels at each site respect the site’s network resource limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' With several promis- ing avenues of improvement to explore, we expect that this computing model will become even more capable and easier to use in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Author Contributions All authors contributed to the study conception and design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Material preparation, data collection and analysis were performed by Tejin Cai, Ken- neth Herner, and Tingjun Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The first draft of the manuscript was prepared by Tejin Cai, Maria Acosta Flechas, Kenneth Herner, Kevin Pedro, Nhan Tran, and Tingjun Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' All authors read and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Acknowledgments We acknowledge the Fast Machine Learning collec- tive as an open community of multi-domain experts and collaborators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' This community was important for the development of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' We acknowl- edge the DUNE collaboration for providing the ProtoDUNE-SP code base and data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' The Springer Nature 2021 LATEX template GPUaaS in ProtoDUNE data 11 analysis is enabled in part by the Digital Research Alliance of Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Declarations Competing Interests The authors have no competing interests to declare that are relevant to the content of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Data Availability The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Funding MF, KH, BH, KP, NT, MW, and TY are sup- ported by Fermi Research Alliance, LLC under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' DE-AC02-07CH11359 with the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of High Energy Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' NT is partially supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Department of Energy Early Career Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' KP is partially supported by the High Velocity Artificial Intelligence grant as part of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Department of Energy High Energy Physics Computational HEP program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' PH is supported by NSF grants #1934700, #193146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Cloud credits for this study were provided by Internet2 man- aged Exploring 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', Garonne, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', di Girolamo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', Goossens, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', Guan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', Guenther, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', Javurek, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', Kuhn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', Lassnig, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', Lopez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', Magini, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', Molfetas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', Nairz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=', Ould- Saada, F.' metadata={'source': 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Wegner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=': Rucio: Scientific data manage- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' Computing and Software for Big Sci- ence 3(1), 11 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} +page_content='1007/ s41781-019-0026-3' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE3T4oBgHgl3EQfogoT/content/2301.04633v1.pdf'} diff --git a/4tAyT4oBgHgl3EQfcPfW/vector_store/index.pkl 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b/7dFJT4oBgHgl3EQfmywg/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6e09c32e3d16707d8b172c42554cac5ceaf19491fc6a061d4af855bfad9e779f +size 283341 diff --git a/7tE2T4oBgHgl3EQflAfG/content/tmp_files/2301.03985v1.pdf.txt b/7tE2T4oBgHgl3EQflAfG/content/tmp_files/2301.03985v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a0b1bab3333258c597428735a1b50090a4341370 --- /dev/null +++ b/7tE2T4oBgHgl3EQflAfG/content/tmp_files/2301.03985v1.pdf.txt @@ -0,0 +1,606 @@ +Frascati Physics Series Vol. 73 (2022) +LFC22: Strong interactions from QCD to new strong dynamics at LHC and Future Colliders +August 29 - September 2, 2022 +Non commutativity between massless and soft limit in processes with heavy quarks +Andrea Ghira +Dipartimento di Fisica, Universit`a degli Studi di Genova and INFN, Via Dodecaneso 33, 16146, Italy +Abstract +Processes involving heavy quarks can be computed in perturbation theory in two different ways: we +can adopt a scheme in which the mass of the quark is considered only as a regulator of the collinear +divergences because of the fact that the hard scale of the process is far bigger or we can consider the +quark as a massive particle. Each picture has its own advantages and drawbacks: we investigate the +differences between the two approaches with particular attention to the soft logarithmic structure. We +examine the origin of this difference, focusing on different processes involving the Higgs boson . Finally +we perform the threshold resummation of the Higgs boson decay rate into a b¯b pair at NLL accuracy in +the massive scheme. +1 +Introduction +Quarks appear in the Quantum Chromo-Dynamics (QCD) lagrangian in different species, named flavours. +From the point of view of strong interactions, different flavours are distinguished purely on the basis of the +value of their masses. It is therefore natural to classify quark flavours according to their masses, compared +to ΛQCD ≃ 300MeV. The masses of up, down and strange quarks, relevant for ordinary matter, are much +smaller than ΛQCD, and can be taken to be zero for most applications in high-energy physics, on the +other hand charm (c) and especially bottom (b) are heavy according to this definition. Heavy-flavour +production cross-sections can be calculated in perturbative QCD because the mass of the b and c quarks +sets the value of the coupling in the perturbative region and regulates collinear singularities. In order to +compute processes involving heavy flavour two main approaches are employed. In the so-called massive +scheme, the final-state heavy quarks are considered massive particles and we can compute order by order +in perturbation theory the scattering amplitude. Within this approach the kinematics is treated correctly +arXiv:2301.03985v1 [hep-ph] 10 Jan 2023 + +h(q) +b(p1) +¯b(p2) +g(k) +1 +h(q) +b(p1) +¯b(p2) +g(k) +1 +Figure 1: Real-emission contributions to the decay of the Higgs boson into a b¯b pair at O (αs). +but calculations become cumbersome at higher and higher perturbative orders. Another drawback is that +large mass logarithms which arise due to the fact that the mass of the heavy quark is far smaller than +hard scale of the process spoil the convergence of the perturbative series. Therefore another framework +is employed which is the so called massless scheme. In the massless scheme, we treat the mass of the +particle only as a regulator of the collinear divergences. Consequently we do not have control on the +kinematics outside the collinear region, i.e. we consider only radiation emitted at small angle. This +approach exploits the factorization theorem: the differential cross section can be written as a convolution +product of a process dependent function times a fragmentation function, which is process independent +and fulfills a first order linear equation that allows us to resum the mass logarithms (DGLAP). The +initial condition of the DGLAP evolution equation is set at a scale µ2 +0 ≃ m2 +c,b ≫ Λ2 +QCD and therefore it +is in the perturbative domain and it can be determined by matching the factorisation theorem with the +massive scheme. It was determined to NLO in QCD for the b quark fragmentation function in +1, 2) +and to NNLO in 3, 4). The initial condition is affected by soft logarithms, that should be resummed +to all-orders too 5, 6). The main problem we want to focus on is that the structure of soft logarithms +in the initial condition of the fragmentation function cannot be always recovered by the massless limit +of a massive-framework calculation: this strongly depends both on the considered process and on the +specific observable that is computed. We will show this particular behaviour using a simple process as +an example which is the decay of a Higgs boson in a b¯b pair. Secondly, we want to derive a resummed +expression of the differential decay rate at NLL accuracy that fully take into account the heavy quark +mass and outline also in this case the non commutativity of the massless and soft limit. +2 +Interplay between soft and massless limit in H → b¯b +In order to explain the aforementioned non commutativity of the limits we focus on the decay of the +Higgs boson at NLO keeping the mass of the quarks: +h(q) → b(p1) + ¯b(p2) + g(k) +p2 +1 = p2 +2 = m2, k2 = 0. +(1) +We compute the differential decay rate dΓ +dx, where x = 2p1·q +q2 +is the energy of the quark in the CoM reference +frame, and we are interested in the small mass limit necessary for the massless scheme ( m2 +|q2| ≡ ξ → 0) + +and in the soft limit (x → 1). Performing the soft limit and the massless in two different orders we find: +lim +ξ→0 lim +x→1 +1 +Γ0 +dΓ +dx = −2αsCF +π +�1 + log ξ +1 − x ++ O(ξ0) + O +� +(1 − x)0�� +, +(2) +lim +x→1 lim +ξ→0 +1 +Γ0 +dΓ +dx = −αsCF +π +� log ξ +1 − x + log(1 − x) +1 − x ++ 7 +4 +1 +1 − x + O(ξ0) + O +� +(1 − x)0�� +, +where Γ0 is the Born level decay rate: +Γ0 = +� +2q2GF m2β3NC +8π +, +β = +� +1 − 4ξ, +(3) +with GF is the Fermi constant. In order to analyze the logarithmic structure of the previous equation, +we introduce the Mellin transformation: +M{f(x)}(N) = +� 1 +0 +xN−1f(x) dx +(4) +We notice that in the first case of equation (2) we have a mass logarithm multiplied by a soft one +( +1 +1−x ↔ log N in Mellin space) whereas in the second one we have an additional term which corresponds +to a log2 N after the Mellin transformation. We note also that the overall coefficient is halved in the +second limit, as if the log(1 − x) contribution in the second line of (2) is playing the role of a mass +logarithm. +We would like to provide a physical interpretation to this fact: a measurment of x fixes +the invariant mass (p2 + k)2 = m2 +g¯b thus screening one of the collinear (mass) logs and preventing the +anti-quark propagator to go on-shell. In order to analyse the actual origin of the double logarithms, we +have to look at the quark propagator: if we integrate it over the angle between the gluon and the quark +in the ⃗p2 + ⃗k = 0 frame we find +� 1 +−1 +1 +1 − β1 cos θ dcos θ = log +x2 +ξ(1 − x) + O +� +(1 − x)0� +, +β1 = x +� +1 − 4ξ/x2 +x − 2ξ +, +(5) +where β1 is the quark velocity in that reference frame. In this limit, collinear logarithms appear in two +distinct ways: as explicit logarithm of the quark mass m or as logarithms of 1 − x. This consideration +brings us to formulate a more general statement about double soft logs in processes with heavy quark. We +expect this behaviour to arise if look at a differential distribution which is directly related to the virtuality +of one of the propagators, here m2 +g¯b. Let us consider the differential distribution in ¯x = (p1+p2)2 +q2 +→ 1 as +k → 0. Performing an explicit calculation: +lim +ξ→0 lim +¯x→1 +1 +Γ0 +dΓ +d¯x = lim +¯x→1 lim +ξ→0 +1 +Γ0 +dΓ +d¯x = −2αsCF +π +1 + log ξ +1 − ¯x ++ O(ξ0) + O +� +(1 − x)0� +, +(6) +In this case we have only a single logarithmic enhancement and the two limits commute. +2.1 +Higgs Production and Higgs DIS +We test our statement by studying other processes related by crossing symmetry to the Higgs boson +decay, i.e Higgs boson production and Higgs DIS. In the Higgs production b(p1) + ¯b(p2) → h(q) + g(k), +we are differential in τ = (p1+p2)2 +q2 +, which is not related to the virtuality of the propagators. In this case +we find that the limits commute, as expected: +lim +τ→1 lim +ξ→0 +1 +σ0 +dσ +dτ = lim +ξ→0 lim +τ→1 +1 +σ0 +dσ +dτ = −2αsCF +π +1 + log ξ +1 − τ ++ O(ξ0) + O +� +(1 − τ)0� +, +(7) +σ0 = +√ +2GF m2βπNC +18s +. + +Finally we study the differential distribution +dσ +dxB with xB = +−q2 +2p1·q for the real emission corrections to the +process b(p1) + h(q) → b(p2) + g(k). Due to the fact that xB is related to the virtuality of one of the +propagator we expect that the limit do not commute. Indeed we find: +lim +xB→1 lim +ξ→0 +1 +¯σ0 +dσ +dxB += −αsCF +π +� log ξ +1 − xB ++ log(1 − xB) +1 − xB ++ 7 +4 +1 +1 − xB ++ O(ξ0) + O +� +(1 − xB)0�� +, +(8) +lim +ξ→0 lim +xB→1 +1 +¯σ0 +dσ +dxB += −2αsCF +π +1 + log ξ +1 − xB ++ +O(ξ0) + O +� +(1 − xB)0� +, +¯σ0 = π +√ +2GF m2NCη +−3q2 +, +η = +� +1 + 4ξ. +3 +Soft Resummation in the Massive Scheme +In this section we want to give an explicit expression for the all-order soft resummation of the Higgs decay +rate in a b¯b pair at NLL accuracy in the massive scheme. Since we look at the differential distribution +over x, we are in class of process with the so called single-particle inclusive kinematics (see 7)). The +main result of +7) is that the resummed expression can be factorized as a product of a soft function +times a hard function times a jet function for every massles particle n the final state. In our case the +resummation formula simplifies considerably there are not massless particles. The resummed result of +7) at NLL, adapted to the process we are considering, reads1 +�Γ(N, ξ) = +� +1 + αs +π C(1)(ξ) + O +� +α2 +s +�� +e +−2 +� 1 +1/ ¯ +N +dz +z +� +αs(z2q2) +π +γ(0) +soft(β)+ +� +αs(z2q2) +π +�2 +γ(1) +soft(β)+O(α3 +s) +� ++ O +� 1 +N +� +, +(9) +with ¯N = NeγE and γsoft the massive soft anomalous dimension. To this logarithmic accuracy we need +the two loops expression of the running coupling, the coefficients γ(0) +soft, γ(1) +soft and C(1). The first order soft +anomalous dimension can be obtained from the calculation of one gluon emission in the eikonal limit: +γ(0) +soft(β) = CF +�1 + β2 +2β +log +�1 + β +1 − β +� +− 1 +� +, +(10) +while the second order was presented in 8)2: +γ(1) +soft = +�K +2 + CA +2 +� +−1 +3 log2 1 − β +1 + β + log 1 − β +1 + β − ζ2 +� ++(1 + β2) +4β +CA +� +Li2 +�(1 − β)2 +(1 + β)2 +� ++ 1 +3 log2 1 − β +1 + β + ζ2 +�� +γ(0) +soft(β) ++ CFCA +�1 +2 + 1 +2 log 1 − β +1 + β + 1 +3 log2 1 − β +1 + β − (1 + β2)2 +8β2 +� +−Li3 +�(1 − β)2 +(1 + β)2 +� ++ ζ3 +� +− (1 + β2) +2β +� +log 1 − β +1 + β log (1 + β)2 +4β +− 1 +6 log2 1 − β +1 + β − Li2 +�(1 − β)2 +(1 + β)2 +��� +, +(11) +1We are not so sure about the argument of the running coupling, since in 7) αs(z2q2) is used, on the +other hand it seems that in 8) αs(z2m2) is used. +2It is worth to mention that there is a mismatch in the literature between 8) and 9) + +with K = CA +� 67 +18 − ζ2 +� +− 5nf +9 . The coefficient C(1) is instead process-dependent, as it receives contri- +butions from both the end-point of the real emission and from the virtual corrections (computed in the +on-shell scheme). Writing the real emission differential decay rate as: +dΓ(R) +dx += αsCF +π +Γ(d) +0 +fε +� +x, ξ, q2 +µ2 +� +(1 − x)1+2ϵ , +Γ(d) +0 += Γ0 +π +5−d +2 +2d−3Γ +� d−1 +2 +� +� 4µ2 +q2β2 +� 4−d +2 +, +(12) +the coefficient C(1) can be determined using the fact that virtual corrections are proportional to δ(1 − x) +and the identity between distributions: +fε +� +x, ξ, q2 +µ2 +� +(1 − x)1+2ε = δ(1 − x) +� +−f0(1, ξ) +2ε ++ f0(1, ξ) log(1 − 2 +� +ξ) − 1 +2 +d +dεfε +� +1, ξ, q2 +µ2 +� ��� +ε=0 +� ++ f0(x, ξ) +(1 − x)+ ++ O(ε) . +(13) +Summing up virtual and real contributions we obtain: +C(1)(ξ) = CF +2 +� +− 2γ(0) +soft(β) +CF +� +−2 log +� +1 − +� +1 − β2 +� ++ log m2 +q2 + log +�1 − β2 +4 +� ++ 1 +� +− 2 ++ 2L(β) +�1 − β2 +β +� ++ 1 + β2 +β +� +1 +2L(β) log +�1 − β2 +4 +� ++ 2L(β)(1 − log β) + 2Li2 +�1 − β +1 + β +� ++ L(β)2 + L(β) log 1 − β +2 ++ 2 +3π2 − 1 +2 +� +Li2 +� +4β +(1 + β)2 +� +− Li2 +� +−4β +(1 − β)2 +�� �� +, +(14) +with L(β) = log +� +1+β +1−β +� +. We note that the non commutativity of the soft and massless limits has conse- +quences for the resummed expression in the massive scheme: In the small ξ limit we find: +αsC(1)(ξ) = αsCF +�1 +2 log2 ξ + log ξ + O(ξ0) +� +. +We have a double log of the mass in disagreement with DGLAP evolution equation. The problem is that +equation (13) does not hold if we perform the massless limit because in this limit f0(1, ξ) is not defined. +In a certain way we can say that double mass logarithms in the soft limit of the massive calculation and +double soft logarithms of the massless scheme are connected. A well defined expression in the massless +limit can be obtained rewriting the differential decay rate as: +1 +Γ0 +dΓ +dx = δ(1 − x) + αs +π +� +CF +�f0(x, ξ) +1 − x +� ++ ++ A(ξ) δ(1 − x) +� +, +(15) +The delta coefficient has an expected behaviour for ξ → 0 +A(ξ) = CF +3 +2 log ξ + O(ξ0). +(16) +4 +Conclusions +We have considered observables with different kinematics in processes involving heavy quarks, and in all +processes we have computed NLO corrections taking into account the mass dependence of the square +amplitude. +We have underlined that soft and massless do not always commute, in particular in the + +massless limit the structure of the distributions can radically change because of the presence of double +logs of N. We have traced back the origin of this particular behaviour to the interplay between the +observable we are computing and the fermionic propagators in the scattering amplitudes. Finally, we +have focused on the massive scheme resummation of the process H → b¯b in the soft limit and we +have found that within this approach double logarithms of the mass may appear, and the origin of this +surprising behaviour can be lead back again to the non commutativity between the large N and small +mass limit. +An interesting phenomenological study, in the context of heavy-quark calculations, would be com- +bine the massive scheme with the massless one where also soft logarithms are resummed. The merging +of the two becomes far from trivial because of the lack of commutativity of the limits. One would like +to design an all-order matching scheme that takes into account both the different logarithmic behaviour +that arises in the two cases. +5 +Acknowledgements +We thank Simone Marzani and Giovanni Ridolfi for the aid in the drafting of this proceeding, which is +entirely based on 10). +References +1. B. Mele and P. Nason, Nucl. Phys. B 361 (1991), 626-644 [erratum: Nucl. Phys. B 921 (2017), +841-842] doi:10.1016/0550-3213(91)90597-Q +2. B. Mele and P. Nason, Phys. Lett. B 245 (1990), 635-639 doi:10.1016/0370-2693(90)90704-A +3. K. Melnikov and A. Mitov, Phys. Rev. D 70 (2004), 034027 doi:10.1103/PhysRevD.70.034027 +[arXiv:hep-ph/0404143 [hep-ph]]. +4. A. Mitov, Phys. Rev. D 71 (2005), 054021 doi:10.1103/PhysRevD.71.054021 [arXiv:hep-ph/0410205 +[hep-ph]]. +5. M. Cacciari and S. Catani, Nucl. Phys. B 617 (2001), 253-290 doi:10.1016/S0550-3213(01)00469-2 +[arXiv:hep-ph/0107138 [hep-ph]]. +6. F. Maltoni, G. Ridolfi, M. Ubiali and M. Zaro, JHEP 10 (2022), 027 doi:10.1007/JHEP10(2022)027 +[arXiv:2207.10038 [hep-ph]]. +7. E. Laenen, G. Oderda and G. F. Sterman, Phys. Lett. B 438 (1998), 173-183 doi:10.1016/S0370- +2693(98)00960-5 [arXiv:hep-ph/9806467 [hep-ph]]. +8. N. +Kidonakis, +Phys. +Rev. +Lett. +102 +(2009), +232003 +doi:10.1103/PhysRevLett.102.232003 +[arXiv:0903.2561 [hep-ph]]. +9. A. von Manteuffel, R. M. Schabinger and H. X. Zhu, Phys. Rev. D 92 (2015) no.4, 045034 +doi:10.1103/PhysRevD.92.045034 [arXiv:1408.5134 [hep-ph]]. +10. D. Gaggero, A. Ghira, S. Marzani and G. Ridolfi, JHEP 09 (2022), 058 doi:10.1007/JHEP09(2022)058 +[arXiv:2207.13567 [hep-ph]]. + diff --git a/7tE2T4oBgHgl3EQflAfG/content/tmp_files/load_file.txt b/7tE2T4oBgHgl3EQflAfG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..db9fc3e1b75919cddde67e794a8cc506ae5fabb6 --- /dev/null +++ b/7tE2T4oBgHgl3EQflAfG/content/tmp_files/load_file.txt @@ -0,0 +1,180 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf,len=179 +page_content='Frascati Physics Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' 73 (2022) LFC22: Strong interactions from QCD to new strong dynamics at LHC and Future Colliders August 29 - September 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' 2022 Non commutativity between massless and soft limit in processes with heavy quarks Andrea Ghira Dipartimento di Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Universit`a degli Studi di Genova and INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Via Dodecaneso 33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' 16146,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Italy Abstract Processes involving heavy quarks can be computed in perturbation theory in two different ways: we can adopt a scheme in which the mass of the quark is considered only as a regulator of the collinear divergences because of the fact that the hard scale of the process is far bigger or we can consider the quark as a massive particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Each picture has its own advantages and drawbacks: we investigate the differences between the two approaches with particular attention to the soft logarithmic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' We examine the origin of this difference, focusing on different processes involving the Higgs boson .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Finally we perform the threshold resummation of the Higgs boson decay rate into a b¯b pair at NLL accuracy in the massive scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' 1 Introduction Quarks appear in the Quantum Chromo-Dynamics (QCD) lagrangian in different species, named flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' From the point of view of strong interactions, different flavours are distinguished purely on the basis of the value of their masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' It is therefore natural to classify quark flavours according to their masses, compared to ΛQCD ≃ 300MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' The masses of up, down and strange quarks, relevant for ordinary matter, are much smaller than ΛQCD, and can be taken to be zero for most applications in high-energy physics, on the other hand charm (c) and especially bottom (b) are heavy according to this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Heavy-flavour production cross-sections can be calculated in perturbative QCD because the mass of the b and c quarks sets the value of the coupling in the perturbative region and regulates collinear singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' In order to compute processes involving heavy flavour two main approaches are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' In the so-called massive scheme, the final-state heavy quarks are considered massive particles and we can compute order by order in perturbation theory the scattering amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Within this approach the kinematics is treated correctly arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content='03985v1 [hep-ph] 10 Jan 2023 h(q) b(p1) ¯b(p2) g(k) 1 h(q) b(p1) ¯b(p2) g(k) 1 Figure 1: Real-emission contributions to the decay of the Higgs boson into a b¯b pair at O (αs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' but calculations become cumbersome at higher and higher perturbative orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Another drawback is that large mass logarithms which arise due to the fact that the mass of the heavy quark is far smaller than hard scale of the process spoil the convergence of the perturbative series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Therefore another framework is employed which is the so called massless scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' In the massless scheme, we treat the mass of the particle only as a regulator of the collinear divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Consequently we do not have control on the kinematics outside the collinear region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' we consider only radiation emitted at small angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' This approach exploits the factorization theorem: the differential cross section can be written as a convolution product of a process dependent function times a fragmentation function, which is process independent and fulfills a first order linear equation that allows us to resum the mass logarithms (DGLAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' The initial condition of the DGLAP evolution equation is set at a scale µ2 0 ≃ m2 c,b ≫ Λ2 QCD and therefore it is in the perturbative domain and it can be determined by matching the factorisation theorem with the massive scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' It was determined to NLO in QCD for the b quark fragmentation function in 1, 2) and to NNLO in 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' The initial condition is affected by soft logarithms, that should be resummed to all-orders too 5, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' The main problem we want to focus on is that the structure of soft logarithms in the initial condition of the fragmentation function cannot be always recovered by the massless limit of a massive-framework calculation: this strongly depends both on the considered process and on the specific observable that is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' We will show this particular behaviour using a simple process as an example which is the decay of a Higgs boson in a b¯b pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Secondly, we want to derive a resummed expression of the differential decay rate at NLL accuracy that fully take into account the heavy quark mass and outline also in this case the non commutativity of the massless and soft limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' 2 Interplay between soft and massless limit in H → b¯b In order to explain the aforementioned non commutativity of the limits we focus on the decay of the Higgs boson at NLO keeping the mass of the quarks: h(q) → b(p1) + ¯b(p2) + g(k) p2 1 = p2 2 = m2, k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' (1) We compute the differential decay rate dΓ dx, where x = 2p1·q q2 is the energy of the quark in the CoM reference frame, and we are interested in the small mass limit necessary for the massless scheme ( m2 |q2| ≡ ξ → 0) and in the soft limit (x → 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Performing the soft limit and the massless in two different orders we find: lim ξ→0 lim x→1 1 Γ0 dΓ dx = −2αsCF π �1 + log ξ 1 − x + O(ξ0) + O � (1 − x)0�� , (2) lim x→1 lim ξ→0 1 Γ0 dΓ dx = −αsCF π � log ξ 1 − x + log(1 − x) 1 − x + 7 4 1 1 − x + O(ξ0) + O � (1 − x)0�� , where Γ0 is the Born level decay rate: Γ0 = � 2q2GF m2β3NC 8π , β = � 1 − 4ξ, (3) with GF is the Fermi constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' In order to analyze the logarithmic structure of the previous equation, we introduce the Mellin transformation: M{f(x)}(N) = � 1 0 xN−1f(x) dx (4) We notice that in the first case of equation (2) we have a mass logarithm multiplied by a soft one ( 1 1−x ↔ log N in Mellin space) whereas in the second one we have an additional term which corresponds to a log2 N after the Mellin transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' We note also that the overall coefficient is halved in the second limit, as if the log(1 − x) contribution in the second line of (2) is playing the role of a mass logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' We would like to provide a physical interpretation to this fact: a measurment of x fixes the invariant mass (p2 + k)2 = m2 g¯b thus screening one of the collinear (mass) logs and preventing the anti-quark propagator to go on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' In order to analyse the actual origin of the double logarithms, we have to look at the quark propagator: if we integrate it over the angle between the gluon and the quark in the ⃗p2 + ⃗k = 0 frame we find � 1 −1 1 1 − β1 cos θ dcos θ = log x2 ξ(1 − x) + O � (1 − x)0� , β1 = x � 1 − 4ξ/x2 x − 2ξ , (5) where β1 is the quark velocity in that reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' In this limit, collinear logarithms appear in two distinct ways: as explicit logarithm of the quark mass m or as logarithms of 1 − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' This consideration brings us to formulate a more general statement about double soft logs in processes with heavy quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' We expect this behaviour to arise if look at a differential distribution which is directly related to the virtuality of one of the propagators, here m2 g¯b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Let us consider the differential distribution in ¯x = (p1+p2)2 q2 → 1 as k → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Performing an explicit calculation: lim ξ→0 lim ¯x→1 1 Γ0 dΓ d¯x = lim ¯x→1 lim ξ→0 1 Γ0 dΓ d¯x = −2αsCF π 1 + log ξ 1 − ¯x + O(ξ0) + O � (1 − x)0� , (6) In this case we have only a single logarithmic enhancement and the two limits commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content='1 Higgs Production and Higgs DIS We test our statement by studying other processes related by crossing symmetry to the Higgs boson decay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content='e Higgs boson production and Higgs DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' In the Higgs production b(p1) + ¯b(p2) → h(q) + g(k), we are differential in τ = (p1+p2)2 q2 , which is not related to the virtuality of the propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' In this case we find that the limits commute, as expected: lim τ→1 lim ξ→0 1 σ0 dσ dτ = lim ξ→0 lim τ→1 1 σ0 dσ dτ = −2αsCF π 1 + log ξ 1 − τ + O(ξ0) + O � (1 − τ)0� , (7) σ0 = √ 2GF m2βπNC 18s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Finally we study the differential distribution dσ dxB with xB = −q2 2p1·q for the real emission corrections to the process b(p1) + h(q) → b(p2) + g(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Due to the fact that xB is related to the virtuality of one of the propagator we expect that the limit do not commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Indeed we find: lim xB→1 lim ξ→0 1 ¯σ0 dσ dxB = −αsCF π � log ξ 1 − xB + log(1 − xB) 1 − xB + 7 4 1 1 − xB + O(ξ0) + O � (1 − xB)0�� , (8) lim ξ→0 lim xB→1 1 ¯σ0 dσ dxB = −2αsCF π 1 + log ξ 1 − xB + +O(ξ0) + O � (1 − xB)0� , ¯σ0 = π √ 2GF m2NCη −3q2 , η = � 1 + 4ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' 3 Soft Resummation in the Massive Scheme In this section we want to give an explicit expression for the all-order soft resummation of the Higgs decay rate in a b¯b pair at NLL accuracy in the massive scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Since we look at the differential distribution over x, we are in class of process with the so called single-particle inclusive kinematics (see 7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' The main result of 7) is that the resummed expression can be factorized as a product of a soft function times a hard function times a jet function for every massles particle n the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' In our case the resummation formula simplifies considerably there are not massless particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' The resummed result of 7) at NLL, adapted to the process we are considering, reads1 �Γ(N, ξ) = � 1 + αs π C(1)(ξ) + O � α2 s �� e −2 � 1 1/ ¯ N dz z � αs(z2q2) π γ(0) soft(β)+ � αs(z2q2) π �2 γ(1) soft(β)+O(α3 s) � + O � 1 N � , (9) with ¯N = NeγE and γsoft the massive soft anomalous dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' To this logarithmic accuracy we need the two loops expression of the running coupling, the coefficients γ(0) soft, γ(1) soft and C(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' The first order soft anomalous dimension can be obtained from the calculation of one gluon emission in the eikonal limit: γ(0) soft(β) = CF �1 + β2 2β log �1 + β 1 − β � − 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' (10) while the second order was presented in 8)2: γ(1) soft = �K 2 + CA 2 � −1 3 log2 1 − β 1 + β + log 1 − β 1 + β − ζ2 � +(1 + β2) 4β CA � Li2 �(1 − β)2 (1 + β)2 � + 1 3 log2 1 − β 1 + β + ζ2 �� γ(0) soft(β) + CFCA �1 2 + 1 2 log 1 − β 1 + β + 1 3 log2 1 − β 1 + β − (1 + β2)2 8β2 � −Li3 �(1 − β)2 (1 + β)2 � + ζ3 � − (1 + β2) 2β � log 1 − β 1 + β log (1 + β)2 4β − 1 6 log2 1 − β 1 + β − Li2 �(1 − β)2 (1 + β)2 ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' (11) 1We are not so sure about the argument of the running coupling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' since in 7) αs(z2q2) is used,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' on the other hand it seems that in 8) αs(z2m2) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' 2It is worth to mention that there is a mismatch in the literature between 8) and 9) with K = CA � 67 18 − ζ2 � − 5nf 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' The coefficient C(1) is instead process-dependent, as it receives contri- butions from both the end-point of the real emission and from the virtual corrections (computed in the on-shell scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Writing the real emission differential decay rate as: dΓ(R) dx = αsCF π Γ(d) 0 fε � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' q2 µ2 � (1 − x)1+2ϵ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Γ(d) 0 = Γ0 π 5−d 2 2d−3Γ � d−1 2 � � 4µ2 q2β2 � 4−d 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' (12) the coefficient C(1) can be determined using the fact that virtual corrections are proportional to δ(1 − x) and the identity between distributions: fε � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' q2 µ2 � (1 − x)1+2ε = δ(1 − x) � −f0(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' ξ) 2ε + f0(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' ξ) log(1 − 2 � ξ) − 1 2 d dεfε � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' q2 µ2 � ��� ε=0 � + f0(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' ξ) (1 − x)+ + O(ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' (13) Summing up virtual and real contributions we obtain: C(1)(ξ) = CF 2 � − 2γ(0) soft(β) CF � −2 log � 1 − � 1 − β2 � + log m2 q2 + log �1 − β2 4 � + 1 � − 2 + 2L(β) �1 − β2 β � + 1 + β2 β � 1 2L(β) log �1 − β2 4 � + 2L(β)(1 − log β) + 2Li2 �1 − β 1 + β � + L(β)2 + L(β) log 1 − β 2 + 2 3π2 − 1 2 � Li2 � 4β (1 + β)2 � − Li2 � −4β (1 − β)2 �� �� , (14) with L(β) = log � 1+β 1−β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' We note that the non commutativity of the soft and massless limits has conse- quences for the resummed expression in the massive scheme: In the small ξ limit we find: αsC(1)(ξ) = αsCF �1 2 log2 ξ + log ξ + O(ξ0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' We have a double log of the mass in disagreement with DGLAP evolution equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' The problem is that equation (13) does not hold if we perform the massless limit because in this limit f0(1, ξ) is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' In a certain way we can say that double mass logarithms in the soft limit of the massive calculation and double soft logarithms of the massless scheme are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' A well defined expression in the massless limit can be obtained rewriting the differential decay rate as: 1 Γ0 dΓ dx = δ(1 − x) + αs π � CF �f0(x, ξ) 1 − x � + + A(ξ) δ(1 − x) � , (15) The delta coefficient has an expected behaviour for ξ → 0 A(ξ) = CF 3 2 log ξ + O(ξ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' (16) 4 Conclusions We have considered observables with different kinematics in processes involving heavy quarks, and in all processes we have computed NLO corrections taking into account the mass dependence of the square amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' We have underlined that soft and massless do not always commute, in particular in the massless limit the structure of the distributions can radically change because of the presence of double logs of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' We have traced back the origin of this particular behaviour to the interplay between the observable we are computing and the fermionic propagators in the scattering amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' Finally, we have focused on the massive scheme resummation of the process H → b¯b in the soft limit and we have found that within this approach double logarithms of the mass may appear, and the origin of this surprising behaviour can be lead back again to the non commutativity between the large N and small mass limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' An interesting phenomenological study, in the context of heavy-quark calculations, would be com- bine the massive scheme with the massless one where also soft logarithms are resummed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' The merging of the two becomes far from trivial because of the lack of commutativity of the limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' One would like to design an all-order matching scheme that takes into account both the different logarithmic behaviour that arises in the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' 5 Acknowledgements We thank Simone Marzani and Giovanni Ridolfi for the aid in the drafting of this proceeding, which is entirely based on 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQflAfG/content/2301.03985v1.pdf'} +page_content=' 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a/AdE4T4oBgHgl3EQfEwwN/content/tmp_files/2301.04879v1.pdf.txt b/AdE4T4oBgHgl3EQfEwwN/content/tmp_files/2301.04879v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7fdd10a2871678ebb8d762429bfce15373ebe5d5 --- /dev/null +++ b/AdE4T4oBgHgl3EQfEwwN/content/tmp_files/2301.04879v1.pdf.txt @@ -0,0 +1,2238 @@ +January 13, 2023 +Tidal deformations of a binary system +induced by an external Kerr black hole +Filippo Camilloni†, Gianluca Grignani†, Troels Harmark‡, +Roberto Oliveri∗, Marta Orselli† ‡, Daniele Pica† ‡ +† Dipartimento di Fisica e Geologia, Universit`a di Perugia, I.N.F.N. Sezione di Perugia, +Via Pascoli, I-06123 Perugia, Italy +‡ Niels Bohr Institute, Copenhagen University, +Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark +∗ LUTH, Laboratoire Univers et Th´eories, Observatoire de Paris, +CNRS, Universit´e PSL, Universit´e Paris Cit´e, +5 place Jules Janssen, 92190 Meudon, France +Abstract +The dynamics of a binary system moving in the background of a black hole is affected by +tidal forces. In this work, for the Kerr black hole, we derive the electric and magnetic +tidal moments at quadrupole order, where the latter are computed for the first time in +full generality. +We make use of these moments in the scenario of a hierarchical triple +system made of a Kerr black hole and an extreme-mass ratio binary system consisting of +a Schwarzschild black hole and a test particle. We study how the secular dynamics of +the test particle in the binary system is distorted by the presence of tidal forces from a +much larger Kerr black hole. Our treatment includes strong gravitational effects beyond +the post-Newtonian approximation both for the binary system and for the tidal forces since +the binary system is allowed to be close to the event horizon of the Kerr black hole. We +compute the shifts in the physical quantities for the secular dynamics of the test particle +and show that they are gauge-invariant. +In particular, we apply our formalism to the +innermost stable circular orbit for the test particle and to the case of the photon sphere. +Our results are relevant for the astrophysical situation in which the binary system is in the +vicinity of a supermassive black hole. +arXiv:2301.04879v1 [gr-qc] 12 Jan 2023 + +Contents +1 +Introduction +1 +2 +Tidal moments induced by a Kerr black hole +3 +2.1 +Carter’s tetrad +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Marck’s tetrad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.3 +Tidal tensors +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.4 +Electric and magnetic quadrupole moments . . . . . . . . . . . . . . . . . . . . . +7 +3 +Hierarchical triple system +8 +3.1 +Tidally deformed Schwarzschild spacetime +. . . . . . . . . . . . . . . . . . . . . +9 +3.2 +Tidal moments in spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . +10 +4 +Secular dynamics of binary system +13 +4.1 +Secular Hamiltonian of test particle in binary system . . . . . . . . . . . . . . . +13 +4.2 +Special case of circular equatorial geodesic in Kerr background . . . . . . . . . . +15 +5 +Secular shifts for ISCO and photon sphere +16 +5.1 +Gauge invariance of secular observables . . . . . . . . . . . . . . . . . . . . . . . +18 +5.2 +Tidal effects around the ISCO orbit . . . . . . . . . . . . . . . . . . . . . . . . . +19 +5.3 +Tidal effects around the photon sphere . . . . . . . . . . . . . . . . . . . . . . . +20 +6 +Conclusions and outlook +21 +1 +Introduction +The detection of gravitational waves from coalescing binary systems by the LIGO-Virgo-Kagra +collaboration [1–3] has unsealed a new powerful and fascinating way of exploring our universe in +a regime of strong gravitational field. This has made it increasingly relevant to investigate new +types of strong gravitational phenomena analytically, to prepare for future experimental results. +Indeed, with the next generation detectors such as the ground-based Einstein Telescope [4] +and Cosmic Explorer [5], as well as the space-based LISA [6] and TianQin [7], the sensitivity +and frequency band will be greatly expanded. This will make it possible to use black hole binary +systems also as probes of their surrounding environment (see Ref. [8] for a comprehensive review). +Examples of the effect of the environment include the presence of various types of energy and +matter, such as an accretion disc [9–11] or dark matter [12–18]. Another example, relevant for +this paper, is the presence of a third body, such as a nearby supermassive black hole [19–26] +bound to the binary system. +Moreover, the expansion in sensitivity and frequency band will make it possible to detect +signals from new types of sources, such as for example extreme-mass-ratio (EMR) inspiraling +systems. Among these systems, the ones that will typically be detectable in the LISA band +[27, 28], are made of a stellar mass compact object of mass m and a black hole with a much +larger mass M ≫ m, with mass ratios m/M ranging from 10−4 to 10−6. +In this paper we are interested in the dynamical effects of having a binary black hole system +immersed in a curved background spacetime. To access a scenario that at the same time is +realistic, has strong gravitational effects included, and can be treated analytically, we consider +the case of an EMR binary system, i.e. a black hole and a test particle, in the background of a +third, larger black hole, affecting the binary system through tidal forces. +We take the curved background spacetime to be the general case of a Kerr black hole of +mass M∗. Instead the EMR binary system will consist of a Schwarzschild black hole of mass M +1 + +with a test particle of mass m, enabling us to use the tidally deformed Schwarzschild metric of +Refs. [29,30] to describe the EMR binary system. For the test particle we consider it to move +on a geodesic, neglecting higher order effects in m/M such as the self-force. As the size of the +binary system will be set by the scale M, we need M ≪ R where R is the curvature length +scale set by the background Kerr black hole. This ensures that the effects of the background +can be described through tidal forces, with the condition M ≪ R known as the small-tide +approximation [30]. +We will consider the quadrupole approximation to the tidal forces, being the leading order +in M/R. This means we can consider the EMR binary as moving on a geodesic of the Kerr +black hole geometry. A particularly interesting regime is when M∗ ≫ M thus corresponding +to a hierarchical three body system. In this case, the binary system can be close to the event +horizon of the Kerr black hole, even while the small-tide approximation is respected. +Our setup is inspired by that of Ref. [25], while at the same time being a significant ex- +tension. Their setup was restricted to a Schwarzschild black hole as the third body, and the +EMR binary system was assumed to be at a large distance. Instead, we are able to consider the +strong gravitational effects on the binary system when it moves in close vicinity to a Kerr black +hole. This also means that we need to consider more carefully the relative orientation of the +EMR binary system relative to the Kerr black hole. This is accomplished by introducing two +independent rotation angles. Moreover, it is important to note that in our setup we are able to +capture strong gravitational effects arising from curved spacetime, in contrast with most of the +extensive literature on three body systems [31–35], as those works employ the approximation +that all three bodies are small relative to their mutual distances. +A significant part of our paper concerns the careful computation of the general quadrupole +tidal forces due to the Kerr black hole, as these constitute the forces that can affect the binary +system in our setup. These forces are described by the tidal tensors Cij and Cijk. The rank-2 +tidal tensors Cij were previously computed for a generic value of the Kerr angle ˆθ in a seminal +paper by Marck [36], where he constructed the orthonormal tetrad that is parallel-transported +along an arbitrary time-like geodesic in the Kerr spacetime. From the rank-2 tidal tensors Cij +one can then compute the “electric” quadrupole moments Eij, which can be considered as “mass +moments” produced by gravitational forces external to a certain region. +A primary result of this paper, is the derivation of the general form of the rank-3 tidal +tensors Cijk for all values of the angle ˆθ in the Kerr spacetime. This generalizes the results of +Ref. [37] (later confirmed in Ref. [38]), where the tidal tensors Cijk were obtained only for the +specific value ˆθ = π/2, namely in the equatorial plane of the Kerr spacetime. From the rank-3 +tidal tensors Cijk we moreover derive the “magnetic” quadrupole moments Bij, which can be +considered as external “current moments” and generate velocity-dependent tidal forces on test +bodies. This is another original result of this paper. +We apply these tidal electric and magnetic quadrupole moments to the case described above, +with an EMR binary system following a geodesic in the Kerr background. The effects induced by +the tidal fields can be studied by computing the Hamiltonian of a test particle (the object of mass +m) in the tidally deformed Schwarzschild spacetime. Specifically, starting from a circular orbit +in the unperturbed Schwarzschild spacetime, we find that the geodesics in the tidally deformed +spacetime acquire a small eccentricity proportional to the deformation parameter. The quasi- +circular dynamics in the perturbed spacetime is governed by a secular Hamiltonian, which keeps +into account the effects of the tidal deformation on circular orbits. It can be written as a sum of +the unperturbed Hamiltonian in the Schwarzschild spacetime and an interaction term of order +∼ M/M∗, which allows for example to compute perturbatively the effects of tides on the location +and properties of the Innermost Stable Circular Orbit (ISCO) and of the photon sphere. +Using the tidal moments we computed, we derive the effects of tides on the frequency, radius, +energy and angular momentum of the ISCO of the binary system, by computing the shifts +2 + +induced by the small tides on these physical quantities. 1 +The case of tides generated by a +Schwarzschild black hole was studied in Ref. [25, 40]. Here we derive the shifts in the case of +tides induced by the Kerr geometry and we derive the expression of the parameter η entering +these shifts. We find that η depends on the spin of the Kerr black hole, the Carter constant +K, the Kerr angle ˆθ and the Boyer-Lindquist radius ˆr at which the black hole of mass M is +located in the Kerr spacetime geometry. More generally, our result does not rely on the specific +nature of the third body responsible for the tides. Indeed, the tidal parameter η in the secular +Hamiltonian is shown to be proportional to the secular average of the scalar part of the electric +tidal moment. This result holds in the quadrupole and in the secular approximation. We provide +an expression for η in terms of arbitrary tides and specialize it to the case of a Kerr black hole. +The paper is organized as follows. In Sec 2, we compute the tidal moments induced by a Kerr +black hole. Following Ref. [36], we first recover the already known expression for the electric +tidal moments and then we derive the most general expressions for the magnetic components of +the tidal moments, generalising the computation done in Ref. [37]. In Sec. 3, we introduce the +hierarchical triple system that we analyse in this paper. We write down the metric for a tidally +deformed Schwarzschild black hole up to the quadrupole order. We moreover write down the +explicit expression for the quadrupole electric and magnetic moments and we introduce the Euler +angles which allow us to study any possible orientation of the binary system. In Sec. 4, we focus +on the secular dynamics of the binary system in order to understand how the parameters which +specify the orbits of the test particle around the Schwarzschild black hole, such as energy and +angular momentum, are shifted by the tidal fields. In Sec. 5, we apply the results of the previous +sections to the case in which the test particle is moving along the ISCO of the Schwarzschild +black hole. +In addition, we extend our computation also to the case of a massless particle +studying how the photon sphere is deformed by the tidal fields. We furthermore discuss the +gauge invariance of our results. Finally, Sec 6 contains our concluding remarks. +Notation: +Throughout this paper Greek indices run from 0 to 3, Latin lower-case indices +(i, j, k, ...) run from 1 to 3, Latin upper-case indices (A, B, C, ...) label spherical coordinates. +Indices in round brackets ((a), (b), (c), ...) label tensor components in the Carter’s tetrad. Sym- +metric and tracefree (STF) tensors are denoted by angular brackets over their indices, e.g., +T⟨ij⟩ = T(ij) − 1 +3δijTklδkl. Hatted coordinates (ˆt, ˆr, ˆθ, ˆφ) are employed for the Kerr spacetime. +Schwarzschild coordinates, used for the binary system, are instead denoted as (t, r, θ, φ). We use +geometrized units with G = c = 1 and the Minkowski metric signature is η = diag(−1, 1, 1, 1). +2 +Tidal moments induced by a Kerr black hole +In this section we derive the general quadrupole tidal moments for geodesic motion around a +Kerr black hole which we will use in Sections 3-5. In Sec. 2.1 we define the Carter’s tetrad, +in terms of which the curvature tensor simplifies. In Sec. 2.2 we present an alternative inertial +frame [36], parallel-transported along a generic geodesic in the Kerr spacetime, here called the +Marck’s tetrad. +This is the most suitable reference frame in which it is possible to extract +analytic information concerning the tidal effects induced by the Kerr geometry on a system +moving along its geodesics. The tidal effects are encoded in the rank-2 and rank-3 tidal tensors +and in the set of electric and magnetic tidal moments, explicitly given in Sec. 2.3 and 2.4 at the +quadrupole order. The expressions of the rank-3 tidal tensor and of the magnetic quadrupole +moments outside the Kerr equatorial plane are derived here for the first time. +1See Ref. [39] for similar treatments in the context of the self-force approximation. +3 + +2.1 +Carter’s tetrad +The Kerr metric for a rotating black hole of mass M∗ and spin J∗, in Boyer-Lindquist (BL) +coordinates ˆxµ = (ˆt, ˆr, ˆθ, ˆφ) takes the form +dˆs2 = − +� +1 − 2M∗ˆr +Σ +� +dˆt2 − 4M∗ˆr +Σ +a sin2 ˆθ dˆt dˆφ + A +Σ sin2 ˆθ dˆφ2 + Σ +∆dˆr2 + Σdˆθ2 , +(2.1) +where a = J∗/M∗ is the specific angular momentum and +Σ = ˆr2 + a2 cos2 ˆθ, +∆ = ˆr2 − 2M∗ˆr + a2, +A = (ˆr2 + a2)2 − a2∆ sin2 ˆθ . +(2.2) +We are interested in considering time-like geodesics around a Kerr black hole, specified by +three constants of motion: the energy per unit mass ˆE, the angular momentum per unit mass +ˆL and the Carter constant K. More specifically, the first integrals of the equations of motion +read [41] +˙ˆt = A ˆE − 2M∗ˆraˆL +∆Σ +, +˙ˆr2 = +� ˆE(ˆr2 + a2) − aˆL +Σ +�2 +− ∆ +Σ2(ˆr2 + K) , +˙ˆθ2 = 1 +Σ2 +� +K − a2 cos ˆθ − +� +a ˆE sin ˆθ − +ˆL +sin ˆθ +�2� +, +˙ˆφ = 1 +∆ +� +2M∗ˆra ˆE +Σ ++ +� +1 − 2M∗ˆr +Σ +� +ˆL +sin2 ˆθ +� +, +(2.3) +where the dot denotes differentiation with respect to the proper time τ. +A convenient tetrad for the Kerr geometry (2.1), such that dˆs2 = η(a)(b)ω(a)ω(b), was intro- +duced in Ref. [42] and reads +ω(0) = +� +∆ +Σ +� +dˆt − a sin2 ˆθdˆφ +� +, +ω(1) = +� +Σ +∆dˆr , +ω(2) = +√ +Σdˆθ , +ω(3) = sin ˆθ +√ +Σ +� +adˆt − (ˆr2 + a2)dˆφ +� +. +(2.4) +We dub this tetrad, the Carter’s tetrad. The curvature 2-form +Ω(a)(b) = 1 +2C(a)(b)(c)(d)ω(c) ∧ ω(d) , +(2.5) +with C(a)(b)(c)(d) being the components of the Weyl tensor, (Cµνρσ = Rµνρσ for the Kerr geometry +(2.1)), projected along the Carter’s tetrad with the inverses of Eq. (2.4), ωµ +(a), C(a)(b)(c)(d) = +Cµνρσ ωµ +(a)ων +(b)ωρ +(c)ωσ +(d), reads [36,43] +Ω(0)(1) = 2I1 ω(0) ∧ ω(1) + 2I2 ω(2) ∧ ω(3) , +Ω(0)(2) = −I1 ω(0) ∧ ω(2) + I2 ω(1) ∧ ω(3) , +Ω(0)(3) = −I1 ω(0) ∧ ω(3) − I2 ω(1) ∧ ω(2) , +Ω(1)(2) = −I1 ω(1) ∧ ω(2) + I2 ω(0) ∧ ω(3) , +Ω(1)(3) = −I1 ω(1) ∧ ω(3) − I2 ω(0) ∧ ω(2) , +Ω(2)(3) = 2I1 ω(2) ∧ ω(3) − 2I2 ω(0) ∧ ω(1) , +(2.6) +where +I1 = M∗ˆr +Σ3 +� +ˆr2 − 3a2 cos2 ˆθ +� +, +I2 = aM∗ cos ˆθ +Σ3 +� +3ˆr2 − a2 cos2 ˆθ +� +. +(2.7) +4 + +2.2 +Marck’s tetrad +The orthonormal tetrad λ(a) = +� +λ(a) +0 , λ(a) +1 , λ(a) +2 , λ(a) +3 +� +that is parallel-transported along an arbi- +trary time-like geodesic was constructed in Ref. [36]. The tetrad component λ(a) +0 +is a time-like +unit vector tangent to the geodesics and λ(a) +i +are space-like unit vectors. They satisfy the fol- +lowing conditions +η(a)(b) λ(a) +α λ(b) +β = ηαβ , +λµ +0∇µλν +α = 0 , +(2.8) +where λµ +α = ωµ +(a)λ(a) +α +and α, β = {0, 1, 2, 3} are the labels of the components of the tetrad. The +first relation in Eq. (2.8) is the orthonormal condition, the second one is the parallel-transported +requirement that implies the tetrad frame is inertial. Their explicit expressions in terms of the +metric functions and the constants of motion are [36] 2 +λ(a) +0 += +� +1 +√ +∆Σ +� +ˆE(ˆr2 + a2) − aˆL +� +, +� +Σ +∆ +˙ˆr, +√ +Σ ˙ˆθ, +1 +√ +Σ +� +a ˆE sin ˆθ − +ˆL +sin ˆθ +�� +, +λ(a) +1 += ˜λ(a) +1 cos Ψ − ˜λ(a) +2 sin Ψ , +λ(a) +2 += ˜λ(a) +1 sin Ψ + ˜λ(a) +2 cos Ψ , +λ(a) +3 += +1 +√ +K +� +a cos ˆθλ(1) +0 , a cos ˆθλ(0) +0 , −ˆrλ(3) +0 , ˆrλ(2) +0 ) +� +, +(2.9) +where +˜λ(a) +1 += +1 +√ +K +� +T +S +� +ˆrλ(1) +0 , ˆrλ(0) +0 , S +T a cos ˆθλ(3) +0 , −S +T a cos ˆθλ(2) +0 +� +, +˜λ(a) +2 += +� +T +S +� +λ(0) +0 , λ(1) +0 , S +T λ(2) +0 , S +T λ(3) +0 +� +, +(2.10) +and +S = ˆr2 + K , +T = K − a2 cos2 ˆθ . +(2.11) +Notice the identity Σ = S − T. +In the second and third tetrad component of Eq. (2.9), we rotated the vectors ˜λ(a) +1 +and ˜λ(a) +2 +of an angle Ψ. This is necessary in order to ensure that the tetrad λ(a) = +� +λ(a) +0 , λ(a) +1 , λ(a) +2 , λ(a) +3 +� +is parallel-transported along the geodesic motion [36]. In particular Ψ is an angle depending on +the proper time along the Kerr geodesic. The equation satisfied by Ψ was derived in Ref. [36] +and reads +˙Ψ = +√ +K +Σ +� ˆE(ˆr2 + a2) − aˆL +S ++ a +ˆL − a ˆE sin2 ˆθ +T +� +. +(2.12) +A solution for this first order differential equation was provided in Ref. [36] and, more explicitly +in terms of the Mino time, in Ref. [44]. +2.3 +Tidal tensors +Tidal effects on test particles moving in the neighborhood of a geodesic in Kerr spacetime are best +computed by evaluating the Weyl tensor on the parallel-transported tetrad λ(a) (see Eq. (2.9)) +with λ(a) +0 +being the four-velocity. The explicit expressions for the tidal tensors are obtained once +the Weyl tensor Cµνρσ is evaluated on the Kerr geodesic. In order to compute the electric and +2We rename λ(a) +2 +and ˜λ(a) +3 +in Ref. [36] with our λ(a) +3 +and ˜λ(a) +2 , respectively. It is also important to stress that +all the components of the space-like vectors λ(a) +i +can be written in terms of λ(a) +0 . +5 + +magnetic quadrupole moments, we first need the following components of the rank-2 and rank-3 +tidal tensors in the basis of the tetrad λ(a) [30,36] +Cij ≡ C(a)(b)(c)(d)λ(a) +0 λ(b) +i λ(c) +0 λ(d) +j +, +Cijk ≡ C(a)(b)(c)(d)λ(a) +0 λ(b) +i λ(c) +j λ(d) +k +, +(2.13) +where we recall that C(a)(b)(c)(d) = Cµνρσ ωµ(a)ων(b)ωρ(c)ωσ(d). Note that, as a consequence of +the symmetries of the Weyl tensor, Cij is an STF tensor, whereas Cijk is trace-free and anti- +symmetric in (j, k) by definition. +Morevoer, it obeys the condition Cijk + Cjki + Ckij = 0, +implying that Cijk − Cjik = −Ckij and Cijk − Ckji = −Cjki. +We compute now the explicit expression for the components of the Weyl tensor that are +relevant for the calculations of the electric and magnetic quadrupole moments. Our expressions +are valid for arbitrary time-like geodesics in the Kerr black hole spacetime. The Cij read +C11 = +� +1 − 3ST +KΣ2(ˆr2 − a2 cos2 ˆθ) cos2 Ψ +� +I1 + 6ST +KΣ2aˆr cos ˆθ cos2 ΨI2 , +C12 = 3ST +KΣ2 +� +− +� +ˆr2 − a2 cos2 ˆθ +� +I1 + 2aˆr cos ˆθI2 +� +sin Ψ cos Ψ , +C13 = −3 +√ +ST +KΣ2 +� +aˆr cos ˆθ(S + T)I1 + +� +ˆr2T − a2S cos2 ˆθ +� +I2 +� +cos Ψ , +C22 = +� +1 − 3ST +KΣ2(ˆr2 − a2 cos2 θ) sin2 Ψ +� +I1 + 6ST +KΣ2aˆr cos ˆθ sin2 ΨI2 , +C23 = −3 +√ +ST +KΣ2 +� +aˆr cos ˆθ(S + T)I1 + +� +ˆr2T − a2S cos2 ˆθ +� +I2 +� +sin Ψ , +C33 = +� +1 + +3 +KΣ2(ˆr2T 2 − a2S2 cos2 ˆθ) +� +I1 − 6ST +KΣ2aˆr cos ˆθI2 . +(2.14) +Note that Cij was already computed in Ref. [36] (with the label 2 renamed with 3 in this paper). +As a new result, we provide also the general expression for the non-vanishing components of +the rank-3 tidal tensor Cijk that enter the calculation of the magnetic moments which will be +done in the next subsection. The non-vanishing components are given by +C112 = 3 +√ +ST +KΣ2 +�� +ˆr2T − a2S cos2 ˆθ +� +I1 − aˆr cos ˆθ(S + T)I2 +� +cos Ψ , +C113 = 3ST +KΣ2 +� +2aˆr cos ˆθI1 + +� +ˆr2 − a2 cos2 ˆθ +� +I2 +� +sin Ψ cos Ψ , +C123 = − 6ST +KΣ2aˆr cos ˆθ cos2 ΨI1 ++ +1 +KΣ2 +�� +ˆr2T + a2S cos2 ˆθ +� +(S − T) − 3ST +� +ˆr2 − a2 cos2 ˆθ +� +cos2 Ψ +� +I2 , +C212 = 3 +√ +ST +KΣ2 +�� +ˆr2T − a2S cos2 ˆθ +� +I1 − aˆr cos ˆθ(S + T)I2 +� +sin Ψ , +C213 = 6ST +KΣ2aˆr cos ˆθ sin2 ΨI1 ++ +1 +KΣ2 +� +ˆr2T(2S + T) − a2 cos2 ˆθS(S + 2T) − 3ST +� +ˆr2 − a2 cos2 ˆθ +� +cos2 Ψ +� +I2, +C312 = 6ST +KΣ2aˆr cos ˆθI1 + +1 +KΣ2 +� +ˆr2T(S + 2T) − a2 cos2 ˆθS(2S + T) +� +I2 . +(2.15) +In addition, we observe that C223 = −C113, C312 = C213 − C123, C313 = −C212, C323 = C112. +6 + +If we specialize to geodesics in the equatorial plane ˆθ = π/2 of the Kerr black hole, the explicit +expressions for the tidal tensors simplify considerably. We get, in agreement with Refs. [36,37,45], +C11 = +� +1 − 3 +� +1 + K +ˆr2 +� +cos2 Ψ +� M∗ +ˆr3 , +C22 = +� +1 − 3 +� +1 + K +ˆr2 +� +sin2 Ψ +� M∗ +ˆr3 , +C12 = −3 +� +1 + K +ˆr2 +� M∗ +ˆr3 cos Ψ sin Ψ , +C33 = +� +1 + 3K +ˆr2 +� M∗ +ˆr3 , +(2.16) +and, for the rank-3 tidal tensor (in agreement with Ref. [37] and Ref. [38]), +C121 = −C112 = C332 = −C323 = −3M∗ +√ +K +ˆr4 +� +1 + K +ˆr2 cos Ψ , +C221 = −C212 = C313 = −C331 = −3M∗ +√ +K +ˆr4 +� +1 + K +ˆr2 sin Ψ , +(2.17) +where, for geodesics in the equatorial plane of the Kerr spacetime, the following expressions +hold [46] +ˆE = +ˆr3/2 − 2M∗ˆr1/2 + σaM 1/2 +∗ +ˆr3/4 +� +ˆr3/2 − 3M∗ˆr1/2 + 2σaM 1/2 +∗ +, +ˆL = +σM 1/2 +∗ +� +ˆr2 + a2 − 2σa M 1/2 +∗ +ˆr1/2� +ˆr3/4 +� +ˆr3/2 − 3M∗ˆr1/2 + 2σaM 1/2 +∗ +, +K = +� +a ˆE − ˆL +�2 +, +˙Ψ = +√ +K +ˆr2 + K +� +ˆE − +a +a ˆE − ˆL +� += σ +� +M∗ +ˆr3 . +(2.18) +Above we introduced the parameter σ = ±1 that allows one to distinguish between prograde +(+) and retrograde (−) orbits. A thorough analysis of the dynamics in the equatorial plane will +be given in Sec. 4.2. +2.4 +Electric and magnetic quadrupole moments +The electric and magnetic quadrupole moments in Cartesian coordinates are defined as [30] +Eij ≡ Cij , +Bij ≡ −1 +2ϵkl⟨iC +kl +j⟩ +, +(2.19) +with ϵijk the three-dimensional Levi-Civita symbol with ϵ123 = +1. We raise and lower Cartesian +indices (i, j, k, ...) with the Kronecker delta δij. Being STF tensors, both the electric Eij and +the magnetic Bij tensors have each five independent components thus, together, they account +for the ten independent components of the Weyl tensor. In particular, the magnetic quadrupole +moments in terms of the components of the rank-3 tidal tensor, read +B11 = −C123 , +B12 = C113 , +B13 = −C112 , +B22 = C213 , +B23 = −C212 , +B33 = C123 − C213 , +(2.20) +7 + +where we used that C223 = −C113, C312 = C213 − C123, C313 = −C212 and C323 = C112. +It is far more useful to decompose the rank-2 and rank-3 tensors by means of their irreducible +representations of SO(3). Following Ref. [30], we introduce the radial unit vector Ωi ≡ xi/r, +with r = +� +δijxixj being the Euclidean radius representing the distance from the geodesic. The +projector to the space orthogonal to Ωi is given by γij = δij − ΩiΩj. The electric quadrupole +moment Eij decomposes as follows +Eij = Eq +� +ΩiΩj − 1 +2γij +� ++ 2Eq +(iΩj) + 1 +2Eq +⟨ij⟩ , +(2.21) +where the scalar Eq, the transverse vector Eq +i (i.e. ΩiEq +i = 0) and the transverse STF tensor Eq +⟨ij⟩ +are given by +Eq ≡ ΩiEijΩj = −γijEij , +Eq +i ≡ γ j +i EjkΩk , +Eq +⟨ij⟩ ≡ 2γ k +i γ l +j Ekl − Eklγklγij = 2γ k +i γ l +j Ekl + Eqγij . +(2.22) +Similarly, for the magnetic quadrupole moment Bij, one has 3 +Bij = ϵlk +(i +� +Bq +l +� +Ωj)Ωk − γj)k +� ++ 1 +4 +� +Bq +⟨j)l⟩Ωk − Bq +⟨j)k⟩Ωl +�� +, +(2.25) +with symmetrization w.r.t. the indices (i, j) and STF w.r.t. the indices ⟨jl⟩ and ⟨jk⟩. The +transverse vector Bq +i and the transverse STF tensor Bq +⟨ij⟩ are +Bq +i ≡ ϵijkΩjBk +lΩl , +Bq +⟨ij⟩ ≡ 2ϵkl(iγm +j)ΩkBl +m . +(2.26) +3 +Hierarchical triple system +In this section we apply the formalism introduced in Sec. 2 to an EMR binary system moving +in the background of a Kerr black hole. The EMR binary system consists of a Schwarzschild +black hole of mass M along with a test-particle of mass m ≪ M. We assume that the black +hole with mass M∗ moves slowly relatively to the EMR binary system (M, m) and that one can +describe the effect on the binary system to a good approximation by taking into account only +the quadrupole tidal moments induced by M∗. This is valid provided +M 2 ≪ +ˆr3 +M + M∗ +, +(3.1) +where ˆr is the Boyer-Lindquist radius at which M is located in the Kerr spacetime geometry +induced by M∗ [30]. This arises from having two widely separated scales: one scale is the length +scale of the Schwarzschild black hole M, the other is the curvature length scale R induced by +3We used the decomposition of the rank-3 tidal tensor +Cijk = Bq +k (ΩiΩj − γij) − Bq +j (ΩiΩk − γik) + 1 +2 +� +Bq +⟨ik⟩Ωj − Bq +⟨ij⟩Ωk +� +, +(2.23) +with the inverse relations given by +Bq +i = CjkiΩjΩk , +Bq +⟨ij⟩ = 2ΩkClk(iγl +j) . +(2.24) +8 + +the Kerr black hole M∗ at the location of M. We then require M ≪ R. This is called small-tide +approximation [30] and it makes it possible to describe the motion of the binary system (M, m) +in the external Kerr geometry, ensuring that the tidal deformation is weak. We can therefore +describe the influence of M∗ on the binary system (M, m) using, to a first approximation, the +quadrupole tidal moments induced by the Kerr black hole itself. Since R ∼ +� +ˆr3/(M + M∗) +this, combined with the condition M ≪ R, gives the condition (3.1). +One natural way to achieve the condition (3.1) is that M is much smaller than M∗, here +called the hierarchical regime +M ≪ M∗ . +(3.2) +This implies (3.1) since ˆr ≳ M∗. In this case we have a hierarchical triple system of black holes +m ≪ M ≪ M∗ (note that one could imagine both M and M∗ being a supermassive black hole, +but with a mass hierarchy). The hierarchical triple system is the case that we shall primarily +consider in this paper, since the dynamics of the triple system in general will depend on the full +expressions of the quadrupole tidal moments of the Kerr black hole M∗. +Another way to achieve the condition (3.1) is the case where M and M∗ are widely separated, +here called the weak field regime +M∗ ≪ ˆr , +(3.3) +assuming as well that M ≲ M∗. This means one can consider two black holes M and M∗ of +similar magnitude. In this case the expression of the tidal moments induced by the Kerr black +hole simplifies considerably [25] due to the fact that frame-dragging effects induced by the Kerr +black hole can be neglected (see discussion around and below Eq. (4.8) for further detail). +It is also important to consider the time scales involved in our approximation. For simplicity, +we consider the binary system having an orbit of m around the Schwarzschild black hole of mass +M such that r = O(M). Then the time scale of the binary system is simply τbinary = O(M). +Assuming ˆr = O(M∗) the time-scale associated with the orbit around the Kerr black hole of mass +M∗ is τkerr = O(M∗). Indeed, one can see explicitly from Eq. (2.12) that we have ˙Ψ = O(1/M∗), +which sets the rate of change of the angle Ψ. Thus, in the hierarchical regime (3.2), we have +τkerr ≫ τbinary, which means that we can assume that the quadrupole moments and Ψ do not +vary with time. Moreover, in the weak field regime (3.3), the time scale for the orbit around +the Kerr black hole is even larger τkerr ≫ M∗ as the velocity will be non-relativistic. Thus, even +if M is of same order as M∗, we find that τkerr ≫ τbinary, and we can again neglect the time +dependence of Ψ and of the quadrupole moments.4 +3.1 +Tidally deformed Schwarzschild spacetime +We can describe the black hole with mass M in the binary system using the tidally deformed +Schwarzschild metric [30]. Concretely, we add to the background metric ¯gµν a tidal perturbation +hµν +ds2 = ¯gµνdxµdxν + hµνdxµdxν , +(3.4) +where the tidal perturbation hµν is computed up to the first order in the small-tide approxima- +tion. The background geometry (in spherical coordinates) is +¯gµνdxµdxν = −fdt2 + dr2 +f ++ r2ΩABdθAdθB , +(3.5) +with f = 1 − 2M/r and M being the black hole mass, θA = (θ, φ) and ΩABdθAdθB = dθ2 + +sin2 θdφ2 being the metric of the unit sphere. By only retaining the quadrupole order terms in +4A more general analysis can also take into account the regime M ≪ r ≪ R for which τbinary = O( +� +r3/M). +9 + +the tidal deformation hµν, one gets +hµνdxµdxν = −r2Eq (fdt + dr)2 − 4 +3r3 (Eq +A − Bq +A) (fdt + dr) dθA +− 1 +3r4 +�� +1 − 2M 2 +r2 +� +Eq +AB − +� +1 − 6M 2 +r2 +� +Bq +AB +� +dθAdθB. +(3.6) +The quadrupole moments are decomposed into the scalar Eq, vector Eq +A, Bq +A and tensor Eq +AB, +Bq +AB components, following the decomposition in Eqs. (2.21)-(2.25), and are written in spherical +coordinates. 5 +For an accurate description of our triple system, it is useful to identify the relative orientation +between the orbital plane of the Kerr black hole – responsible for the tidal deformation – and +the orbital plane where the dynamics of the EMR binary system (M, m) takes place; see Fig. 1 +illustrating four possible configurations in the special case when M∗ is a Schwarzschild black hole +and the binary system is moving on a circular geodesic. To describe an arbitrary configuration, +one first introduces the unit directional vector +Ωi = (cos φ sin θ, sin φ sin θ, cos θ) , +(3.7) +centered in the Schwarzschild black hole of mass M, and attached to the reference frame of +the EMR system (M, m). +One then sets, without loss of generality, the polar angle in the +Schwarzschild reference system θ = π/2: this is because the orbital motion takes place on an +orbital plane and we set it to be the equatorial plane. Any arbitrary orientation is therefore +given by performing a rotation on the unit vector in Eq. (3.7), namely, +⃗Ω′ = RχRβRα · ⃗Ω , +(3.8) +with the Euler rotational matrices +Rα = +� +� +cos α +sin α +0 +− sin α +cos α +0 +0 +0 +1 +� +� , +Rβ = +� +� +1 +0 +0 +0 +cos β +sin β +0 +− sin β +cos β +� +� , +Rχ = +� +� +cos χ +sin χ +0 +− sin χ +cos χ +0 +0 +0 +1 +� +� . +(3.9) +Note that Eq. (3.8) is only one among the 12 possible permutations of Euler matrices. Further- +more, since we aim at describing an equatorial orbit in the binary system, it turns out that one +of the Euler angle – α in our convention – can always be reabsorbed by a redefinition of the +Schwarzschild azimuthal angle φ → φ + α. As a consequence, any orientation of a Schwarzschild +orbit with respect to the Kerr perturber is specified only by the two angles β and χ. +3.2 +Tidal moments in spherical coordinates +The tidal moments also depend on the relative configuration between the binary system (M, m) +and the Kerr pertuber. Here, we compute the explicit expression of the tidal quadrupole moments +5For the sake of completeness, we write the change of coordinates from Cartesian to spherical coordinates: +Eq +i dxi = ∂xi +∂xA Eq +i dxA = Eq +θ (rdθ) + Eq +φ(rdφ) , +Eq +⟨ij⟩dxi ⊗ dxj = ∂xi +∂xA +∂xj +∂xB Eq +⟨ij⟩dxA ⊗ dxB = Eq +θθ(rdθ)2 + 2Eq +θφr2dθdφ + Eq +φφ(rdφ)2 . +Similar considerations apply to the magnetic multipole moments Bq +i and Bq +⟨ij⟩. +10 + +I. Orthogonal Configuration +β = 0, χ = 0 +II. Radial Configuration +β = π +2, χ = − π +2 +III. Tangential Configuration +β = − π +2, χ = 0 +IV. Arbitrary Configuration +β = − π +4, χ = 5π +6 +Figure 1: For illustrative purposes, we show four possible configurations for a hierarchical +three-body system M∗ ≫ M ≫ m in the special case for which the perturber +M∗ is a Schwarzschild black hole and the EMR binary system (M, m) is parallel- +transported around a circular geodesic around M∗, whose orbital plane is depicted +in gray and terminates at the ISCO. These configurations are altered significantly +in more general cases with a Kerr perturber or non-circular geodesics. The names +of the configurations refer to the orientation of the orbital angular momentum +L of the binary system with respect to the gray orbital plane. The grey curve +represents the orbit around M∗. The blue orbit marks a conventional “initial” +orthogonal configuration for the binary system reference frame, with the Cartesian +axis oriented according to the parallel transported tetrad (panel I). The red orbits +in panels II, III and IV are obtained by Euler rotations with angles written in the +bottom-left of each panel. +associated to an arbitrary configuration. We recall that we set θ = π/2 because we start with an +equatorial orbit around the Schwarzschild black hole. In Fig. 1 we have illustrated this and other +configurations obtained by Euler rotations in the special case for which M∗ is a Schwarzschild +black hole and the binary system moves on a circular geodesic. In spherical coordinates, the +decomposition of the electric quadrupole moment in its scalar, transverse vector and STF tensor +components is given by Eq. (2.22), where the unit directional vector Ωi is now replaced by the +more general Ω′i defined in Eq. (3.8). +11 + +M* +M +M, +M*The electric quadrupole moments read as +Eq = −1 +8 +� +C33 + T + +2 + T + +4 +� ++ 1 +8 +� +4T + +3 sin 2φ − +� +3(C33 + T + +2 ) − T + +4 +� +cos 2φ +� +, +Eq +θ = 1 +4 +� +2T − +3 cos φ − T − +4 sin φ +� +, +Eq +φ = 1 +8 +� +4T + +3 cos 2φ + +� +3(C33 + T + +2 ) − T + +4 +� +sin 2φ +� +, +Eq +θθ = −Eq +φφ = Eq + 1 +2 +� +C33 + T + +2 + T + +4 +� +, +Eq +θφ = −1 +2 +� +2T − +3 sin φ + T − +4 cos φ +� +, +(3.10) +where we defined the following rotations around χ of the components of Cij +T + +1 = C23 cos χ + C13 sin χ , +T − +1 = C23 sin χ − C13 cos χ , +T + +2 = 2C12 sin 2χ + (2C22 + C33) cos 2χ , +T − +2 = 2C12 cos 2χ − (2C22 + C33) sin 2χ +(3.11) +and the rotations around β of T ± +1,2 +T + +3 = 2T − +1 sin β + T − +2 cos β , +T − +3 = 2T − +1 cos β − T − +2 sin β , +T + +4 = 4T + +1 sin 2β + (3C33 − T + +2 ) cos 2β , +T − +4 = 4T + +1 cos 2β − (3C33 − T + +2 ) sin 2β . +(3.12) +Similarly for the magnetic quadrupole moments, whose decomposition is given in Eq. (2.26), +we find that +Bq +θ = 1 +8 +� +4S+ +3 cos 2φ + +� +3(C312 − S+ +2 ) − S+ +4 +� +sin 2φ +� +, +Bq +φ = −1 +4 +� +2S− +3 cos φ − S− +4 sin φ +� +, +Bq +θθ = −Bq +φφ = −1 +2 +� +2S− +3 sin φ + S− +4 cos φ +� +, +Bq +θφ = −3 +8 +� +C312 − S+ +2 + S+ +4 +� +− 1 +8 +� +4S+ +3 sin 2φ − +� +3(C312 − S+ +2 ) − S+ +4 +� +cos 2φ +� +, +(3.13) +where we defined the rotations around χ of the components of Cijk +S+ +1 = C212 cos χ + C112 sin χ , +S− +1 = C212 sin χ − C112 cos χ , +S+ +2 = 2C113 sin 2χ + (C123 + C213) cos 2χ , +S− +2 = 2C113 cos 2χ − (C123 + C213) sin 2χ +(3.14) +and the rotations around β of S± +1,2 +S+ +3 = 2S− +1 sin β − S− +2 cos β , +S− +3 = 2S− +1 cos β + S− +2 sin β , +S+ +4 = 4S+ +1 sin 2β + (3C312 + S+ +2 ) cos 2β , +S− +4 = 4S+ +1 cos 2β − (3C312 + S+ +2 ) sin 2β . +(3.15) +12 + +The structure of the tidal quadrupole moments (3.10) and (3.13) is the following: the tidal +deformations sourced by a generic third body over the EMR binary system (M, m) are fully +encoded in the tidal tensors Cij and Cijk, while the angles β and χ, parametrizing the relative +orientation between the third body and the binary system, affect the tidal effects over the binary +system. We remark that the above expressions of the tidal quadrupole moments are general, and +can also be employed to model environmental effects in numerical works. In the specific case of +a Kerr black hole as a third body responsible for the tidal deformations, the explicit expressions +of the tidal tensors Cij and Cijk are given, respectively, in Eqs. (2.14) and (2.15). +We anticipate here another property of the tidal quadrupole moments. As we shall see in +the next section, it is often useful to define the secular average over the azimuthal angle φ. The +explicit dependence of the tidal quadrupole moments (3.10) and (3.13) implies that only Eq (and +Eq +θθ = −Eq +φφ) as well as Bq +θφ are relevant for physical observables upon secular averaging. +4 +Secular dynamics of binary system +In this section we focus on the secular dynamics of the binary system (M, m), i.e. the dynamics +of the binary system after a large number of orbits of the test particle of mass m, and analyze +how it is affected by the tidal fields induced by the Kerr perturber of mass M∗, in the hierarchical +regime m ≪ M ≪ M∗. More specifically our goal is to understand how the orbital parameters +of the test particle around the Schwarzschild black hole, such as the energy or the angular +momentum, are shifted by the presence of an external tidal field. +4.1 +Secular Hamiltonian of test particle in binary system +Following the setup of the previous section, we focus on the orbital motion of the object of mass +m, approximated as a test particle, taking place on the equatorial plane of the Schwarzschild +black hole. This amounts to set θ = π/2. We approximate the four-velocity as +uµ ≃ ¯uµ + uµ +(1) , +(4.1) +where ¯uµ is the 4-velocity of the unperturbed bound orbit, that can be taken as circular or elliptic, +and uµ +(1) is the leading correction due to the tidal perturbation hµν. In this work, we focus on +perturbations of circular orbits ¯uµ = ( ¯E/f, 0, 0, ¯L/r2) on the Schwarzschild background metric +¯gµν. Here ¯E = −¯uµ¯gµν(∂t)ν and ¯L = ¯uµ¯gµν(∂φ)ν are the conserved energy and angular momentum +of the test particle in the Schwarzschild geometry. Tidal deformations to the four-velocity affect +the gauge-independent photon red-shift measurements [47] (∼ ut +(1)), are responsible for radial +deviations (∼ ur +(1)), tilt the orbital plane (∼ uθ +(1)), and shift the orbital frequency (∼ uφ +(1)). +The Hamiltonian of a test particle moving around a tidally deformed Schwarzschild black +hole (see Eq. (3.4)) is given by +H = 1 +2uµuνgµν ≃ 1 +2 ¯uµ � +¯uν + 2uµ +(1) +� +¯gµν + 1 +2 ¯uµ¯uνhµν . +(4.2) +In the specific case of a circular orbit ¯uµ in the Schwarzschild background metric ¯gµν, radial and +polar deviations affects the dynamics only at higher order [25,48]. Moreover, from Eq. (4.2), the +tidal perturbations that enter the Hamiltonian are htt ∝ Eq, htφ ∝ Eq +φ, Bq +φ, and hφφ ∝ Eq +φφ, Bq +φφ. +A further simplification, that is very common in celestial mechanics, is the secular averaging +over a timescale much bigger than the orbital timescale. The effective dynamics of a test particle +which follows a tidally-deformed geodesic γ′ at the first order in hµν can be well captured by +replacing the physical trajectory γ′ with an averaged circular trajectory γ in the perturbed +13 + +spacetime. +The averaged geodesic γ can be interpreted as a secular orbit in the perturbed +background. We introduce the secular average of a quantity A as [25] +⟨A⟩ = 1 +2π +� 2π +0 +A +�� +γ dφ , +(4.3) +where γ is the averaged circular orbit on gµν. In particular, if γ′ is quasi-circular, the averaged +secular geodesic γ deviates from the physical orbit γ′ only starting from second order in hµν in +the Hamiltonian (4.2). +After averaging, from Eqs. (3.10) and (3.13), we get 6 +⟨htt⟩ = −r2f 2⟨Eq⟩, +(4.4) +⟨htφ⟩ = 0, +(4.5) +⟨hφφ⟩ = −r4 +� +1 − 2M 2 +r2 +� +⟨Eq⟩. +(4.6) +and therefore the secular average of the Hamiltonian (4.2) up to quadrupole order can be recast +as 7 +⟨H⟩ ≃ −1 +2 +�⟨E⟩2 +f +− ⟨L⟩2 +r2 +� +− η +� +⟨E⟩2 + +� +1 − 2M 2 +r2 +� ⟨L⟩2 +r2 +� r2 +M 2 , +(4.7) +where η is a parameter that encodes all the effects of the tidal deformations at the quadrupole +order. E = −uµgµν(∂t)ν and L = −uµgµν(∂φ)ν are, respectively, the energy and angular mo- +mentum with respect to the perturbed spacetime and the symbol ⟨·⟩ stands for secular average. +We stress that ⟨E⟩ and ⟨L⟩ encode the kinematics (including the secular effects on the orbits), +while the parameter η effectively depends on the secular tidal deformations (∝ Cij) and on the +orientation (β, χ) of the binary system. More explicitly, we find that the tidal parameter η is +proportional to the secular average of the electric scalar tidal field +η = −M 2 +2 ⟨Eq⟩ = M 2 +16 +� +C33 (1 + 3 cos 2β) + 4 (C13 sin χ + C23 cos χ) sin 2β ++ [2C12 sin 2χ + (2C22 + C33) cos 2χ] (1 − cos 2β) +� +. +(4.8) +Notice that this expression for η can also be used for other tidal tensors Cij than the one induced +by the Kerr black hole in this paper. In fact, it is a general result for any EMR binary system +consisting of a Schwarzschild black hole of mass M and a test particle of mass m, under the +assumptions that: 1) it is immersed in a tidal environment, 2) only the quadrupole order is +retained and 3) the secular approximation is valid. +If we specialize Eq. (4.8) to the tidal tensors of a Kerr perturber that we presented in Sec. 2 in +Eq. (2.14), it can be shown that the Marck’s angle Ψ appearing in the Cij’s, which is a constant +in this approximation, can be reabsorbed by a simple shift of the angle χ , χ → χ − Ψ so that +η is explicitly given by +η = I1M 2 +16KΣ2 +� +3ST(ˆr2 − a2 cos2 ˆθ)(1 − 4 sin2 β sin2 χ) + 6 cos 2β +� +ˆr2T 2 − a2S2 cos2 ˆθ +� +−3a cos ˆθ +� +aS2 cos ˆθ + 4ˆr sin 2β +√ +ST(S + T) sin χ +� ++ KΣ2 + 3ˆr2T 2� +(4.9) ++ 3I2M 2√ +ST +4KΣ2 +�� +a2S cos2 ˆθ − ˆr2T +� +sin 2β sin χ − 2aˆr +√ +ST cos ˆθ +� +cos2 β − sin2 β sin2 χ +�� +, +6Our result differs from the one in Ref. [25] where ⟨htφ⟩ ̸= 0. +7Notice that we used that ⟨uµuνgµν⟩ ≃ ⟨uµ⟩⟨uν⟩⟨gµν⟩ including corrections of order hµν. +14 + +where K is the Carter constant, and I1, I2, S and T are defined in Eqs. (2.7) and (2.11). +In the weak field regime, where M⋆ ≪ ˆr, the leading order part of η is given by +η = M 2 +4K +M⋆ +ˆr3 +� +3T(cos2 β − sin2 β sin2 χ) − K +� +2 − 3 sin2 β +� +− 3a +√ +T cos ˆθ sin χ sin 2β +� +. +(4.10) +In the equatorial plane of the Kerr pertuber ˆθ = π/2, the parameter η takes the simpler form +η = M 2 +4 +M⋆ +ˆr3 +� +1 − 3 sin2 β sin2 χ +� +, +(4.11) +that depends only on the two Euler angles χ and β and not on the spin parameter a, so one +cannot distinguish the effect of the tidal forces from the case of a Schwarzschild perturber (a = 0). +This is reasonable in the sense that if one goes at large distances on the equatorial plane, one +cannot feel the effect of the spin of the Kerr black hole. For χ = π/2, in particular, Eq. (4.11) +coincides with the result of Ref. [25], provided one identifies β as the angle between the tidal +symmetry axis, parallel to z, and the orbital plane: η = M2M⋆ +4ˆr3 +� +1 − 3 sin2 β +� +. +4.2 +Special case of circular equatorial geodesic in Kerr background +We emphasize that neither the construction of the tidal quadrupole moments in Sec. 2, nor the +discussion about the secular dynamics of the Schwarzschild binary system in the current section +rely on any assumption concerning the geodesic motion followed by the Schwarzschild black hole +of mass M around the Kerr black hole of mass M∗ ≫ M. +However, in order to simplify the discussion, we now focus on solutions of the geodesic +equations (2.3) describing circular ( ˙ˆr = 0) and equatorial geodesics (ˆθ = π/2 and ˙ˆθ = 0) in +the Kerr spacetime. Under these assumptions, the parameters that characterise the geodesic – +namely the energy, the angular momentum and the Carter’s constant – are written explicitly +in Eq. (2.18). In this case the effective parameter η given in Eq. (4.9) reduces to the simple +expression +η = M∗M 2 +16ˆr3 +� +1 + 3K +ˆr2 − 3 +�K +ˆr2 + +� +1 + K +ˆr2 +� +sin2 χ +� +sin2 β +� +. +(4.12) +Note that this is a general result, valid beyond the weak-field regime (M⋆ ≪ ˆr). +For a circular equatorial geodesic it is moreover easy to express the Carter constant K in +terms of the Kerr parameters (a, M∗) and the orbital radius ˆr, by means of the following relation +K +ˆr2 = −1 +2 +� +1 − ˆr2 − ˆrM∗ − 2σa√ˆrM∗ + 2a2 +ˆr2 − 3ˆrM∗ + 2σa√ˆrM∗ +� +. +(4.13) +We recall that σ = ±1 distinguishes whether a circular orbit is co-rotating or counter-rotating +with respect to the Kerr black hole angular momentum. +An intriguing observation is that, from the expression (4.12), one can see that there exist +certain configurations for the EMR binary system (M, m) on the Kerr equatorial plane, such +that η = 0, namely such that the dynamical contribution of the tidal effects vanishes in the +secular approximation. For a given angle χ, this holds when the angle β = β∗(χ) with +sin2 β∗(χ) = +1 + 3K/ˆr2 +3 +� +K/ˆr2 + (1 + K/ˆr2) sin2 χ +� . +(4.14) +In the weak-field limit this relation reduces to sin2 β∗(χ) = (3 sin2 χ)−1, thus generalising the +result obtained in Ref. [25], which is valid only for χ = π/2. Instead, the above result goes +beyond the weak-field regime, and can be used also for circular geodesics close to the event +horizon of Kerr. +15 + +Among all the time-like equatorial circular orbits, the Innermost Stable Circular Orbit +(ISCO) stands out for its relevance in black hole astrophysics. We recall that two ISCOs ex- +ist in the equatorial plane of a Kerr black hole, one which is co-rotating (σ = +1) and the +other counter-rotating (σ = −1). As an illustrative example, previously not considered in the +literature concerning hierarchical three-body systems, one can analyse the case where the circu- +lar equatorial orbit, in which the binary system is located, is given by the Kerr ISCOs. More +specifically, in the following we set +ˆr ≡ ˆrσ +ISCO = M∗ +� +3 + Z2 − σ +� +(3 − Z1)(3 + Z1 + 2Z2) +� +, +(4.15) +where +Z1 = 1 + +� +1 − a2 +M 2 +∗ +�1/3 �� +1 + a +M∗ +�1/3 ++ +� +1 − a +M∗ +�1/3� +, +Z2 = +� +Z2 +1 + 3 a2 +M 2 +∗ +. +(4.16) +It is possible to show that the following relation implicitly defines the ISCOs in terms of the +conserved Killing energy [46] +ˆE2 +ISCO = 1 − 2 +3 +M∗ +ˆrσ +ISCO +, +(4.17) +so that, by combining the expression above with K = (a ˆE − ˆL)2 as in Eq. (2.18), one obtains +that the Carter constant at the ISCOs takes the value K = 1/3 (ˆrσ +ISCO)2. The expression for η +in this limit considerably simplifies and it is given by +η = +M 2M∗ +2 (ˆrσ +ISCO)3 +� +1 − 1 +2(1 + 4 sin2 χ) sin2 β +� +. +(4.18) +Notice that, even if ˆrσ +ISCO ∼ O(M∗), the small tide approximation Eq. (3.1) is still valid since +M ≪ M∗. This means that one can still legitimately consider the quadrupole approximation +for a hierarchical three-body system in which the binary system (M, m) is orbiting on the ISCO +of the Kerr black hole of mass M⋆. It is interesting to notice that in the expression (4.18) the +dependence on the spin parameter of the Kerr perturber is only contained in the prefactor, +whereas the part inside square brackets specifies the configuration of the binary system. A plot +of the prefactor showing the dependence on the spin of the Kerr black hole is shown in Fig. 2 +for different values of the ratio M/M∗. +It is also interesting to observe that the expression for η at the ISCO remains well-defined +even when the Kerr black holes is rotating close to extremality, namely for a → M∗. In this case +one has ˆr+ +ISCO → M∗, so that the prefactor only depends on the ratio M 2/M 2 +∗. It is also evident +by means of the plot in Fig. 2 that the extreme case represents the maximum value of η at the +ISCO for a given configuration of the binary system. +For the EMR binary system moving on the ISCO in the Kerr black hole spacetime, we can +get the angle β = β∗(χ), as function of the angle χ, for which η = 0, at which the tidal effects +vanish from the secular dynamics of the binary system. Using that K/(ˆrσ +ISCO)2 = 1/3, one gets +sin2 β∗(χ) = +2 +1 + 4 sin2 χ . +(4.19) +In Fig. 3 we show the admissible values of β∗(χ) when the binary system is at the ISCO. +5 +Secular shifts for ISCO and photon sphere +In this section we investigate how the tidal deformations affect the secular motion of the charac- +teristic orbits of a test-particle around a Schwarzschild black hole using the Hamiltonian given +16 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-12 +-10 +-8 +-6 +-4 +a/M∗ +log10 +� +M2M∗ +2(ˆrσ +isco) +3 +� +Figure 2: The picture represents how η, when evaluated at the ISCO ˆr ≡ ˆrσ +isco, depends on +the black hole spin a. The logarithm of the prefactor in Eq. (4.18) is considered +in order to have a clear distinction for the curves. Colours are used to represent +different magnitudes for the ratio µ = M/M∗. In particular µ = 10−2 in blue, +µ = 10−3 in purple, µ = 10−4 in red and µ = 10−5 in orange. Solid lines are +representative for the co-rotating ISCO σ = 1, whereas dashed lines for counter- +rotating ISCO σ = −1. +0 +π +6 +π +2 +5 π +6 +π +7 π +6 +3 π +2 +11 π +6 +2 π +0 +π +4 +π +2 +χ +β∗(χ) +Figure 3: The red line identifies the configurations β∗(χ) for which the secular effect of tidal +deformations vanishes under the assumption ˆr ≡ ˆrσ +ISCO. The gray areas represent +exclusion zones, namely values of the angle χ in which the relation (4.19) cannot +be satisfied. More specifically, these corresponds to values of χ that would lead +| sin2 β∗| > 1. +17 + +in Eq. (4.7). In particular, we consider two specific orbits in the case of general configurations of +the three-body system, namely the ISCO and the photon sphere in the perturbed Schwarzschild +spacetime. Before computing tidal effects on the orbital motion, we address the issue of gauge +invariance of such effects. +5.1 +Gauge invariance of secular observables +We start by recalling that the energy E can be expressed in terms of the Killing vector ∂t, +namely +E = −uµgµνT ν , +(5.1) +where in our coordinates T = ∂t and gµν and uν are the metric and four-velocity including tidal +perturbations. Given that T is a Killing vector field, dE/dτ = 0 in any coordinate system when +evaluated on a geodesic. Therefore, E is conserved and gauge-invariant. +The angular momentum can be covariantly written as +L = uµgµνJν , +(5.2) +where in our coordinates J = ∂φ. However, as J is not a Killing vector field for the full metric +gµν, L is not conserved along geodesics. The strategy here is to get a conserved quantity and +show that it is also gauge-invariant. We assume that the angular momentum L can be expanded +as +L ≃ ¯L + ηL1 , +(5.3) +where ¯L is the conserved angular momentum in the Schwarzschild background, while L1 is +the correction induced by the tidal fields at the quadrupole order, which in general it is not +conserved. +The key observation is that the averaged metric field ⟨gµν⟩ does not depend on +φ = φ(τ), implying that ⟨L⟩ is now a conserved quantity along the secular geodesic. Therefore, +for a quasi-circular orbit we can write +⟨L⟩ ≃ +� 2π +0 +�¯L + ηL1 +� +|γdφ = 2π ¯L + η +� 2π +0 +L1|γdφ . +(5.4) +We now consider a coordinate transformation which, up to the quadrupole order, is of the +form +φ → ˜φ ≃ φ + ηχ(φ) , +(5.5) +such that χ is a periodic function of φ with a period of 2π, namely χ(φ) = χ(φ + 2π). Under +this gauge transformation, the first term in Eq. (5.4) reads as +� 2π +0 +¯L|γd˜φ → +� 2π +0 +¯L|γdφ + η +� 2π +0 +¯L|γdχ = 2π ¯L , +(5.6) +where we used the periodicity of χ and the fact that ¯L does not depend on φ. The second term +in Eq. (5.4), under the gauge transformation in (5.5), transforms as +� 2π +0 +L1|γd˜φ → +� 2π +0 +L1|γdφ + η +� 2π +0 +L1|γdχ . +(5.7) +The second integral in the expression above does not vanish in general, since L1 depends on φ. +However, we can neglect it because the second integral will be multiplied by η2 and therefore it +is of higher order. Putting the pieces together we have +⟨L⟩ ≃ +� 2π +0 +�¯L + ηL1 +� +|γd˜φ → 2π ¯L + η +� 2π +0 +L1|γdφ , +(5.8) +18 + +thus ⟨L⟩ is gauge-invariant under coordinate transformations of order O(η) which are 2π-periodic +in φ. +Along the same line of reasoning, one can prove the gauge invariance of ⟨uφ⟩ and ⟨ut⟩. Since +the orbital frequency for a quasi-circular orbit is defined by +Ω = uφ +ut , +(5.9) +we conclude that ⟨Ω⟩ is also gauge-invariant under coordinate transformations of order O(η) +which are 2π-periodic in φ. +As a side remark, we could extend the reasoning for the gauge invariance of secular quantities +to certain classes of gauge transformations. For example, we can consider the case where the +coordinate transformation involves a radial function +˜φ ≃ φ + ηA (r) χ (φ) , +(5.10) +where χ is still a function of φ with period 2π. In the averaging procedure, we would also have +an integral over r that vanishes because the secular geodesic is circular. Another example is a +gauge transformation depending on the polar coordinate θ, namely +˜φ ≃ φ + ηA (θ) χ (φ) . +(5.11) +Once again, being any shift in θ of order O(η) and being the function A multiplied by η, we can +neglect any contribution of A (θ) to the averaging procedure that goes beyond the first order in +η. +5.2 +Tidal effects around the ISCO orbit +The innermost stable circular orbit (ISCO) for massive test-particles is completely characterised +by three parameters: its radius, energy and angular momentum. It is defined as an extreme +point of the Hamiltonian (4.7), namely +⟨H⟩|r=rISCO = −1 +2 , +d⟨H⟩ +dr +���� +r=rISCO += 0 , +∂2⟨H⟩ +∂r2 +���� +r=rISCO += 0 . +(5.12) +Using these conditions and keeping only terms proportional to η, it is possible to compute the +secular effects caused by the tidal perturbations to the energy, angular momentum and radius +of the Schwarzschild ISCO. +We assume that observables are expanded around their unperturbed values. Physically, this +is equivalent to assume that tidal (secular) effects are all proportional to the tidal parameter η. 8 +This assumption also defines the numerical values of the tidal corrections. Tidal corrections to +the radius,9 the averaged energy and angular momentum read as 10 +rISCO ≃ r0 + η r1 , +EISCO ≃ E0 + η E1 , +LISCO ≃ L0 + η L1 . +(5.13) +By solving Eqs. (5.12) at leading order one can determine the value of (r0, E0, L0), respectively +the value for the radius, the energy and the angular momentum of the ISCO for an unperturbed +Schwarzschild black hole. They are +r0 = 6 M , +E0 = 2 +√ +2 +3 +, +L0 = 2 +√ +3 M . +(5.14) +8We recall that we consider only up to first order contributions in the small-tide approximation. +9which is not a gauge-invariant quantity; see discussion at the end of this section. +10From now on, we will drop the symbol of the secular average ⟨·⟩ for the sake of presentation. +19 + +At the first order in η, the first corrections to the ISCO quantities are given by +r1 = 3072 M , +E1 = −152 +√ +2 +3 +, +L1 = −348 +√ +3 M . +(5.15) +Note that we fixed our conventions for η in order to precisely reproduce the same numerical +values of (r1, E1, L1) previously obtained in Ref. [25]. However, while the results of Ref. [25] are +only valid in the weak-field approximation where ˆr ≫ M⋆ and on the equatorial plane ˆθ = π/2, +our results are more general and hold for any value of ˆr and ˆθ, as we discussed earlier in Sec. 4. +It is also possible to compute the shift in the ISCO orbital frequency. In general, for quasi- +circular orbits, the orbital frequency can be determined by means of the ratio [25,47,49] +Ω2 = +�uφ +ut +�2 += +1 +2r2 +�2M +r +− (r − 3M) uµuν∂r⟨hµν⟩ +� +, +(5.16) +where uµ are the components of the four-velocity (4.1). To first order in η, we obtain +ΩISCO ≃ Ω0 + η Ω1 , +(5.17) +where 11 +M Ω0 = +1 +6 +√ +6, +M Ω1 = − +� +2 +3 +491 +6 +. +(5.18) +This gives the shift induced by the tidal fields in the orbital frequency of the ISCO. +Following Ref. [47], the angular frequency Ω can be used to compute a gauge-independent +measure of the radial separation between the Schwarzschild black hole and the test particle. One +defines +RΩ = +�M +Ω2 +�1/3 +, +(5.19) +so that according to Eqs. (5.17) and (5.18) +RΩ ≃ 22/3M +Ω2/3 +0 +� +1 − 2 +3ηΩ1 +Ω0 +� += 6M + 3928η M . +(5.20) +We notice that this gives a different radial shift than in Eq. (5.15). However, this is not surprising +as the radial shift of Eq. (5.15), unlike the above, is not gauge-invariant. +5.3 +Tidal effects around the photon sphere +The photon sphere around a Schwarzschild black hole is composed by the last stable circular +orbits for massless test-particles. Differently from the case of the ISCO, this orbit is only specified +by two parameters: the photon sphere radius and the impact parameter b = L/E. A previous +analysis of the photon sphere properties in a tidal environment can be found in Ref. [40], under +more limited assumptions than the ones considered in this paper. +From the secular Hamiltonian (4.7), one enforces the conditions +⟨H⟩|r=rPS = 0 , +d⟨H⟩ +dr +���� +r=rPS += 0 . +(5.21) +11Notice that this result agrees with Ref. [40] (but not with Ref. [25]), after a rescaling of -1/2 of the η parameter. +For the ease of comparison, our radial configuration (see Fig. 1) is called polar companion configuration in +Ref. [40]: this can be obtained in the weak-field limit ˆr ≫ M∗ and for β = π/2 and χ = −π/2. +20 + +By expanding the kinematic quantities in the tidal parameter η to retain only the leading +contribution of the tidal secular effects in the small-tide approximation, one obtains +rPS ≃ r0 + η r1 , +bPS ≃ b0 + η b1 , +(5.22) +where the unperturbed values for the Schwarzschild black hole are obtained by solving (5.21) at +the leading order +r0 = 3 M , +b0 = 3 +√ +3 M . +(5.23) +Similarly, the tidal corrections are given by +r1 = −30 M , +b1 = 30 +√ +3 M . +(5.24) +This results generalize the one obtained in Ref. [40] for the special configuration of polar com- +panions (equivalent to our radial configuration), after a rescaling of η. +Again, the orbital frequency at the photon sphere at first order in the tidal corrections can +be computed in general from +Ω = uφ +ut = 1 +b , +(5.25) +which at first order in η yields to +ΩPS ≃ Ω0 + η Ω1 . +(5.26) +By means of Eqs. (5.23) and (5.24), one directly obtains the shift in the frequency of the photon +sphere, given by +M Ω0 = +1 +3 +√ +3 , +M Ω1 = − 10 +3 +√ +3 . +(5.27) +6 +Conclusions and outlook +We conclude by summarising our new results and discussing further developments. +In Sec. 2, we retraced the computation performed in Ref. [36] for the construction of the +Marck’s tetrad, defining a local inertial frame which is parallel-transported around a time-like +geodesic in Kerr spacetime. Tidal effects induced by a Kerr black hole are obtained by projecting +the Weyl tensor on certain components of the Marck’s tetrad. While the components of the rank- +2 tensor Cij were computed in Marck’s paper [36], the components of the rank-3 tensor Cijk were +previously known only on the equatorial plane of a Kerr black hole [37,38]. This paper therefore +fills the gap in the literature: the explicit expressions for Cijk are given in Eq. (2.15). Our result +is valid for generic angles ˆθ and for arbitrary time-like geodesics in the Kerr spacetime. +In Sec. 3, we found a natural application of the tidal tensors computed in the previous sec- +tion in the modeling of a hierarchical three-body system in General Relativity. We considered +a 3-body system describing a supermassive rotating black hole of mass M∗ and an EMR bi- +nary system, made of a non-rotating black hole of mass M ≪ M∗ and a smaller companion +of mass m ≪ M, which gravitates around the supermassive black hole. In order to go be- +yond the post-Newtonian approximation, in which the three bodies are sufficiently distant from +each other to be treated as point-like masses, and capture strong general relativistic effects, +one can model the region around the non-rotating black hole in terms of a tidally-deformed +Schwarzschild spacetime. To this aim, it is convenient to decompose the tidal tensor in terms of +irreducible representations of the rotation group, so as to construct “electric” E and “magnetic” +B quadrupole tidal moments, that encode the leading-order deformations to the Schwzarschild +metric immersed in a generic tidal environment [30]. By approximating the motion of the small- +est body as that of a test-mass, it is possible to take into account all the possible configurations +21 + +of the binary system by introducing two Euler’s angles. Another new result obtained in this +work is the explicit expressions for the electric and magnetic quadrupole tidal moments given +in Eqs. (3.10)-(3.13), that take into account arbitrary orientations of the binary system with +respect to the source of the tidal deformations. We remark that these expressions are valid for +arbitrary sources of tidal effects. This can be of interest for numerical simulations and analytical +study of binary systems immersed in a tidal environment. For the case of a supermassive Kerr +black hole, the tidal moments (3.10) and (3.13) together with our result in Sec. 2 allow us to +analytically compute tidal effects induced by a Kerr black hole in full generality. +The hierarchy of masses makes it natural to study the dynamics of the binary system in the +secular approximation. As first pointed out in Ref. [25], the tidal effects perturb the secular +Hamiltonian for the binary system. Remarkably, at the quadrupole approximation, the tidal +perturbation can be recast into an effective perturbative parameter η. The main result of Sec. 4 +is a general expression for η given in Eq. (4.8). It holds at the quadrupole order in the small- +tide regime and in the secular approximation, and it models the deformed secular dynamics +of a binary system. Our η generalises results obtained in Ref. [25] and Ref. [40] to arbitrary +orientations of the binary system and tidal effects induced by a rotating black hole, including +the strong gravity regime. +Tidal deformations induce changes in certain gauge-invariant quantities characterising the +EMR binary systems, such as the orbital frequency. Such tidal deformations induced by the +environment are completely encoded in the effective perturbative parameter η. +We devoted +Sec. 5 to the study of such shifts in the case of marginally stable orbits for massive (ISCO shifts) +and massless (photon sphere shifts) test-particles. We also addressed the issue of the gauge +invariance of the shifts in the secular approximation. While we focus on the case of a Kerr +black hole as the perturber, one can also use our expressions with general tidal moments. For a +Kerr perturber, the expression for η (see Eq. (4.9)) shows the rich phenomenology of the triple +system: it combines the parameters of the background Kerr metric (M∗ and a), the location of +the geodesic where the binary system is located (ˆr, ˆθ, K), and the Euler angles that capture the +geometric orientation of the binary system with respect to the Kerr perturber (β and χ). Our +parameter η includes strong general relativistic effects of an EMR binary system which is affected +by the presence of a large Kerr black hole, and considerably generalises the setup considered +in Ref. [25] and Ref. [40] beyond the weak-field regime and for arbitrary configurations. As an +example of a regime which was previously overlooked in the literature, in Sec. 4.2, we focused +on the case in which the EMR system is placed on the ISCO of the Kerr background. We +also derived configurations of the EMR system for which the tidal effects vanish in the secular +approximation, generalising the findings of Ref. [25]. +There is a number of directions in which this work can be further extended, and for which +the results obtained here can be of interest. In this paper, we analyze triple systems whose +dynamics is stationary in time and restricted to circular orbits. This implies that we do not +have gravitational waves in our setup. We also work in the leading quadrupole approximation +for the tidal effects. The setup in this paper, though simplified, is useful to get analytic results +and it should be considered as a first step towards a more realistic scenario that can be relevant +for astrophysical interest. +An extension of this work would include higher-order effects beyond the quadrupole approx- +imation [50] and the stationary regime. It would be interesting to further develop waveforms +from triple hierarchical systems [51,52] and approaches to effective description thereof [53,54]. +Another natural development would be extending this study where the primary companion of +the EMR is a Kerr black hole. The metric for a rotating black hole deformed by tidal effects has +been derived in full generality in Ref. [55] by solving the Teukolsky equation and using metric +reconstruction techniques. Due to the very complicated structure of that metric, a simplified +version obtained in the small-spin regime has been obtained in Ref. [56], explicitly written +22 + +in terms of tidal quadrupole moments. +This is sufficient to capture all the main important +features of spacetimes with non-vanishing angular momentum, and can lead to an even richer +phenomenology – including couplings between the spins of the two black holes – possibly already +at the level of the secular dynamics. +A third interesting direction concerns the analysis of eccentric binary systems subject to +tidal deformations. For this specific case it is probably more convenient to use the action-angle +variables formalism [57–60]. This would allow us not only to extend our computation to the +case of elliptic orbits for the test particle in the binary system, but also to study the precession +of the orbits around the Schwarzschild black hole and the presence of possible resonances in the +binary system [61,62]. +Acknowledgments +We thank P. S. Cole, B. Liu and J. Samsing for interesting discussions. We thank V. 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D 91 (2015), no. 8 083001 [1501.07728]. +27 + diff --git a/AdE4T4oBgHgl3EQfEwwN/content/tmp_files/load_file.txt b/AdE4T4oBgHgl3EQfEwwN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5d02797f4b672eec96c39ebbc0123a370853d1fe --- /dev/null +++ b/AdE4T4oBgHgl3EQfEwwN/content/tmp_files/load_file.txt @@ -0,0 +1,1479 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf,len=1478 +page_content='January 13, 2023 Tidal deformations of a binary system induced by an external Kerr black hole Filippo Camilloni†, Gianluca Grignani†, Troels Harmark‡, Roberto Oliveri∗, Marta Orselli† ‡, Daniele Pica† ‡ † Dipartimento di Fisica e Geologia, Universit`a di Perugia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Sezione di Perugia, Via Pascoli, I-06123 Perugia, Italy ‡ Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark ∗ LUTH, Laboratoire Univers et Th´eories, Observatoire de Paris, CNRS, Universit´e PSL, Universit´e Paris Cit´e, 5 place Jules Janssen, 92190 Meudon, France Abstract The dynamics of a binary system moving in the background of a black hole is affected by tidal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In this work, for the Kerr black hole, we derive the electric and magnetic tidal moments at quadrupole order, where the latter are computed for the first time in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We make use of these moments in the scenario of a hierarchical triple system made of a Kerr black hole and an extreme-mass ratio binary system consisting of a Schwarzschild black hole and a test particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We study how the secular dynamics of the test particle in the binary system is distorted by the presence of tidal forces from a much larger Kerr black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Our treatment includes strong gravitational effects beyond the post-Newtonian approximation both for the binary system and for the tidal forces since the binary system is allowed to be close to the event horizon of the Kerr black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We compute the shifts in the physical quantities for the secular dynamics of the test particle and show that they are gauge-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In particular, we apply our formalism to the innermost stable circular orbit for the test particle and to the case of the photon sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Our results are relevant for the astrophysical situation in which the binary system is in the vicinity of a supermassive black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='04879v1 [gr-qc] 12 Jan 2023 Contents 1 Introduction 1 2 Tidal moments induced by a Kerr black hole 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1 Carter’s tetrad .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 15 5 Secular shifts for ISCO and photon sphere 16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1 Gauge invariance of secular observables .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2 Tidal effects around the ISCO orbit .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='3 Tidal effects around the photon sphere .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 20 6 Conclusions and outlook 21 1 Introduction The detection of gravitational waves from coalescing binary systems by the LIGO-Virgo-Kagra collaboration [1–3] has unsealed a new powerful and fascinating way of exploring our universe in a regime of strong gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This has made it increasingly relevant to investigate new types of strong gravitational phenomena analytically, to prepare for future experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Indeed, with the next generation detectors such as the ground-based Einstein Telescope [4] and Cosmic Explorer [5], as well as the space-based LISA [6] and TianQin [7], the sensitivity and frequency band will be greatly expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This will make it possible to use black hole binary systems also as probes of their surrounding environment (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [8] for a comprehensive review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Examples of the effect of the environment include the presence of various types of energy and matter, such as an accretion disc [9–11] or dark matter [12–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Another example, relevant for this paper, is the presence of a third body, such as a nearby supermassive black hole [19–26] bound to the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Moreover, the expansion in sensitivity and frequency band will make it possible to detect signals from new types of sources, such as for example extreme-mass-ratio (EMR) inspiraling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Among these systems, the ones that will typically be detectable in the LISA band [27, 28], are made of a stellar mass compact object of mass m and a black hole with a much larger mass M ≫ m, with mass ratios m/M ranging from 10−4 to 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In this paper we are interested in the dynamical effects of having a binary black hole system immersed in a curved background spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' To access a scenario that at the same time is realistic, has strong gravitational effects included, and can be treated analytically, we consider the case of an EMR binary system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' a black hole and a test particle, in the background of a third, larger black hole, affecting the binary system through tidal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We take the curved background spacetime to be the general case of a Kerr black hole of mass M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Instead the EMR binary system will consist of a Schwarzschild black hole of mass M 1 with a test particle of mass m, enabling us to use the tidally deformed Schwarzschild metric of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [29,30] to describe the EMR binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' For the test particle we consider it to move on a geodesic, neglecting higher order effects in m/M such as the self-force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' As the size of the binary system will be set by the scale M, we need M ≪ R where R is the curvature length scale set by the background Kerr black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This ensures that the effects of the background can be described through tidal forces, with the condition M ≪ R known as the small-tide approximation [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We will consider the quadrupole approximation to the tidal forces, being the leading order in M/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This means we can consider the EMR binary as moving on a geodesic of the Kerr black hole geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' A particularly interesting regime is when M∗ ≫ M thus corresponding to a hierarchical three body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In this case, the binary system can be close to the event horizon of the Kerr black hole, even while the small-tide approximation is respected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Our setup is inspired by that of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [25], while at the same time being a significant ex- tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Their setup was restricted to a Schwarzschild black hole as the third body, and the EMR binary system was assumed to be at a large distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Instead, we are able to consider the strong gravitational effects on the binary system when it moves in close vicinity to a Kerr black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This also means that we need to consider more carefully the relative orientation of the EMR binary system relative to the Kerr black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This is accomplished by introducing two independent rotation angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Moreover, it is important to note that in our setup we are able to capture strong gravitational effects arising from curved spacetime, in contrast with most of the extensive literature on three body systems [31–35], as those works employ the approximation that all three bodies are small relative to their mutual distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' A significant part of our paper concerns the careful computation of the general quadrupole tidal forces due to the Kerr black hole, as these constitute the forces that can affect the binary system in our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' These forces are described by the tidal tensors Cij and Cijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The rank-2 tidal tensors Cij were previously computed for a generic value of the Kerr angle ˆθ in a seminal paper by Marck [36], where he constructed the orthonormal tetrad that is parallel-transported along an arbitrary time-like geodesic in the Kerr spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' From the rank-2 tidal tensors Cij one can then compute the “electric” quadrupole moments Eij, which can be considered as “mass moments” produced by gravitational forces external to a certain region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' A primary result of this paper, is the derivation of the general form of the rank-3 tidal tensors Cijk for all values of the angle ˆθ in the Kerr spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This generalizes the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [37] (later confirmed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [38]), where the tidal tensors Cijk were obtained only for the specific value ˆθ = π/2, namely in the equatorial plane of the Kerr spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' From the rank-3 tidal tensors Cijk we moreover derive the “magnetic” quadrupole moments Bij, which can be considered as external “current moments” and generate velocity-dependent tidal forces on test bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This is another original result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We apply these tidal electric and magnetic quadrupole moments to the case described above, with an EMR binary system following a geodesic in the Kerr background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The effects induced by the tidal fields can be studied by computing the Hamiltonian of a test particle (the object of mass m) in the tidally deformed Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Specifically, starting from a circular orbit in the unperturbed Schwarzschild spacetime, we find that the geodesics in the tidally deformed spacetime acquire a small eccentricity proportional to the deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The quasi- circular dynamics in the perturbed spacetime is governed by a secular Hamiltonian, which keeps into account the effects of the tidal deformation on circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' It can be written as a sum of the unperturbed Hamiltonian in the Schwarzschild spacetime and an interaction term of order ∼ M/M∗, which allows for example to compute perturbatively the effects of tides on the location and properties of the Innermost Stable Circular Orbit (ISCO) and of the photon sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Using the tidal moments we computed, we derive the effects of tides on the frequency, radius, energy and angular momentum of the ISCO of the binary system, by computing the shifts 2 induced by the small tides on these physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 1 The case of tides generated by a Schwarzschild black hole was studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [25, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Here we derive the shifts in the case of tides induced by the Kerr geometry and we derive the expression of the parameter η entering these shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We find that η depends on the spin of the Kerr black hole, the Carter constant K, the Kerr angle ˆθ and the Boyer-Lindquist radius ˆr at which the black hole of mass M is located in the Kerr spacetime geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' More generally, our result does not rely on the specific nature of the third body responsible for the tides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Indeed, the tidal parameter η in the secular Hamiltonian is shown to be proportional to the secular average of the scalar part of the electric tidal moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This result holds in the quadrupole and in the secular approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We provide an expression for η in terms of arbitrary tides and specialize it to the case of a Kerr black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In Sec 2, we compute the tidal moments induced by a Kerr black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [36], we first recover the already known expression for the electric tidal moments and then we derive the most general expressions for the magnetic components of the tidal moments, generalising the computation done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 3, we introduce the hierarchical triple system that we analyse in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We write down the metric for a tidally deformed Schwarzschild black hole up to the quadrupole order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We moreover write down the explicit expression for the quadrupole electric and magnetic moments and we introduce the Euler angles which allow us to study any possible orientation of the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 4, we focus on the secular dynamics of the binary system in order to understand how the parameters which specify the orbits of the test particle around the Schwarzschild black hole, such as energy and angular momentum, are shifted by the tidal fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 5, we apply the results of the previous sections to the case in which the test particle is moving along the ISCO of the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In addition, we extend our computation also to the case of a massless particle studying how the photon sphere is deformed by the tidal fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We furthermore discuss the gauge invariance of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Finally, Sec 6 contains our concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Notation: Throughout this paper Greek indices run from 0 to 3, Latin lower-case indices (i, j, k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=') run from 1 to 3, Latin upper-case indices (A, B, C, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=') label spherical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Indices in round brackets ((a), (b), (c), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=') label tensor components in the Carter’s tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Sym- metric and tracefree (STF) tensors are denoted by angular brackets over their indices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=', T⟨ij⟩ = T(ij) − 1 3δijTklδkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Hatted coordinates (ˆt, ˆr, ˆθ, ˆφ) are employed for the Kerr spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Schwarzschild coordinates, used for the binary system, are instead denoted as (t, r, θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We use geometrized units with G = c = 1 and the Minkowski metric signature is η = diag(−1, 1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2 Tidal moments induced by a Kerr black hole In this section we derive the general quadrupole tidal moments for geodesic motion around a Kerr black hole which we will use in Sections 3-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1 we define the Carter’s tetrad, in terms of which the curvature tensor simplifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2 we present an alternative inertial frame [36], parallel-transported along a generic geodesic in the Kerr spacetime, here called the Marck’s tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This is the most suitable reference frame in which it is possible to extract analytic information concerning the tidal effects induced by the Kerr geometry on a system moving along its geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The tidal effects are encoded in the rank-2 and rank-3 tidal tensors and in the set of electric and magnetic tidal moments, explicitly given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='4 at the quadrupole order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The expressions of the rank-3 tidal tensor and of the magnetic quadrupole moments outside the Kerr equatorial plane are derived here for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 1See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [39] for similar treatments in the context of the self-force approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1 Carter’s tetrad The Kerr metric for a rotating black hole of mass M∗ and spin J∗, in Boyer-Lindquist (BL) coordinates ˆxµ = (ˆt, ˆr, ˆθ, ˆφ) takes the form dˆs2 = − � 1 − 2M∗ˆr Σ � dˆt2 − 4M∗ˆr Σ a sin2 ˆθ dˆt dˆφ + A Σ sin2 ˆθ dˆφ2 + Σ ∆dˆr2 + Σdˆθ2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1) where a = J∗/M∗ is the specific angular momentum and Σ = ˆr2 + a2 cos2 ˆθ, ∆ = ˆr2 − 2M∗ˆr + a2, A = (ˆr2 + a2)2 − a2∆ sin2 ˆθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2) We are interested in considering time-like geodesics around a Kerr black hole, specified by three constants of motion: the energy per unit mass ˆE, the angular momentum per unit mass ˆL and the Carter constant K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' More specifically, the first integrals of the equations of motion read [41] ˙ˆt = A ˆE − 2M∗ˆraˆL ∆Σ , ˙ˆr2 = � ˆE(ˆr2 + a2) − aˆL Σ �2 − ∆ Σ2(ˆr2 + K) , ˙ˆθ2 = 1 Σ2 � K − a2 cos ˆθ − � a ˆE sin ˆθ − ˆL sin ˆθ �2� , ˙ˆφ = 1 ∆ � 2M∗ˆra ˆE Σ + � 1 − 2M∗ˆr Σ � ˆL sin2 ˆθ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='3) where the dot denotes differentiation with respect to the proper time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' A convenient tetrad for the Kerr geometry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1), such that dˆs2 = η(a)(b)ω(a)ω(b), was intro- duced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [42] and reads ω(0) = � ∆ Σ � dˆt − a sin2 ˆθdˆφ � , ω(1) = � Σ ∆dˆr , ω(2) = √ Σdˆθ , ω(3) = sin ˆθ √ Σ � adˆt − (ˆr2 + a2)dˆφ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='4) We dub this tetrad, the Carter’s tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The curvature 2-form Ω(a)(b) = 1 2C(a)(b)(c)(d)ω(c) ∧ ω(d) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='5) with C(a)(b)(c)(d) being the components of the Weyl tensor, (Cµνρσ = Rµνρσ for the Kerr geometry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1)), projected along the Carter’s tetrad with the inverses of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='4), ωµ (a), C(a)(b)(c)(d) = Cµνρσ ωµ (a)ων (b)ωρ (c)ωσ (d), reads [36,43] Ω(0)(1) = 2I1 ω(0) ∧ ω(1) + 2I2 ω(2) ∧ ω(3) , Ω(0)(2) = −I1 ω(0) ∧ ω(2) + I2 ω(1) ∧ ω(3) , Ω(0)(3) = −I1 ω(0) ∧ ω(3) − I2 ω(1) ∧ ω(2) , Ω(1)(2) = −I1 ω(1) ∧ ω(2) + I2 ω(0) ∧ ω(3) , Ω(1)(3) = −I1 ω(1) ∧ ω(3) − I2 ω(0) ∧ ω(2) , Ω(2)(3) = 2I1 ω(2) ∧ ω(3) − 2I2 ω(0) ∧ ω(1) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='6) where I1 = M∗ˆr Σ3 � ˆr2 − 3a2 cos2 ˆθ � , I2 = aM∗ cos ˆθ Σ3 � 3ˆr2 − a2 cos2 ˆθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='7) 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2 Marck’s tetrad The orthonormal tetrad λ(a) = � λ(a) 0 , λ(a) 1 , λ(a) 2 , λ(a) 3 � that is parallel-transported along an arbi- trary time-like geodesic was constructed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The tetrad component λ(a) 0 is a time-like unit vector tangent to the geodesics and λ(a) i are space-like unit vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' They satisfy the fol- lowing conditions η(a)(b) λ(a) α λ(b) β = ηαβ , λµ 0∇µλν α = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='8) where λµ α = ωµ (a)λ(a) α and α, β = {0, 1, 2, 3} are the labels of the components of the tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The first relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='8) is the orthonormal condition, the second one is the parallel-transported requirement that implies the tetrad frame is inertial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Their explicit expressions in terms of the metric functions and the constants of motion are [36] 2 λ(a) 0 = � 1 √ ∆Σ � ˆE(ˆr2 + a2) − aˆL � , � Σ ∆ ˙ˆr, √ Σ ˙ˆθ, 1 √ Σ � a ˆE sin ˆθ − ˆL sin ˆθ �� , λ(a) 1 = ˜λ(a) 1 cos Ψ − ˜λ(a) 2 sin Ψ , λ(a) 2 = ˜λ(a) 1 sin Ψ + ˜λ(a) 2 cos Ψ , λ(a) 3 = 1 √ K � a cos ˆθλ(1) 0 , a cos ˆθλ(0) 0 , −ˆrλ(3) 0 , ˆrλ(2) 0 ) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='9) where ˜λ(a) 1 = 1 √ K � T S � ˆrλ(1) 0 , ˆrλ(0) 0 , S T a cos ˆθλ(3) 0 , −S T a cos ˆθλ(2) 0 � , ˜λ(a) 2 = � T S � λ(0) 0 , λ(1) 0 , S T λ(2) 0 , S T λ(3) 0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='10) and S = ˆr2 + K , T = K − a2 cos2 ˆθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='11) Notice the identity Σ = S − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In the second and third tetrad component of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='9), we rotated the vectors ˜λ(a) 1 and ˜λ(a) 2 of an angle Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This is necessary in order to ensure that the tetrad λ(a) = � λ(a) 0 , λ(a) 1 , λ(a) 2 , λ(a) 3 � is parallel-transported along the geodesic motion [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In particular Ψ is an angle depending on the proper time along the Kerr geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The equation satisfied by Ψ was derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [36] and reads ˙Ψ = √ K Σ � ˆE(ˆr2 + a2) − aˆL S + a ˆL − a ˆE sin2 ˆθ T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='12) A solution for this first order differential equation was provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [36] and, more explicitly in terms of the Mino time, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='3 Tidal tensors Tidal effects on test particles moving in the neighborhood of a geodesic in Kerr spacetime are best computed by evaluating the Weyl tensor on the parallel-transported tetrad λ(a) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='9)) with λ(a) 0 being the four-velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The explicit expressions for the tidal tensors are obtained once the Weyl tensor Cµνρσ is evaluated on the Kerr geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In order to compute the electric and 2We rename λ(a) 2 and ˜λ(a) 3 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [36] with our λ(a) 3 and ˜λ(a) 2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' It is also important to stress that all the components of the space-like vectors λ(a) i can be written in terms of λ(a) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 5 magnetic quadrupole moments, we first need the following components of the rank-2 and rank-3 tidal tensors in the basis of the tetrad λ(a) [30,36] Cij ≡ C(a)(b)(c)(d)λ(a) 0 λ(b) i λ(c) 0 λ(d) j , Cijk ≡ C(a)(b)(c)(d)λ(a) 0 λ(b) i λ(c) j λ(d) k , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='13) where we recall that C(a)(b)(c)(d) = Cµνρσ ωµ(a)ων(b)ωρ(c)ωσ(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Note that, as a consequence of the symmetries of the Weyl tensor, Cij is an STF tensor, whereas Cijk is trace-free and anti- symmetric in (j, k) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Morevoer, it obeys the condition Cijk + Cjki + Ckij = 0, implying that Cijk − Cjik = −Ckij and Cijk − Ckji = −Cjki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We compute now the explicit expression for the components of the Weyl tensor that are relevant for the calculations of the electric and magnetic quadrupole moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Our expressions are valid for arbitrary time-like geodesics in the Kerr black hole spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The Cij read C11 = � 1 − 3ST KΣ2(ˆr2 − a2 cos2 ˆθ) cos2 Ψ � I1 + 6ST KΣ2aˆr cos ˆθ cos2 ΨI2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' C12 = 3ST KΣ2 � − � ˆr2 − a2 cos2 ˆθ � I1 + 2aˆr cos ˆθI2 � sin Ψ cos Ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' C13 = −3 √ ST KΣ2 � aˆr cos ˆθ(S + T)I1 + � ˆr2T − a2S cos2 ˆθ � I2 � cos Ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' C22 = � 1 − 3ST KΣ2(ˆr2 − a2 cos2 θ) sin2 Ψ � I1 + 6ST KΣ2aˆr cos ˆθ sin2 ΨI2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' C23 = −3 √ ST KΣ2 � aˆr cos ˆθ(S + T)I1 + � ˆr2T − a2S cos2 ˆθ � I2 � sin Ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' C33 = � 1 + 3 KΣ2(ˆr2T 2 − a2S2 cos2 ˆθ) � I1 − 6ST KΣ2aˆr cos ˆθI2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='14) Note that Cij was already computed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [36] (with the label 2 renamed with 3 in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' As a new result, we provide also the general expression for the non-vanishing components of the rank-3 tidal tensor Cijk that enter the calculation of the magnetic moments which will be done in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The non-vanishing components are given by C112 = 3 √ ST KΣ2 �� ˆr2T − a2S cos2 ˆθ � I1 − aˆr cos ˆθ(S + T)I2 � cos Ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' C113 = 3ST KΣ2 � 2aˆr cos ˆθI1 + � ˆr2 − a2 cos2 ˆθ � I2 � sin Ψ cos Ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' C123 = − 6ST KΣ2aˆr cos ˆθ cos2 ΨI1 + 1 KΣ2 �� ˆr2T + a2S cos2 ˆθ � (S − T) − 3ST � ˆr2 − a2 cos2 ˆθ � cos2 Ψ � I2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' C212 = 3 √ ST KΣ2 �� ˆr2T − a2S cos2 ˆθ � I1 − aˆr cos ˆθ(S + T)I2 � sin Ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' C213 = 6ST KΣ2aˆr cos ˆθ sin2 ΨI1 + 1 KΣ2 � ˆr2T(2S + T) − a2 cos2 ˆθS(S + 2T) − 3ST � ˆr2 − a2 cos2 ˆθ � cos2 Ψ � I2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' C312 = 6ST KΣ2aˆr cos ˆθI1 + 1 KΣ2 � ˆr2T(S + 2T) − a2 cos2 ˆθS(2S + T) � I2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='15) In addition, we observe that C223 = −C113, C312 = C213 − C123, C313 = −C212, C323 = C112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 6 If we specialize to geodesics in the equatorial plane ˆθ = π/2 of the Kerr black hole, the explicit expressions for the tidal tensors simplify considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We get, in agreement with Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [36,37,45], C11 = � 1 − 3 � 1 + K ˆr2 � cos2 Ψ � M∗ ˆr3 , C22 = � 1 − 3 � 1 + K ˆr2 � sin2 Ψ � M∗ ˆr3 , C12 = −3 � 1 + K ˆr2 � M∗ ˆr3 cos Ψ sin Ψ , C33 = � 1 + 3K ˆr2 � M∗ ˆr3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='16) and, for the rank-3 tidal tensor (in agreement with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [37] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [38]), C121 = −C112 = C332 = −C323 = −3M∗ √ K ˆr4 � 1 + K ˆr2 cos Ψ , C221 = −C212 = C313 = −C331 = −3M∗ √ K ˆr4 � 1 + K ˆr2 sin Ψ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='17) where, for geodesics in the equatorial plane of the Kerr spacetime, the following expressions hold [46] ˆE = ˆr3/2 − 2M∗ˆr1/2 + σaM 1/2 ∗ ˆr3/4 � ˆr3/2 − 3M∗ˆr1/2 + 2σaM 1/2 ∗ , ˆL = σM 1/2 ∗ � ˆr2 + a2 − 2σa M 1/2 ∗ ˆr1/2� ˆr3/4 � ˆr3/2 − 3M∗ˆr1/2 + 2σaM 1/2 ∗ , K = � a ˆE − ˆL �2 , ˙Ψ = √ K ˆr2 + K � ˆE − a a ˆE − ˆL � = σ � M∗ ˆr3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='18) Above we introduced the parameter σ = ±1 that allows one to distinguish between prograde (+) and retrograde (−) orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' A thorough analysis of the dynamics in the equatorial plane will be given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='4 Electric and magnetic quadrupole moments The electric and magnetic quadrupole moments in Cartesian coordinates are defined as [30] Eij ≡ Cij , Bij ≡ −1 2ϵkl⟨iC kl j⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='19) with ϵijk the three-dimensional Levi-Civita symbol with ϵ123 = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We raise and lower Cartesian indices (i, j, k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=') with the Kronecker delta δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Being STF tensors, both the electric Eij and the magnetic Bij tensors have each five independent components thus, together, they account for the ten independent components of the Weyl tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In particular, the magnetic quadrupole moments in terms of the components of the rank-3 tidal tensor, read B11 = −C123 , B12 = C113 , B13 = −C112 , B22 = C213 , B23 = −C212 , B33 = C123 − C213 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='20) 7 where we used that C223 = −C113, C312 = C213 − C123, C313 = −C212 and C323 = C112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' It is far more useful to decompose the rank-2 and rank-3 tensors by means of their irreducible representations of SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [30], we introduce the radial unit vector Ωi ≡ xi/r, with r = � δijxixj being the Euclidean radius representing the distance from the geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The projector to the space orthogonal to Ωi is given by γij = δij − ΩiΩj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The electric quadrupole moment Eij decomposes as follows Eij = Eq � ΩiΩj − 1 2γij � + 2Eq (iΩj) + 1 2Eq ⟨ij⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='21) where the scalar Eq, the transverse vector Eq i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' ΩiEq i = 0) and the transverse STF tensor Eq ⟨ij⟩ are given by Eq ≡ ΩiEijΩj = −γijEij , Eq i ≡ γ j i EjkΩk , Eq ⟨ij⟩ ≡ 2γ k i γ l j Ekl − Eklγklγij = 2γ k i γ l j Ekl + Eqγij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='22) Similarly, for the magnetic quadrupole moment Bij, one has 3 Bij = ϵlk (i � Bq l � Ωj)Ωk − γj)k � + 1 4 � Bq ⟨j)l⟩Ωk − Bq ⟨j)k⟩Ωl �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='25) with symmetrization w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' the indices (i, j) and STF w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' the indices ⟨jl⟩ and ⟨jk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The transverse vector Bq i and the transverse STF tensor Bq ⟨ij⟩ are Bq i ≡ ϵijkΩjBk lΩl , Bq ⟨ij⟩ ≡ 2ϵkl(iγm j)ΩkBl m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='26) 3 Hierarchical triple system In this section we apply the formalism introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2 to an EMR binary system moving in the background of a Kerr black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The EMR binary system consists of a Schwarzschild black hole of mass M along with a test-particle of mass m ≪ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We assume that the black hole with mass M∗ moves slowly relatively to the EMR binary system (M, m) and that one can describe the effect on the binary system to a good approximation by taking into account only the quadrupole tidal moments induced by M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This is valid provided M 2 ≪ ˆr3 M + M∗ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1) where ˆr is the Boyer-Lindquist radius at which M is located in the Kerr spacetime geometry induced by M∗ [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This arises from having two widely separated scales: one scale is the length scale of the Schwarzschild black hole M, the other is the curvature length scale R induced by 3We used the decomposition of the rank-3 tidal tensor Cijk = Bq k (ΩiΩj − γij) − Bq j (ΩiΩk − γik) + 1 2 � Bq ⟨ik⟩Ωj − Bq ⟨ij⟩Ωk � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='23) with the inverse relations given by Bq i = CjkiΩjΩk , Bq ⟨ij⟩ = 2ΩkClk(iγl j) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='24) 8 the Kerr black hole M∗ at the location of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We then require M ≪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This is called small-tide approximation [30] and it makes it possible to describe the motion of the binary system (M, m) in the external Kerr geometry, ensuring that the tidal deformation is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We can therefore describe the influence of M∗ on the binary system (M, m) using, to a first approximation, the quadrupole tidal moments induced by the Kerr black hole itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Since R ∼ � ˆr3/(M + M∗) this, combined with the condition M ≪ R, gives the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' One natural way to achieve the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1) is that M is much smaller than M∗, here called the hierarchical regime M ≪ M∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2) This implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1) since ˆr ≳ M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In this case we have a hierarchical triple system of black holes m ≪ M ≪ M∗ (note that one could imagine both M and M∗ being a supermassive black hole, but with a mass hierarchy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The hierarchical triple system is the case that we shall primarily consider in this paper, since the dynamics of the triple system in general will depend on the full expressions of the quadrupole tidal moments of the Kerr black hole M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Another way to achieve the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1) is the case where M and M∗ are widely separated, here called the weak field regime M∗ ≪ ˆr , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='3) assuming as well that M ≲ M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This means one can consider two black holes M and M∗ of similar magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In this case the expression of the tidal moments induced by the Kerr black hole simplifies considerably [25] due to the fact that frame-dragging effects induced by the Kerr black hole can be neglected (see discussion around and below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='8) for further detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' It is also important to consider the time scales involved in our approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' For simplicity, we consider the binary system having an orbit of m around the Schwarzschild black hole of mass M such that r = O(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Then the time scale of the binary system is simply τbinary = O(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Assuming ˆr = O(M∗) the time-scale associated with the orbit around the Kerr black hole of mass M∗ is τkerr = O(M∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Indeed, one can see explicitly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='12) that we have ˙Ψ = O(1/M∗), which sets the rate of change of the angle Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Thus, in the hierarchical regime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2), we have τkerr ≫ τbinary, which means that we can assume that the quadrupole moments and Ψ do not vary with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Moreover, in the weak field regime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='3), the time scale for the orbit around the Kerr black hole is even larger τkerr ≫ M∗ as the velocity will be non-relativistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Thus, even if M is of same order as M∗, we find that τkerr ≫ τbinary, and we can again neglect the time dependence of Ψ and of the quadrupole moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1 Tidally deformed Schwarzschild spacetime We can describe the black hole with mass M in the binary system using the tidally deformed Schwarzschild metric [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Concretely, we add to the background metric ¯gµν a tidal perturbation hµν ds2 = ¯gµνdxµdxν + hµνdxµdxν , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='4) where the tidal perturbation hµν is computed up to the first order in the small-tide approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The background geometry (in spherical coordinates) is ¯gµνdxµdxν = −fdt2 + dr2 f + r2ΩABdθAdθB , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='5) with f = 1 − 2M/r and M being the black hole mass, θA = (θ, φ) and ΩABdθAdθB = dθ2 + sin2 θdφ2 being the metric of the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' By only retaining the quadrupole order terms in 4A more general analysis can also take into account the regime M ≪ r ≪ R for which τbinary = O( � r3/M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 9 the tidal deformation hµν, one gets hµνdxµdxν = −r2Eq (fdt + dr)2 − 4 3r3 (Eq A − Bq A) (fdt + dr) dθA − 1 3r4 �� 1 − 2M 2 r2 � Eq AB − � 1 − 6M 2 r2 � Bq AB � dθAdθB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='6) The quadrupole moments are decomposed into the scalar Eq, vector Eq A, Bq A and tensor Eq AB, Bq AB components, following the decomposition in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='21)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='25), and are written in spherical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 5 For an accurate description of our triple system, it is useful to identify the relative orientation between the orbital plane of the Kerr black hole – responsible for the tidal deformation – and the orbital plane where the dynamics of the EMR binary system (M, m) takes place;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 1 illustrating four possible configurations in the special case when M∗ is a Schwarzschild black hole and the binary system is moving on a circular geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' To describe an arbitrary configuration, one first introduces the unit directional vector Ωi = (cos φ sin θ, sin φ sin θ, cos θ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='7) centered in the Schwarzschild black hole of mass M, and attached to the reference frame of the EMR system (M, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' One then sets, without loss of generality, the polar angle in the Schwarzschild reference system θ = π/2: this is because the orbital motion takes place on an orbital plane and we set it to be the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Any arbitrary orientation is therefore given by performing a rotation on the unit vector in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='7), namely, ⃗Ω′ = RχRβRα · ⃗Ω , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='8) with the Euler rotational matrices Rα = � � cos α sin α 0 − sin α cos α 0 0 0 1 � � , Rβ = � � 1 0 0 0 cos β sin β 0 − sin β cos β � � , Rχ = � � cos χ sin χ 0 − sin χ cos χ 0 0 0 1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='9) Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='8) is only one among the 12 possible permutations of Euler matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Further- more, since we aim at describing an equatorial orbit in the binary system, it turns out that one of the Euler angle – α in our convention – can always be reabsorbed by a redefinition of the Schwarzschild azimuthal angle φ → φ + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' As a consequence, any orientation of a Schwarzschild orbit with respect to the Kerr perturber is specified only by the two angles β and χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2 Tidal moments in spherical coordinates The tidal moments also depend on the relative configuration between the binary system (M, m) and the Kerr pertuber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Here, we compute the explicit expression of the tidal quadrupole moments 5For the sake of completeness, we write the change of coordinates from Cartesian to spherical coordinates: Eq i dxi = ∂xi ∂xA Eq i dxA = Eq θ (rdθ) + Eq φ(rdφ) , Eq ⟨ij⟩dxi ⊗ dxj = ∂xi ∂xA ∂xj ∂xB Eq ⟨ij⟩dxA ⊗ dxB = Eq θθ(rdθ)2 + 2Eq θφr2dθdφ + Eq φφ(rdφ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Similar considerations apply to the magnetic multipole moments Bq i and Bq ⟨ij⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 10 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Orthogonal Configuration β = 0, χ = 0 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Radial Configuration β = π 2, χ = − π 2 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Tangential Configuration β = − π 2, χ = 0 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Arbitrary Configuration β = − π 4, χ = 5π 6 Figure 1: For illustrative purposes, we show four possible configurations for a hierarchical three-body system M∗ ≫ M ≫ m in the special case for which the perturber M∗ is a Schwarzschild black hole and the EMR binary system (M, m) is parallel- transported around a circular geodesic around M∗, whose orbital plane is depicted in gray and terminates at the ISCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' These configurations are altered significantly in more general cases with a Kerr perturber or non-circular geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The names of the configurations refer to the orientation of the orbital angular momentum L of the binary system with respect to the gray orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The grey curve represents the orbit around M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The blue orbit marks a conventional “initial” orthogonal configuration for the binary system reference frame, with the Cartesian axis oriented according to the parallel transported tetrad (panel I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The red orbits in panels II, III and IV are obtained by Euler rotations with angles written in the bottom-left of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' associated to an arbitrary configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We recall that we set θ = π/2 because we start with an equatorial orbit around the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 1 we have illustrated this and other configurations obtained by Euler rotations in the special case for which M∗ is a Schwarzschild black hole and the binary system moves on a circular geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In spherical coordinates, the decomposition of the electric quadrupole moment in its scalar, transverse vector and STF tensor components is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='22), where the unit directional vector Ωi is now replaced by the more general Ω′i defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 11 M* M M, M*The electric quadrupole moments read as Eq = −1 8 � C33 + T + 2 + T + 4 � + 1 8 � 4T + 3 sin 2φ − � 3(C33 + T + 2 ) − T + 4 � cos 2φ � , Eq θ = 1 4 � 2T − 3 cos φ − T − 4 sin φ � , Eq φ = 1 8 � 4T + 3 cos 2φ + � 3(C33 + T + 2 ) − T + 4 � sin 2φ � , Eq θθ = −Eq φφ = Eq + 1 2 � C33 + T + 2 + T + 4 � , Eq θφ = −1 2 � 2T − 3 sin φ + T − 4 cos φ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='10) where we defined the following rotations around χ of the components of Cij T + 1 = C23 cos χ + C13 sin χ , T − 1 = C23 sin χ − C13 cos χ , T + 2 = 2C12 sin 2χ + (2C22 + C33) cos 2χ , T − 2 = 2C12 cos 2χ − (2C22 + C33) sin 2χ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='11) and the rotations around β of T ± 1,2 T + 3 = 2T − 1 sin β + T − 2 cos β , T − 3 = 2T − 1 cos β − T − 2 sin β , T + 4 = 4T + 1 sin 2β + (3C33 − T + 2 ) cos 2β , T − 4 = 4T + 1 cos 2β − (3C33 − T + 2 ) sin 2β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='12) Similarly for the magnetic quadrupole moments, whose decomposition is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='26), we find that Bq θ = 1 8 � 4S+ 3 cos 2φ + � 3(C312 − S+ 2 ) − S+ 4 � sin 2φ � , Bq φ = −1 4 � 2S− 3 cos φ − S− 4 sin φ � , Bq θθ = −Bq φφ = −1 2 � 2S− 3 sin φ + S− 4 cos φ � , Bq θφ = −3 8 � C312 − S+ 2 + S+ 4 � − 1 8 � 4S+ 3 sin 2φ − � 3(C312 − S+ 2 ) − S+ 4 � cos 2φ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='13) where we defined the rotations around χ of the components of Cijk S+ 1 = C212 cos χ + C112 sin χ , S− 1 = C212 sin χ − C112 cos χ , S+ 2 = 2C113 sin 2χ + (C123 + C213) cos 2χ , S− 2 = 2C113 cos 2χ − (C123 + C213) sin 2χ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='14) and the rotations around β of S± 1,2 S+ 3 = 2S− 1 sin β − S− 2 cos β , S− 3 = 2S− 1 cos β + S− 2 sin β , S+ 4 = 4S+ 1 sin 2β + (3C312 + S+ 2 ) cos 2β , S− 4 = 4S+ 1 cos 2β − (3C312 + S+ 2 ) sin 2β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='15) 12 The structure of the tidal quadrupole moments (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='13) is the following: the tidal deformations sourced by a generic third body over the EMR binary system (M, m) are fully encoded in the tidal tensors Cij and Cijk, while the angles β and χ, parametrizing the relative orientation between the third body and the binary system, affect the tidal effects over the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We remark that the above expressions of the tidal quadrupole moments are general, and can also be employed to model environmental effects in numerical works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In the specific case of a Kerr black hole as a third body responsible for the tidal deformations, the explicit expressions of the tidal tensors Cij and Cijk are given, respectively, in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We anticipate here another property of the tidal quadrupole moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' As we shall see in the next section, it is often useful to define the secular average over the azimuthal angle φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The explicit dependence of the tidal quadrupole moments (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='13) implies that only Eq (and Eq θθ = −Eq φφ) as well as Bq θφ are relevant for physical observables upon secular averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 4 Secular dynamics of binary system In this section we focus on the secular dynamics of the binary system (M, m), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' the dynamics of the binary system after a large number of orbits of the test particle of mass m, and analyze how it is affected by the tidal fields induced by the Kerr perturber of mass M∗, in the hierarchical regime m ≪ M ≪ M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' More specifically our goal is to understand how the orbital parameters of the test particle around the Schwarzschild black hole, such as the energy or the angular momentum, are shifted by the presence of an external tidal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1 Secular Hamiltonian of test particle in binary system Following the setup of the previous section, we focus on the orbital motion of the object of mass m, approximated as a test particle, taking place on the equatorial plane of the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This amounts to set θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We approximate the four-velocity as uµ ≃ ¯uµ + uµ (1) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1) where ¯uµ is the 4-velocity of the unperturbed bound orbit, that can be taken as circular or elliptic, and uµ (1) is the leading correction due to the tidal perturbation hµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In this work, we focus on perturbations of circular orbits ¯uµ = ( ¯E/f, 0, 0, ¯L/r2) on the Schwarzschild background metric ¯gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Here ¯E = −¯uµ¯gµν(∂t)ν and ¯L = ¯uµ¯gµν(∂φ)ν are the conserved energy and angular momentum of the test particle in the Schwarzschild geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Tidal deformations to the four-velocity affect the gauge-independent photon red-shift measurements [47] (∼ ut (1)), are responsible for radial deviations (∼ ur (1)), tilt the orbital plane (∼ uθ (1)), and shift the orbital frequency (∼ uφ (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The Hamiltonian of a test particle moving around a tidally deformed Schwarzschild black hole (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='4)) is given by H = 1 2uµuνgµν ≃ 1 2 ¯uµ � ¯uν + 2uµ (1) � ¯gµν + 1 2 ¯uµ¯uνhµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2) In the specific case of a circular orbit ¯uµ in the Schwarzschild background metric ¯gµν, radial and polar deviations affects the dynamics only at higher order [25,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Moreover, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2), the tidal perturbations that enter the Hamiltonian are htt ∝ Eq, htφ ∝ Eq φ, Bq φ, and hφφ ∝ Eq φφ, Bq φφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' A further simplification, that is very common in celestial mechanics, is the secular averaging over a timescale much bigger than the orbital timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The effective dynamics of a test particle which follows a tidally-deformed geodesic γ′ at the first order in hµν can be well captured by replacing the physical trajectory γ′ with an averaged circular trajectory γ in the perturbed 13 spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The averaged geodesic γ can be interpreted as a secular orbit in the perturbed background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We introduce the secular average of a quantity A as [25] ⟨A⟩ = 1 2π � 2π 0 A �� γ dφ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='3) where γ is the averaged circular orbit on gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In particular, if γ′ is quasi-circular, the averaged secular geodesic γ deviates from the physical orbit γ′ only starting from second order in hµν in the Hamiltonian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' After averaging, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='13), we get 6 ⟨htt⟩ = −r2f 2⟨Eq⟩, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='4) ⟨htφ⟩ = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='5) ⟨hφφ⟩ = −r4 � 1 − 2M 2 r2 � ⟨Eq⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='6) and therefore the secular average of the Hamiltonian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2) up to quadrupole order can be recast as 7 ⟨H⟩ ≃ −1 2 �⟨E⟩2 f − ⟨L⟩2 r2 � − η � ⟨E⟩2 + � 1 − 2M 2 r2 � ⟨L⟩2 r2 � r2 M 2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='7) where η is a parameter that encodes all the effects of the tidal deformations at the quadrupole order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' E = −uµgµν(∂t)ν and L = −uµgµν(∂φ)ν are, respectively, the energy and angular mo- mentum with respect to the perturbed spacetime and the symbol ⟨·⟩ stands for secular average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We stress that ⟨E⟩ and ⟨L⟩ encode the kinematics (including the secular effects on the orbits), while the parameter η effectively depends on the secular tidal deformations (∝ Cij) and on the orientation (β, χ) of the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' More explicitly, we find that the tidal parameter η is proportional to the secular average of the electric scalar tidal field η = −M 2 2 ⟨Eq⟩ = M 2 16 � C33 (1 + 3 cos 2β) + 4 (C13 sin χ + C23 cos χ) sin 2β + [2C12 sin 2χ + (2C22 + C33) cos 2χ] (1 − cos 2β) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='8) Notice that this expression for η can also be used for other tidal tensors Cij than the one induced by the Kerr black hole in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In fact, it is a general result for any EMR binary system consisting of a Schwarzschild black hole of mass M and a test particle of mass m, under the assumptions that: 1) it is immersed in a tidal environment, 2) only the quadrupole order is retained and 3) the secular approximation is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' If we specialize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='8) to the tidal tensors of a Kerr perturber that we presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='14), it can be shown that the Marck’s angle Ψ appearing in the Cij’s, which is a constant in this approximation, can be reabsorbed by a simple shift of the angle χ , χ → χ − Ψ so that η is explicitly given by η = I1M 2 16KΣ2 � 3ST(ˆr2 − a2 cos2 ˆθ)(1 − 4 sin2 β sin2 χ) + 6 cos 2β � ˆr2T 2 − a2S2 cos2 ˆθ � −3a cos ˆθ � aS2 cos ˆθ + 4ˆr sin 2β √ ST(S + T) sin χ � + KΣ2 + 3ˆr2T 2� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='9) + 3I2M 2√ ST 4KΣ2 �� a2S cos2 ˆθ − ˆr2T � sin 2β sin χ − 2aˆr √ ST cos ˆθ � cos2 β − sin2 β sin2 χ �� , 6Our result differs from the one in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [25] where ⟨htφ⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 7Notice that we used that ⟨uµuνgµν⟩ ≃ ⟨uµ⟩⟨uν⟩⟨gµν⟩ including corrections of order hµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 14 where K is the Carter constant, and I1, I2, S and T are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In the weak field regime, where M⋆ ≪ ˆr, the leading order part of η is given by η = M 2 4K M⋆ ˆr3 � 3T(cos2 β − sin2 β sin2 χ) − K � 2 − 3 sin2 β � − 3a √ T cos ˆθ sin χ sin 2β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='10) In the equatorial plane of the Kerr pertuber ˆθ = π/2, the parameter η takes the simpler form η = M 2 4 M⋆ ˆr3 � 1 − 3 sin2 β sin2 χ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='11) that depends only on the two Euler angles χ and β and not on the spin parameter a, so one cannot distinguish the effect of the tidal forces from the case of a Schwarzschild perturber (a = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This is reasonable in the sense that if one goes at large distances on the equatorial plane, one cannot feel the effect of the spin of the Kerr black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' For χ = π/2, in particular, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='11) coincides with the result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [25], provided one identifies β as the angle between the tidal symmetry axis, parallel to z, and the orbital plane: η = M2M⋆ 4ˆr3 � 1 − 3 sin2 β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2 Special case of circular equatorial geodesic in Kerr background We emphasize that neither the construction of the tidal quadrupole moments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2, nor the discussion about the secular dynamics of the Schwarzschild binary system in the current section rely on any assumption concerning the geodesic motion followed by the Schwarzschild black hole of mass M around the Kerr black hole of mass M∗ ≫ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' However, in order to simplify the discussion, we now focus on solutions of the geodesic equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='3) describing circular ( ˙ˆr = 0) and equatorial geodesics (ˆθ = π/2 and ˙ˆθ = 0) in the Kerr spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Under these assumptions, the parameters that characterise the geodesic – namely the energy, the angular momentum and the Carter’s constant – are written explicitly in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In this case the effective parameter η given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='9) reduces to the simple expression η = M∗M 2 16ˆr3 � 1 + 3K ˆr2 − 3 �K ˆr2 + � 1 + K ˆr2 � sin2 χ � sin2 β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='12) Note that this is a general result, valid beyond the weak-field regime (M⋆ ≪ ˆr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' For a circular equatorial geodesic it is moreover easy to express the Carter constant K in terms of the Kerr parameters (a, M∗) and the orbital radius ˆr, by means of the following relation K ˆr2 = −1 2 � 1 − ˆr2 − ˆrM∗ − 2σa√ˆrM∗ + 2a2 ˆr2 − 3ˆrM∗ + 2σa√ˆrM∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='13) We recall that σ = ±1 distinguishes whether a circular orbit is co-rotating or counter-rotating with respect to the Kerr black hole angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' An intriguing observation is that, from the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='12), one can see that there exist certain configurations for the EMR binary system (M, m) on the Kerr equatorial plane, such that η = 0, namely such that the dynamical contribution of the tidal effects vanishes in the secular approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' For a given angle χ, this holds when the angle β = β∗(χ) with sin2 β∗(χ) = 1 + 3K/ˆr2 3 � K/ˆr2 + (1 + K/ˆr2) sin2 χ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='14) In the weak-field limit this relation reduces to sin2 β∗(χ) = (3 sin2 χ)−1, thus generalising the result obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [25], which is valid only for χ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Instead, the above result goes beyond the weak-field regime, and can be used also for circular geodesics close to the event horizon of Kerr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 15 Among all the time-like equatorial circular orbits, the Innermost Stable Circular Orbit (ISCO) stands out for its relevance in black hole astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We recall that two ISCOs ex- ist in the equatorial plane of a Kerr black hole, one which is co-rotating (σ = +1) and the other counter-rotating (σ = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' As an illustrative example, previously not considered in the literature concerning hierarchical three-body systems, one can analyse the case where the circu- lar equatorial orbit, in which the binary system is located, is given by the Kerr ISCOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' More specifically, in the following we set ˆr ≡ ˆrσ ISCO = M∗ � 3 + Z2 − σ � (3 − Z1)(3 + Z1 + 2Z2) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='15) where Z1 = 1 + � 1 − a2 M 2 ∗ �1/3 �� 1 + a M∗ �1/3 + � 1 − a M∗ �1/3� , Z2 = � Z2 1 + 3 a2 M 2 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='16) It is possible to show that the following relation implicitly defines the ISCOs in terms of the conserved Killing energy [46] ˆE2 ISCO = 1 − 2 3 M∗ ˆrσ ISCO , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='17) so that, by combining the expression above with K = (a ˆE − ˆL)2 as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='18), one obtains that the Carter constant at the ISCOs takes the value K = 1/3 (ˆrσ ISCO)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The expression for η in this limit considerably simplifies and it is given by η = M 2M∗ 2 (ˆrσ ISCO)3 � 1 − 1 2(1 + 4 sin2 χ) sin2 β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='18) Notice that, even if ˆrσ ISCO ∼ O(M∗), the small tide approximation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1) is still valid since M ≪ M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This means that one can still legitimately consider the quadrupole approximation for a hierarchical three-body system in which the binary system (M, m) is orbiting on the ISCO of the Kerr black hole of mass M⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' It is interesting to notice that in the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='18) the dependence on the spin parameter of the Kerr perturber is only contained in the prefactor, whereas the part inside square brackets specifies the configuration of the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' A plot of the prefactor showing the dependence on the spin of the Kerr black hole is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2 for different values of the ratio M/M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' It is also interesting to observe that the expression for η at the ISCO remains well-defined even when the Kerr black holes is rotating close to extremality, namely for a → M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In this case one has ˆr+ ISCO → M∗, so that the prefactor only depends on the ratio M 2/M 2 ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' It is also evident by means of the plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2 that the extreme case represents the maximum value of η at the ISCO for a given configuration of the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' For the EMR binary system moving on the ISCO in the Kerr black hole spacetime, we can get the angle β = β∗(χ), as function of the angle χ, for which η = 0, at which the tidal effects vanish from the secular dynamics of the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Using that K/(ˆrσ ISCO)2 = 1/3, one gets sin2 β∗(χ) = 2 1 + 4 sin2 χ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='19) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 3 we show the admissible values of β∗(χ) when the binary system is at the ISCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 5 Secular shifts for ISCO and photon sphere In this section we investigate how the tidal deformations affect the secular motion of the charac- teristic orbits of a test-particle around a Schwarzschild black hole using the Hamiltonian given 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='0 12 10 8 6 4 a/M∗ log10 � M2M∗ 2(ˆrσ isco) 3 � Figure 2: The picture represents how η, when evaluated at the ISCO ˆr ≡ ˆrσ isco, depends on the black hole spin a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The logarithm of the prefactor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='18) is considered in order to have a clear distinction for the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Colours are used to represent different magnitudes for the ratio µ = M/M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In particular µ = 10−2 in blue, µ = 10−3 in purple, µ = 10−4 in red and µ = 10−5 in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Solid lines are representative for the co-rotating ISCO σ = 1, whereas dashed lines for counter- rotating ISCO σ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 0 π 6 π 2 5 π 6 π 7 π 6 3 π 2 11 π 6 2 π 0 π 4 π 2 χ β∗(χ) Figure 3: The red line identifies the configurations β∗(χ) for which the secular effect of tidal deformations vanishes under the assumption ˆr ≡ ˆrσ ISCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The gray areas represent exclusion zones, namely values of the angle χ in which the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='19) cannot be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' More specifically, these corresponds to values of χ that would lead | sin2 β∗| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 17 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In particular, we consider two specific orbits in the case of general configurations of the three-body system, namely the ISCO and the photon sphere in the perturbed Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Before computing tidal effects on the orbital motion, we address the issue of gauge invariance of such effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1 Gauge invariance of secular observables We start by recalling that the energy E can be expressed in terms of the Killing vector ∂t, namely E = −uµgµνT ν , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1) where in our coordinates T = ∂t and gµν and uν are the metric and four-velocity including tidal perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Given that T is a Killing vector field, dE/dτ = 0 in any coordinate system when evaluated on a geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Therefore, E is conserved and gauge-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The angular momentum can be covariantly written as L = uµgµνJν , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2) where in our coordinates J = ∂φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' However, as J is not a Killing vector field for the full metric gµν, L is not conserved along geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The strategy here is to get a conserved quantity and show that it is also gauge-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We assume that the angular momentum L can be expanded as L ≃ ¯L + ηL1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='3) where ¯L is the conserved angular momentum in the Schwarzschild background, while L1 is the correction induced by the tidal fields at the quadrupole order, which in general it is not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The key observation is that the averaged metric field ⟨gµν⟩ does not depend on φ = φ(τ), implying that ⟨L⟩ is now a conserved quantity along the secular geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Therefore, for a quasi-circular orbit we can write ⟨L⟩ ≃ � 2π 0 �¯L + ηL1 � |γdφ = 2π ¯L + η � 2π 0 L1|γdφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='4) We now consider a coordinate transformation which, up to the quadrupole order, is of the form φ → ˜φ ≃ φ + ηχ(φ) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='5) such that χ is a periodic function of φ with a period of 2π, namely χ(φ) = χ(φ + 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Under this gauge transformation, the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='4) reads as � 2π 0 ¯L|γd˜φ → � 2π 0 ¯L|γdφ + η � 2π 0 ¯L|γdχ = 2π ¯L , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='6) where we used the periodicity of χ and the fact that ¯L does not depend on φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='4), under the gauge transformation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='5), transforms as � 2π 0 L1|γd˜φ → � 2π 0 L1|γdφ + η � 2π 0 L1|γdχ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='7) The second integral in the expression above does not vanish in general, since L1 depends on φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' However, we can neglect it because the second integral will be multiplied by η2 and therefore it is of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Putting the pieces together we have ⟨L⟩ ≃ � 2π 0 �¯L + ηL1 � |γd˜φ → 2π ¯L + η � 2π 0 L1|γdφ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='8) 18 thus ⟨L⟩ is gauge-invariant under coordinate transformations of order O(η) which are 2π-periodic in φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Along the same line of reasoning, one can prove the gauge invariance of ⟨uφ⟩ and ⟨ut⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Since the orbital frequency for a quasi-circular orbit is defined by Ω = uφ ut , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='9) we conclude that ⟨Ω⟩ is also gauge-invariant under coordinate transformations of order O(η) which are 2π-periodic in φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' As a side remark, we could extend the reasoning for the gauge invariance of secular quantities to certain classes of gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' For example, we can consider the case where the coordinate transformation involves a radial function ˜φ ≃ φ + ηA (r) χ (φ) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='10) where χ is still a function of φ with period 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In the averaging procedure, we would also have an integral over r that vanishes because the secular geodesic is circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Another example is a gauge transformation depending on the polar coordinate θ, namely ˜φ ≃ φ + ηA (θ) χ (φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='11) Once again, being any shift in θ of order O(η) and being the function A multiplied by η, we can neglect any contribution of A (θ) to the averaging procedure that goes beyond the first order in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2 Tidal effects around the ISCO orbit The innermost stable circular orbit (ISCO) for massive test-particles is completely characterised by three parameters: its radius, energy and angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' It is defined as an extreme point of the Hamiltonian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='7), namely ⟨H⟩|r=rISCO = −1 2 , d⟨H⟩ dr ���� r=rISCO = 0 , ∂2⟨H⟩ ∂r2 ���� r=rISCO = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='12) Using these conditions and keeping only terms proportional to η, it is possible to compute the secular effects caused by the tidal perturbations to the energy, angular momentum and radius of the Schwarzschild ISCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We assume that observables are expanded around their unperturbed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Physically, this is equivalent to assume that tidal (secular) effects are all proportional to the tidal parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 8 This assumption also defines the numerical values of the tidal corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Tidal corrections to the radius,9 the averaged energy and angular momentum read as 10 rISCO ≃ r0 + η r1 , EISCO ≃ E0 + η E1 , LISCO ≃ L0 + η L1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='13) By solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='12) at leading order one can determine the value of (r0, E0, L0), respectively the value for the radius, the energy and the angular momentum of the ISCO for an unperturbed Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' They are r0 = 6 M , E0 = 2 √ 2 3 , L0 = 2 √ 3 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='14) 8We recall that we consider only up to first order contributions in the small-tide approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 9which is not a gauge-invariant quantity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' see discussion at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 10From now on, we will drop the symbol of the secular average ⟨·⟩ for the sake of presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 19 At the first order in η, the first corrections to the ISCO quantities are given by r1 = 3072 M , E1 = −152 √ 2 3 , L1 = −348 √ 3 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='15) Note that we fixed our conventions for η in order to precisely reproduce the same numerical values of (r1, E1, L1) previously obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' However, while the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [25] are only valid in the weak-field approximation where ˆr ≫ M⋆ and on the equatorial plane ˆθ = π/2, our results are more general and hold for any value of ˆr and ˆθ, as we discussed earlier in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' It is also possible to compute the shift in the ISCO orbital frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In general, for quasi- circular orbits, the orbital frequency can be determined by means of the ratio [25,47,49] Ω2 = �uφ ut �2 = 1 2r2 �2M r − (r − 3M) uµuν∂r⟨hµν⟩ � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='16) where uµ are the components of the four-velocity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' To first order in η, we obtain ΩISCO ≃ Ω0 + η Ω1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='17) where 11 M Ω0 = 1 6 √ 6, M Ω1 = − � 2 3 491 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='18) This gives the shift induced by the tidal fields in the orbital frequency of the ISCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [47], the angular frequency Ω can be used to compute a gauge-independent measure of the radial separation between the Schwarzschild black hole and the test particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' One defines RΩ = �M Ω2 �1/3 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='19) so that according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='17) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='18) RΩ ≃ 22/3M Ω2/3 0 � 1 − 2 3ηΩ1 Ω0 � = 6M + 3928η M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='20) We notice that this gives a different radial shift than in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' However, this is not surprising as the radial shift of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='15), unlike the above, is not gauge-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='3 Tidal effects around the photon sphere The photon sphere around a Schwarzschild black hole is composed by the last stable circular orbits for massless test-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Differently from the case of the ISCO, this orbit is only specified by two parameters: the photon sphere radius and the impact parameter b = L/E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' A previous analysis of the photon sphere properties in a tidal environment can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [40], under more limited assumptions than the ones considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' From the secular Hamiltonian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='7), one enforces the conditions ⟨H⟩|r=rPS = 0 , d⟨H⟩ dr ���� r=rPS = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='21) 11Notice that this result agrees with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [40] (but not with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [25]), after a rescaling of -1/2 of the η parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' For the ease of comparison, our radial configuration (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 1) is called polar companion configuration in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [40]: this can be obtained in the weak-field limit ˆr ≫ M∗ and for β = π/2 and χ = −π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 20 By expanding the kinematic quantities in the tidal parameter η to retain only the leading contribution of the tidal secular effects in the small-tide approximation, one obtains rPS ≃ r0 + η r1 , bPS ≃ b0 + η b1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='22) where the unperturbed values for the Schwarzschild black hole are obtained by solving (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='21) at the leading order r0 = 3 M , b0 = 3 √ 3 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='23) Similarly, the tidal corrections are given by r1 = −30 M , b1 = 30 √ 3 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='24) This results generalize the one obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [40] for the special configuration of polar com- panions (equivalent to our radial configuration), after a rescaling of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Again, the orbital frequency at the photon sphere at first order in the tidal corrections can be computed in general from Ω = uφ ut = 1 b , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='25) which at first order in η yields to ΩPS ≃ Ω0 + η Ω1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='26) By means of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='23) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='24), one directly obtains the shift in the frequency of the photon sphere, given by M Ω0 = 1 3 √ 3 , M Ω1 = − 10 3 √ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='27) 6 Conclusions and outlook We conclude by summarising our new results and discussing further developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2, we retraced the computation performed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [36] for the construction of the Marck’s tetrad, defining a local inertial frame which is parallel-transported around a time-like geodesic in Kerr spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Tidal effects induced by a Kerr black hole are obtained by projecting the Weyl tensor on certain components of the Marck’s tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' While the components of the rank- 2 tensor Cij were computed in Marck’s paper [36], the components of the rank-3 tensor Cijk were previously known only on the equatorial plane of a Kerr black hole [37,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This paper therefore fills the gap in the literature: the explicit expressions for Cijk are given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Our result is valid for generic angles ˆθ and for arbitrary time-like geodesics in the Kerr spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 3, we found a natural application of the tidal tensors computed in the previous sec- tion in the modeling of a hierarchical three-body system in General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We considered a 3-body system describing a supermassive rotating black hole of mass M∗ and an EMR bi- nary system, made of a non-rotating black hole of mass M ≪ M∗ and a smaller companion of mass m ≪ M, which gravitates around the supermassive black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In order to go be- yond the post-Newtonian approximation, in which the three bodies are sufficiently distant from each other to be treated as point-like masses, and capture strong general relativistic effects, one can model the region around the non-rotating black hole in terms of a tidally-deformed Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' To this aim, it is convenient to decompose the tidal tensor in terms of irreducible representations of the rotation group, so as to construct “electric” E and “magnetic” B quadrupole tidal moments, that encode the leading-order deformations to the Schwzarschild metric immersed in a generic tidal environment [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' By approximating the motion of the small- est body as that of a test-mass, it is possible to take into account all the possible configurations 21 of the binary system by introducing two Euler’s angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Another new result obtained in this work is the explicit expressions for the electric and magnetic quadrupole tidal moments given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='10)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='13), that take into account arbitrary orientations of the binary system with respect to the source of the tidal deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We remark that these expressions are valid for arbitrary sources of tidal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This can be of interest for numerical simulations and analytical study of binary systems immersed in a tidal environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' For the case of a supermassive Kerr black hole, the tidal moments (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='13) together with our result in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2 allow us to analytically compute tidal effects induced by a Kerr black hole in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The hierarchy of masses makes it natural to study the dynamics of the binary system in the secular approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' As first pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [25], the tidal effects perturb the secular Hamiltonian for the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Remarkably, at the quadrupole approximation, the tidal perturbation can be recast into an effective perturbative parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The main result of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 4 is a general expression for η given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' It holds at the quadrupole order in the small- tide regime and in the secular approximation, and it models the deformed secular dynamics of a binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Our η generalises results obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [25] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [40] to arbitrary orientations of the binary system and tidal effects induced by a rotating black hole, including the strong gravity regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Tidal deformations induce changes in certain gauge-invariant quantities characterising the EMR binary systems, such as the orbital frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Such tidal deformations induced by the environment are completely encoded in the effective perturbative parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We devoted Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 5 to the study of such shifts in the case of marginally stable orbits for massive (ISCO shifts) and massless (photon sphere shifts) test-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We also addressed the issue of the gauge invariance of the shifts in the secular approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' While we focus on the case of a Kerr black hole as the perturber, one can also use our expressions with general tidal moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' For a Kerr perturber, the expression for η (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='9)) shows the rich phenomenology of the triple system: it combines the parameters of the background Kerr metric (M∗ and a), the location of the geodesic where the binary system is located (ˆr, ˆθ, K), and the Euler angles that capture the geometric orientation of the binary system with respect to the Kerr perturber (β and χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Our parameter η includes strong general relativistic effects of an EMR binary system which is affected by the presence of a large Kerr black hole, and considerably generalises the setup considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [25] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [40] beyond the weak-field regime and for arbitrary configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' As an example of a regime which was previously overlooked in the literature, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='2, we focused on the case in which the EMR system is placed on the ISCO of the Kerr background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We also derived configurations of the EMR system for which the tidal effects vanish in the secular approximation, generalising the findings of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' There is a number of directions in which this work can be further extended, and for which the results obtained here can be of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' In this paper, we analyze triple systems whose dynamics is stationary in time and restricted to circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This implies that we do not have gravitational waves in our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We also work in the leading quadrupole approximation for the tidal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The setup in this paper, though simplified, is useful to get analytic results and it should be considered as a first step towards a more realistic scenario that can be relevant for astrophysical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' An extension of this work would include higher-order effects beyond the quadrupole approx- imation [50] and the stationary regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' It would be interesting to further develop waveforms from triple hierarchical systems [51,52] and approaches to effective description thereof [53,54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Another natural development would be extending this study where the primary companion of the EMR is a Kerr black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The metric for a rotating black hole deformed by tidal effects has been derived in full generality in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [55] by solving the Teukolsky equation and using metric reconstruction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Due to the very complicated structure of that metric, a simplified version obtained in the small-spin regime has been obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [56], explicitly written 22 in terms of tidal quadrupole moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This is sufficient to capture all the main important features of spacetimes with non-vanishing angular momentum, and can lead to an even richer phenomenology – including couplings between the spins of the two black holes – possibly already at the level of the secular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' A third interesting direction concerns the analysis of eccentric binary systems subject to tidal deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' For this specific case it is probably more convenient to use the action-angle variables formalism [57–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' This would allow us not only to extend our computation to the case of elliptic orbits for the test particle in the binary system, but also to study the precession of the orbits around the Schwarzschild black hole and the presence of possible resonances in the binary system [61,62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Acknowledgments We thank P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Cole, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Liu and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Samsing for interesting discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' We thank V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Car- doso for useful comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' acknowledge support from Fondo Ricerca di Base 2020 (MOSAICO) and 2021 (MEGA) of the University of Perugia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The work of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' is supported in part by the project “Towards a deeper understanding of black holes with non-relativistic holography” of the Independent Research Fund Denmark (grant number DFF-6108-00340).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' The work of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' is supported by the R´egion ˆIle-de-France within the DIM ACAV+ project SYMONGRAV (Sym´etries asymptotiques et ondes gravitationnelles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' thank the Niels Bohr Institute for hospitality at different stages of this project.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Kavanagh and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Nichols, Measuring the dark matter environments of black hole binaries with gravitational waves, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' D 105 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 4 043009 [2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='04154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Cole, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Bertone, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Coogan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Gaggero, T.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Tomaselli, Disks, spikes, and clouds: distinguishing environmental effects on BBH gravitational waveforms, 2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='01362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Bonetti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Haardt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Sesana and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Barausse, Post-Newtonian evolution of massive black hole triplets in galactic nuclei – I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Numerical implementation and tests, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Soc.' metadata={'source': 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E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Barausse, Post-Newtonian evolution of massive black hole triplets in galactic nuclei – II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Survey of the parameter space, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 477 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 3 3910–3926 [1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='06088].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Bonetti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Sesana, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Barausse and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Haardt, Post-Newtonian evolution of massive black hole triplets in galactic nuclei – III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' A robust lower limit to the nHz stochastic background of gravitational waves, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 477 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 2 2599–2612 [1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='06095].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Bonetti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Sesana, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Haardt, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Barausse and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Colpi, Post-Newtonian evolution of massive black hole triplets in galactic nuclei – IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Implications for LISA, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 486 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' 3 4044–4060 [1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content='01011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' [23] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQfEwwN/content/2301.04879v1.pdf'} +page_content=' Yunes, M.' 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a/BdE2T4oBgHgl3EQf8gnM/content/tmp_files/2301.04220v1.pdf.txt b/BdE2T4oBgHgl3EQf8gnM/content/tmp_files/2301.04220v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d90f8f2c0fabdf84b031709496beb91bf22fc4a8 --- /dev/null +++ b/BdE2T4oBgHgl3EQf8gnM/content/tmp_files/2301.04220v1.pdf.txt @@ -0,0 +1,3831 @@ +Correlative mapping of local hysteresis properties in VO2 +Melissa Alzate Banguero,1 Sayan Basak,2, 3 Nicolas Raymond,1 Forrest Simmons,2, 3 Pavel Salev,4, 5 +Ivan K. Schuller,5 Lionel Aigouy,1, ∗ Erica W. Carlson,2, 3, 1, † and Alexandre Zimmers1, ‡ +1Laboratoire de Physique et d’´Etude des Mat´eriaux, ESPCI Paris, +PSL Universit´e, CNRS, Sorbonne Universit´e, 75005 Paris, France +2Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA +3Purdue Quantum Science and Engineering Institute, West Lafayette, IN 47907, USA +4Department of Physics and Astronomy, University of Denver, Denver, Colorado 80208, USA +5Department of Physics and Center for Advanced Nanoscience, +University of California San Diego, La Jolla, California 92093, USA +(Dated: Thursday 12th January, 2023) +We have developed a new optical microscopy technique able to track micron-sized surface clusters +as temperature is varied. Potential candidates for study include phase separated metal-insulator +materials, ferroelectrics, and porous structures. Several key techniques (including autofocus, step +motor/cross correlation alignments, single-pixel thresholding, pair connectivity correlation length +and image convolution) were implemented in order to obtain a time series of thresholded images. +Here, we apply this new method to probe the archetypal phase separated insulator-metal transition in +VO2. A precise time and temperature series of the insulator-metal transition was achieved, allowing +us to construct for the first time in this material spatial maps of the transition temperature Tc. +These maps reveal the formation of micron-sized patterns that are reproducible through multiple +temperature sweeps within ∼0.6°C, although a few isolated patches showed Tc deviations up to +±2°C. We also derive maps of the local hysteresis widths ∆Tc and local transition widths δTc. +The hysteresis width maps show an average width of ∆Tc =4.3°C, consistent with macroscopic +transport measurements, with, however, small regions as low as ∆Tc∼[0°C-1°C], and as high as +8°C. The transition width δTc maps shows an average of 2.8°C and vary greatly (from 0°C to +8°C), confirming the strong inhomogeneities of Tc in the subpixel structure. A positive correlation +between Tc value and hysteresis width ∆Tc is observed by comparing the spatial distributions of each +map. Finally, individual pixels with unique transition characteristics are identified and put forward. +This unprecedented knowledge of the local properties of each spot along with the behavior of the +entire network paves the way to novel electronics applications enabled by, e.g., addressing specific +regions with desired memory and/or switching characteristics, as well as detailed explorations of +open questions in the theory of hysteresis. +I. +INTRODUCTION +Electronic phase separation commonly emerges in a +wide variety of quantum materials such as high-Tc su- +perconductors [1], colossal magnetoresistance mangan- +ites [2], insulator-metal transition (IMT) materials [3], +multilayer rhombohedral graphene [4],etc. An archety- +pal example of a phase-separated material is vanadium +dioxide, VO2, which undergoes a 1st order IMT at Tc +∼68°C [5] (i.e., just above room temperature) accompa- +nied by an abrupt several-order-of-magnitude resistivity +decrease and monoclinic-to-tetragonal structural change. +The exact nature of the transition, whether it is a Peierls +transition driven by electron-phonon interactions or a +Mott-Hubbard transition driven by electron-electron in- +teractions, is still under debate [6]. In the vicinity of the +transition, VO2 exhibits a spatial coexistence of metal +and insulator domains that form intricate patterns [7]. +Analyzing the shape, characteristic size and scaling prop- +erties of those patterns can yield valuable information +∗ lionel.aigouy@espci.fr +† ewcarlson@purdue.edu +‡ azimmers@espci.fr +about the fundamental interactions that drive the tran- +sition [8]. Therefore, understanding and controlling the +phase-separate state in quantum materials has become a +major research field in recent years [9]. +Currently, phase separation imaging in quantum mate- +rials reported in the literature mostly comes from scan- +ning probe techniques such as STM [1, 2] and s-SNIM +[7, 8]. +While these methods have a very high spatial +resolution, fine temporal resolution remains hard to im- +plement since scanning probes are very time-consuming. +Moreover, STM lacks resolution at room temperature +and loses registry as the temperature is changed [10]. To +solve this we have developed a new microscopy method +to map out clear and stabilized images of the IMT. This +optical method allows the precise filming of the transi- +tion with hundreds or even thousands of images taken +in quick succession (∼10 seconds per final image). This +allows us to not only follow fine details in the time evo- +lution of the metal-insulating patches but also to filter +out thermal noise if needed. We first describe the sam- +ple preparation and optical response. We then describe +the experimental steps necessary to achieve this map- +ping. +While most steps are straightforward, four new +crucial steps were keys to this study: “Height z focusing”, +“Single pixel time traces”, “Pair connectivity correlation +arXiv:2301.04220v1 [cond-mat.str-el] 10 Jan 2023 + +2 +length” and “Time domain convolution”. These techni- +cal developments allowed us to acquire accurate spatial +maps of transition temperature distribution, from which +the phase separation patterns can be easily obtained at +any given temperature. The Tc maps reveal multiple in- +teresting features including the presence of spots with an +extremely large or nearly absent hysteresis of the IMT, a +positive correlation between the Tc value and the hystere- +sis width, and high cycle-to-cycle reproducibility of the +transition. The detailed knowledge of local properties is +the necessary ingredient to develop and test basic phase +separation and hysteresis theories, as well as to gain mi- +croscopic understanding of the device performance for +practical applications of quantum materials. +II. +METHODS +A. +VO2 thin film epitaxy, resistivity, and +reflectivity +Vanadium dioxide thin films were prepared by reactive +RF magnetron sputtering of a V2O3 target (>99.7%, ACI +Alloys, Inc.) on an r-cut sapphire substrate. Sample A is +130nm thick and sample B is 300nm thick. A mixture of +ultrahigh purity (UHP) argon and UHP oxygen was used +for sputtering. The total pressure during deposition was +4mTorr, and the oxygen partial pressure was optimized +to 0.1mTorr (2.5% of the total pressure). The substrate +temperature during deposition was 600oC while the RF +magnetron power was kept at 100W. Grain size in these +films is typically found to be 40-130nm in 100-150nm +films [11]. Grain size is expected to typically be slightly +larger in the 300nm film. The sample is found to have +a relative 27% optical change in the visible range when +passing the IMT (see SI Sec.S1 for details). Gold elec- +trodes were deposited on top of the film, separated by +10µm (sample A) and 30µm (sample B). Both samples +showed a clear IMT (see Fig. S1) above 68oC as evidenced +by a drop in resistivity of 4 orders of magnitude [12]. +B. +Image/temperature recording +The optical experimental setup consists of a VO2 thin +film sample placed on a Peltier heater or a Linkam +Thms350V temperature controller inside a Nikon opti- +cal microscope in epi configuration (both the illumina- +tion and reflection of light travel through the same objec- +tive). Illumination in the visible range was used (halogen +lamp, no filters) [13]. Two surface sample images (sample +A 10µm×50µm and sample B 30µm×35µm) were mea- +sured around the focal point of 1mm in the visible range +using a ×150 magnification dry Olympus objective lens +with an optical aperture of NA = 0.9. The theoretical +lateral resolution is estimated to be δr= 1.22λ/(2 NA) = +370nm in the visible range using the Rayleigh criterion +[14]. +Temperature was measured using a Pt100 glued +next to the sample. Temperature sweeps (35oC≪Tc to +82oC≫Tc and back) spanning the entire IMT were per- +formed multiple times at a rate of 1°C/min, temperature +swept linearly, with temperature and images recorded ev- +ery ∼0.17°C. +C. +Height z focusing and x-y drift correction +Inevitable temperature dilation of the experimental +system during temperature sweeps brings the sample out +of focus during temperature sweeps. In order to com- +pensate for this z drift, we employ a “fuzzy focusing” +technique as follows. During the experiment, the sam- +ple was continually moved up and down 10µm every 10 +seconds by a piezoelectric crystal placed under it, in or- +der to bring the sample in and out of focus. +A stack +of 120 images was recorded this way for each tempera- +ture. Over the years, various metrics have been evaluated +for selecting the sharpest image in such a stack [16–18]. +Some studies focus explicitly on images that don’t have +sharp contrast [19], like the raw images acquired here (see +Fig. 2(m)). Most metrics reported perform well in select- +ing the focused image. We have first chosen one using the +compression rate of the recorded images [20]. This one is +based on the intuitive idea that, when very out of focus, +the sample surface will look homogeneously gray due to +blurring. In this case, the raw recorded Bitmap (BMP) +image can be highly compressed in lossless Tiff format +using a standard Lempel-Ziv-Welch (LZW) compression +protocol [21, 22], since nearly every pixel is the same. +On the contrary, when the sample is in focus, the image +contains much more information (since most pixels are +different from their neighbors), and the raw BMP im- +age cannot be compressed as much. Using this method, +one can determine the most sharply focused image in the +stack by selecting the one with the largest Tiff file size +[23, 24]. Among the 62,000 images of sample A acquired +during the 14 hour experiment (consisting of 3 major +temperature loops and 10 subloops [25]), we retain the +894 images that are in focus within 80nm. +A recent update of the microscope has allowed us to +select the best focused image of sample B during the +experiment. In the live selection process we have used +a computationally faster method based on image gra- +dient using the Tenengrad function [19]. Both metrics +cited above were vetted using micron-sized gold disks +lithographed on a glass substrate where the sharpest im- +age can be defined as the image with the sharpest step +function (gold to substrate). +Using the focusing stack +technique, we have also compared the image height on +the sample four corners. This allowed us to correct the +tilt of the sample (due to sample positioning using ther- +mal paste). The updated setup also uses a piezoelectric +PI Pifoc PD72Z1x to move the objective up and down +rather than moving the sample placed inside the Linkam +stage. The current setup can thus output an image ev- +ery 10s in focus on the full field of view as a function of + +3 +FIG. 1. +Schematics of the microscope and image analysis created specifically to measure spatial maps of clusters in VO2 +during the IMT while recording resistivity R(T) simultaneously. The sample was positioned on a Peltier heater or Linkam +Thms350V temperature controller to apply temperature ramps (bottom left). The sample height was varied by steps of 80nm +via a piezoelectric actuator placed under it. The best-focused images were chosen post-experiment using an image compression +method and Tenengrad function (described in Sec. II C). The height focus of the sample was thus controlled within 80nm +throughout the experiment. Fine xy plane drift correction within a single pixel was performed post-experiment (described in +Sec.II C). Camera sensitivity was normalized throughout the recording (described in Sec.S3 of the SI). Using this fully stabilized +image series, black and white thresholds were applied for each pixel individually, accurately determining if it is in the metallic or +insulating state (described in Sec. II D). We use this information to construct spatial maps of the local transition temperature +Tc, hysteresis width ∆Tc and transition width δTc. +temperature. +As the temperature is cycled repeatedly, in addition +to drifts along z-axis (perpendicular to the film), there +are also drifts in the xy plane (the plane of the film). +These thermal drifts were compensated: (i) live within +1µm using step xy motors below the sample and (ii) post +experiment using cross correlation to track and realign +part of the gold leads which contain imperfections (spots) +and rough edges with VO2 (see Fig. 5 (a)). Although +the lateral image resolution is limited by diffraction and +is estimated to be 370nm, the drift compensation tracks +each pixel (≈ 37nm wide) on the sample throughout the +whole experiment. +The remaining spatial variations we observe in re- +flected intensity from the VO2 region are primarily due +to changes in local reflectivity due to the IMT. However, +there can be other contributions to this spatial varia- +tion, including effects such as surface height variations +from sample warping, variations in film thickness, minor +surface defects, and even shadows cast from the 150nm +thick gold leads. There can even be differences in pixel +sensitivity in the camera itself. +Because each of these +contributions is independent of temperature (i.e. +con- +stant in time), their effects can be distinguished from +that of the temperature driven IMT, as described in the +next section. +D. +Single pixel scaled and binary thresholded +images +In order to isolate the changes in local reflectivity +which are due to the IMT, we introduce two novel image +processing techniques. We use single pixel time traces to +generate single pixel scaled images (panel (n) of Fig. 2), +as well as binary thresholded images (panel (o) of Fig. 2, +discussed in the following subsections). Both types of im- +ages begin by considering a full warming or cooling sweep +(i.e. from fully insulating to fully metallic, or vice versa) +to follow the intensity and analyze each pixel individu- +ally. As an example, Fig. 2 (a-l) shows the raw optical +intensity time/frame traces of 12 different pixels during +a cooling sweep. See S6 for the time traces of 1600 pix- +els from the center of the sample. In order to construct + +CcD camera +Height z focusing +x-y drift correction +Light +Microscope +Image sensitivity drift correction +source +R(T) +Single pixel intensity time trace +z piezoelectric +heater +Single pixel thresholded image +stage +VO2 +Temperature ramp +9 +80 +Temperature ( +60 +40 +△Tc map +STcmap +1 +2 +3 +Tc map +4 +Time (Hrs)4 +FIG. 2. Single pixel intensity normalization and thresholding process. (a-l) Representative single-pixel turn-on functions in +sample A during cooling. Blue traces are the raw intensity in 8-bit grayscale where 0 is black and 255 is white. The orange +traces are smoothed versions of the blue traces, in which we have applied an 11-point Gaussian convolution (σ=2.5). Purple +curves are the difference between the raw (blue) curve and the smoothed version (orange curve). The green curve is a numerical +derivative of the blue curve (discussed and used in SI Sec. S4), taken via a finite difference with a 10-point stencil [15]. (m) +Raw optical image (frame 847) partway through cooling for VO2 sample A. (n) The same image after the intensity is scaled, +pixel-by-pixel, such that light pixels are in the insulating phase and dark pixels are in the metallic phase. (o) The same image, +with metal and insulator domains, clearly delineated as black and white. Images are 7.3µm wide. +a single pixel scaled image, we normalize each individual +pixel’s 8-bit grayscale intensity time trace with respect to +itself, such that its maximum intensity is scaled to 1, and +its minimum intensity is scaled to 0. The resulting single +pixel scaled image is shown in Fig. 2(n). This type of im- +age is a relatively quick way to study the temperature de- +pendent IMT, as it eliminates temperature-independent +spatial variations that are not due to the IMT. +In order to construct a binary thresholded image which +clearly delineates metal and insulator domains, we must +define a criterion for when each pixel changes from metal +to insulator or vice versa. The orange curve in each of +the panels (a-l) in Fig. 2 is a Gaussian-smoothed version +of the raw time trace, using an 11-point Gaussian convo- +lution (σ=2.5). We use this smoothed time trace of the +intensity in order to determine the midway point inten- +sity for each individual pixel (shown by the red horizon- +tal dotted lines). We use the pair connectivity correla- +tion length to justify setting the threshold at midway, as +described in the following subsections (Secs. II D 1 and +II D 2). +This allows us to construct binary black and +white images of the metal and insulator domains at each +measured temperature, as shown in Fig. 2(o). Different +pixels go through the midway point at different frame +numbers, and therefore at different temperatures. We use +this information to construct spatial maps of the local +transition temperature Tc recorded at each pixel reveal- +ing the highly spatially-textured nature of the IMT in +VO2 [7, 8]. These Tc maps, as well as hysteresis width +∆Tc maps and transition width δTc maps, are presented +in the experimental results Sec. III. +1. +Pair Connectivity Correlation Length +As can be seen in the single pixel time traces shown in +Fig. 2 (see SI Figures. S6 for many more examples), each +pixel experiences a definite switch from metal to insula- +tor or vice versa, consistent with the Ising-type model we +have previously developed to describe the IMT in VO2 +thin films [8, 26]. While the Ising model was originally +developed to describe magnetic domains of orientation +“up” or “down”, here we map “up” and “down” to metal +and insulator domains. While the metal-insulator tran- +sition is first order, this transition ends in a critical point +as a function of quenched disorder. The influence of that +critical point is felt throughout a critical region, which +includes part of the first order line in the vicinity of the +critical end point.[8] We use the correlation length of the +pair connectivity correlation function to determine the +threshold between metal and insulator domains. +Dur- +ing the IMT, VO2 metal-insulator domains form intri- + +Horizontal pixel location +[40] +[80] +[120] +150 +Raw +Convolved (11pt) +[400] +Derivative (1lpt) +a) +b) +Convolved-Raw +c) +75 +(Min+Max)/2 +Max Slope +0 +150 +Vertical pixel location +[300] +Pixel intensity +d) +e) +f) +75 +0 +150 +[200] +g) +h) +75 +0 +150 +[100] +j) +k) +D) +75 +0. +840 +880 +840 +880 +840 +880 +Frame NumberRaw +Single Pixel Scaled +Single Pixel Threshold +Image +[Min,Max]-->[0,1] +(Min+Max)/2 +m) +h +a +b +400 +300 +9 +h +200 +100 +40 +80 +120 +40 +80 +120 +40 +80 +1205 +FIG. 3. Pair connectivity correlation length ξpair vs. temper- +ature during the warming branch of an extremal hysteresis +loop, as a function of different threshold values for determin- +ing metal and insulator domains in sample A. The correlation +length diverges when the system is closest to criticality. +cate patterns, often becoming fractal due to proximity +to a critical point [8]. At criticality, correlation lengths +diverge. Away from criticality, the divergence is muted, +although the correlation length still displays a maximum +at the point of closest approach to criticality. For exam- +ple, changing the interaction strength between metal and +insulator domains to be farther away from criticality, or +changing the strength of various types of disorder farther +from criticality causes the correlation length to go down. +Similarly, changing the intensity threshold by which we +identify metal and insulator domains also changes this +correlation length. In disordered systems, setting an un- +physical threshold will not move the system toward crit- +icality, but only away. +Therefore, one way to set the +proper threshold between metal and insulator domains is +to maximize the correlation length. +The pair connectivity correlation function is familiar +from percolation models, where the corresponding pair +connectivity correlation length diverges at the critical +point [27]. Coniglio and coworkers showed that the pair +connectivity correlation length also diverges at the criti- +cal temperature in the two-dimensional Ising model [28]. +We have recently shown that the pair connectivity corre- +lation length also diverges at other Ising critical points, +including that of the two-dimensional random field Ising +model [29], as well as on slices of three dimensional mod- +els at criticality, including the clean Ising model [30] and +the random field Ising model [29]. Near a critical point, +the correlation function is power law at distances less +than the correlation length, in this case ξpair. This pair +correlation length can be calculated directly from an im- +age via [31]: +ξ2 +pair = +� +i,j r2 +i,jpf +i,j +� +i,j pf +i,j +(1) +FIG. 4. +(a) Single pixel time trace of intensity. +The blue +curve is the raw time trace of the measured optical intensity +of pixel (127,734) in sample B. The orange curve is a Gaussian +convolution (σ=2.5) of the same time trace over 3 frames. The +double crossing at the midway is eliminated in the smoothed +data set. (b) Binary black and white image (frame 260) of the +sample generated by thresholding at midway the single pixel +time traces as presented in (a). +(c) Smoothed out binary +black and white image (frame 260) of the sample generated +by thresholding at midway the 3 frame convoluted single pixel +time traces as presented in (a). +where pf +i,j is the likelihood that i and j are in the same +finite cluster. Another way to view this is as: +ξpair = +� +⟨R2 +G⟩f +(2) +where RG is the radius of gyration of each connected +cluster, and the average is taken over the finite clusters. +This quantity diverges at the percolation threshold as: +ξpair ∝ +1 +|p − pc|νpair . +(3) +It diverges at clean Ising transitions as: +ξpair ∝ +1 +|T − Tc|νpair , +(4) +and it diverges at random field Ising transitions as: +ξpair ∝ +1 +|R − Rc|νpair . +(5) + +2.5 +Threshold +-10% +2.0 +-7.5% +Correlation Length [μm] +-5% +-2.5% +1.5 +Midway ++2.5% ++5% +1.0 ++7.5% ++10% +0.5 +0.0 +45 +50 +55 +60 +65 +70 +75 +Temperature [oC]a +Raw +90 +Conv (3pt) +(Min+Max)/2 +85 +80 +Pixel Intensity +75 +70 +65 +60 +55 +0 +100 +200 +300 +400 +500 +600 +Frame Number +3 point convoluted +Raw binary image +binary image6 +2. +Setting Thresholds of Metal and Insulator Signal in +Optical Data +In order to know at what intensity to set the threshold +between metal and insulator in each pixel, we calculate +the pair connectivity correlation length in a series of im- +ages, as a function of different intensity thresholds. For +this we use the single pixel scaled images as described in +the previous subsection. In Fig. 3, we plot the evolution +of the pair connectivity correlation length (Eqn. 1) during +the warming branch of a hysteresis loop. The blue circles +in Fig. 3 have each pixel’s threshold set at the midway +point of that particular pixel’s intensity. The black circles +have each pixel’s threshold set higher by an amount that +is +10% of the difference between the saturated metal +and saturated insulator values of intensity. The pink cir- +cles have each pixel’s threshold set higher by only +7.5%, +and similarly for other colors as denoted in the figure leg- +end. Similar to the way the theoretical threshold was set +in Ref. [8], we set the threshold according to the longest +correlation lengths. Since in Fig. 3 the longest correla- +tion length happens for a threshold equal to the average +between metal and insulator intensity (the blue circles +in Fig. 3) we use this midway threshold throughout the +paper. +E. +Time domain convolution +One of the strong points of obtaining a series of 100- +1000 images via this autofocus optical microscope is the +possibility of filtering out high frequency noise. A simi- +lar technique is used in resistivity experiments that probe +samples thousands of times per second. Fig. 4 (a) com- +pares a raw single pixel time trace to a smoothed ver- +sion in which a 3-point Gaussian convolution (σ=2.5) +has been applied in the time domain. In this example, +the raw single pixel time trace crosses the midway point +twice, whereas the 3-point convolved curve passes the +midway point only once. Notice that this procedure of +filtering high frequency noise in the time domain greatly +suppresses the white noise evident in the spatial domain +near the metal-insulator boundaries derived from the raw +time traces (see Fig. 4 (b) and (c) for comparison). This +smoothing is useful for studying spatial correlations from +frame to frame. However, if filtering is not necessary, raw +data is used throughout the analysis. This is the case for +Tc maps in the section below and ramp reversal memory +maps presented elsewhere [25]. High frequency noise was +filtered in the temperature data taken using the Pt100 +by fitting a linear slope through the large temperature +sweeps. This matched the internal temperature sensor +slope of the Linkam Thms350V temperature controller. +III. +RESULTS +Having described the various key steps in the previ- +ous sections (including autofocusing, step motor/cross +correlation aligning, single pixel scaling and threshold- +ing, pair connectivity correlation length analysis, and +time domain convolution) we now present the detailed +spatially-resolved study of the IMT in VO2 films using +our new optical mapping method. +Maps +Transition Temperature Tc maps: Fig. 5 (c) re- +ports the local critical temperature Tc map in VO2 sam- +ple B. These maps show a large spatial variation in Tc, +with rich pattern formation over tens of microns, similar +to s-SNIM sub-micron measurements [7], but acquired +with a much faster procedure that allows for much finer +time and temperature resolution. This large scale spatial +variation, along with detailed spatial knowledge of the lo- +cation of these variations, can potentially be exploited to +optimize memory elements by addressing specific regions +of the sample. +Reproducibility of Tc maps: +Previous reports on +avalanches in this material showed jumps in resistivity +randomly appearing during the transition in macroscopic +transport measurements [33]. +This suggested that the +metal-insulator patterns could be appearing randomly +during each temperature sweep. +At first glance, this +appears to be at odds with the optical data reported +in this study, where we find that the metal and insu- +lator patterns are highly repeatable globally (occurring +at the same location and with the same shape) during +successive temperature sweeps (see Fig 6). The repeata- +bility suggests that the patterns are strongly influenced +by an underlying random field present in the thin film +or its substrate [8, 26, 34]. The observed stochasticity +of resistance jumps in transport measurements [33] could +arise from small variations in the exact time at which +avalanches are triggered. In addition, small changes in +optical maps can potentially create large changes in re- +sistance, when tiny “shorts” connect pre-existing larger +metallic clusters. +Transition Width δTc maps: The transition width +δTc of each pixel can be accessed by fitting single pixel +scaled intensity time traces to a hyperbolic tangent: +− 1 +2(tanh( T−Tc +δTc )-1). Because Tc is known from our time +trace analysis, there is only one fitting parameter. The +map of δTc distribution is shown in Fig. 5 (e). The aver- +age transition width of the pixels as measured in optics +is 2.8 ± 1.1°C with extremes from 0°C to 8°C. Moreover, +a small number of pixels show more than one step dur- +ing a transition (see for example first pixel (305,300) in +Fig. S6). These cases could arise from an overlap be- +tween multiple metal or insulator domains affecting a +single pixel. This could be due to information from sur- +rounding pixels affecting the signal at one pixel, since the + +7 +FIG. 5. (a) Optical image of VO2 sample B during the insulator (light gray) to metal (dark gray) transition (warming cycle), +two gold leads are seen at the top and bottom. +These electrodes also display some structure (spots) due to gold surface +imperfections. Contrary to VO2 IMT structures seen in this image, gold imperfections do not change with time (see online +movie [32]). Usually these imperfections are purposely washed away using strong image brightness. Here, on the contrary, +brightness was set low to see and use these imperfections to autoalign within a pixel the images and thus compensate xy +thermal drifts. +Sapphire substrate is the dark surface. +One can easily see the metal dark patches appearing. +Scale bar +is 10µm. (b) Single pixel intensity curve defining critical temperature Tc, hysteresis width ∆Tc and transition width δTc. +Tc were determined at midways as explained in the main text. Hysteresis width was determine by taking the temperature +differences between heating and cooling cycles Tc +up-Tc +down. Transition width was determined by fitting (smooth curve) the +single time trace to a hyperbolic tangent: − 1 +2(tanh( T −Tc +δTc )-1). (c) Local critical temperature Tc map, (d) ∆Tc maps, (e) δTc +map (presented here for the temperature ramping up branch). Image are 27.6µm high. Histograms (with mean and standard +deviation of maps a), b) and c) are shown in Fig. 7 +pixel size is ∼10 times smaller than the resolution. Or, +it could arise from structures that are smaller than the +pixel size. Indeed, s-SNIM has clearly observed inhomo- +geneities on smaller length scales than the optical maps +presented here [7, 8]. Interestingly, the standard devia- +tion of local Tc’s across the sample, σTc(1.2°C), is smaller +than the average transition width of pixels δTc(2.8°C). It +remains an open question whether the self-similar metal- +insulator domain patterns discussed in Ref. [8] could be +the source of this difference. +Hysteresis Width ∆Tc maps: By subtracting Tcup- +Tcdown (see the caption of Fig.5 (b) for the definition) one +can construct a hysteresis width ∆Tc map. The hystere- +sis width ∆Tc map is shown in Fig. 5 (d) for sample B. +The average width is found to be 4.3 ± 1.1°C as seen in +macroscopic transport measurements. However, certain +small regions have small ∆Tc, in the range [0°C - 1°C] +(small blue clusters in Fig. 5 (d)). Probing these region +with other local probes could shed light on whether this is +an intrinsic property of these regions. These hysteresis- +free patches could be very useful in multiple switching +applications such as optical electronic devices. Indeed it +has been shown that the presence of a large hysteresis +in VO2 greatly complicates using it as an optical sensor +[35]. +Correlations between maps +With all of the maps above, one can check for cor- +relations between these quantities. Fig. 8 plots Tc vs. +∆Tc, ∆Tc vs. δTc and Tc vs. δTc for each pixel. A few +horizontal and diagonal lines appear in these plots. The +horizontal lines come from multiple pixels (spatially close +by) switching at the same temperature (upon warming). +The diagonal lines come from multiple pixels (spatially +close by) switching at the same temperature (upon cool- +ing). Although this is typically what one would expect + +1.2 +b) +a + Single pixel scaled intensity +1.0 +Single pixel scaled +0.8 +intensity time trace +0.6 +△T +T,down +dn + 0.4 +0.2 +0.0 +2 STc +-0.2 +50 +60 +70 +80 +Temperature °C +T_map +△T_map +ST_map +c) +T [°C] +d) +T [°C] +e) +T [°C] +C +C +72 +8 +12 +7 +71 +10 +70 +5 +8 +69 +4 +68 +6 +3 +67 +2 +4 +66 +2 +65 +0 +0 +648 +FIG. 6. a) Three Tc maps while cycling through the IMT (warming) at 1°C/min. b) Difference maps between cycles. Global +patterns are generally reproducible (σTc/Tc = 0.6°C/68°C= 1%). However some small regions present deviations up to ±2°C. +Full histograms (with mean and standard deviation) of maps in b) are shown in Fig. 7. Difference map between Tc3 and Tc1 +(the most separated, time wise, temperature sweeps in this study) and the corresponding histogram are presented in SI Fig. S4. +Images are 33.6µm x 27.6µm. +from avalanches, further analysis is needed to extract the +full dynamics occurring. In the three correlation maps, +no trend is seen in the last two, but Tc vs. ∆Tc shows a +slight positive correlation. This means that pixels with +low Tc tend to have low ∆Tc (i.e. close to zero) and vice +versa. The positive correlation in Fig. 8(a) is not to be +confused with the few diagonal lines present in this panel +explained just above. +Hand picking specific hysteric properties +The wide range of behaviors contained in the three +maps presented in the section above (Fig. 5 c, d and e), +gives us the unprecedented opportunity to find individual +pixels with desired properties. Fig. 9 shows the Tc map of +the sample with six different types of pixels selected. The +pixel labeled “std” for standard has a rounded transition +with values of Tc, ∆Tc and δTc which are close to the +average values found in the distribution of these three +quantities (see Fig. 7 a, b and c). +Pixels A and B show the most common type of local +characteristics found in the maps: when Tc is high, ∆Tc +is high; when Tc is low, ∆Tc is low. This positive correla- +tion is evident at a global level in Fig. 8 (a). However, on +a local level, individual pixels can have a large deviation +from the global average behavior. Indeed pixel E shows +a possibility of finding ∆Tc very low (0.3°C) with a Tc +(66.3°C) low but closer to the mean value of the map. +Pixels C and D illustrate the case where the width δTc +of the transition is very sharp (0.5°C) or very wide (5°C). +Pixel C shows a representative sharp pixel, where within +the temperature steps of 0.17°C, the transition occurs in +a sharp, avalanche mode. Further analysis to see where +and how these avalanches occur will be pursued in future +work. +Finally pixel E shows a case where ∆Tc is within the +lower values [0°C-1°C]. As mentioned previously, small +hysteresis could be useful in opto-electronic devices or +neuromorphic devices. In the first case, small hysteresis +avoids optical detectors getting stuck in subloops [35]; +in the second case, small hysteresis allows lowering the +voltage threshold needed for spiking [36]. +General remarks on the pixel selection procedure: (i) +as mentioned previously in the δTc section above, some +pixels in the map clearly present two steps during the +IMT. These two-step pixels can potentially be detected +in an automated way from their anomalously high error +on the fit to the hyperbolic tangent function; (ii) the fea- +tures put forward in these 6 pixels above are not unique +to the 37nm square pixel location. These features usu- +ally also hold for many pixels around the xy coordinates +reported. + +Cycle # 2 - T.2map +72 +a +71 +69 +68 +67 +65 +64 +c2 +2 +b +T [°C] +3 +2 +1 +0 +-1 +-2 +39 +FIG. 7. Histograms of maps presented in in Fig. 5 and 6. (a) Tc maps (upon warming); (b) ∆Tc map; (c) δTc map and (d) +and (e) two difference maps Tc2-Tc1 and Tc3-Tc2 +FIG. 8. Correlations between Tc (upon warming), ∆Tc and δTc. Each of the 666,000 pixels (900x740) is represented. Only +Tc vs. ∆Tc (panel (a) shows a slight diagonal trend meaning that pixels with low Tc tend to have low ∆Tc (i.e. close to zero) +and vice versa. +IV. +CONCLUSIONS +We have reported the first Tc maps derived from sin- +gle pixel optical imaging on VO2. Multiple new exper- +imental steps were needed to align, focus and calibrate +the raw grayscale images recorded. These experimental +achievements allowed us to accurately track the spatial +distribution of metal and insulator clusters. Binary black +and white images, time traces, Tc maps, ∆Tc maps, and +δTc maps were plotted and discussed. The sample shows +micron-sized patterns that are found to be mostly repro- +ducible through multiple temperature sweeps. The ∆Tc +hysteresis width map exhibits, on average, the same av- +erage hysteresis width of 4.3°C as macroscopic resistiv- +ity hysteresis, but exhibits strong variation on a local +scale, down to ∼[0°C-1°C] in certain small regions and +as large as ∼ 8°C in other regions. These findings open +an exciting opportunity to access local properties of VO2 +by, e.g., contacting specific parts of the sample electri- +cally in order to select unique parameter combinations + +20 +80 +80 +b) +a) +c) +75 +75 - +15 +P0% +p +70 +70 +10 +ooo +65 +65 +5 +60 +60 +0 : +55 +55 +0 +5 +10 +15 +20 +0 +10 +12 +14 +2 +4 +6 +8 +10 +12 +14 +2 +4 +6 +8 +0 +△T, [°C] +[°C] +ST +ST +Cx104 +x104 +a) +c) +μ= 2.8 [°C] +μ= 68.2 [°C] +μ= 4.3 [°℃] +6 +5. += 1.1 [C] + = 1.2 [°℃] + = 1.1 [°℃] +4 + pixels +5 +4 +4 +3 +3 +Number +3 +2 +2 +2 +1 +1. +1 +0 +0 +64　66　68　70　7274 +2 +４6 +234５ +62 +0 +8 +10 +0 +678 +△T, [°C] +T,[°C] +ST,[°C] +x104 +x104 +d) +e) +10- +μ= 0.0 [°℃] +μ= 0.0 [C] + = 0.6 [°C] += 0.6 [℃C] +8 +8. +Number of pixels +6 +6 +4 +4. +2 +2 +0 +0+ +1234 +-4-3 -2 -1 0 +-4-3 -2 -1 +１234 +T.,-T., [C] +.[°C] +c210 +FIG. 9. Tc map with six pixels chosen to illustrate specific characteristics in the hysteresis loops. The table shows the numerical +values of Tc, ∆Tc and δTc for each pixel. The numbers in bold highlight the unique characteristic of each pixel. +for specific applications in electrical and optoelectronic +devices. +The observation of a positive correlation be- +tween Tc value and hysteresis width could enable a new +approach for tailoring the material’s response to exter- +nal drives, in addition to providing a new perspective in +studying open questions in the theory of hysteresis. +ACKNOWLEDGEMENTS +We thank M. J. Carlson for technical assistance with +image stabilization, and acknowledge helpful conversa- +tions with K. A. Dahmen. S.B., F.S., and E.W.C. ac- +knowledge support from NSF Grant No. DMR-2006192 +and the Research Corporation for Science Advancement +Cottrell SEED Award. S.B. acknowledges support from +a Bilsland Dissertation Fellowship. +E.W.C. acknowl- +edges support from a Fulbright Fellowship, and thanks +the Laboratoire de Physique et d’´Etude des Mat´eriaux +(LPEM) at ´Ecole Sup´erieure de Physique et de Chimie +Industrielles de la Ville de Paris (ESPCI) for hospital- +ity. This research was supported in part through com- +putational resources provided by Research Computing +at Purdue, West Lafayette, Indiana [37]. The work at + +D +0.8 +0.6 +std +Single pixel scaled intensity +0.4 +0.8 +0.2 +0.0 +0.0 +0.4 +50 +60 +70 +80 +40 +50 +09 +70 +Temperature [°C] +Temperature [°C] +0.2 +0.0 +40 +50 +70 +700 +60 +T_[C] +Temperature [°C] +72 +600 +71 +70 +500 +B +69 +400 +68 +300 +67 +0.4 +66 +0.0 +200 +50 +60 +0 +65 +Temperature [°C] +100 + 64 +A +01 +0.6 +200 +300 +500 +600 +100 +400 +700 +800 +900 +0.4 +E +0.0 +50 +40 +60 +Temperature [°C] + 0.2 +0.0 +40 +70 +50 +60 +Temperature [°C] +Label +(x,y) position +Specific +T. 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Polyakov, +and A. Shabalov, +Journal of Applied Physics 106, 104504 (2009). +[36] P. Maffezzoni, L. Daniel, N. Shukla, S. Datta, +and +A. Raychowdhury, IEEE Transactions on Circuits and + +12 +Systems I: Regular Papers 62, 2207 (2015). +[37] G. McCartney, +T. Hacker and B. Yang, +Educause +Review, 2014; https://er.educause.edu/articles/2014/7/ +empowering-faculty-a-campus-cyberinfrastructure- +strategy-for-research-communities. +[38] M. M. Qazilbash, A. A. Schafgans, K. S. Burch, S. J. +Yun, B. G. Chae, B. J. Kim, H. T. Kim, +and D. N. +Basov, Physical Review B 77, 115121 (2008). +[39] A. Zimmers, J. M. Tomczak, R. P. S. M. Lobo, N. Bon- +temps, C. P. Hill, M. C. Barr, Y. Dagan, R. L. Greene, +A. J. Millis, +and C. C. Homes, Europhysics Letters +(EPL) 70, 225 (2005). + +13 +SUPPORTING INFORMATION: CORRELATIVE +MAPPING OF LOCAL HYSTERESIS +PROPERTIES IN VO2 +S1. +VO2 Reflectivity +The fact that the metallic reflectivity of VO2 is lower +than that of the insulating phase in the visible range is +counterintuitive. +This is due to a subtle combination +of a Drude response as well as intraband and interband +transitions and thin film interferences in this material. +The largest reported spectra in VO2 was measured by +ellipsometry [38]. +Using the reported real part of the +optical conductivity σ1, we have calculated the reflectiv- +ity of the insulator and metallic states (see Fig. S2 and +S3). +This clearly shows that, as one would expect in +the infrared, the sample becomes highly reflective when +metallic. +Above the plasma frequency (∼12000cm−1), +interband transitions and spectral weight conservation +make the reflectivity curves cross, leading to the metallic +state having a lower reflectivity than the insulating state +in this range. The relative optical contrast in the visible +range (27%), is still more than sufficient in our setup +to identify both states clearly (as seen in a raw image +Fig. S1 (a)). +S2. +Key steps making this study possible +The key step that have allowed us completing this +study comes from the unique qualities of the VO2 ma- +terial : +- The IMT is above room temperature, which allows +close optical microscopy (strong objective ×150 with a +high numerical aperture 0.9 brought to 1mm focus above +the sample surface). This setup would be much harder +to achieve if cryogenic cooling (i.e. +a cryostat with a +window between the sample and objective) was needed. +- Phase separation was observed by s-SNIM at sub- +micron scales in this material [7, 8]. The fact that this +phase separation is still found up to 30µm makes these +optical microscopy surface maps possible. +- In the visible range, a relative 27% drop in the thin +film reflectivity is found in the metallic state Measuring +in the visible range gave us results with a 400nm reso- +lution. In the infrared, the contrast between metal and +insulator is much larger, as expected, but only allows +optical resolution up to the IR wavelength, i.e. 1-10µm. +FIG. S1. (a) 35µm wide etched VO2 sample B image with +30µm separated gap gold leads. The white square represents +the 33.6µm x 27.6µm region where Tc maps (Fig.s 5). Scale +bar is 10µm.(a) R(T) measurement of the IMT + +(a) +Gold +Substrate +vO2 +1000 +(b) +100 +Resistance (kΩ2) +10 +0.1 +0.01 +40 +50 +60 +70 +80 +Temperature (°C)14 +FIG. S2. +Simulated optical reflectivity of the insulating and metallic states in bulk VO2. +Optical functions were derived +by fitting standard Drude-Lorentz functions to ellipsometry measurements reporting the raw σ1 response in a large spectral +range at low and high temperatures [38]. This procedure [39] allows other optical functions to be deduced, such as reflectivity, +transmission, absorption, or dielectric constant. Reflectivities in this figure are not reported below 1000cm−1 as the fitting +procedure was not precise enough in this low frequency/high σ1 region. On the other hand, reflectivities in the visible region +(∼14000cm−1 to ∼25000cm−1) are in the middle of the spectral range and can be found with confidence. +FIG. S3. Simulated optical reflectivity of the insulating and metallic states of a 130nm VO2 thin film on an r-cut sapphire +substrate. Optical functions were found as described in Fig. S2. In contrast with the bulk reflectivity, a pronounced oscillation +can be seen in the blue insulating spectrum. This is due to interference in the 130nm thin film (for example, constructive thin +film interference creates a peak at ∼6700cm−1). Reflectivities are not reported below 1000cm−1 as the fitting procedure was +not precise enough in this low frequency/high σ1 region. On the other hand, reflectivities in the visible region (∼14000cm−1 +to ∼25000cm−1) are in the middle of the spectral range and can be found with confidence. + +photon energy (eV) +0 +1 +2 +3 +4 +1.0 +Metal +T=360K ab0ve T +0.8 +Insulator T=295K below T +G +Reflectivity +0.6 +0.4 +Microscope spectral range +0.2 +0.0 +0 +10000 +20000 +30000 +Wavenumber (cm-1)photon energy (eV) +0 +1 +2 +3 +4 +1.0 +Metal (thin film) +T=360K ab0ve T +0.8 +Insulator (thin film) +T=295K below +Reflectivity +0.6 +0.4 +Microscope spectral range +0.2 +0.0 +0 +10000 +20000 +30000 +Wavenumber (cm-1)15 +S3. +Image sensitivity drift correction +Whereas the relative average intensity of VO2 increases +almost 30% in changing from metal to insulator, the +change in sapphire reflectance in this temperature range +is negligible. We have used this fact to correct for any +changes in incident light or CCD detector sensitivity +throughout the experiment by dividing the average in- +tensity in the VO2 region by the intensity in the sapphire +region of the sample. +Details: We assume that the input intensity is a func- +tion of time I0(t) but spatially uniform. The reflected in- +tensity from any region is IR(t, x, y) = I0(t) × R(t, x, y). +Since the Sapphire’s reflectance does not vary signifi- +cantly over the range of temperature the sample went +through, it is assumed to be a constant. Let the spatially +averaged sapphire reflectivity be RS. Then, the spatial +average reflected intensity from the sapphire region is: +IS +R(t) = I0(t) × RS Any region of VO2 has a reflected +intensity: IV +R (t, x, y) = I0(t) × RV (t, x, y). +Therefore, +the ratio of reflected intensities from Sapphire and VO2 +is independent of input intensity: +IV +R (t, x, y)/IS +R(t) = +RV (t, x, y)/RS. We will use IS +R(t) as a reference to cor- +rect IV +R (t) for any variation due to fluctuation of ambi- +ent light. The quantity independent of input intensity: +RV (t, x, y) = RSIV +R (t, x, y)/IS +R(t), Hence, setting the ref- +erence input intensity I0(0), the corrected reflected in- +tensity from VO2 would be: +˜IV +R (t, x, y) = I0(0)RV (t, x, y) = IV +R (t, x, y) +IS +R(t)/IS +R(0) +S4. +Single pixel thresholded images: inflection +point +In the main text, we have set the threshold between +metal and insulator domains at the midway point of +the intensity, based on the pair connectivity correlation +length criterion described in Sec. II D. We also tested +another method of setting the threshold based on the +inflection point of the single pixel time traces. +The +green curves in panels (a-l) of Fig. 2 show a smoothed +derivative of the raw time traces, achieved by using a +finite difference with a 11-point Gaussian convolution +(σ=2.5) [15]. The vertical dotted green line shows the +extremum of this derivative, which locates the inflection +point of the orange curves. +Since the pixel switching +curves (orange and blue traces) exhibit a relatively rapid +change from metal to insulator, this inflection point at +which the pixel brightness is changing most rapidly is +the most natural place to assign a change from insulator +to metal and vice versa. Because we have used a stencil +with even number of 10, the inflection point happens +between frames, and allows us to clearly identify frames +which precede the inflection point (which are metallic) +from frames which come after the inflection point (which +are insulating). Notice that the frame number at which +the solid orange curves cross the dotted orange lines +coincides with the inflection point for each pixel. This +means that both methods are equivalent for determining +the frame number at which a pixel switches from metal +to insulator or vice versa. + +16 +FIG. S4. Map and histogram of the difference between Tc3 and Tc1. Although these Tc maps are the most separated, time +wise, in this study, they remain similar (mean and σ) to Tc2-Tc1 and Tc3-Tc2 presented in Fig.6. +FIG. S5. Online movie[32] screenshot of the ∼1500 in focus consecutive spatial maps of a 33.6µm x 27.6µm VO2 surface. +Central panels: raw, scaled and thresholded surface image (sample B) using “Single pixel scaled image” and “Single pixel +intensity time trace and threshold” methods. Left panels: corresponding histogram changes during temperature ramps. Top +right panel: average sample intensity (raw, scaled, thresholded) vs. frame number. Middle right panel: average sample intensity +(raw, scaled, thresholded) vs. sample temperature. Bottom right panel: Temperature protocol - 3 major temperature loop +spanning the entire IMT (36oC - 82oC), + +[。l 1 +c3 +X104 +3 +10 +μ= 0.0 [°C] + = 0.6 [℃C] +Z +8 +Number of pixels +6 +0 +4 +-1 +2 +-2 +0 +¥-3-2 +-1 +0 +1 +2 +3 +4 +4 +-3 +T.,-T.,[°C] +C3×105 +raw +raw +1.0 +2.5 +1.0 +80 +0.8 +2.0 +0.8 +75 +0.6 +1.5 +70 +1.0 +0.4 +65 +0.2 +0.2 +0.5 - +60 +0.0 +0.0 +0.0 +40 +50 +60 +70 +80 +90 +0 +200 +400 +600 +800 +1000 +12001400 +Framenumber +×104 +scaled +scaled +3.5 +1.0 +1.0 +80 +3.0 +0.8 +0.8 +2.5. +75 +2.0 +0.6 +70 +1.5 +0.4 +65 +1.0 - +0.2 +田 +raw +0.2 +0.5 +田 + scaled +60 +0.0 +thresholded +0.0 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +40 +50 +60 +70 +80 +Temperature[°C] +×105 +thresholded +thresholded +1.0 +80 +6 +0.8 +5 +70 +4 +0.6 +60 +3 +0.4 +2 +50 +0.2 +1 +40 +0 +0.0 +0 +0 +200 +400 +800 +1000 +1200 +1400 +Framenumber17 +0 +50 +100 +150 +Intensity +(305,300) +Raw +11pt Conv +(min+max)/2 +|Max Slope| +(305,301) +(305,302) +(305,303) +(305,304) +(305,305) +(305,306) +(305,307) 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+880 +Frame Index +(344,338) +820 +840 +860 +880 +Frame Index +(344,339) +11pt Slope +Raw - 11pt Conv +Max Slope Frame +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +0 +20 +40 +Slope/Noise +FIG. S6. ML3 time trace of sample A in a patch of 40×40 pixels in the middle of the sample. Each pixel coordinates are +indicated above the time trace. Description of the four curves in each mini panel is the same as the main text Figure 2. + diff --git a/BdE2T4oBgHgl3EQf8gnM/content/tmp_files/load_file.txt b/BdE2T4oBgHgl3EQf8gnM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..344875fac7077ce778e8771b5ecbccdecc9d5e4a --- /dev/null +++ b/BdE2T4oBgHgl3EQf8gnM/content/tmp_files/load_file.txt @@ -0,0 +1,2887 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf,len=2886 +page_content='Correlative mapping of local hysteresis properties in VO2 Melissa Alzate Banguero,1 Sayan Basak,2, 3 Nicolas Raymond,1 Forrest Simmons,2, 3 Pavel Salev,4, 5 Ivan K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Schuller,5 Lionel Aigouy,1, ∗ Erica W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Carlson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' † and Alexandre Zimmers1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' ‡ 1Laboratoire de Physique et d’´Etude des Mat´eriaux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' ESPCI Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' PSL Universit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Sorbonne Universit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' France 2Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Purdue University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' West Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' IN 47907,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' USA 3Purdue Quantum Science and Engineering Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' West Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' IN 47907,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' USA 4Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' University of Denver,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Denver,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Colorado 80208,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' USA 5Department of Physics and Center for Advanced Nanoscience,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' University of California San Diego,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' La Jolla,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' California 92093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' USA (Dated: Thursday 12th January,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 2023) We have developed a new optical microscopy technique able to track micron-sized surface clusters as temperature is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Potential candidates for study include phase separated metal-insulator materials, ferroelectrics, and porous structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Several key techniques (including autofocus, step motor/cross correlation alignments, single-pixel thresholding, pair connectivity correlation length and image convolution) were implemented in order to obtain a time series of thresholded images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Here, we apply this new method to probe the archetypal phase separated insulator-metal transition in VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' A precise time and temperature series of the insulator-metal transition was achieved, allowing us to construct for the first time in this material spatial maps of the transition temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' These maps reveal the formation of micron-sized patterns that are reproducible through multiple temperature sweeps within ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6°C, although a few isolated patches showed Tc deviations up to ±2°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We also derive maps of the local hysteresis widths ∆Tc and local transition widths δTc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The hysteresis width maps show an average width of ∆Tc =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='3°C, consistent with macroscopic transport measurements, with, however, small regions as low as ∆Tc∼[0°C-1°C], and as high as 8°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The transition width δTc maps shows an average of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='8°C and vary greatly (from 0°C to 8°C), confirming the strong inhomogeneities of Tc in the subpixel structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' A positive correlation between Tc value and hysteresis width ∆Tc is observed by comparing the spatial distributions of each map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Finally, individual pixels with unique transition characteristics are identified and put forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This unprecedented knowledge of the local properties of each spot along with the behavior of the entire network paves the way to novel electronics applications enabled by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=', addressing specific regions with desired memory and/or switching characteristics, as well as detailed explorations of open questions in the theory of hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' INTRODUCTION Electronic phase separation commonly emerges in a wide variety of quantum materials such as high-Tc su- perconductors [1], colossal magnetoresistance mangan- ites [2], insulator-metal transition (IMT) materials [3], multilayer rhombohedral graphene [4],etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' An archety- pal example of a phase-separated material is vanadium dioxide, VO2, which undergoes a 1st order IMT at Tc ∼68°C [5] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=', just above room temperature) accompa- nied by an abrupt several-order-of-magnitude resistivity decrease and monoclinic-to-tetragonal structural change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The exact nature of the transition, whether it is a Peierls transition driven by electron-phonon interactions or a Mott-Hubbard transition driven by electron-electron in- teractions, is still under debate [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In the vicinity of the transition, VO2 exhibits a spatial coexistence of metal and insulator domains that form intricate patterns [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Analyzing the shape, characteristic size and scaling prop- erties of those patterns can yield valuable information ∗ lionel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='aigouy@espci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='fr † ewcarlson@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='edu ‡ azimmers@espci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='fr about the fundamental interactions that drive the tran- sition [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Therefore, understanding and controlling the phase-separate state in quantum materials has become a major research field in recent years [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Currently, phase separation imaging in quantum mate- rials reported in the literature mostly comes from scan- ning probe techniques such as STM [1, 2] and s-SNIM [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' While these methods have a very high spatial resolution, fine temporal resolution remains hard to im- plement since scanning probes are very time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Moreover, STM lacks resolution at room temperature and loses registry as the temperature is changed [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' To solve this we have developed a new microscopy method to map out clear and stabilized images of the IMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This optical method allows the precise filming of the transi- tion with hundreds or even thousands of images taken in quick succession (∼10 seconds per final image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This allows us to not only follow fine details in the time evo- lution of the metal-insulating patches but also to filter out thermal noise if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We first describe the sam- ple preparation and optical response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We then describe the experimental steps necessary to achieve this map- ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' While most steps are straightforward, four new crucial steps were keys to this study: “Height z focusing”, “Single pixel time traces”, “Pair connectivity correlation arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='04220v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='str-el] 10 Jan 2023 2 length” and “Time domain convolution”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' These techni- cal developments allowed us to acquire accurate spatial maps of transition temperature distribution, from which the phase separation patterns can be easily obtained at any given temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The Tc maps reveal multiple in- teresting features including the presence of spots with an extremely large or nearly absent hysteresis of the IMT, a positive correlation between the Tc value and the hystere- sis width, and high cycle-to-cycle reproducibility of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The detailed knowledge of local properties is the necessary ingredient to develop and test basic phase separation and hysteresis theories, as well as to gain mi- croscopic understanding of the device performance for practical applications of quantum materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' VO2 thin film epitaxy, resistivity, and reflectivity Vanadium dioxide thin films were prepared by reactive RF magnetron sputtering of a V2O3 target (>99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='7%, ACI Alloys, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=') on an r-cut sapphire substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Sample A is 130nm thick and sample B is 300nm thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' A mixture of ultrahigh purity (UHP) argon and UHP oxygen was used for sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The total pressure during deposition was 4mTorr, and the oxygen partial pressure was optimized to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='1mTorr (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5% of the total pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The substrate temperature during deposition was 600oC while the RF magnetron power was kept at 100W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Grain size in these films is typically found to be 40-130nm in 100-150nm films [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Grain size is expected to typically be slightly larger in the 300nm film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The sample is found to have a relative 27% optical change in the visible range when passing the IMT (see SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='S1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Gold elec- trodes were deposited on top of the film, separated by 10µm (sample A) and 30µm (sample B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Both samples showed a clear IMT (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S1) above 68oC as evidenced by a drop in resistivity of 4 orders of magnitude [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Image/temperature recording The optical experimental setup consists of a VO2 thin film sample placed on a Peltier heater or a Linkam Thms350V temperature controller inside a Nikon opti- cal microscope in epi configuration (both the illumina- tion and reflection of light travel through the same objec- tive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Illumination in the visible range was used (halogen lamp, no filters) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Two surface sample images (sample A 10µm×50µm and sample B 30µm×35µm) were mea- sured around the focal point of 1mm in the visible range using a ×150 magnification dry Olympus objective lens with an optical aperture of NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The theoretical lateral resolution is estimated to be δr= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='22λ/(2 NA) = 370nm in the visible range using the Rayleigh criterion [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Temperature was measured using a Pt100 glued next to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Temperature sweeps (35oC≪Tc to 82oC≫Tc and back) spanning the entire IMT were per- formed multiple times at a rate of 1°C/min, temperature swept linearly, with temperature and images recorded ev- ery ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='17°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Height z focusing and x-y drift correction Inevitable temperature dilation of the experimental system during temperature sweeps brings the sample out of focus during temperature sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In order to com- pensate for this z drift, we employ a “fuzzy focusing” technique as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' During the experiment, the sam- ple was continually moved up and down 10µm every 10 seconds by a piezoelectric crystal placed under it, in or- der to bring the sample in and out of focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' A stack of 120 images was recorded this way for each tempera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Over the years, various metrics have been evaluated for selecting the sharpest image in such a stack [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Some studies focus explicitly on images that don’t have sharp contrast [19], like the raw images acquired here (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 2(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Most metrics reported perform well in select- ing the focused image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We have first chosen one using the compression rate of the recorded images [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This one is based on the intuitive idea that, when very out of focus, the sample surface will look homogeneously gray due to blurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In this case, the raw recorded Bitmap (BMP) image can be highly compressed in lossless Tiff format using a standard Lempel-Ziv-Welch (LZW) compression protocol [21, 22], since nearly every pixel is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' On the contrary, when the sample is in focus, the image contains much more information (since most pixels are different from their neighbors), and the raw BMP im- age cannot be compressed as much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Using this method, one can determine the most sharply focused image in the stack by selecting the one with the largest Tiff file size [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Among the 62,000 images of sample A acquired during the 14 hour experiment (consisting of 3 major temperature loops and 10 subloops [25]), we retain the 894 images that are in focus within 80nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' A recent update of the microscope has allowed us to select the best focused image of sample B during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In the live selection process we have used a computationally faster method based on image gra- dient using the Tenengrad function [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Both metrics cited above were vetted using micron-sized gold disks lithographed on a glass substrate where the sharpest im- age can be defined as the image with the sharpest step function (gold to substrate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Using the focusing stack technique, we have also compared the image height on the sample four corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This allowed us to correct the tilt of the sample (due to sample positioning using ther- mal paste).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The updated setup also uses a piezoelectric PI Pifoc PD72Z1x to move the objective up and down rather than moving the sample placed inside the Linkam stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The current setup can thus output an image ev- ery 10s in focus on the full field of view as a function of 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Schematics of the microscope and image analysis created specifically to measure spatial maps of clusters in VO2 during the IMT while recording resistivity R(T) simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The sample was positioned on a Peltier heater or Linkam Thms350V temperature controller to apply temperature ramps (bottom left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The sample height was varied by steps of 80nm via a piezoelectric actuator placed under it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The best-focused images were chosen post-experiment using an image compression method and Tenengrad function (described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' II C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The height focus of the sample was thus controlled within 80nm throughout the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Fine xy plane drift correction within a single pixel was performed post-experiment (described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='II C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Camera sensitivity was normalized throughout the recording (described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='S3 of the SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Using this fully stabilized image series, black and white thresholds were applied for each pixel individually, accurately determining if it is in the metallic or insulating state (described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' II D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We use this information to construct spatial maps of the local transition temperature Tc, hysteresis width ∆Tc and transition width δTc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' As the temperature is cycled repeatedly, in addition to drifts along z-axis (perpendicular to the film), there are also drifts in the xy plane (the plane of the film).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' These thermal drifts were compensated: (i) live within 1µm using step xy motors below the sample and (ii) post experiment using cross correlation to track and realign part of the gold leads which contain imperfections (spots) and rough edges with VO2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 5 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Although the lateral image resolution is limited by diffraction and is estimated to be 370nm, the drift compensation tracks each pixel (≈ 37nm wide) on the sample throughout the whole experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The remaining spatial variations we observe in re- flected intensity from the VO2 region are primarily due to changes in local reflectivity due to the IMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' However, there can be other contributions to this spatial varia- tion, including effects such as surface height variations from sample warping, variations in film thickness, minor surface defects, and even shadows cast from the 150nm thick gold leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' There can even be differences in pixel sensitivity in the camera itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Because each of these contributions is independent of temperature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' con- stant in time), their effects can be distinguished from that of the temperature driven IMT, as described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Single pixel scaled and binary thresholded images In order to isolate the changes in local reflectivity which are due to the IMT, we introduce two novel image processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We use single pixel time traces to generate single pixel scaled images (panel (n) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 2), as well as binary thresholded images (panel (o) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 2, discussed in the following subsections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Both types of im- ages begin by considering a full warming or cooling sweep (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' from fully insulating to fully metallic, or vice versa) to follow the intensity and analyze each pixel individu- ally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' As an example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 2 (a-l) shows the raw optical intensity time/frame traces of 12 different pixels during a cooling sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' See S6 for the time traces of 1600 pix- els from the center of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In order to construct CcD camera Height z focusing x-y drift correction Light Microscope Image sensitivity drift correction source R(T) Single pixel intensity time trace z piezoelectric heater Single pixel thresholded image stage VO2 Temperature ramp 9 80 Temperature ( 60 40 △Tc map STcmap 1 2 3 Tc map 4 Time (Hrs)4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Single pixel intensity normalization and thresholding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (a-l) Representative single-pixel turn-on functions in sample A during cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Blue traces are the raw intensity in 8-bit grayscale where 0 is black and 255 is white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The orange traces are smoothed versions of the blue traces, in which we have applied an 11-point Gaussian convolution (σ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Purple curves are the difference between the raw (blue) curve and the smoothed version (orange curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The green curve is a numerical derivative of the blue curve (discussed and used in SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S4), taken via a finite difference with a 10-point stencil [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (m) Raw optical image (frame 847) partway through cooling for VO2 sample A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (n) The same image after the intensity is scaled, pixel-by-pixel, such that light pixels are in the insulating phase and dark pixels are in the metallic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (o) The same image, with metal and insulator domains, clearly delineated as black and white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Images are 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='3µm wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' a single pixel scaled image, we normalize each individual pixel’s 8-bit grayscale intensity time trace with respect to itself, such that its maximum intensity is scaled to 1, and its minimum intensity is scaled to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The resulting single pixel scaled image is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 2(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This type of im- age is a relatively quick way to study the temperature de- pendent IMT, as it eliminates temperature-independent spatial variations that are not due to the IMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In order to construct a binary thresholded image which clearly delineates metal and insulator domains, we must define a criterion for when each pixel changes from metal to insulator or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The orange curve in each of the panels (a-l) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 2 is a Gaussian-smoothed version of the raw time trace, using an 11-point Gaussian convo- lution (σ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We use this smoothed time trace of the intensity in order to determine the midway point inten- sity for each individual pixel (shown by the red horizon- tal dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We use the pair connectivity correla- tion length to justify setting the threshold at midway, as described in the following subsections (Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' II D 1 and II D 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This allows us to construct binary black and white images of the metal and insulator domains at each measured temperature, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 2(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Different pixels go through the midway point at different frame numbers, and therefore at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We use this information to construct spatial maps of the local transition temperature Tc recorded at each pixel reveal- ing the highly spatially-textured nature of the IMT in VO2 [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' These Tc maps, as well as hysteresis width ∆Tc maps and transition width δTc maps, are presented in the experimental results Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Pair Connectivity Correlation Length As can be seen in the single pixel time traces shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 2 (see SI Figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S6 for many more examples), each pixel experiences a definite switch from metal to insula- tor or vice versa, consistent with the Ising-type model we have previously developed to describe the IMT in VO2 thin films [8, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' While the Ising model was originally developed to describe magnetic domains of orientation “up” or “down”, here we map “up” and “down” to metal and insulator domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' While the metal-insulator tran- sition is first order, this transition ends in a critical point as a function of quenched disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The influence of that critical point is felt throughout a critical region, which includes part of the first order line in the vicinity of the critical end point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' [8] We use the correlation length of the pair connectivity correlation function to determine the threshold between metal and insulator domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Dur- ing the IMT, VO2 metal-insulator domains form intri- Horizontal pixel location [40] [80] [120] 150 Raw Convolved (11pt) [400] Derivative (1lpt) a) b) Convolved-Raw c) 75 (Min+Max)/2 Max Slope 0 150 Vertical pixel location [300] Pixel intensity d) e) f) 75 0 150 [200] g) h) 75 0 150 [100] j) k) D) 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 840 880 840 880 840 880 Frame NumberRaw Single Pixel Scaled Single Pixel Threshold Image [Min,Max]-->[0,1] (Min+Max)/2 m) h a b 400 300 9 h 200 100 40 80 120 40 80 120 40 80 1205 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Pair connectivity correlation length ξpair vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' temper- ature during the warming branch of an extremal hysteresis loop, as a function of different threshold values for determin- ing metal and insulator domains in sample A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The correlation length diverges when the system is closest to criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' cate patterns, often becoming fractal due to proximity to a critical point [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' At criticality, correlation lengths diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Away from criticality, the divergence is muted, although the correlation length still displays a maximum at the point of closest approach to criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' For exam- ple, changing the interaction strength between metal and insulator domains to be farther away from criticality, or changing the strength of various types of disorder farther from criticality causes the correlation length to go down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Similarly, changing the intensity threshold by which we identify metal and insulator domains also changes this correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In disordered systems, setting an un- physical threshold will not move the system toward crit- icality, but only away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Therefore, one way to set the proper threshold between metal and insulator domains is to maximize the correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The pair connectivity correlation function is familiar from percolation models, where the corresponding pair connectivity correlation length diverges at the critical point [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Coniglio and coworkers showed that the pair connectivity correlation length also diverges at the criti- cal temperature in the two-dimensional Ising model [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We have recently shown that the pair connectivity corre- lation length also diverges at other Ising critical points, including that of the two-dimensional random field Ising model [29], as well as on slices of three dimensional mod- els at criticality, including the clean Ising model [30] and the random field Ising model [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Near a critical point, the correlation function is power law at distances less than the correlation length, in this case ξpair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This pair correlation length can be calculated directly from an im- age via [31]: ξ2 pair = � i,j r2 i,jpf i,j � i,j pf i,j (1) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (a) Single pixel time trace of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The blue curve is the raw time trace of the measured optical intensity of pixel (127,734) in sample B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The orange curve is a Gaussian convolution (σ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5) of the same time trace over 3 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The double crossing at the midway is eliminated in the smoothed data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (b) Binary black and white image (frame 260) of the sample generated by thresholding at midway the single pixel time traces as presented in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (c) Smoothed out binary black and white image (frame 260) of the sample generated by thresholding at midway the 3 frame convoluted single pixel time traces as presented in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' where pf i,j is the likelihood that i and j are in the same finite cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Another way to view this is as: ξpair = � ⟨R2 G⟩f (2) where RG is the radius of gyration of each connected cluster, and the average is taken over the finite clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This quantity diverges at the percolation threshold as: ξpair ∝ 1 |p − pc|νpair .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (3) It diverges at clean Ising transitions as: ξpair ∝ 1 |T − Tc|νpair , (4) and it diverges at random field Ising transitions as: ξpair ∝ 1 |R − Rc|νpair .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5 Threshold 10% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5% Correlation Length [μm] 5% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5 Midway +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5% +5% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5% +10% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 45 50 55 60 65 70 75 Temperature [oC]a Raw 90 Conv (3pt) (Min+Max)/2 85 80 Pixel Intensity 75 70 65 60 55 0 100 200 300 400 500 600 Frame Number 3 point convoluted Raw binary image binary image6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Setting Thresholds of Metal and Insulator Signal in Optical Data In order to know at what intensity to set the threshold between metal and insulator in each pixel, we calculate the pair connectivity correlation length in a series of im- ages, as a function of different intensity thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' For this we use the single pixel scaled images as described in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 3, we plot the evolution of the pair connectivity correlation length (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 1) during the warming branch of a hysteresis loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The blue circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 3 have each pixel’s threshold set at the midway point of that particular pixel’s intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The black circles have each pixel’s threshold set higher by an amount that is +10% of the difference between the saturated metal and saturated insulator values of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The pink cir- cles have each pixel’s threshold set higher by only +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5%, and similarly for other colors as denoted in the figure leg- end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Similar to the way the theoretical threshold was set in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' [8], we set the threshold according to the longest correlation lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Since in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 3 the longest correla- tion length happens for a threshold equal to the average between metal and insulator intensity (the blue circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 3) we use this midway threshold throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Time domain convolution One of the strong points of obtaining a series of 100- 1000 images via this autofocus optical microscope is the possibility of filtering out high frequency noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' A simi- lar technique is used in resistivity experiments that probe samples thousands of times per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 4 (a) com- pares a raw single pixel time trace to a smoothed ver- sion in which a 3-point Gaussian convolution (σ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5) has been applied in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In this example, the raw single pixel time trace crosses the midway point twice, whereas the 3-point convolved curve passes the midway point only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Notice that this procedure of filtering high frequency noise in the time domain greatly suppresses the white noise evident in the spatial domain near the metal-insulator boundaries derived from the raw time traces (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 4 (b) and (c) for comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This smoothing is useful for studying spatial correlations from frame to frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' However, if filtering is not necessary, raw data is used throughout the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This is the case for Tc maps in the section below and ramp reversal memory maps presented elsewhere [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' High frequency noise was filtered in the temperature data taken using the Pt100 by fitting a linear slope through the large temperature sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This matched the internal temperature sensor slope of the Linkam Thms350V temperature controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' RESULTS Having described the various key steps in the previ- ous sections (including autofocusing, step motor/cross correlation aligning, single pixel scaling and threshold- ing, pair connectivity correlation length analysis, and time domain convolution) we now present the detailed spatially-resolved study of the IMT in VO2 films using our new optical mapping method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Maps Transition Temperature Tc maps: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 5 (c) re- ports the local critical temperature Tc map in VO2 sam- ple B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' These maps show a large spatial variation in Tc, with rich pattern formation over tens of microns, similar to s-SNIM sub-micron measurements [7], but acquired with a much faster procedure that allows for much finer time and temperature resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This large scale spatial variation, along with detailed spatial knowledge of the lo- cation of these variations, can potentially be exploited to optimize memory elements by addressing specific regions of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Reproducibility of Tc maps: Previous reports on avalanches in this material showed jumps in resistivity randomly appearing during the transition in macroscopic transport measurements [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This suggested that the metal-insulator patterns could be appearing randomly during each temperature sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' At first glance, this appears to be at odds with the optical data reported in this study, where we find that the metal and insu- lator patterns are highly repeatable globally (occurring at the same location and with the same shape) during successive temperature sweeps (see Fig 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The repeata- bility suggests that the patterns are strongly influenced by an underlying random field present in the thin film or its substrate [8, 26, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The observed stochasticity of resistance jumps in transport measurements [33] could arise from small variations in the exact time at which avalanches are triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In addition, small changes in optical maps can potentially create large changes in re- sistance, when tiny “shorts” connect pre-existing larger metallic clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Transition Width δTc maps: The transition width δTc of each pixel can be accessed by fitting single pixel scaled intensity time traces to a hyperbolic tangent: − 1 2(tanh( T−Tc δTc )-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Because Tc is known from our time trace analysis, there is only one fitting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The map of δTc distribution is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 5 (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The aver- age transition width of the pixels as measured in optics is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='1°C with extremes from 0°C to 8°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Moreover, a small number of pixels show more than one step dur- ing a transition (see for example first pixel (305,300) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' These cases could arise from an overlap be- tween multiple metal or insulator domains affecting a single pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This could be due to information from sur- rounding pixels affecting the signal at one pixel, since the 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (a) Optical image of VO2 sample B during the insulator (light gray) to metal (dark gray) transition (warming cycle), two gold leads are seen at the top and bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' These electrodes also display some structure (spots) due to gold surface imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Contrary to VO2 IMT structures seen in this image, gold imperfections do not change with time (see online movie [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Usually these imperfections are purposely washed away using strong image brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Here, on the contrary, brightness was set low to see and use these imperfections to autoalign within a pixel the images and thus compensate xy thermal drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Sapphire substrate is the dark surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' One can easily see the metal dark patches appearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Scale bar is 10µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (b) Single pixel intensity curve defining critical temperature Tc, hysteresis width ∆Tc and transition width δTc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Tc were determined at midways as explained in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Hysteresis width was determine by taking the temperature differences between heating and cooling cycles Tc up-Tc down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Transition width was determined by fitting (smooth curve) the single time trace to a hyperbolic tangent: − 1 2(tanh( T −Tc δTc )-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (c) Local critical temperature Tc map, (d) ∆Tc maps, (e) δTc map (presented here for the temperature ramping up branch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Image are 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6µm high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Histograms (with mean and standard deviation of maps a), b) and c) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 7 pixel size is ∼10 times smaller than the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Or, it could arise from structures that are smaller than the pixel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Indeed, s-SNIM has clearly observed inhomo- geneities on smaller length scales than the optical maps presented here [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Interestingly, the standard devia- tion of local Tc’s across the sample, σTc(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2°C), is smaller than the average transition width of pixels δTc(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='8°C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' It remains an open question whether the self-similar metal- insulator domain patterns discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' [8] could be the source of this difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Hysteresis Width ∆Tc maps: By subtracting Tcup- Tcdown (see the caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5 (b) for the definition) one can construct a hysteresis width ∆Tc map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The hystere- sis width ∆Tc map is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 5 (d) for sample B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The average width is found to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='1°C as seen in macroscopic transport measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' However, certain small regions have small ∆Tc, in the range [0°C - 1°C] (small blue clusters in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 5 (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Probing these region with other local probes could shed light on whether this is an intrinsic property of these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' These hysteresis- free patches could be very useful in multiple switching applications such as optical electronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Indeed it has been shown that the presence of a large hysteresis in VO2 greatly complicates using it as an optical sensor [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Correlations between maps With all of the maps above, one can check for cor- relations between these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 8 plots Tc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' ∆Tc, ∆Tc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' δTc and Tc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' δTc for each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' A few horizontal and diagonal lines appear in these plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The horizontal lines come from multiple pixels (spatially close by) switching at the same temperature (upon warming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The diagonal lines come from multiple pixels (spatially close by) switching at the same temperature (upon cool- ing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Although this is typically what one would expect 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2 b) a Single pixel scaled intensity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 Single pixel scaled 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='8 intensity time trace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6 △T T,down dn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 2 STc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2 50 60 70 80 Temperature °C T_map △T_map ST_map c) T [°C] d) T [°C] e) T [°C] C C 72 8 12 7 71 10 70 5 8 69 4 68 6 3 67 2 4 66 2 65 0 0 648 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' a) Three Tc maps while cycling through the IMT (warming) at 1°C/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' b) Difference maps between cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Global patterns are generally reproducible (σTc/Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6°C/68°C= 1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' However some small regions present deviations up to ±2°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Full histograms (with mean and standard deviation) of maps in b) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Difference map between Tc3 and Tc1 (the most separated, time wise, temperature sweeps in this study) and the corresponding histogram are presented in SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Images are 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6µm x 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' from avalanches, further analysis is needed to extract the full dynamics occurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In the three correlation maps, no trend is seen in the last two, but Tc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' ∆Tc shows a slight positive correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This means that pixels with low Tc tend to have low ∆Tc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' close to zero) and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The positive correlation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 8(a) is not to be confused with the few diagonal lines present in this panel explained just above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Hand picking specific hysteric properties The wide range of behaviors contained in the three maps presented in the section above (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 5 c, d and e), gives us the unprecedented opportunity to find individual pixels with desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 9 shows the Tc map of the sample with six different types of pixels selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The pixel labeled “std” for standard has a rounded transition with values of Tc, ∆Tc and δTc which are close to the average values found in the distribution of these three quantities (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 7 a, b and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Pixels A and B show the most common type of local characteristics found in the maps: when Tc is high, ∆Tc is high;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' when Tc is low, ∆Tc is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This positive correla- tion is evident at a global level in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 8 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' However, on a local level, individual pixels can have a large deviation from the global average behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Indeed pixel E shows a possibility of finding ∆Tc very low (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='3°C) with a Tc (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='3°C) low but closer to the mean value of the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Pixels C and D illustrate the case where the width δTc of the transition is very sharp (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5°C) or very wide (5°C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Pixel C shows a representative sharp pixel, where within the temperature steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='17°C, the transition occurs in a sharp, avalanche mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Further analysis to see where and how these avalanches occur will be pursued in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Finally pixel E shows a case where ∆Tc is within the lower values [0°C-1°C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' As mentioned previously, small hysteresis could be useful in opto-electronic devices or neuromorphic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In the first case, small hysteresis avoids optical detectors getting stuck in subloops [35];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' in the second case, small hysteresis allows lowering the voltage threshold needed for spiking [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' General remarks on the pixel selection procedure: (i) as mentioned previously in the δTc section above, some pixels in the map clearly present two steps during the IMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' These two-step pixels can potentially be detected in an automated way from their anomalously high error on the fit to the hyperbolic tangent function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (ii) the fea- tures put forward in these 6 pixels above are not unique to the 37nm square pixel location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' These features usu- ally also hold for many pixels around the xy coordinates reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Cycle # 2 - T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2map 72 a 71 69 68 67 65 64 c2 2 b T [°C] 3 2 1 0 1 2 39 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Histograms of maps presented in in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (a) Tc maps (upon warming);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (b) ∆Tc map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (c) δTc map and (d) and (e) two difference maps Tc2-Tc1 and Tc3-Tc2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Correlations between Tc (upon warming), ∆Tc and δTc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Each of the 666,000 pixels (900x740) is represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Only Tc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' ∆Tc (panel (a) shows a slight diagonal trend meaning that pixels with low Tc tend to have low ∆Tc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' close to zero) and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' CONCLUSIONS We have reported the first Tc maps derived from sin- gle pixel optical imaging on VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Multiple new exper- imental steps were needed to align, focus and calibrate the raw grayscale images recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' These experimental achievements allowed us to accurately track the spatial distribution of metal and insulator clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Binary black and white images, time traces, Tc maps, ∆Tc maps, and δTc maps were plotted and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The sample shows micron-sized patterns that are found to be mostly repro- ducible through multiple temperature sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The ∆Tc hysteresis width map exhibits, on average, the same av- erage hysteresis width of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='3°C as macroscopic resistiv- ity hysteresis, but exhibits strong variation on a local scale, down to ∼[0°C-1°C] in certain small regions and as large as ∼ 8°C in other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' These findings open an exciting opportunity to access local properties of VO2 by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=', contacting specific parts of the sample electri- cally in order to select unique parameter combinations 20 80 80 b) a) c) 75 75 - 15 P0% p 70 70 10 ooo 65 65 5 60 60 0 : 55 55 0 5 10 15 20 0 10 12 14 2 4 6 8 10 12 14 2 4 6 8 0 △T, [°C] [°C] ST ST Cx104 x104 a) c) μ= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='8 [°C] μ= 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2 [°C] μ= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='3 [°℃] 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='1 [C] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2 [°℃] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='1 [°℃] 4 pixels 5 4 4 3 3 Number 3 2 2 2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 1 0 0 64 66 68 70 7274 2 ４6 234５ 62 0 8 10 0 678 △T, [°C] T,[°C] ST,[°C] x104 x104 d) e) 10- μ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 [°℃] μ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 [C] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6 [°C] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6 [℃C] 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Number of pixels 6 6 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 2 2 0 0+ 1234 4-3 -2 -1 0 4-3 -2 -1 １234 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=',-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=', [C] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' [°C] c210 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Tc map with six pixels chosen to illustrate specific characteristics in the hysteresis loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The table shows the numerical values of Tc, ∆Tc and δTc for each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The numbers in bold highlight the unique characteristic of each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' for specific applications in electrical and optoelectronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The observation of a positive correlation be- tween Tc value and hysteresis width could enable a new approach for tailoring the material’s response to exter- nal drives, in addition to providing a new perspective in studying open questions in the theory of hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Carlson for technical assistance with image stabilization, and acknowledge helpful conversa- tions with K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Dahmen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=', and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' ac- knowledge support from NSF Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' DMR-2006192 and the Research Corporation for Science Advancement Cottrell SEED Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' acknowledges support from a Bilsland Dissertation Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' acknowl- edges support from a Fulbright Fellowship, and thanks the Laboratoire de Physique et d’´Etude des Mat´eriaux (LPEM) at ´Ecole Sup´erieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI) for hospital- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This research was supported in part through com- putational resources provided by Research Computing at Purdue, West Lafayette, Indiana [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The work at D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6 std Single pixel scaled intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='4 50 60 70 80 40 50 09 70 Temperature [°C] Temperature [°C] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 40 50 70 700 60 T_[C] Temperature [°C] 72 600 71 70 500 B 69 400 68 300 67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='4 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 200 50 60 0 65 Temperature [°C] 100 64 A 01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6 200 300 500 600 100 400 700 800 900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='4 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 50 40 60 Temperature [°C] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 40 70 50 60 Temperature [°C] Label (x,y) position Specific T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' [°℃C] △T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' [°℃] STc[°C] characteristic std (85 , 285) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6 Tc,△Tc, STc (standard) close to mean value A (34, 135) Low T / Low △Tc 64.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5 c (506 ,440) Low oT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='4 D (670, 547) High T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The work at ESPCI (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=') was supported by Cofund AI4theSciences hosted by PSL University, through the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sk�lodowska-Curie Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 945304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' McElroy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='edu/articles/2014/7/ empowering-faculty-a-campus-cyberinfrastructure- strategy-for-research-communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Qazilbash, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Schafgans, K.' metadata={'source': 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Zimmers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Tomczak, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Lobo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Bon- temps, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Hill, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Barr, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Dagan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Greene, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Millis, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Homes, Europhysics Letters (EPL) 70, 225 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 13 SUPPORTING INFORMATION: CORRELATIVE MAPPING OF LOCAL HYSTERESIS PROPERTIES IN VO2 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' VO2 Reflectivity The fact that the metallic reflectivity of VO2 is lower than that of the insulating phase in the visible range is counterintuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This is due to a subtle combination of a Drude response as well as intraband and interband transitions and thin film interferences in this material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The largest reported spectra in VO2 was measured by ellipsometry [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Using the reported real part of the optical conductivity σ1, we have calculated the reflectiv- ity of the insulator and metallic states (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S2 and S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This clearly shows that, as one would expect in the infrared, the sample becomes highly reflective when metallic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Above the plasma frequency (∼12000cm−1), interband transitions and spectral weight conservation make the reflectivity curves cross, leading to the metallic state having a lower reflectivity than the insulating state in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The relative optical contrast in the visible range (27%), is still more than sufficient in our setup to identify both states clearly (as seen in a raw image Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S1 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Key steps making this study possible The key step that have allowed us completing this study comes from the unique qualities of the VO2 ma- terial : The IMT is above room temperature, which allows close optical microscopy (strong objective ×150 with a high numerical aperture 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='9 brought to 1mm focus above the sample surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This setup would be much harder to achieve if cryogenic cooling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' a cryostat with a window between the sample and objective) was needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Phase separation was observed by s-SNIM at sub- micron scales in this material [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The fact that this phase separation is still found up to 30µm makes these optical microscopy surface maps possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In the visible range, a relative 27% drop in the thin film reflectivity is found in the metallic state Measuring in the visible range gave us results with a 400nm reso- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In the infrared, the contrast between metal and insulator is much larger, as expected, but only allows optical resolution up to the IR wavelength, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 1-10µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (a) 35µm wide etched VO2 sample B image with 30µm separated gap gold leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The white square represents the 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6µm x 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6µm region where Tc maps (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='s 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Scale bar is 10µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' (a) R(T) measurement of the IMT (a) Gold Substrate vO2 1000 (b) 100 Resistance (kΩ2) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='01 40 50 60 70 80 Temperature (°C)14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Simulated optical reflectivity of the insulating and metallic states in bulk VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Optical functions were derived by fitting standard Drude-Lorentz functions to ellipsometry measurements reporting the raw σ1 response in a large spectral range at low and high temperatures [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This procedure [39] allows other optical functions to be deduced, such as reflectivity, transmission, absorption, or dielectric constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Reflectivities in this figure are not reported below 1000cm−1 as the fitting procedure was not precise enough in this low frequency/high σ1 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' On the other hand, reflectivities in the visible region (∼14000cm−1 to ∼25000cm−1) are in the middle of the spectral range and can be found with confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Simulated optical reflectivity of the insulating and metallic states of a 130nm VO2 thin film on an r-cut sapphire substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Optical functions were found as described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' In contrast with the bulk reflectivity, a pronounced oscillation can be seen in the blue insulating spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This is due to interference in the 130nm thin film (for example, constructive thin film interference creates a peak at ∼6700cm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Reflectivities are not reported below 1000cm−1 as the fitting procedure was not precise enough in this low frequency/high σ1 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' On the other hand, reflectivities in the visible region (∼14000cm−1 to ∼25000cm−1) are in the middle of the spectral range and can be found with confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' photon energy (eV) 0 1 2 3 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 Metal T=360K ab0ve T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='8 Insulator T=295K below T G Reflectivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='4 Microscope spectral range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 0 10000 20000 30000 Wavenumber (cm-1)photon energy (eV) 0 1 2 3 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 Metal (thin film) T=360K ab0ve T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='8 Insulator (thin film) T=295K below Reflectivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='4 Microscope spectral range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='0 0 10000 20000 30000 Wavenumber (cm-1)15 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Image sensitivity drift correction Whereas the relative average intensity of VO2 increases almost 30% in changing from metal to insulator, the change in sapphire reflectance in this temperature range is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We have used this fact to correct for any changes in incident light or CCD detector sensitivity throughout the experiment by dividing the average in- tensity in the VO2 region by the intensity in the sapphire region of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Details: We assume that the input intensity is a func- tion of time I0(t) but spatially uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The reflected in- tensity from any region is IR(t, x, y) = I0(t) × R(t, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Since the Sapphire’s reflectance does not vary signifi- cantly over the range of temperature the sample went through, it is assumed to be a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Let the spatially averaged sapphire reflectivity be RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Then, the spatial average reflected intensity from the sapphire region is: IS R(t) = I0(t) × RS Any region of VO2 has a reflected intensity: IV R (t, x, y) = I0(t) × RV (t, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Therefore, the ratio of reflected intensities from Sapphire and VO2 is independent of input intensity: IV R (t, x, y)/IS R(t) = RV (t, x, y)/RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We will use IS R(t) as a reference to cor- rect IV R (t) for any variation due to fluctuation of ambi- ent light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The quantity independent of input intensity: RV (t, x, y) = RSIV R (t, x, y)/IS R(t), Hence, setting the ref- erence input intensity I0(0), the corrected reflected in- tensity from VO2 would be: ˜IV R (t, x, y) = I0(0)RV (t, x, y) = IV R (t, x, y) IS R(t)/IS R(0) S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Single pixel thresholded images: inflection point In the main text, we have set the threshold between metal and insulator domains at the midway point of the intensity, based on the pair connectivity correlation length criterion described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' II D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' We also tested another method of setting the threshold based on the inflection point of the single pixel time traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The green curves in panels (a-l) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 2 show a smoothed derivative of the raw time traces, achieved by using a finite difference with a 11-point Gaussian convolution (σ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='5) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' The vertical dotted green line shows the extremum of this derivative, which locates the inflection point of the orange curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Since the pixel switching curves (orange and blue traces) exhibit a relatively rapid change from metal to insulator, this inflection point at which the pixel brightness is changing most rapidly is the most natural place to assign a change from insulator to metal and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Because we have used a stencil with even number of 10, the inflection point happens between frames, and allows us to clearly identify frames which precede the inflection point (which are metallic) from frames which come after the inflection point (which are insulating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Notice that the frame number at which the solid orange curves cross the dotted orange lines coincides with the inflection point for each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' This means that both methods are equivalent for determining the frame number at which a pixel switches from metal to insulator or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' 16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Map and histogram of the difference between Tc3 and Tc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Although these Tc maps are the most separated, time wise, in this study, they remain similar (mean and σ) to Tc2-Tc1 and Tc3-Tc2 presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Online movie[32] screenshot of the ∼1500 in focus consecutive spatial maps of a 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6µm x 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='6µm VO2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Central panels: raw, scaled and thresholded surface image (sample B) using “Single pixel scaled image” and “Single pixel intensity time trace and threshold” methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Left panels: corresponding histogram changes during temperature ramps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Top right panel: average sample intensity (raw, scaled, thresholded) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' frame number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Middle right panel: average sample intensity (raw, scaled, thresholded) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' sample temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Bottom right panel: Temperature protocol - 3 major temperature loop spanning the entire IMT (36oC - 82oC), [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='l 1 c3 X104 3 10 μ= 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='319) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='320) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='321) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='322) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='323) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='324) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='325) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='326) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='327) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='328) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='329) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='336) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='337) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='338) 820 840 860 880 Frame Index (344,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='339) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='11pt Slope ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='Raw - 11pt Conv ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' ML3 time trace of sample A in a patch of 40×40 pixels in the middle of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Each pixel coordinates are indicated above the time trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} +page_content=' Description of the four curves in each mini panel is the same as the main text Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE2T4oBgHgl3EQf8gnM/content/2301.04220v1.pdf'} diff --git a/CdE4T4oBgHgl3EQfeQ2g/vector_store/index.faiss b/CdE4T4oBgHgl3EQfeQ2g/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..394d65333effa1af21c5286451ed46ec456e9768 --- /dev/null +++ b/CdE4T4oBgHgl3EQfeQ2g/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4d7e3725a4ba1928b231b6f8a48d55dd9180bee0a27f165daf467695a24e1c2f +size 5242925 diff --git a/DtE0T4oBgHgl3EQfggGb/content/tmp_files/2301.02419v1.pdf.txt b/DtE0T4oBgHgl3EQfggGb/content/tmp_files/2301.02419v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d289d62e985d9f7d2c1702ec25cecbf049e7dc9 --- /dev/null +++ b/DtE0T4oBgHgl3EQfggGb/content/tmp_files/2301.02419v1.pdf.txt @@ -0,0 +1,1908 @@ +Published in Transactions on Machine Learning Research (08/2022) +Exploring Efficient Few-shot Adaptation for Vision Trans- +formers +Chengming Xu +cmxu18@fudan.edu.cn +School of Data Science, Fudan University +Siqian Yang +seasonsyang@tencent.com +Yabiao Wang +caseywang@tencent.com +Youtu Lab, Tencent +Zhanxiong Wang +maxzxwang@tencent.com +Tencent +Yanwei Fu∗ +yanweifu@fudan.edu.cn +Xiangyang Xue +xiangyangxue@fudan.edu.cn +School of Data Science, Fudan University +Reviewed on OpenReview: https: // openreview. net/ forum? id= n3qLz4eL1l +Abstract +The task of Few-shot Learning (FSL) aims to do the inference on novel categories containing +only few labeled examples, with the help of knowledge learned from base categories containing +abundant labeled training samples. While there are numerous works into FSL task, Vision +Transformers (ViTs) have rarely been taken as the backbone to FSL with few trials (Hu +et al., 2022; Evci et al., 2022; Abnar et al.) focusing on naïve finetuning of whole backbone +or classification layer. +Essentially, despite ViTs have been shown to enjoy comparable +or even better performance on other vision tasks, it is still very nontrivial to efficiently +finetune the ViTs in real-world FSL scenarios. To this end, we propose a novel efficient +Transformer Tuning (eTT) method that facilitates finetuning ViTs in the FSL tasks. The +key novelties come from the newly presented Attentive Prefix Tuning (APT) and Domain +Residual Adapter (DRA) for the task and backbone tuning, individually. Specifically, in APT, +the prefix is projected to new key and value pairs that are attached to each self-attention +layer to provide the model with task-specific information. Moreover, we design the DRA in +the form of learnable offset vectors to handle the potential domain gaps between base and +novel data. To ensure the APT would not deviate from the initial task-specific information +much, we further propose a novel prototypical regularization, which maximizes the similarity +between the projected distribution of prefix and initial prototypes, regularizing the update +procedure. Our method receives outstanding performance on the challenging Meta-Dataset. +We conduct extensive experiments to show the efficacy of our model. Our code is available +at https://github.com/loadder/eTT_TMLR2022. +1 +Introduction +Modern computer vision models such as ResNet (He et al., 2016) and Faster R-CNN (Ren et al., 2015) +are trained on large-scale training sets, and not well generalize to handle the long tail categories with few +∗This paper is supported by the project NSFC(62076067). +1 +arXiv:2301.02419v1 [cs.CV] 6 Jan 2023 + +Published in Transactions on Machine Learning Research (08/2022) +labeled samples. Few-shot Learning (FSL) has thus been studied to make inference on insufficiently-labeled +novel categories typically with the transferable knowledge learned from base categories which are provided +with abundant labeled training samples. Essentially, the FSL can be taken as representation learning, as its +backbones should ideally extract features representative and generalizable to various novel tasks. Currently +Convolutional Neural Networks (CNNs), especially ResNet, are the predominant backbone and widely utilized +in most existing FSL works (Ravi & Larochelle, 2017; Finn et al., 2017; Nichol et al., 2018; Li et al., 2017; +Sun et al., 2019). +Recently, by taking the merits of Multi-headed Self-Attention (MSA) mechanism and Feed Forward Net- +work (FFN), the transformers have been widely used in the recognition (Alexey et al.; Liu et al., 2021b), +detection (Beal et al., 2020) and image editing (Cao et al., 2021). The general pipeline of Pretrain-(Meta- +train)-Finetune has been explored in few ViTs on FSL (Hu et al., 2022; Evci et al., 2022; Abnar et al.), +recently. Particularly, the ViT models are first pretrained/meta-trained on a large-scale dataset. Then +a test-time finetune procedure is set up for each target task on novel data. The finetuning strategy can +be generally categorized into linear classifier probing and backbone tuning: the former one optimizes the +reasonable decision boundaries by the fixed embeddings, while the latter one considers the adaptation of +both embedding space and classifier. +In this paper we focus on the backbone tuning method. (Hu et al., 2022) shows that the naïve Pretrain-Meta- +train-Finetune (P>M>F) baseline can generally have satisfactory performance in FSL. Unfortunately, it +involves heavy computations and potential overfitting in FSL setting. Particularly, (1) It typically demands +extraordinary computing power to formulate episodes from a large number of support classes to update +the whole network parameters. Thus it is less efficient in many real-case applications. For example, the +edge devices such as mobiles donot have enough computational power to adapt all model parameters by +personalized/specialized data collected on these devices. (2) It is very subtle and difficult to directly fine-tune +trained deep models on one or two labeled instances per class, as such few-shot models will suffer from severe +overfitting (Snell et al., 2017; Fei-Fei et al., 2006; Brian et al.). By contrast, humans have the ability of +conducting few-shot recognition from even single example of unseen novel category with very high confidence. +Such problems may be the culprit of the phenomenon that their proposed finetune strategy only works on +part of datasets and has less effect to the others. This suggests their limited usage of ViT backbone for any +potential FSL applications. An alternative choice is to finetune specific layers in a ViT model with much +smaller tunable parameters (ViT-s block in Fig. 1(a)). Such a strategy nevertheless can only finetune either +low-level or high-level features, leading to inferior performance in many cases. Therefore it is desirable to +have an efficient and light-weighted ViT tuning method that shall not only avoid overfitting to small training +samples, but also achieve high performance of FSL. +In this paper, we present a novel efficient Transformer Tuning (eTT) for few-shot learning task, which adopts +a pretrain-finetune pipeline. To pretrain our transformer, we advocate utilizing the recent self-supervised +method – DINO (Caron et al., 2021). Our key novelties are in the finetuning stage. As illustrated in Fig. 1(b), +we propose Attentive Prefix Tuning (APT) and Domain Residual Adapter (DRA) as the key components to +our eTT, to efficiently learn the newly-introduced tunable parameters over novel support sets. Specifically, we +formulate the attentive prototypes by aggregating patch embeddings with the corresponding attention weights +of the class token for each image, so as to provide the model with abundant task-specific information and +guide each self-attention layer to aggregate more class-related features. To encourage the prefix to keep the +prior knowledge from initial prototypes, we further propose a novel prototypical regularization which restricts +the relationship between the prefix and prototypes by optimizing the similarity of their projected distributions. +Moreover, we propose to additionally adopt a light-weighted domain residual adapter in the form of learnable +offset to deal with the potential failure of APT on large domain gaps. Extensive experiments are conducted to +evaluate our eTT: we use the ViT-tiny and ViT-small backbones on the large-scale Meta-Dataset (Triantafillou +et al., 2019) consisting of ten sub-datasets from different domains; and the results show that our model can +achieve outstanding performance with comparable or even much fewer model parameters. Thus our eTT is a +promising method on efficiently finetuning ViTs on the FSL tasks. +Our paper has the following contributions. +1. In order to solve the problem of inefficiency and make better use of ViT in FSL, we propose a novel +2 + +Published in Transactions on Machine Learning Research (08/2022) +Domain residual adapter +ViT block +… +Attentive +prototypes +Visual +prefix +Domain residual adapter +ViT block +Few-shot +episodes +(a) Tunable parameters in Backbone Finetuning +(b) Attentive Prefix Tuning in Task Tuning +Support images +Initialize +Key/value +pairs +project +plug +Figure 1: (a) Comparing with other backbones, we propose the Domain Residual Adapter (DRA) to tune much +less parameters in our efficient Transformer Tuning (eTT); and effective for large-scale FSL. (b) The few-shot +support samples are first processed into attentive prototypes which are used to initialize the task-specific +visual prefix. Then the prefix together with the domain adapter are attached to each layer of the ViT to +finetune our ViTs. +finetuning method named efficient Transformer Tuning (eTT). +2. Inspired by recent advance in language model, a novel attentive prefix tuning is presented utilizing the +attentive prototypes to embed the task-specific knowledge into pretrained ViT model. Particularly, we propose +a new initialization strategy tailored for FSL by leveraging prototypical information from the self-attention +layers. Moreover, a novel domain residual adapter is repurposed to handle the various domain gaps between +training and testing data. +3. We introduce a prototypical regularization term which can constrain the update procedure of prefix during +finetuning to maintain the initial task-specific knowledge. +4. +By utilizing the proposed eTT, our ViT models receive remarkable performance on Meta-Dataset, +overpassing the existing ResNet-based methods without using additional training data. More importantly, +both of the model scale and efficiency of our method are comparable with the other competitors, indicating +the promising application of ViTs in FSL. +2 +Related Works +Few-shot recognition. FSL learns transferable knowledge from base classes and adapt it to a disjoint set +(novel classes) with limited training data. Among those FSL tasks, few-shot image recognition is the one with +most focus and researches. Existing works can be grouped into two main categories. One is optimization-based +methods (Ravi & Larochelle, 2017; Finn et al., 2017; Nichol et al., 2018; Li et al., 2017; Sun et al., 2019), +which learn parameters that can be better finetuned on few-shot support sets. The other is metric-based +methods such as ProtoNet (Snell et al., 2017), RelationNet (Sung et al., 2018), CAN (Hou et al., 2019), +DMF (Xu et al., 2021), COSOC (Luo et al., 2021) and CTX (Doersch et al., 2020), which solve FSL by +applying an existing or learned metric on the extracted features of images. Particularly, CTX (Doersch et al., +2020) builds up a cross attention module which interacts between query and support images to adaptively +aggregate better prototypes than simply averaging all support features. While these methods perform well on +classical few-shot learning settings, most of them adopt convnet as backbone, especially ResNet (He et al., +2016). We, on the opposite, try to make full use of another widely-applied structure, i.e. ViT, in FSL, which +requires extra design for training and finetuning strategy. +Transformer in vision tasks. Transformers widely utilize the self-attention mechanism which originally +are employed to process the feature sequence in Vaswani et al. (2017). Then large scale transformers become +increasingly popular in NLP tasks to build complex language models, and also extend to vision tasks (Alexey +3 + +Published in Transactions on Machine Learning Research (08/2022) +et al.; Yuan et al., 2021; Liu et al., 2021b) by formulating the token sequence with image patches processed +with position embedding. It has been shown the efficacy in various applications, such as (Liu et al., 2021a) for +image caption, (Sun et al., 2020) for multiple object tracking and (Esser et al., 2021; Cao et al., 2021) for image +inpainting and editing. Critically, ViTs is typically trained by very large-scale dataset, and few effort has +been dedicated in training or finetuning on few-shot supervision. We follow the pretrain-meta-train-finetune +pipeline (Hu et al., 2022), while their method finetune the whole ViTs on few-shot examples, and thus has +less efficiency and can easily overfit. In contrast, our proposed eTT has the key components of DRA and +APT, demanding much less tunable parameters with much better performance. +Finetuning algorithm for ViT. The idea of finetuning ViTs on small-scale datasets has been partly +investigated in Natural Language Processing (NLP) communities. Houlsby et al. (2019) proposed to attach +two learnable bottleneck adapters to each transformer layer. Other works (Xiang & Percy; Brian et al.) +make use of the prompt which trains a small task-specific prompt for each task so that the prompt can guide +the model with knowledge corresponding to the task. Such a prompting idea from NLP is inherited and +repurposed to finetune a learnable prefix for each novel episode in this paper. However, these works (Xiang +& Percy; Brian et al.; Houlsby et al., 2019) initialize the prefix or prompt with word embeddings which is +not available in our problem. Instead, we propose an attentive prototype with regularization initializing the +visual prefix with object-centric embeddings. Additionally, we notice that a very good concurrent technical +report (Jia et al., 2022) also studies finetuning visual prompt for pretrained ViTs in downstream tasks. We +highlight the two key differences from our eTT. The first is about the initialization. While initialization +strategy does not matter in their method and the corresponding tasks, we show in our experiments that +randomly initializing prefix does lead to sub-optimal performance in FSL, which leads to the necessity of a +well-designed initialization. The second is that we further propose a regularization term to restrict the prefix, +which has never been studied in existing works. +Task-specific Adapter. The idea of task-specific adapter has been explored in several works like (Li et al., +2022; Rebuffi et al., 2017) to adapt CNNs to learn the whole information from support set. Besides, (Requeima +et al., 2019; Bateni et al., 2020) adopt Feature-wise Linear Modulation (FiLM) layers (Perez et al., 2018) to +adapt task-specific information into networks. In contrast, we repurpose the adapter as the domain residual +to update transformer blocks in a more light-weighted way with less learnable parameters. Beyond different +structures, our proposed DRA intrinsically serves as the domain adapter rather than meta-learner for the +FSL in Rusu et al. (2018); Sun et al. (2019); Requeima et al. (2019). While these previous works require +meta-training to optimize their adaptation modules, our method simply utilizes the novel support data to +learn the DRA, thus reducing the training cost. Furthermore, our DRA is mostly tuned to bridge the visual +domain gap between base and novel categories, thus improving the learning of APT on each episode task. +3 +Methodology +3.1 +Problem Setup +We formulate few-shot learning in the meta-learning paradigm. In general, we have two sets of data, namely +meta-train set Ds = {(Ii, yi) , yi ∈ Cs} and meta-test set Dt = {(Ii, yi) , yi ∈ Ct} which contain the base and +novel data respectively and are possibly collected from different domains. Cs and Ct (Cs ∩ Ct = ∅) denote +base and novel category sets. FSL aims to train a model on Ds which is generalizable enough on Dt. In the +testing phase, the model can learn from few labelled data from each category of Ct. +While most previous FSL works (Snell et al., 2017; Sung et al., 2018) utilize the setting of N-way K-shot in +mini-ImageNet, i.e., K training samples from N class, we follow CTX (Doersch et al., 2020) to adopt the +setting on the large-scale Meta-Dataset (Triantafillou et al., 2019). In each episode T , N is first uniformly +sampled from [5, Nmax] where Nmax equals to min(50, |Ct|) or min(50, |Cs|) on training or testing stage, +accordingly. N is supposed to be accessible knowledge during both training and testing. In the most naïve +case, one can get N by directly counting the number of support classes. From each of the sampled category, +M query samples per category are randomly selected, and thus constructing the query set Q = {(Iq +i , yq +i )}NQ +i=1. +After that random amount of samples are taken from the rest of samples belonging to these categories to +form the support set S = {(Isupp +i +, ysupp +i +)}NS +i=1. Note that compared to the classical N-way K-shot setting, +4 + +Published in Transactions on Machine Learning Research (08/2022) +Patch embedding +layer +… +transformer layer +… +transformer layer +Patch +embeddings +Linear +ProtoNet +aggregate +ෝ𝒚 +𝜽𝒑 +෡𝑨 +MSA ++ +LN +FFN +LN ++ +projector +attention +Q +K +V +𝜽𝒌 𝜽𝒗 +MSA +𝜹𝒇 +𝜹𝒂 +𝜽𝒑 +Image +embedding +Q +K +𝜽𝒌 +V +𝜽𝒗 +𝒈 +Figure 2: Schematic illustration of our proposed model. For each image, we first fetch its patch embedding +sequence and the attention score with regard to the last layer’s class token, from which the image embedding +can be computed. Then the visual prefix is initialized as the attentive prototypes of image embeddings. The +prefix, together with the proposed domain residual adapter are attached to the model. The final features are +processed with an extra linear transformation layer and predicted with ProtoNet. Dashed arrows denote +forward propagation before test-time finetuning. +such a setting generates class-imbalanced support sets, and different episodes contain different numbers of +support samples. This is much more challenging to the model and learning algorithms, as they shall handle +both extremely large and small support sets. +3.2 +Overview of Our Method +To handle the optimization of various episodes on large-scale dataset, we present our novel finetuning model – +efficient Transformer Tuning (eTT) as shown in Fig. 2. Our eTT follows the pipeline in Hu et al. (2022), and +has key stages of the pretraining and finetuning. We employ DINO as pretraining, and conduct the task +tuning by attentive prefix tuning (Sec. 3.4), and backbone tuning with domain residual adapter (Sec. 3.5). +Pre-training. As previous work (Hu et al., 2022) shows the importance of self-supervised pre-training to +learning vision transformer models, we adopt the same principle and introduce the self-supervised learning +model to pre-train our ViT backbone on base data. Specifically, we utilize the recent State-of-the-art +self-supervised ViT models – DINO (Caron et al., 2021) to pretrain our model. DINO builds up supervision +based on a self-distillation framework by using the multi-crop strategy (Caron et al., 2020). As we will show +in our experiments, such a pre-trained model shall have good cluster property even among cross domain +images, potentially benefiting our following FSL stages. Note that different from (Hu et al., 2022) which +takes an off-the-shelf model pretrained with DINO on full ImageNet, we strictly follow the FSL protocols to +retrain the DINO models on the meta-train split in the target dataset to avoid the abuse of extra data. +One would ask whether it is necessary to make use of the annotations for base data, since supervised pretrain +has been proven to be effective in many previous FSL works (Ye et al., 2020; Hou et al., 2019). As we will +show in the experiments, an additional finetuning with image labels on base data cannot bring consistent +improvement and even makes it worse on most datasets, which may be caused by the overfitting on the +image labels leads to less generalization ability across different domains. Moreover, compared with vanilla +supervised training, the attention maps for models trained by DINO contain more semantic information, +which we will utilize in the following context. +3.3 +Preliminary: Vanilla Test-time Finetuning +Before fully developing our fine-tuning contributions, we review the simple and effective finetuning method +named LT+NCC (Li et al., 2021). The novel modules proposed by us in the following context are all adopted +together with this simple baseline method. Given a ViT backbone fθ that is parameterized by θ and an +episode T , the support features {xsupp +i +}NS +i=1, where xsupp +i += fθ(Isupp +i +), are extracted from the support set +5 + +Published in Transactions on Machine Learning Research (08/2022) +{Isupp +i +}NS +i=1. Then, a learnable linear transformation φ is added to the model to realize the adaptation, which +results in the final support features used for classification {ˆxsupp +i +}NS +i=1, where ˆxsupp +i += φ(xsupp +i +). The prediction +of these support images can thus be calculated based on the similarity between the transformed features and +the aggregated prototypes as, +¯xc = +1 +�Ns +i=1 1ysupp +i +=c +Ns +� +i=1 +ˆxsupp +i +1ysupp +i +=c +ˆysupp +i +(c) = +exp(d(ˆxsupp +i +, ¯xc)) +�N +c=1 exp(d(ˆxsupp +i +, ¯xc)) +(1) +where d denotes cosine similarity, i.e., d(a, b) = +aT b +∥a∥∥b∥. We fix all of the parameters in the original backbone, +and adopt the cross entropy loss to optimize the transformation φ. Precisely speaking, for each support image +Isupp together with its annotation ysupp, the objective function is as following: +ℓCE = −ysupp · log ˆysupp +(2) +After finetuning, φ is applied to query features and the same procedure as above is performed between the +processed query features {ˆxq +i } and prototypes {¯xc}N +c=1 for the inference of each episode. +3.4 +Task Tuning by Attentive Prefix Tuning +We finetune the pre-trained ViT with support set via an attentive prefix tuning strategy. Specifically, a prefix +matrix θP ∈ RNP ×d is first initialized, where NP denotes the number of prefix. Then a bottleneck g is added +upon θP to produce ˆθP ∈ RNP ×(2Ld), where L denotes the number of backbone layers. The g plays the same +role as the projector in each self-attention layer, except that all layers share the same module. The produced +ˆθP can be reshaped and seen as L value and key pairs {θl +v, θl +k}L +l=1, θl +v, θl +k ∈ RNP ×d. The MSA block in the +L-th layer can then be reformed by attaching these new pairs to the original key and value sequences: +Al = Attn(Q, +� +K; θl +k +� +) +output = Al � +V ; θl +v +� +(3) +where [·; ·] denotes concatenation, Attn denotes the calculation of MSA matrices. In this way, the prefix can +affect the attention matrix Al and result in different output features from the original ones. +Remark. Compared with the naive strategy that finetunes specific layers in ViT (ViT-s block in Fig. 1(a)) +which can only adjust part of blocks, the prefix can evenly adapt each layer’s image embedding with almost +the same parameter size as one transformer layer, as shown in Tab. 1(a). By fixing the model parameters and +optimizing the prefix θP and the transformation module g, the support knowledge can be smoothly embedded +into the prefix, which further helps the task adaptation. +Attentive Prototype. The initialization of the prefix is very important to our APT, as it greatly boosts +the performance. Critically, quite different from the prefix or prompt tuning in NLP and visual-context tasks +that have task-specific instructions explicitly as word embedding sequences, each episode in our FSL only +has the few support images and their labels. Thus, rather than steering the model with ’what should be +done’ as in Xiang & Percy, our APT shall provide the model with ’what we have globally’ by leveraging the +class-specific information. Thus, the attentive prototype is presented to aggregate the image embeddings +with object-centric attention, as the initialization of the prefix. Particularly, each support image Isupp is first +transformed to a patch embebdding sequence {˜xsupp +m +}P +m=1 with the starting patch embedding layer, +˜xsupp +m += fθpe(Isupp +m +) + Epos +m +(4) +where m = 1, · · · , P 2 is the patch index; fθpe denotes the patch embedding layer which is typically a +convolutional layer whose kernel size equals to patch size; and Epos indicates the position embedding. +Meanwhile, we can get unnormalized attention score A ∈ Rh×P between the class token and image patches +from the last MSA layer, where h denotes number of heads in each MSA module. Such an attention vector +can focus on the foreground in the image, especially for models trained with DINO (Caron et al., 2021), with +each head indicating a particular part or an object. We can thus get the initial image-level representation +ˆA = σ(A) +˜xsupp = 1 +h +h +� +n=1 +P 2 +� +m=1 +ˆAnm˜xsupp +m +(5) +6 + +Published in Transactions on Machine Learning Research (08/2022) +where σ is softmax function. Compared with simply averaging all patch embeddings, the attentive embeddings +can highlight the objects of interest and suppress the background information. Then the prototypes ¯x can +be calculated by averaging the attentive image embeddings belonging to each support category. We set the +number of prefix as N, which is available during testing for each episode, and initialize the prefix with ¯x. +Remark. In this way, commonly-used prototypes can provide the model with comprehensive information +about the episode. Also such a first-order statistics is comparable with the normal patch features among +the layers. This can benefit the training with more stability. When N is large, more prefix are required +to fully learn the information included by each episode. On the other hand, when N is small so that the +episode is relatively easy, fewer prefix can handle the support knowledge without trouble while decreasing the +computing debt. +3.5 +Backbone Tuning by Domain Residual Adapter +Finetuning few-shot tasks by APT will make a good balance between performance and efficiency. To further +improve the model generalization ability on different domains, we further propose the backbone tuning by +leveraging the Domain Residual Adapters (DRA), as illustrated in Fig. 2. Specifically, for the l-th transformer +layer, we attach two learnable offset vectors δl +a, δl +f ∈ Rd to the MSA and FFN. After features are processed +with MSA and FFN, the corresponding offsets are added to them so that the extreme domain gap can be +neutralized. These offsets are expected to represent the gap between source and target domains, and transfer +the original manifold to a more appropriate one. +3.6 +Loss Functions +Prototypical Regularization. In addition to the cross entropy loss in Eq. 2, we propose a novel prototypical +regularization to ensure the class-specific information, which is embedded in the prefix via initialization, +can be maintained during update. The knowledge in attentive prototypes is distilled to the prefix during +finetuning. Concretely, in each iteration, the prototypes ¯x and prefix θP are first projected to a latent space +via a projector module ψ, which produces ¯x′ and θ′ +P respectively. Then the distillation loss is computed using +these two embeddings as, +ℓdist = 1 +N +N +� +n=1 +H(¯x′n, θ′n +P ) +(6) +where H(a, b) = −a log b. The above objective function can ensure the prototype of each category and the +corresponding prefix contain consistent information, which is indicated by the similarity between distributions +after projection. To make training more stable and avoid collapse, for each episode we maintain an exponential +moving average (EMA) of ¯x′ as the center variable ccenter. Before calculating ℓdist, we standardize ¯x′ as +σ( ¯x′−xcenter +τ +), where σ denotes softmax function and τ is the temperature typically set as 0.04. +Once having both of the above losses calculated, we can optimize the model parameters including the DRA, +the prefix together with the transformation g and the projector ψ, with the following objective function: +L = ℓCE + λℓdist +(7) +where the scalar weight λ controls the strength of the regularization. +Remarks. For a ViT with L layers, nh heads and d feature dimension, the size of trainable parameters is +(N + d′ + dproj + d)d + 2(d′ + 1)Ld, where d′ is the hidden dimension for transformation module g and dproj +denotes output dimension for the projector ψ, which is much smaller than that of the whole backbone model. +Specifically, the learnable modules during finetuning have only about 9% parameters with regard to the whole +transformer model when using ViT-small and ViT-tiny. +7 + +Published in Transactions on Machine Learning Research (08/2022) +4 +Experiments +4.1 +Experimental Setup +Datasets. We use Meta-Dataset (Triantafillou et al., 2019) – the most comprehensive and challenging +large-scale FSL benchmark. It has 10 sub-datasets such as ImageNet (Deng et al., 2009) and Omniglot (Lake +et al., 2015), with various domain gaps. Our experiments are conducted under the single training source +setting, i.e. only ImageNet is used for training, and the meta-test split of all ten datasets for evaluation. Some +of the test datasets such as CUB share similar or highly-related categories with ImageNet, while the others +have greater domain gaps. Note that Hu et al. (2022) claims pretraining on all images in the training set of +ImageNet is reasonable for introducing extra data and boosting the performance. However, such a strategy +utilizes much more training samples (1.28M images, 1000 classes in ImageNet v.s. 0.9M images, 712 classes +in meta-train split of ImageNet). Empirically so many additional images can greatly benefit generalization +ability of self-supervised learning methods. Therefore to make a more fair comparison, we strictly follow +the experiment protocol used in CTX (Doersch et al., 2020) and shall not use any extra data even in the +unsupervised pretraining stage. We resize all images to 224 × 224 for ViT-small and 84 × 84 for ViT-tiny. +Implementation details. We set the patch size as 8 for ViT-tiny (as it has small input image size), and +keep the other hyper-parameters as default. We adopt standard ViT-small with 12 layers, 6 attention heads, +feature dimension as 384 and patch size as 16. We strictly follow the hyper-parameter setting and data +augmentation in DINO (Caron et al., 2021) for pretraining. In test-time finetuning, we empirically set the +hidden dimension d′ of the transformation module as d/2, and output dimension dproj of the projector as 64 +for all datasets. We utilize AdamW optimizer finetuning, with learning rate set as 1e − 3 for TrafficSign and +5e − 4 for other datasets. λ is set as 0.1. For simplicity, the selection of hyper-parameters is conducted on +the meta-validation set of ImageNet, which is the only within-domain setting in Meta-Dataset. +Evaluation benchmark. We report the accuracy of randomly sampled 600 episodes for each dataset and +the average accuracy when comparing with the existing methods. The comprehensive comparison of both +accuracy and 95% confidence interval is in Appendix. +Backbone +Image size +Params(M) +FLOPs(G) +Res18 +84×84 +11.69 +1.82 +ViT-tiny +84×84 +5.38 +0.72 +Res34 +224×224 +21.80 +3.68 +ViT-small +224×224 +21.97 +4.61 +Table 1: +Comparison of parameter size and FLOPs between different backbones. +4.2 +Comparison with State-of-the-art Methods +Before the comprehensive comparison, it is necessary to show the comparison between different backbone +is fair enough since our backbone model is not the same as the existing method. Therefore we present the +comparison of size of model parameters and FLOPs in Tab. 1, in which the FLOPs of all models are computed +by fvcore1. The results show that (1) compared with Res18, ViT-tiny is a much smaller and efficient model, +and (2) ViT-small is approximately comparable to Res34. In this way, the comparison of our proposed +method with state-of-the-art methods is reasonable and fair. +We compare our model with ProtoNet(Snell et al., 2017), CTX (Doersch et al., 2020), TSA (Li et al., 2022), +etc. These methods take the backbones of ResNet18 or ResNet34. Also, the pretrain-meta-train-finetune +baseline (P>M>F) (Hu et al., 2022) is not considered in computing average rank since extra data is used. As +in Tab. 2, when using ViT-small as backbone whose parameter size is comparable to that of ResNet34, our +model receives 1.6 average rank on all dataset. Specifically, on Texture and Fungi, our model outperforms the +strongest competitors CTX and TSA by about 8% and 10%, while on other datasets the performance of our +model is still comparable with or slight better than that of the existing methods. We notice that our model +1https://github.com/facebookresearch/fvcore +8 + +Published in Transactions on Machine Learning Research (08/2022) +Model +Backbone ILSVRC Omni Acraft +CUB +DTD +QDraw Fungi Flower +Sign +COCO +Avg +Rank +Finetune +Res18 +45.78 +60.85 +68.69 +57.31 +69.05 +42.60 +38.20 +85.51 +66.79 +34.86 +56.96 +10.2 +Proto +50.50 +59.98 +53.10 +68.79 +66.56 +48.96 +39.71 +85.27 +47.12 +41.00 +56.10 +10.5 +Relation +34.69 +45.35 +40.73 +49.51 +52.97 +43.30 +30.55 +68.76 +33.67 +29.15 +42.87 +14.6 +P-MAML +49.53 +63.37 +55.95 +68.66 +66.49 +51.52 +39.96 +87.15 +48.83 +43.74 +57.52 +9.2 +BOHB +51.92 +67.57 +54.12 +70.69 +68.34 +50.33 +41.38 +87.34 +51.80 +48.03 +59.15 +8.2 +TSA +59.50 +78.20 +72.20 +74.90 +77.30 +67.60 +44.70 +90.90 +82.50 +59.00 +70.68 +4.3 +Ours +ViT-t +56.40 +72.52 +72.84 +73.79 +77.57 +67.97 +51.23 +93.30 +84.09 +55.68 +70.54 +4.1 +Proto +Res34 +53.70 +68.50 +58.00 +74.10 +68.80 +53.30 +40.70 +87.00 +58.10 +41.70 +60.39 +7.4 +CTX +62.76 +82.21 +79.49 +80.63 +75.57 +72.68 +51.58 +95.34 +82.65 +59.90 +74.28 +2.8 +TSA +63.73 +82.58 80.13 +83.39 +79.61 +71.03 +51.38 +94.05 +81.71 +61.67 +74.93 +2.5 +P>M>F∗ +74.69 +80.68 +76.78 +85.04 +86.63 +71.25 +54.78 +94.57 +88.33 +62.57 +77.53 +— +Ours +ViT-s +67.37 +78.11 +79.94 +85.93 87.62 +71.34 +61.80 +96.57 +85.09 +62.33 +77.61 +1.6 +Table 2: +Test accuracies and average rank on Meta-Dataset. Note that different backbones are adopted by +these methods. * denotes using extra data for training. The bolded items are the best ones with highest +accuracies. +is inferior to the best ones in Omniglot, while this is reasonable. Since Omniglot images represent simple +characters with monotone color patterns, each image patches contain less information than images in other +datasets. Vanilla ViTs have less efficiency in dealing with these image patches due to limited interaction +among patch embeddings. This problem can be solved with much sophisticated variants of ViT like Swin (Liu +et al., 2021b), and will be taken as future works. Moreover, our proposed method is better than P>M>F, +which not only utilizes extra data for training but also finetunes all model parameters during testing, on more +than half of the datasets, which strongly indicates the effectiveness of the proposed finetuning strategy in this +paper. As for using ViT-tiny which has much less parameter than Res18, our model is still comparable to the +state-of-the-art methods and outperforms many popular baselines. Particularly, compared with ProtoNet +which is one of the most famous and efficient methods in FSL, our eTT shows significant boost on Aircraft +by 19.74% and TrafficSign by 36.97%. The reason of the inferior results on several datasets against TSA can +be two folds. Firstly, the ViT-tiny intrinsically has smaller capacity than Res18. On the other hand, while it +is common to train ViT with large scale images and patches so that the images are splitted into abundant +patches and each patch-level token can receive enough information. In contrast, we adopt 84 × 84 images +with 8 × 8 patch size for ViT-tiny so that the comparison with Res18 is fair, which lead to less patches with +smaller size and may have negative influence on the performance. In general, the results indicate that our +proposed eTT can make ViT models a desirable choice for large scale FSL problems. +4.3 +Model Analysis +To further validate the effectiveness of our method, we conduct a series of ablation studies on Meta-Dataset +using ViT-small below. +4.3.1 +Design of Each Module +Can finetuning on meta-train set boost the performance? One would ask whether it is necessary to +make use of base annotations, as supervised pretraining is also effective in many FSL works (Ye et al., 2020; +Hou et al., 2019). To verify it, we finetune DINO-pre-trained ViT-small on meta-train split of ImageNet, in +which the options of all hyper-parameters and data augmentations follow DeiT (Touvron et al., 2021) using +either way of class token features or averaged patch features as image representations. After such a supervised +finetuning, we test the models both with the basic test-time finetuning method as in Sec. 3.3, which we +denote as LT+NCC, and with our proposed eTT. The results are shown in Fig. 3, from which we find out +that (1) Supervised finetuning does improve test accuracies on ImageNet, CUB and MSCOCO. Particularly, +the token finetune model receives 89.83% accuracy on CUB when testing with our eTT, which is remarkably +better than any other models. This is reasonable as similar images between ImageNet and these datasets. By +9 + +Published in Transactions on Machine Learning Research (08/2022) +Model +ILSVRC +Omni +Acraft +CUB +DTD +QDraw +Fungi +Flower +Sign +COCO +Avg +Proto +63.37 +65.86 +45.11 +72.01 +83.50 +60.88 +51.02 +92.39 +49.23 +54.99 +63.84 +LT+NCC +65.96 +67.62 +64.03 +77.10 +83.46 +63.88 +57.79 +93.13 +66.91 +56.04 +69.59 +Last +66.32 +71.04 +78.04 +86.25 +86.67 +64.22 +55.69 +94.44 +65.55 +55.94 +72.42 +First +61.54 +50.46 +69.23 +79.17 +83.10 +68.69 +49.93 +93.50 +54.28 +58.45 +66.84 +LN +66.22 +70.45 +69.41 +81.29 +86.37 +66.28 +58.38 +96.25 +71.09 +59.57 +72.53 +APT +66.75 +75.16 +75.41 +84.25 +86.47 +69.55 +60.03 +96.38 +78.20 +61.10 +75.33 +Adapter +66.53 +72.31 +73.75 +83.73 +86.86 +66.74 +58.49 +96.15 +82.65 +62.40 +74.93 +eTT +67.37 +78.11 +79.94 +85.93 +87.62 +71.34 +61.80 +96.57 +85.09 +62.33 +77.61 +Random +66.12 +76.33 +78.35 +84.77 +86.78 +70.13 +59.25 +96.00 +82.28 +59.59 +75.96 +Avg +66.11 +75.06 +77.07 +85.16 +87.35 +70.72 +61.79 +96.54 +84.28 +62.18 +76.73 +Sampling +67.81 +76.72 +77.96 +85.79 +87.25 +70.19 +60.73 +96.27 +83.72 +62.17 +76.86 +Full +67.37 +78.11 +79.94 +85.93 +87.62 +71.34 +61.80 +96.57 +85.09 +62.33 +77.61 +Linear +66.35 +74.26 +79.42 +83.65 +86.02 +71.11 +55.73 +95.89 +82.73 +59.90 +75.51 +Bottleneck +67.29 +76.06 +79.72 +85.60 +87.21 +70.59 +61.59 +96.15 +85.00 +62.02 +77.12 +FiLM +66.91 +75.32 +78.26 +85.78 +86.83 +70.29 +61.65 +96.50 +84.48 +61.75 +76.78 +Offset +67.37 +78.11 +79.94 +85.93 +87.62 +71.34 +61.80 +96.57 +85.09 +62.33 +77.61 +w/o PR +66.72 +74.20 +78.42 +85.06 +87.01 +70.34 +61.64 +96.51 +84.23 +61.08 +76.52 +w PR +67.37 +78.11 +79.94 +85.93 +87.62 +71.34 +61.80 +96.57 +85.09 +62.33 +77.61 +w/o Stand +67.09 +76.42 +78.87 +83.10 +86.50 +70.09 +61.02 +96.33 +82.88 +61.33 +76.36 +w Stand +67.37 +78.11 +79.94 +85.93 +87.62 +71.34 +61.80 +96.57 +85.09 +62.33 +77.61 +Table 3: +Test accuracies on Meta-Dataset of different variants of our proposed method. The bolded items +are the best ones with highest accuracies. +training on the image annotations of ImageNet, the model learns class-specific knowledge which cannot be +obtained during self-supervised learning. Since the categories are highly correlated and overlapped among +these datasets, the learned knowledge can also benefit the recognition on these novel datasets even though the +specific novel classes do not appear in the meta-train set. (2) Despite the improvement on the three datasets, +models with supervised finetuning degrade on the other datasets, especially on Traffic Sign and VGG Flower. +This is due to fitting class labels weakens the effect of these features and makes it harder to generalize to +novel domains. When taking into account the performance of all datasets, pretraining with DINO is generally +the much more desirable choice for better generalization over different domains. (3) The improvement of +our propose method against the basic LT+NCC is not consistent among three different kinds of pretraining +strategy. For example, while our method can boost the performance of DINO pre-trained model by 9.47% on +Aircraft and 4.83% on CUB, it can only bring much less advantage on models with supervised finetuning. +Effectiveness of APT and DRA. We test the DINO pre-trained model with different kinds of testing +strategies including (1) Proto: Directly generating predictions based on ProtoNet. The prototypes are +computed using averaged class token features from each category. +(2) LT+NCC: The basic test-time +finetuning method in Sec. 3.3. (3) Last: Finetuning the last transformer layer during testing, together with +LT+NCC. which has similar parameter size to our method. (4) First: Finetuning the first transformer layer +during testing, together with LT+NCC. which has similar parameter size to our method. (5) LN: We try +to finetune the affinity parameter in each layer normalization as an alternative finetune strategy, which is +utilized in many cross-domain FSL works (Tseng et al.; Tsutsui et al., 2022). (6) APT: The model is finetuned +using APT together with LT+NCC, using cross entropy loss and the proposed prototypical regularization. +(7) Adapter: The model is finetuned using DRA together with LT+NCC, using cross entropy loss. (8) eTT: +The model is finetuned using our proposed APT, DRA and LT+NCC. The results in Tab. 3 show that while +LT+NCC can fundamentally improve the model which indicates the importance of test-time finetuning, +adding our proposed modules to the finetuning procedure can consistently bring higher performance. Also, +finetuning specific transformer layer can only bring limited improvement on few datasets: finetuning the last +10 + +Published in Transactions on Machine Learning Research (08/2022) +Figure 3: Test accuracy of different training strategy if testing with (a) LT+NCC or (b) our eTT. +Figure 4: Visualization of feature embeddings from a randomly sampled episode of TrafficSign. +layer leads to good performance on Aircraft, CUB and Texture, while updating the first layer leads to good +performance on Quickdraw and MSCOCO. However, this simple finetuning strategy cannot bring consistent +improvement on all datasets. This indicates that different data requires different levels of adaptation, and the +improvement is much smaller than that of our method. Moreover, we give the tSNE visualization of feature +embeddings of a randomly sampled episode from TrafficSign in Fig. 4, which demonstrates that utilizing our +proposed method can better regulate the feature embeddings into proper clusters. +Is prototypical initialization necessary? One of the most important parts of our APT is the attentive +prototypical initialization in which we use attentively aggregated patch embeddings to initialize the prefix +matrix. To verify this strategy, we compare several different choices of initialization, including (1) Random: +random initialization from normal distribution. (2) Avg: simply averaging all patch embeddings from each +category. (3) Sampling: randomly sampling one image for each category, and then initializing the prefix +matrix with the averaged patch embeddings of each image. (4) Full: computing prototypes with our proposed +attentive prototype. Results in Tab. 3 show that random initialization performs the worst, which can be +resulted from insufficient task-specific information provided by the prefix in this way. Meanwhile, among all +other strategies, using the attention map to aggregate patch embeddings as in Eq. 5 is better than simply +averaging, leading to about 1% improvement on average. +Do we need a more complex adapter structure? One would argue that our DRA structure is too simple +to learn the complex knowledge from support images. In Tab. 3 we compare three different instantiations of +adapters including (1) Linear: As in Li et al. (2022), we use a linear layer for each adapter, whose output +are than added to the original features in the MSA and FFN. (2) Bottleneck: We expand the linear layer +11 + +token +token +90 +pool +90 +pool +DINO +DINO +Test Accuracy +80 +Test Accuracy +80 +70 +70 +60 +60 +50 +50 +40 +40 +craft +oraw +Fung +raw +(a)TestAccuracy when using LT+NCC +(b)TestAccuracywhenusing eTTPublished in Transactions on Machine Learning Research (08/2022) +to a bottleneck structure where two linear layers are used for each adapters. (3) FiLM: We compare DRA +with a FiLM-like variant, in which we add a scaling vector for each adapter as in FiLM layer Perez et al. +(2018). Note that such a method is similar to MTL (Sun et al., 2019) in FSL. The difference lies in that we +still use the original way to directly tune the parameters on the novel support sets, instead of using another +meta-trained module to generate the parameters. (4) Offset: Only an offset vector is adopted for each adapter. +The results reveals that the linear adapter performs the worst, which means such a structure is improper for +ViT in finetuning. Moreover, we also find that using the bottleneck adapter will result in a dilemma. If we +use small initial value for the adapter, the weights of each layer can only achieve gradient with extremely +small values. As the result, these weights, except the bias term of the last layer, can hardly be updated based +on the support knowledge, which means such an architecture almost equals to our design where only an offset +vector is utilized. On the other hand, if large initial values are adopted to avoid gradient diminishing, then +the output features from the adapters can make the predictions severely deviate from those without adapters, +thus leading to worse performance. As for the FiLM-like DRA, it is worse than offset DRA by about 0.8% on +average, while it doubles the parameter size based on offset DRA, leading to no significant additional efficacy. +Effectiveness of prototypical regularization. We also validate this regularization. In Tab. 3 we present +the test accuracy when finetuning with and without this loss function. We can find that by applying this +objective function, the model can have higher results on most datasets. Besides, as described in Sec 3.6, we +use a standardization technique when computing the prototypical regularization. To verify its efficacy, we +compare the model with and without such a standardization. The results are shown in Tab. 3. When not +using standardization, the results are generally worse given comparable confidence intervals (Tab. 11). The +results verify that this strategy can help the model with more stable finetuning procedure. +dproj +ILSVRC +Omni +Acraft +CUB +DTD +QDraw +Fungi +Flower +Sign +COCO +Avg +64 +67.18 +75.30 +78.88 +86.20 +87.09 +69.82 +61.61 +96.31 +82.24 +62.14 +76.68 +96 +66.23 +75.69 +78.26 +85.67 +87.28 +70.25 +61.97 +96.59 +84.10 +62.17 +76.82 +128 +67.31 +76.83 +78.81 +85.77 +87.36 +70.16 +60.81 +96.53 +84.29 +62.12 +77.00 +256 +66.83 +78.04 +78.38 +84.60 +86.68 +70.43 +61.03 +96.23 +85.33 +62.10 +76.97 +192 +67.37 +78.11 +79.94 +85.93 +87.62 +71.34 +61.80 +96.57 +85.09 +62.33 +77.61 +Table 4: +Test accuracies on Meta-Dataset of different variants of our proposed method. The bolded items +are the best ones with highest accuracies. +4.3.2 +Comparison among Different Hyper-parameter Settings +In additional to the ablation study about the proposed module, We further verify different choices of hyper- +parameters in our model. Especially, dproj for the transformation module in APT and λ for the prototypical +regularization are tested in Tab. 4 and Tab. 5 in the Appendix. In general, the improvement is not consistent. +For dproj, we can find that using 192-d hidden dimension can get globally better results, which indicates +that such a choice can make a good balance between the model capacity and scale so that the finetuning +can be conducted both efficiently and effectively. As for λ, 0.1 seems to be a desirable choice. Intuitively, +smaller λ leads to less control of the prefix from the proposed prototypical regularization. Therefore, the +prefix may lose the desired information during the optimization on the support set. On the other hand, when +λ is too large, the regularization overwhelms the label supervision, and thus the model can hardly adapt to +the support knowledge, leading to worse performance especially on Omniglot and Aircraft. +5 +Conclusion +We propose a novel finetuning method named efficient Transformer Tuning (eTT) for few-shot learning with +ViT as our backbone. By fixing the parameters in the backbone and utilizing attentive prefix tuning and +domain residual adapter, our method can guide the ViT model with comprehensive task-specific information, +which leads to better representations and performance. This is demonstrated by the fact that we establish +new state-of-the-arts on the large-scale benchmark Meta-Dataset. +12 + +Published in Transactions on Machine Learning Research (08/2022) +References +Samira Abnar, Mostafa Dehghani, Behnam Neyshabur, and Hanie Sedghi. Exploring the limits of large scale +pre-training. 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(2) A flexible finetuning algorithm such as (Lee et al.) may have +better generalization ability when facing episodes with various shots and ways, than the commonly-used +methods that adopt fixed test-time finetuning iterations. However, it is non-trivial to directly merge such +methods with our proposed eTT due to different network structures and tuning strategies. It can be taken as +the future work to properly utilize these flexible finetuning algorithm to further improve the performance of +ViT in FSL. +15 + +Published in Transactions on Machine Learning Research (08/2022) +A.2 +Additional Experiment Results +A.2.1 +Full Comparison with state-of-the-art methods +We show the accuracies together with confidence interval in Tab. 6. Beyond the accuracies which is analyzed +in the main context, the confidence interval of our eTT on both ViT-tiny and ViT-small is comparable with +the other competitors, which reflects that our method is stable and robust enough among different testing +episodes. +A.2.2 +Influence of Training set +As we have stated in the main context, our eTT is trained on the meta-train split of ImageNet to make fair +comparison with other methods. To show to what extent training on the full ImageNet training set instead of +the meta-train set can impact the performance, we test our eTT using off-the-shelf DINO ViT-s model. The +results are shown in Tab. 7. We can find that (1) For those datasets on which DINO meta-train performs +better than P>M>F, using full ImageNet to train DINO can bring further improvement. (2) With the help of +more data, our eTT overpasses P>M>F on ILSVRC and MSCOCO. (3) While more data does improve the +results on Omniglot and TrafficSign, the final results are still worse than those of P>M>F, which we think +may be correlated with the limitations of our method as analyzed above. Given all these results, as a lighter +model in that no meta-training phase is utilized and only few parameters are engaged in the test-time tuning, +our method can still enjoy comparable performance with P>M>F when training on same amount of data. +A.2.3 +Influence of DINO +The DINO pretrain procedure is an important part of our method. To verify the effectiveness of DINO +pretrain so that the comparison with other methods is fair enough, we present in Tab. 8 the results of TSA +using DINO-pretrained ResNet-34 and eTT using supervised pretrained ViT-s. We can find that (1) The +effect of DINO is not consistent on two backbones. While DINO benifits our eTT with about 5% accuracy on +average, it severely weakens the performance of TSA with a large margin. It means that for FSL, different +backbones require different pretrain strategy respectively. (2) While supervised pretrained ViT-s performs +worse on most datasets, it is better on CUB and COCO, which indicates learning label information from +ImageNet can help the model understanding novel knowledge from these two datasets. +A.2.4 +Verification of potential overfitting in finetuning +As we have stated in Sec. 1, finetuning the whole backbone model with few support data will meet potential +overfitting problem. To reveal if such a problem exists in Meta-Dataset, we conduct an experiment as follow: +during a normal testing phase, we select all episodes whose minimum shot (minimum number of support +images for each class) is no larger than 2 (extremely small number of labelled instances), and compare the +average accuracies of eTT and simple finetuning based on these episodes. Tab. 9 and Tab. 10 show the +accuracies on support sets and query sets respectively. We can find that most of the datasets both methods +receive nearly 100% accuracy, which means these two methods can well learn the support data. Given this +fact, finetuning is much worse than eTT in terms of query accuracies. The overfitting can be reflected given +high training accuracies and worse testing performance, and to some extent our proposed eTT can fix this +problem. +A.2.5 +More Visualization +We visualize the self-attention map from models with and without DRA on ILSVRC and TraffignSign in +Fig. 5. Specifically, we randomly sample an episode from each dataset and use our eTT to tune the model +based on the support samples. Then we calculate the self-attention map of the last layer’s class token and +highlight the areas with top 20% attention scores. We can find that for the in-domain ILSVRC episode, the +model can attend to similar regions no matter whether DRA is used. In contrast, the model without DRA +can easily attend to background regions with less valuable information, which reveals a potential reason that +these two models has similar accuracies on ILSVRC but large performance gap on TrafficSign. +16 + +Published in Transactions on Machine Learning Research (08/2022) +Futhermore, we visualize more episodes from Aircraft, TrafficSign and MSCOCO in Fig. 6, Fig. 7 and Fig. 8, +which shows that our porposed eTT can remarkably improve the embedding space after test-time finetuning. +A.3 +Broader Impact +Our paper presents a more efficient and practical FSL pipeline utilizing ViT. We hope this work can shed +light on the broader usage of ViT in FSL tasks. On the other hand, the proposed method can provide +researchers with alternative choice for FSL applications in real-case scenarios with large-scale meta-train set +and challenging various test episodes. +dproj +ILSVRC +Omni +Acraft +CUB +DTD +QDraw +Fungi +Flower +Sign +COCO +Avg +0.01 +67.01 +76.56 +78.34 +85.53 +86.96 +70.03 +61.20 +96.17 +85.00 +62.67 +76.95 +0.05 +66.49 +77.40 +78.92 +85.80 +87.54 +70.23 +60.78 +96.28 +84.95 +62.38 +77.08 +0.5 +66.88 +77.73 +78.65 +86.00 +87.15 +70.48 +61.64 +96.23 +84.39 +63.39 +77.25 +0.9 +67.03 +76.55 +77.89 +85.78 +87.04 +70.08 +62.45 +96.20 +84.44 +62.83 +77.03 +0.1 +67.37 +78.11 +79.94 +85.93 +87.62 +71.34 +61.80 +96.57 +85.09 +62.33 +77.61 +Table 5: +Test accuracies on Meta-Dataset of different variants of our proposed method. The bolded items +are the best ones with highest accuracies. +Model +Backbone +ILSVRC +Omni +Acraft +CUB +DTD +QDraw +Fungi +Flower +Sign +COCO +Rank +Finetune +Res18 +45.781.10 +60.851.58 +68.691.26 +57.311.26 +69.050.90 +42.601.17 +38.201.02 +85.510.68 +66.791.31 +34.860.97 +10.2 +Proto +50.501.08 +59.981.35 +53.101.00 +68.791.01 +66.560.83 +48.961.08 +39.711.11 +85.270.77 +47.121.10 +41.001.10 +10.5 +Relation +34.691.01 +45.351.36 +40.730.83 +49.511.05 +52.970.69 +43.301.08 +30.551.04 +68.760.83 +33.671.05 +29.151.01 +14.6 +P-MAML +49.531.05 +63.371.33 +55.950.99 +68.660.96 +66.490.83 +51.521.00 +39.961.14 +87.150.69 +48.831.09 +43.741.12 +9.2 +BOHB +51.921.05 +67.571.21 +54.120.90 +70.690.90 +68.340.76 +50.331.04 +41.381.12 +87.340.59 +51.801.04 +48.030.99 +8.2 +TSA +59.501.10 +78.201.20 +72.201.00 +74.900.90 +77.300.70 +67.600.90 +44.701.00 +90.900.60 +82.500.80 +59.001.00 +4.3 +Ours +ViT-t +56.401.13 +72.521.36 +72.841.04 +73.791.09 +77.570.84 +67.970.88 +51.231.15 +93.300.57 +84.091.07 +55.681.05 +4.1 +Proto +Res34 +53.701.07 +68.501.27 +58.000.96 +74.100.92 +68.800.77 +53.301.06 +40.701.15 +87.000.73 +58.101.05 +41.701.08 +7.4 +CTX +62.760.99 +82.211.00 +79.490.89 +80.630.88 +75.570.64 +72.680.82 +51.581.11 +95.340.37 +82.650.76 +59.901.02 +2.8 +TSA +63.730.99 +82.581.11 +80.131.01 +83.390.80 +79.610.68 +71.030.84 +51.381.17 +94.050.45 +81.710.95 +61.670.95 +2.5 +Ours +ViT-s +67.370.97 +78.111.22 +79.941.06 +85.930.91 +87.620.57 +71.340.87 +61.801.06 +96.570.46 +85.090.90 +62.330.99 +1.6 +Table 6: +Test accuracies, confidence interval and average rank on Meta-Dataset. Note that different +backbones are adopted by these methods. The bolded items are the best ones with highest accuracies. +Model +Train Set +ILSVRC +Omni +Acraft +CUB +DTD +QDraw +Fungi +Flower +Sign +COCO +Avg +eTT +meta-train +67.37 +78.11 +79.94 +85.93 +87.62 +71.34 +61.80 +96.57 +85.09 +62.33 +77.61 +eTT +full +74.76 +78.73 +80.10 +86.99 +87.72 +71.20 +61.95 +96.66 +85.83 +64.25 +78.82 +P>M>F +full +74.69 +80.68 +76.78 +85.04 +86.63 +71.25 +54.78 +94.57 +88.33 +62.57 +77.53 +Table 7: +Test accuracies on Meta-Dataset of different variants of our proposed method. The bolded items +are the best ones with highest accuracies. The highlighted rows denote the final model in our main paper. +17 + +Published in Transactions on Machine Learning Research (08/2022) +Model +Pretrain +ILSVRC +Omni +Acraft +CUB +DTD +QDraw +Fungi +Flower +Sign +COCO +Avg +TSA +Sup. +59.50 +78.20 +72.20 +74.90 +77.30 +67.60 +44.70 +90.90 +82.50 +59.00 +70.68 +TSA +DINO +48.18 +64.94 +56.74 +45.49 +69.06 +59.51 +31.13 +81.01 +48.70 +26.18 +53.09 +eTT +Sup. +65.17 +67.47 +73.30 +87.71 +84.50 +67.46 +55.51 +92.55 +64.08 +63.68 +72.14 +eTT +DINO +67.37 +78.11 +79.94 +85.93 +87.62 +71.34 +61.80 +96.57 +85.09 +62.33 +77.61 +Table 8: +Test accuracies on Meta-Dataset of different variants of our proposed method. The bolded items +are the best ones with highest accuracies. The highlighted rows denote the final model in our main paper. +Model +ILSVRC +Omni +Acraft +CUB +DTD +QDraw +Fungi +Flower +Sign +COCO +Avg +eTT +100.00 +99.99 +100.00 +100.00 +100.00 +100.00 +99.79 +100.00 +100.00 +99.15 +99.89 +FT +100.00 +99.87 +100.00 +100.00 +100.00 +100.00 +96.95 +100.00 +100.00 +95.20 +99.20 +Table 9: +Support set accuracies of eTT and Finetune on testing episodes whose minimum shots is no larger +than 2. +Model +ILSVRC +Omni +Acraft +CUB +DTD +QDraw +Fungi +Flower +Sign +COCO +Avg +FT +29.19 +54.54 +35.10 +41.54 +53.66 +43.37 +38.53 +76.76 +72.90 +41.21 +48.68 +eTT +40.22 +64.79 +41.33 +55.11 +66.20 +49.14 +56.33 +85.03 +75.29 +56.19 +58.96 +Table 10: +Query set accuracies of eTT and Finetune on testing episodes whose minimum shots is no larger +than 2. The bolded items are the best ones with highest accuracies. +Model +ILSVRC +Omni +Acraft +CUB +DTD +QDraw +Fungi +Flower +Sign +COCO +w/o stand +1.06 +1.25 +1.05 +0.89 +0.64 +0.92 +1.05 +0.38 +0.96 +0.96 +w stand +0.97 +1.22 +1.06 +0.91 +0.57 +0.87 +1.06 +0.46 +0.90 +0.99 +Table 11: +The corresponding confidence intervals of models in ablation study on standardization. +18 + +Published in Transactions on Machine Learning Research (08/2022) +w DRA +w DRA +w/o DRA +w/o DRA +Figure 5: Visualization of self-attention from model with and without DRA on ILSVRC (left) and TrafficSign +(right). The white regions are those with top 20% attention scores +19 + ++0STOPPublished in Transactions on Machine Learning Research (08/2022) +Figure 6: More visualization of feature embeddings from a randomly sampled episode of TrafficSign. +20 + +Published in Transactions on Machine Learning Research (08/2022) +Figure 7: More visualization of feature embeddings from a randomly sampled episode of Aircraft. +21 + +Published in Transactions on Machine Learning Research (08/2022) +Figure 8: More visualization of feature embeddings from a randomly sampled episode of MSCOCO. +22 + diff --git a/DtE0T4oBgHgl3EQfggGb/content/tmp_files/load_file.txt b/DtE0T4oBgHgl3EQfggGb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e2970607e7907ba997466b871dee471727cad462 --- /dev/null +++ b/DtE0T4oBgHgl3EQfggGb/content/tmp_files/load_file.txt @@ -0,0 +1,1639 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf,len=1638 +page_content='Published in Transactions on Machine Learning Research (08/2022) Exploring Efficient Few-shot Adaptation for Vision Trans- formers Chengming Xu cmxu18@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='cn School of Data Science, Fudan University Siqian Yang seasonsyang@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='com Yabiao Wang caseywang@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='com Youtu Lab, Tencent Zhanxiong Wang maxzxwang@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='com Tencent Yanwei Fu∗ yanweifu@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='cn Xiangyang Xue xiangyangxue@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='cn School of Data Science, Fudan University Reviewed on OpenReview: https: // openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' net/ forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' id= n3qLz4eL1l Abstract The task of Few-shot Learning (FSL) aims to do the inference on novel categories containing only few labeled examples, with the help of knowledge learned from base categories containing abundant labeled training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' While there are numerous works into FSL task, Vision Transformers (ViTs) have rarely been taken as the backbone to FSL with few trials (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Evci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Abnar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=') focusing on naïve finetuning of whole backbone or classification layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Essentially, despite ViTs have been shown to enjoy comparable or even better performance on other vision tasks, it is still very nontrivial to efficiently finetune the ViTs in real-world FSL scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' To this end, we propose a novel efficient Transformer Tuning (eTT) method that facilitates finetuning ViTs in the FSL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The key novelties come from the newly presented Attentive Prefix Tuning (APT) and Domain Residual Adapter (DRA) for the task and backbone tuning, individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Specifically, in APT, the prefix is projected to new key and value pairs that are attached to each self-attention layer to provide the model with task-specific information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Moreover, we design the DRA in the form of learnable offset vectors to handle the potential domain gaps between base and novel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' To ensure the APT would not deviate from the initial task-specific information much, we further propose a novel prototypical regularization, which maximizes the similarity between the projected distribution of prefix and initial prototypes, regularizing the update procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Our method receives outstanding performance on the challenging Meta-Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We conduct extensive experiments to show the efficacy of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Our code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='com/loadder/eTT_TMLR2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 1 Introduction Modern computer vision models such as ResNet (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2016) and Faster R-CNN (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2015) are trained on large-scale training sets, and not well generalize to handle the long tail categories with few ∗This paper is supported by the project NSFC(62076067).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='02419v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='CV] 6 Jan 2023 Published in Transactions on Machine Learning Research (08/2022) labeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Few-shot Learning (FSL) has thus been studied to make inference on insufficiently-labeled novel categories typically with the transferable knowledge learned from base categories which are provided with abundant labeled training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Essentially, the FSL can be taken as representation learning, as its backbones should ideally extract features representative and generalizable to various novel tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Currently Convolutional Neural Networks (CNNs), especially ResNet, are the predominant backbone and widely utilized in most existing FSL works (Ravi & Larochelle, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Nichol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Recently, by taking the merits of Multi-headed Self-Attention (MSA) mechanism and Feed Forward Net- work (FFN), the transformers have been widely used in the recognition (Alexey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021b), detection (Beal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2020) and image editing (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The general pipeline of Pretrain-(Meta- train)-Finetune has been explored in few ViTs on FSL (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Evci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Abnar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' ), recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Particularly, the ViT models are first pretrained/meta-trained on a large-scale dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Then a test-time finetune procedure is set up for each target task on novel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The finetuning strategy can be generally categorized into linear classifier probing and backbone tuning: the former one optimizes the reasonable decision boundaries by the fixed embeddings, while the latter one considers the adaptation of both embedding space and classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In this paper we focus on the backbone tuning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022) shows that the naïve Pretrain-Meta- train-Finetune (P>M>F) baseline can generally have satisfactory performance in FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Unfortunately, it involves heavy computations and potential overfitting in FSL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Particularly, (1) It typically demands extraordinary computing power to formulate episodes from a large number of support classes to update the whole network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Thus it is less efficient in many real-case applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' For example, the edge devices such as mobiles donot have enough computational power to adapt all model parameters by personalized/specialized data collected on these devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2) It is very subtle and difficult to directly fine-tune trained deep models on one or two labeled instances per class, as such few-shot models will suffer from severe overfitting (Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Fei-Fei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Brian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' By contrast, humans have the ability of conducting few-shot recognition from even single example of unseen novel category with very high confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Such problems may be the culprit of the phenomenon that their proposed finetune strategy only works on part of datasets and has less effect to the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' This suggests their limited usage of ViT backbone for any potential FSL applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' An alternative choice is to finetune specific layers in a ViT model with much smaller tunable parameters (ViT-s block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Such a strategy nevertheless can only finetune either low-level or high-level features, leading to inferior performance in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Therefore it is desirable to have an efficient and light-weighted ViT tuning method that shall not only avoid overfitting to small training samples, but also achieve high performance of FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In this paper, we present a novel efficient Transformer Tuning (eTT) for few-shot learning task, which adopts a pretrain-finetune pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' To pretrain our transformer, we advocate utilizing the recent self-supervised method – DINO (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Our key novelties are in the finetuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 1(b), we propose Attentive Prefix Tuning (APT) and Domain Residual Adapter (DRA) as the key components to our eTT, to efficiently learn the newly-introduced tunable parameters over novel support sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Specifically, we formulate the attentive prototypes by aggregating patch embeddings with the corresponding attention weights of the class token for each image, so as to provide the model with abundant task-specific information and guide each self-attention layer to aggregate more class-related features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' To encourage the prefix to keep the prior knowledge from initial prototypes, we further propose a novel prototypical regularization which restricts the relationship between the prefix and prototypes by optimizing the similarity of their projected distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Moreover, we propose to additionally adopt a light-weighted domain residual adapter in the form of learnable offset to deal with the potential failure of APT on large domain gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Extensive experiments are conducted to evaluate our eTT: we use the ViT-tiny and ViT-small backbones on the large-scale Meta-Dataset (Triantafillou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2019) consisting of ten sub-datasets from different domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' and the results show that our model can achieve outstanding performance with comparable or even much fewer model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Thus our eTT is a promising method on efficiently finetuning ViTs on the FSL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Our paper has the following contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In order to solve the problem of inefficiency and make better use of ViT in FSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' we propose a novel 2 Published in Transactions on Machine Learning Research (08/2022) Domain residual adapter ViT block … Attentive prototypes Visual prefix Domain residual adapter ViT block Few-shot episodes (a) Tunable parameters in Backbone Finetuning (b) Attentive Prefix Tuning in Task Tuning Support images Initialize Key/value pairs project plug Figure 1: (a) Comparing with other backbones,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' we propose the Domain Residual Adapter (DRA) to tune much less parameters in our efficient Transformer Tuning (eTT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' and effective for large-scale FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (b) The few-shot support samples are first processed into attentive prototypes which are used to initialize the task-specific visual prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Then the prefix together with the domain adapter are attached to each layer of the ViT to finetune our ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' finetuning method named efficient Transformer Tuning (eTT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Inspired by recent advance in language model, a novel attentive prefix tuning is presented utilizing the attentive prototypes to embed the task-specific knowledge into pretrained ViT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Particularly, we propose a new initialization strategy tailored for FSL by leveraging prototypical information from the self-attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Moreover, a novel domain residual adapter is repurposed to handle the various domain gaps between training and testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We introduce a prototypical regularization term which can constrain the update procedure of prefix during finetuning to maintain the initial task-specific knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' By utilizing the proposed eTT, our ViT models receive remarkable performance on Meta-Dataset, overpassing the existing ResNet-based methods without using additional training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' More importantly, both of the model scale and efficiency of our method are comparable with the other competitors, indicating the promising application of ViTs in FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 2 Related Works Few-shot recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' FSL learns transferable knowledge from base classes and adapt it to a disjoint set (novel classes) with limited training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Among those FSL tasks, few-shot image recognition is the one with most focus and researches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Existing works can be grouped into two main categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' One is optimization-based methods (Ravi & Larochelle, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Nichol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2019), which learn parameters that can be better finetuned on few-shot support sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The other is metric-based methods such as ProtoNet (Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2017), RelationNet (Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2018), CAN (Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2019), DMF (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021), COSOC (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021) and CTX (Doersch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2020), which solve FSL by applying an existing or learned metric on the extracted features of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Particularly, CTX (Doersch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2020) builds up a cross attention module which interacts between query and support images to adaptively aggregate better prototypes than simply averaging all support features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' While these methods perform well on classical few-shot learning settings, most of them adopt convnet as backbone, especially ResNet (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We, on the opposite, try to make full use of another widely-applied structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' ViT, in FSL, which requires extra design for training and finetuning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Transformer in vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Transformers widely utilize the self-attention mechanism which originally are employed to process the feature sequence in Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Then large scale transformers become increasingly popular in NLP tasks to build complex language models, and also extend to vision tasks (Alexey 3 Published in Transactions on Machine Learning Research (08/2022) et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021b) by formulating the token sequence with image patches processed with position embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' It has been shown the efficacy in various applications, such as (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021a) for image caption, (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2020) for multiple object tracking and (Esser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021) for image inpainting and editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Critically, ViTs is typically trained by very large-scale dataset, and few effort has been dedicated in training or finetuning on few-shot supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We follow the pretrain-meta-train-finetune pipeline (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022), while their method finetune the whole ViTs on few-shot examples, and thus has less efficiency and can easily overfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In contrast, our proposed eTT has the key components of DRA and APT, demanding much less tunable parameters with much better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Finetuning algorithm for ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The idea of finetuning ViTs on small-scale datasets has been partly investigated in Natural Language Processing (NLP) communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2019) proposed to attach two learnable bottleneck adapters to each transformer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Other works (Xiang & Percy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Brian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=') make use of the prompt which trains a small task-specific prompt for each task so that the prompt can guide the model with knowledge corresponding to the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Such a prompting idea from NLP is inherited and repurposed to finetune a learnable prefix for each novel episode in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' However, these works (Xiang & Percy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Brian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2019) initialize the prefix or prompt with word embeddings which is not available in our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Instead, we propose an attentive prototype with regularization initializing the visual prefix with object-centric embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Additionally, we notice that a very good concurrent technical report (Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022) also studies finetuning visual prompt for pretrained ViTs in downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We highlight the two key differences from our eTT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The first is about the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' While initialization strategy does not matter in their method and the corresponding tasks, we show in our experiments that randomly initializing prefix does lead to sub-optimal performance in FSL, which leads to the necessity of a well-designed initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The second is that we further propose a regularization term to restrict the prefix, which has never been studied in existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Task-specific Adapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The idea of task-specific adapter has been explored in several works like (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Rebuffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2017) to adapt CNNs to learn the whole information from support set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Besides, (Requeima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Bateni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2020) adopt Feature-wise Linear Modulation (FiLM) layers (Perez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2018) to adapt task-specific information into networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In contrast, we repurpose the adapter as the domain residual to update transformer blocks in a more light-weighted way with less learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Beyond different structures, our proposed DRA intrinsically serves as the domain adapter rather than meta-learner for the FSL in Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Requeima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' While these previous works require meta-training to optimize their adaptation modules, our method simply utilizes the novel support data to learn the DRA, thus reducing the training cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Furthermore, our DRA is mostly tuned to bridge the visual domain gap between base and novel categories, thus improving the learning of APT on each episode task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3 Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='1 Problem Setup We formulate few-shot learning in the meta-learning paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In general, we have two sets of data, namely meta-train set Ds = {(Ii, yi) , yi ∈ Cs} and meta-test set Dt = {(Ii, yi) , yi ∈ Ct} which contain the base and novel data respectively and are possibly collected from different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Cs and Ct (Cs ∩ Ct = ∅) denote base and novel category sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' FSL aims to train a model on Ds which is generalizable enough on Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In the testing phase, the model can learn from few labelled data from each category of Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' While most previous FSL works (Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2018) utilize the setting of N-way K-shot in mini-ImageNet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', K training samples from N class, we follow CTX (Doersch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2020) to adopt the setting on the large-scale Meta-Dataset (Triantafillou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In each episode T , N is first uniformly sampled from [5, Nmax] where Nmax equals to min(50, |Ct|) or min(50, |Cs|) on training or testing stage, accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' N is supposed to be accessible knowledge during both training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In the most naïve case, one can get N by directly counting the number of support classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' From each of the sampled category, M query samples per category are randomly selected, and thus constructing the query set Q = {(Iq i , yq i )}NQ i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' After that random amount of samples are taken from the rest of samples belonging to these categories to form the support set S = {(Isupp i , ysupp i )}NS i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Note that compared to the classical N-way K-shot setting, 4 Published in Transactions on Machine Learning Research (08/2022) Patch embedding layer … transformer layer … transformer layer Patch embeddings Linear ProtoNet aggregate ෝ𝒚 𝜽𝒑 \u0de1𝑨 MSA + LN FFN LN + projector attention Q K V 𝜽𝒌 𝜽𝒗 MSA 𝜹𝒇 𝜹𝒂 𝜽𝒑 Image embedding Q K 𝜽𝒌 V 𝜽𝒗 𝒈 Figure 2: Schematic illustration of our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' For each image, we first fetch its patch embedding sequence and the attention score with regard to the last layer’s class token, from which the image embedding can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Then the visual prefix is initialized as the attentive prototypes of image embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The prefix, together with the proposed domain residual adapter are attached to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The final features are processed with an extra linear transformation layer and predicted with ProtoNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Dashed arrows denote forward propagation before test-time finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' such a setting generates class-imbalanced support sets, and different episodes contain different numbers of support samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' This is much more challenging to the model and learning algorithms, as they shall handle both extremely large and small support sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='2 Overview of Our Method To handle the optimization of various episodes on large-scale dataset, we present our novel finetuning model – efficient Transformer Tuning (eTT) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Our eTT follows the pipeline in Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2022), and has key stages of the pretraining and finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We employ DINO as pretraining, and conduct the task tuning by attentive prefix tuning (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='4), and backbone tuning with domain residual adapter (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' As previous work (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022) shows the importance of self-supervised pre-training to learning vision transformer models, we adopt the same principle and introduce the self-supervised learning model to pre-train our ViT backbone on base data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Specifically, we utilize the recent State-of-the-art self-supervised ViT models – DINO (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021) to pretrain our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' DINO builds up supervision based on a self-distillation framework by using the multi-crop strategy (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' As we will show in our experiments, such a pre-trained model shall have good cluster property even among cross domain images, potentially benefiting our following FSL stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Note that different from (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022) which takes an off-the-shelf model pretrained with DINO on full ImageNet, we strictly follow the FSL protocols to retrain the DINO models on the meta-train split in the target dataset to avoid the abuse of extra data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' One would ask whether it is necessary to make use of the annotations for base data, since supervised pretrain has been proven to be effective in many previous FSL works (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' As we will show in the experiments, an additional finetuning with image labels on base data cannot bring consistent improvement and even makes it worse on most datasets, which may be caused by the overfitting on the image labels leads to less generalization ability across different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Moreover, compared with vanilla supervised training, the attention maps for models trained by DINO contain more semantic information, which we will utilize in the following context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='3 Preliminary: Vanilla Test-time Finetuning Before fully developing our fine-tuning contributions, we review the simple and effective finetuning method named LT+NCC (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The novel modules proposed by us in the following context are all adopted together with this simple baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Given a ViT backbone fθ that is parameterized by θ and an episode T , the support features {xsupp i }NS i=1, where xsupp i = fθ(Isupp i ), are extracted from the support set 5 Published in Transactions on Machine Learning Research (08/2022) {Isupp i }NS i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Then, a learnable linear transformation φ is added to the model to realize the adaptation, which results in the final support features used for classification {ˆxsupp i }NS i=1, where ˆxsupp i = φ(xsupp i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The prediction of these support images can thus be calculated based on the similarity between the transformed features and the aggregated prototypes as, ¯xc = 1 �Ns i=1 1ysupp i =c Ns � i=1 ˆxsupp i 1ysupp i =c ˆysupp i (c) = exp(d(ˆxsupp i , ¯xc)) �N c=1 exp(d(ˆxsupp i , ¯xc)) (1) where d denotes cosine similarity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', d(a, b) = aT b ∥a∥∥b∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We fix all of the parameters in the original backbone, and adopt the cross entropy loss to optimize the transformation φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Precisely speaking, for each support image Isupp together with its annotation ysupp, the objective function is as following: ℓCE = −ysupp · log ˆysupp (2) After finetuning, φ is applied to query features and the same procedure as above is performed between the processed query features {ˆxq i } and prototypes {¯xc}N c=1 for the inference of each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='4 Task Tuning by Attentive Prefix Tuning We finetune the pre-trained ViT with support set via an attentive prefix tuning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Specifically, a prefix matrix θP ∈ RNP ×d is first initialized, where NP denotes the number of prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Then a bottleneck g is added upon θP to produce ˆθP ∈ RNP ×(2Ld), where L denotes the number of backbone layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The g plays the same role as the projector in each self-attention layer, except that all layers share the same module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The produced ˆθP can be reshaped and seen as L value and key pairs {θl v, θl k}L l=1, θl v, θl k ∈ RNP ×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The MSA block in the L-th layer can then be reformed by attaching these new pairs to the original key and value sequences: Al = Attn(Q, � K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' θl k � ) output = Al � V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' θl v � (3) where [·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' ·] denotes concatenation, Attn denotes the calculation of MSA matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In this way, the prefix can affect the attention matrix Al and result in different output features from the original ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Compared with the naive strategy that finetunes specific layers in ViT (ViT-s block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 1(a)) which can only adjust part of blocks, the prefix can evenly adapt each layer’s image embedding with almost the same parameter size as one transformer layer, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' By fixing the model parameters and optimizing the prefix θP and the transformation module g, the support knowledge can be smoothly embedded into the prefix, which further helps the task adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Attentive Prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The initialization of the prefix is very important to our APT, as it greatly boosts the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Critically, quite different from the prefix or prompt tuning in NLP and visual-context tasks that have task-specific instructions explicitly as word embedding sequences, each episode in our FSL only has the few support images and their labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Thus, rather than steering the model with ’what should be done’ as in Xiang & Percy, our APT shall provide the model with ’what we have globally’ by leveraging the class-specific information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Thus, the attentive prototype is presented to aggregate the image embeddings with object-centric attention, as the initialization of the prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Particularly, each support image Isupp is first transformed to a patch embebdding sequence {˜xsupp m }P m=1 with the starting patch embedding layer, ˜xsupp m = fθpe(Isupp m ) + Epos m (4) where m = 1, · · · , P 2 is the patch index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' fθpe denotes the patch embedding layer which is typically a convolutional layer whose kernel size equals to patch size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' and Epos indicates the position embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Meanwhile, we can get unnormalized attention score A ∈ Rh×P between the class token and image patches from the last MSA layer, where h denotes number of heads in each MSA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Such an attention vector can focus on the foreground in the image, especially for models trained with DINO (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021), with each head indicating a particular part or an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We can thus get the initial image-level representation ˆA = σ(A) ˜xsupp = 1 h h � n=1 P 2 � m=1 ˆAnm˜xsupp m (5) 6 Published in Transactions on Machine Learning Research (08/2022) where σ is softmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Compared with simply averaging all patch embeddings, the attentive embeddings can highlight the objects of interest and suppress the background information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Then the prototypes ¯x can be calculated by averaging the attentive image embeddings belonging to each support category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We set the number of prefix as N, which is available during testing for each episode, and initialize the prefix with ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In this way, commonly-used prototypes can provide the model with comprehensive information about the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Also such a first-order statistics is comparable with the normal patch features among the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' This can benefit the training with more stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' When N is large, more prefix are required to fully learn the information included by each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' On the other hand, when N is small so that the episode is relatively easy, fewer prefix can handle the support knowledge without trouble while decreasing the computing debt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='5 Backbone Tuning by Domain Residual Adapter Finetuning few-shot tasks by APT will make a good balance between performance and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' To further improve the model generalization ability on different domains, we further propose the backbone tuning by leveraging the Domain Residual Adapters (DRA), as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Specifically, for the l-th transformer layer, we attach two learnable offset vectors δl a, δl f ∈ Rd to the MSA and FFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' After features are processed with MSA and FFN, the corresponding offsets are added to them so that the extreme domain gap can be neutralized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' These offsets are expected to represent the gap between source and target domains, and transfer the original manifold to a more appropriate one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='6 Loss Functions Prototypical Regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In addition to the cross entropy loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 2, we propose a novel prototypical regularization to ensure the class-specific information, which is embedded in the prefix via initialization, can be maintained during update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The knowledge in attentive prototypes is distilled to the prefix during finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Concretely, in each iteration, the prototypes ¯x and prefix θP are first projected to a latent space via a projector module ψ, which produces ¯x′ and θ′ P respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Then the distillation loss is computed using these two embeddings as, ℓdist = 1 N N � n=1 H(¯x′n, θ′n P ) (6) where H(a, b) = −a log b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The above objective function can ensure the prototype of each category and the corresponding prefix contain consistent information, which is indicated by the similarity between distributions after projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' To make training more stable and avoid collapse, for each episode we maintain an exponential moving average (EMA) of ¯x′ as the center variable ccenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Before calculating ℓdist, we standardize ¯x′ as σ( ¯x′−xcenter τ ), where σ denotes softmax function and τ is the temperature typically set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Once having both of the above losses calculated, we can optimize the model parameters including the DRA, the prefix together with the transformation g and the projector ψ, with the following objective function: L = ℓCE + λℓdist (7) where the scalar weight λ controls the strength of the regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' For a ViT with L layers, nh heads and d feature dimension, the size of trainable parameters is (N + d′ + dproj + d)d + 2(d′ + 1)Ld, where d′ is the hidden dimension for transformation module g and dproj denotes output dimension for the projector ψ, which is much smaller than that of the whole backbone model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Specifically, the learnable modules during finetuning have only about 9% parameters with regard to the whole transformer model when using ViT-small and ViT-tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 7 Published in Transactions on Machine Learning Research (08/2022) 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='1 Experimental Setup Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We use Meta-Dataset (Triantafillou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2019) – the most comprehensive and challenging large-scale FSL benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' It has 10 sub-datasets such as ImageNet (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2009) and Omniglot (Lake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2015), with various domain gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Our experiments are conducted under the single training source setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' only ImageNet is used for training, and the meta-test split of all ten datasets for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Some of the test datasets such as CUB share similar or highly-related categories with ImageNet, while the others have greater domain gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Note that Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2022) claims pretraining on all images in the training set of ImageNet is reasonable for introducing extra data and boosting the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' However, such a strategy utilizes much more training samples (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='28M images, 1000 classes in ImageNet v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='9M images, 712 classes in meta-train split of ImageNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Empirically so many additional images can greatly benefit generalization ability of self-supervised learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Therefore to make a more fair comparison, we strictly follow the experiment protocol used in CTX (Doersch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2020) and shall not use any extra data even in the unsupervised pretraining stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We resize all images to 224 × 224 for ViT-small and 84 × 84 for ViT-tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We set the patch size as 8 for ViT-tiny (as it has small input image size), and keep the other hyper-parameters as default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We adopt standard ViT-small with 12 layers, 6 attention heads, feature dimension as 384 and patch size as 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We strictly follow the hyper-parameter setting and data augmentation in DINO (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021) for pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In test-time finetuning, we empirically set the hidden dimension d′ of the transformation module as d/2, and output dimension dproj of the projector as 64 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We utilize AdamW optimizer finetuning, with learning rate set as 1e − 3 for TrafficSign and 5e − 4 for other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' λ is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' For simplicity, the selection of hyper-parameters is conducted on the meta-validation set of ImageNet, which is the only within-domain setting in Meta-Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Evaluation benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We report the accuracy of randomly sampled 600 episodes for each dataset and the average accuracy when comparing with the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The comprehensive comparison of both accuracy and 95% confidence interval is in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Backbone Image size Params(M) FLOPs(G) Res18 84×84 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='82 ViT-tiny 84×84 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='72 Res34 224×224 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='68 ViT-small 224×224 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='97 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='61 Table 1: Comparison of parameter size and FLOPs between different backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='2 Comparison with State-of-the-art Methods Before the comprehensive comparison, it is necessary to show the comparison between different backbone is fair enough since our backbone model is not the same as the existing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Therefore we present the comparison of size of model parameters and FLOPs in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 1, in which the FLOPs of all models are computed by fvcore1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The results show that (1) compared with Res18, ViT-tiny is a much smaller and efficient model, and (2) ViT-small is approximately comparable to Res34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In this way, the comparison of our proposed method with state-of-the-art methods is reasonable and fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We compare our model with ProtoNet(Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2017), CTX (Doersch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2020), TSA (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' These methods take the backbones of ResNet18 or ResNet34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Also, the pretrain-meta-train-finetune baseline (P>M>F) (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022) is not considered in computing average rank since extra data is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' As in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 2, when using ViT-small as backbone whose parameter size is comparable to that of ResNet34, our model receives 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='6 average rank on all dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Specifically, on Texture and Fungi, our model outperforms the strongest competitors CTX and TSA by about 8% and 10%, while on other datasets the performance of our model is still comparable with or slight better than that of the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We notice that our model 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='com/facebookresearch/fvcore 8 Published in Transactions on Machine Learning Research (08/2022) Model Backbone ILSVRC Omni Acraft CUB DTD QDraw Fungi Flower Sign COCO Avg Rank Finetune Res18 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='78 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='85 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='69 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='31 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='05 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='60 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='20 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='51 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='79 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='86 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='96 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='2 Proto 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='50 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='98 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='10 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='79 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='56 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='96 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='71 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='27 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='12 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='5 Relation 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='69 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='35 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='73 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='51 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='97 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='30 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='55 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='76 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='67 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='15 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='87 14.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='5 P>M>F∗ 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='69 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='68 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='78 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='04 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='63 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='25 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='78 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='57 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='33 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='57 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='53 — Ours ViT-s 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='37 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='11 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='94 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='93 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='62 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='34 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='80 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='57 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='09 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='33 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='6 Table 2: Test accuracies and average rank on Meta-Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Note that different backbones are adopted by these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' * denotes using extra data for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The bolded items are the best ones with highest accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' is inferior to the best ones in Omniglot, while this is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Since Omniglot images represent simple characters with monotone color patterns, each image patches contain less information than images in other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Vanilla ViTs have less efficiency in dealing with these image patches due to limited interaction among patch embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' This problem can be solved with much sophisticated variants of ViT like Swin (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021b), and will be taken as future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Moreover, our proposed method is better than P>M>F, which not only utilizes extra data for training but also finetunes all model parameters during testing, on more than half of the datasets, which strongly indicates the effectiveness of the proposed finetuning strategy in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' As for using ViT-tiny which has much less parameter than Res18, our model is still comparable to the state-of-the-art methods and outperforms many popular baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Particularly, compared with ProtoNet which is one of the most famous and efficient methods in FSL, our eTT shows significant boost on Aircraft by 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='74% and TrafficSign by 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='97%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The reason of the inferior results on several datasets against TSA can be two folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Firstly, the ViT-tiny intrinsically has smaller capacity than Res18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' On the other hand, while it is common to train ViT with large scale images and patches so that the images are splitted into abundant patches and each patch-level token can receive enough information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In contrast, we adopt 84 × 84 images with 8 × 8 patch size for ViT-tiny so that the comparison with Res18 is fair, which lead to less patches with smaller size and may have negative influence on the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In general, the results indicate that our proposed eTT can make ViT models a desirable choice for large scale FSL problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='3 Model Analysis To further validate the effectiveness of our method, we conduct a series of ablation studies on Meta-Dataset using ViT-small below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='1 Design of Each Module Can finetuning on meta-train set boost the performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' One would ask whether it is necessary to make use of base annotations, as supervised pretraining is also effective in many FSL works (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' To verify it, we finetune DINO-pre-trained ViT-small on meta-train split of ImageNet, in which the options of all hyper-parameters and data augmentations follow DeiT (Touvron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2021) using either way of class token features or averaged patch features as image representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' After such a supervised finetuning, we test the models both with the basic test-time finetuning method as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='3, which we denote as LT+NCC, and with our proposed eTT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3, from which we find out that (1) Supervised finetuning does improve test accuracies on ImageNet, CUB and MSCOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Particularly, the token finetune model receives 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='83% accuracy on CUB when testing with our eTT, which is remarkably better than any other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' This is reasonable as similar images between ImageNet and these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' By 9 Published in Transactions on Machine Learning Research (08/2022) Model ILSVRC Omni Acraft CUB DTD QDraw Fungi Flower Sign COCO Avg Proto 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='37 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='86 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='11 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='01 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='50 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='88 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='02 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='39 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='23 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='99 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='84 LT+NCC 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='96 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='62 64.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='57 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='09 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='33 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='61 Table 3: Test accuracies on Meta-Dataset of different variants of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The bolded items are the best ones with highest accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' training on the image annotations of ImageNet, the model learns class-specific knowledge which cannot be obtained during self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Since the categories are highly correlated and overlapped among these datasets, the learned knowledge can also benefit the recognition on these novel datasets even though the specific novel classes do not appear in the meta-train set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2) Despite the improvement on the three datasets, models with supervised finetuning degrade on the other datasets, especially on Traffic Sign and VGG Flower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' This is due to fitting class labels weakens the effect of these features and makes it harder to generalize to novel domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' When taking into account the performance of all datasets, pretraining with DINO is generally the much more desirable choice for better generalization over different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (3) The improvement of our propose method against the basic LT+NCC is not consistent among three different kinds of pretraining strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' For example, while our method can boost the performance of DINO pre-trained model by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='47% on Aircraft and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='83% on CUB, it can only bring much less advantage on models with supervised finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Effectiveness of APT and DRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We test the DINO pre-trained model with different kinds of testing strategies including (1) Proto: Directly generating predictions based on ProtoNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The prototypes are computed using averaged class token features from each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2) LT+NCC: The basic test-time finetuning method in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (3) Last: Finetuning the last transformer layer during testing, together with LT+NCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' which has similar parameter size to our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (4) First: Finetuning the first transformer layer during testing, together with LT+NCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' which has similar parameter size to our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (5) LN: We try to finetune the affinity parameter in each layer normalization as an alternative finetune strategy, which is utilized in many cross-domain FSL works (Tseng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Tsutsui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (6) APT: The model is finetuned using APT together with LT+NCC, using cross entropy loss and the proposed prototypical regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (7) Adapter: The model is finetuned using DRA together with LT+NCC, using cross entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (8) eTT: The model is finetuned using our proposed APT, DRA and LT+NCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3 show that while LT+NCC can fundamentally improve the model which indicates the importance of test-time finetuning, adding our proposed modules to the finetuning procedure can consistently bring higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Also, finetuning specific transformer layer can only bring limited improvement on few datasets: finetuning the last 10 Published in Transactions on Machine Learning Research (08/2022) Figure 3: Test accuracy of different training strategy if testing with (a) LT+NCC or (b) our eTT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Figure 4: Visualization of feature embeddings from a randomly sampled episode of TrafficSign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' layer leads to good performance on Aircraft, CUB and Texture, while updating the first layer leads to good performance on Quickdraw and MSCOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' However, this simple finetuning strategy cannot bring consistent improvement on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' This indicates that different data requires different levels of adaptation, and the improvement is much smaller than that of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Moreover, we give the tSNE visualization of feature embeddings of a randomly sampled episode from TrafficSign in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 4, which demonstrates that utilizing our proposed method can better regulate the feature embeddings into proper clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Is prototypical initialization necessary?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' One of the most important parts of our APT is the attentive prototypical initialization in which we use attentively aggregated patch embeddings to initialize the prefix matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' To verify this strategy, we compare several different choices of initialization, including (1) Random: random initialization from normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2) Avg: simply averaging all patch embeddings from each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (3) Sampling: randomly sampling one image for each category, and then initializing the prefix matrix with the averaged patch embeddings of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (4) Full: computing prototypes with our proposed attentive prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3 show that random initialization performs the worst, which can be resulted from insufficient task-specific information provided by the prefix in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Meanwhile, among all other strategies, using the attention map to aggregate patch embeddings as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 5 is better than simply averaging, leading to about 1% improvement on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Do we need a more complex adapter structure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' One would argue that our DRA structure is too simple to learn the complex knowledge from support images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3 we compare three different instantiations of adapters including (1) Linear: As in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2022), we use a linear layer for each adapter, whose output are than added to the original features in the MSA and FFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2) Bottleneck: We expand the linear layer 11 token token 90 pool 90 pool DINO DINO Test Accuracy 80 Test Accuracy 80 70 70 60 60 50 50 40 40 craft oraw Fung raw (a)TestAccuracy when using LT+NCC (b)TestAccuracywhenusing eTTPublished in Transactions on Machine Learning Research (08/2022) to a bottleneck structure where two linear layers are used for each adapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (3) FiLM: We compare DRA with a FiLM-like variant, in which we add a scaling vector for each adapter as in FiLM layer Perez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Note that such a method is similar to MTL (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', 2019) in FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The difference lies in that we still use the original way to directly tune the parameters on the novel support sets, instead of using another meta-trained module to generate the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (4) Offset: Only an offset vector is adopted for each adapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The results reveals that the linear adapter performs the worst, which means such a structure is improper for ViT in finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Moreover, we also find that using the bottleneck adapter will result in a dilemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' If we use small initial value for the adapter, the weights of each layer can only achieve gradient with extremely small values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' As the result, these weights, except the bias term of the last layer, can hardly be updated based on the support knowledge, which means such an architecture almost equals to our design where only an offset vector is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' On the other hand, if large initial values are adopted to avoid gradient diminishing, then the output features from the adapters can make the predictions severely deviate from those without adapters, thus leading to worse performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' As for the FiLM-like DRA, it is worse than offset DRA by about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='8% on average, while it doubles the parameter size based on offset DRA, leading to no significant additional efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Effectiveness of prototypical regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We also validate this regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3 we present the test accuracy when finetuning with and without this loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We can find that by applying this objective function, the model can have higher results on most datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Besides, as described in Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='6, we use a standardization technique when computing the prototypical regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' To verify its efficacy, we compare the model with and without such a standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' When not using standardization, the results are generally worse given comparable confidence intervals (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The results verify that this strategy can help the model with more stable finetuning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' dproj ILSVRC Omni Acraft CUB DTD QDraw Fungi Flower Sign COCO Avg 64 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='18 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='30 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='88 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='20 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='09 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='82 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='61 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='31 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='24 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='14 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='68 96 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='23 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='69 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='26 85.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='34 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='80 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='57 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='09 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='33 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='61 Table 4: Test accuracies on Meta-Dataset of different variants of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The bolded items are the best ones with highest accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='2 Comparison among Different Hyper-parameter Settings In additional to the ablation study about the proposed module, We further verify different choices of hyper- parameters in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Especially, dproj for the transformation module in APT and λ for the prototypical regularization are tested in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 4 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 5 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In general, the improvement is not consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' For dproj, we can find that using 192-d hidden dimension can get globally better results, which indicates that such a choice can make a good balance between the model capacity and scale so that the finetuning can be conducted both efficiently and effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' As for λ, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='1 seems to be a desirable choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Intuitively, smaller λ leads to less control of the prefix from the proposed prototypical regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Therefore, the prefix may lose the desired information during the optimization on the support set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' On the other hand, when λ is too large, the regularization overwhelms the label supervision, and thus the model can hardly adapt to the support knowledge, leading to worse performance especially on Omniglot and Aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 5 Conclusion We propose a novel finetuning method named efficient Transformer Tuning (eTT) for few-shot learning with ViT as our backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' By fixing the parameters in the backbone and utilizing attentive prefix tuning and domain residual adapter, our method can guide the ViT model with comprehensive task-specific information, which leads to better representations and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' This is demonstrated by the fact that we establish new state-of-the-arts on the large-scale benchmark Meta-Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 12 Published in Transactions on Machine Learning Research (08/2022) References Samira Abnar, Mostafa Dehghani, Behnam Neyshabur, and Hanie Sedghi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Exploring the limits of large scale pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In The Tenth International Conference on Learning Representations, ICLR 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Dosovitskiy Alexey, Beyer Lucas, Kolesnikov Alexander, Weissenborn Dirk, Zhai Xiaohua, Unterthiner Thomas, Dehghani Mostafa, Minderer Matthias, Heigold Georg, Gelly 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Few-shot learning via embedding adaptation with set-to-set functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 8808–8817, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Li Yuan, Yunpeng Chen, Tao Wang, Weihao Yu, Yujun Shi, Zi-Hang Jiang, Francis EH Tay, Jiashi Feng, and Shuicheng Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Tokens-to-token vit: Training vision transformers from scratch on imagenet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 558–567, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='1 Limitations and Future Work Despite the marginal effectiveness and efficiency of our proposed eTT, we mainly notice two points that should be explored in the future: (1) The plain ViT backbone utilized in this paper, may not be the best choice to the simple dataset, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=', Omniglot, while a well-designed ViT backbone may potentially better improve the efficacy of our method on such dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2) A flexible finetuning algorithm such as (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=') may have better generalization ability when facing episodes with various shots and ways, than the commonly-used methods that adopt fixed test-time finetuning iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' However, it is non-trivial to directly merge such methods with our proposed eTT due to different network structures and tuning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' It can be taken as the future work to properly utilize these flexible finetuning algorithm to further improve the performance of ViT in FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 15 Published in Transactions on Machine Learning Research (08/2022) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='2 Additional Experiment Results A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='1 Full Comparison with state-of-the-art methods We show the accuracies together with confidence interval in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Beyond the accuracies which is analyzed in the main context, the confidence interval of our eTT on both ViT-tiny and ViT-small is comparable with the other competitors, which reflects that our method is stable and robust enough among different testing episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='2 Influence of Training set As we have stated in the main context, our eTT is trained on the meta-train split of ImageNet to make fair comparison with other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' To show to what extent training on the full ImageNet training set instead of the meta-train set can impact the performance, we test our eTT using off-the-shelf DINO ViT-s model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We can find that (1) For those datasets on which DINO meta-train performs better than P>M>F, using full ImageNet to train DINO can bring further improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2) With the help of more data, our eTT overpasses P>M>F on ILSVRC and MSCOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (3) While more data does improve the results on Omniglot and TrafficSign, the final results are still worse than those of P>M>F, which we think may be correlated with the limitations of our method as analyzed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Given all these results, as a lighter model in that no meta-training phase is utilized and only few parameters are engaged in the test-time tuning, our method can still enjoy comparable performance with P>M>F when training on same amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='3 Influence of DINO The DINO pretrain procedure is an important part of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' To verify the effectiveness of DINO pretrain so that the comparison with other methods is fair enough, we present in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 8 the results of TSA using DINO-pretrained ResNet-34 and eTT using supervised pretrained ViT-s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We can find that (1) The effect of DINO is not consistent on two backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' While DINO benifits our eTT with about 5% accuracy on average, it severely weakens the performance of TSA with a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' It means that for FSL, different backbones require different pretrain strategy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' (2) While supervised pretrained ViT-s performs worse on most datasets, it is better on CUB and COCO, which indicates learning label information from ImageNet can help the model understanding novel knowledge from these two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='4 Verification of potential overfitting in finetuning As we have stated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 1, finetuning the whole backbone model with few support data will meet potential overfitting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' To reveal if such a problem exists in Meta-Dataset, we conduct an experiment as follow: during a normal testing phase, we select all episodes whose minimum shot (minimum number of support images for each class) is no larger than 2 (extremely small number of labelled instances), and compare the average accuracies of eTT and simple finetuning based on these episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 9 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 10 show the accuracies on support sets and query sets respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We can find that most of the datasets both methods receive nearly 100% accuracy, which means these two methods can well learn the support data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Given this fact, finetuning is much worse than eTT in terms of query accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The overfitting can be reflected given high training accuracies and worse testing performance, and to some extent our proposed eTT can fix this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='5 More Visualization We visualize the self-attention map from models with and without DRA on ILSVRC and TraffignSign in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Specifically, we randomly sample an episode from each dataset and use our eTT to tune the model based on the support samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Then we calculate the self-attention map of the last layer’s class token and highlight the areas with top 20% attention scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We can find that for the in-domain ILSVRC episode, the model can attend to similar regions no matter whether DRA is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' In contrast, the model without DRA can easily attend to background regions with less valuable information, which reveals a potential reason that these two models has similar accuracies on ILSVRC but large performance gap on TrafficSign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 16 Published in Transactions on Machine Learning Research (08/2022) Futhermore, we visualize more episodes from Aircraft, TrafficSign and MSCOCO in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 6, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 8, which shows that our porposed eTT can remarkably improve the embedding space after test-time finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='3 Broader Impact Our paper presents a more efficient and practical FSL pipeline utilizing ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' We hope this work can shed light on the broader usage of ViT in FSL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' On the other hand, the proposed method can provide researchers with alternative choice for FSL applications in real-case scenarios with large-scale meta-train set and challenging various test episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' dproj ILSVRC Omni Acraft CUB DTD QDraw Fungi Flower Sign COCO Avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='01 67.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='09 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='33 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='61 Table 5: Test accuracies on Meta-Dataset of different variants of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The bolded items are the best ones with highest accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Model Backbone ILSVRC Omni Acraft CUB DTD QDraw Fungi Flower Sign COCO Rank Finetune Res18 45.' metadata={'source': 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average rank on Meta-Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Note that different backbones are adopted by these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The bolded items are the best ones with highest accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Model Train Set ILSVRC Omni Acraft CUB DTD QDraw Fungi Flower Sign COCO Avg eTT meta-train 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='37 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='11 79.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='68 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='78 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='04 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='63 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='25 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='78 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='57 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='33 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='57 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='53 Table 7: Test accuracies on Meta-Dataset of different variants of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The bolded items are the best ones with highest accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The highlighted rows denote the final model in our main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 17 Published in Transactions on Machine Learning Research (08/2022) Model Pretrain ILSVRC Omni Acraft CUB DTD QDraw Fungi Flower Sign COCO Avg TSA Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='50 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='20 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='20 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='90 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='30 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='60 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='70 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='90 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='50 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='68 TSA DINO 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='18 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='94 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='74 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='49 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='06 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='51 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='13 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='01 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='70 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='18 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='09 eTT Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='17 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='47 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='30 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='71 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='50 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='46 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='51 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='55 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='08 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='68 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='14 eTT DINO 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='37 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='11 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='94 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='93 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='62 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='34 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='80 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='57 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='09 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='33 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='61 Table 8: Test accuracies on Meta-Dataset of different variants of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The bolded items are the best ones with highest accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The highlighted rows denote the final model in our main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Model ILSVRC Omni Acraft CUB DTD QDraw Fungi Flower Sign COCO Avg eTT 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='99 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='79 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='15 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='89 FT 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='87 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='95 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='00 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='20 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='20 Table 9: Support set accuracies of eTT and Finetune on testing episodes whose minimum shots is no larger than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Model ILSVRC Omni Acraft CUB DTD QDraw Fungi Flower Sign COCO Avg FT 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='19 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='54 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='10 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='54 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='66 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='37 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='53 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='76 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='90 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='21 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='68 eTT 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='22 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='79 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='33 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='11 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='20 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='14 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='33 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='03 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='29 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='19 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='96 Table 10: Query set accuracies of eTT and Finetune on testing episodes whose minimum shots is no larger than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The bolded items are the best ones with highest accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' Model ILSVRC Omni Acraft CUB DTD QDraw Fungi Flower Sign COCO w/o stand 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='96 w stand 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content='99 Table 11: The corresponding confidence intervals of models in ablation study on standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 18 Published in Transactions on Machine Learning Research (08/2022) w DRA w DRA w/o DRA w/o DRA Figure 5: Visualization of self-attention from model with and without DRA on ILSVRC (left) and TrafficSign (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' The white regions are those with top 20% attention scores 19 +0STOPPublished in Transactions on Machine Learning Research (08/2022) Figure 6: More visualization of feature embeddings from a randomly sampled episode of TrafficSign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 20 Published in Transactions on Machine Learning Research (08/2022) Figure 7: More visualization of feature embeddings from a randomly sampled episode of Aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 21 Published in Transactions on Machine Learning Research (08/2022) Figure 8: More visualization of feature embeddings from a randomly sampled episode of MSCOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE0T4oBgHgl3EQfggGb/content/2301.02419v1.pdf'} diff --git a/E9AzT4oBgHgl3EQfUPzF/content/tmp_files/2301.01264v1.pdf.txt b/E9AzT4oBgHgl3EQfUPzF/content/tmp_files/2301.01264v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1409f5efa62775e2e45226b5557b2324457e4bb9 --- /dev/null +++ b/E9AzT4oBgHgl3EQfUPzF/content/tmp_files/2301.01264v1.pdf.txt @@ -0,0 +1,2590 @@ +Tunable intracellular transport on converging microtubule morphologies +Niranjan Sarpangala,1 Brooke Randell,2 Ajay Gopinathan,1 and Oleg Kogan2 +1University of California, Merced, CA, 95343 +2California Polytechnic State University, San Luis Obispo, CA, 93407 +A common type of cytoskeletal morphology involves multiple converging microbutubules with their +minus ends collected and stabilized by a microtubule organizing center (MTOC) in the interior of the +cell. This arrangement enables the ballistic transport of cargo bound to microtubules, both dynein +mediated transport towards the MTOC and kinesin mediated transport away from it, interspersed +with diffusion for unbound cargo-motor complexes. Spatial and temporal positioning of the MTOC +allows for bidirectional transport towards and away from specific organelles and locations within the +cell and also the sequestering and subsequent dispersal of dynein transported cargo. The general +principles governing dynamics, efficiency and tunability of such transport in the MTOC vicinity is +not fully understood. To address this, we develop a one-dimensional model that includes advective +transport towards an attractor (such as the MTOC), and diffusive transport that allows particles +to reach absorbing boundaries (such as cellular membranes). We calculated the mean first passage +time (MFPT) for cargo to reach the boundaries as a measure of the effectiveness of sequestering +(large MFPT) and diffusive dispersal (low MFPT). We show that the MFPT experiences a dramatic +growth in magnitude, transitioning from a low to high MFPT regime (dispersal to sequestering) over +a window of cargo attachment/detachment rates that is close to in vivo values. Furthermore, we +find that increasing either the attachment or detachment rate, while fixing the other, can result in +optimal dispersal when the attractor is placed asymmetrically. Finally, we also describe a regime of +rare events where the MFPT scales exponentially with advective velocity towards the attractor and +the escape location becomes exponentially sensitive to the attractor positioning. Taken together, +our results suggest that structures such as the MTOC allow for the sensitive control of the spatial +and temporal features of transport and corresponding function under physiological conditions. +Introduction +The transport of material within eukaryotic cells is a +critically important physiological process that cannot be +achieved by passive diffusion alone. In these cells, cargo, +including vesicles and organelles, are dragged along by +a variety of molecular motors which utilize energy from +ATP hydrolysis to power their directed stepping motion +along cytoskeletal protein filaments with a well-defined +polarity [1]. Motors from different families such as ki- +nesins and myosins step along different filaments (micro- +tubules and actin respectively) and others such as dynein +move along the same microtubule filaments as kinesins +but in the opposite direction. Transport at the cellular +scale is therefore a complex process that involves phases +of multiple motors effecting directed transport along cy- +toskeletal filament networks interspersed with passive dif- +fusion of the cargo [2, 3]. This process is essential for +the transport of a variety of cargo between specific lo- +cations and organelles within the cell. Examples include +the transport of cargo in cilia [4], between the plasma +membrane and Golgi apparatus [5], [6], between Endo- +plasmic Reticulum and Golgi [7], [3], transport of viruses +towards replication sites [8], [9], and the transport of +many other vesicles and organelles for various functional +purposes (see review [3]), [10]. +Much like the design of road networks affect traf- +fic flow, the morphologies of the cytoskeletal networks +in cells have been shown to have a significant effect +on intracellular transport [11–14]. +This is particularly +important as, even a single type of cytoskeletal fila- +ment such as microtubules exhibit a wide diversity of +morphologies within different cell types to enable dif- +ferent functions[15]. +In some situations, such as in +melanophores microtubules have a strongly orderly (in +this case radial) - organization [16]. In others, the orien- +tation or polarity of microtubule (MT) morphology can +be broadly distributed. In pancreatic β cells, for exam- +ple, MTs are arranged with both an orientational and +polarity disorder [17], although there is an average po- +larity. On the other hand, MTs in neuronal dendrites +are essentially aligned with the long direction of the den- +drite, but their polarity is not uniform [18] resulting in +junctions of plus or minus ends along the dendrite. +A common structural feature that governs these micro- +tubule morphologies is the microtubule organizing center +(MTOC) that is responsible for growing MTs and local- +izing and stabilizing their minus ends leading to multi- +ple MTs converging with their minus ends at the MTOC +[15]. Dynein-driven transport along MTs will move cargo +to the vicinity of MTOC, while kinesin mediated trans- +port moves cargo away from it. These ballistic phases are +interspersed with isotropic diffusion for unbound cargo- +motor complexes. The spatial and temporal positioning +of the MTOC therefore allows for bidirectional transport +towards and away from specific organelles that can act +as MTOCs as well as locations within the cell in the +vicinity of the MTOC. Examples in which MTOC facil- +itates direct transport to the destination of interest in- +clude transport of cargo such as secretory vesicles away +from the Golgi apparatus toward the cell membrane and +endocytic vesicles towards the Golgi which is known to +perform as an MTOC in many mammalian cells [5], [6]. +arXiv:2301.01264v1 [q-bio.CB] 3 Jan 2023 + +2 ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ +(a) +(b) +FIG. 1: (a) A model of a cell in which microtubules have +a strong central organization, with minus ends at the cen- +trosome. A dark circle represents an organelle. Dynein mo- +tors are shown moving on microtubules. (b) One dimensional +morphology found in dentrites. Here the ends of the same po- +larity from different microtubules can face each other. This +schematic is based on [19]. +The dynein mediated transport of some viruses toward +the nuclear envelope is also enabled by the presence of a +MTOC in the vicinity of the nucleus [8, 9]. +In some cases, cargo need to traverse regions with +convergent MT morphologies. Such cases occur in den- +dritic processes of neuronal cells that have been shown +to have regions of alternating polarity of MTs [18]. Di- +rected transport of dynein (kinesin) carrying cargo at a +junction of minus(plus) ends will have to overcome what +is essentially a trap to maintain observed unidirectional +transport towards or away from the main cell body [18]. +Finally, the location of MTOCs can also be tuned over +time to accommodate different cellular functions such +as sequestering and dispersal of cargo. For example, in +melanophores [16, 20], a perinuclear MTOC produces a +radial MT structure with minus ends in towards the nu- +cleus and plus ends out toward the membrane. +Cells +achieve color change by aggregating and sequestering pig- +ment containing melanosomes near the nucleus via bal- +listic dynein mediated transport. Upon hormonal stimu- +lation they can switch to a superdiffusive dispersal phase +powered by a combination of kinesin and actin. Another +example occurs in lymphocytes that enable cytotoxicity +by secreting the contents of lysosomes (lytic granules) at +the immunological synapse to kill the target cell. Here, +dynein dependent sequestering of the lytic granules at +the MTOC occurs rapidly followed by the gradual move- +ment of the MTOC towards the synapse with subsequent +secretion [21, 22]. +In all these cases, it is important to understand +the dynamics of the transport and its sensitivity to +biological parameters in order to understand functional +efficiency and robustness. In particular, given the wide +variety of functional contexts in which the converging +MT geometry facilitates transport, +it is critical to +understand the general principles governing dynamics, +efficiency and tunability of such transport in the MTOC +vicinity. +To address this gap, +we develop a simple one- +dimensional model that includes advective transport to- +wards an attractor (such as the MTOC), and diffusive +transport that allows particles to reach absorbing bound- +aries (such as cellular membranes). This can be viewed +as a 2-layer model consisting of an advective layer en- +dowed with an attractor, a diffusive layer, and absorb- +ing boundaries along the perimeter of the domain. We +take the mean first passage time (MFPT) for cargo to +reach the boundaries as a measure of the effectiveness +of sequestering or directed transport (large MFPT) and +diffusive dispersal (low MFPT). The number of indepen- +dent control parameters in this problem can be reduced +to four. These are the rates of attachment to and de- +tachment from microtubules, advective velocity, and the +placement of the attractor within the domain. +Using this model we were able to make a series of tan- +talizing predictions - on which we report here. A cen- +tral calculation here is the residence time, or what is +commonly called in the literature the mean first passage +time (MFPT). Thus, given an initial location of the cargo +within the domain (determined by organelle placement), +this quantity tells the average time to reach either of the +absorbing boundaries (i.e. escape the domain), or a spe- +cific boundary (in one dimension, left or right). Another +relevant quantity is the probability of escape through one +or the other domain. +Symmetric, or nearly symmetric attractor positions +can give rise to a dramatic increase in the value of MFPT +within a certain window of dimensionless coupling rates +between the layers. Concurrently with this dramatic rise +of MFPT, the probability to escape purely diffusively +goes to zero in the same range of (dimensionless) cou- +pling values. This means that for larger coupling values, +any cargo particle will have to experience at least one +episode of motion on microtubules. Crucially, we found +that biophysical parameters in cells correspond precisely + +3 +to this range of dimensionless coupling rates. This sug- +gests that parameter values in cells are optimized for the +greatest sensitivity to small changes. With such parame- +ters, a cell can achieve the largest change in functionality +with smallest changes in parameter values. +Second, we predict the existence of optimal coupling +rates that minimize the MFPT. This minimal MFPT +happens when the attractor is positioned asymmetrically +(off center) in the domain. A similar phenomenon has +been predicted in the study of diffusion with stochastic +reset [23], [24]. Indeed, attachment to the microtubule, +followed by a rapid transport to the attractor, followed +by detachment from the microtubule back to diffusion in +the cytoplasm is effectively a reset. +When the coupling rates are much larger than all other +rates in the problem, the model reduces to effectively +one-layer. Here we demonstrate that even a slight asym- +metry in the position of the attractor can lead to a very +strong amplification of the preferred exit end. This pro- +vides another example of sensitivity to small parameter +changes - in this case asymmetric of the attractor place- +ment. This effect happens at sufficiently large advective +velocity, and corresponds to rare event physics. In the +regime of rare events, a small fraction of particles escape +quickly, while the majority advect to the attractor, and +form a quasi-stationary distribution around it. They stay +in the vicinity of the attractor for a time that scales expo- +nentially with advective velocity (or inverse of diffusion +coefficient). +Methods +Model +We consider the minimal model in a one-dimensional +domain of length L. It contains an advective layer (AL) +that represents motion along microtubules, and a dif- +fusive layer (DL) that represents diffusion in the cyto- +plasm. We assume that attachment to and detachment +from microtubules are Poisson processes, endowed with +rates α and β respectively. This means, for example, that +a motor spends on average a time 1/β since attaching to +a microtubule. While advecting, particles move with a +uniform velocity towards the attractor - which is an at- +tracting fixed point located at some coordinate x = X0 +between x = 0 and x = L. Letting ρ(x) and θ(x) be +probability densities of particles in the AL and DL re- +spectively, the model reads +∂ρ +∂t = − ∂ +∂x (v(x)ρ) + αθ − βρ +(1) +∂θ +∂t = −αθ + βρ + D ∂2θ +∂x2 +(2) +on 0 ≤ x ≤ L. The velocity field is given by +v(x) = +� ++v0 ... x < X0 +−v0 ... x > X0 +(3) +The parameters are rates α and β, the diffusion coeffi- +cient D, the advective velocity on microtubules v0, and +the location of the attractor X0. +There are absorbing +BCs at x = 0 and x = L, i.e. +ρ(0) = θ(0) = 0 and +ρ(L) = θ(L) = 0. All together, there are six physical +parameters. +We will switch to dimensionless variables by rescaling +the lengths by L and times by L2/D. Thus, x′ = x/L +and t′ = tD/L2. The resulting equations will be +∂ρ +∂t′ = − ∂ +∂x′ (v′(x)ρ) + aθ − bρ +(4) +∂θ +∂t′ = −aθ + bρ + ∂2θ +∂x′2 +(5) +on 0 < x′ < 1, with ρ(0) = θ(0) = 0 and ρ(1) = θ(1) = 0, +the velocity field +v′(x) = +� ++v ... x < X +−v ... x > X +where X = X0/L and v = v0L +D , and coupling rates a = +αL2 +D +and b = βL2 +D . From now on, we will drop primes. +The model is depicted schematically in Fig. 2. +𝐷 = 1 +𝑥 = 1 +FIG. 2: One-dimensional model with dimensionless parame- +ters. +Range of parameters +Here we review the values of parameters from litera- +ture. Both adsorption rate α and desorption rate β are +expected to be of the order of 1 per second. For example, +[11] cites α = 5 s−1 and β = 1 s−1. Microtubule lengths +typically fall in the range of 1−10 µm [11]. However, the +length of advective path may be much larger. For exam- +ple, in neurons, a cargo that needs to be delivered from +the soma to synapses on the ends of axons will travel a +length of the order of a meter [3]. The velocity of molec- +ular motors on MTs is on the order of 1 µm/s [3], [11], +although this quantity also has a degree of variability +[25]. Diffusion coefficient of vesicular organelles in the +cytoplasm fall in the range 10−3 − 10−1 µm2/s [3]. +Given these physical parameters, our dimensionless pa- +rameters a and b will take on values in the range [10, 105], +and parameter v will take on values in the range [10, 104]. + +v(x) +Advective layer +rate a +rate b +Diffusive layer +D +Junction point +x=0 +x=L +at x = X4 +There are four timescales in the problem: 1/a, 1/b, +the advective timescale 1/v, and the diffusive timescale +(which is of order 1 in dimensionless units). +Different +special cases or behavioral regimes emerge when one of +these timescales differs significantly from others. +The limit that is particularly amenable to analysis is +one in which 1/a and 1/b are both much smaller than +the advective time (which is of order 1/v in dimension- +less units) and diffusive time (which is of order 1 in di- +mensionless units). We will formally call it the a, b → ∞ +limit. In this regime, the model reduces to an advection- +diffusion process in one single layer, which amenable to +many analytical results. +Analytical approach in the one-layer limit +A very important special case is a = b. As a = b → ∞, +the model reduces to an effeective one-layer model: +∂P +∂t = − ∂ +∂x +� +v(x)P(x) − ∂P +∂x +� +where P(x, t) is the probability density (i.e. P describes +both θ and ρ, which become identical). A general solution +will be written as an eigenfunction expansion +P(x, t) = +� +n +cnpn(x)eσnt, +(6) +where pn(x) and σn is nth eigenfunction and eigenvalue, +which satisfy Opn = σnpn, with the operator O given by +O = − ∂ +∂x +� +v(x) − ∂ +∂x +� +, +(7) +with +v(x) = +� ++v ... x < X +−v ... x > X +(8) +and a constant v. Thus, the one-layer model contains +two parameters: dimensionless advective velocity v and +dimensionless position of the attractor X, which can take +on values between 0 and 1. +The computation of eigenvalues σn and eigenfunctions +pn(x) of the operator O, as well as the computation of +the eigenfunctions qn(x) of the operator O† is given in +Appendix B. +Starting from the initial condition P(x, t = 0) = δ(x − +x0), the probability density will be given by +P(x, t; x0) = +� +n +q∗ +n(x0)pn(x) +� 1 +0 q∗n(x′)pn(x′) dx′ eσnt +(9) +Everything that we need to compute MFPT can be ex- +tracted from this probability density. +To calculate the MFPT τ(x0), we notice that the mag- +nitude of the flux through the boundary is given by +f(t) = +�� ∂P +∂x +�� +bdry in dimensionless units. Then f(t)dt gives +the fraction of initial particles that cross the boundary +in [t, t + dt] = probability of crossing that boundary in +[t, t + dt], since the initial condition is normalized to 1. +So, p = +� ∞ +0 +f(t) dt gives the probability of ever leav- +ing through that boundary, +f(t)dt +p +gives the probability +that particles that leave through that boundary do so +in [t, t + dt], and finally τ = +� ∞ +0 +t f(t) +p dt is the average +time to leave through that boundary. In this problem, +there are two boundaries, with τl and τr denoting MFPT +to exit through the left and the right boundary respec- +tively. +We expect τl → 0 as x0 → 0 and τr → 0 as +x0 → 1. Finally, MFPT in general - without condition- +ing on a specific boundary - is the weighted average of +the two: τ = τlpl+τrpr, which matches predictions using +other methods [26]. +Analytical approach in the general case +Analogously to the one-layer approach, we again seek +a general solution to Eqs. (4)-(5) via an eigenfunction +expansion of the form +� +ρ(x, t) +θ(x, t) +� += +� +n +cn +� +Rn(x) +Θn(x) +� +e−σnt +(10) +(we found it convenient to factor out the negative sign +from σ here), where +� +Rn +Θn +� +and σn is the nth (vector) +eigenfunction and eigenvalue, which satisfy O +� +Rn +Θn +� += +σn +� +Rn +Θn +� +, with the operator O given by +O = +� +∂ +∂xv(x) + b +−a +−b +a − +∂2 +∂x2 +� +(11) +with v(x) given by Eq. (8). The full model contains four +parameters: dimensionless advective velocity v, dimen- +sionless rates a and b, and dimensionless position of the +attractor X, which can take on values between 0 and +1. The computation of eigenvalues and eigenfunctions is +given in Appendix A. +Remarkably, there are only a finite number of eigen- +functions and eigenvalues. In other words, the eigenset is +not complete. As a = b → ∞, this number goes to infin- +ity, while the lower-lying eigenvalues and eigenfunctions +approach those of the one-layer model. The completeness +is not guaranteed, since the operator O is not Hermitian. +Thus, an expansion such as in Eq. (10) is of limited use, +and cannot be used to fit a solution for an arbitrary initial +condition - including a point-like δ function initial con- +dition. This also implies that we cannot compute escape +currents and MFPT from such initial conditions. +However, we can always compute the ground state +eigenvalue, σ1. +Then the time 1/σ1, while not a true +MFPT, is an estimate of a characteristic time for escape. + +5 +We found that this time alone agrees with MFPT com- +puted in simulations quite well, so we will make MFPT +arguments based on this estimate. +Monte Carlo simulation method +We considered a simple one dimensional computational +model to simulate the transport process in a domain of +length L with attractor formed by oppositely oriented mi- +crotubules. Our computational model involves two lay- +ers, an advective layer (AL) where the particle undergoes +active transport and a diffusive layer where it does one +dimensional random walk. We consider one particle at +a time. To begin, we initialize the particle at position +x = x0 within the domain x ∈ [0, L = 1] either in the +diffusive or in advective layer as required. We consider +that the particle can switch from diffusive layer to ad- +vective layer with a rate a and from advective layer to +diffusive layer with a rate b. When a particle switches to +diffusive layer, a time td is drawn from the exponential +distribution e−at and the particle is allowed to diffuse for +n = td/∆t number of steps. ∆t is the time step in the +simulation. In each step the position is updated as +x(t + ∆t) = x(t) + r∆x, +(12) +where r is drawn from the set {−1, 0, 1} with the proba- +bility p = 1/3. ∆x is the step size which is chosen such +that the diffusion constant of the particle D = p∆x2 +∆t +is +1. Right after finishing a diffusive portion of a simula- +tion run, the particle switches from diffusive to advective +layer. In the advective layer, the particle stays for a time +ta drawn from e−bt, i.e. n = ta/∆t number of steps. In +the advective layer, the position of the particle is updated +as +x(t + ∆t) = x(t) + v(x)∆t, +(13) +where v(x) is the advective velocity given by Eq. (8). +These alternative portions of a simulation run in diffu- +sive and advective layers are continued until the particle +reaches one of the boundaries (x = 0 or x = 1) or until +maximum simulation time, Tmax is reached. We then re- +peat with N particles to get enough statistics to calculate +the overall MFPT, probabilities and MFPTs to exit out +of specific boundaries, and other quantities. +Trajectories +To get the trajectories, we record the data of the x po- +sition and the layer in which particle is located at regular +time intervals during each simulation run. An example +of trajectories is shown in Fig. 3. +Diffusion +1D Random Walk +Molecular Motor +Based Transport +FIG. 3: A sample trajectory generated by the Monte Carlo +simulation. +Diffusive motion is indicated with orange line, +and advective motion with a magenta line. Grey colored lines +indicate more sample trajectories. Here X = 0.5. +Computation of Net MFPT +To compute the net MFPT for a given parameter set, +we perform simulation runs until the particle exits out of +one of the boundaries (x = 0 or x = 1). We record the +time of exit for each run and then compute the mean and +standard error of the mean for all N runs. +Computation of Conditional MFPT and escape probability +To compute the MFPT for exit specifically through +the left (or the right) boundary, we record the time as +well as the boundary through which the particle exits. +Then we filter out only those simulation runs where a +particle exited out of the left (or right) boundary. Then +we compute mean and standard error of the mean for +those runs. We compute the escape probability through +left (or right) boundary as the fraction of runs that exited +out of the left (or right) boundary. +Statistics of visits to the AL +We measure the fraction of simulation runs in which +a particle that started on the DL ended up making at +least one visit to the AL. In each simulation run, we +also compute the number of visits to the advection layer +before exiting. To do this, we update a counter every +time the particle switches its layer to get the number of +times it switches layers prior to exiting the domain. We +then compute the average over N runs. + +6 +Results and Discussion +Variation of coupling rates can change escape times +by orders of magnitude +We begin our presentation of results with the symmet- +ric case, X = 1/2. For simplicity we will set the particles’ +initial placement at x0 = 1/2 - this is the initial condi- +tion (IC) in analytical calculations - and let a = b for +now. Figure 4 (a) displays the mean first passage time +(MFPT) as a function of a = b at different advective +speeds v. To help understand the physics of the process, +we also plot the fraction of times that particles visit the +advective layer in panel (b) (for particles initially placed +on the DL), as well as the number of times they do so in +panel (c) (also when starting on DL). +Two crossovers are evident from the plot of MFPT vs +a (= b). The first crossover takes place around a = 10−2. +As suggested by the plot of the fraction of visits to the +AL, at this coupling rate the probability of visiting the +AL becomes non-zero; below this crossover, the advective +layer is not visited and the MFPT is a purely diffusive +time ≈ 0.12. For a above this crossover value, the prob- +ability of visiting the AL grows with increasing a. While +the fraction of particles visiting the AL grows ∝ a, the +time to remain in the AL (the longest time scale in this +range of a) decreases ∝ 1/a, resulting in the plateau of +MFPT vs. a. +Because the probability (or fraction) to +visits to the AL is less than 1 (for particles startin in the +DL), a particle has a chance to escape purely diffusively +for as in this plateau region. +The MFPT is in dimensionless time units; to convert to +time in seconds, multiply by L2/D expressed in physical +units. For example, for L = 1 µm and D = 10−2 µm2/s, +the MFPT of 10 dimensionless time units corresponds +to 103 seconds. The MFPT for diffusive transport on a +domain with two absorbing boundaries and a midpoint +initial condition is 0.125 (in dimensionless time units), +which is half of the first plateau value, and much lower +than plateaus after the second crossover for v > 1. +We continue our discussion of Fig. 4. The probabil- +ity of visiting the AL (for particles starting in the DL) +eventually reaches 1 at larger a; particles are now certain +to visit the AL at least once. In other words, the prob- +ability of a purely diffusive escape reaches zero and we +encounter the second crossover. For v = 20, for example, +this second crossover happens around a = b = 10, but its +location - defined by the point of inflection - varies some- +what with v. This crossover is broad - it can be several +decades wide - and marked by a drastic growth of the +MFPT, especially at larger v. In this second crossover +regime, each particle experiences intermittent advection, +punctuated by periods of diffusion. In other words, on a +typical run from an initial location to one of the bound- +aries, a particle’s trajectory will include multiple episodes +of advection and diffusion following each other. Eventu- +ally, we reach the second plateau, when the switching +between the layers is so rapid that we now reach an ef- +FIG. 4: Symmetric case: X = 0.5, the initial location of +particles is also at x0 = 0.5. (a) MFPT vs. a = b. Dots: IC on +the DL; crosses: IC on the AL. The solid curves are analytical +estimations of MFPT given by 1/σ1, where σ1 is the ground +state eigenvalue. The MFPT is in dimensionless time units; +to convert to time in seconds, multiply by L2/D expressed in +physical units. The dashed horizontal line has a value 0.25. +The last two points (a = 105 and 106) required a smaller +dt = 10−6 +3 +; dt = 10−4 +3 +was sufficient for the rest. Therefore, +we used N = 103 for the last two points to optimize simulation +time, and N = 104 for the rest. (b) Fraction of simulation +runs that visit the AL at least once after starting in the DL. +The dashed line is a fit, of the form 0.079a. Here N = 104 +and dt = +10−4 +3 +. +(c) Average number of visits for particles +starting in the DL. Here N = 103, dt = +10−4 +3 +(circles), and +10−6 +3 +(diamonds). The x-axis is the same in all three plots; the +plots are aligned. The shading guides the eye to the second +crossover region. + +7 +fectively one-layer regime. This regime will be studied +in the next section, where we examie a one layer model +with advection and diffusion taking place simultaneously. +MFPTs predicted by that model match the high a = b +plateaus. Interestingly, there is a strong velocity depen- +dence in the one-layer regime, but not in the range of +a = b in the plateau below the second crossover. +For a = b < 1/(simulation time), particles with IC in +the advective layer (plus symbols in Fig. 4) will never +enter the DL and therefore will not escape. MFPT will +simply be limited by the simulation time - this is mani- +fested in the saturation at MFPT = 500, since this was +the simulation time. +Appendix C displays examples of particle trajectories +for a broad range of a = b that cover all of the behavioral +regimes shown in Fig. 4. These figures demonstrate the +change in the character of trajectories - from the types +that contain advective periods long enough to arrive to +the attractor at low a = b, to intermittent behavior in the +second crossover region, to very rapid switching between +layers for a = b beyond the second crossover - when the +model is effectively in the one-layer regime. +The region of the most sensitive behavioral tuning matches +the biological parameters +We now turn our attention to the biological significance +of these results. +Note that the second crossover takes +place between a = 10 and a = 104. Remarkably, this +is precisely the range of these parameters found in cells +- see “Range of parameters” above. This might imply +that these parameters evolved to have such values for +an easy tunability. Indeed, the second crossover region +is precisely where a change in the rates gives rise to the +largest change in the outcome - especially at larger values +of v. +There is an optimal coupling rate between advective +and diffusive behavior +Placing the attractor asymmetrically can give rise to +a decrease in MFPT with increasing coupling rates - see +Fig. 5. This effect is only seen at larger v. The decrease +happens over a range of 1/a that is comparable to the +advective time, ∼ 1/v. +For example, for v = 20, the +time scale to travel advectively to the attractor is ∼ 0.05, +while the decrease is seen for a between 1 and 100, which +corresponds to the time scale between 1 and 0.01. +We think that this decrease in the MFPT happens be- +cause an increase in the interlayer coupling causes more +material to congregate at the attractor, which is close to +one of the ends - thus leading to an overall decrease in +the MFPT. +Fig. 6 shows an example of this phenomenon due to +only the parameter a varied at fixed b. We mentioned in +the discussion of the analytical approach in the general +5 +1 +FIG. 5: Asymmetric case: X = 0.85. In this particular case, +x0 = 0.7, but such dips are also present at other x0. +case that a complete eigenset in the two-layer model does +not exist, so the exact solution cannot be obtained as a +sum of the modes. However, the MFPT can be estimated +as τ = 1/σ1, where σ1 is the ground state eigenvalue. +𝑎 +𝑣 = 13 +𝑏 = 169 +𝑿 = 𝟏/𝟐𝟔 +𝑣 = 13 +𝑏 = 169 +𝑿 = 𝟏/𝟐 +Net MFPT +𝑎 +Net MFPT +(a) +(b) +Out[ ]= +0.01 +100 +0.1 +10 +1000 +105 +Out[ ]= +0.01 +100 +0.1 +1 +10 +100 +1000 +FIG. 6: τ(a) at fixed b = 169. (a) X = 0.5, (b) X = 1/26. +v = 13 for both. Lines: theory, dots: simulation. Red dots - +IC on the diffusive layer, blue dots - IC on the advective layer. +The numbers for the two types of initial conditions are not +identical, but the difference is almost invisible. The analytical +prediction is 1/σ1 - the inverse of the ground state eigenvalue, +which is not a true MFPT. The IC in the simulation was at +x0 = 0.5. The simulation time was 1000, which is the reason +for flattening of the simulation data at large a in panel (a). + +8 +𝑏 +Net MFPT +𝑏 +Net MFPT +(a) +(b) +Out[ ]= +0.1 +1 +10 +100 +1000 +104 +105 +0.1 +1 +10 +100 +1000 +Out[ ]= +0.01 +100 +0.1 +1 +10 +100 +1000 +FIG. 7: τ(b) at fixed a = 169. (a) X = 0.5, (b) X = 1/26. +v = 13 for both. Lines: theory, dots: simulation. Red dots +- IC on the diffusive layer, blue dots - IC on the advective +layer. The analytical prediction is 1/σ1 - the inverse of the +ground state eigenvalue, which is not a true MFPT. The IC +in the simulation was at x0 = 0.5. We again see saturation of +simulation results at low b at the simulation time (here, 1000 +time units). +The solid lines in Fig. 6 are values of 1/σ1. +This es- +timation should become more accurate as escape events +become rare (MFPT ≫ than all other time scales); this is +because higher eigenmodes contribute little to the prob- +ability current in the rare event limit. Moreover, while +this calculation does not give IC dependence, MFPT loses +this dependence as escape events become rare. Some dis- +cussion of rare events can be found in the next section, +and a much more in-depth discussion will appear in [27]. +The dips in Fig. 6 happen, again, because increasing a +causes particles to return back to the attractor, thus min- +imizing the chance for them to wander too far to the right +while diffusively exploring the long part of the domain. +On the other hand, increasing a even further tends to +keep the particles in the AL and therefore prevents them +from escaping (particles cannot move in the direction of +the ends when they are in the AL due to the advective +flow being directed towards the attractor). +These dips are somewhat counter-intuitive - an overall +escape time is lowered by increasing the tendency to go +towards the attractor inside the domain - as long as the +attractor is placed asymmetrically. +A similar phenomenon has been reported in connection +to the problem of mean first passage time with a reset +[23], [24], [28]. Here, in addition to diffusion, a particle +experiences a reset back to some location, and resets form +a Poisson process, endowed with a reset rate r. The au- +thors of these sources found there exists an optimal rate, +r∗ which minimizes the MFPT out of the semi-infinite +domain. +We note, however that these sources appear +to return the particle back to the reset location once it +has hit the absorbing end of the semi-infinite domain, +thereby conserving the probability. This is different from +our problem, in which the total probability inside the +domain decreases with time, because once particles have +reached one of the two absorbing ends, they are not re- +turned back into the domain. +This difference aside, the problem that we are ana- +lyzing can be viewed as a version of a reset problem, +although the time to reset is not instantaneous. More- +over, the reset location is not necessarily the location +of the attractor x = X, since a particle has a chance +to return to the diffusive layer before reaching the at- +tractor. The limit of infinite v would correspond to the +instantaneous reset to the attractor, and the limit b → 0 +would cause the resetting to take particles back to x = X, +i.e. approximating the standard reset problem (although, +again, without returning of particles that have reached +either of the domain ends). +The dip phenomenon is also observed when b is varied +at fixed a, see Fig. 7. At low b, MFPT is dominated by +the waiting time 1/b to return from the attractor to the +DL. A large b asymptote (for b ≫ a) is the regime of +purely diffusive motion - the particles are forced into the +DL. Evidently, having some acccess to the AL leads to +a lowering of MFPT because it allows more material to +congregate close to one end. +It is interesting to ask what effect increasing the ad- +vective velocity would have. The intuition - supported +by the physics of the one-layer model - is that higher v +should lead to either an increase of the MFPT or the +disappearance of the dip, because with sufficiently large +velocity, the density will be more and more localized near +the attractor; so, even though the attractor is closer to +one end than the other, it is no longer close to this end +𝑎 +1/𝜎! +Out[ ]= +0.01 +0.10 +1 +10 +100 +1000 +104 +0.02 +0.05 +0.10 +0.20 +v=20, 40, 60, 80 top to bottom +X=0.5 +FIG. 8: Top to bottom: v = 20, 40, 60, 80. Here X = 0.5, +and b = 169. + +9 +Out[ ]= +0.01 +0.10 +1 +10 +100 +1000 +104 +0.10 +1 +10 +100 +X=1/26, 2/26, 3/26, from bottom to top +V=20 +1/𝜎! +𝑎 +FIG. 9: Top to bottom: X = 3/26, 2/26, 1/26. Here v = 20, +and b = 169. +in comparison to the width of the density distribution. +However, analytical calculations in fact predict the de- +crease in the value of 1/σ1 at a fixed a with increasing v, +see Fig. 8. +An in-depth study of density distributions, which will +be published elsewhere [27], sheds light on the reason for +this counter-intuitive prediction. While the density pro- +file in both layers does become more localized with larger +velocity (as expected), the part of the profile between the +attractor and the close end is not affected; the decrease in +the spread is due to the other side of the profile. There- +fore, as velocity is increased, more and more material is +localized near the close end, while the chance of escap- +ing through this end does not diminish - resulting in the +overall decrease of escape time. +We also study the effect of varying X in Fig. 9. Here +the results conform to the intuitive expectation that a de- +crease in asymmetry will lead to a decrease in the mag- +nitude of the dip (with no dip at all in a completely +symmetric geometry). An attractor placed much closer +to the left end than the right one, for example, has two +effects. First, it lowers the MFPT overall, since there is +less distance to travel during the escape. Second, pre- +venting particles from wandering too far to the right (by +increasing a, and thus the reset rate) causes the particles +to congregate closer to the left end in the more asymmet- +ric situation, leading to a lower MFPT. +One-layer limit +Dynamics of probability density +The analytical approach in the one-layer limit is out- +lined in the Methods section, with details in Appendix +B. These predictions are verified by simulations (see Ap- +pendix D). Here we present results of analytical calcula- +tions. +In Fig. 10 we show several snapshots in the evolution of +the probability density profiles for a specific placement of +!! +"(!) +𝑣 = 1 +!! +"(!) +𝑣 = 20 +FIG. 10: X = 0.85, x0 = 0.35. The distributions are shown +for t = 1.3×10−5, t = 1.3×10−4, t = 1.3×10−3, t = 1.3×10−2, +t = 1.3 × 10−1. Top: v = 20, bottom: v = 1. For v = 1, the +distributions never reach an asymptotic form that is centered +on x0 = X. +the attractor and specific initial condition, for two values +of the advective velocity. Following a δ-function initial +condition, there is a quick diffusive spread. While this +spread is happening, the center of the distribution is also +advected towards the attractor. Note that in the v = 1 +case, the average position of particles reaches 1/2. On +the other hand, for the case of stronger at v = 20, the +center of the distribution reaches the attractor at x = X. +At v = 20 we begin to see the emergence of large- +time asymptotic profile centered on the attractor. +At +large times, the distribution reaches a stationary limit- +ing form. As this profile develops, diffusive spread of the +density profile is followed by a contraction, as particles +congregate around the attractor and σx decreases. At +t ≈ 0.06 the width stops evolving, and the cusp-shaped +profile is established in the vicinity of the attractor. After +that, the probability to remain in the domain continues +to decrease (the area under the curve will continue to +decrease), although the shape of the profile remains sta- +tionary. We will call this limiting profile the large-time +distribution or the limiting distribution. The width of +this cusp-shaped limiting distribution decreases with in- + +80 +60 +40 +20 +0.2 +0.4 +0.6 +0.8 +1.080 +60 +40 +20 +0.2 +0.4 +0.6 +0.8 +1.010 +creasing v. At lower v, the width also saturates to a con- +stant value at large times, and the limiting distribution +also emerges, but it is not centered on the attractor. +Thus, the picture is this: the attractor captures some +particles and pulls them in to its vicinity at larger v, +whereas at lower v, most of the particles escape be- +fore this happens. The decay rate also decreases - as v +grows ever larger, the large-time limiting profile localized +around the attractor will decay ever slower, its rate of de- +cay decreasing exponentially with v (this is for sufficiently +large v, i.e. it is an asymptotic scaling). In this large v +regime, the profile that develops after an initial rapid re- +laxation may be called quasistationary - as it decays on a +time scale much smaller than all other time scales in the +problem. This is the regime of rare events, and we now +discuss the scaling of MFPT and escape probabilities in +this limiting regime. +0.2 +0.4 +0.6 +0.8 +1.0 +x0 +0.2 +0.4 +0.6 +0.8 +pr +0.2 +0.4 +0.6 +0.8 +1.0 +x0 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +� +0.2 +0.4 +0.6 +0.8 +1.0 +x0 +0.2 +0.4 +0.6 +0.8 +pr +0.2 +0.4 +0.6 +0.8 +1.0 +x0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +� +0.2 +0.4 +0.6 +0.8 +1.0 +x0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +�r +0.2 +0.4 +0.6 +0.8 +1.0 +x0 +0.2 +0.4 +0.6 +0.8 +pr +0.2 +0.4 +0.6 +0.8 +1.0 +x0 +10 +20 +30 +40 +50 +�r +0.2 +0.4 +0.6 +0.8 +1.0 +x0 +10 +20 +30 +40 +50 +� +0.2 +0.4 +0.6 +0.8 +1.0 +x0 +0.2 +0.4 +0.6 +0.8 +pr +0.2 +0.4 +0.6 +0.8 +1.0 +x0 +0.1 +0.2 +0.3 +0.4 +�r +(𝑎) +0.2 +0.4 +0.6 +0.8 +1.0 +x0 +50000 +100000 +150000 +200000 +250000 +�r +0.2 +0.4 +0.6 +0.8 +1.0 +x0 +50000 +100000 +150000 +200000 +250000 +� +𝑝 +𝜏 +𝜏 +(𝑏) +(𝑐) +(𝑑) +Right +Left +𝑝 +𝜏 +𝜏 +𝑝 +𝜏 +𝜏 +𝑝 +𝜏 +𝜏 +𝑥! +𝑥! +𝑥! +𝑥! +𝑥! +𝑥! +𝑥! +𝑥! +𝑥! +𝑥! +𝑥! +𝑥! +FIG. 11: Escape probability and MFPT through both ends +versus the location x0 of the IC. The attractor is located at +X = 0.51. (a) v = 5, (b) v = 10, (c) v = 20, (d) v = 40. The +aberrations at the edge are numerical artifacts. +Scaling of MFPT in the rare event limit +In this regime, various functions of x0 - such as the +escape probability and escape time - develop character- +istic distinctions between a boundary layer and interior + +C11 +regions. This is shown in Fig. 11. As v increases, the +MFPT to exit increases, and eventually this time be- +comes much larger than all the other characteristic time +scales of the problem. In this large v regime, escape be- +comes a rare event. Starting from an initial condition x0, +a particle will, with overwhelming probability drift to- +wards the fixed point, and fluctuate around it for a time +that scales exponentially with v as stated above. There- +fore, the initial condition will be forgotten. This effect +is manifested in Fig. 11 by distinct plateaus, that show +the absence of dependence on x0. We show the compar- +ison between such analytical predictions and simulation +results of the one-layer regime in Appendix D. +Escape rates in these plateaus will follow the usual +Arrhenius scaling 1/τ ∼ e−∆Ueff /D in physical units. +The effective barrier to escape to the left will be vX = +∆Ul and the effective barrier to escape to the right will +be v(1 − X) = ∆Ur. A small difference between X and +(1−X) will be exponentially amplified by large v. Thus, +for 0.5 < X < 1, the dominant factor will be v(1 − X), +and therefore, τ ∼ ev(1−X)/D, or in dimensionless units, +simply +τ ∼ ev(1−X). +(14) +A more detailed analysis [27] predicts the prefactor as +well, so the asymptotic expression (i.e. in the rare event +regime) is given by τ = 4v−2ev(1−X). +One comment regarding MFPT results is in order. We +notice that the overall MFPT τ in Fig. 11 is ≈ 2 times +smaller than the a = b → ∞ limit in Fig. 4 (see v = 10 +and v = 20 graphs). While a small difference is due to +slightly different X (0.51 in Fig. 11 vs. 0.5 in Fig. 4), +the main reason for this difference is that in the two- +layer problem, the advection and diffusion take turns, +while they take place simulataneously in the two-layer +model. Thus, all timescales are slowed down by exactly +a factor of two in the two-layer model than its truly one- +layer equivalent. +In other words, to make the proper +comparison, we must multiply the one layer result by 2 +to match the a = b → ∞ limit of the two-layer model. +Small asymmetry leads to a large bias in the exit location +One prominent feature of Fig. 11 is the amplification in +the asymmetry in results (for example pl and pr - proba- +bilities to escape through the left and right ends respec- +tively) due to a small asymmetry in the placement of +the attractor. Note that pr = ae−∆Ur and pl = ae−∆Ul, +where a is some constant. +We can find this constant +from the fact that pr + pl = 1 (a particle definitely +exists through one of the two ends eventually). Thus, +a = +� +e−∆Ur + e−∆Ul�−1, altogether giving +pr − pl = tanh [(X − 1/2)v] +(15) +We overlay this prediction on top of ∆p obtained from +the analytic results (depicted in Fig. 11) in Fig. 12 (a). +! +∆" +! +∆" +(a) +(b) +FIG. 12: (a) ∆p vs. v. Top (blue): X = 0.55, bottom (or- +ange): X = 0.51. Dots - full theory, solid curves - Eqn. (15). +(b) ∆p vs. X, given by Eqn. (15). Top (orange): v = 20, +bottom (blue): v = 10. +Conclusion +In this paper, we looked at a one-dimensional model of +intracellular transport via a combination of advection on +microtubules and diffusion in the cytoplasm. This one- +dimensional model was motivated by a scenario involv- +ing an attractor in the interior of the cell - for example, +MTOC. There are other situations where attractors may +arise. Consider, the β cell example from the Introduc- +tion. Here motors transport insulin granules along MTs. +Due to orientational disorder [29], several MTs can meet +with ends of the same polarity facing each other, forming +an aster-like morphological trap (or attractor) for mo- +tors that would all congregate at this junction [11]. It +is meaningful to talk about the domain of attraction of +such a trap in the following sense. A molecular motor +that attaches to a MT anywhere within this domain will +be taken towards the attractor, while a motor that at- +taches to a mirotubule outside of the domain has a non- +zero probability to be taken away from the trap. When +placed inside such a domain - where advective motion +along microtubules tends to only attract particles - they +can nevertheless escape the domain of attraction of the +attractor by desorbing from MTs and diffusing within +the cytoplasm until they end up outside of the domain. + +1.0 +0.8 +0.6 +0.4 +0.2 +10 +20 +30 +40 +50 +601.0 +0.8 +0.6 +0.4 +0.2 +0.6 +0.7 +0.8 +0.9 +1.012 +Naturally, a question about the time to be stuck in the +vicinity of the attractor arises - along with the question +of how formation of such traps affects the functioning +of the cell and the overall transport of insulin granules +across it. +Using our one-dimensional model, We calculated es- +cape probability through each end, pl(x0) and pr(x0), +and overall p(x0). We also calculated the mean first pas- +sage time (MFPT) to escape the domain through each +end, τl(x0) and τr(x0), and overall τ(x0). The initial lo- +cation inside the cell is determined by the organelles pro- +ducing the cargo. The other parameters in the problem +were the dimensionless location of the attractor toward +which the advective motion is directed, and the dimen- +sionless advective velocity v. +In situations like these, when there is either orienta- +tional or polarity disorder, we can think of cells as being +divided into domains. +We made several predictions. When the attractor is +placed symmetrically and a and b are finite, there is a +crossover between τ ∼ 0.1 - diffusive timescale to τ that +grows exponentially in v. The range of a = b over which +this crossover happens is wide - a couple of orders of +magnitude, but it corresponds to the values of a and b +actually found in cells. This served as our first example +of “fine-tuning” that allows cells to achieve the biggest +change in the functionality with the smallest change in +parameter. +For a = b significantly below the crossover, a particle +that was released into the diffusive layer has a chance to +escape the domain purely diffusively without ever visiting +the AL. For a = b around the crossover value, the proba- +bility of this goes to zero - every particle will be advected +towards the attractor for at least some of the time. For +a = b significantly above the crossover, the transport en- +ters the effective one-layer regime and exhibits rare event +physics. +Asymmetric placement of the attractor gives rise to an +interesting phenomenon of an optimal coupling. Thus, +we found that it is possible to minimize the residence +time in the domain by increasing the coupling, because +that will lower the diffusive spread, and bring particles +close to one end of the domain. +We discussed the effective one-layer regime that re- +sults at sufficiently large couplings. We also discussed +rare event physics that happens at large dimensionless +advective velocities. In such a rare event regime, a por- +tion of particles will be localized in the vicinity of the +attractor for a time exponentially long in v. +We pro- +vide an explicit formula formula for the overall MFPT +- including not only the exponential part, but also the +prefactor, which scales as v−2. +The idea of exponential sensitivity, and phenomena +such as strong amplification of the preferred exit end +due to a slight asymmetry is tantalizing. Extrapolating +this finding to two dimensions suggests that in complex, +crowded environments that allow for multiple trap-like +morphologies (for example, asters), the distribution of +cargo around the cell will be non-homogeneous. This re- +mains to be verified in the future, by extending our model +two two dimensions. +Our work is complementary to prior theoretical models +of transport that involves a combination of diffusion and +advection along microtubules [30] and [31], as neither of +these sources are focusing on questions of residence time +or the role of asymmetry. +To continue our current work, we would like to study +models with reflecting-reflecting or absorbing-reflecting +boundary conditions, or models in which the source is +on one end and the target is on the other. Such mod- +els would be better suited for transport of cargo in cilia +[4], transport between the plasma membrane and Golgi +apparatus [5], [6], or between Endoplasmic Reticulum +and Golgi [7], [3], transport of viruses towards replication +sites [8], [9], and other intracellular transport situations +[3], [10]. +This +work +was +supported +by +the +National +Sci- +ence Foundation (NSF-DMS-1616926 to AG) and NSF- +CREST: Center for Cellular and Bio-molecular Ma- +chines at UC Merced (NSF-HRD-1547848 and 2112675 +to AG). AG and NS also acknowledge partial sup- +port from the NSF Center for Engineering Mechanobi- +ology grant CMMI-154857 and computing time on the +Multi-Environment Computer for Exploration and Dis- +covery (MERCED) cluster at UC Merced (NSF-ACI- +1429783). +NS acknowledges Graduate Student Oppor- +tunity Program Fellowship from the University of Cal- +ifornia, Merced. +BR acknowledges the support of the +William and Linda Cal Poly Frost fund for undergradu- +ate research. + +13 +A: Details of the two-layer calculations +We start with the full one-dimensional, two-layer model in dimensionless form (primes have been omitted for clarity): +∂ρ +∂t = − ∂ +∂x (v(x)ρ) + aθ − bρ +(16) +∂θ +∂t = −aθ + bρ + ∂2θ +∂x2 +(17) +Here a and b are respectively the rates of adsorption to and desorption from microtubules, v is the dimensionless +velocity profile, ρ is the density of particles on microtubules, and θ is the density of particles diffusing in the cytoplasm. +We seek modal solutions (or eigensolutions) of the form +� +ρ +θ +� += +� +R(x) +Θ(x) +� +e−σt. +(18) +The vector +� +R(x) +Θ(x) +� +is an eigenvector of the operator (see Eq. (11) of the text) that represents minus the right hand +side of Eqs. (16)-(17), and σ is an eigenvalue of this operator. +However, due to the mass accumulation at the attractor, we must also include a δ-function term to accommodate +for this mathematically. The mass will not accumulate at the junction point due to the diffusive term that acts on the +diffusive layer density. Note also that the δ- function in the advective layer acts like a point source for the diffusive +layer. When we study a simple diffusive problem with a δ-function source plus absorbing boundaries, and seek a +steady-state (time independent) solution, the density profile does not acquire a δ-function response - the diffusion +acts infinitely quickly to dissipate such a singularity. With this in mind, we must augment the above formula to +� +ρ +θ +� += +� +R(x) +Θ(x) +� +e−σt + +� +1 +0 +� � +M0e−σt� +δ(x − X). +(19) +Substituting this back to Eqs. (16)-(17), and setting Q = dΘ +dx , we get +d +dx +� +� +R +Q +Θ +� +� = +� +� +(− b +v + σ +v ) 0 +a +v +−b +0 (a − σ) +0 +1 +0 +� +� +� +� +R +Q +Θ +� +� +(20) +for 0 ≤ x < X (call it Region-I) and +d +dx +� +� +R +Q +Θ +� +� = +� +� +( b +v − σ +v ) 0 +− a +v +−b +0 (a − σ) +0 +1 +0 +� +� +� +� +R +Q +Θ +� +� , +(21) +for X < x ≤ 1 (call it Region-II). The solutions, will take the form: +� +� +RI +QI +ΘI +� +� = A +� +� +v1 +R +v1 +Q +v1 +Θ +� +� eλ1x + B +� +� +v2 +R +v1 +Q +v2 +Θ +� +� eλ2x + C +� +� +v3 +R +v3 +Q +v3 +Θ +� +� eλ3x +(22) +in Region-I and +� +� +RII +QII +ΘII +� +� = D +� +� +w1 +R +w1 +Q +w1 +Θ +� +� eµ1x + E +� +� +w2 +R +w1 +Q +w2 +Θ +� +� eµ2x + F +� +� +w3 +R +w3 +Q +w3 +Θ +� +� eµ3x, +(23) +in Region-II. The ⃗vs and λs are eigenvectors and eigenvalues of the matrix in Eq. (20), while ⃗ws and µs are eigenvectors +and eigenvalues of the matrix in Eq. (21). The λs satisfy the equation +−λ3 + +�σ − b +v +� +λ2 + (a − σ)λ + σ2 − σ(a + b) +v += 0, +(24) + +14 +and the µs satisfy the equation +−µ3 − +�σ − b +v +� +µ2 + (a − σ)µ − σ2 − σ(a + b) +v += 0. +(25) +The eigenvectors have the structure +⃗v = +� +� +−λ2+a−σ +b +λ +1 +� +� , +(26) +and +⃗w = +� +� +−µ2+a−σ +b +µ +1 +� +� . +(27) +The functions on both sides of the attractor are different, and they need to be stitched correctly. The stitching +is determined by the boundary conditions, so we now discuss these. The boundary conditions will determine the +eigenvalues σn. We note that there are seven unknowns: coefficients A - F (see Eqs. (22)-(23)), and the mass growth +rate M0 (see Eq. (19), so we need seven constraints (or conditions). +First, there are absorbing boundary conditions at each end, which require that R(x = 0) = Θ(x = 0) = 0 and +R(x = 1) = Θ(x = 1) = 0. The additional three conditions come from the location of stitching, i.e. the attractor +location at x = X. The diffusive layer density must be continuous to avoid infinite currents. Thus, ΘI(X) = ΘII(X). +The remaining two boundary conditions come from mass conservation. To extract these, we integrate Eqs. (16)-(17) +through the junction point, i.e. from x − ϵ to x + ϵ for arbitrarily small ϵ. Performing this on Eq. (16) gives +−σM0 = −bM0 − (vIIRII(X) − vIRI(X)) = −bM0 + v (RII(X) + RI(X)) . +(28) +Note that the temporal terms would not be absent if the δ-function component of ρ was not proportional to e−σt. +This equation says that the rate of growth of the advective layer mass at x = X (i.e. of the strength of the δ-function) +is driven by the inflow from this layer, and outflow into the diffusive layer. Performing the integration on Eq. (5) +gives +bM0 = dΘI +dx +���� +x=X +− dΘII +dx +���� +x=X +. +(29) +This equation says that any difference in the outflow rates (i.e. +different slopes of the diffusive layer density) is +balanced by the inflow from the advective layer. +We now implement these boundary conditions algebraically. We have: +1. Absorbing boundary condition at x = 0 in the advective layer: RI(x = 0) = 0 +A +�−λ2 +1 + a − σ +b +� ++ B +�−λ2 +2 + a − σ +b +� ++ C +�−λ2 +3 + a − σ +b +� += 0 +(30) +2. Absorbing boundary condition at x = 0 in the diffusive layer: ΘI(x = 0) = 0 +A + B + C = 0 +(31) +3. Absorbing boundary condition at x = 1 in the advective layer: RII(x = 1) = 0 +D +�−µ2 +1 + a − σ +b +� +eµ1 + E +�−µ2 +2 + a − σ +b +� +eµ2 + F +�−µ2 +3 + a − σ +b +� +eµ3 = 0 +(32) +4. Absorbing boundary condition at x = 1 in the diffusive layer: ΘII(x = 1) = 0 +Deµ1 + Eeµ2 + Feµ3 = 0 +(33) +5. Continuity at x = X in the diffusive layer (to prevent infinite diffusive currents): ΘI(x = X) = ΘII(x = X) +Aeλ1X + Beλ2X + Ceλ3X = Deµ1X + Eeµ2X + Feµ3X +(34) + +15 +6. Mass conserving boundary condition in advective layer: RII(x = X) + RI(x = X) = b−σ +v M0 +D +�−µ2 +1 + a − σ +b +� +eµ1X + E +�−µ2 +2 + a − σ +b +� +eµ2X + F +�−µ2 +3 + a − σ +b +� +eµ3X ++ A +�−λ2 +1 + a − σ +b +� +eλ1X + B +�−λ2 +2 + a − σ +b +� +eλ2X + C +�−λ2 +3 + a − σ +b +� +eλ3X = b − σ +v +M0 +(35) +7. Mass conserving boundary condition in diffusive layer: +dΘI +dx +�� +x=X − dΘII +dx +�� +x=X = bM0 +Aλ1eλ1X + Bλ2eλ2X + Cλ3eλ3X − Dµ1eµ1X − Eµ2eµ2X − Fµ3eµ3X = bM0. +(36) +We can write all these seven equations in the compact matrix form: +� +� +� +� +� +� +� +� +−λ2 +1+a−σ +b +−λ2 +2+a−σ +b +−λ2 +3+a−σ +b +0 +0 +0 +0 +1 +1 +1 +0 +0 +0 +0 +0 +0 +0 +−µ2 +1+a−σ +b +eµ1 +−µ2 +2+a−σ +b +eµ2 +−µ2 +3+a−σ +b +eµ3 +0 +0 +0 +0 +eµ1 +eµ2 +eµ3 +0 +eλ1X +eλ2X +eλ3X +−eµ1X +−eµ2X +−eµ3X +0 +−λ2 +1+a−σ +b +eλ1X +−λ2 +2+a−σ +b +eλ2X +−λ2 +3+a−σ +b +eλ3X +−µ2 +1+a−σ +b +eµ1X +−µ2 +2+a−σ +b +eµ2X +−µ2 +3+a−σ +b +eµ3X +σ−b +v +λ1eλ1X +λ2eλ2X +λ3eλ3X +−µ1eµ1X +−µ2eµ2X +−µ3eµ3X +−b +� +� +� +� +� +� +� +� +� +� +� +� +� +A +B +C +D +E +F +M0 +� +� +� +� +� += +� +� +� +� +� +0 +0 +0 +0 +0 +0 +0 +� +� +� +� +� +. +(37) +Because of the structure of this equation, we see that (i) the determinant must be non-zero for a non-trivial solution +and (ii) the nontrivial solution is non-unique - it has at least one degree of freedom. For instance, we are free to +choose one of the variables, or we are free to choose the normalization. Making use of this freedom, we chose to set +M0 = 1. These equations were then used to solve for the remaining coefficients A, B, C, D, E, and F. +Thus, calling the matrix involved in Eq. (37), M, Det(M) = 0 should provide an algebraic equation for σ. +Expanding determinant in terms of minors, we have +0 = bDet (m77) + +�σ − b +v +� +Det (m67) +(38) +where the minor mij is a matrix obtained by removing ith row and jthe column from M. +Once M0 is chosen, the coefficients (A, ..., F) must be unique. This means that both Det (m77) and Det (m67) must +both be non-zero. If Det (m77) is non-zero, then the solution (A, ..., F) obtained from the first six equations can be +found with the inverse of m77, and is unique. This implies that Det (m67) must also be non-zero (otherwise, the +solution (A, ..., F) obtained from the first five and the seventh equation is non-unique, leading to a contradiction). +Thus, the kind of a zero of Det(M) that we want is one in which Det (m77) and Det (m67) are both non-zero. +Therefore, we’re interested in the zeros of the following quantity: +Det′ = b + +�σ − b +v +� Det (m67) +Det (m77). +(39) +It is the zeros of this determinant that gives us σ in terms of (a, b, v, X). +We were primarily interested in the lowest (ground state) eigenvalue σ1, and the inverse 1/σ1 that serves as a +characteristic measure of the escape time[33]. Because the set of eigenfunctions and eigenvalues turned out to be +finite, they are of limited value in being able to construct a solution that fits the δ-function initial condition, and +thereby to properly compute MFPT. + +16 +B: One-layer theory +We now discuss the computation of the eigenfunctions p(x). The subscript n will be dropped to lighten the notation. +Recall that 0 < x < X is Region-I, and that X < x < 1 is Region-II. The eignfunctions satisfy +σp = −v dp +dx + d2p +dx2 +(40) +in Region-I, and +σp = v dp +dx + d2p +dx2 +(41) +in Region-II. The solution in Region-I is pI = aIeλ+x + bIeλ−x, where the λs satisfy +λ± = v ± +√ +v2 + 4σ +2 +. +(42) +The solution in Region-II is pII = aIIeµ+x + bIIeµ−x, where the µs satisfy +µ± = −v ± +√ +v2 + 4σ +2 +. +(43) +The coefficients a and b will be fixed with the following four boundary conditions (BCs). +The first two are the +absorbing BCs at the ends, pI(0) = pII(1) = 0. The third boundary condition is the continuity of the solution +pI(X) = pII(X). A discontinuous solution is unphysical due to the diffusion term. In a one-layer theory, there will +not be an accumulation of mass at the trap, i.e. there will be no term like δ(x − X). Any such density would be +immediately smoothed out by the action of the diffusion. Note that in the full, two-layer theory, such term existed +only in the advective layer, but not in the diffusive layer. In the absence of a δ-function-like accumulation of mass at +x = X, the currents across x = X will be continuous. This gives us the fourth boundary condition that enforces the +continuity of currents at the junction: vpI(X) − dpI +dx +��� +x=X = −vpII(X) − dpII +dx +��� +x=X. +Applying these four boundary conditions leads to four equations: +aI + bI += 0 +(44) +aIIeµ+ + bIIeµ− += 0 +(45) +aIeλ+X + bIeλ−X += aIIeµ+X + bIIeµ−X +(46) +v � +aIeλ+X + bIeλ−X� +− � +λ+aIeλ+X + λ−bIeλ−X� += −v � +aIIeµ+X + bIIeµ−X� +− � +µ+aIIeµ+X + µ−bIIeµ−X� +(47) +Substituting the first two into the last two gives +aI +� +eλ+X − eλ−X� += aII +� +eµ+X − eµ+−µ−eµ−X� +vaI +� +eλ+X − eλ−X� +− aI +� +λ+eλ+X − λ−eλ−X� += −vaII +� +eµ+X − eµ+−µ−eµ−X� +− aII +� +µ+eµ+X − µ−eµ+−µ−eµ−X� +Using the first of these, and substituting into the second we obtain +v � +eλ+X − eλ−X� +−� +λ+eλ+X − λ−eλ−X� +−� +−v � +eµ+X − eµ+−µ−eµ−X� +− � +µ+eµ+X − µ−eµ+−µ−eµ−X��� +eλ+X − eλ−X +eµ+X − eµ+−µ−eµ−X +� += 0, +(48) +where λs and µs are given by Eqs. (42) and (43) respectively. Eq. (48) is an equation for eigenvalues σ as a function +of v and X. Moreover, +pI = +� +eλ+x − eλ−x� +, +(49) +and +pII = +� +eλ+X − eλ−X +eµ+X − eµ+−µ−eµ−X +� � +eµ+x − eµ+−µ−eµ−x� +(50) +The modes given this way are not normalized; they will be normalized below. We will see below that eigenvalues turn +out to be real. + +17 +The coefficients cn are determined as usual by the initial condition, P(x, t = 0) = � +n cnpn(x). Because the operator +O is non-Hermitian, eigenfunctions are generally non-orthogonal, i.e. +� 1 +0 p∗ +n(x)pm(x) dx ̸= 0, so we can’t compute cm +with the help of an inner product +� 1 +0 P(x, 0)pm(x) dx. However, eigenfunctions of the adjoint operator O† have the +property that they are either orthogonal to the eigenfunctions of O, or otherwise have eigenvalues that are complex +conjugates of each other. +Therefore, in order to be able to express initial conditions, we need to compute a set of eigenfunctions and eigenvalues +of O†. Even after this, there is no guarantee that we will be able to express any initial condition, because there’s also +no guarantee of completeness, due to operators being non-Hermitian. +The adjoint of O is given by +O† = v(x) d +dx + d2 +dx2 . +(51) +To find the eigenfunctions of this operator, it helps to look back at the original equation with operator O. We note +that both Eqs. (40)-(41) can be written compactly as one equation +σp = d +dx +�dU +dx p + d2p +dx2 +� +, +(52) +where the potential (in analogy with physics) U is given by +U(x) = +� +v(X − x) x ≤ X, +v(x − X) x ≥ X, +(53) +or, more compactly, v(x) = − dU +dx . +Now, let p = q(x)e−U(x) - we can always do this. Substituting this ansatz we find that q(x) obeys +σ1q = −dU +dx +dq +dx + d2q +dx2 += v(x) dq +dx + d2q +dx2 . +(54) +That is, q = p(x)eU(x) is the eigenfunction of the adjoint operator that we were seeking! Moreover, it has the same +eigenvalue as the operator O. The modes given this way are not normalized; they will be normalized below. +For operators with a finite dimensional eigenspace, eigenvalues of an adjoint operator O† are complex conjugates +of the eigenvalues of the operator O. In such cases, equality of the two sets of eigenvalues implies that they are real. +In our case the eigenspace is not guaranteed to be finite (in fact, we hope that it isn’t, if there is any chance at +completeness). However, our numerical investigation revealed that eigenvalues σ are always real (and negative). +Next, we give an example of the result of several hundred low-lying eigenvalues. The first example is for X = 0.85 +and v = 1. We observe an interesting feature that eigenvalues appear in groups. The second example is for X = 0.6 +20 +40 +60 +80 +100Mode +-100000 +-80000 +-60000 +-40000 +-20000 +σ +FIG. 13: Lowest 100 eigenvalues for X = 0.85 and v = 1. +and v = 1. We notice that the size of groups has changed. There is no obvious relation between X the the size of + +18 +20 +40 +60 +80 +100Mode +-150000 +-100000 +-50000 +σ +FIG. 14: Lowest 100 eigenvalues for X = 0.6 and v = 1. +groups - for instance, for X = 0.55 the groups are again increased in size. +We verified numerically the orthogonality of several eigenfunctions belonging to different eigenvalues, and found it +to hold true. Eigenfunctions were also normalized by multiplying by the following factors: +ap = +1 +�� 1 +0 p∗n(x)pn(x) +, +Aq = +1 +�� 1 +0 q∗n(x)qn(x) +, +where ps and qs are given by Eqs. (49)-(50) and q = p(x)eU(x). The resultant modes came out to be either purely +real or purely imaginary. In this latter case, they can be made real by multiplying by −i. +The following are examples of eigenfunctions. +The discontinuity in the slope of ps - but not of qs - is clearly visible +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +x +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Im[p1] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +x +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Im[q1] +0.2 +0.4 +0.6 +0.8 +1.0 +x +-1.5 +-1.0 +-0.5 +0.5 +1.0 +1.5 +Im[p20] +0.2 +0.4 +0.6 +0.8 +1.0 +x +-1.5 +-1.0 +-0.5 +0.5 +1.0 +1.5 +Im[q20] +Mode 1 +Mode 20 +𝑞! +𝑝! +𝑞"# +𝑝"# +FIG. 15: The first and the twentieth modes for X = 0.85, v = 1. + +19 +0.2 +0.4 +0.6 +0.8 +1.0 +x +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Re[p1] +0.2 +0.4 +0.6 +0.8 +1.0 +x +0.5 +1.0 +1.5 +Re[q1] +0.2 +0.4 +0.6 +0.8 +1.0 +x +-3 +-2 +-1 +1 +2 +3 +Im[p20] +0.2 +0.4 +0.6 +0.8 +1.0 +x +-3 +-2 +-1 +1 +2 +3 +4 +Im[q20] +Mode 1 +Mode 20 +𝑞! +𝑝! +𝑞"# +𝑝"# +FIG. 16: The first and the twentieth modes for X = 0.85, v = 10. +in the first mode. We can understand this by substituting the form p(x) = q(x)e−U(x) into the fourth boundary +condition on p (i.e. vpI(X) − dpI +dx +��� +x=X = −vpII(X) − dpII +dx +��� +x=X), and find that dq +dx is continuous across the junction, +i.e. +qI +dx +�� +x=X = dqII +dx +��� +x=X. The other three boundary conditions for q are the same as for p. +With all this information, we conclude that the set of functions {qn} is then sufficient for us to be able to find the +coefficients cn in the series P(x, t) = � +n cnpn(x)eσnt - as long as there is completeness. The coefficients are given by +cn = +� 1 +0 P(x, t = 0)q∗ +n(x) dx +� 1 +0 q∗n(x)pn(x) dx +(55) +Completeness is not guaranteed, but unlike the two-layer case, we found that the method works, provided enough +modes are used. We will not discuss convergence properies of the series here. +In relation to the mean first passage time problem, we are interested in the δ-function initial condition, P(x, t = +0) = δ(x − x0), in which case the coefficients are given by +cn = +q∗ +n(x0) +� 1 +0 q∗n(x)pn(x) dx +. +(56) + +20 +C: Trajectory examples +Fig. 4 and the subsequent discussion in our main text discussed several regimes of MFPT, depending on the value +of a = b, for symmetric trap placement. We now show trajectories in each of those regimes. +First, we show trajectories in the plateau regime that precedes the second crossover. This takes place for a roughly +in the range [10−2, 10]. This is shown in Fig. 17. +𝑎 = 0.1 +𝑎 = 1 +𝑎 = 10 +Time +Time +FIG. 17: Nine trajectories at lower as. All particles are placed initially at x0 = 0.5 on the DL. Here X = 0.5 and v = 20. The +right panels show a smaller window of time. +We can clearly see that as a increases, thee probability of switching into the AL increases. Once a particle switches +to the AL, it will move towards the attractor. +As a increases further, the likelihood of the advective motion towards the attractor all in one ride on the AL +decreases. Instead, a typical particle will experience episodes of a little bit of advective motion, followed by a little +bit of diffusive motion, and so on - see Fig. 18. This happens in the second crossover regime that begins for a ≈ 10 +and continues for several decades. + +..21 +Time +Time +𝑎 = 100 +𝑎 = 1000 +FIG. 18: Nine trajectories at intermediate as. All particles are placed initially at x0 = 0.5. Here X = 0.5, v = 20. The right +panels show a smaller window of time. +For a even larger - the system enters the second plateau, when any further increase in a does not increase MFPT. +This means that the system behaves in accordance to the one-layer model [34]. The the episodes of diffusion and +advection become even shorter. Trajectories in such a regime are shown in Fig. 19, for progressively narrower windows +of time, from left to right. + +22 +𝑎 = 𝑏 = 10! +𝑎 = 𝑏 = 10" +𝑎 = 𝑏 = 10# +Time (m.u.) +Time (m.u.) +Time (m.u.) +Time (m.u.) +FIG. 19: Trajectories for a between 104 to 106 in powers of 10. Here again X = 0.5 and v = 20. Leftmost column has 10 +trajectories, while the other columns show one trajectory for progressively narrower windows of time, from left to right. In +these right three columns, the red color indicates advective portions of trajectories, while grey are diffusive portions. + +0.70 +1: +0.65 +0.60 +0.55 +X 0.50 +0.45 +0.40 +0.35- +0.30 +20 +40 +60 +80 +100 0 +2 +4 +6 +8 +10 0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +Time (m. u.) +Time (m. u.) +Time (m. u.)0.70 +0.65 +0.60 +0.55 +X 0.50111 +0.45 +0.40 +0.35- +0.30 +20 +40 +60 +80 +100 0 +2 +8 +10 0.0 +0.2 +0.4 +0.6 +0 +6 +0.8 +1.0 +Time (m. u.) +Time (m. u.) +Time (m. u.)0.70 +0.65 +0.60- +0.55 +X 0.50 +0.45 +0.40 +0.35- +0.30 +0 +20 +40 +60 +80 +100 0 +2 +6 +8 +10 0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Time (m. u.) +Time (m. u.) +Time (m. u.)0.70 +0.65 +0.60 +0.55 +X 0.50 +0.45 +0.40 +0.35 +0.30 +0 +2 +4 +6 +8 +10 +Time (m. u.)0.70 +0.65 +0.60 +0.55 +X 0.50 +0.45 +0.40 +0.35 +0.30 +0 +2 +4 +6 +8 +10 +Time (m. u.)0.70 +0.65 +0.60 +0.55 +X 0.50 +0.45 +0.40 +0.35 +0.30 +0 +2 +4 +6 +8 +10 +Time (m. u.)23 +D: Theory and simulation comparison - one-layer limit +In this section we show the comparison between the one-layer analytical predictions of pl, pr, τl, τr, and τ with +results of simulations of the two-layer model. +𝑥 +𝑥 +𝑥 +𝑝 +𝜏 +𝜏 +Right +Left +FIG. 20: Comparison between analytical quantities (open circles) and simulation results (filled circles - Monte Carlo simulation +as described in the main paper, filled triangles - forward flux sampling algorithm [32]). Left column: probabilities to escape +through the left end (blue) pl and right end (orange) pr. Middle column: escape time conditioned on the left exit (blue) τl and +right exit (orange) τr. Right column: net MFPT τ. The growing discrepancy between simulation and analytical results is due +to the diffusive approximation of the latter; the details will be discussed in the coming publication [27]. Here X = 0.5. Top +row: v = 0.1, middle row v = 5, bottom row v = 20. + +.24 +[1] J. Howard and R. Clark, Appl. Mech. Rev. 55, B39 +(2002). +[2] J. L. Ross, M. Y. Ali, and D. M. Warshaw, Current Opin- +ion in Cell Biology 20, 41 (2008), ISSN 0955-0674, cell +structure and dynamics. +[3] S. S. Mogre, A. I. Brown, and E. F. Koslover, Physical +Biology 17, 061003 (2020). +[4] A. Chien, S. M. Shih, R. Bower, D. Tritschler, M. E. +Porter, and A. Yildiz, Elife 6, e28606 (2017). +[5] S. Yadav and A. D. Linstedt, Cold Spring Harbor per- +spectives in biology 3, a005322 (2011). +[6] F. Mascanzoni, R. 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Majumdar, Journal of Physics A: +Mathematical and Theoretical 44, 435001 (2011). +[25] J. Xu, Z. Shu, S. J. King, and S. P. Gross, Traffic 13, +1198 (2012). +[26] C. W. Gardiner et al., Handbook of stochastic methods, +vol. 3 (Springer, Berlin, 1985). +[27] N. Sarpangala, B. Randell, A. Gopinathan, and O. Ko- +gan, To appear (2023). +[28] R. D. Schumm and P. C. Bressloff, Journal of Physics A: +Mathematical and Theoretical 54, 404004 (2021). +[29] X. Zhu, R. Hu, M. Brissova, R. W. Stein, A. C. Pow- +ers, G. Gu, and I. Kaverina, Developmental cell 34, 656 +(2015). +[30] F. N´ed´elec, T. Surrey, and A. Maggs, Physical Review +Letters 86, 3192 (2001). +[31] S. Klumpp, T. M. Nieuwenhuizen, and R. Lipowsky, Bio- +physical Journal 88, 3118 (2005). +[32] R. J. Allen, D. Frenkel, and P. R. ten Wolde, The Journal +of chemical physics 124, 024102 (2006). +[33] This is especially true in the rare event regime that devel- +ops at sufficiently large v - when σ1 should be separated +from the rest of σs by a gap that grows exponentially in +v - while there is no such gap between the rest of the +eigenvalues. +[34] This is not what makes escape events rare. The signa- +ture of the rarity of escape events (that is MFPT is +much greater than all other time scales) is the exponen- +tial growth of MFPT with v. + diff --git a/E9AzT4oBgHgl3EQfUPzF/content/tmp_files/load_file.txt b/E9AzT4oBgHgl3EQfUPzF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b556a667e78e56da966139678923d50ea6a286ea --- /dev/null +++ b/E9AzT4oBgHgl3EQfUPzF/content/tmp_files/load_file.txt @@ -0,0 +1,1436 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf,len=1435 +page_content='Tunable intracellular transport on converging microtubule morphologies Niranjan Sarpangala,1 Brooke Randell,2 Ajay Gopinathan,1 and Oleg Kogan2 1University of California, Merced, CA, 95343 2California Polytechnic State University, San Luis Obispo, CA, 93407 A common type of cytoskeletal morphology involves multiple converging microbutubules with their minus ends collected and stabilized by a microtubule organizing center (MTOC) in the interior of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This arrangement enables the ballistic transport of cargo bound to microtubules, both dynein mediated transport towards the MTOC and kinesin mediated transport away from it, interspersed with diffusion for unbound cargo-motor complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Spatial and temporal positioning of the MTOC allows for bidirectional transport towards and away from specific organelles and locations within the cell and also the sequestering and subsequent dispersal of dynein transported cargo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The general principles governing dynamics, efficiency and tunability of such transport in the MTOC vicinity is not fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' To address this, we develop a one-dimensional model that includes advective transport towards an attractor (such as the MTOC), and diffusive transport that allows particles to reach absorbing boundaries (such as cellular membranes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We calculated the mean first passage time (MFPT) for cargo to reach the boundaries as a measure of the effectiveness of sequestering (large MFPT) and diffusive dispersal (low MFPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We show that the MFPT experiences a dramatic growth in magnitude, transitioning from a low to high MFPT regime (dispersal to sequestering) over a window of cargo attachment/detachment rates that is close to in vivo values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Furthermore, we find that increasing either the attachment or detachment rate, while fixing the other, can result in optimal dispersal when the attractor is placed asymmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Finally, we also describe a regime of rare events where the MFPT scales exponentially with advective velocity towards the attractor and the escape location becomes exponentially sensitive to the attractor positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Taken together, our results suggest that structures such as the MTOC allow for the sensitive control of the spatial and temporal features of transport and corresponding function under physiological conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Introduction The transport of material within eukaryotic cells is a critically important physiological process that cannot be achieved by passive diffusion alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In these cells, cargo, including vesicles and organelles, are dragged along by a variety of molecular motors which utilize energy from ATP hydrolysis to power their directed stepping motion along cytoskeletal protein filaments with a well-defined polarity [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Motors from different families such as ki- nesins and myosins step along different filaments (micro- tubules and actin respectively) and others such as dynein move along the same microtubule filaments as kinesins but in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Transport at the cellular scale is therefore a complex process that involves phases of multiple motors effecting directed transport along cy- toskeletal filament networks interspersed with passive dif- fusion of the cargo [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This process is essential for the transport of a variety of cargo between specific lo- cations and organelles within the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Examples include the transport of cargo in cilia [4], between the plasma membrane and Golgi apparatus [5], [6], between Endo- plasmic Reticulum and Golgi [7], [3], transport of viruses towards replication sites [8], [9], and the transport of many other vesicles and organelles for various functional purposes (see review [3]), [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Much like the design of road networks affect traf- fic flow, the morphologies of the cytoskeletal networks in cells have been shown to have a significant effect on intracellular transport [11–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This is particularly important as, even a single type of cytoskeletal fila- ment such as microtubules exhibit a wide diversity of morphologies within different cell types to enable dif- ferent functions[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In some situations, such as in melanophores microtubules have a strongly orderly (in this case radial) - organization [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In others, the orien- tation or polarity of microtubule (MT) morphology can be broadly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In pancreatic β cells, for exam- ple, MTs are arranged with both an orientational and polarity disorder [17], although there is an average po- larity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' On the other hand, MTs in neuronal dendrites are essentially aligned with the long direction of the den- drite, but their polarity is not uniform [18] resulting in junctions of plus or minus ends along the dendrite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' A common structural feature that governs these micro- tubule morphologies is the microtubule organizing center (MTOC) that is responsible for growing MTs and local- izing and stabilizing their minus ends leading to multi- ple MTs converging with their minus ends at the MTOC [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Dynein-driven transport along MTs will move cargo to the vicinity of MTOC, while kinesin mediated trans- port moves cargo away from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' These ballistic phases are interspersed with isotropic diffusion for unbound cargo- motor complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The spatial and temporal positioning of the MTOC therefore allows for bidirectional transport towards and away from specific organelles that can act as MTOCs as well as locations within the cell in the vicinity of the MTOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Examples in which MTOC facil- itates direct transport to the destination of interest in- clude transport of cargo such as secretory vesicles away from the Golgi apparatus toward the cell membrane and endocytic vesicles towards the Golgi which is known to perform as an MTOC in many mammalian cells [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='01264v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='CB] 3 Jan 2023 2 + + + + + + + + + + + + + (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 1: (a) A model of a cell in which microtubules have a strong central organization, with minus ends at the cen- trosome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' A dark circle represents an organelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Dynein mo- tors are shown moving on microtubules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (b) One dimensional morphology found in dentrites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here the ends of the same po- larity from different microtubules can face each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This schematic is based on [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The dynein mediated transport of some viruses toward the nuclear envelope is also enabled by the presence of a MTOC in the vicinity of the nucleus [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In some cases, cargo need to traverse regions with convergent MT morphologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Such cases occur in den- dritic processes of neuronal cells that have been shown to have regions of alternating polarity of MTs [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Di- rected transport of dynein (kinesin) carrying cargo at a junction of minus(plus) ends will have to overcome what is essentially a trap to maintain observed unidirectional transport towards or away from the main cell body [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Finally, the location of MTOCs can also be tuned over time to accommodate different cellular functions such as sequestering and dispersal of cargo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For example, in melanophores [16, 20], a perinuclear MTOC produces a radial MT structure with minus ends in towards the nu- cleus and plus ends out toward the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Cells achieve color change by aggregating and sequestering pig- ment containing melanosomes near the nucleus via bal- listic dynein mediated transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Upon hormonal stimu- lation they can switch to a superdiffusive dispersal phase powered by a combination of kinesin and actin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Another example occurs in lymphocytes that enable cytotoxicity by secreting the contents of lysosomes (lytic granules) at the immunological synapse to kill the target cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here, dynein dependent sequestering of the lytic granules at the MTOC occurs rapidly followed by the gradual move- ment of the MTOC towards the synapse with subsequent secretion [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In all these cases, it is important to understand the dynamics of the transport and its sensitivity to biological parameters in order to understand functional efficiency and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In particular, given the wide variety of functional contexts in which the converging MT geometry facilitates transport, it is critical to understand the general principles governing dynamics, efficiency and tunability of such transport in the MTOC vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' To address this gap, we develop a simple one- dimensional model that includes advective transport to- wards an attractor (such as the MTOC), and diffusive transport that allows particles to reach absorbing bound- aries (such as cellular membranes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This can be viewed as a 2-layer model consisting of an advective layer en- dowed with an attractor, a diffusive layer, and absorb- ing boundaries along the perimeter of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We take the mean first passage time (MFPT) for cargo to reach the boundaries as a measure of the effectiveness of sequestering or directed transport (large MFPT) and diffusive dispersal (low MFPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The number of indepen- dent control parameters in this problem can be reduced to four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' These are the rates of attachment to and de- tachment from microtubules, advective velocity, and the placement of the attractor within the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Using this model we were able to make a series of tan- talizing predictions - on which we report here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' A cen- tral calculation here is the residence time, or what is commonly called in the literature the mean first passage time (MFPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Thus, given an initial location of the cargo within the domain (determined by organelle placement), this quantity tells the average time to reach either of the absorbing boundaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' escape the domain), or a spe- cific boundary (in one dimension, left or right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Another relevant quantity is the probability of escape through one or the other domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Symmetric, or nearly symmetric attractor positions can give rise to a dramatic increase in the value of MFPT within a certain window of dimensionless coupling rates between the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Concurrently with this dramatic rise of MFPT, the probability to escape purely diffusively goes to zero in the same range of (dimensionless) cou- pling values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This means that for larger coupling values, any cargo particle will have to experience at least one episode of motion on microtubules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Crucially, we found that biophysical parameters in cells correspond precisely 3 to this range of dimensionless coupling rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This sug- gests that parameter values in cells are optimized for the greatest sensitivity to small changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' With such parame- ters, a cell can achieve the largest change in functionality with smallest changes in parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Second, we predict the existence of optimal coupling rates that minimize the MFPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This minimal MFPT happens when the attractor is positioned asymmetrically (off center) in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' A similar phenomenon has been predicted in the study of diffusion with stochastic reset [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Indeed, attachment to the microtubule, followed by a rapid transport to the attractor, followed by detachment from the microtubule back to diffusion in the cytoplasm is effectively a reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' When the coupling rates are much larger than all other rates in the problem, the model reduces to effectively one-layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here we demonstrate that even a slight asym- metry in the position of the attractor can lead to a very strong amplification of the preferred exit end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This pro- vides another example of sensitivity to small parameter changes - in this case asymmetric of the attractor place- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This effect happens at sufficiently large advective velocity, and corresponds to rare event physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In the regime of rare events, a small fraction of particles escape quickly, while the majority advect to the attractor, and form a quasi-stationary distribution around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' They stay in the vicinity of the attractor for a time that scales expo- nentially with advective velocity (or inverse of diffusion coefficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Methods Model We consider the minimal model in a one-dimensional domain of length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' It contains an advective layer (AL) that represents motion along microtubules, and a dif- fusive layer (DL) that represents diffusion in the cyto- plasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We assume that attachment to and detachment from microtubules are Poisson processes, endowed with rates α and β respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This means, for example, that a motor spends on average a time 1/β since attaching to a microtubule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' While advecting, particles move with a uniform velocity towards the attractor - which is an at- tracting fixed point located at some coordinate x = X0 between x = 0 and x = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Letting ρ(x) and θ(x) be probability densities of particles in the AL and DL re- spectively, the model reads ∂ρ ∂t = − ∂ ∂x (v(x)ρ) + αθ − βρ (1) ∂θ ∂t = −αθ + βρ + D ∂2θ ∂x2 (2) on 0 ≤ x ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The velocity field is given by v(x) = � +v0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' x < X0 −v0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' x > X0 (3) The parameters are rates α and β, the diffusion coeffi- cient D, the advective velocity on microtubules v0, and the location of the attractor X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' There are absorbing BCs at x = 0 and x = L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' ρ(0) = θ(0) = 0 and ρ(L) = θ(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' All together, there are six physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We will switch to dimensionless variables by rescaling the lengths by L and times by L2/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Thus, x′ = x/L and t′ = tD/L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The resulting equations will be ∂ρ ∂t′ = − ∂ ∂x′ (v′(x)ρ) + aθ − bρ (4) ∂θ ∂t′ = −aθ + bρ + ∂2θ ∂x′2 (5) on 0 < x′ < 1, with ρ(0) = θ(0) = 0 and ρ(1) = θ(1) = 0, the velocity field v′(x) = � +v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' x < X −v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' x > X where X = X0/L and v = v0L D , and coupling rates a = αL2 D and b = βL2 D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' From now on, we will drop primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The model is depicted schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝐷 = 1 𝑥 = 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 2: One-dimensional model with dimensionless parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Range of parameters Here we review the values of parameters from litera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Both adsorption rate α and desorption rate β are expected to be of the order of 1 per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For example, [11] cites α = 5 s−1 and β = 1 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Microtubule lengths typically fall in the range of 1−10 µm [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' However, the length of advective path may be much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For exam- ple, in neurons, a cargo that needs to be delivered from the soma to synapses on the ends of axons will travel a length of the order of a meter [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The velocity of molec- ular motors on MTs is on the order of 1 µm/s [3], [11], although this quantity also has a degree of variability [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Diffusion coefficient of vesicular organelles in the cytoplasm fall in the range 10−3 − 10−1 µm2/s [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Given these physical parameters, our dimensionless pa- rameters a and b will take on values in the range [10, 105], and parameter v will take on values in the range [10, 104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' v(x) Advective layer rate a rate b Diffusive layer D Junction point x=0 x=L at x = X4 There are four timescales in the problem: 1/a, 1/b, the advective timescale 1/v, and the diffusive timescale (which is of order 1 in dimensionless units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Different special cases or behavioral regimes emerge when one of these timescales differs significantly from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The limit that is particularly amenable to analysis is one in which 1/a and 1/b are both much smaller than the advective time (which is of order 1/v in dimension- less units) and diffusive time (which is of order 1 in di- mensionless units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We will formally call it the a, b → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In this regime, the model reduces to an advection- diffusion process in one single layer, which amenable to many analytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Analytical approach in the one-layer limit A very important special case is a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' As a = b → ∞, the model reduces to an effeective one-layer model: ∂P ∂t = − ∂ ∂x � v(x)P(x) − ∂P ∂x � where P(x, t) is the probability density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' P describes both θ and ρ, which become identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' A general solution will be written as an eigenfunction expansion P(x, t) = � n cnpn(x)eσnt, (6) where pn(x) and σn is nth eigenfunction and eigenvalue, which satisfy Opn = σnpn, with the operator O given by O = − ∂ ∂x � v(x) − ∂ ∂x � , (7) with v(x) = � +v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' x < X −v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' x > X (8) and a constant v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Thus, the one-layer model contains two parameters: dimensionless advective velocity v and dimensionless position of the attractor X, which can take on values between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The computation of eigenvalues σn and eigenfunctions pn(x) of the operator O, as well as the computation of the eigenfunctions qn(x) of the operator O† is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Starting from the initial condition P(x, t = 0) = δ(x − x0), the probability density will be given by P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' x0) = � n q∗ n(x0)pn(x) � 1 0 q∗n(x′)pn(x′) dx′ eσnt (9) Everything that we need to compute MFPT can be ex- tracted from this probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' To calculate the MFPT τ(x0), we notice that the mag- nitude of the flux through the boundary is given by f(t) = �� ∂P ∂x �� bdry in dimensionless units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Then f(t)dt gives the fraction of initial particles that cross the boundary in [t, t + dt] = probability of crossing that boundary in [t, t + dt], since the initial condition is normalized to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' So, p = � ∞ 0 f(t) dt gives the probability of ever leav- ing through that boundary, f(t)dt p gives the probability that particles that leave through that boundary do so in [t, t + dt], and finally τ = � ∞ 0 t f(t) p dt is the average time to leave through that boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In this problem, there are two boundaries, with τl and τr denoting MFPT to exit through the left and the right boundary respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We expect τl → 0 as x0 → 0 and τr → 0 as x0 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Finally, MFPT in general - without condition- ing on a specific boundary - is the weighted average of the two: τ = τlpl+τrpr, which matches predictions using other methods [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Analytical approach in the general case Analogously to the one-layer approach, we again seek a general solution to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (4)-(5) via an eigenfunction expansion of the form � ρ(x, t) θ(x, t) � = � n cn � Rn(x) Θn(x) � e−σnt (10) (we found it convenient to factor out the negative sign from σ here), where � Rn Θn � and σn is the nth (vector) eigenfunction and eigenvalue, which satisfy O � Rn Θn � = σn � Rn Θn � , with the operator O given by O = � ∂ ∂xv(x) + b −a −b a − ∂2 ∂x2 � (11) with v(x) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The full model contains four parameters: dimensionless advective velocity v, dimen- sionless rates a and b, and dimensionless position of the attractor X, which can take on values between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The computation of eigenvalues and eigenfunctions is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Remarkably, there are only a finite number of eigen- functions and eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In other words, the eigenset is not complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' As a = b → ∞, this number goes to infin- ity, while the lower-lying eigenvalues and eigenfunctions approach those of the one-layer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The completeness is not guaranteed, since the operator O is not Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Thus, an expansion such as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (10) is of limited use, and cannot be used to fit a solution for an arbitrary initial condition - including a point-like δ function initial con- dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This also implies that we cannot compute escape currents and MFPT from such initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' However, we can always compute the ground state eigenvalue, σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Then the time 1/σ1, while not a true MFPT, is an estimate of a characteristic time for escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 5 We found that this time alone agrees with MFPT com- puted in simulations quite well, so we will make MFPT arguments based on this estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Monte Carlo simulation method We considered a simple one dimensional computational model to simulate the transport process in a domain of length L with attractor formed by oppositely oriented mi- crotubules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Our computational model involves two lay- ers, an advective layer (AL) where the particle undergoes active transport and a diffusive layer where it does one dimensional random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We consider one particle at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' To begin, we initialize the particle at position x = x0 within the domain x ∈ [0, L = 1] either in the diffusive or in advective layer as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We consider that the particle can switch from diffusive layer to ad- vective layer with a rate a and from advective layer to diffusive layer with a rate b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' When a particle switches to diffusive layer, a time td is drawn from the exponential distribution e−at and the particle is allowed to diffuse for n = td/∆t number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' ∆t is the time step in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In each step the position is updated as x(t + ∆t) = x(t) + r∆x, (12) where r is drawn from the set {−1, 0, 1} with the proba- bility p = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' ∆x is the step size which is chosen such that the diffusion constant of the particle D = p∆x2 ∆t is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Right after finishing a diffusive portion of a simula- tion run, the particle switches from diffusive to advective layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In the advective layer, the particle stays for a time ta drawn from e−bt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' n = ta/∆t number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In the advective layer, the position of the particle is updated as x(t + ∆t) = x(t) + v(x)∆t, (13) where v(x) is the advective velocity given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' These alternative portions of a simulation run in diffu- sive and advective layers are continued until the particle reaches one of the boundaries (x = 0 or x = 1) or until maximum simulation time, Tmax is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We then re- peat with N particles to get enough statistics to calculate the overall MFPT, probabilities and MFPTs to exit out of specific boundaries, and other quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Trajectories To get the trajectories, we record the data of the x po- sition and the layer in which particle is located at regular time intervals during each simulation run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' An example of trajectories is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Diffusion 1D Random Walk Molecular Motor Based Transport FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 3: A sample trajectory generated by the Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Diffusive motion is indicated with orange line, and advective motion with a magenta line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Grey colored lines indicate more sample trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Computation of Net MFPT To compute the net MFPT for a given parameter set, we perform simulation runs until the particle exits out of one of the boundaries (x = 0 or x = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We record the time of exit for each run and then compute the mean and standard error of the mean for all N runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Computation of Conditional MFPT and escape probability To compute the MFPT for exit specifically through the left (or the right) boundary, we record the time as well as the boundary through which the particle exits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Then we filter out only those simulation runs where a particle exited out of the left (or right) boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Then we compute mean and standard error of the mean for those runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We compute the escape probability through left (or right) boundary as the fraction of runs that exited out of the left (or right) boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Statistics of visits to the AL We measure the fraction of simulation runs in which a particle that started on the DL ended up making at least one visit to the AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In each simulation run, we also compute the number of visits to the advection layer before exiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' To do this, we update a counter every time the particle switches its layer to get the number of times it switches layers prior to exiting the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We then compute the average over N runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 6 Results and Discussion Variation of coupling rates can change escape times by orders of magnitude We begin our presentation of results with the symmet- ric case, X = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For simplicity we will set the particles’ initial placement at x0 = 1/2 - this is the initial condi- tion (IC) in analytical calculations - and let a = b for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Figure 4 (a) displays the mean first passage time (MFPT) as a function of a = b at different advective speeds v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' To help understand the physics of the process, we also plot the fraction of times that particles visit the advective layer in panel (b) (for particles initially placed on the DL), as well as the number of times they do so in panel (c) (also when starting on DL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Two crossovers are evident from the plot of MFPT vs a (= b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The first crossover takes place around a = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' As suggested by the plot of the fraction of visits to the AL, at this coupling rate the probability of visiting the AL becomes non-zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' below this crossover, the advective layer is not visited and the MFPT is a purely diffusive time ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For a above this crossover value, the prob- ability of visiting the AL grows with increasing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' While the fraction of particles visiting the AL grows ∝ a, the time to remain in the AL (the longest time scale in this range of a) decreases ∝ 1/a, resulting in the plateau of MFPT vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Because the probability (or fraction) to visits to the AL is less than 1 (for particles startin in the DL), a particle has a chance to escape purely diffusively for as in this plateau region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The MFPT is in dimensionless time units;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' to convert to time in seconds, multiply by L2/D expressed in physical units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For example, for L = 1 µm and D = 10−2 µm2/s, the MFPT of 10 dimensionless time units corresponds to 103 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The MFPT for diffusive transport on a domain with two absorbing boundaries and a midpoint initial condition is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='125 (in dimensionless time units), which is half of the first plateau value, and much lower than plateaus after the second crossover for v > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We continue our discussion of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The probabil- ity of visiting the AL (for particles starting in the DL) eventually reaches 1 at larger a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' particles are now certain to visit the AL at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In other words, the prob- ability of a purely diffusive escape reaches zero and we encounter the second crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For v = 20, for example, this second crossover happens around a = b = 10, but its location - defined by the point of inflection - varies some- what with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This crossover is broad - it can be several decades wide - and marked by a drastic growth of the MFPT, especially at larger v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In this second crossover regime, each particle experiences intermittent advection, punctuated by periods of diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In other words, on a typical run from an initial location to one of the bound- aries, a particle’s trajectory will include multiple episodes of advection and diffusion following each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Eventu- ally, we reach the second plateau, when the switching between the layers is so rapid that we now reach an ef- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 4: Symmetric case: X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5, the initial location of particles is also at x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (a) MFPT vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Dots: IC on the DL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' crosses: IC on the AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The solid curves are analytical estimations of MFPT given by 1/σ1, where σ1 is the ground state eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The MFPT is in dimensionless time units;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' to convert to time in seconds, multiply by L2/D expressed in physical units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The dashed horizontal line has a value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The last two points (a = 105 and 106) required a smaller dt = 10−6 3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' dt = 10−4 3 was sufficient for the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Therefore, we used N = 103 for the last two points to optimize simulation time, and N = 104 for the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (b) Fraction of simulation runs that visit the AL at least once after starting in the DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The dashed line is a fit, of the form 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='079a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here N = 104 and dt = 10−4 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (c) Average number of visits for particles starting in the DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here N = 103, dt = 10−4 3 (circles), and 10−6 3 (diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The x-axis is the same in all three plots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' the plots are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The shading guides the eye to the second crossover region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 7 fectively one-layer regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This regime will be studied in the next section, where we examie a one layer model with advection and diffusion taking place simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' MFPTs predicted by that model match the high a = b plateaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Interestingly, there is a strong velocity depen- dence in the one-layer regime, but not in the range of a = b in the plateau below the second crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For a = b < 1/(simulation time), particles with IC in the advective layer (plus symbols in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 4) will never enter the DL and therefore will not escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' MFPT will simply be limited by the simulation time - this is mani- fested in the saturation at MFPT = 500, since this was the simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Appendix C displays examples of particle trajectories for a broad range of a = b that cover all of the behavioral regimes shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' These figures demonstrate the change in the character of trajectories - from the types that contain advective periods long enough to arrive to the attractor at low a = b, to intermittent behavior in the second crossover region, to very rapid switching between layers for a = b beyond the second crossover - when the model is effectively in the one-layer regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The region of the most sensitive behavioral tuning matches the biological parameters We now turn our attention to the biological significance of these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Note that the second crossover takes place between a = 10 and a = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Remarkably, this is precisely the range of these parameters found in cells see “Range of parameters” above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This might imply that these parameters evolved to have such values for an easy tunability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Indeed, the second crossover region is precisely where a change in the rates gives rise to the largest change in the outcome - especially at larger values of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' There is an optimal coupling rate between advective and diffusive behavior Placing the attractor asymmetrically can give rise to a decrease in MFPT with increasing coupling rates - see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This effect is only seen at larger v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The decrease happens over a range of 1/a that is comparable to the advective time, ∼ 1/v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For example, for v = 20, the time scale to travel advectively to the attractor is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='05, while the decrease is seen for a between 1 and 100, which corresponds to the time scale between 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We think that this decrease in the MFPT happens be- cause an increase in the interlayer coupling causes more material to congregate at the attractor, which is close to one of the ends - thus leading to an overall decrease in the MFPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 6 shows an example of this phenomenon due to only the parameter a varied at fixed b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We mentioned in the discussion of the analytical approach in the general 5 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 5: Asymmetric case: X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In this particular case, x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='7, but such dips are also present at other x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' case that a complete eigenset in the two-layer model does not exist, so the exact solution cannot be obtained as a sum of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' However, the MFPT can be estimated as τ = 1/σ1, where σ1 is the ground state eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑎 𝑣 = 13 𝑏 = 169 𝑿 = 𝟏/𝟐𝟔 𝑣 = 13 𝑏 = 169 𝑿 = 𝟏/𝟐 Net MFPT 𝑎 Net MFPT (a) (b) Out[\uf750 ]= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='01 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1 10 1000 105 Out[\uf750 ]= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='01 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1 1 10 100 1000 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 6: τ(a) at fixed b = 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (a) X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5, (b) X = 1/26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' v = 13 for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Lines: theory, dots: simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Red dots - IC on the diffusive layer, blue dots - IC on the advective layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The numbers for the two types of initial conditions are not identical, but the difference is almost invisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The analytical prediction is 1/σ1 - the inverse of the ground state eigenvalue, which is not a true MFPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The IC in the simulation was at x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The simulation time was 1000, which is the reason for flattening of the simulation data at large a in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 8 𝑏 Net MFPT 𝑏 Net MFPT (a) (b) Out[\uf750 ]= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1 1 10 100 1000 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1 1 10 100 1000 Out[\uf750 ]= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='01 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1 1 10 100 1000 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 7: τ(b) at fixed a = 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (a) X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5, (b) X = 1/26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' v = 13 for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Lines: theory, dots: simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Red dots IC on the diffusive layer, blue dots - IC on the advective layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The analytical prediction is 1/σ1 - the inverse of the ground state eigenvalue, which is not a true MFPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The IC in the simulation was at x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We again see saturation of simulation results at low b at the simulation time (here, 1000 time units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 6 are values of 1/σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This es- timation should become more accurate as escape events become rare (MFPT ≫ than all other time scales);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' this is because higher eigenmodes contribute little to the prob- ability current in the rare event limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Moreover, while this calculation does not give IC dependence, MFPT loses this dependence as escape events become rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Some dis- cussion of rare events can be found in the next section, and a much more in-depth discussion will appear in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The dips in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 6 happen, again, because increasing a causes particles to return back to the attractor, thus min- imizing the chance for them to wander too far to the right while diffusively exploring the long part of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' On the other hand, increasing a even further tends to keep the particles in the AL and therefore prevents them from escaping (particles cannot move in the direction of the ends when they are in the AL due to the advective flow being directed towards the attractor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' These dips are somewhat counter-intuitive - an overall escape time is lowered by increasing the tendency to go towards the attractor inside the domain - as long as the attractor is placed asymmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' A similar phenomenon has been reported in connection to the problem of mean first passage time with a reset [23], [24], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here, in addition to diffusion, a particle experiences a reset back to some location, and resets form a Poisson process, endowed with a reset rate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The au- thors of these sources found there exists an optimal rate, r∗ which minimizes the MFPT out of the semi-infinite domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We note, however that these sources appear to return the particle back to the reset location once it has hit the absorbing end of the semi-infinite domain, thereby conserving the probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This is different from our problem, in which the total probability inside the domain decreases with time, because once particles have reached one of the two absorbing ends, they are not re- turned back into the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This difference aside, the problem that we are ana- lyzing can be viewed as a version of a reset problem, although the time to reset is not instantaneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' More- over, the reset location is not necessarily the location of the attractor x = X, since a particle has a chance to return to the diffusive layer before reaching the at- tractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The limit of infinite v would correspond to the instantaneous reset to the attractor, and the limit b → 0 would cause the resetting to take particles back to x = X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' approximating the standard reset problem (although, again, without returning of particles that have reached either of the domain ends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The dip phenomenon is also observed when b is varied at fixed a, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' At low b, MFPT is dominated by the waiting time 1/b to return from the attractor to the DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' A large b asymptote (for b ≫ a) is the regime of purely diffusive motion - the particles are forced into the DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Evidently, having some acccess to the AL leads to a lowering of MFPT because it allows more material to congregate close to one end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' It is interesting to ask what effect increasing the ad- vective velocity would have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The intuition - supported by the physics of the one-layer model - is that higher v should lead to either an increase of the MFPT or the disappearance of the dip, because with sufficiently large velocity, the density will be more and more localized near the attractor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' so, even though the attractor is closer to one end than the other, it is no longer close to this end 𝑎 1/𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Out[\uf750 ]= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='10 1 10 100 1000 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='20 v=20, 40, 60, 80 top to bottom X=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 8: Top to bottom: v = 20, 40, 60, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5, and b = 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 9 Out[\uf750 ]= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='10 1 10 100 1000 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='10 1 10 100 X=1/26, 2/26, 3/26, from bottom to top V=20 1/𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑎 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 9: Top to bottom: X = 3/26, 2/26, 1/26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here v = 20, and b = 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' in comparison to the width of the density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' However, analytical calculations in fact predict the de- crease in the value of 1/σ1 at a fixed a with increasing v, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' An in-depth study of density distributions, which will be published elsewhere [27], sheds light on the reason for this counter-intuitive prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' While the density pro- file in both layers does become more localized with larger velocity (as expected), the part of the profile between the attractor and the close end is not affected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' the decrease in the spread is due to the other side of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' There- fore, as velocity is increased, more and more material is localized near the close end, while the chance of escap- ing through this end does not diminish - resulting in the overall decrease of escape time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We also study the effect of varying X in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here the results conform to the intuitive expectation that a de- crease in asymmetry will lead to a decrease in the mag- nitude of the dip (with no dip at all in a completely symmetric geometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' An attractor placed much closer to the left end than the right one, for example, has two effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' First, it lowers the MFPT overall, since there is less distance to travel during the escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Second, pre- venting particles from wandering too far to the right (by increasing a, and thus the reset rate) causes the particles to congregate closer to the left end in the more asymmet- ric situation, leading to a lower MFPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' One-layer limit Dynamics of probability density The analytical approach in the one-layer limit is out- lined in the Methods section, with details in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' These predictions are verified by simulations (see Ap- pendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here we present results of analytical calcula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 10 we show several snapshots in the evolution of the probability density profiles for a specific placement of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' "(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=') 𝑣 = 1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' "(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=') 𝑣 = 20 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 10: X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='85, x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The distributions are shown for t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='3×10−5, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='3×10−4, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='3×10−3, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='3×10−2, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='3 × 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Top: v = 20, bottom: v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For v = 1, the distributions never reach an asymptotic form that is centered on x0 = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' the attractor and specific initial condition, for two values of the advective velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Following a δ-function initial condition, there is a quick diffusive spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' While this spread is happening, the center of the distribution is also advected towards the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Note that in the v = 1 case, the average position of particles reaches 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' On the other hand, for the case of stronger at v = 20, the center of the distribution reaches the attractor at x = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' At v = 20 we begin to see the emergence of large- time asymptotic profile centered on the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' At large times, the distribution reaches a stationary limit- ing form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' As this profile develops, diffusive spread of the density profile is followed by a contraction, as particles congregate around the attractor and σx decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' At t ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='06 the width stops evolving, and the cusp-shaped profile is established in the vicinity of the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' After that, the probability to remain in the domain continues to decrease (the area under the curve will continue to decrease), although the shape of the profile remains sta- tionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We will call this limiting profile the large-time distribution or the limiting distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The width of this cusp-shaped limiting distribution decreases with in- 80 60 40 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='080 60 40 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='010 creasing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' At lower v, the width also saturates to a con- stant value at large times, and the limiting distribution also emerges, but it is not centered on the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Thus, the picture is this: the attractor captures some particles and pulls them in to its vicinity at larger v, whereas at lower v, most of the particles escape be- fore this happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The decay rate also decreases - as v grows ever larger, the large-time limiting profile localized around the attractor will decay ever slower, its rate of de- cay decreasing exponentially with v (this is for sufficiently large v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' it is an asymptotic scaling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In this large v regime, the profile that develops after an initial rapid re- laxation may be called quasistationary - as it decays on a time scale much smaller than all other time scales in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This is the regime of rare events, and we now discuss the scaling of MFPT and escape probabilities in this limiting regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 x0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='4 �r (𝑎) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 x0 50000 100000 150000 200000 250000 �r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 x0 50000 100000 150000 200000 250000 � 𝑝 𝜏 𝜏 (𝑏) (𝑐) (𝑑) Right Left 𝑝 𝜏 𝜏 𝑝 𝜏 𝜏 𝑝 𝜏 𝜏 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 11: Escape probability and MFPT through both ends versus the location x0 of the IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The attractor is located at X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (a) v = 5, (b) v = 10, (c) v = 20, (d) v = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The aberrations at the edge are numerical artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Scaling of MFPT in the rare event limit In this regime, various functions of x0 - such as the escape probability and escape time - develop character- istic distinctions between a boundary layer and interior C11 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' As v increases, the MFPT to exit increases, and eventually this time be- comes much larger than all the other characteristic time scales of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In this large v regime, escape be- comes a rare event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Starting from an initial condition x0, a particle will, with overwhelming probability drift to- wards the fixed point, and fluctuate around it for a time that scales exponentially with v as stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' There- fore, the initial condition will be forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This effect is manifested in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 11 by distinct plateaus, that show the absence of dependence on x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We show the compar- ison between such analytical predictions and simulation results of the one-layer regime in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Escape rates in these plateaus will follow the usual Arrhenius scaling 1/τ ∼ e−∆Ueff /D in physical units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The effective barrier to escape to the left will be vX = ∆Ul and the effective barrier to escape to the right will be v(1 − X) = ∆Ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' A small difference between X and (1−X) will be exponentially amplified by large v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Thus, for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5 < X < 1, the dominant factor will be v(1 − X), and therefore, τ ∼ ev(1−X)/D, or in dimensionless units, simply τ ∼ ev(1−X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (14) A more detailed analysis [27] predicts the prefactor as well, so the asymptotic expression (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' in the rare event regime) is given by τ = 4v−2ev(1−X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' One comment regarding MFPT results is in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We notice that the overall MFPT τ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 11 is ≈ 2 times smaller than the a = b → ∞ limit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 4 (see v = 10 and v = 20 graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' While a small difference is due to slightly different X (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='51 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 11 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 4), the main reason for this difference is that in the two- layer problem, the advection and diffusion take turns, while they take place simulataneously in the two-layer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Thus, all timescales are slowed down by exactly a factor of two in the two-layer model than its truly one- layer equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In other words, to make the proper comparison, we must multiply the one layer result by 2 to match the a = b → ∞ limit of the two-layer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Small asymmetry leads to a large bias in the exit location One prominent feature of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 11 is the amplification in the asymmetry in results (for example pl and pr - proba- bilities to escape through the left and right ends respec- tively) due to a small asymmetry in the placement of the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Note that pr = ae−∆Ur and pl = ae−∆Ul, where a is some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We can find this constant from the fact that pr + pl = 1 (a particle definitely exists through one of the two ends eventually).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Thus, a = � e−∆Ur + e−∆Ul�−1, altogether giving pr − pl = tanh [(X − 1/2)v] (15) We overlay this prediction on top of ∆p obtained from the analytic results (depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 11) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 12 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' ∆" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' ∆" (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 12: (a) ∆p vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Top (blue): X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='55, bottom (or- ange): X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Dots - full theory, solid curves - Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (b) ∆p vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' X, given by Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Top (orange): v = 20, bottom (blue): v = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Conclusion In this paper, we looked at a one-dimensional model of intracellular transport via a combination of advection on microtubules and diffusion in the cytoplasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This one- dimensional model was motivated by a scenario involv- ing an attractor in the interior of the cell - for example, MTOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' There are other situations where attractors may arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Consider, the β cell example from the Introduc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here motors transport insulin granules along MTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Due to orientational disorder [29], several MTs can meet with ends of the same polarity facing each other, forming an aster-like morphological trap (or attractor) for mo- tors that would all congregate at this junction [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' It is meaningful to talk about the domain of attraction of such a trap in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' A molecular motor that attaches to a MT anywhere within this domain will be taken towards the attractor, while a motor that at- taches to a mirotubule outside of the domain has a non- zero probability to be taken away from the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' When placed inside such a domain - where advective motion along microtubules tends to only attract particles - they can nevertheless escape the domain of attraction of the attractor by desorbing from MTs and diffusing within the cytoplasm until they end up outside of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2 10 20 30 40 50 601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='012 Naturally, a question about the time to be stuck in the vicinity of the attractor arises - along with the question of how formation of such traps affects the functioning of the cell and the overall transport of insulin granules across it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Using our one-dimensional model, We calculated es- cape probability through each end, pl(x0) and pr(x0), and overall p(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We also calculated the mean first pas- sage time (MFPT) to escape the domain through each end, τl(x0) and τr(x0), and overall τ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The initial lo- cation inside the cell is determined by the organelles pro- ducing the cargo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The other parameters in the problem were the dimensionless location of the attractor toward which the advective motion is directed, and the dimen- sionless advective velocity v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In situations like these, when there is either orienta- tional or polarity disorder, we can think of cells as being divided into domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We made several predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' When the attractor is placed symmetrically and a and b are finite, there is a crossover between τ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1 - diffusive timescale to τ that grows exponentially in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The range of a = b over which this crossover happens is wide - a couple of orders of magnitude, but it corresponds to the values of a and b actually found in cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This served as our first example of “fine-tuning” that allows cells to achieve the biggest change in the functionality with the smallest change in parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For a = b significantly below the crossover, a particle that was released into the diffusive layer has a chance to escape the domain purely diffusively without ever visiting the AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For a = b around the crossover value, the proba- bility of this goes to zero - every particle will be advected towards the attractor for at least some of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For a = b significantly above the crossover, the transport en- ters the effective one-layer regime and exhibits rare event physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Asymmetric placement of the attractor gives rise to an interesting phenomenon of an optimal coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Thus, we found that it is possible to minimize the residence time in the domain by increasing the coupling, because that will lower the diffusive spread, and bring particles close to one end of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We discussed the effective one-layer regime that re- sults at sufficiently large couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We also discussed rare event physics that happens at large dimensionless advective velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In such a rare event regime, a por- tion of particles will be localized in the vicinity of the attractor for a time exponentially long in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We pro- vide an explicit formula formula for the overall MFPT including not only the exponential part, but also the prefactor, which scales as v−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The idea of exponential sensitivity, and phenomena such as strong amplification of the preferred exit end due to a slight asymmetry is tantalizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Extrapolating this finding to two dimensions suggests that in complex, crowded environments that allow for multiple trap-like morphologies (for example, asters), the distribution of cargo around the cell will be non-homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This re- mains to be verified in the future, by extending our model two two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Our work is complementary to prior theoretical models of transport that involves a combination of diffusion and advection along microtubules [30] and [31], as neither of these sources are focusing on questions of residence time or the role of asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' To continue our current work, we would like to study models with reflecting-reflecting or absorbing-reflecting boundary conditions, or models in which the source is on one end and the target is on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Such mod- els would be better suited for transport of cargo in cilia [4], transport between the plasma membrane and Golgi apparatus [5], [6], or between Endoplasmic Reticulum and Golgi [7], [3], transport of viruses towards replication sites [8], [9], and other intracellular transport situations [3], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This work was supported by the National Sci- ence Foundation (NSF-DMS-1616926 to AG) and NSF- CREST: Center for Cellular and Bio-molecular Ma- chines at UC Merced (NSF-HRD-1547848 and 2112675 to AG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' AG and NS also acknowledge partial sup- port from the NSF Center for Engineering Mechanobi- ology grant CMMI-154857 and computing time on the Multi-Environment Computer for Exploration and Dis- covery (MERCED) cluster at UC Merced (NSF-ACI- 1429783).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' NS acknowledges Graduate Student Oppor- tunity Program Fellowship from the University of Cal- ifornia, Merced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' BR acknowledges the support of the William and Linda Cal Poly Frost fund for undergradu- ate research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 13 A: Details of the two-layer calculations We start with the full one-dimensional, two-layer model in dimensionless form (primes have been omitted for clarity): ∂ρ ∂t = − ∂ ∂x (v(x)ρ) + aθ − bρ (16) ∂θ ∂t = −aθ + bρ + ∂2θ ∂x2 (17) Here a and b are respectively the rates of adsorption to and desorption from microtubules, v is the dimensionless velocity profile, ρ is the density of particles on microtubules, and θ is the density of particles diffusing in the cytoplasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We seek modal solutions (or eigensolutions) of the form � ρ θ � = � R(x) Θ(x) � e−σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (18) The vector � R(x) Θ(x) � is an eigenvector of the operator (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (11) of the text) that represents minus the right hand side of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (16)-(17), and σ is an eigenvalue of this operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' However, due to the mass accumulation at the attractor, we must also include a δ-function term to accommodate for this mathematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The mass will not accumulate at the junction point due to the diffusive term that acts on the diffusive layer density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Note also that the δ- function in the advective layer acts like a point source for the diffusive layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' When we study a simple diffusive problem with a δ-function source plus absorbing boundaries, and seek a steady-state (time independent) solution, the density profile does not acquire a δ-function response - the diffusion acts infinitely quickly to dissipate such a singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' With this in mind, we must augment the above formula to � ρ θ � = � R(x) Θ(x) � e−σt + � 1 0 � � M0e−σt� δ(x − X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (19) Substituting this back to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (16)-(17), and setting Q = dΘ dx , we get d dx � � R Q Θ � � = � � (− b v + σ v ) 0 a v −b 0 (a − σ) 0 1 0 � � � � R Q Θ � � (20) for 0 ≤ x < X (call it Region-I) and d dx � � R Q Θ � � = � � ( b v − σ v ) 0 − a v −b 0 (a − σ) 0 1 0 � � � � R Q Θ � � , (21) for X < x ≤ 1 (call it Region-II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The solutions, will take the form: � � RI QI ΘI � � = A � � v1 R v1 Q v1 Θ � � eλ1x + B � � v2 R v1 Q v2 Θ � � eλ2x + C � � v3 R v3 Q v3 Θ � � eλ3x (22) in Region-I and � � RII QII ΘII � � = D � � w1 R w1 Q w1 Θ � � eµ1x + E � � w2 R w1 Q w2 Θ � � eµ2x + F � � w3 R w3 Q w3 Θ � � eµ3x, (23) in Region-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The ⃗vs and λs are eigenvectors and eigenvalues of the matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (20), while ⃗ws and µs are eigenvectors and eigenvalues of the matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The λs satisfy the equation −λ3 + �σ − b v � λ2 + (a − σ)λ + σ2 − σ(a + b) v = 0, (24) 14 and the µs satisfy the equation −µ3 − �σ − b v � µ2 + (a − σ)µ − σ2 − σ(a + b) v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (25) The eigenvectors have the structure ⃗v = � � −λ2+a−σ b λ 1 � � , (26) and ⃗w = � � −µ2+a−σ b µ 1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (27) The functions on both sides of the attractor are different, and they need to be stitched correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The stitching is determined by the boundary conditions, so we now discuss these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The boundary conditions will determine the eigenvalues σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We note that there are seven unknowns: coefficients A - F (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (22)-(23)), and the mass growth rate M0 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (19), so we need seven constraints (or conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' First, there are absorbing boundary conditions at each end, which require that R(x = 0) = Θ(x = 0) = 0 and R(x = 1) = Θ(x = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The additional three conditions come from the location of stitching, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' the attractor location at x = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The diffusive layer density must be continuous to avoid infinite currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Thus, ΘI(X) = ΘII(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The remaining two boundary conditions come from mass conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' To extract these, we integrate Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (16)-(17) through the junction point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' from x − ϵ to x + ϵ for arbitrarily small ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Performing this on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (16) gives −σM0 = −bM0 − (vIIRII(X) − vIRI(X)) = −bM0 + v (RII(X) + RI(X)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (28) Note that the temporal terms would not be absent if the δ-function component of ρ was not proportional to e−σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This equation says that the rate of growth of the advective layer mass at x = X (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' of the strength of the δ-function) is driven by the inflow from this layer, and outflow into the diffusive layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Performing the integration on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (5) gives bM0 = dΘI dx ���� x=X − dΘII dx ���� x=X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (29) This equation says that any difference in the outflow rates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' different slopes of the diffusive layer density) is balanced by the inflow from the advective layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We now implement these boundary conditions algebraically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We have: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Absorbing boundary condition at x = 0 in the advective layer: RI(x = 0) = 0 A �−λ2 1 + a − σ b � + B �−λ2 2 + a − σ b � + C �−λ2 3 + a − σ b � = 0 (30) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Absorbing boundary condition at x = 0 in the diffusive layer: ΘI(x = 0) = 0 A + B + C = 0 (31) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Absorbing boundary condition at x = 1 in the advective layer: RII(x = 1) = 0 D �−µ2 1 + a − σ b � eµ1 + E �−µ2 2 + a − σ b � eµ2 + F �−µ2 3 + a − σ b � eµ3 = 0 (32) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Absorbing boundary condition at x = 1 in the diffusive layer: ΘII(x = 1) = 0 Deµ1 + Eeµ2 + Feµ3 = 0 (33) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Continuity at x = X in the diffusive layer (to prevent infinite diffusive currents): ΘI(x = X) = ΘII(x = X) Aeλ1X + Beλ2X + Ceλ3X = Deµ1X + Eeµ2X + Feµ3X (34) 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Mass conserving boundary condition in advective layer: RII(x = X) + RI(x = X) = b−σ v M0 D �−µ2 1 + a − σ b � eµ1X + E �−µ2 2 + a − σ b � eµ2X + F �−µ2 3 + a − σ b � eµ3X + A �−λ2 1 + a − σ b � eλ1X + B �−λ2 2 + a − σ b � eλ2X + C �−λ2 3 + a − σ b � eλ3X = b − σ v M0 (35) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Mass conserving boundary condition in diffusive layer: dΘI dx �� x=X − dΘII dx �� x=X = bM0 Aλ1eλ1X + Bλ2eλ2X + Cλ3eλ3X − Dµ1eµ1X − Eµ2eµ2X − Fµ3eµ3X = bM0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='(36) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='We can write all these seven equations in the compact matrix form: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1+a−σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2+a−σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='3+a−σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1+a−σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eµ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2+a−σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eµ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='3+a−σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eµ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eµ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eµ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eµ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eλ1X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eλ2X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eλ3X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−eµ1X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−eµ2X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−eµ3X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1+a−σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eλ1X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2+a−σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eλ2X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='3+a−σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eλ3X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1+a−σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eµ1X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2+a−σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eµ2X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='3+a−σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eµ3X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='σ−b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='λ1eλ1X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='λ2eλ2X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='λ3eλ3X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−µ1eµ1X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−µ2eµ2X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−µ3eµ3X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='−b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='M0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (37) Because of the structure of this equation, we see that (i) the determinant must be non-zero for a non-trivial solution and (ii) the nontrivial solution is non-unique - it has at least one degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For instance, we are free to choose one of the variables, or we are free to choose the normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Making use of this freedom, we chose to set M0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' These equations were then used to solve for the remaining coefficients A, B, C, D, E, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Thus, calling the matrix involved in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (37), M, Det(M) = 0 should provide an algebraic equation for σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Expanding determinant in terms of minors, we have 0 = bDet (m77) + �σ − b v � Det (m67) (38) where the minor mij is a matrix obtained by removing ith row and jthe column from M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Once M0 is chosen, the coefficients (A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=', F) must be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This means that both Det (m77) and Det (m67) must both be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' If Det (m77) is non-zero, then the solution (A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=', F) obtained from the first six equations can be found with the inverse of m77, and is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This implies that Det (m67) must also be non-zero (otherwise, the solution (A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=', F) obtained from the first five and the seventh equation is non-unique, leading to a contradiction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Thus, the kind of a zero of Det(M) that we want is one in which Det (m77) and Det (m67) are both non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Therefore, we’re interested in the zeros of the following quantity: Det′ = b + �σ − b v � Det (m67) Det (m77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (39) It is the zeros of this determinant that gives us σ in terms of (a, b, v, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We were primarily interested in the lowest (ground state) eigenvalue σ1, and the inverse 1/σ1 that serves as a characteristic measure of the escape time[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Because the set of eigenfunctions and eigenvalues turned out to be finite, they are of limited value in being able to construct a solution that fits the δ-function initial condition, and thereby to properly compute MFPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 16 B: One-layer theory We now discuss the computation of the eigenfunctions p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The subscript n will be dropped to lighten the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Recall that 0 < x < X is Region-I, and that X < x < 1 is Region-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The eignfunctions satisfy σp = −v dp dx + d2p dx2 (40) in Region-I, and σp = v dp dx + d2p dx2 (41) in Region-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The solution in Region-I is pI = aIeλ+x + bIeλ−x, where the λs satisfy λ± = v ± √ v2 + 4σ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (42) The solution in Region-II is pII = aIIeµ+x + bIIeµ−x, where the µs satisfy µ± = −v ± √ v2 + 4σ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (43) The coefficients a and b will be fixed with the following four boundary conditions (BCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The first two are the absorbing BCs at the ends, pI(0) = pII(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The third boundary condition is the continuity of the solution pI(X) = pII(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' A discontinuous solution is unphysical due to the diffusion term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In a one-layer theory, there will not be an accumulation of mass at the trap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' there will be no term like δ(x − X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Any such density would be immediately smoothed out by the action of the diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Note that in the full, two-layer theory, such term existed only in the advective layer, but not in the diffusive layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In the absence of a δ-function-like accumulation of mass at x = X, the currents across x = X will be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This gives us the fourth boundary condition that enforces the continuity of currents at the junction: vpI(X) − dpI dx ��� x=X = −vpII(X) − dpII dx ��� x=X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='Applying these four boundary conditions leads to four equations: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='aI + bI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='= 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='(44) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='aIIeµ+ + bIIeµ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='= 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='(45) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='aIeλ+X + bIeλ−X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='= aIIeµ+X + bIIeµ−X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='(46) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='v � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='aIeλ+X + bIeλ−X� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='− � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='λ+aIeλ+X + λ−bIeλ−X� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='= −v � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='aIIeµ+X + bIIeµ−X� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='− � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='µ+aIIeµ+X + µ−bIIeµ−X� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='(47) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='Substituting the first two into the last two gives ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='aI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eλ+X − eλ−X� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='= aII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eµ+X − eµ+−µ−eµ−X� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='vaI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eλ+X − eλ−X� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='− aI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='λ+eλ+X − λ−eλ−X� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='= −vaII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='eµ+X − eµ+−µ−eµ−X� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='− aII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='µ+eµ+X − µ−eµ+−µ−eµ−X� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='Using the first of these,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' and substituting into the second we obtain v � eλ+X − eλ−X� −� λ+eλ+X − λ−eλ−X� −� −v � eµ+X − eµ+−µ−eµ−X� − � µ+eµ+X − µ−eµ+−µ−eµ−X��� eλ+X − eλ−X eµ+X − eµ+−µ−eµ−X � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (48) where λs and µs are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (42) and (43) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (48) is an equation for eigenvalues σ as a function of v and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Moreover, pI = � eλ+x − eλ−x� , (49) and pII = � eλ+X − eλ−X eµ+X − eµ+−µ−eµ−X � � eµ+x − eµ+−µ−eµ−x� (50) The modes given this way are not normalized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' they will be normalized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We will see below that eigenvalues turn out to be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 17 The coefficients cn are determined as usual by the initial condition, P(x, t = 0) = � n cnpn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Because the operator O is non-Hermitian, eigenfunctions are generally non-orthogonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' � 1 0 p∗ n(x)pm(x) dx ̸= 0, so we can’t compute cm with the help of an inner product � 1 0 P(x, 0)pm(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' However, eigenfunctions of the adjoint operator O† have the property that they are either orthogonal to the eigenfunctions of O, or otherwise have eigenvalues that are complex conjugates of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Therefore, in order to be able to express initial conditions, we need to compute a set of eigenfunctions and eigenvalues of O†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Even after this, there is no guarantee that we will be able to express any initial condition, because there’s also no guarantee of completeness, due to operators being non-Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The adjoint of O is given by O† = v(x) d dx + d2 dx2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (51) To find the eigenfunctions of this operator, it helps to look back at the original equation with operator O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We note that both Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (40)-(41) can be written compactly as one equation σp = d dx �dU dx p + d2p dx2 � , (52) where the potential (in analogy with physics) U is given by U(x) = � v(X − x) x ≤ X, v(x − X) x ≥ X, (53) or, more compactly, v(x) = − dU dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Now, let p = q(x)e−U(x) - we can always do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Substituting this ansatz we find that q(x) obeys σ1q = −dU dx dq dx + d2q dx2 = v(x) dq dx + d2q dx2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (54) That is, q = p(x)eU(x) is the eigenfunction of the adjoint operator that we were seeking!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Moreover, it has the same eigenvalue as the operator O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The modes given this way are not normalized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' they will be normalized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For operators with a finite dimensional eigenspace, eigenvalues of an adjoint operator O† are complex conjugates of the eigenvalues of the operator O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In such cases, equality of the two sets of eigenvalues implies that they are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In our case the eigenspace is not guaranteed to be finite (in fact, we hope that it isn’t, if there is any chance at completeness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' However, our numerical investigation revealed that eigenvalues σ are always real (and negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Next, we give an example of the result of several hundred low-lying eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The first example is for X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='85 and v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We observe an interesting feature that eigenvalues appear in groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The second example is for X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 20 40 60 80 100Mode 100000 80000 60000 40000 20000 σ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 13: Lowest 100 eigenvalues for X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='85 and v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' and v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We notice that the size of groups has changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' There is no obvious relation between X the the size of 18 20 40 60 80 100Mode 150000 100000 50000 σ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 14: Lowest 100 eigenvalues for X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 and v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' groups - for instance, for X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='55 the groups are again increased in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We verified numerically the orthogonality of several eigenfunctions belonging to different eigenvalues, and found it to hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Eigenfunctions were also normalized by multiplying by the following factors: ap = 1 �� 1 0 p∗n(x)pn(x) , Aq = 1 �� 1 0 q∗n(x)qn(x) , where ps and qs are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (49)-(50) and q = p(x)eU(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The resultant modes came out to be either purely real or purely imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In this latter case, they can be made real by multiplying by −i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The following are examples of eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The discontinuity in the slope of ps - but not of qs - is clearly visible 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑞"# 𝑝"# FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 15: The first and the twentieth modes for X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='85, v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 Re[p1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5 Re[q1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 x 3 2 1 1 2 3 Im[p20] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='0 x 3 2 1 1 2 3 4 Im[q20] Mode 1 Mode 20 𝑞!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑝!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑞"# 𝑝"# FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 16: The first and the twentieth modes for X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='85, v = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' in the first mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We can understand this by substituting the form p(x) = q(x)e−U(x) into the fourth boundary condition on p (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' vpI(X) − dpI dx ��� x=X = −vpII(X) − dpII dx ��� x=X), and find that dq dx is continuous across the junction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' qI dx �� x=X = dqII dx ��� x=X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The other three boundary conditions for q are the same as for p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' With all this information, we conclude that the set of functions {qn} is then sufficient for us to be able to find the coefficients cn in the series P(x, t) = � n cnpn(x)eσnt - as long as there is completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The coefficients are given by cn = � 1 0 P(x, t = 0)q∗ n(x) dx � 1 0 q∗n(x)pn(x) dx (55) Completeness is not guaranteed, but unlike the two-layer case, we found that the method works, provided enough modes are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We will not discuss convergence properies of the series here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In relation to the mean first passage time problem, we are interested in the δ-function initial condition, P(x, t = 0) = δ(x − x0), in which case the coefficients are given by cn = q∗ n(x0) � 1 0 q∗n(x)pn(x) dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' (56) 20 C: Trajectory examples Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 4 and the subsequent discussion in our main text discussed several regimes of MFPT, depending on the value of a = b, for symmetric trap placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We now show trajectories in each of those regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' First, we show trajectories in the plateau regime that precedes the second crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This takes place for a roughly in the range [10−2, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1 𝑎 = 1 𝑎 = 10 Time Time FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 17: Nine trajectories at lower as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' All particles are placed initially at x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5 on the DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5 and v = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The right panels show a smaller window of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' We can clearly see that as a increases, thee probability of switching into the AL increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Once a particle switches to the AL, it will move towards the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' As a increases further, the likelihood of the advective motion towards the attractor all in one ride on the AL decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Instead, a typical particle will experience episodes of a little bit of advective motion, followed by a little bit of diffusive motion, and so on - see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This happens in the second crossover regime that begins for a ≈ 10 and continues for several decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='.21 Time Time 𝑎 = 100 𝑎 = 1000 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 18: Nine trajectories at intermediate as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' All particles are placed initially at x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5, v = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The right panels show a smaller window of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' For a even larger - the system enters the second plateau, when any further increase in a does not increase MFPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' This means that the system behaves in accordance to the one-layer model [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The the episodes of diffusion and advection become even shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Trajectories in such a regime are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 19, for progressively narrower windows of time, from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 22 𝑎 = 𝑏 = 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑎 = 𝑏 = 10" 𝑎 = 𝑏 = 10# Time (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=') Time (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=') Time (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=') Time (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=') FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 19: Trajectories for a between 104 to 106 in powers of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here again X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5 and v = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Leftmost column has 10 trajectories, while the other columns show one trajectory for progressively narrower windows of time, from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' In these right three columns, the red color indicates advective portions of trajectories, while grey are diffusive portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='70 1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='55 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='40 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='30 0 2 4 6 8 10 Time (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' )23 D: Theory and simulation comparison - one-layer limit In this section we show the comparison between the one-layer analytical predictions of pl, pr, τl, τr, and τ with results of simulations of the two-layer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 𝑥 𝑥 𝑥 𝑝 𝜏 𝜏 Right Left FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 20: Comparison between analytical quantities (open circles) and simulation results (filled circles - Monte Carlo simulation as described in the main paper, filled triangles - forward flux sampling algorithm [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Left column: probabilities to escape through the left end (blue) pl and right end (orange) pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Middle column: escape time conditioned on the left exit (blue) τl and right exit (orange) τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Right column: net MFPT τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The growing discrepancy between simulation and analytical results is due to the diffusive approximation of the latter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' the details will be discussed in the coming publication [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Here X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Top row: v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='1, middle row v = 5, bottom row v = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content='24 [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Howard and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Clark, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 55, B39 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Ross, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Ali, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Warshaw, Current Opin- ion in Cell Biology 20, 41 (2008), ISSN 0955-0674, cell structure and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Mogre, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Brown, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Koslover, Physical Biology 17, 061003 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' Chien, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' 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+page_content=' ten Wolde, The Journal of chemical physics 124, 024102 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' [33] This is especially true in the rare event regime that devel- ops at sufficiently large v - when σ1 should be separated from the rest of σs by a gap that grows exponentially in v - while there is no such gap between the rest of the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' [34] This is not what makes escape events rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} +page_content=' The signa- ture of the rarity of escape events (that is MFPT is much greater than all other time scales) is the exponen- tial growth of MFPT with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AzT4oBgHgl3EQfUPzF/content/2301.01264v1.pdf'} diff --git a/E9FRT4oBgHgl3EQfBzcS/content/tmp_files/2301.13466v1.pdf.txt b/E9FRT4oBgHgl3EQfBzcS/content/tmp_files/2301.13466v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..97e2e00475c3f5b727674918a8f42f1a60dd6224 --- /dev/null +++ b/E9FRT4oBgHgl3EQfBzcS/content/tmp_files/2301.13466v1.pdf.txt @@ -0,0 +1,1139 @@ +First passage time statistics of non-Markovian random walker: Onsager’s +regression hypothesis approach +Yuta Sakamoto and Takahiro Sakaue∗ +Department of Physical Sciences, Aoyama Gakuin University, +5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Japan +First passage time plays a fundamental role in dynamical characterization of stochastic processes. +Crucially, our current understanding on the problem is almost entirely relies on the theoretical +formulations, which assume the processes under consideration are Markovian, despite abundant non- +Markovian dynamics found in complex systems. Here we introduce a simple and physically appealing +analytical framework to calculate the first passage time statistics of non-Markovian walkers grounded +in a fundamental principle of nonequilibrium statistical physics that connects the fluctuations in +stochastic system to the macroscopic law of relaxation. Pinpointing a crucial role of the memory +in the first passage time statistics, our approach not only allows us to confirm the non-trivial +scaling conjectures for fractional Brownian motion, but also provides a formula of the first passage +time distribution in the entire time scale, and establish the quantitative description of the position +probability distribution of non-Markovian walkers in the presence of absorbing boundary. +How long does it take for a random walker to reach +a destination? Such a question on the first passage +time (FPT) is relevant to a broad range of situa- +tions in science, technology and every-day life applica- +tions as encountered, for instance, in diffusion-limited +reactions [1–3], barrier crossing [4–7], target search +processes [8, 9], cyclization of DNA molecule [10– +13], price fluctuation in market [2] and spread of dis- +eases [14]. Today, the concept of the FPT and its im- +portance in the study of stochastic processes are well +recognized, and theoretical methods for its computa- +tion are standardized [1, 2]. However, most of them +are devised for Markovian random walkers, whose de- +cision making does not depend on its past history, thus +not applicable to non-Markovian walkers despite their +ubiquitousness. +Indeed, a growing body of evidence suggests that +the non-Markovian dynamics is found quite gener- +ally in rheologically complex matters typically, but +not exclusively, with viscoelastic properties. Classi- +cal examples are found in the diffusion of interact- +ing particles in narrow channels [15] and the motion +of tagged monomers in long polymer chain [16, 17]. +Other notable representatives include colloidal parti- +cles in polymer solutions [18] or nematic solvents [19], +lipids molecules and cholesterols in cellular mem- +brane [20], proteins in crowded media [21], and chro- +mosome loci [22] as well as membraneless organelles +in living cells [23]. Such systems commonly exhibit a +slow dynamics in the form of sub-diffusion MSD(t) ∼ +tα characterized by the anomalous exponent α < 1, +where MSD(t) stands for the mean-square displace- +ment of the observed particle during the time scale t +. Here the physical mechanism at work is the inter- +action of observed degree of freedom with the collec- +tive modes with broad range of time scales underly- +ing complex environment. Because of its importance +in e.g. +intracellular transport, the theoretical tools +to describe/diagnose such anomalous diffusion phe- +nomenology have been well developed in the last few +decades [24]. +However, most of them rely on MSD +and related quantities, while much less attention has +been paid to the FPT, despite its fundamental and +practical importance to characterize the underlying +stochastic process. This is particularly true for sys- +tems possessing memory, as nontrivial information on +the history dependence of the system is encoded in +the FPT statistics [25]. It has long been known that +the anomalous transport properties affect the rates +of chemical and biochemical reactions [26], and such +reactions are initiated by the encounter of reactant +molecules, so precisely quantified by means of the FTP +statistics. +Unfortunately, our current understanding on the +FPT of non-Markovian walker lags far behind that +of Markovian counterpart, where the difficulty is +largely associated to the lack of appropriate theoret- +ical foothold [25, 27, 28]. +While the Fokker-Planck +equation and its related methods play a key role to +analyze the time evolution of the probability distri- +bution of the Markovian walkers, their careless ap- +plication is problematic for walkers with memory, a +defining property of the non-Markovian process. At +𝑡 = 𝜏 +(a) +(b) +𝑡 = 𝜏 +𝜏 +FIG. 1. Regression hypothesis applied to non-Markovian +walkers. +(a) Example trajectory of fBM with α = 0.5 +starting from the initial position x = x0. Before (after) +the first hitting on absorbing boundary at x = 0, the +trajectory is drawn by solid (dotted) curve. +First pas- +sage event can be viewed as a large fluctuation to create +a non-equilibrium state at t = τ. (b) After the first pas- +sage (t > τ), the process follows, on average, the macro- +scopic relaxation law, for sub-diffusive fBM, represented +by the harmonic restoring force, whose spring constant +gets smaller algebraically in longer time scales. +arXiv:2301.13466v1 [cond-mat.stat-mech] 31 Jan 2023 + +Absorbing wall +Time +0 +0 +o +PositionAbsorbing wall +Potential +0 +o +Position2 +present, available results are quite limited with no- +table examples being the perturbative and scaling ar- +guments to estimate the asymptotic exponents charac- +terizing the distribution of FPT and related quantities +in unbounded domain [25, 29–31], some approxima- +tion schemes to calculate the mean FPT of polymer +looping process [3, 10–13], and more recent analytical +treatment to compute the mean FPT in confined do- +mains [28]. However, neither of the full distribution of +FPT or position distribution of non-Markovian walk- +ers in the presence of boundary are available, making +the computation of these quantities in non-Markovian +processes fundamental challenge. +In this Letter, we provide a simple and physically +appealing method to calculate the FPT statistics of +non-Markovian walkers by identifying the moment of +first passage (t = τ) as an initial condition for the re- +laxation process afterwards (t > τ), see Fig. 1. Our +argument is thus rooted in a non-Markovian exten- +sion of the regression hypothesis of Onsager, a corner +stone for the development in the nonequilibrium sta- +tistical physics [32]. We obtain an exact integral equa- +tion for the FPT distribution, the analysis of which +yields, in addition to its asymptotic decay exponent, +full functional form in leading order over entire time +scales, and also the walker’s probability distribution +function. +Importantly, our formalism allows one to +unveil how and why the textbook standard “method +of image” [2, 33] breaks down by pinpointing the role +of memory built up during the first passage process. +Here we focus on the sub-diffusive fractional Brownian +motion [34] (fBM with α < 1), an important class of +non-Markovian walkers found in widespread complex +systems including living cells and nuclei [20–23]. +FIG. 2. Illustration of the method of image. For Marko- +vian walkers (α += +1), Q(x, t; 1) can be constructed +by the method of image. +Integrating Eqs. (2) over +the entire space (including negative domain), one finds +S(t; 1) = 1 − +� ∞ +−∞ Q(x, t; 1)dx, where the surviving proba- +bility S(t; 1) = +� ∞ +0 +P+(x, t; 1)dx is denoted by the hatched +area. Equivalent to the above relation is +� ∞ +0 +Q(x, t; 1)dx = +(1−S(t; 1))/2 thanks to the reversal symmetry of Q(x, t; 1) +with respect to x = 0, producing a factor 1/2. The same +relation is obtained by integrating Eq. (3) over the positive +x domain with ⟨x(t)⟩FPT=τ = 0. +Generalized Langevin equation and power-law mem- +ory kernel +– As a paradigm, consider a random +walker in one dimensional half space with an absorb- +ing boundary at origin. A walker is initially positioned +at x = x0(> 0) at t = 0, and evolves according to the +following generalized Langevin equation: +dx(t) +dt += +� t +0 +µ(t − t′)f(t′)dt′ + η(t) +(1) +where +f(t) +and +η(t) +are, +respectively, +a +time- +dependent external force and the noise acting on the +walker [17]. +The latter is assumed to be Gaussian +with zero mean and its auto-correlation is related to +the mobility kernel via the fluctuation-dissipation re- +lation ⟨η(t)η(t′)⟩ = Tµ(|t − t′|) with T being the +noise strength. The memory effect is encoded in µ(t), +for which we assume for large t the power-law de- +cay µ(t) ≃ −T −1Dαtα−2 (0 < α < 1) in addition +to instantaneous response µ(t) = 2γ−1δ(t) at short +time, where γ is a bare friction coefficient. Finally, +we require on physical ground +� ∞ +0 +µ(t)dt = 0 such +that Eq. (1) describes the sub-diffusive fBM with the +MSD exponent α. +This sum rule is a consequence +of the relaxation nature of the sub-diffusive fBM, +which is caused by the visco-elastic effect [17]. For +a free walker (f = 0) in free space (no boundrary), +its position probability distribution P(x, t; x0) is sim- +ply given by N(x, x0, 2Dαtα), where N(x, A, B) = +(2πB)−1/2e(x−A)2/2B denotes Gaussian distribution +of x with the average A and the variance B. +Process after first-passage – We now set a stage by +introducing an absorbing boundary at the origin x = 0 +such that the walker performs fBM in half space x > 0 +with the same initial condition as before. Using the +free space propagator P(x, t; x0), the walker’s position +probability P+(x, t; x0) is now represented as +P+(x, t; x0) = P(x, t; x0) − Q(x, t; x0) +(2) +where Q(x, t; x0) is the position distribution of dead +walker, who touched the absorbing boundary by this +moment. +Note that while one usually looks at the +walker’s behavior in physical domain (x ≥ 0) up to the +absorption (t ≤ τ) in the context of FPT, Eq. (2) holds +in entire space and time domains in a sprit similar to +[28]; the absorbing boundary at x = 0 necessitates +P(x, t; x0) = Q(x, t; x0) for x ≤ 0. Using the FPT +distribution F(τ; x0), Q(x, t; x0) is represented as +Q(x, t; x0) = +� t +0 +F(τ; x0) P(x, t; x0|FPT = τ)dτ (3) +where P(x, t; x0|FPT = τ) is the conditional proba- +bility of the walker’s position at time t after its first +passage at time τ. Being the Gaussian process, one +expects the form +P(x, t; x0|FPT = τ) = N(x, ⟨x(t)⟩FPT=τ, 2Dα(t − τ)α) +. +(4) +In the absence of memory effect, ⟨x(t)⟩FPT=τ = 0 ir- +respective of the starting position x0. Then, by not- +ing +� t +0 F(t′; x0)dt′ = 1 − S(t; x0), integrating Eq. (2) +over half space leads to a classical result of the +survival probability S(t; x0) ≡ +� ∞ +0 +P+(x, t; x0)dx = +erf(x0/√4D1t) for Markovian case, see Fig. 2. +Al- +though not applicable to non-Markovian walker, the + +0.5 +Q(c, t; 1) +t=1 +P(c, t; 1) +0.4 +P(c,t; - 1) +0.3 +0.2 +0.1 +0.0 +-3 +-2 +-1 +0 +1 +2 +33 +above calculation highlights ⟨x(t)⟩FPT=τ, which gen- +erally depends on x0, as a central quantity to account +for the memory effect in the first passage statistics. +History-dependent relaxation: regression hypothesis +view – A key idea to quantify ⟨x(t)⟩FPT=τ comes from +the fundamental connection between fluctuation and +response in nonequilibrium statistical physics. In his +seminal paper, Onsager pointed out that the decay of +mesoscopic fluctuations follow, on average, the macro- +scopic law of relaxation [32]. Applying this so-called +regression hypothesis to our problem, we view the pro- +cess after the first passage t > τ as a relaxation pro- +cess, whose “initial” condition x(τ) = 0 can be pre- +pared either naturally (by fluctuation) or artificially +(by external force), see Fig. 1. In the latter scenario, +we take the sub-ensemble of walkers whose FPT is τ, +and describe their average time evolution using Eq. (1) +with the constant force f(t) = f0 for t < τ. This leads +to +⟨ ˙x(t)⟩FPT=τ = f0 +� t +0 +µ(t′)dt′ +(t < τ) +(5) +then, identifying ⟨ ˙x(τ)⟩FPT=τ ≃ −x0/τ, we find +f0 ≃ −Tx0 +Dα +τ −α. +(6) +Now the desired non-equilibrium state is prepared at +t = τ, at which we switch off the force. +The sub- +sequent relaxation is described, again using Eq. (1), +by +⟨ ˙x(t)⟩FPT=τ = f0 +� t +t−τ +µ(t′)dt′, +(t > τ) +(7) +whose +integral +with +respect +to +time +leads +to +⟨x(t)⟩FPT=τ, where a numerical coefficient implicit in +Eq. (6) is fixed by requiring ⟨x(t)⟩FPT=τ → x0 for +t/τ ≫ 1 as a consequence of the sum rule. Collecting +all together, our analytical formulation is summarized +as the following integral equation [35]: +1 − erf +� +1 +√ +2tα +� += +� t +0 +F(τ; 1) [1 − erf(h(t, τ))] dτ +(8) +with the memory function +h(t, τ) = +1 +� +2(t − τ)α +� +1 + +� t +τ − 1 +�α +− +� t +τ +�α� +.(9) +From here onwards, we measure the length and the +time in unit of x0 and τx0 = (x2 +0/2Dα)1/α, respec- +tively, which are the sole characteristic length and +time scales in the problem, making the initial posi- +tion x0 = 1 upon rescaling. +First passage time distribution – We now determine +the leading order solution of Eq. (8) in the form +F(τ; 1) = Cα exp +� +− +� 1 +2τ +�ω� +τ −(1+p) +(10) +where Cα is a normalization constant. This function, +a generalization of the Markovian result [2] ω = 1, +(a) +(b) +FIG. 3. +FPT distribution of non-Markovian walk- +ers. +(a) FPT distribution F(τ; 1) for sub-diffusive fBM +(α = 0.8, 0.5). Inset shows the double logarithmic plot of +large τ regime, where the asymptotic slope p+1 = 2−α/2 +is clearly visible. The data for α = 0.8 is shifted downward +(×10−2) for visual clarity. Both in main panel and inset, +symbols represent simulation results and the curves corre- +spond to the analytical formula (10) with p = 1−α/2 and +ω given by Eq. (11). The error bars represent 95 % CI. +(b) Exponent ω as a function of α, which characterizes the +early time regime in FPT distribution. Blue solid circles +are obtained by fitting the numerical simulation data for +several α values (two of them shown in Fig. 2(a)) with the +formula (10). Fitting these data with Eq. (11) fixes the +parameter c1 = 0.1. +p = 1/2, exhibits a peak at τ = τ ∗ = (1/2)(ω/(1 + +p))1/ω and develops a power-law tail F(τ; 1) ∼ τ −(1+p) +at τ ≫ τ ∗. With this in mind, we plug the ansatz +(10) into Eq. (8) and perform the asymptotic analysis, +which yields p = 1 − α/2 in agreement with previous +scaling argument [25, 29]. In addition, our formulation +allows us to obtain the exponent ω, which satisfies the +relation +(2 − α)2ω(2 + α)α +(2ω)α += +�3 +2 +�ω +cω(α−1) +1 +(11) +with a numerical constant c1 of order unity [35]. +In Fig. 3 , we compare our analytical formula for +F(τ; 1) with the results obtained from numerical sim- +ulation [35]. +As shown, the agreement is excellent, +encompassing the short time singularity to the peak, +and the eventual long time power-law tail, which are +characterized by the exponents ω and p, respectively. +The peak position τ ∗ is rather sensitive to the value +of ω. This is particularly true for small ω, which is +the case for the small α, shifting the peak position τ ∗ +vanishingly small in the limit α → 0. +Probability distributions of dead and survived walk- +ers – We are now in a position to take a close look +at Q(x, t; 1) that is the distribution of walkers af- +ter their first passage. +From Eqs. (3) and (4), we +immediately find the memory effect in the form of +restoring force represented by nonzero ⟨x(t)⟩FPT=τ +breaks the reversal symmetry with respect to x = 0, +i.e., Q(x, t; 1) ̸= Q(−x, t; 1) that clearly manifests the +breakdown of the image method (Figs. 2, 4) [35]. The +value of ⟨x(t)⟩FPT=τ corresponds to the peak position +of P(x, t; 1|FPT = τ), which is zero initially (t = τ), +and slowly evolves with time towards x = x0. Such +a distribution P(x, t; 1|FPT = τ) characterizes the + +4 +O +α= 0.5 +0 +0 +α= 0.8 +3 +F(T) +2 +1 +0 +0 +0.2 +0.0 +0.4 +0.6 +0.8 +1.0 +T1.0 +Eq. (11) +0.8 +w=α +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0101 +10-1, +10-3 +10-5 +10-7 +10-1 +100 +101 +102 +103 +1044 +FIG. 4. +Probability distribution Q(x, t; 1) of the position of absorbed sub-diffusive walkers. Plots of Q(x, t; 1) for sub- +diffusive fBM (a)-(c) with α = 0.8 and (d)-(f) with α = 0.5 at early, middle and late times (t = 0.2, 1, 10, respectively). +Analytical prediction (green solid curve) is obtained using Eqs. (3), (4) and (10), which quantitatively reproduces the +numerical simulation results (red circles). The error bar evaluated as 95 % CI is smaller than the size of symbol. Blue +dashed curve represent the free space propagator P(x, t; 1). The asymmetry in Q(x, t; 1) grows with the memory effect, +which is stronger for smaller α. +(a) +~ +(b) +~ +FIG. 5. Probability distribution P+(x, t; 1) of the position +of survived sub-diffusive walkers. +Plots of the normal- +ized position probability ˜P+(x, t; 1) ≡ P+(x, t; 1)/S(x; 1) +for sub-diffusive fBM with (a) α = 0.8 and (b) α = 0.5 +at early, middle and late times (t = 0.2, 1, 10, respec- +tively). Analytical prediction (dashed curve) is obtained +using Eq. (2), which reproduces the numerical simulation +results (symbols). Error bars represent 95 % CI. +subensemble of walkers with fixed FPT, whose super- +imposition with the weight F(τ; 1) results in Q(x, t; 1), +see Eq. (3). As Fig. 4 shows, our analytical predic- +tion of Q(x, t; 1) quantitatively captures the results +obtained by numerical simulations. +In Fig. 5, we plot the normalized position prob- +ability ˜P+(x, t; 1) ≡ P+(x, t; 1)/S(x; 1) of the sur- +vival walker from Eq. (2). +Again, our prediction +captures all the salient features seen in numerical +simulations. +One notable feature here is that the +slope (∂ ˜P+(x, t; 1)/∂x)x→0 at the boundary is van- +ishingly small [36]. Such an anomalous behavior of +˜P+(x, t; 1) ∼ xδ close to the boundary with non-trivial +exponent δ can be quantified from our expression for +Q(x, t; 1) as follows. Note first that in long time limit +t ≫ 1 ( ⇔ +x2 +0/Dαtα ≪ 1 in original unit), the +asymptotic behavior of ˜P+(x, t; 1) is obtained by tak- +ing x0 → 0 limit [30]. +For the walker absorbed at +time τ, its characteristic travel distance during the +subsequent time interval s = t − τ is evaluated as +∆x(s) ∼ sα/2. This indicates that, for a given loca- +tion x, the walker only starts substantially contribut- +ing to Q(x, t; 1) after the time t(x) = x2/α. From Eq. +(3), we thus find +Q(x, t; 1) ∼ +� t−τ ∗ +t(x) +(t − s)−(2−α/2) s−α/2 ds +∼ t−α/2 � +1 − t−(2−α)x(2−α)/α� +(12) +The first term cancels the free space propaga- +tor +P(x, t; 1) +∼ +t−α/2, +leaving +P+(x, t; 1) +∼ +t−(2−α/2)x(2−α)/α, +or +equivalently, +˜P+(x, t; 1) +∼ +t−1x(2−α)/α. The predicted exponent δ = (2 − α)/α +agrees with that obtained from heuristic scaling argu- +ment [30]. +For the Markovian case α = 1, the slope at the +boundary is finite (δ = 1), which multiplied by dif- +fusion coefficient is the outgoing flux. The peculiar +nature of the flux for α ̸= 1 case implies the break- +down of the Fick’s law, and makes the implementation +of a reflective boundary non-trivial. This rephrases a +fact that there is no diffusion (more generally Fokker- + +(f) +Q(c,t= 10; 1) +0.6 +P(c,t; 1) +α=0.5 +0.5 +Q(αc,t; 1) +0.4 +0.3 +0.2 +0.1 +0.0 +.4 +-2 +0 +2 +4 +6(d) +Q(α,t=0.2; 1) +0.6 +P(c, t; 1) +α=0.5 +0.5 +Q(αc,t; 1) +0.4 +0.3 +0.2 +0.1 +0.0 +-2 +0 +2 +4 +6(e) +Q(α,t=1; 1) +0.6 +P(c,t; 1) +α=0.5 +0.5 +Q(c,t; 1) +0.4 +0.3 +0.2 +0.1 +0.0 +0 +2 +-2 +4 +6(c) +Q(α,t= 10; 1) +0.5 +P(c,t; 1) +α=0.8 +0.4 +Q(c,t; 1) +0.3 +0.2 +0.1 +0.0 +-2 +0 +2 +4 +4 +6 +8(a) +Q(α,t=0.2; 1) +0.5 +P(c, t; 1) +α = 0.8 +0.4 +Q(αc,t; 1) +0.3 +0.2 +0.1 +0.0 +-6 +-2 +0 +2 +4 +6 +8(b) +Q(α,t=1; 1) +0.5 +P(c,t; 1) +α=0.8 +0.4 +Q(αc,t; 1) +0.3 +0.2 +0.1 +0.0 +-4 +-6 +-2 +0 +2 +4 +6 +8P+(α,t; 1) +1.0 +t=0.2 +0.8 +t=1 +t=10 +0.6 +α= 0.8 +0.4 +0.2 +0.0 +0 +2 +3 +4 +5 +1 +6 +7 +8P+(α,t; 1) +1.0 +t=0.2 +0.8 +t=1 +t=10 +0.6 +α= 0.5 +0.4 +0.2 +0.0 +0 +2 +3 +4 +5 +6 +7 +8 +75 +Planck) equation for non-Markovian walkers in the +ordinary sense. +In conclusion, we have provided a natural frame- +work with which the first passage process of non- +Markovian walkers can be analyzed. 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E 76, 061121 +(2007). + +Supplementary Material +Yuta Sakamoto and Takahiro Sakaue∗ +Department of Physical Sciences, Aoyama Gakuin University, +5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Japan +1 +arXiv:2301.13466v1 [cond-mat.stat-mech] 31 Jan 2023 + +DERIVATION OF INTEGRAL EQUATION +We start with Eq. (2) in the main text; +P+(x, t; 1) = P(x, t; 1) − Q(x, t; 1) +(1) +Here the walker’s initial position x0 > 0 is a sole length scale in the problem, and we measure +the length in unit of x0. Similarly, we introduce the unit of time τx0 = (x2 +0/2Dα)1/α, which +corresponds to the time scale for a walker to diffuse over the length scale x0. Note the rescaled +initial position x0 = 1, and +P(x, t; 1) = +1 +√ +2πtαe− (x−1)2 +2tα +(2) +Q(x, t; 1) = +� t +0 +F(τ; 1) P(x, t; 1|FPT = τ)dτ += +� t +0 +F(τ; 1) +1 +� +2π(t − τ)αe− {x−⟨x(t)⟩FPT=τ }2 +2(t−τ)α +dτ +(3) +where +⟨x(t)⟩FPT=τ = 1 + +� t +τ − 1 +�α +− +� t +τ +�α +(t ≥ τ) +(4) +is the average trajectory of the walkers after the first-passage at t = τ, which is calculated by +applying the regression hypothesis idea of Onsager as explained in the main text. +The integral of Eq. (1) over the half space (x ≥ 0) leads to +S(t; 1) = 1 +2 +� +erf +� +1 +√ +2tα +� ++ 1 +� +− 1 +2 +� t +0 +F(τ; 1) erf +� +⟨x(t)⟩FPT=τ +� +2(t − τ)α +� +dτ +(5) +where S(t; 1) is the survival probability. Noting the relation S(t; 1) = 1 − +� t +0 F(τ; 1)dτ, the +above equation is transformed to +1 − erf +� +1 +√ +2tα +� += +� t +0 +F(τ; 1) [1 − erf(h(t, τ))] dτ +(6) +with the memory function h(t, τ) = ⟨x(t)⟩FPT=τ +√ +2(t−τ)α , which is an exact integral equation to determine +F(τ, 1) (Eq. (8) in the main text). +ANALYSIS OF INTEGRAL EQUATION +To analyze the integral equation (6), we first rewrite the memory function as +h(t, τ) = t−α/2 +√ +2 g(u) +(7) +2 + +with +g(u) = (1 − u)−α/2(1 − u−α) + (1 − u)α/2u−α +(8) +where u ≡ τ/t. The error function in the integrand is expanded as +erf(h(t, τ)) = erf +�t−α/2 +√ +2 +� ++ +� +2 +πt−α/2(g(u) − 1) + O(t−3α/2) +(9) +Neglecting higher order terms O(t−3α/2), Eq. (6) becomes +S(t; 1) +� +1 − erf +�t−α/2 +√ +2 +�� +≃ +� +2 +π t1−α/2 +� 1 +0 +F(τ(u); 1) {1 − g(u)} du +(10) +Motivated by the known analytical solution +F(τ; 1) = C1 exp +� +− +� 1 +2τ +�� +τ −3/2 +(11) +for the Markovian case (α = 1), where C1 is a normalization constant, we seek for the solution +in the form +F(τ; 1) = Cα exp +� +− +� 1 +2τ +�ω� +τ −(1+p) += Cαt−(1+p) exp +� +− +� 1 +2tu +�ω� +u−(1+p) +(12) +Substituting the above ansatz in Eq. (10), we obtain +S(t; 1) +� +1 − erf +�t−α/2 +√ +2 +�� +≃ +� +2 +πCα t−(p+α/2) +� 1 +0 +e−( +1 +2tu) +ω � +αu−(α+p)(1 + O(u)) − α +2 u−p(1 + O(u)) +� +du +(13) +To evaluate the above integral, we note the following: +� 1 +0 +e−( +1 +2tu) +ω +u−κdu ≃ +� 1 +u∗ u−κdu +(14) +where u∗ = c1t−1(ω/κ)1/ω with c1 being a numerical constant of order unity. +Then, at leading order in 1/t, Eq. (13) becomes +S(t; 1) ≃ +� +2 +π Cαt−(1−α/2) +α +α + p − 1 +� +c1 +� +ω +α + p +�1/ω�1−α−p +(15) +which is asymptotically correct at large t. Calculating −dS(t; 1)/dt and comparing it with the +assumed form of F(t; 1), we find the persistence exponent +p = 1 − α +2 +(16) +3 + +in agreement with earlier scaling argument [1]. In addition, by comparing two expressions of +prefactor, we find a relation between ω and α; +(2 − α) +�2 + α +2ω +�α/(2ω) +c−α/2 +1 += c2 +(17) +where we introduce another numerical constant c2 of order unity to make the evaluated relation +equality. Since we know ω = 1 for the Markovian limit α = 1, one of the numerical constants +can be eliminated through +c2 = +�3 +2 +�1/2 +c−1/2 +1 +(18) +This leads to Eq. (11) in main text with one fitting parameter c1, which should be determined +through the comparison with numerical simulation data. As discussed in the main text, we +found c1 = 0.1 describes the simulation results well. The resultant dependence of ω on α is +shown in Fig. 3(b) in the main text. Apparently, the relation is close to ω = α, but the value of +ω is slightly larger than α in a systematic way. We note that, while irrelevant to the long time +asymptotic power-law behavior, the short time behavior is highly sensitive to this ω exponent. +For example, we show in Fig. S1 the short time part of the FPT distribution F(τ) for the case +of α = 0.4 and 0.5, where our formula for ω(α), but not ω = α, provides satisfactory fittings. +FIG. 1: +Short time part of FPT distribution of non-Markovian walkers. Plot of F(τ) for +the case (a) α = 0.4 and (b) α = 0.5. The best fit values are ω = 0.45 for α = 0.4 and ω = 0.544 for +α = 0.5 , which are included in the plot of Fig. 3(b) in the main text. +FAILURE OF THE METHOD OF IMAGE +The effect of the persistent memory in fBM becomes stronger with the departure from the +Markovian limit α = 1. This is seen, for instance, in the spatial profile of Q(x, t; 1) shown in +Fig. 4 in the main text, where the degree of the asymmetry Q(x, t; 1) ̸= Q(−x, t; 1), a hallmark +of the memory effect, clearly shows up in α = 0.5 case, but less evident in α = 0.8 case. In +4 + +6 +w= 0.5 +5 +w = 0.544 +4 +α = 0.5 +F(T) +3 +2 +1 +xxxxxxxxxxxxx +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +T25 +w= 0.4 +20 +w = 0.45 +α=0.4 +F(T) +15 +10 +5 +0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Tsuch a situation, one may expect that the method of image, a standard method used in the +Markovian system, might provide an acceptable approximate solution. In Fig. S2, we show the +probability of the survival walkers P+(x, t; 1) for α = 0.8, 0.5 cases, where the comparison is +made for our solution and that constructed by the method of image. Clearly, the method of +image fails to capture the profile even qualitatively. In contrast, our method is capable of a +quantitative description. +FIG. 2: +Failure of the method of image. Plot of P+(x, t; 1) for (a) α = 0.8 and (b) α = 0.5. Solid +curves are obtained from our theory, which quantitatively describe the numerical simulation result +(symbols). In contrast, the method of image provide qualitatively wrong profiles (dashed curves). +NUMERICAL SIMULATION +To simulate fBM trajectories {x0, x1, · · · , xN} of length N, we numerically integrated the +discretized version of Eq. (1) in main text with f = 0. +The Gaussian variables ηi, called +fractional Gaussian noise, have temporal correlation, whose long time part is characterized by +the power-law memory as described after Eq. (1) in main text. To generate the fractional +Gaussian noise, we employed the Davies and Harte algorithm [2], and generated m samples +of length N for each α. From these simulations, we calculated the standard deviation of the +walker’s displacement ∆xN ≡ +� +⟨(xN − x0)2⟩ after N steps. To analyze the FPT statistics, we +placed the hypothetical absorbing wall at x = x0 − ˜c ∆xN such that the initial separation from +the walker to the boundary is ˜c ∆x. We then reanalyzed each of m trajectories to find its first +arrival at the wall, and constructed the FPT distribution and the walkers’ distribution after +the FPT. We adopted N = 105, m = 105 and ˜c = 1 except for the FPT distribution data for +long time regime (Fig. 2 (a) inset), where we adopted N = 106 and m = 104 and ˜c = 0.5. +∗ corresponding author, sakaue@phys.aoyama.ac.jp +5 + +P+(α,t=1;1) +0.4 +α= 0.5 +0.3 +0.2 +0.1 +0.0 +0 +2 +3 +4P+(α,t=l; 1) +0.4 +α= 0.8 +0.3 +0.2 +0.1 +0.0 +0 +2 +3[1] J. Krug, H. Kallabis, S. N. Majumdar, S. J. Cornell, A. J. Bray, and C. Sire, Phys. Rev. E 56, +2702 (1997), URL https://link.aps.org/doi/10.1103/PhysRevE.56.2702. +[2] R. Davies and D. Harte, Biometrika 74, 95 (1987). +6 + diff --git a/E9FRT4oBgHgl3EQfBzcS/content/tmp_files/load_file.txt b/E9FRT4oBgHgl3EQfBzcS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6304159f199153d4ff61af591fd326c53fa5121b --- /dev/null +++ b/E9FRT4oBgHgl3EQfBzcS/content/tmp_files/load_file.txt @@ -0,0 +1,707 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf,len=706 +page_content='First passage time statistics of non-Markovian random walker: Onsager’s regression hypothesis approach Yuta Sakamoto and Takahiro Sakaue∗ Department of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Japan First passage time plays a fundamental role in dynamical characterization of stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Crucially, our current understanding on the problem is almost entirely relies on the theoretical formulations, which assume the processes under consideration are Markovian, despite abundant non- Markovian dynamics found in complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Here we introduce a simple and physically appealing analytical framework to calculate the first passage time statistics of non-Markovian walkers grounded in a fundamental principle of nonequilibrium statistical physics that connects the fluctuations in stochastic system to the macroscopic law of relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Pinpointing a crucial role of the memory in the first passage time statistics, our approach not only allows us to confirm the non-trivial scaling conjectures for fractional Brownian motion, but also provides a formula of the first passage time distribution in the entire time scale, and establish the quantitative description of the position probability distribution of non-Markovian walkers in the presence of absorbing boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' How long does it take for a random walker to reach a destination?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Such a question on the first passage time (FPT) is relevant to a broad range of situa- tions in science, technology and every-day life applica- tions as encountered, for instance, in diffusion-limited reactions [1–3], barrier crossing [4–7], target search processes [8, 9], cyclization of DNA molecule [10– 13], price fluctuation in market [2] and spread of dis- eases [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Today, the concept of the FPT and its im- portance in the study of stochastic processes are well recognized, and theoretical methods for its computa- tion are standardized [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' However, most of them are devised for Markovian random walkers, whose de- cision making does not depend on its past history, thus not applicable to non-Markovian walkers despite their ubiquitousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Indeed, a growing body of evidence suggests that the non-Markovian dynamics is found quite gener- ally in rheologically complex matters typically, but not exclusively, with viscoelastic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Classi- cal examples are found in the diffusion of interact- ing particles in narrow channels [15] and the motion of tagged monomers in long polymer chain [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Other notable representatives include colloidal parti- cles in polymer solutions [18] or nematic solvents [19], lipids molecules and cholesterols in cellular mem- brane [20], proteins in crowded media [21], and chro- mosome loci [22] as well as membraneless organelles in living cells [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Such systems commonly exhibit a slow dynamics in the form of sub-diffusion MSD(t) ∼ tα characterized by the anomalous exponent α < 1, where MSD(t) stands for the mean-square displace- ment of the observed particle during the time scale t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Here the physical mechanism at work is the inter- action of observed degree of freedom with the collec- tive modes with broad range of time scales underly- ing complex environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Because of its importance in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' intracellular transport, the theoretical tools to describe/diagnose such anomalous diffusion phe- nomenology have been well developed in the last few decades [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' However, most of them rely on MSD and related quantities, while much less attention has been paid to the FPT, despite its fundamental and practical importance to characterize the underlying stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' This is particularly true for sys- tems possessing memory, as nontrivial information on the history dependence of the system is encoded in the FPT statistics [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' It has long been known that the anomalous transport properties affect the rates of chemical and biochemical reactions [26], and such reactions are initiated by the encounter of reactant molecules, so precisely quantified by means of the FTP statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Unfortunately, our current understanding on the FPT of non-Markovian walker lags far behind that of Markovian counterpart, where the difficulty is largely associated to the lack of appropriate theoret- ical foothold [25, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' While the Fokker-Planck equation and its related methods play a key role to analyze the time evolution of the probability distri- bution of the Markovian walkers, their careless ap- plication is problematic for walkers with memory, a defining property of the non-Markovian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' At 𝑡 = 𝜏 (a) (b) 𝑡 = 𝜏 𝜏 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Regression hypothesis applied to non-Markovian walkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (a) Example trajectory of fBM with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 starting from the initial position x = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Before (after) the first hitting on absorbing boundary at x = 0, the trajectory is drawn by solid (dotted) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' First pas- sage event can be viewed as a large fluctuation to create a non-equilibrium state at t = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (b) After the first pas- sage (t > τ), the process follows, on average, the macro- scopic relaxation law, for sub-diffusive fBM, represented by the harmonic restoring force, whose spring constant gets smaller algebraically in longer time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='13466v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='stat-mech] 31 Jan 2023 Absorbing wall Time 0 0 o PositionAbsorbing wall Potential 0 o Position2 present,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' available results are quite limited with no- table examples being the perturbative and scaling ar- guments to estimate the asymptotic exponents charac- terizing the distribution of FPT and related quantities in unbounded domain [25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 29–31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' some approxima- tion schemes to calculate the mean FPT of polymer looping process [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 10–13],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' and more recent analytical treatment to compute the mean FPT in confined do- mains [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' However, neither of the full distribution of FPT or position distribution of non-Markovian walk- ers in the presence of boundary are available, making the computation of these quantities in non-Markovian processes fundamental challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' In this Letter, we provide a simple and physically appealing method to calculate the FPT statistics of non-Markovian walkers by identifying the moment of first passage (t = τ) as an initial condition for the re- laxation process afterwards (t > τ), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Our argument is thus rooted in a non-Markovian exten- sion of the regression hypothesis of Onsager, a corner stone for the development in the nonequilibrium sta- tistical physics [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' We obtain an exact integral equa- tion for the FPT distribution, the analysis of which yields, in addition to its asymptotic decay exponent, full functional form in leading order over entire time scales, and also the walker’s probability distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Importantly, our formalism allows one to unveil how and why the textbook standard “method of image” [2, 33] breaks down by pinpointing the role of memory built up during the first passage process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Here we focus on the sub-diffusive fractional Brownian motion [34] (fBM with α < 1), an important class of non-Markovian walkers found in widespread complex systems including living cells and nuclei [20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Illustration of the method of image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' For Marko- vian walkers (α = 1), Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) can be constructed by the method of image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Integrating Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (2) over the entire space (including negative domain), one finds S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) = 1 − � ∞ −∞ Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1)dx, where the surviving proba- bility S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) = � ∞ 0 P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1)dx is denoted by the hatched area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Equivalent to the above relation is � ∞ 0 Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1)dx = (1−S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1))/2 thanks to the reversal symmetry of Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) with respect to x = 0, producing a factor 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The same relation is obtained by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (3) over the positive x domain with ⟨x(t)⟩FPT=τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Generalized Langevin equation and power-law mem- ory kernel – As a paradigm, consider a random walker in one dimensional half space with an absorb- ing boundary at origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' A walker is initially positioned at x = x0(> 0) at t = 0, and evolves according to the following generalized Langevin equation: dx(t) dt = � t 0 µ(t − t′)f(t′)dt′ + η(t) (1) where f(t) and η(t) are, respectively, a time- dependent external force and the noise acting on the walker [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The latter is assumed to be Gaussian with zero mean and its auto-correlation is related to the mobility kernel via the fluctuation-dissipation re- lation ⟨η(t)η(t′)⟩ = Tµ(|t − t′|) with T being the noise strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The memory effect is encoded in µ(t), for which we assume for large t the power-law de- cay µ(t) ≃ −T −1Dαtα−2 (0 < α < 1) in addition to instantaneous response µ(t) = 2γ−1δ(t) at short time, where γ is a bare friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Finally, we require on physical ground � ∞ 0 µ(t)dt = 0 such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (1) describes the sub-diffusive fBM with the MSD exponent α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' This sum rule is a consequence of the relaxation nature of the sub-diffusive fBM, which is caused by the visco-elastic effect [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' For a free walker (f = 0) in free space (no boundrary), its position probability distribution P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0) is sim- ply given by N(x, x0, 2Dαtα), where N(x, A, B) = (2πB)−1/2e(x−A)2/2B denotes Gaussian distribution of x with the average A and the variance B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Process after first-passage – We now set a stage by introducing an absorbing boundary at the origin x = 0 such that the walker performs fBM in half space x > 0 with the same initial condition as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Using the free space propagator P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0), the walker’s position probability P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0) is now represented as P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0) = P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0) − Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0) (2) where Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0) is the position distribution of dead walker, who touched the absorbing boundary by this moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Note that while one usually looks at the walker’s behavior in physical domain (x ≥ 0) up to the absorption (t ≤ τ) in the context of FPT, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (2) holds in entire space and time domains in a sprit similar to [28];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' the absorbing boundary at x = 0 necessitates P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0) = Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0) for x ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Using the FPT distribution F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0), Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0) is represented as Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0) = � t 0 F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0) P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0|FPT = τ)dτ (3) where P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0|FPT = τ) is the conditional proba- bility of the walker’s position at time t after its first passage at time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Being the Gaussian process, one expects the form P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0|FPT = τ) = N(x, ⟨x(t)⟩FPT=τ, 2Dα(t − τ)α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (4) In the absence of memory effect, ⟨x(t)⟩FPT=τ = 0 ir- respective of the starting position x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Then, by not- ing � t 0 F(t′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0)dt′ = 1 − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0), integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (2) over half space leads to a classical result of the survival probability S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0) ≡ � ∞ 0 P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' x0)dx = erf(x0/√4D1t) for Markovian case, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Al- though not applicable to non-Markovian walker, the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 Q(c, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) t=1 P(c, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 P(c,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' - 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 3 2 1 0 1 2 33 above calculation highlights ⟨x(t)⟩FPT=τ, which gen- erally depends on x0, as a central quantity to account for the memory effect in the first passage statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' History-dependent relaxation: regression hypothesis view – A key idea to quantify ⟨x(t)⟩FPT=τ comes from the fundamental connection between fluctuation and response in nonequilibrium statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' In his seminal paper, Onsager pointed out that the decay of mesoscopic fluctuations follow, on average, the macro- scopic law of relaxation [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Applying this so-called regression hypothesis to our problem, we view the pro- cess after the first passage t > τ as a relaxation pro- cess, whose “initial” condition x(τ) = 0 can be pre- pared either naturally (by fluctuation) or artificially (by external force), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' In the latter scenario, we take the sub-ensemble of walkers whose FPT is τ, and describe their average time evolution using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (1) with the constant force f(t) = f0 for t < τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' This leads to ⟨ ˙x(t)⟩FPT=τ = f0 � t 0 µ(t′)dt′ (t < τ) (5) then, identifying ⟨ ˙x(τ)⟩FPT=τ ≃ −x0/τ, we find f0 ≃ −Tx0 Dα τ −α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (6) Now the desired non-equilibrium state is prepared at t = τ, at which we switch off the force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The sub- sequent relaxation is described, again using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (1), by ⟨ ˙x(t)⟩FPT=τ = f0 � t t−τ µ(t′)dt′, (t > τ) (7) whose integral with respect to time leads to ⟨x(t)⟩FPT=τ, where a numerical coefficient implicit in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (6) is fixed by requiring ⟨x(t)⟩FPT=τ → x0 for t/τ ≫ 1 as a consequence of the sum rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Collecting all together, our analytical formulation is summarized as the following integral equation [35]: 1 − erf � 1 √ 2tα � = � t 0 F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) [1 − erf(h(t, τ))] dτ (8) with the memory function h(t, τ) = 1 � 2(t − τ)α � 1 + � t τ − 1 �α − � t τ �α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (9) From here onwards, we measure the length and the time in unit of x0 and τx0 = (x2 0/2Dα)1/α, respec- tively, which are the sole characteristic length and time scales in the problem, making the initial posi- tion x0 = 1 upon rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' First passage time distribution – We now determine the leading order solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (8) in the form F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) = Cα exp � − � 1 2τ �ω� τ −(1+p) (10) where Cα is a normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' This function, a generalization of the Markovian result [2] ω = 1, (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' FPT distribution of non-Markovian walk- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (a) FPT distribution F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) for sub-diffusive fBM (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Inset shows the double logarithmic plot of large τ regime, where the asymptotic slope p+1 = 2−α/2 is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The data for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 is shifted downward (×10−2) for visual clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Both in main panel and inset, symbols represent simulation results and the curves corre- spond to the analytical formula (10) with p = 1−α/2 and ω given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The error bars represent 95 % CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (b) Exponent ω as a function of α, which characterizes the early time regime in FPT distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Blue solid circles are obtained by fitting the numerical simulation data for several α values (two of them shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 2(a)) with the formula (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Fitting these data with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (11) fixes the parameter c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' p = 1/2, exhibits a peak at τ = τ ∗ = (1/2)(ω/(1 + p))1/ω and develops a power-law tail F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) ∼ τ −(1+p) at τ ≫ τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' With this in mind, we plug the ansatz (10) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (8) and perform the asymptotic analysis, which yields p = 1 − α/2 in agreement with previous scaling argument [25, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' In addition, our formulation allows us to obtain the exponent ω, which satisfies the relation (2 − α)2ω(2 + α)α (2ω)α = �3 2 �ω cω(α−1) 1 (11) with a numerical constant c1 of order unity [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 3 , we compare our analytical formula for F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) with the results obtained from numerical sim- ulation [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' As shown, the agreement is excellent, encompassing the short time singularity to the peak, and the eventual long time power-law tail, which are characterized by the exponents ω and p, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The peak position τ ∗ is rather sensitive to the value of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' This is particularly true for small ω, which is the case for the small α, shifting the peak position τ ∗ vanishingly small in the limit α → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Probability distributions of dead and survived walk- ers – We are now in a position to take a close look at Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) that is the distribution of walkers af- ter their first passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (3) and (4), we immediately find the memory effect in the form of restoring force represented by nonzero ⟨x(t)⟩FPT=τ breaks the reversal symmetry with respect to x = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=', Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) ̸= Q(−x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) that clearly manifests the breakdown of the image method (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 2, 4) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The value of ⟨x(t)⟩FPT=τ corresponds to the peak position of P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1|FPT = τ), which is zero initially (t = τ), and slowly evolves with time towards x = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Such a distribution P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1|FPT = τ) characterizes the 4 O α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 0 0 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 3 F(T) 2 1 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 w=α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0101 10-1, 10-3 10-5 10-7 10-1 100 101 102 103 1044 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Probability distribution Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) of the position of absorbed sub-diffusive walkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Plots of Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) for sub- diffusive fBM (a)-(c) with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 and (d)-(f) with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 at early, middle and late times (t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2, 1, 10, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Analytical prediction (green solid curve) is obtained using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (3), (4) and (10), which quantitatively reproduces the numerical simulation results (red circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The error bar evaluated as 95 % CI is smaller than the size of symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Blue dashed curve represent the free space propagator P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The asymmetry in Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) grows with the memory effect, which is stronger for smaller α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (a) ~ (b) ~ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Probability distribution P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) of the position of survived sub-diffusive walkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Plots of the normal- ized position probability ˜P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) ≡ P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1)/S(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) for sub-diffusive fBM with (a) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 and (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 at early, middle and late times (t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2, 1, 10, respec- tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Analytical prediction (dashed curve) is obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (2), which reproduces the numerical simulation results (symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Error bars represent 95 % CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' subensemble of walkers with fixed FPT, whose super- imposition with the weight F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) results in Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 4 shows, our analytical predic- tion of Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) quantitatively captures the results obtained by numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 5, we plot the normalized position prob- ability ˜P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) ≡ P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1)/S(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) of the sur- vival walker from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Again, our prediction captures all the salient features seen in numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' One notable feature here is that the slope (∂ ˜P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1)/∂x)x→0 at the boundary is van- ishingly small [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Such an anomalous behavior of ˜P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) ∼ xδ close to the boundary with non-trivial exponent δ can be quantified from our expression for Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Note first that in long time limit t ≫ 1 ( ⇔ x2 0/Dαtα ≪ 1 in original unit), the asymptotic behavior of ˜P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) is obtained by tak- ing x0 → 0 limit [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' For the walker absorbed at time τ, its characteristic travel distance during the subsequent time interval s = t − τ is evaluated as ∆x(s) ∼ sα/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' This indicates that, for a given loca- tion x, the walker only starts substantially contribut- ing to Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) after the time t(x) = x2/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (3), we thus find Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) ∼ � t−τ ∗ t(x) (t − s)−(2−α/2) s−α/2 ds ∼ t−α/2 � 1 − t−(2−α)x(2−α)/α� (12) The first term cancels the free space propaga- tor P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) ∼ t−α/2, leaving P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) ∼ t−(2−α/2)x(2−α)/α, or equivalently, ˜P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) ∼ t−1x(2−α)/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The predicted exponent δ = (2 − α)/α agrees with that obtained from heuristic scaling argu- ment [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' For the Markovian case α = 1, the slope at the boundary is finite (δ = 1), which multiplied by dif- fusion coefficient is the outgoing flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The peculiar nature of the flux for α ̸= 1 case implies the break- down of the Fick’s law, and makes the implementation of a reflective boundary non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' This rephrases a fact that there is no diffusion (more generally Fokker- (f) Q(c,t= 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='6 P(c,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 Q(αc,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 2 0 2 4 6(d) Q(α,t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='6 P(c, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 Q(αc,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 2 0 2 4 6(e) Q(α,t=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='6 P(c,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 Q(c,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 0 2 2 4 6(c) Q(α,t= 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 P(c,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 Q(c,t;' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 P(c, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 Q(αc,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 6 2 0 2 4 6 8(b) Q(α,t=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 P(c,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 Q(αc,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 4 6 2 0 2 4 6 8P+(α,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 t=1 t=10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='6 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 0 2 3 4 5 1 6 7 8P+(α,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 t=1 t=10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='6 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 0 2 3 4 5 6 7 8 75 Planck) equation for non-Markovian walkers in the ordinary sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' In conclusion, we have provided a natural frame- work with which the first passage process of non- Markovian walkers can be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' It is very sim- ple, yet has a quantitative predictability as we have demonstrated here for the system with persistent memory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=', sub-diffusive fBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' We expect that the proposed method with suitable extension and general- ization will find versatile applicability to explore rich FPT problems in non-Markovian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Acknowledgements We thank E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Carlon for fruitful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' This work is supported by JSPS KAKANHI (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' JP18H05529 and JP21H05759).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' ∗ corresponding author, sakaue@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='aoyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='jp [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Kampen, Stochastic processes in physics and chemistry (North Holland, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Redner, A guide to first-passage processes (Cam- bridge University Press, Cambridge, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Szabo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Schulten, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Schulten, The Journal of Chemical Physics 72, 4350 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' [4] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Kramers, Physica 7, 284 (1940).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' H¨anggi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Talkner, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Borkovec, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 62, 251 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Carlon, H.' 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[36] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Kantor and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Kardar, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' E 76, 061121 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Supplementary Material Yuta Sakamoto and Takahiro Sakaue∗ Department of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Japan 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='13466v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='stat-mech] 31 Jan 2023 DERIVATION OF INTEGRAL EQUATION We start with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (2) in the main text;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) = P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) − Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) (1) Here the walker’s initial position x0 > 0 is a sole length scale in the problem, and we measure the length in unit of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Similarly, we introduce the unit of time τx0 = (x2 0/2Dα)1/α, which corresponds to the time scale for a walker to diffuse over the length scale x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Note the rescaled initial position x0 = 1, and P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) = 1 √ 2πtαe− (x−1)2 2tα (2) Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) = � t 0 F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) P(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1|FPT = τ)dτ = � t 0 F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 1 � 2π(t − τ)αe− {x−⟨x(t)⟩FPT=τ }2 2(t−τ)α dτ (3) where ⟨x(t)⟩FPT=τ = 1 + � t τ − 1 �α − � t τ �α (t ≥ τ) (4) is the average trajectory of the walkers after the first-passage at t = τ, which is calculated by applying the regression hypothesis idea of Onsager as explained in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (1) over the half space (x ≥ 0) leads to S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) = 1 2 � erf � 1 √ 2tα � + 1 � − 1 2 � t 0 F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) erf � ⟨x(t)⟩FPT=τ � 2(t − τ)α � dτ (5) where S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) is the survival probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Noting the relation S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) = 1 − � t 0 F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1)dτ, the above equation is transformed to 1 − erf � 1 √ 2tα � = � t 0 F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) [1 − erf(h(t, τ))] dτ (6) with the memory function h(t, τ) = ⟨x(t)⟩FPT=τ √ 2(t−τ)α , which is an exact integral equation to determine F(τ, 1) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (8) in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' ANALYSIS OF INTEGRAL EQUATION To analyze the integral equation (6), we first rewrite the memory function as h(t, τ) = t−α/2 √ 2 g(u) (7) 2 with g(u) = (1 − u)−α/2(1 − u−α) + (1 − u)α/2u−α (8) where u ≡ τ/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The error function in the integrand is expanded as erf(h(t, τ)) = erf �t−α/2 √ 2 � + � 2 πt−α/2(g(u) − 1) + O(t−3α/2) (9) Neglecting higher order terms O(t−3α/2), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (6) becomes S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) � 1 − erf �t−α/2 √ 2 �� ≃ � 2 π t1−α/2 � 1 0 F(τ(u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) {1 − g(u)} du (10) Motivated by the known analytical solution F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) = C1 exp � − � 1 2τ �� τ −3/2 (11) for the Markovian case (α = 1), where C1 is a normalization constant, we seek for the solution in the form F(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) = Cα exp � − � 1 2τ �ω� τ −(1+p) = Cαt−(1+p) exp � − � 1 2tu �ω� u−(1+p) (12) Substituting the above ansatz in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (10), we obtain S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) � 1 − erf �t−α/2 √ 2 �� ≃ � 2 πCα t−(p+α/2) � 1 0 e−( 1 2tu) ω � αu−(α+p)(1 + O(u)) − α 2 u−p(1 + O(u)) � du (13) To evaluate the above integral, we note the following: � 1 0 e−( 1 2tu) ω u−κdu ≃ � 1 u∗ u−κdu (14) where u∗ = c1t−1(ω/κ)1/ω with c1 being a numerical constant of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Then, at leading order in 1/t, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (13) becomes S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) ≃ � 2 π Cαt−(1−α/2) α α + p − 1 � c1 � ω α + p �1/ω�1−α−p (15) which is asymptotically correct at large t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Calculating −dS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1)/dt and comparing it with the assumed form of F(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1), we find the persistence exponent p = 1 − α 2 (16) 3 in agreement with earlier scaling argument [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' In addition, by comparing two expressions of prefactor, we find a relation between ω and α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (2 − α) �2 + α 2ω �α/(2ω) c−α/2 1 = c2 (17) where we introduce another numerical constant c2 of order unity to make the evaluated relation equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Since we know ω = 1 for the Markovian limit α = 1, one of the numerical constants can be eliminated through c2 = �3 2 �1/2 c−1/2 1 (18) This leads to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (11) in main text with one fitting parameter c1, which should be determined through the comparison with numerical simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' As discussed in the main text, we found c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='1 describes the simulation results well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The resultant dependence of ω on α is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Apparently, the relation is close to ω = α, but the value of ω is slightly larger than α in a systematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' We note that, while irrelevant to the long time asymptotic power-law behavior, the short time behavior is highly sensitive to this ω exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' For example, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' S1 the short time part of the FPT distribution F(τ) for the case of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5, where our formula for ω(α), but not ω = α, provides satisfactory fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1: Short time part of FPT distribution of non-Markovian walkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Plot of F(τ) for the case (a) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 and (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The best fit values are ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='45 for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 and ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='544 for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 , which are included in the plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' FAILURE OF THE METHOD OF IMAGE The effect of the persistent memory in fBM becomes stronger with the departure from the Markovian limit α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' This is seen, for instance, in the spatial profile of Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 4 in the main text, where the degree of the asymmetry Q(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) ̸= Q(−x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1), a hallmark of the memory effect, clearly shows up in α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 case, but less evident in α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' In 4 6 w= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 5 w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='544 4 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 F(T) 3 2 1 xxxxxxxxxxxxx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 T25 w= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 20 w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='45 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 F(T) 15 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 Tsuch a situation, one may expect that the method of image, a standard method used in the Markovian system, might provide an acceptable approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' S2, we show the probability of the survival walkers P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 cases, where the comparison is made for our solution and that constructed by the method of image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Clearly, the method of image fails to capture the profile even qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' In contrast, our method is capable of a quantitative description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 2: Failure of the method of image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Plot of P+(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) for (a) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 and (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Solid curves are obtained from our theory, which quantitatively describe the numerical simulation result (symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' In contrast, the method of image provide qualitatively wrong profiles (dashed curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' NUMERICAL SIMULATION To simulate fBM trajectories {x0, x1, · · · , xN} of length N, we numerically integrated the discretized version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (1) in main text with f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' The Gaussian variables ηi, called fractional Gaussian noise, have temporal correlation, whose long time part is characterized by the power-law memory as described after Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' (1) in main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' To generate the fractional Gaussian noise, we employed the Davies and Harte algorithm [2], and generated m samples of length N for each α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' From these simulations, we calculated the standard deviation of the walker’s displacement ∆xN ≡ � ⟨(xN − x0)2⟩ after N steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' To analyze the FPT statistics, we placed the hypothetical absorbing wall at x = x0 − ˜c ∆xN such that the initial separation from the walker to the boundary is ˜c ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' We then reanalyzed each of m trajectories to find its first arrival at the wall, and constructed the FPT distribution and the walkers’ distribution after the FPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' We adopted N = 105, m = 105 and ˜c = 1 except for the FPT distribution data for long time regime (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 2 (a) inset), where we adopted N = 106 and m = 104 and ˜c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' ∗ corresponding author, sakaue@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='aoyama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='jp 5 P+(α,t=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 0 2 3 4P+(α,t=l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='4 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content='0 0 2 3[1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Krug, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Kallabis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Majumdar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Cornell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Bray, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Sire, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Rev.' 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+page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Davies and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' Harte, Biometrika 74, 95 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} +page_content=' 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FRT4oBgHgl3EQfBzcS/content/2301.13466v1.pdf'} diff --git a/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf b/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6957c239642b34e54c5f64413706411d499aeeed --- /dev/null +++ b/ENE4T4oBgHgl3EQffQ0N/content/2301.05105v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Statistical inference for situations where there is both +multi-way dependence and cluster heterogeneity has thus far been an open issue. Existing theory +for multi-way clustering inference requires identical distributions across clusters (implied by the +so-called separate exchangeability assumption). Yet no such homogeneity requirement is needed +in the existing theory for one-way clustering. The new result therefore theoretically justifies the +view that multi-way clustering is a more robust version of one-way clustering, consistent with +applied practice. The result is applied to linear regression, where it is shown that a standard +plug-in variance estimator is valid for inference. +1 +Introduction +Clustering standard errors on multiple dimensions is common and attractive in applied econometrics +because it allows observations to be dependent whenever they share a cluster on any dimension.1 +The variance estimator proposed by Cameron et al. (2011) (henceforth CGM) has thus been widely +applied to contexts with multi-way dependence. Existing justification for the asymptotic validity +of the CGM estimator and other inference procedures in multi-way clustering relies on separate +∗Department of Economics, Princeton University. Email: lyap@princeton.edu. +1E.g., Dube et al. (2010) clustered on state and border segment when studying the effect of minimum wages on +employment; Nunn and Wantchekon (2011) clustered on ethnic groups and district when studying the effect of slave +trade on trust; Michalopoulos and Papaioannou (2013) clustered on country and ethnolinguistic family when studying +the effect of pre-colonial institutions on development. +1 + +exchangeability, which implies the homogeneity of clusters. This paper provides general conditions +such that the plug-in mean estimator is asymptotically normal, and the CGM variance estima- +tor is consistent, even when clusters are heterogeneous. These conditions do not include separate +exchangeability, and they mimic the conditions in one-way clustering: the only substantive assump- +tion is that two observations are independent when they do not share any cluster. Since asymptotic +normality and consistent variance estimation are sufficient for valid inference, the results in this +paper provide sufficient general conditions for valid inference in multi-way clustering. +An environment with multi-way clustering permits dependence whenever observations share at +least one cluster. To fix ideas, suppose observations can be partitioned on two different dimen- +sions — state and industry. Observations in the same state or in the same industry are plausibly +correlated, but two observations in different states and different industries are assumed to be inde- +pendent.2 The CGM variance estimator accommodates such dependence, and subsequent literature +provided a theoretical basis for its validity (e.g., Davezies et al. (2021); MacKinnon et al. (2021)). +Menzel (2021) also showed the validity of a bootstrap procedure for multi-way clustering that is +robust to asymptotic non-normalities.3 The theoretical basis for inference thus far relies on sepa- +rate exchangeability, the assumption that random variables are exchangeable on either clustering +dimension, though not necessarily both. +However, as noted by MacKinnon et al. (2021), separate exchangeability implies identical marginal +distributions. Since exchangeability implies identical distribution, separate exchangeability in the +state-industry example implies that random variable in Alaska and California must be drawn +from the same distribution. In contrast, existing asymptotic theory on one-way clustering (e.g., +Hansen and Lee (2019); Djogbenou et al. (2019)) allows the distribution of the random variable to +be heterogeneous over clusters. The only substantive assumption is that observations that do not +share any cluster are independent. Since the only available conditions for the validity of multi-way +clustering require separate exchangeability, the literature lacks general conditions for multi-way +clustering that generalize one-way clustering and permit heterogeneity over clusters. This paper +fills the gap, and thus justifies multi-way clustering as a more robust version of one-way clustering. +2This setting permits more general dependence structures than one-way clustering. +If there is one-way clustering +by state, then two observations from different states are automatically independent. +In two-way clustering, two +observations from different states are not necessarily independent because they may share the same industry. +3Menzel (2021) pointed out that a purely interactive data-generating processes unique to multi-way dependence has +an asymptotic distribution that is not normal. Section 2 will consider this process and show how the assumptions of +this paper rules it out. +2 + +Example 1. To illustrate separate exchangeability, consider an additive random effects model. +Individual i who belongs to cluster g(i) on the G dimension and cluster h(i) on the H dimension +has random variable Wi generated from Wi = αg(i) + γh(i) + εi, where cluster-specific αg, γh and +individual-specific εi are independent of each other. If we assume separate exchangeability, then αg, +γh, and εi are iid.4 In contrast, under one-way cluster asymptotics, the cluster-specific error αg +is allowed to be heteroskedastic. General conditions provided in this paper permits valid inference +even when αg, γh, εi are heteroskedastic in this model. +The main result is a central limit theorem for multi-way clustering with heterogeneous cluster sizes +and distributions. I apply the theorem to a simple setting of a linear regression, but it is more +broadly applicable to many other econometric procedures that exhibit a similar clustering structure. +2 +Setting and Main Result +Consider a setup with two-way clustering on dimensions G and H for random vectors {Wi}n +i=1, +where Wi := (Wi1, Wi2, · · · , WiK)′ ∈ RK and i is the unit of observation, for a sequence of pop- +ulations of size n.5 For example, G could denote states and H denote industries. This section +establishes a central limit theorem (CLT) for a weighted sum of the random vector i.e., � +i ωiWi, +where ωi are nonstochastic scalar weights, as n → ∞. For C ∈ {G, H}, let N C +c +denote the set of +observations in cluster c on dimension C — this partitions the population on the C dimension. +Let g(i) and h(i) denote the cluster that observation i belongs to on the G and H dimensions +respectively. +These cluster identities are nonstochastic and observed. +Let N C +c += |N C +c | denote +the cluster size for C ∈ {G, H} and Ngh := |N G +g ∩ N H +h |. These cluster sizes are allowed to be +heterogeneous in a way that will be formalized in the assumptions below. Wi is assumed to be +independent of any Wj when j /∈ N G +g(i) ∪ N H +h(i) =: Ni, i.e., when i and j do not share a cluster on +either dimension. Hence, Ni is the set of observations plausibly dependent with i. This environment +is stated as Assumption 1, the main substantive assumption. +Assumption 1. Wi ⊥⊥ Wj if g(i) ̸= g(j) and h(i) ̸= h(j). +4To see this, for individuals i and j where g(i) ̸= g(j), h(i) = h(j) = h, separate exchangeability implies αg(i)+γh+εi +d= +αg(j) + γh + εj. Since αg, γh and εi are independent, εi +d= εj and αg +d= αg′. +5Clustering in more than two dimensions is possible, and derivations are entirely analogous. +3 + +Assumption 1 is agnostic about the dependence structure when Wi and Wj share at least one cluster. +It also allows the data generating process to be arbitrarily heterogeneous across different clusters, +mimicking the heterogeneity permitted in one-way clustering (e.g., Hansen and Lee (2019)). Since +one-way clustering is a special case of two-way clustering where everyone is in their own H cluster, +the result here generalizes existing results in one-way clustering. In contrast, existing literature +in multi-way clustering assumes separate exchangeability that additionally imposes identical dis- +tribution over clusters, so they do not immediately generalize one-way clustering. {Wi}n +i=1 being +separately exchangeable implies Assumption 1 but the converse is not true.6 +Observations that share a cluster are allowed to be dependent, but they need not be. +Hence, +let Aij := 1[Wi ̸⊥⊥ Wj] be a 0-1 indicator for whether Wi and Wj are actually dependent, so +Aij = Aji, and Aii = 1.7 +This notation allows a particular form of misspecification where the +researcher is conservative and clusters on dimension G when it is not required. Every observation +Wi is weighted by nonstochastic scalar ωi. For positive definite matrix Q, let λmin(Q) denote the +smallest eigenvalue of Q. Then, let Qn := V ar (�n +i=1 ωiWi) denote the variance of the sum and +λn := λmin(Qn) denote its smallest eigenvalue. For example, when K = 1 and equal weights are +placed on all observations, Wi is a scalar and λn = Qn = V ar(� +i ωiWi). K0 is used throughout +the paper to denote an arbitrary constant. +Assumption 2. For C ∈ {G, H}, and k ∈ {1, 2, · · · , K}, there exists K0 < ∞ such that: +1. E[W 4 +ik] ≤ K0 for all i. +2. +1 +λn maxc +�� +i∈N C +c |ωi| +�2 +→ 0. +3. +1 +λn +� +c +� +i,j∈N C +c Aij|ωiωj| ≤ K0. +Assumption 2.1 requires the fourth moment to be bounded, which is stronger than the moment +condition in one-way clustering.8 +The proof in one-way clustering usually verifies a Lindeberg +6To illustrate this, let Ngh = 1 and Wgh denote the observation in cluster g and h on the respective dimensions. +Due to Kallenberg (2005), {Wgh}g≥1,h≥1 is separately exchangeable if and only if there exists a representation +Wgh = f(αg, γh, εgh), where (αg, γh, εgh) +iid +∼ U[0, 1]. Then, it is obvious that Wgh ⊥⊥ Wg′h′ for g ̸= g′, h ̸= h′. +A counterexample for the converse is some Wgh = −Wgh′. +These random variables are allowed to be perfectly +correlated since they share a cluster under Assumption 1. However, we cannot find a representation f(.), because +that representation implies E[Wgh|αg]⊥⊥ E[Wgh′|αg]. +7It is insufficient to define the indicator as Aij := 1[Cov(Wi, Wj) ̸= 0], since the proof contains third and fourth +moments. +For K = 1, zero covariance between a pair of observations is insufficient to ensure objects such as +E[WiWjWk] and E[WiWjWkWl] − E[WiWk]E[WjWl] are zero. +8See equation (7) of Hansen and Lee (2019) for the condition in one-way clustering. +4 + +condition because blocks of observations are independent of each other. With multi-way depen- +dence, we no longer have independent blocks because each cluster can have observations that are +dependent with observations from a different cluster when these observations share a cluster on +a different dimension. Hence, a different proof strategy is required. The proof in this paper uses +Stein’s method, which requires stronger moment restrictions, but provides a non-asymptotic bound +on the approximation error — details are in Subsection 2.1. +Assumption 2.2 requires the contribution of the cluster with the largest weight to be small relative +to the total variance. In the special case where everyone is equally weighted with ωi = 1, the +condition is simply (1/λn) maxc(N C +c )2 → 0. +Intuitively, this condition is required so that the +removal of a cluster does not change the variance substantively. This assumption allows the ratio +of any two cluster sizes to diverge to infinity. It is identical to equation (12) of Hansen and Lee +(2019) when C = G = H. Assumption 2.2 also rules out having components that are perfectly +negatively correlated: if the components of the vector were perfectly negatively correlated, λn = 0. +Assumption 2.3 is fairly unrestrictive about the convergence rate. +To aid exposition, suppose +ωi = 1 ∀i, K = 1, and C is taken to be the clustering dimension that λn ≍ � +c +� +i,j∈N C +c Aij.9 With +strong dependence, Aij = 1 for all i, j ∈ N C +c , so λn ≍ � +c(N C +c )2. However, if the researcher were +conservative and clustered on C when the data is indeed iid, then Aij = 1 if and only if i = j, so +λn ≍ n. Assumption 2.3 has implications on λn, which then determines how strong Assumption 2.2 +is. Namely, when λn ≍ n, Assumption 2.2 requires maxc(N C +c )2/n → 0. When λn ≍ � +c(N C +c )2, then +Assumption 2.2 only requires maxc(N C +c )2/(� +c′(N C +c′ )2) → 0. The weaker version of Assumption 2.2 +permits balanced panels where the unit and time dimensions increase at the same rate, while the +stronger version does not.10 The assumption that (1/λn) � +c(N C +c )2 ≤ K0 matches equation (11) of +Hansen and Lee (2019). +Remark 1. Assumption 2.3 rules out the following purely interactive model. As pointed out by +Menzel (2021), this model has an asymptotic distribution that is non-normal, and there is no +analog in one-way clustering. For g ∈ {1, · · · , M}, h ∈ {1, · · · , M} and Ngh = 1, we observe +Wgh = αgγh, where αg, γh are iid with mean zero and variances σ2 +α and σ2 +γ respectively, so there are +M2 observations. Then, � +g,h Wgh/M = +�� +g αg/ +√ +M +� �� +h γh/ +√ +M +� d−→ Z1Z2, where Z1 and Z2 +9To be clear about the notation, a ≍ b if and only if there exists K0 < ∞ such that a/b, b/a ∈ [−K0, K0]. Since E[W 2 +i ] +is bounded, λn ≍ maxC∈{G,H} +� +c +� +i,j∈N C +c Aij. +10To +see +this, +let +M +denote +the +number +of +units +and +time +periods, +so +there +are +M 2 +observations. +maxc(N C +c )2/(� +c′(N C +c′ )2) = M 2/M 3 = 1/M → 0, but maxc(N C +c )2/n = M 2/M 2 = 1 ̸= o(1). +5 + +are independent standard normal distributions. This limiting distribution is also known as Gaussian +chaos. � +g(N G +g )2/λn = M3/(M2σ2 +ασ2 +γ) = M/σ2 +ασ2 +γ → ∞ violates Assumption 2.3. +Theorem 1. Under Assumption 1 and 2, Q−1/2 +n +�n +i=1 ωi(Wi − E[Wi]) d−→ N(0, IK). Further, +1. If E[Wi] = 0 ∀i, then Q−1 +n ˆQn +p−→ IK, where ˆQn := � +i +� +j∈Ni ωiωjWiW ′ +j. +2. If E[Wi] = µ ∀i and +1 +λn +� +c +� +i,j∈N C +c |ωiωj| ≤ K0 for some K0 < ∞, then, for +¯W = +(� +i ωiWi)/(� +j ωj) and ˆQn := � +i +� +j∈Ni ωiωj(Wi− ¯W)(Wj− ¯W)′, ¯W +p−→ µ and Q−1 +n ˆQn +p−→ IK. +The theorem tells us that, under the aforementioned conditions, Q−1/2 +n +�n +i=1 ωi(Wi − E[Wi]) is +asymptotically standard normal and the plug-in variance estimator proposed by CGM is consistent +for multi-way clustering. One-way clustering is a special case of this theorem when one dimension +is weakly nested within the other: examples include G = H so both dimensions are identical, or if +we cluster by county and state (as counties are nested in states), or if everyone is in their own H +cluster. A sufficient condition for consistent variance estimation is E[Wi] = 0, similiar to theorem 3 +of Hansen and Lee (2019). This assumption is sufficient in many applications: for example, linear +regressions considered in Section 3 are identified by requiring the expectation of the residual term +to be zero. Additionally, the condition E[Wi] = µ matches theorem 4 of Hansen and Lee (2019) for +consistent variance estimation. Theorem 1.2 uses a stronger form of Assumption 2.3 where Aij = 1 +for all i, j ∈ N C +c . +Remark 2. If E[Wi] ̸= 0, then the variance estimator need not be consistent. Unlike one-way +clustering, it may not even be conservative. Suppose E[Wi] ̸= 0 for some i, and define ˜Wi := Wi − +E[Wi]. Then, Q−1 +n +� +i +� +j∈Ni WiW ′ +j = Q−1 +n +�� +i +� +j∈Ni ˜Wi ˜W ′ +j +� ++ Q−1 +n +�� +i +� +j∈Ni E[Wi]E[Wj]′� +. +Since Q−1 +n +�� +i +� +j∈Ni ˜Wi ˜W ′ +j +� += oP (1) by Theorem 1.1, and Qn is positive semidefinite, whether +the asymptotic variance is over or under estimated depends on whether � +i +� +j∈Ni E[Wi]E[Wj]′ is +positive semidefinite. Let K = 1 for exposition. In one-way clustering, the variance is weakly over- +estimated, so inference is conservative. To see this, let W G +g denote the vector of Wi such that g(i) = +g. � +i +� +j∈Ni E[Wi]E[Wj] = � +g +� +i,j∈N G +g E[Wi]E[Wj] = � +g 1′E[W G +g ]E[W G +g ]′1 ≥ 0. In two-way +clustering, � +i +� +j∈Ni E[Wi]E[Wj] can be negative. An example is where n = 3: cov(W1, W3) = 0 +but cov(W1, W2) ̸= 0 and cov(W2, W3) ̸= 0, so W1 and W2 share a cluster in one dimension and W2 +and W3 share a cluster on a different dimension. Further, E[W2] = −1 and E[W1] = E[W3] = 1. +Then, � +i +� +j∈Ni E[Wi]E[Wj] = −1. +6 + +2.1 +Proof Sketch +The proof of Theorem 1 proceeds by first proving a CLT for a scalar random variable, then applying +the Cramer-Wold device to obtain the multivariate CLT. The scalar CLT is proven using Stein’s +method. I adapt the proof strategy from Ross (2011) to obtain an upper bound on the Wasserstein +distance between a pivotal statistic and the standard normal random variable. By exploiting the +multi-way clustering structure, the upper bound on the distance can be shown to converge to zero. +All details are in Appendix A. +For ease of exposition, consider a simpler environment where K = 1, ωi = 1 for all i, and Aij = 1 +whenever c(i) = c(j) for some c, and E[Wi] = 0. Lemma 4 in Appendix A provides an explicit +bound on the Wasserstein distance. With dW (.) denoting the Wasserstein distance, σ2 +n := Qn and +R = � +i Xi/σn, +dW (R, Z) ≤ 1 +σ3n +n +� +i=1 +������ +� +j,k∈Ni +E[WiWjWk] +������ ++ +√ +2 +√πσ2n +� +� +� +� +�V ar + + +n +� +i=1 +� +j∈Ni +WiWj + + +At this point, my proof departs from the proofs in existing statistical literature that employ Stein’s +method (e.g., Chen and Shao (2004)). Let Ni := |Ni|. Holder’s inequality is employed on objects +such as � +i | � +j,k∈Ni E[WiWjWk]|. Existing literature uses the L1 norm of moments E[W 3 +i ] and +L∞ norm of Ni, resulting in (maxm Nm)2 � +i E[W 3 +i ]. In contrast, my proof uses the L∞ norm of +E[W 3 +i ] and L1 norm of Ni, resulting in maxm E[W 3 +m] � +i N 2 +i . Hence, +1 +σ3n +n +� +i=1 +������ +� +j,k∈Ni +E[WiWjWk] +������ +≤ 1 +σ3n +max +m E[W 3 +m] +� +i +N 2 +i +Since maxm E[W 3 +m] is bounded by Assumption 2.1, it suffices to show � +i N 2 +i /σ3 +n → 0. Due to +Assumption 1, Ni ≤ N G +g(i) + N H +h(i), so +1 +σ3n +� +i +N 2 +i ≤ 1 +σ3n +� +i +(N G +g(i) + N H +h(i))2 ≤ 1 +σ3n +max +g,h (N G +g + N H +h ) +� +i +(Ng(i) + Nh(i)) +≤ +� 1 +σn +max +g,h (N G +g + N H +h ) +� 1 +σ2n +�� +g +(N G +g )2 + +� +h +(N H +h )2 +� +7 + +Since λn = σn when K = 1, maxg,h(N G +g +N H +h )/σn → 0 by Assumption 2.2 and +�� +g(N G +g )2 + � +h(N H +h )2� +/σ2 +n +is bounded by Assumption 2.3. Hence, the term is o(1). +A similar argument is made for the fourth moment that features in V ar +��n +i=1 +� +j∈Ni WiWj +� +. +To complete the proof for variance estimation, observe that since the fourth moments exist, the +consistency of the plug-in variance estimator can be proven by using Chebyshev’s inequality and +existing intermediate results. +Remark 3. Due to the proof strategy, the intermediate results are informative about the quality +of the normal approximation. With dK(.) denoting the Kolmogorov distance, proposition 1.2 from +Ross (2011) implies that dK(R, Z) ≤ (2/π)1/4� +dW (R, Z). Since Z is standard normal in the proof +of CLT, the bound on dW (.) also places a bound on the Kolmogorov distance dK(.). This is then +informative of the maximum distance between the pivotal statistic and the standard normal. +3 +Application +This section applies Theorem 1 to linear regressions, showing that using the normal approximation +with the CGM estimator is valid. Consider a linear model where scalar outcome Yi is generated by +Yi = Diθ + W ′ +iγ + ui =: X′ +iβ + ui +Di ∈ R is the regressor of interest, Wi ∈ RK−1 is a vector of controls that may include the +intercept, and let Xi = (Xi1, Xi2, · · · , XiK)′ := (Di, W ′ +i)′ ∈ RK. We are interested in estimating +θ. The coefficient vector β := (θ, γ′)′ ∈ RK is the same for all individuals. The stochastic residual +term ui satisfies E[ui|Xi] = 0 for all i, and is allowed to be multi-way clustered. The standard OLS +estimator is +ˆβ = +� n +� +i=1 +XiX′ +i +�−1 � n +� +i=1 +XiYi +� += β + +� n +� +i=1 +XiX′ +i +�−1 � n +� +i=1 +Xiui +� +This object is assumed to be well-defined in that �n +i=1 XiX′ +i is invertible. Using an equivalent +representation with data matrices, the model is Y = Dθ + Wγ + u = Xβ + u. +Let MW = +I − W(W ′W)−1W ′ denote the annihilator matrix. Let ˜D := MW D be the D with W’s partialled +8 + +out, and define ˜Y , ˜u in a similar manner. By the Frisch-Waugh-Lovell theorem (FWL), +ˆθ = ( ˜D′ ˜D)−1 ˜D′ ˜Y = θ + ( ˜D′ ˜D)−1 ˜D′˜u = θ + +�� +i +˜D2 +i +�−1 �� +i +˜Di˜ui +� += ˆβ1 +where ˜Di is the ith component of ˜D, so � +i ˜Di˜ui = ˜D′˜u = D′MWu = � +i ˜Diui. Let σ2 +n := V ar(ˆθ) = +V ar +�� +i ˜Diui/(� +i′ ˜D2 +i′) +� +and ˆσ2 +n := +�� +i +� +j∈Ni ˆuiˆuj ˜Di ˜Dj +� +/ +�� +i ˜D2 +i +�2 +. Estimated residuals are +ˆui := Yi − Xi ˆβ = ui − Xi(ˆβ − β). Due to FWL, ˆui = ˜Yi − ˜Diˆθ = ui − ˜Di(ˆθ − θ). +Inference for ˆθ, depends on whether we are conditioning on X: the conditions for asymptotic +normality differ slightly between random and nonrandom X. I consider each of them in turn. +3.1 +Fixed Regressors +First, consider regressions where the X’s are nonrandom. An example might be when the object of +interest is the difference between male and female wages. Their unobserved error may be correlated +by state and industry conditional on X, but the gender status Di is fixed. This can be viewed as +inference on a descriptive object. +With ui’s having a multi-way clustered structure, we can apply Theorem 1 on +�� +i ˜D2 +i +�−1 � +i ˜Diui, +where scalar weights are given by ωi = ˜Di/(� +i′ ˜D2 +i′). +Assumption 3. For C ∈ {G, H} and nonstochastic ˜Di, there exists K0 < ∞ such that: +1. E[u4 +i ] ≤ K0, E[ui] = 0. +2. +maxc +�� +i∈N C +c | ˜Di| +�2 +� +c′ +�� +j∈N C +c′ | ˜Dj| +�2 → 0. +3. +� +c′ +� +i,j∈N C +c | ˜Di ˜Dj| +V ar( +� +i ˜Diui) +≤ K0. +4. ui ⊥⊥ uj if g(i) ̸= g(j) and h(i) ̸= h(j). +Proposition 1. Under Assumption 3, (ˆθ − θ)/σn +d−→ N(0, 1), and ˆσ2 +n/σ2 +n +p−→ 1. +Assumption 3 works in the environment where there is no misspecification, so Aij = 1 whenever i, j +share at least one cluster. Hence, σ2 +n ≍ maxC∈{G,H} +� +c +� +i,j∈N C +c |ωiωj|, satisfying the conditions +9 + +of Theorem 1. Consequently, instead of making an assumption on the contribution of the cluster +with the largest weight on the total variance, a leverage condition in the form of Assumption +3.2 can be obtained. +This condition is also empirically verifiable: the researcher can calculate +LC := maxc +�� +i∈N C +c | ˜Di| +�2 +/ +�� +c′ +�� +j∈N C +c′ | ˜Dj| +�2� +, and check if it is small. As a benchmark, +when observations are not clustered and all weights ˜Di are the same, LC = 1/n. Hence, if we +believe that n = 30 is sufficiently large for asymptotics in the iid case, then LC < 1/30 may be +acceptable. +Proposition 1 implies that the usual inference procedure is still valid even when the unobserved +component is arbitrarily heterogeneous across different clusters. In contrast, the separate exchange- +ability of ui requires ui to be identically distributed across different clusters (e.g., the unobserved +component of wages for women is identically distributed across states) — it is a strong assumption +that is no longer required here. If there are fixed effects in the model, the vector of indicators can +be collected in W and the argument proceeds as usual.11 +3.2 +Stochastic Regressors +Next, consider stochastic X. This is the relevant case when considering causal regressions. For +example, we may be interested in the effect of a randomly assigned opportunity to participate in +a job training program Di on wages Yi. Both Xi and ui are plausibly correlated within state and +within industry. Although ˆθ = ˆβ1, we can no longer apply Theorem 1 to � +i ˜Diui because the +multi-way dependence structure breaks once Xi’s are random. +Define Sn := �n +i=1 E[XiX′ +i] and Qn := V ar (�n +i=1 Xiui), and denote their sample analogs as ˆSn = +� +i XiX′ +i and ˆQn := � +i +� +j∈Ni ˆuiˆujXiX′ +j. Let the smallest eigenvalue of Qn be λn := λmin(Qn). +The asymptotic variance of ˆβ and its sample analog are V (ˆβ) := S−1 +n QnS−1 +n +and ˆV (ˆβ) := ˆS−1 +n +ˆQn ˆS−1 +n +respectively. +Assumption 4 provides sufficient conditions for asymptotic normality of the estimator ˆβ and con- +sistency of the CGM variance estimator. The conditions mimic Assumption 2 so that Theorem 1 +is applicable to the random vector Xiui. The new condition is a weak regularity condition that +λmin (Sn/n) ≥ K1 > 0, mimicking to the rank condition in OLS. +11Fixed effects account for a shift in the unobserved component, so separate exchangeability still makes a restriction +on the distribution of the remaining unobserved component. +10 + +Assumption 4. For C ∈ {G, H}, and k ∈ {1, 2, · · · , K}, there exists K0 < ∞ and K1 > 0: +1. E[u4 +i |Xi] ≤ K0, E[X4 +ik] ≤ K0, E[ui|Xi] = 0 for all i. +2. +1 +λn maxc(N C +c )2 → 0. +3. +1 +λn +� +c(N C +c )2 ≤ K0. +4. (X′ +i, ui)′ ⊥⊥ (X′ +j, uj)′ if g(i) ̸= g(j) and h(i) ̸= h(j). +5. λmin +� 1 +nSn +� +≥ K1. +Proposition 2. Under Assumption 4, Q−1/2 +n +Sn(ˆβ−β) d−→ N(0, IK), and [S−1 +n QnS−1 +n ]−1[ ˆS−1 +n +ˆQn ˆS−1 +n ] +p−→ +IK. +Proposition 2 is useful for doing F tests on a subvector of β. The proof of Proposition 2 proceeds +by applying Theorem 1 to � +i Xiui, and showing that S−1 +n ˆSn +p−→ IK. The latter requires the rank +condition of Assumption 4.5. It then remains to show that the remainder terms are asymptotically +negligible. Nonetheless, if we are only interested in θ, using the residualized objects ˆθ and variance +estimator for the residualized object ˆσ2 +n is still valid. This follows from FWL, and the refinement +of FWL for variance estimators in Ding (2021). +Corollary 1. Under Assumption 4, (ˆθ − θ)/σn +d−→ N(0, 1), and ˆσn/σn +p−→ 1. +The practitioner’s takeaway from Proposition 2 is that the existing CGM variance estimator can be +used for valid inference with multi-way clustering. With Corollary 1, ˆθ and ˆσ2 +n can be used as the +mean and variance estimators respectively. These results provide the formal theoretical guarantee +for using the estimator, under weaker conditions that permits heterogeneity across clusters. +Besides the application mentioned, Theorem 1 also has implications on the conditions required for +valid inference when the random variable is multi-way clustered in many other econometric models, +including design-based settings and instrument variables models. Inference for estimators based +on moment conditions can be done by straightforward application of Theorem 1 as in the linear +regression case. +11 + +A +Proof of Theorem 1 +The proof strategy is as follows. I first prove Lemma 1, which is a central limit theorem (CLT) for +scalars that permits weights on the random variable. The proof of Lemma 1 relies on Lemmas 2 to +7. Lemmas 2 to 4 derive an upper bound on the Wasserstein distance between a pivotal statistic +and standard normal Z. Lemmas 5 to 7 then show that the derived upper bound is o(1). With +Lemma 1, the multivariate CLT of Theorem 1 is obtained by using the Cramer-Wold device. The +remainder of the proof proceeds in the following order: (i) introduce definitions and notation, (ii) +state Lemma 1, (iii) state and prove Lemmas 2 to 7, (iv) prove Lemma 1, (v) state and prove Lemma +8 that is required for consistent variance estimation, then (vi) complete the proof of Theorem 1. +The following definitions and notations are used throughout the proof. Let dW (X, Y ) denote the +Wasserstein distance between random variables X and Y , so dW(X, Y ) = 0 if and only if the +distributions of X and Y are identical. The norms of functions are defined as the sup norm i.e., +||f|| = supx∈D |f(x)|. For vector a, ||a|| = (a′a)1/2 is the Euclidean norm, and for positive semi- +definite matrix A and λmax(A) denoting the largest eigenvalue, ||A|| = +� +λmax(A′A) denotes the +spectral norm, and A1/2 denotes the symmetric matrix such that A1/2A1/2 = A. � +i∈N G +g +� +j∈N G +g is +abbreviated as � +i,j∈N G +g . The dependency neighborhood of i, Ni ⊆ {1, · · · , n}, is defined as the set +of observations where i ∈ Ni and Xi is independent of {Xj}j̸=Ni, and Ni := |Ni| is the number of +observations in i’s dependency neighborhood. In the rest of this proof, Xi denotes a scalar random +variable while Wi ∈ RK as stated in the main text is a random vector. +Every scalar random variable Xi is weighted by nonstochastic ωi. Denote the variance of the sum +as σ2 +n := V ar (�n +i=1 ωiXi). We are interested in the asymptotic distribution of (1/σn) �n +i=1 ωiXi. +If all observations are equally weighted, ωi = 1 ∀i. +Assumption 5. For C ∈ {G, H}, there exists K0 < ∞ such that: +1. E[Xi] = 0 and E[X4 +i ] ≤ K0 < ∞ for all i. +2. +1 +σ2n maxc +�� +i∈N C +c |ωi| +�2 +→ 0 +3. +1 +σ2n +� +c +� +i,j∈N C +c Aij|ωiωj| ≤ K0 < ∞ +4. Xi ⊥⊥ Xj if g(i) ̸= g(j) and h(i) ̸= h(j). +12 + +Lemma 1. Under Assumption 5, (1/σn) �n +i=1 ωiXi +d−→ N(0, 1), where σ2 +n := V ar (�n +i=1 ωiXi). +Further, using feasible estimator ˆσ2 +n := � +i +� +j∈Ni ωiωjXiXj, ˆσ2 +n/σ2 +n +p−→ 1. +Lemma 2. If R is a random variable and Z has a standard normal distribution, and we define +the family of functions F = {f : ||f||, ||f ′′|| ≤ 2, ||f ′|| ≤ +√ +2π}, then dW (R, Z) ≤ supf∈F |E[f ′(R) − +Rf(R)]|. +Proof. See Ross (2011) theorem 3.1. +Lemma 3. Let X1, · · · , Xn be random variables such that E[Xi] = 0, σ2 +n = V ar(� +i Xi), and define +R = � +i Xi/σn. If Ri := � +j̸=Ni Xj/σn, then +E[Rf(R)] = E +� +1 +σn +n +� +i=1 +Xi(f(R) − f(Ri) − (R − Ri)f ′(R)) +� ++ E +� +1 +σn +n +� +i=1 +Xi(R − Ri)f ′(R) +� +Proof. Start from right hand side: +E +� +1 +σn +n +� +i=1 +Xi(f(R) − f(Ri) − (R − Ri)f ′(R)) +� ++ E +� +1 +σn +n +� +i=1 +Xi(R − Ri)f ′(R) +� += E +� +1 +σn +n +� +i=1 +Xi(f(R) − f(Ri)) +� += E +� +1 +σn +n +� +i=1 +Xif(R) +� ++ E +� +1 +σn +n +� +i=1 +Xif(Ri) +� += E +� +1 +σn +n +� +i=1 +Xif(R) +� += E[Rf(R)] +The first equality in the final line comes from the fact that Ri is independent of Xi based on how +dependency neighborhoods are defined. Hence, E[Xif(Ri)] = 0. +Lemma 4. Let X1, · · · , Xn be random variables such that, E[Xi] = 0, σ2 +n = V ar(� +i Xi), and define +R = � +i Xi/σn. Let the collection (X1, · · · , Xn) have dependency neighborhoods Ni, i = 1, · · · , n. +Then for Z a standard normal random variable, +dW(R, Z) ≤ 1 +σ3n +n +� +i=1 +������ +� +j,k∈Ni +E[XiXjXk] +������ ++ +√ +2 +√πσ2n +� +� +� +� +�V ar + + +n +� +i=1 +� +j∈Ni +XiXj + + +(1) +Proof. Due to Lemma 2, to bound dW (R, Z) from above, it is sufficient to bound |E[f ′(R)−Rf(R)]|, +13 + +where ||f||, ||f ′′|| ≤ 2, ||f ′|| ≤ +� +2/π. Define Ri := � +j̸=Ni Xj/σn, so Xi is independent of Ri. +|E[f ′(R) − Rf(R)]| = |E[f ′(R)] − E[Rf(R)]| +≤ +�����E[f ′(R)] − E +� +1 +σn +n +� +i=1 +Xi(f(R) − f(Ri) − (R − Ri)f ′(R)) +� +− E +� +1 +σn +n +� +i=1 +Xi(R − Ri)f ′(R) +������ +≤ +�����E +� +1 +σn +n +� +i=1 +Xi(f(R) − f(Ri) − (R − Ri)f ′(R)) +������ + +�����E +� +f ′(R) +� +1 − 1 +σn +n +� +i=1 +Xi(R − Ri) +������� +The first inequality applies Lemma 3, and the second inequality applies the triangle inequality. +Consequently, it is sufficient to show that the first term is bounded by the corresponding first term +of Equation (1), and the second term is bounded by the corresponding second term. +Consider the first term. By Taylor expansion of f(Ri) around f(R), and the triangle inequality, +the term that generates the third moment is: +|E +� +1 +σn +n +� +i=1 +Xi(f(R) − f(Ri) − (R − Ri)f ′(R)) +� +| ≤ ||f ′′|| +2σn +����� +n +� +i=1 +E[Xi(R − Ri)2] +����� += 1 +σ3n +������ +n +� +i=1 +E + +Xi + + � +j∈Ni +Xj + + +2 + +������ += 1 +σ3n +������ +n +� +i=1 +� +j,k∈Ni +E[XiXjXk] +������ +≤ 1 +σ3n +n +� +i=1 +������ +� +j,k∈Ni +E[XiXjXk] +������ +Turning now to the second term, +�����E +� +f ′(R) +� +1 − 1 +σn +n +� +i=1 +Xi(R − Ri) +������� ≤ ||f ′|| +σ2n +�����E +� +σ2 +n − σn +n +� +i=1 +Xi(R − Ri) +������ +≤ ||f ′|| +σ2n +E +������ +σ2 +n − +n +� +i=1 +Xi + + � +j∈Ni +Xj + + +������ +≤ ||f ′|| +σ2n +E + + + +σ2 +n − +n +� +i=1 +Xi + + � +j∈Ni +Xj + + + + +2 + +1/2 +11/2 +≤ +√ +2 +√πσ2n +� +� +� +� +�V ar + + +n +� +i=1 +� +j∈Ni +XiXj + + +Lemma 5. E[|XiXjXk|] ≤ maxm E[|Xm|3], E[|XiXjXkXl|] ≤ maxm E[|Xm|4], and |E[XiXk]E[XjXl]| ≤ +maxm E[|Xm|4]. +14 + +Proof. By the arithmetic mean — geometric mean (AM-GM) inequality, +E|XiXjXk| ≤ 1 +3 +� +E|Xi|3 + E|Xj|3 + E|Xk|3� +≤ max +m E[|Xm|3] +A similar argument yields E[|XiXjXkXl|] ≤ maxm E[|Xm|4]. For the final result, first observe that +E[XiXk]2 ± 2E[XiXk]E[XjXl] + E[XjXl]2 = (E[XiXk] ± E[XjXl])2 ≥ 0. Hence, +|E[XiXk]E[XjXl]| ≤ 1 +2(E[XiXk]2 + E[XjXl]2) ≤ 1 +2(E[X2 +i X2 +k] + E[X2 +j X2 +l ]) +≤ 1 +4(E[X4 +i ] + E[X4 +j ] + E[X4 +k] + E[X4 +l ]) ≤ max +m E[X4 +m] +Lemma 6. Under Assumption 5, +1 +σ3n +�n +i=1 +���� +j,k∈Ni E[ωiωjωkXiXjXk] +��� → 0. +Proof. Note that E[XiXjXk] = 0 whenever one of {Xi, Xj, Xk} is independent of the other two, +so E[ωiωjωkXiXjXk] is nonzero only if Aij, Aik, or Ajk is nonzero. Apply the triangle inequality +and push the absolute value into the expectation. +1 +σ3n +n +� +i=1 +������ +� +j,k∈Ni +E[ωiωjωkXiXjXk] +������ +≤ 1 +σ3n +n +� +i=1 +������ +� +j,k∈Ni +(Aij + Ajk + Aik)E[ωiωjωkXiXjXk] +������ +≤ 1 +σ3n +n +� +i=1 +� +j,k∈Ni +(Aij + Ajk + Aik)|ωiωjωk|E[|XiXjXk|] +≤ maxm E[|Xm|3] +σ3n +n +� +i=1 +� +j,k∈Ni +|ωiωjωk|(Aij + Ajk + Aik) +The last inequality applies Lemma 5. Observe maxm E[|Xm|3] ≤ K0 since the 4th moment exists, +so it remains to show that the remaining terms are o(1). +1 +σ3n +n +� +i=1 +� +j,k∈Ni +(Aij + Ajk + Aik)|ωiωjωk| ≤ 1 +σ3n +n +� +i=1 + + + +� +j,k∈N G +g(i) ++ +� +j,k∈N H +h(i) + + + (Aij + Ajk + Aik)|ωiωjωk| +15 + +It is sufficient to consider the G dimension as the H dimension is analogous. +1 +σ3n +n +� +i=1 +� +j,k∈N G +g(i) +(Aij + Ajk + Aik)|ωiωjωk| = 3 +σ3n +� +g +� +i,j,k∈N G +g +Aij|ωiωjωk| +1 +σ3n +� +g +� +i,j,k∈N G +g +Aij|ωiωj||ωk| ≤ +�maxg +� +k∈N G +g |ωk| +σn +� +1 +σ2n +� +g +� +i,j∈N G +g +Aij|ωiωj| = o(1) +Convergence occurs because (1/σ2 +n) � +g +� +i,j∈N G +g Aij|ωiωj| < ∞ by Assumption 5.3 and maxg +� +k∈N G +g |ωk|/σn = +� +maxg +�� +k∈N G +g |ωk| +�2 +/σ2 +n +�1/2 += o(1) by Assumption 5.2. +Lemma 7. Under Assumption 5, +1 +σ4n V ar +��n +i=1 +� +j∈Ni ωiωjXiXj +� += o(1). +Proof. +1 +σ4n +V ar + +� +i +� +j∈Ni +ωiωjXiXj + + = 1 +σ4n +E + + + +� +i +� +j∈Ni +ωiωjXiXj + + +2 + − 1 +σ4n + +� +i +� +j∈Ni +E[ωiωjXiXj] + + +2 += 1 +σ4n +� +i +� +j +� +k∈Ni +� +l∈Nj +(E[ωiωjωkωlXiXjXkXl] − E[ωiωkXiXk]E[ωjωlXjXl]) += 1 +σ4n +� +i +� +j +� +k∈Ni +� +l∈Nj +ωiωjωkωl(E[XiXjXkXl] − E[XiXk]E[XjXl]) +When (Xi, Xk) ⊥⊥ (Xj, Xl), E[XiXjXkXl] = E[XiXj]E[XkXl]. Hence, we only have to consider +where there is at least one pair that is correlated i.e., when Aij, Ail, Akj, or Akl is not zero. As +before, with finite 4th moment and Lemma 5, it is sufficient to show +1 +σ4n +� +i +� +j +� +k∈Ni +� +l∈Nj +|ωiωjωkωl|(Aij + Ail + Akj + Akl) = o(1) +It is sufficient to consider the Aij term because everything else is analogous. +� +i +� +j +� +k∈Ni +� +l∈Nj +|ωiωjωkωl|Aij ≤ +� +i + + + +� +j∈N G +g(i) ++ +� +j∈N H +h(i) + + + + + + +� +k∈N G +g(i) ++ +� +k∈N H +h(i) + + + + + + +� +l∈N G +g(j) ++ +� +l∈N H +h(j) + + + |ωiωjωkωl|Aij +16 + +The first and last terms of the summation take the form: +� +i +� +j∈N G +g(i) +� +k∈N G +g(i) +� +l∈N G +g(j) +|ωiωjωkωl|Aij = +� +g +� +i,j,k,l∈N G +g +|ωiωjωkωl|Aij ≤ + +max +g +� +k,l∈N G +g +|ωk||ωl| + + � +g +� +i,j∈N G +g +|ωiωj|Aij +Since +1 +σ2n maxh +� +i,k∈N H +h |ωi||ωk| = o(1) and +1 +σ2n +� +g +� +i,j∈N G +g |ωiωj|Aij < ∞ by Assumption 5, these +terms are o(1) when divided by σ4 +n. +The interactive terms have the form: +� +i +� +j∈N G +g(i) +� +k∈N G +g(i) +� +l∈N H +h(j) +|ωiωjωkωl|Aij += +� +i,j,k +� +g +1[i ∈ N G +g ]1[j ∈ N G +g ]1[k ∈ N G +g ] +� +l +� +h +1[j ∈ N H +h ]1[l ∈ N H +h ]|ωiωjωkωl|Aij += +� +j +� +i,k +� +g +1[i ∈ N G +g ]1[j ∈ N G +g ]1[k ∈ N G +g ]Aij|ωiωjωk| +� +h +� +l +1[j ∈ N H +h ]1[l ∈ N H +h ]|ωl| +≤ +� +max +j +� +h +� +l +1[j ∈ N H +h ]1[l ∈ N H +h ]|ωl| +�  +� +g +� +i,j,k∈N G +g +|ωiωjωk|Aij + + += + +max +h +� +l∈N H +h +|ωl| + + + +� +g +� +i,j,k∈N G +g +|ωiωjωk|Aij + + += + +max +h +� +l∈N H +h +|ωl| + + + +max +g +� +k∈N G +g +|ωk| + + + +� +g +� +i,j∈N G +g +|ωiωj|Aij + + +Since � +g +� +i,j∈N G +g |ωiωj|Aij/σ2 +n ≤ K0 and maxg +� +k∈N G +g |ωk|/σn = o(1), +1 +σ4n +� +i +� +j∈N G +g(i) +� +k∈N G +g(i) +� +l∈N H +h(j) +|ωiωjωkωl|Aij +≤ + + 1 +σn +max +h +� +l∈N H +h +|ωl| + + + + 1 +σn +max +g +� +k∈N G +g +|ωk| + + + + 1 +σ2n +� +g +� +i,j∈N G +g +|ωiωj|Aij + + = o(1) +17 + +Proof of Lemma 1. Apply Lemma 4 on random variable ωiXi to obtain: +dW (R, Z) ≤ 1 +σ3n +n +� +i=1 +������ +� +j,k∈Ni +E[ωiωjωkXiXjXk] +������ ++ +√ +2 +√πσ2n +� +� +� +� +�V ar + + +n +� +i=1 +� +j∈Ni +ωiωjXiXj + + +Applying Lemma 6 and 7 on each of the two terms, dW (R, Z) = o(1). Proof for consistency of the +variance estimator is equivalent to proving that (ˆσ2 +n − σ2 +n)/σ2 +n = oP(1). By Chebyshev’s inequality, +P +� ˆσ2 +n − σ2 +n +σ2n +> ǫ +� +≤ 1 +ǫ2 +1 +σ4n +E[(ˆσ2 +n − σ2 +n)2] = +V ar +�� +i +� +j∈Ni ωiωjXiXj +� +ǫ2σ4n += oP (1) +The convergence in the last step occurs by Lemma 7. +Lemma 8. Under Assumption 1, 2.1 and 2.2, ∀i, ||(1/(� +i ωi)) � +i ωi(Wi − E[Wi])|| +p−→ 0. +Proof. It suffices to show convergence elementwise. Let Xi denote a scalar components of Wi, i.e., +Xi = Wim, where m ∈ {1, 2, · · · , K}. By Chebyshev’s inequality, and maxm,k E[W 2 +mk] < K0, +P +� +1 +� +i ωi +� +i +ωi(Xi − E[Xi]) > ǫ +� +≤ 1 +ǫ2 +1 +(� +i ωi)2 E + +� +i +� +j∈Ni +ωiωj(Xi − E[Xi])(Yi − E[Yi]) + + +≤ +K0 +ǫ2 +�� +j ωj +�2 +� +i +� +j∈Ni +ωiωj +Hence, it suffices to show (� +i +� +j∈Ni ωiωj)/ +�� +j ωj +�2 += o(1). Observe +� +i +� +j∈Ni ωiωj +�� +j ωj +�2 +≤ +maxi +� +j∈Ni |ωj| +�� +j ωj +� +�� +j ωj +� +�� +j ωj +� +so it suffices to show maxi +� +j∈Ni |ωj|/ +�� +j ωj +� += o(1). Since λn ≤ � +i +� +j∈Ni |ωiωj| maxm E[W 2 +mk] ≤ +�� +j |ωj| +�2 +maxm E[W 2 +mk], +� +maxi +� +j∈Ni |ωj| +�2 +�� +j ωj +�2 += +� +maxi +� +j∈Ni |ωj| +�2 +maxm E[W 2 +mk] +�� +j ωj +�2 +maxm E[W 2 +mk] +≤ max +m E[W 2 +mk] +� +maxi +� +j∈Ni |ωj| +�2 +λn += o(1) +18 + +Convergence occurs due to Assumption 2.2 and maxm E[W 2 +mk] < K0. +Proof of Theorem 1. To show that Q−1/2 +n +�n +i=1 ωi(Wi − E[Wi]) +d−→ N(0, IK), due to the Cramer- +Wold device, it suffices to show that ∀l ∈ RK, l′Q−1/2 +n +�n +i=1 ωi(Wi − E[Wi]) +d−→ l′N(0, IK). If l +is a vector of zeroes, then l′Q−1/2 +n +�n +i=1 ωi(Wi − E[Wi]) +d−→ l′N(0, IK) is immediate. For ||l|| > +0, it suffices to show (1/||l||)l′Q−1/2 +n +�n +i=1 ωi(Wi − E[Wi]) +d−→ (1/||l||)l′N(0, IK) = N(0, 1). +For +all nonstochastic l ∈ RK\{0}, let σ2 +n(l) := V ar +�� +i(l/||l||)′ (Qn/λn)−1/2 ωi(Wi − E[Wi]) +� +, so the +following hold: +1. E +��� +l +||l|| +�′ � +1 +λn Qn +�−1/2 +(Wi − E[Wi]) +�� += 0 and E +��� +l +||l|| +�′ � +1 +λn Qn +�−1/2 +(Wi − E[Wi]) +�4� +≤ +K0 for all i. +2. +1 +σ2n(l) maxc +�� +i∈N C +c |ωi| +�2 +→ 0. +3. +1 +σ2n(l) +� +c +� +i,j∈N C +c Aij|ωiωj| ≤ K0. +4. +�� +l +||l|| +�′ � +1 +λn Qn +�−1/2 +(Wi − E[Wi]) +� +⊥⊥ +�� +l +||l|| +�′ � +1 +λn Qn +�−1/2 +Wj +� +if g(i) ̸= g(j) and h(i) ̸= +h(j). +For item 1, since λn := λmin(Qn), all eigenvalues of Qn/λn must be at least 1. Hence, all eigen- +values of (Qn/λn)−1/2 are bounded above by 1. +This implies |(l/||l||)′(Qn/λn)−1/2| ≤ K1 for +some arbitrary constant K1 < ∞. +Item 1 then follows from Assumption 2.1. +Observe that +σ2 +n(l) = (l/||l||)′(Qn/λn)−1/2Qn(Qn/λn)−1/2(l/||l||) = 1/λn. +Then, Assumption 2.2 yields item +2 and Assumption 2.3 yields item 3. Item 4 is immediate from Assumption 1. By applying Lemma +1, (1/σn(l))(l/||l||)′(Qn/λn)−1/2 �n +i=1 ωi(Wi − E[Wi]) +d−→ N(0, 1). By using σ2 +n(l) = 1/λn, this is +equivalent to (l/||l||)′Q−1/2 +n +�n +i=1 ωi(Wi − E[Wi]) d−→ N(0, 1) as required. +Proof of Theorem 1.1 +Turning to consistent variance estimation, it suffices to show that for all l ∈ RK such that ||l|| = 1, +P(l′Q−1 +n ( ˆQn − Qn)l > ǫ) → 0. Now, impose the assumption that E[Wi] = 0. +P(l′Q−1 +n ( ˆQn − Qn)l > ǫ) ≤ 1 +ǫ2 E +�� +l′(Q−1 +n ( ˆQn − Qn)) +�2� += 1 +ǫ2 E + + +� +l′ +� 1 +λn +Qn +�−1 1 +λn +( ˆQn − Qn) +�2 + ≤ 1 +ǫ2 E +�� +l′ +0 +1 +λn +( ˆQn − Qn) +�2� +19 + +where l0 is a vector whose entries are all bounded above by some arbitrary constant K1 < ∞ by +a similar argument as before. Hence, it suffices to show that (1/λn)( ˆQn − Qn) +p−→ 0K×K, where +0K×K is a K × K matrix of zeroes. Since ˆQn − Qn = � +i +� +j∈Ni ωiωjWiW ′ +j − E[ωiωjWiW ′ +j], it +suffices to show convergence elementwise. +Let Xi and Yi denote scalar components of Wi, i.e., +Xi = Wim, Yi = Wip, where m, p ∈ {1, 2, · · · , K}. +P + + 1 +λn +� +i +� +j∈Ni +ωiωj(XiYj − E[XiYj]) > ǫ + + ≤ 1 +ǫ2 +1 +λ2n +V ar + +� +i +� +j∈Ni +ωiωjXiYj + + +≤ +1 +ǫ2λ2n +� +i +� +j +� +k∈Ni +� +l∈Nj +|E[ωiωjωkωlXiXjYkYl] − E[ωiωkXiYk]E[ωjωlXjYl]| +≤ K0 +λ2n +� +i +� +j +� +k∈Ni +� +l∈Nj +|ωiωjωkωl|(Aij + Ail + Akj + Akl) = o(1) +The inequality in the last line is obtained due to Holder’s inequality and finite moments. +An +argument similar to that of Lemma 7 yields the o(1) equality. +Proof of Theorem 1.2 +Now assume E[Wi] = µ. Using Lemma 8, ¯W +p−→ µ is immediate, i.e., ¯W = µ + oP(1). To ease +notation, let ˜Wi := Wi − µ. Hence, Qn = � +i +� +j∈Ni ωiωjE[ ˜Wi ˜W ′ +j]. +ˆQn = +� +i +� +j∈Ni +ωiωj(Wi − ¯W)(Wj − ¯W)′ = +� +i +� +j∈Ni +ωiωj( ˜Wi + oP(1))( ˜ +Wj + oP (1))′ += +� +i +� +j∈Ni +ωiωj ˜Wi ˜W ′ +j + 2 +� +i +� +j∈Ni +ωiωj ˜Wi1′ +KoP(1) + +� +i +� +j∈Ni +ωiωj1K1′ +KoP (1) +Since Q−1 +n +� +i +� +j∈Ni ωiωj ˜Wi ˜W ′ +j = 1 + oP (1) by Theorem 1.1, it then remains to show that each of +the two remaining terms are oP (1) when pre-multiplied by Q−1 +n . +� 1 +λ nQn +�−1 1 +λ n +� +i +� +j∈Ni +ωiωj1K1′ +K ≤ K0 +� 1 +λ nQn +�−1 +1K1′ +K = O(1)1K1′ +K +The first inequality is due to the assumption that (1/λn) � +i +� +j∈Ni |ωiωj| ≤ K0, and the O(1) term +occurs due to the eigenvalues of (Qn/λn)−1 being bounded above by 1. Take some component ˜Xi +20 + +of ˜Wi. For all ǫ > 0, there exists Mǫ = K2 +0/ǫ < ∞ such that: +P + + +������ +1 +λn +� +i +� +j∈Ni +ωiωj ˜Xi +������ +≥ Mǫ + + ≤ +1 +λnMǫ +E + + +������ +� +i +� +j∈Ni +ωiωj ˜Xi +������ + + +≤ +1 +Mǫ +max +i +E[| ˜Xi|] 1 +λn +� +i +� +j∈Ni +|ωiωj| ≤ +K0 +K2 +0/ǫ = ǫ +Hence, Q−1 +n +� +i +� +j∈Ni ωiωj ˜Wi1′ +K = 1K1′ +KOP (1). Since OP (1)oP (1) = oP (1), the result is obtained. +B +Proof of Propositions +Proof of Proposition 1. We have ˆθ−θ = +�� +i ˜D2 +i +�−1 �� +i ˜Diui +� += � +i ωiui, where ωi := ˜Di/(� +j ˜D2 +j). +Let σ2 +n := V ar(ωiui). Apply Theorem 1 with K = 1 to � +i ωiui. Assumption 1 and Assumption +2.1 are automatically satisfied for clustered random variable ui and weight ωi. Assumption 2.2 is +satisfied because +1 +σ2n +max +c + + � +i∈N C +c +|ωi| + + +2 +≤ +1 +( +� +i ˜D2 +i ) +2 maxc +�� +i∈N C +c | ˜Di| +�2 +K0 +1 +( +� +i ˜D2 +i ) +2 +� +c′ +�� +j∈N C +c′ | ˜Dj| +�2 = +1 +( +� +i ˜D2 +i ) +2 maxc +�� +i∈N C +c | ˜Di| +�2 +1 +( +� +i ˜D2 +i ) +2 +� +c′ +�� +j∈N C +c′ | ˜Dj| +�2 → 0 +where the first inequality comes from Assumption 3.3 and convergence occurs due to Assumption +3.2. Assumption 2.3 is satisfied because +1 +σ2n +� +c +� +i,j∈N C +c +Aij|ωiωj| = +1 +( +� +i ˜D2 +i ) +2 +� +c′ +� +i,j∈N C +c | ˜Di ˜Dj| +1 +( +� +i ˜D2 +i ) +2V ar +�� +i ˜Diui +� +< ∞ +Hence, Theorem 1 yields (ˆθ − θ)/σn +d−→ N(0, 1). +To prove consistent variance estimation, it suffices to show (ˆσ2 +n − σ2 +n)/σ2 +n = oP (1). +ˆσ2 +n = +� +i +� +j∈Ni +ωiuiωjuj − 2 + +� +i +� +j∈Ni +ω2 +i ωjuj + + (ˆθ − θ) + + +� +i +� +j∈Ni +ω2 +i ω2 +j + + (ˆθ − θ)2 +21 + +By Theorem 1, +�� +i +� +j∈Ni ωiuiωjuj − σ2 +n +� +/σ2 +n = oP (1). Since (ˆθ − θ)2/σ2 +n +d−→ Z2 = χ2 +1, +�� +i +� +j∈Ni ω2 +i ω2 +j +� +(ˆθ − θ)2 +σ2n += + +� +i +� +j∈Ni +ω2 +i ω2 +j + + OP (1) +�� +i +� +j∈Ni ˜D2 +i ˜D2 +j +� +�� +i ˜D2 +i +�4 +≤ +� +maxi +� +j∈Ni ˜D2 +j +� +�� +i ˜D2 +i +�3 +� +i ˜D2 +i +� +i ˜D2 +i +≤ +� +maxi +� +j∈Ni ˜D2 +j +� +�� +i ˜D2 +i +�2 +O(1) +≤ + + + +maxg +�� +j∈N G +g | ˜Dj| +�2 +� +g′ +�� +j∈N G +g′ | ˜Dj| +�2 + +maxh +�� +j∈N H +h | ˜Dj| +�2 +� +h′ +�� +j∈N H +h′ | ˜Dj| +�2 + + + O(1) = o(1) +Convergence occurs due to Assumption 3.2, so +�� +i +� +j∈Ni ω2 +i ω2 +j +� +(ˆθ − θ)2/σ2 +n = oP (1). Finally, +�� +i +� +j∈Ni ω2 +i ωjuj +� +(ˆθ − θ) +σ2n += +�� +i +� +j∈Ni ˜D2 +i ˜Djuj +� +σn +�� +i ˜D2 +i +�3 +OP (1) +Applying Markov and Minkowski inequalities, +P + + + +��� +� +i +� +j∈Ni ˜D2 +i ˜Djuj +��� +�� +i ˜D2 +i +�3 +σn +> ǫ + + + ≤ 1 +ǫ +1 +�� +i ˜D2 +i +�3 +σn +E + + +������ +� +i +� +j∈Ni +˜D2 +i ˜Djuj +������ + + +≤ 1 +ǫ +1 +�� +i ˜D2 +i +�3 +σn +� +i +� +j∈Ni +E| ˜D2 +i ˜Djuj| ≤ 1 +ǫ +maxi +� +j∈Ni E| ˜Djuj| +�� +i ˜D2 +i +�2 +σn +� +i ˜D2 +i +� +i ˜D2 +i += o(1) +Convergence occurs because +maxi +�� +j∈Ni E| ˜Djuj| +�2 +�� +i ˜D2 +i +�2 +≤ +maxj E|uj|2 maxi +�� +j∈Ni | ˜Dj| +�2 +�� +i ˜D2 +i +�2 += o(1) +For Proposition 2, I first prove a consistency result. +Lemma 9. Under Assumption 1, 2.1 and 2.2, and E[Wi] = 0 ∀i, ||(1/(� +i ωi)) � +i ωi(WiW ′ +i − +22 + +E[WiW ′ +i])|| +p−→ 0. +Proof. It suffices to show convergence elementwise. Let Xi and Yi denote scalar components of +Wi, i.e., Xi = Wim, Yi = Wip, where m, p ∈ {1, 2, · · · , K}. +By Chebyshev’s inequality, and +maxm,k E[W 4 +mk] < K0, +P +� +1 +� +i ωi +� +i +ωi(XiYi − E[XiYi]) > ǫ +� +≤ 1 +ǫ2 +1 +(� +i ωi)2 E + +� +i +� +j∈Ni +ωiωj(XiYi − E[XiYi])(XjYj − E[XjYj]) + + ≤ +K0 +ǫ2 +�� +j ωj +�2 +� +i +� +j∈Ni +ωiωj +Hence, it suffices to show (� +i +� +j∈Ni ωiωj)/ +�� +j ωj +�2 += o(1). This follows from a similar argument +as Lemma 8. +Proof of Proposition 2. E[ui|Xi] = 0 implies E[Xiui] = 0 by law of iterated expectations. Since +E[u4 +i |Xi] ≤ K0, E[u4 +i X4 +ik] = E[E[u4 +i |Xi]X4 +ik] ≤ K0E[X4 +ik] ≤ K2 +0 is bounded. +By Theorem 1, +Q−1/2 +n +�n +i=1 Xiui +d−→ N(0, IK). +To complete the normality result, I show that S−1 +n ˆSn +p−→ IK, which is the same as showing that +||S−1 +n ( ˆSn − Sn)|| +p−→ 0. By applying Lemma 9 with ωi = 1, (1/n)( ˆSn − Sn) = (1/n) � +i(XiX′ +i − +E[XiX′ +i]) = oP (1). Hence, it suffices that (Sn/n)−1 has bounded eigenvalues, i.e., λmin(Sn/n) ≥ +K1 > 0, which is true by Assumption 4.5. +Since ˆβ − β = ˆS−1 +n +� +i Xiui, by Slutsky’s lemma, +Q−1/2 +n +Sn(ˆβ − β) d−→ N(0, IK). +Next, proceed to consistent variance estimation. Showing that ||Q−1 +n +ˆQn−IK|| = oP (1) is equivalent +to showing that, ∀l ∈ RK, l′ � +Q−1 +n ( ˆQn − Qn) +� +l = oP (1). +ˆQn := +� +i +� +j∈Ni +ˆuiˆujXiX′ +j = +� +i +� +j∈Ni +(ui − X′ +i(ˆβ − β))(uj − X′ +j(ˆβ − β))XiX′ +j += +� +i +� +j∈Ni +uiujXiX′ +j − 2 + +� +i +� +j∈Ni +uiX′ +j(ˆβ − β)XiX′ +j + + + + +� +i +� +j∈Ni +X′ +i(ˆβ − β)X′ +j(ˆβ − β)XiX′ +j + + +23 + +By Theorem 1, l′Q−1 +n (� +i +� +j∈Ni uiujXiX′ +j − Qn)l = oP (1). Hence, it remains to show: +������ +������ +Q−1 +n + +−2 + +� +i +� +j∈Ni +uiX′ +j(ˆβ − β)XiX′ +j + + + + +� +i +� +j∈Ni +X′ +i(ˆβ − β)X′ +j(ˆβ − β)XiX′ +j + + + + +������ +������ += oP (1) +Observe that X′ +i(ˆβ−β) = +� +X′ +iS−1 +n Q1/2 +n +� � +Q−1/2 +n +Sn(ˆβ − β) +� += +� +X′ +iS−1 +n Q1/2 +n +� +(ZK+1KoP (1)), where +1K is a K-vector of ones. Hence, addressing the second term, +X′ +i(ˆβ − β)X′ +j(ˆβ − β) = +� +X′ +iS−1 +n Q1/2 +n +� +(ZK + 1KoP (1))(ZK + 1KoP (1))′ � +X′ +jS−1 +n Q1/2 +n +�′ += +� +X′ +iS−1 +n Q1/2 +n +� +(IKOP (1) + oP (1)) +� +X′ +jS−1 +n Q1/2 +n +�′ += X′ +iS−1 +n QnS−1 +n XjOP (1) +This implies +Q−1 +n + +� +i +� +j∈Ni +X′ +i(ˆβ − β)X′ +j(ˆβ − β)XiX′ +j + + = Q−1 +n + +� +i +� +j∈Ni +� +X′ +iS−1 +n QnS−1 +n Xj +� +XiX′ +j + + OP (1) += 1 +n2 +� 1 +λn +Qn +�−1 + +� +i +� +j∈Ni +� +X′ +i +� 1 +nSn +�−1 � 1 +λn +Qn +� � 1 +nSn +�−1 +Xj +� +XiX′ +j + + OP (1) +The eigenvalues of (Qn/λn) are bounded. To see this, it suffices to show that there exists K0 < ∞ +such that λmax(Qn)/λn ≤ K0. Due to finite moments, Qn := V ar(� +i Xi) ≤ K01K×K +� +c(N C +c )2. +Since (� +c(N C +c )2)/λn ≤ K0 by Assumption 4, λnK0 ≥ � +c(N C +c )2, which implies λn ≥ (� +c(N C +c )2)/K0. +Hence, +λmax(Qn) +λn +≤ +� +c(N C +c )2K0 +� +c(N C +c )2 1 +K0 += K2 +0 +Recall that (Sn/n)−1 has bounded eigenvalues. The proof of Theorem 1 also showed that (Qn/λn)−1 +24 + +has bounded eigenvalues. By using Markov and Minkowski inequalities, +P + + 1 +n2 +������ +l′ +� 1 +λn +Qn +�−1 + +� +i +� +j∈Ni +� +X′ +i +� 1 +nSn +�−1 � 1 +λn +Qn +� � 1 +nSn +�−1 +Xj +� +XiX′ +j + + l +������ +> ǫ + + +≤ +1 +n2ǫE + + +������ +l′ +� 1 +λn +Qn +�−1 + +� +i +� +j∈Ni +� +X′ +i +� 1 +nSn +�−1 � 1 +λn +Qn +� � 1 +nSn +�−1 +Xj +� +XiX′ +j + + l +������ + + +≤ +1 +n2ǫ +� +i +Ni max +m,k E[X4 +mk]K0 ≤ maxi Ni +n +n +nK0 → 0 +where K0 ∈ R is an arbitrary (finite) constant. Convergence occurs due to Assumption 4.2, which +implies maxi Ni/n → 0. This occurs due to the result that maxi +� +j∈Ni |ωj|/ +�� +j ωj +� += o(1) in the +proof of Lemma 8, using ωi = 1. +Going back to the first term, +Q−1 +n +� +i +� +j∈Ni +uiX′ +j(ˆβ − β)XiX′ +j = Q−1 +n +� +i +� +j∈Ni +ui +� +X′ +iS−1 +n Q1/2 +n +� +(ZK + 1KoP (1))XiX′ +j += +1 +n√λn +� 1 +λn +Qn +�−1 � +i +� +j∈Ni +ui +� +X′ +i +� 1 +nSn +�−1 � 1 +λn +Qn +�1/2� +XiX′ +jOP (1) +By using Markov and Minkowski inequalities, +P + + +1 +n√λn +������ +l′ +� 1 +λn +Qn +�−1 � +i +� +j∈Ni +ui +� +X′ +i +� 1 +nSn +�−1 � 1 +λn +Qn +�1/2� +XiX′ +jl +������ +> ǫ + + +≤ +1 +n√λnǫE + + +������ +l′ +� 1 +λn +Qn +�−1 � +i +� +j∈Ni +ui +� +X′ +i +� 1 +nSn +�−1 � 1 +λn +Qn +�1/2� +XiX′ +jl +������ + + +≤ +1 +n√λnǫ +� +i +� +j∈Ni +max +m1,m2,k E +���Xm1kum1X2 +m2 +��� +K0 +≤ +1 +n√λnǫ +� +i +Ni max +m1,m2,k E +� +|Xm1kum1|2�1/2 E +� +|X2 +m2|2�1/2 K0 +≤ maxi Ni +√λn +1 +ǫ +max +m1,m2,k E[X2 +m1ku2 +m1]1/2E[X4 +m2]1/2K0 = o(1) +The penultimate inequality occurs due to Holder’s inequality. Observe that maxi Ni/√λn = o(1) if +and only if maxc(N C +c )2/λn = o(1), which is given by Assumption 4.2. Convergence in the last step +occurs because maxi Ni/√λn = o(1), and finite moments. +25 + +Hence, it has been shown that Q−1 +n ˆQn +p−→ IK. +Then, [S−1 +n QnS−1 +n ]−1[ ˆS−1 +n +ˆQn ˆS−1 +n ] +p−→ IK by the +continuous mapping theorem. +Proof of Corollary 1. By Proposition 2, (ˆβ1 − β1)/[V (ˆβ)]1/2 +11 +d−→ N(0, 1). Since ˆθ = ˆβ1, [V (ˆβ)]11 = +V (ˆβ1) = V (ˆθ) = σ2 +n. Hence, (ˆθ − θ)/σn +d−→ N(0, 1). +A further implication of Proposition 2 is that [ ˆV (ˆβ)]11/[V (ˆβ)]11 +p−→ 1. Using theorem 3 of Ding +(2021), the Liang-Zeger estimators (Liang and Zeger, 1986) are numerically equivalent regardless +of whether the long regression or the residualized regression were used. Since the CGM estimator +is a function of the Liang-Zeger estimators, ˆσ2 +n = [ ˆV (ˆβ)]11. Hence, ˆσ2 +n/σ2 +n +p−→ 1. +References +Cameron, A. C., J. B. Gelbach, and D. L. Miller (2011): “Robust inference with multiway +clustering,” Journal of Business & Economic Statistics, 29, 238–249. +Chen, L. H. and Q.-M. Shao (2004): “Normal approximation under local dependence,” The +Annals of Probability, 32, 1985–2028. +Davezies, L., X. D’Haultfœuille, and Y. Guyonvarch (2021): “Empirical process results +for exchangeable arrays,” The Annals of Statistics, 49, 845–862. +Ding, P. (2021): “The Frisch–Waugh–Lovell theorem for standard errors,” Statistics & Probability +Letters, 168, 108945. +Djogbenou, A. A., J. G. MacKinnon, and M. Ø. Nielsen (2019): “Asymptotic theory and +wild bootstrap inference with clustered errors,” Journal of Econometrics, 212, 393–412. +Dube, A., T. W. Lester, and M. Reich (2010): “Minimum wage effects across state borders: +Estimates using contiguous counties,” The review of economics and statistics, 92, 945–964. +Hansen, B. E. and S. Lee (2019): “Asymptotic theory for clustered samples,” Journal of econo- +metrics, 210, 268–290. +Kallenberg, O. (2005): Probabilistic symmetries and invariance principles, vol. 9, Springer. +26 + +Liang, K.-Y. and S. L. Zeger (1986): “Longitudinal data analysis using generalized linear +models,” Biometrika, 73, 13–22. +MacKinnon, J. G., M. Ø. Nielsen, and M. D. Webb (2021): “Wild bootstrap and asymptotic +inference with multiway clustering,” Journal of Business & Economic Statistics, 39, 505–519. +Menzel, K. (2021): “Bootstrap With Cluster-Dependence in Two or More Dimensions,” Econo- +metrica, 89, 2143–2188. +Michalopoulos, S. and E. Papaioannou (2013): “Pre-colonial ethnic institutions and contem- +porary African development,” Econometrica, 81, 113–152. +Nunn, N. and L. Wantchekon (2011): “The slave trade and the origins of mistrust in Africa,” +American Economic Review, 101, 3221–52. +Ross, N. (2011): “Fundamentals of Stein’s method,” Probability Surveys, 8, 210–293. +27 + diff --git a/HNE2T4oBgHgl3EQfTwcd/content/tmp_files/load_file.txt b/HNE2T4oBgHgl3EQfTwcd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac2d8453776d7e39601038185ea458bfe88a252d --- /dev/null +++ b/HNE2T4oBgHgl3EQfTwcd/content/tmp_files/load_file.txt @@ -0,0 +1,697 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf,len=696 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='03805v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='EM] 10 Jan 2023 General Conditions for Valid Inference in Multi-Way Clustering Luther Yap ∗ January 11, 2023 Abstract This paper proves a new central limit theorem for a sample that exhibits multi-way depen- dence and heterogeneity across clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Statistical inference for situations where there is both multi-way dependence and cluster heterogeneity has thus far been an open issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Existing theory for multi-way clustering inference requires identical distributions across clusters (implied by the so-called separate exchangeability assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Yet no such homogeneity requirement is needed in the existing theory for one-way clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The new result therefore theoretically justifies the view that multi-way clustering is a more robust version of one-way clustering, consistent with applied practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The result is applied to linear regression, where it is shown that a standard plug-in variance estimator is valid for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 Introduction Clustering standard errors on multiple dimensions is common and attractive in applied econometrics because it allows observations to be dependent whenever they share a cluster on any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1 The variance estimator proposed by Cameron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' (2011) (henceforth CGM) has thus been widely applied to contexts with multi-way dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Existing justification for the asymptotic validity of the CGM estimator and other inference procedures in multi-way clustering relies on separate ∗Department of Economics, Princeton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Email: lyap@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', Dube et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' (2010) clustered on state and border segment when studying the effect of minimum wages on employment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Nunn and Wantchekon (2011) clustered on ethnic groups and district when studying the effect of slave trade on trust;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Michalopoulos and Papaioannou (2013) clustered on country and ethnolinguistic family when studying the effect of pre-colonial institutions on development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 exchangeability, which implies the homogeneity of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This paper provides general conditions such that the plug-in mean estimator is asymptotically normal, and the CGM variance estima- tor is consistent, even when clusters are heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' These conditions do not include separate exchangeability, and they mimic the conditions in one-way clustering: the only substantive assump- tion is that two observations are independent when they do not share any cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since asymptotic normality and consistent variance estimation are sufficient for valid inference, the results in this paper provide sufficient general conditions for valid inference in multi-way clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' An environment with multi-way clustering permits dependence whenever observations share at least one cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' To fix ideas, suppose observations can be partitioned on two different dimen- sions — state and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Observations in the same state or in the same industry are plausibly correlated, but two observations in different states and different industries are assumed to be inde- pendent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 The CGM variance estimator accommodates such dependence, and subsequent literature provided a theoretical basis for its validity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', Davezies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' MacKinnon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Menzel (2021) also showed the validity of a bootstrap procedure for multi-way clustering that is robust to asymptotic non-normalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='3 The theoretical basis for inference thus far relies on sepa- rate exchangeability, the assumption that random variables are exchangeable on either clustering dimension, though not necessarily both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' However, as noted by MacKinnon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' (2021), separate exchangeability implies identical marginal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since exchangeability implies identical distribution, separate exchangeability in the state-industry example implies that random variable in Alaska and California must be drawn from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' In contrast, existing asymptotic theory on one-way clustering (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', Hansen and Lee (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Djogbenou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' (2019)) allows the distribution of the random variable to be heterogeneous over clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The only substantive assumption is that observations that do not share any cluster are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since the only available conditions for the validity of multi-way clustering require separate exchangeability, the literature lacks general conditions for multi-way clustering that generalize one-way clustering and permit heterogeneity over clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This paper fills the gap, and thus justifies multi-way clustering as a more robust version of one-way clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 2This setting permits more general dependence structures than one-way clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' If there is one-way clustering by state, then two observations from different states are automatically independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' In two-way clustering, two observations from different states are not necessarily independent because they may share the same industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 3Menzel (2021) pointed out that a purely interactive data-generating processes unique to multi-way dependence has an asymptotic distribution that is not normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Section 2 will consider this process and show how the assumptions of this paper rules it out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 2 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' To illustrate separate exchangeability, consider an additive random effects model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Individual i who belongs to cluster g(i) on the G dimension and cluster h(i) on the H dimension has random variable Wi generated from Wi = αg(i) + γh(i) + εi, where cluster-specific αg, γh and individual-specific εi are independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' If we assume separate exchangeability, then αg, γh, and εi are iid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='4 In contrast, under one-way cluster asymptotics, the cluster-specific error αg is allowed to be heteroskedastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' General conditions provided in this paper permits valid inference even when αg, γh, εi are heteroskedastic in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The main result is a central limit theorem for multi-way clustering with heterogeneous cluster sizes and distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' I apply the theorem to a simple setting of a linear regression, but it is more broadly applicable to many other econometric procedures that exhibit a similar clustering structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 2 Setting and Main Result Consider a setup with two-way clustering on dimensions G and H for random vectors {Wi}n i=1, where Wi := (Wi1, Wi2, · · · , WiK)′ ∈ RK and i is the unit of observation, for a sequence of pop- ulations of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='5 For example, G could denote states and H denote industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This section establishes a central limit theorem (CLT) for a weighted sum of the random vector i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', � i ωiWi, where ωi are nonstochastic scalar weights, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For C ∈ {G, H}, let N C c denote the set of observations in cluster c on dimension C — this partitions the population on the C dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let g(i) and h(i) denote the cluster that observation i belongs to on the G and H dimensions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' These cluster identities are nonstochastic and observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let N C c = |N C c | denote the cluster size for C ∈ {G, H} and Ngh := |N G g ∩ N H h |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' These cluster sizes are allowed to be heterogeneous in a way that will be formalized in the assumptions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Wi is assumed to be independent of any Wj when j /∈ N G g(i) ∪ N H h(i) =: Ni, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', when i and j do not share a cluster on either dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, Ni is the set of observations plausibly dependent with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This environment is stated as Assumption 1, the main substantive assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Wi ⊥⊥ Wj if g(i) ̸= g(j) and h(i) ̸= h(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 4To see this, for individuals i and j where g(i) ̸= g(j), h(i) = h(j) = h, separate exchangeability implies αg(i)+γh+εi d= αg(j) + γh + εj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since αg, γh and εi are independent, εi d= εj and αg d= αg′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 5Clustering in more than two dimensions is possible, and derivations are entirely analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 3 Assumption 1 is agnostic about the dependence structure when Wi and Wj share at least one cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' It also allows the data generating process to be arbitrarily heterogeneous across different clusters, mimicking the heterogeneity permitted in one-way clustering (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', Hansen and Lee (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since one-way clustering is a special case of two-way clustering where everyone is in their own H cluster, the result here generalizes existing results in one-way clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' In contrast, existing literature in multi-way clustering assumes separate exchangeability that additionally imposes identical dis- tribution over clusters, so they do not immediately generalize one-way clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' {Wi}n i=1 being separately exchangeable implies Assumption 1 but the converse is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='6 Observations that share a cluster are allowed to be dependent, but they need not be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, let Aij := 1[Wi ̸⊥⊥ Wj] be a 0-1 indicator for whether Wi and Wj are actually dependent, so Aij = Aji, and Aii = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='7 This notation allows a particular form of misspecification where the researcher is conservative and clusters on dimension G when it is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Every observation Wi is weighted by nonstochastic scalar ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For positive definite matrix Q, let λmin(Q) denote the smallest eigenvalue of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Then, let Qn := V ar (�n i=1 ωiWi) denote the variance of the sum and λn := λmin(Qn) denote its smallest eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For example, when K = 1 and equal weights are placed on all observations, Wi is a scalar and λn = Qn = V ar(� i ωiWi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' K0 is used throughout the paper to denote an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For C ∈ {G, H}, and k ∈ {1, 2, · · · , K}, there exists K0 < ∞ such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' E[W 4 ik] ≤ K0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 λn maxc �� i∈N C c |ωi| �2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 λn � c � i,j∈N C c Aij|ωiωj| ≤ K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1 requires the fourth moment to be bounded, which is stronger than the moment condition in one-way clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='8 The proof in one-way clustering usually verifies a Lindeberg 6To illustrate this, let Ngh = 1 and Wgh denote the observation in cluster g and h on the respective dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Due to Kallenberg (2005), {Wgh}g≥1,h≥1 is separately exchangeable if and only if there exists a representation Wgh = f(αg, γh, εgh), where (αg, γh, εgh) iid ∼ U[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Then, it is obvious that Wgh ⊥⊥ Wg′h′ for g ̸= g′, h ̸= h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' A counterexample for the converse is some Wgh = −Wgh′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' These random variables are allowed to be perfectly correlated since they share a cluster under Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' However, we cannot find a representation f(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' ), because that representation implies E[Wgh|αg]⊥⊥ E[Wgh′|αg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 7It is insufficient to define the indicator as Aij := 1[Cov(Wi, Wj) ̸= 0], since the proof contains third and fourth moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For K = 1, zero covariance between a pair of observations is insufficient to ensure objects such as E[WiWjWk] and E[WiWjWkWl] − E[WiWk]E[WjWl] are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 8See equation (7) of Hansen and Lee (2019) for the condition in one-way clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 4 condition because blocks of observations are independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' With multi-way depen- dence, we no longer have independent blocks because each cluster can have observations that are dependent with observations from a different cluster when these observations share a cluster on a different dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, a different proof strategy is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The proof in this paper uses Stein’s method, which requires stronger moment restrictions, but provides a non-asymptotic bound on the approximation error — details are in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 requires the contribution of the cluster with the largest weight to be small relative to the total variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' In the special case where everyone is equally weighted with ωi = 1, the condition is simply (1/λn) maxc(N C c )2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Intuitively, this condition is required so that the removal of a cluster does not change the variance substantively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This assumption allows the ratio of any two cluster sizes to diverge to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' It is identical to equation (12) of Hansen and Lee (2019) when C = G = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 also rules out having components that are perfectly negatively correlated: if the components of the vector were perfectly negatively correlated, λn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='3 is fairly unrestrictive about the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' To aid exposition, suppose ωi = 1 ∀i, K = 1, and C is taken to be the clustering dimension that λn ≍ � c � i,j∈N C c Aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='9 With strong dependence, Aij = 1 for all i, j ∈ N C c , so λn ≍ � c(N C c )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' However, if the researcher were conservative and clustered on C when the data is indeed iid, then Aij = 1 if and only if i = j, so λn ≍ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='3 has implications on λn, which then determines how strong Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Namely, when λn ≍ n, Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 requires maxc(N C c )2/n → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' When λn ≍ � c(N C c )2, then Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 only requires maxc(N C c )2/(� c′(N C c′ )2) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The weaker version of Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 permits balanced panels where the unit and time dimensions increase at the same rate, while the stronger version does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='10 The assumption that (1/λn) � c(N C c )2 ≤ K0 matches equation (11) of Hansen and Lee (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='3 rules out the following purely interactive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' As pointed out by Menzel (2021), this model has an asymptotic distribution that is non-normal, and there is no analog in one-way clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For g ∈ {1, · · · , M}, h ∈ {1, · · · , M} and Ngh = 1, we observe Wgh = αgγh, where αg, γh are iid with mean zero and variances σ2 α and σ2 γ respectively, so there are M2 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Then, � g,h Wgh/M = �� g αg/ √ M � �� h γh/ √ M � d−→ Z1Z2, where Z1 and Z2 9To be clear about the notation, a ≍ b if and only if there exists K0 < ∞ such that a/b, b/a ∈ [−K0, K0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since E[W 2 i ] is bounded, λn ≍ maxC∈{G,H} � c � i,j∈N C c Aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 10To see this, let M denote the number of units and time periods, so there are M 2 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' maxc(N C c )2/(� c′(N C c′ )2) = M 2/M 3 = 1/M → 0, but maxc(N C c )2/n = M 2/M 2 = 1 ̸= o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 5 are independent standard normal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This limiting distribution is also known as Gaussian chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' � g(N G g )2/λn = M3/(M2σ2 ασ2 γ) = M/σ2 ασ2 γ → ∞ violates Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Under Assumption 1 and 2, Q−1/2 n �n i=1 ωi(Wi − E[Wi]) d−→ N(0, IK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Further, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' If E[Wi] = 0 ∀i, then Q−1 n ˆQn p−→ IK, where ˆQn := � i � j∈Ni ωiωjWiW ′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' If E[Wi] = µ ∀i and 1 λn � c � i,j∈N C c |ωiωj| ≤ K0 for some K0 < ∞, then, for ¯W = (� i ωiWi)/(� j ωj) and ˆQn := � i � j∈Ni ωiωj(Wi− ¯W)(Wj− ¯W)′, ¯W p−→ µ and Q−1 n ˆQn p−→ IK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The theorem tells us that, under the aforementioned conditions, Q−1/2 n �n i=1 ωi(Wi − E[Wi]) is asymptotically standard normal and the plug-in variance estimator proposed by CGM is consistent for multi-way clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' One-way clustering is a special case of this theorem when one dimension is weakly nested within the other: examples include G = H so both dimensions are identical, or if we cluster by county and state (as counties are nested in states), or if everyone is in their own H cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' A sufficient condition for consistent variance estimation is E[Wi] = 0, similiar to theorem 3 of Hansen and Lee (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This assumption is sufficient in many applications: for example, linear regressions considered in Section 3 are identified by requiring the expectation of the residual term to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Additionally, the condition E[Wi] = µ matches theorem 4 of Hansen and Lee (2019) for consistent variance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 uses a stronger form of Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='3 where Aij = 1 for all i, j ∈ N C c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' If E[Wi] ̸= 0, then the variance estimator need not be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Unlike one-way clustering, it may not even be conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Suppose E[Wi] ̸= 0 for some i, and define ˜Wi := Wi − E[Wi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Then, Q−1 n � i � j∈Ni WiW ′ j = Q−1 n �� i � j∈Ni ˜Wi ˜W ′ j � + Q−1 n �� i � j∈Ni E[Wi]E[Wj]′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since Q−1 n �� i � j∈Ni ˜Wi ˜W ′ j � = oP (1) by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1, and Qn is positive semidefinite, whether the asymptotic variance is over or under estimated depends on whether � i � j∈Ni E[Wi]E[Wj]′ is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let K = 1 for exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' In one-way clustering, the variance is weakly over- estimated, so inference is conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' To see this, let W G g denote the vector of Wi such that g(i) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' � i � j∈Ni E[Wi]E[Wj] = � g � i,j∈N G g E[Wi]E[Wj] = � g 1′E[W G g ]E[W G g ]′1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' In two-way clustering, � i � j∈Ni E[Wi]E[Wj] can be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' An example is where n = 3: cov(W1, W3) = 0 but cov(W1, W2) ̸= 0 and cov(W2, W3) ̸= 0, so W1 and W2 share a cluster in one dimension and W2 and W3 share a cluster on a different dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Further, E[W2] = −1 and E[W1] = E[W3] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Then, � i � j∈Ni E[Wi]E[Wj] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1 Proof Sketch The proof of Theorem 1 proceeds by first proving a CLT for a scalar random variable, then applying the Cramer-Wold device to obtain the multivariate CLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The scalar CLT is proven using Stein’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' I adapt the proof strategy from Ross (2011) to obtain an upper bound on the Wasserstein distance between a pivotal statistic and the standard normal random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By exploiting the multi-way clustering structure, the upper bound on the distance can be shown to converge to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' All details are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For ease of exposition, consider a simpler environment where K = 1, ωi = 1 for all i, and Aij = 1 whenever c(i) = c(j) for some c, and E[Wi] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Lemma 4 in Appendix A provides an explicit bound on the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' With dW (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=') denoting the Wasserstein distance, σ2 n := Qn and R = � i Xi/σn, dW (R, Z) ≤ 1 σ3n n � i=1 ������ � j,k∈Ni E[WiWjWk] ������ + √ 2 √πσ2n � � � � �V ar \uf8eb \uf8ed n � i=1 � j∈Ni WiWj \uf8f6 \uf8f8 At this point, my proof departs from the proofs in existing statistical literature that employ Stein’s method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', Chen and Shao (2004)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let Ni := |Ni|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Holder’s inequality is employed on objects such as � i | � j,k∈Ni E[WiWjWk]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Existing literature uses the L1 norm of moments E[W 3 i ] and L∞ norm of Ni, resulting in (maxm Nm)2 � i E[W 3 i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' In contrast, my proof uses the L∞ norm of E[W 3 i ] and L1 norm of Ni, resulting in maxm E[W 3 m] � i N 2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, 1 σ3n n � i=1 ������ � j,k∈Ni E[WiWjWk] ������ ≤ 1 σ3n max m E[W 3 m] � i N 2 i Since maxm E[W 3 m] is bounded by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1, it suffices to show � i N 2 i /σ3 n → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Due to Assumption 1, Ni ≤ N G g(i) + N H h(i), so 1 σ3n � i N 2 i ≤ 1 σ3n � i (N G g(i) + N H h(i))2 ≤ 1 σ3n max g,h (N G g + N H h ) � i (Ng(i) + Nh(i)) ≤ � 1 σn max g,h (N G g + N H h ) � 1 σ2n �� g (N G g )2 + � h (N H h )2 � 7 Since λn = σn when K = 1, maxg,h(N G g +N H h )/σn → 0 by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 and �� g(N G g )2 + � h(N H h )2� /σ2 n is bounded by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, the term is o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' A similar argument is made for the fourth moment that features in V ar ��n i=1 � j∈Ni WiWj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' To complete the proof for variance estimation, observe that since the fourth moments exist, the consistency of the plug-in variance estimator can be proven by using Chebyshev’s inequality and existing intermediate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Due to the proof strategy, the intermediate results are informative about the quality of the normal approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' With dK(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=') denoting the Kolmogorov distance, proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 from Ross (2011) implies that dK(R, Z) ≤ (2/π)1/4� dW (R, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since Z is standard normal in the proof of CLT, the bound on dW (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=') also places a bound on the Kolmogorov distance dK(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This is then informative of the maximum distance between the pivotal statistic and the standard normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 3 Application This section applies Theorem 1 to linear regressions, showing that using the normal approximation with the CGM estimator is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Consider a linear model where scalar outcome Yi is generated by Yi = Diθ + W ′ iγ + ui =: X′ iβ + ui Di ∈ R is the regressor of interest, Wi ∈ RK−1 is a vector of controls that may include the intercept, and let Xi = (Xi1, Xi2, · · · , XiK)′ := (Di, W ′ i)′ ∈ RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' We are interested in estimating θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The coefficient vector β := (θ, γ′)′ ∈ RK is the same for all individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The stochastic residual term ui satisfies E[ui|Xi] = 0 for all i, and is allowed to be multi-way clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The standard OLS estimator is ˆβ = � n � i=1 XiX′ i �−1 � n � i=1 XiYi � = β + � n � i=1 XiX′ i �−1 � n � i=1 Xiui � This object is assumed to be well-defined in that �n i=1 XiX′ i is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Using an equivalent representation with data matrices, the model is Y = Dθ + Wγ + u = Xβ + u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let MW = I − W(W ′W)−1W ′ denote the annihilator matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let ˜D := MW D be the D with W’s partialled 8 out, and define ˜Y , ˜u in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By the Frisch-Waugh-Lovell theorem (FWL), ˆθ = ( ˜D′ ˜D)−1 ˜D′ ˜Y = θ + ( ˜D′ ˜D)−1 ˜D′˜u = θ + �� i ˜D2 i �−1 �� i ˜Di˜ui � = ˆβ1 where ˜Di is the ith component of ˜D, so � i ˜Di˜ui = ˜D′˜u = D′MWu = � i ˜Diui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let σ2 n := V ar(ˆθ) = V ar �� i ˜Diui/(� i′ ˜D2 i′) � and ˆσ2 n := �� i � j∈Ni ˆuiˆuj ˜Di ˜Dj � / �� i ˜D2 i �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Estimated residuals are ˆui := Yi − Xi ˆβ = ui − Xi(ˆβ − β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Due to FWL, ˆui = ˜Yi − ˜Diˆθ = ui − ˜Di(ˆθ − θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Inference for ˆθ, depends on whether we are conditioning on X: the conditions for asymptotic normality differ slightly between random and nonrandom X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' I consider each of them in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1 Fixed Regressors First, consider regressions where the X’s are nonrandom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' An example might be when the object of interest is the difference between male and female wages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Their unobserved error may be correlated by state and industry conditional on X, but the gender status Di is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This can be viewed as inference on a descriptive object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' With ui’s having a multi-way clustered structure, we can apply Theorem 1 on �� i ˜D2 i �−1 � i ˜Diui, where scalar weights are given by ωi = ˜Di/(� i′ ˜D2 i′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For C ∈ {G, H} and nonstochastic ˜Di, there exists K0 < ∞ such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' E[u4 i ] ≤ K0, E[ui] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' maxc �� i∈N C c | ˜Di| �2 � c′ �� j∈N C c′ | ˜Dj| �2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' � c′ � i,j∈N C c | ˜Di ˜Dj| V ar( � i ˜Diui) ≤ K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' ui ⊥⊥ uj if g(i) ̸= g(j) and h(i) ̸= h(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Under Assumption 3, (ˆθ − θ)/σn d−→ N(0, 1), and ˆσ2 n/σ2 n p−→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 3 works in the environment where there is no misspecification, so Aij = 1 whenever i, j share at least one cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, σ2 n ≍ maxC∈{G,H} � c � i,j∈N C c |ωiωj|, satisfying the conditions 9 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Consequently, instead of making an assumption on the contribution of the cluster with the largest weight on the total variance, a leverage condition in the form of Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This condition is also empirically verifiable: the researcher can calculate LC := maxc �� i∈N C c | ˜Di| �2 / �� c′ �� j∈N C c′ | ˜Dj| �2� , and check if it is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' As a benchmark, when observations are not clustered and all weights ˜Di are the same, LC = 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, if we believe that n = 30 is sufficiently large for asymptotics in the iid case, then LC < 1/30 may be acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proposition 1 implies that the usual inference procedure is still valid even when the unobserved component is arbitrarily heterogeneous across different clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' In contrast, the separate exchange- ability of ui requires ui to be identically distributed across different clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', the unobserved component of wages for women is identically distributed across states) — it is a strong assumption that is no longer required here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' If there are fixed effects in the model, the vector of indicators can be collected in W and the argument proceeds as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 Stochastic Regressors Next, consider stochastic X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This is the relevant case when considering causal regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For example, we may be interested in the effect of a randomly assigned opportunity to participate in a job training program Di on wages Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Both Xi and ui are plausibly correlated within state and within industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Although ˆθ = ˆβ1, we can no longer apply Theorem 1 to � i ˜Diui because the multi-way dependence structure breaks once Xi’s are random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Define Sn := �n i=1 E[XiX′ i] and Qn := V ar (�n i=1 Xiui), and denote their sample analogs as ˆSn = � i XiX′ i and ˆQn := � i � j∈Ni ˆuiˆujXiX′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let the smallest eigenvalue of Qn be λn := λmin(Qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The asymptotic variance of ˆβ and its sample analog are V (ˆβ) := S−1 n QnS−1 n and ˆV (ˆβ) := ˆS−1 n ˆQn ˆS−1 n respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 4 provides sufficient conditions for asymptotic normality of the estimator ˆβ and con- sistency of the CGM variance estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The conditions mimic Assumption 2 so that Theorem 1 is applicable to the random vector Xiui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The new condition is a weak regularity condition that λmin (Sn/n) ≥ K1 > 0, mimicking to the rank condition in OLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 11Fixed effects account for a shift in the unobserved component, so separate exchangeability still makes a restriction on the distribution of the remaining unobserved component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 10 Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For C ∈ {G, H}, and k ∈ {1, 2, · · · , K}, there exists K0 < ∞ and K1 > 0: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' E[u4 i |Xi] ≤ K0, E[X4 ik] ≤ K0, E[ui|Xi] = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 λn maxc(N C c )2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 λn � c(N C c )2 ≤ K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' (X′ i, ui)′ ⊥⊥ (X′ j, uj)′ if g(i) ̸= g(j) and h(i) ̸= h(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' λmin � 1 nSn � ≥ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Under Assumption 4, Q−1/2 n Sn(ˆβ−β) d−→ N(0, IK), and [S−1 n QnS−1 n ]−1[ ˆS−1 n ˆQn ˆS−1 n ] p−→ IK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proposition 2 is useful for doing F tests on a subvector of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The proof of Proposition 2 proceeds by applying Theorem 1 to � i Xiui, and showing that S−1 n ˆSn p−→ IK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The latter requires the rank condition of Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' It then remains to show that the remainder terms are asymptotically negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Nonetheless, if we are only interested in θ, using the residualized objects ˆθ and variance estimator for the residualized object ˆσ2 n is still valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This follows from FWL, and the refinement of FWL for variance estimators in Ding (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Under Assumption 4, (ˆθ − θ)/σn d−→ N(0, 1), and ˆσn/σn p−→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The practitioner’s takeaway from Proposition 2 is that the existing CGM variance estimator can be used for valid inference with multi-way clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' With Corollary 1, ˆθ and ˆσ2 n can be used as the mean and variance estimators respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' These results provide the formal theoretical guarantee for using the estimator, under weaker conditions that permits heterogeneity across clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Besides the application mentioned, Theorem 1 also has implications on the conditions required for valid inference when the random variable is multi-way clustered in many other econometric models, including design-based settings and instrument variables models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Inference for estimators based on moment conditions can be done by straightforward application of Theorem 1 as in the linear regression case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 11 A Proof of Theorem 1 The proof strategy is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' I first prove Lemma 1, which is a central limit theorem (CLT) for scalars that permits weights on the random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The proof of Lemma 1 relies on Lemmas 2 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Lemmas 2 to 4 derive an upper bound on the Wasserstein distance between a pivotal statistic and standard normal Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Lemmas 5 to 7 then show that the derived upper bound is o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' With Lemma 1, the multivariate CLT of Theorem 1 is obtained by using the Cramer-Wold device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The remainder of the proof proceeds in the following order: (i) introduce definitions and notation, (ii) state Lemma 1, (iii) state and prove Lemmas 2 to 7, (iv) prove Lemma 1, (v) state and prove Lemma 8 that is required for consistent variance estimation, then (vi) complete the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The following definitions and notations are used throughout the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let dW (X, Y ) denote the Wasserstein distance between random variables X and Y , so dW(X, Y ) = 0 if and only if the distributions of X and Y are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The norms of functions are defined as the sup norm i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', ||f|| = supx∈D |f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For vector a, ||a|| = (a′a)1/2 is the Euclidean norm, and for positive semi- definite matrix A and λmax(A) denoting the largest eigenvalue, ||A|| = � λmax(A′A) denotes the spectral norm, and A1/2 denotes the symmetric matrix such that A1/2A1/2 = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' � i∈N G g � j∈N G g is abbreviated as � i,j∈N G g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The dependency neighborhood of i, Ni ⊆ {1, · · · , n}, is defined as the set of observations where i ∈ Ni and Xi is independent of {Xj}j̸=Ni, and Ni := |Ni| is the number of observations in i’s dependency neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' In the rest of this proof, Xi denotes a scalar random variable while Wi ∈ RK as stated in the main text is a random vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Every scalar random variable Xi is weighted by nonstochastic ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Denote the variance of the sum as σ2 n := V ar (�n i=1 ωiXi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' We are interested in the asymptotic distribution of (1/σn) �n i=1 ωiXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' If all observations are equally weighted, ωi = 1 ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For C ∈ {G, H}, there exists K0 < ∞ such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' E[Xi] = 0 and E[X4 i ] ≤ K0 < ∞ for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 σ2n maxc �� i∈N C c |ωi| �2 → 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 σ2n � c � i,j∈N C c Aij|ωiωj| ≤ K0 < ∞ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Xi ⊥⊥ Xj if g(i) ̸= g(j) and h(i) ̸= h(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 12 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Under Assumption 5, (1/σn) �n i=1 ωiXi d−→ N(0, 1), where σ2 n := V ar (�n i=1 ωiXi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Further, using feasible estimator ˆσ2 n := � i � j∈Ni ωiωjXiXj, ˆσ2 n/σ2 n p−→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' If R is a random variable and Z has a standard normal distribution, and we define the family of functions F = {f : ||f||, ||f ′′|| ≤ 2, ||f ′|| ≤ √ 2π}, then dW (R, Z) ≤ supf∈F |E[f ′(R) − Rf(R)]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' See Ross (2011) theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let X1, · · · , Xn be random variables such that E[Xi] = 0, σ2 n = V ar(� i Xi), and define R = � i Xi/σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' If Ri := � j̸=Ni Xj/σn, then E[Rf(R)] = E � 1 σn n � i=1 Xi(f(R) − f(Ri) − (R − Ri)f ′(R)) � + E � 1 σn n � i=1 Xi(R − Ri)f ′(R) � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Start from right hand side: E � 1 σn n � i=1 Xi(f(R) − f(Ri) − (R − Ri)f ′(R)) � + E � 1 σn n � i=1 Xi(R − Ri)f ′(R) � = E � 1 σn n � i=1 Xi(f(R) − f(Ri)) � = E � 1 σn n � i=1 Xif(R) � + E � 1 σn n � i=1 Xif(Ri) � = E � 1 σn n � i=1 Xif(R) � = E[Rf(R)] The first equality in the final line comes from the fact that Ri is independent of Xi based on how dependency neighborhoods are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, E[Xif(Ri)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let X1, · · · , Xn be random variables such that, E[Xi] = 0, σ2 n = V ar(� i Xi), and define R = � i Xi/σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let the collection (X1, · · · , Xn) have dependency neighborhoods Ni, i = 1, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Then for Z a standard normal random variable, dW(R, Z) ≤ 1 σ3n n � i=1 ������ � j,k∈Ni E[XiXjXk] ������ + √ 2 √πσ2n � � � � �V ar \uf8eb \uf8ed n � i=1 � j∈Ni XiXj \uf8f6 \uf8f8 (1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Due to Lemma 2, to bound dW (R, Z) from above, it is sufficient to bound |E[f ′(R)−Rf(R)]|, 13 where ||f||, ||f ′′|| ≤ 2, ||f ′|| ≤ � 2/π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Define Ri := � j̸=Ni Xj/σn, so Xi is independent of Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' |E[f ′(R) − Rf(R)]| = |E[f ′(R)] − E[Rf(R)]| ≤ �����E[f ′(R)] − E � 1 σn n � i=1 Xi(f(R) − f(Ri) − (R − Ri)f ′(R)) � − E � 1 σn n � i=1 Xi(R − Ri)f ′(R) ������ ≤ �����E � 1 σn n � i=1 Xi(f(R) − f(Ri) − (R − Ri)f ′(R)) ������ + �����E � f ′(R) � 1 − 1 σn n � i=1 Xi(R − Ri) ������� The first inequality applies Lemma 3, and the second inequality applies the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Consequently, it is sufficient to show that the first term is bounded by the corresponding first term of Equation (1), and the second term is bounded by the corresponding second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Consider the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By Taylor expansion of f(Ri) around f(R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' and the triangle inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' the term that generates the third moment is: |E � 1 σn n � i=1 Xi(f(R) − f(Ri) − (R − Ri)f ′(R)) � | ≤ ||f ′′|| 2σn ����� n � i=1 E[Xi(R − Ri)2] ����� = 1 σ3n ������ n � i=1 E \uf8ee \uf8f0Xi \uf8eb \uf8ed � j∈Ni Xj \uf8f6 \uf8f8 2\uf8f9 \uf8fb ������ = 1 σ3n ������ n � i=1 � j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='k∈Ni E[XiXjXk] ������ ≤ 1 σ3n n � i=1 ������ � j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='k∈Ni E[XiXjXk] ������ Turning now to the second term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' �����E � f ′(R) � 1 − 1 σn n � i=1 Xi(R − Ri) ������� ≤ ||f ′|| σ2n �����E � σ2 n − σn n � i=1 Xi(R − Ri) ������ ≤ ||f ′|| σ2n E ������ σ2 n − n � i=1 Xi \uf8eb \uf8ed � j∈Ni Xj \uf8f6 \uf8f8 ������ ≤ ||f ′|| σ2n E \uf8ee \uf8f0 \uf8eb \uf8edσ2 n − n � i=1 Xi \uf8eb \uf8ed � j∈Ni Xj \uf8f6 \uf8f8 \uf8f6 \uf8f8 2\uf8f9 \uf8fb 1/2 11/2 ≤ √ 2 √πσ2n � � � � �V ar \uf8eb \uf8ed n � i=1 � j∈Ni XiXj \uf8f6 \uf8f8 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' E[|XiXjXk|] ≤ maxm E[|Xm|3], E[|XiXjXkXl|] ≤ maxm E[|Xm|4], and |E[XiXk]E[XjXl]| ≤ maxm E[|Xm|4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By the arithmetic mean — geometric mean (AM-GM) inequality, E|XiXjXk| ≤ 1 3 � E|Xi|3 + E|Xj|3 + E|Xk|3� ≤ max m E[|Xm|3] A similar argument yields E[|XiXjXkXl|] ≤ maxm E[|Xm|4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For the final result, first observe that E[XiXk]2 ± 2E[XiXk]E[XjXl] + E[XjXl]2 = (E[XiXk] ± E[XjXl])2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, |E[XiXk]E[XjXl]| ≤ 1 2(E[XiXk]2 + E[XjXl]2) ≤ 1 2(E[X2 i X2 k] + E[X2 j X2 l ]) ≤ 1 4(E[X4 i ] + E[X4 j ] + E[X4 k] + E[X4 l ]) ≤ max m E[X4 m] Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Under Assumption 5, 1 σ3n �n i=1 ���� j,k∈Ni E[ωiωjωkXiXjXk] ��� → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Note that E[XiXjXk] = 0 whenever one of {Xi, Xj, Xk} is independent of the other two, so E[ωiωjωkXiXjXk] is nonzero only if Aij, Aik, or Ajk is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Apply the triangle inequality and push the absolute value into the expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 σ3n n � i=1 ������ � j,k∈Ni E[ωiωjωkXiXjXk] ������ ≤ 1 σ3n n � i=1 ������ � j,k∈Ni (Aij + Ajk + Aik)E[ωiωjωkXiXjXk] ������ ≤ 1 σ3n n � i=1 � j,k∈Ni (Aij + Ajk + Aik)|ωiωjωk|E[|XiXjXk|] ≤ maxm E[|Xm|3] σ3n n � i=1 � j,k∈Ni |ωiωjωk|(Aij + Ajk + Aik) The last inequality applies Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Observe maxm E[|Xm|3] ≤ K0 since the 4th moment exists, so it remains to show that the remaining terms are o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 σ3n n � i=1 � j,k∈Ni (Aij + Ajk + Aik)|ωiωjωk| ≤ 1 σ3n n � i=1 \uf8eb \uf8ec \uf8ed � j,k∈N G g(i) + � j,k∈N H h(i) \uf8f6 \uf8f7 \uf8f8 (Aij + Ajk + Aik)|ωiωjωk| 15 It is sufficient to consider the G dimension as the H dimension is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 σ3n n � i=1 � j,k∈N G g(i) (Aij + Ajk + Aik)|ωiωjωk| = 3 σ3n � g � i,j,k∈N G g Aij|ωiωjωk| 1 σ3n � g � i,j,k∈N G g Aij|ωiωj||ωk| ≤ �maxg � k∈N G g |ωk| σn � 1 σ2n � g � i,j∈N G g Aij|ωiωj| = o(1) Convergence occurs because (1/σ2 n) � g � i,j∈N G g Aij|ωiωj| < ∞ by Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='3 and maxg � k∈N G g |ωk|/σn = � maxg �� k∈N G g |ωk| �2 /σ2 n �1/2 = o(1) by Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Under Assumption 5, 1 σ4n V ar ��n i=1 � j∈Ni ωiωjXiXj � = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 σ4n V ar \uf8eb \uf8ed� i � j∈Ni ωiωjXiXj \uf8f6 \uf8f8 = 1 σ4n E \uf8ee \uf8f0 \uf8eb \uf8ed� i � j∈Ni ωiωjXiXj \uf8f6 \uf8f8 2\uf8f9 \uf8fb − 1 σ4n \uf8eb \uf8ed� i � j∈Ni E[ωiωjXiXj] \uf8f6 \uf8f8 2 = 1 σ4n � i � j � k∈Ni � l∈Nj (E[ωiωjωkωlXiXjXkXl] − E[ωiωkXiXk]E[ωjωlXjXl]) = 1 σ4n � i � j � k∈Ni � l∈Nj ωiωjωkωl(E[XiXjXkXl] − E[XiXk]E[XjXl]) When (Xi, Xk) ⊥⊥ (Xj, Xl), E[XiXjXkXl] = E[XiXj]E[XkXl].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, we only have to consider where there is at least one pair that is correlated i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', when Aij, Ail, Akj, or Akl is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' As before, with finite 4th moment and Lemma 5, it is sufficient to show 1 σ4n � i � j � k∈Ni � l∈Nj |ωiωjωkωl|(Aij + Ail + Akj + Akl) = o(1) It is sufficient to consider the Aij term because everything else is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' � i � j � k∈Ni � l∈Nj |ωiωjωkωl|Aij ≤ � i \uf8eb \uf8ec \uf8ed � j∈N G g(i) + � j∈N H h(i) \uf8f6 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ed � k∈N G g(i) + � k∈N H h(i) \uf8f6 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ed � l∈N G g(j) + � l∈N H h(j) \uf8f6 \uf8f7 \uf8f8 |ωiωjωkωl|Aij 16 The first and last terms of the summation take the form: � i � j∈N G g(i) � k∈N G g(i) � l∈N G g(j) |ωiωjωkωl|Aij = � g � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='l∈N G g |ωiωjωkωl|Aij ≤ \uf8eb \uf8edmax g � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='l∈N G g |ωk||ωl| \uf8f6 \uf8f8 � g � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j∈N G g |ωiωj|Aij Since 1 σ2n maxh � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='k∈N H h |ωi||ωk| = o(1) and 1 σ2n � g � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j∈N G g |ωiωj|Aij < ∞ by Assumption 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' these terms are o(1) when divided by σ4 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The interactive terms have the form: � i � j∈N G g(i) � k∈N G g(i) � l∈N H h(j) |ωiωjωkωl|Aij = � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='k � g 1[i ∈ N G g ]1[j ∈ N G g ]1[k ∈ N G g ] � l � h 1[j ∈ N H h ]1[l ∈ N H h ]|ωiωjωkωl|Aij = � j � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='k � g 1[i ∈ N G g ]1[j ∈ N G g ]1[k ∈ N G g ]Aij|ωiωjωk| � h � l 1[j ∈ N H h ]1[l ∈ N H h ]|ωl| ≤ � max j � h � l 1[j ∈ N H h ]1[l ∈ N H h ]|ωl| � \uf8eb \uf8ed� g � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='k∈N G g |ωiωjωk|Aij \uf8f6 \uf8f8 = \uf8eb \uf8edmax h � l∈N H h |ωl| \uf8f6 \uf8f8 \uf8eb \uf8ed� g � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='k∈N G g |ωiωjωk|Aij \uf8f6 \uf8f8 = \uf8eb \uf8edmax h � l∈N H h |ωl| \uf8f6 \uf8f8 \uf8eb \uf8edmax g � k∈N G g |ωk| \uf8f6 \uf8f8 \uf8eb \uf8ed� g � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j∈N G g |ωiωj|Aij \uf8f6 \uf8f8 Since � g � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j∈N G g |ωiωj|Aij/σ2 n ≤ K0 and maxg � k∈N G g |ωk|/σn = o(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 σ4n � i � j∈N G g(i) � k∈N G g(i) � l∈N H h(j) |ωiωjωkωl|Aij ≤ \uf8eb \uf8ed 1 σn max h � l∈N H h |ωl| \uf8f6 \uf8f8 \uf8eb \uf8ed 1 σn max g � k∈N G g |ωk| \uf8f6 \uf8f8 \uf8eb \uf8ed 1 σ2n � g � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j∈N G g |ωiωj|Aij \uf8f6 \uf8f8 = o(1) 17 Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Apply Lemma 4 on random variable ωiXi to obtain: dW (R, Z) ≤ 1 σ3n n � i=1 ������ � j,k∈Ni E[ωiωjωkXiXjXk] ������ + √ 2 √πσ2n � � � � �V ar \uf8eb \uf8ed n � i=1 � j∈Ni ωiωjXiXj \uf8f6 \uf8f8 Applying Lemma 6 and 7 on each of the two terms, dW (R, Z) = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proof for consistency of the variance estimator is equivalent to proving that (ˆσ2 n − σ2 n)/σ2 n = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By Chebyshev’s inequality, P � ˆσ2 n − σ2 n σ2n > ǫ � ≤ 1 ǫ2 1 σ4n E[(ˆσ2 n − σ2 n)2] = V ar �� i � j∈Ni ωiωjXiXj � ǫ2σ4n = oP (1) The convergence in the last step occurs by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Under Assumption 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2, ∀i, ||(1/(� i ωi)) � i ωi(Wi − E[Wi])|| p−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' It suffices to show convergence elementwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let Xi denote a scalar components of Wi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', Xi = Wim, where m ∈ {1, 2, · · · , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By Chebyshev’s inequality, and maxm,k E[W 2 mk] < K0, P � 1 � i ωi � i ωi(Xi − E[Xi]) > ǫ � ≤ 1 ǫ2 1 (� i ωi)2 E \uf8eb \uf8ed� i � j∈Ni ωiωj(Xi − E[Xi])(Yi − E[Yi]) \uf8f6 \uf8f8 ≤ K0 ǫ2 �� j ωj �2 � i � j∈Ni ωiωj Hence, it suffices to show (� i � j∈Ni ωiωj)/ �� j ωj �2 = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Observe � i � j∈Ni ωiωj �� j ωj �2 ≤ maxi � j∈Ni |ωj| �� j ωj � �� j ωj � �� j ωj � so it suffices to show maxi � j∈Ni |ωj|/ �� j ωj � = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since λn ≤ � i � j∈Ni |ωiωj| maxm E[W 2 mk] ≤ �� j |ωj| �2 maxm E[W 2 mk], � maxi � j∈Ni |ωj| �2 �� j ωj �2 = � maxi � j∈Ni |ωj| �2 maxm E[W 2 mk] �� j ωj �2 maxm E[W 2 mk] ≤ max m E[W 2 mk] � maxi � j∈Ni |ωj| �2 λn = o(1) 18 Convergence occurs due to Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 and maxm E[W 2 mk] < K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' To show that Q−1/2 n �n i=1 ωi(Wi − E[Wi]) d−→ N(0, IK), due to the Cramer- Wold device, it suffices to show that ∀l ∈ RK, l′Q−1/2 n �n i=1 ωi(Wi − E[Wi]) d−→ l′N(0, IK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' If l is a vector of zeroes, then l′Q−1/2 n �n i=1 ωi(Wi − E[Wi]) d−→ l′N(0, IK) is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For ||l|| > 0, it suffices to show (1/||l||)l′Q−1/2 n �n i=1 ωi(Wi − E[Wi]) d−→ (1/||l||)l′N(0, IK) = N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For all nonstochastic l ∈ RK\\{0}, let σ2 n(l) := V ar �� i(l/||l||)′ (Qn/λn)−1/2 ωi(Wi − E[Wi]) � , so the following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' E ��� l ||l|| �′ � 1 λn Qn �−1/2 (Wi − E[Wi]) �� = 0 and E ��� l ||l|| �′ � 1 λn Qn �−1/2 (Wi − E[Wi]) �4� ≤ K0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 σ2n(l) maxc �� i∈N C c |ωi| �2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 1 σ2n(l) � c � i,j∈N C c Aij|ωiωj| ≤ K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' �� l ||l|| �′ � 1 λn Qn �−1/2 (Wi − E[Wi]) � ⊥⊥ �� l ||l|| �′ � 1 λn Qn �−1/2 Wj � if g(i) ̸= g(j) and h(i) ̸= h(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For item 1, since λn := λmin(Qn), all eigenvalues of Qn/λn must be at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, all eigen- values of (Qn/λn)−1/2 are bounded above by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This implies |(l/||l||)′(Qn/λn)−1/2| ≤ K1 for some arbitrary constant K1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Item 1 then follows from Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Observe that σ2 n(l) = (l/||l||)′(Qn/λn)−1/2Qn(Qn/λn)−1/2(l/||l||) = 1/λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Then, Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 yields item 2 and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='3 yields item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Item 4 is immediate from Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By applying Lemma 1, (1/σn(l))(l/||l||)′(Qn/λn)−1/2 �n i=1 ωi(Wi − E[Wi]) d−→ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By using σ2 n(l) = 1/λn, this is equivalent to (l/||l||)′Q−1/2 n �n i=1 ωi(Wi − E[Wi]) d−→ N(0, 1) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1 Turning to consistent variance estimation, it suffices to show that for all l ∈ RK such that ||l|| = 1, P(l′Q−1 n ( ˆQn − Qn)l > ǫ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Now, impose the assumption that E[Wi] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' P(l′Q−1 n ( ˆQn − Qn)l > ǫ) ≤ 1 ǫ2 E �� l′(Q−1 n ( ˆQn − Qn)) �2� = 1 ǫ2 E \uf8ee \uf8f0 � l′ � 1 λn Qn �−1 1 λn ( ˆQn − Qn) �2\uf8f9 \uf8fb ≤ 1 ǫ2 E �� l′ 0 1 λn ( ˆQn − Qn) �2� 19 where l0 is a vector whose entries are all bounded above by some arbitrary constant K1 < ∞ by a similar argument as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, it suffices to show that (1/λn)( ˆQn − Qn) p−→ 0K×K, where 0K×K is a K × K matrix of zeroes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since ˆQn − Qn = � i � j∈Ni ωiωjWiW ′ j − E[ωiωjWiW ′ j], it suffices to show convergence elementwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let Xi and Yi denote scalar components of Wi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', Xi = Wim, Yi = Wip, where m, p ∈ {1, 2, · · · , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' P \uf8eb \uf8ed 1 λn � i � j∈Ni ωiωj(XiYj − E[XiYj]) > ǫ \uf8f6 \uf8f8 ≤ 1 ǫ2 1 λ2n V ar \uf8eb \uf8ed� i � j∈Ni ωiωjXiYj \uf8f6 \uf8f8 ≤ 1 ǫ2λ2n � i � j � k∈Ni � l∈Nj |E[ωiωjωkωlXiXjYkYl] − E[ωiωkXiYk]E[ωjωlXjYl]| ≤ K0 λ2n � i � j � k∈Ni � l∈Nj |ωiωjωkωl|(Aij + Ail + Akj + Akl) = o(1) The inequality in the last line is obtained due to Holder’s inequality and finite moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' An argument similar to that of Lemma 7 yields the o(1) equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 Now assume E[Wi] = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Using Lemma 8, ¯W p−→ µ is immediate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', ¯W = µ + oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' To ease notation, let ˜Wi := Wi − µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, Qn = � i � j∈Ni ωiωjE[ ˜Wi ˜W ′ j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' ˆQn = � i � j∈Ni ωiωj(Wi − ¯W)(Wj − ¯W)′ = � i � j∈Ni ωiωj( ˜Wi + oP(1))( ˜ Wj + oP (1))′ = � i � j∈Ni ωiωj ˜Wi ˜W ′ j + 2 � i � j∈Ni ωiωj ˜Wi1′ KoP(1) + � i � j∈Ni ωiωj1K1′ KoP (1) Since Q−1 n � i � j∈Ni ωiωj ˜Wi ˜W ′ j = 1 + oP (1) by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1, it then remains to show that each of the two remaining terms are oP (1) when pre-multiplied by Q−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' � 1 λ nQn �−1 1 λ n � i � j∈Ni ωiωj1K1′ K ≤ K0 � 1 λ nQn �−1 1K1′ K = O(1)1K1′ K The first inequality is due to the assumption that (1/λn) � i � j∈Ni |ωiωj| ≤ K0, and the O(1) term occurs due to the eigenvalues of (Qn/λn)−1 being bounded above by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Take some component ˜Xi 20 of ˜Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' For all ǫ > 0, there exists Mǫ = K2 0/ǫ < ∞ such that: P \uf8eb \uf8ed ������ 1 λn � i � j∈Ni ωiωj ˜Xi ������ ≥ Mǫ \uf8f6 \uf8f8 ≤ 1 λnMǫ E \uf8ee \uf8f0 ������ � i � j∈Ni ωiωj ˜Xi ������ \uf8f9 \uf8fb ≤ 1 Mǫ max i E[| ˜Xi|] 1 λn � i � j∈Ni |ωiωj| ≤ K0 K2 0/ǫ = ǫ Hence, Q−1 n � i � j∈Ni ωiωj ˜Wi1′ K = 1K1′ KOP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since OP (1)oP (1) = oP (1), the result is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' B Proof of Propositions Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' We have ˆθ−θ = �� i ˜D2 i �−1 �� i ˜Diui � = � i ωiui, where ωi := ˜Di/(� j ˜D2 j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let σ2 n := V ar(ωiui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Apply Theorem 1 with K = 1 to � i ωiui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 1 and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1 are automatically satisfied for clustered random variable ui and weight ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2 is satisfied because 1 σ2n max c \uf8eb \uf8ed � i∈N C c |ωi| \uf8f6 \uf8f8 2 ≤ 1 ( � i ˜D2 i ) 2 maxc �� i∈N C c | ˜Di| �2 K0 1 ( � i ˜D2 i ) 2 � c′ �� j∈N C c′ | ˜Dj| �2 = 1 ( � i ˜D2 i ) 2 maxc �� i∈N C c | ˜Di| �2 1 ( � i ˜D2 i ) 2 � c′ �� j∈N C c′ | ˜Dj| �2 → 0 where the first inequality comes from Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='3 and convergence occurs due to Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='3 is satisfied because 1 σ2n � c � i,j∈N C c Aij|ωiωj| = 1 ( � i ˜D2 i ) 2 � c′ � i,j∈N C c | ˜Di ˜Dj| 1 ( � i ˜D2 i ) 2V ar �� i ˜Diui � < ∞ Hence, Theorem 1 yields (ˆθ − θ)/σn d−→ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' To prove consistent variance estimation, it suffices to show (ˆσ2 n − σ2 n)/σ2 n = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' ˆσ2 n = � i � j∈Ni ωiuiωjuj − 2 \uf8eb \uf8ed� i � j∈Ni ω2 i ωjuj \uf8f6 \uf8f8 (ˆθ − θ) + \uf8eb \uf8ed� i � j∈Ni ω2 i ω2 j \uf8f6 \uf8f8 (ˆθ − θ)2 21 By Theorem 1, �� i � j∈Ni ωiuiωjuj − σ2 n � /σ2 n = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since (ˆθ − θ)2/σ2 n d−→ Z2 = χ2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' �� i � j∈Ni ω2 i ω2 j � (ˆθ − θ)2 σ2n = \uf8eb \uf8ed� i � j∈Ni ω2 i ω2 j \uf8f6 \uf8f8 OP (1) �� i � j∈Ni ˜D2 i ˜D2 j � �� i ˜D2 i �4 ≤ � maxi � j∈Ni ˜D2 j � �� i ˜D2 i �3 � i ˜D2 i � i ˜D2 i ≤ � maxi � j∈Ni ˜D2 j � �� i ˜D2 i �2 O(1) ≤ \uf8eb \uf8ec \uf8ed maxg �� j∈N G g | ˜Dj| �2 � g′ �� j∈N G g′ | ˜Dj| �2 + maxh �� j∈N H h | ˜Dj| �2 � h′ �� j∈N H h′ | ˜Dj| �2 \uf8f6 \uf8f7 \uf8f8 O(1) = o(1) Convergence occurs due to Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2, so �� i � j∈Ni ω2 i ω2 j � (ˆθ − θ)2/σ2 n = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' �� i � j∈Ni ω2 i ωjuj � (ˆθ − θ) σ2n = �� i � j∈Ni ˜D2 i ˜Djuj � σn �� i ˜D2 i �3 OP (1) Applying Markov and Minkowski inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' P \uf8eb \uf8ec \uf8ed ��� � i � j∈Ni ˜D2 i ˜Djuj ��� �� i ˜D2 i �3 σn > ǫ \uf8f6 \uf8f7 \uf8f8 ≤ 1 ǫ 1 �� i ˜D2 i �3 σn E \uf8ee \uf8f0 ������ � i � j∈Ni ˜D2 i ˜Djuj ������ \uf8f9 \uf8fb ≤ 1 ǫ 1 �� i ˜D2 i �3 σn � i � j∈Ni E| ˜D2 i ˜Djuj| ≤ 1 ǫ maxi � j∈Ni E| ˜Djuj| �� i ˜D2 i �2 σn � i ˜D2 i � i ˜D2 i = o(1) Convergence occurs because maxi �� j∈Ni E| ˜Djuj| �2 �� i ˜D2 i �2 ≤ maxj E|uj|2 maxi �� j∈Ni | ˜Dj| �2 �� i ˜D2 i �2 = o(1) For Proposition 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' I first prove a consistency result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Under Assumption 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2, and E[Wi] = 0 ∀i, ||(1/(� i ωi)) � i ωi(WiW ′ i − 22 E[WiW ′ i])|| p−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' It suffices to show convergence elementwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Let Xi and Yi denote scalar components of Wi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', Xi = Wim, Yi = Wip, where m, p ∈ {1, 2, · · · , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By Chebyshev’s inequality, and maxm,k E[W 4 mk] < K0, P � 1 � i ωi � i ωi(XiYi − E[XiYi]) > ǫ � ≤ 1 ǫ2 1 (� i ωi)2 E \uf8eb \uf8ed� i � j∈Ni ωiωj(XiYi − E[XiYi])(XjYj − E[XjYj]) \uf8f6 \uf8f8 ≤ K0 ǫ2 �� j ωj �2 � i � j∈Ni ωiωj Hence, it suffices to show (� i � j∈Ni ωiωj)/ �� j ωj �2 = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This follows from a similar argument as Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' E[ui|Xi] = 0 implies E[Xiui] = 0 by law of iterated expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since E[u4 i |Xi] ≤ K0, E[u4 i X4 ik] = E[E[u4 i |Xi]X4 ik] ≤ K0E[X4 ik] ≤ K2 0 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By Theorem 1, Q−1/2 n �n i=1 Xiui d−→ N(0, IK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' To complete the normality result, I show that S−1 n ˆSn p−→ IK, which is the same as showing that ||S−1 n ( ˆSn − Sn)|| p−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By applying Lemma 9 with ωi = 1, (1/n)( ˆSn − Sn) = (1/n) � i(XiX′ i − E[XiX′ i]) = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, it suffices that (Sn/n)−1 has bounded eigenvalues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', λmin(Sn/n) ≥ K1 > 0, which is true by Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since ˆβ − β = ˆS−1 n � i Xiui, by Slutsky’s lemma, Q−1/2 n Sn(ˆβ − β) d−→ N(0, IK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Next, proceed to consistent variance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Showing that ||Q−1 n ˆQn−IK|| = oP (1) is equivalent to showing that, ∀l ∈ RK, l′ � Q−1 n ( ˆQn − Qn) � l = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' ˆQn := � i � j∈Ni ˆuiˆujXiX′ j = � i � j∈Ni (ui − X′ i(ˆβ − β))(uj − X′ j(ˆβ − β))XiX′ j = � i � j∈Ni uiujXiX′ j − 2 \uf8eb \uf8ed� i � j∈Ni uiX′ j(ˆβ − β)XiX′ j \uf8f6 \uf8f8 + \uf8eb \uf8ed� i � j∈Ni X′ i(ˆβ − β)X′ j(ˆβ − β)XiX′ j \uf8f6 \uf8f8 23 By Theorem 1, l′Q−1 n (� i � j∈Ni uiujXiX′ j − Qn)l = oP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, it remains to show: ������ ������ Q−1 n \uf8ee \uf8f0−2 \uf8eb \uf8ed� i � j∈Ni uiX′ j(ˆβ − β)XiX′ j \uf8f6 \uf8f8 + \uf8eb \uf8ed� i � j∈Ni X′ i(ˆβ − β)X′ j(ˆβ − β)XiX′ j \uf8f6 \uf8f8 \uf8f9 \uf8fb ������ ������ = oP (1) Observe that X′ i(ˆβ−β) = � X′ iS−1 n Q1/2 n � � Q−1/2 n Sn(ˆβ − β) � = � X′ iS−1 n Q1/2 n � (ZK+1KoP (1)), where 1K is a K-vector of ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' addressing the second term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='X′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='i(ˆβ − β)X′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j(ˆβ − β) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='X′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='iS−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n Q1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='(ZK + 1KoP (1))(ZK + 1KoP (1))′ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='X′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='jS−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n Q1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='�′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='X′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='iS−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n Q1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='(IKOP (1) + oP (1)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='X′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='jS−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n Q1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='�′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='= X′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='iS−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n QnS−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n XjOP (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='This implies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='Q−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='\uf8ed� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j∈Ni ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='X′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='i(ˆβ − β)X′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j(ˆβ − β)XiX′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='\uf8f8 = Q−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='\uf8ed� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j∈Ni ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='X′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='iS−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n QnS−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n Xj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='XiX′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='\uf8f8 OP (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='λn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='Qn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='\uf8ed� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j∈Ni ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='X′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='nSn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='�−1 � 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='λn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='Qn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� � 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='nSn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='Xj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='XiX′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='\uf8f8 OP (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='The eigenvalues of (Qn/λn) are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' To see this, it suffices to show that there exists K0 < ∞ such that λmax(Qn)/λn ≤ K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Due to finite moments, Qn := V ar(� i Xi) ≤ K01K×K � c(N C c )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since (� c(N C c )2)/λn ≤ K0 by Assumption 4, λnK0 ≥ � c(N C c )2, which implies λn ≥ (� c(N C c )2)/K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, λmax(Qn) λn ≤ � c(N C c )2K0 � c(N C c )2 1 K0 = K2 0 Recall that (Sn/n)−1 has bounded eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' The proof of Theorem 1 also showed that (Qn/λn)−1 24 has bounded eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By using Markov and Minkowski inequalities, P \uf8eb \uf8ed 1 n2 ������ l′ � 1 λn Qn �−1 \uf8eb \uf8ed� i � j∈Ni � X′ i � 1 nSn �−1 � 1 λn Qn � � 1 nSn �−1 Xj � XiX′ j \uf8f6 \uf8f8 l ������ > ǫ \uf8f6 \uf8f8 ≤ 1 n2ǫE \uf8ee \uf8f0 ������ l′ � 1 λn Qn �−1 \uf8eb \uf8ed� i � j∈Ni � X′ i � 1 nSn �−1 � 1 λn Qn � � 1 nSn �−1 Xj � XiX′ j \uf8f6 \uf8f8 l ������ \uf8f9 \uf8fb ≤ 1 n2ǫ � i Ni max m,k E[X4 mk]K0 ≤ maxi Ni n n nK0 → 0 where K0 ∈ R is an arbitrary (finite) constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Convergence occurs due to Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2, which implies maxi Ni/n → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' This occurs due to the result that maxi � j∈Ni |ωj|/ �� j ωj � = o(1) in the proof of Lemma 8, using ωi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Going back to the first term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Q−1 n � i � j∈Ni uiX′ j(ˆβ − β)XiX′ j = Q−1 n � i � j∈Ni ui � X′ iS−1 n Q1/2 n � (ZK + 1KoP (1))XiX′ j = 1 n√λn � 1 λn Qn �−1 � i � j∈Ni ui � X′ i � 1 nSn �−1 � 1 λn Qn �1/2� XiX′ jOP (1) By using Markov and Minkowski inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' P \uf8eb \uf8ed 1 n√λn ������ l′ � 1 λn Qn �−1 � i � j∈Ni ui � X′ i � 1 nSn �−1 � 1 λn Qn �1/2� XiX′ jl ������ > ǫ \uf8f6 \uf8f8 ≤ 1 n√λnǫE \uf8ee \uf8f0 ������ l′ � 1 λn Qn �−1 � i � j∈Ni ui � X′ i � 1 nSn �−1 � 1 λn Qn �1/2� XiX′ jl ������ \uf8f9 \uf8fb ≤ 1 n√λnǫ � i � j∈Ni max m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='k E ���Xm1kum1X2 m2 ��� K0 ≤ 1 n√λnǫ � i Ni max m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='k E � |Xm1kum1|2�1/2 E � |X2 m2|2�1/2 K0 ≤ maxi Ni √λn 1 ǫ max m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='k E[X2 m1ku2 m1]1/2E[X4 m2]1/2K0 = o(1) The penultimate inequality occurs due to Holder’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Observe that maxi Ni/√λn = o(1) if and only if maxc(N C c )2/λn = o(1), which is given by Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Convergence in the last step occurs because maxi Ni/√λn = o(1), and finite moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' 25 Hence, it has been shown that Q−1 n ˆQn p−→ IK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Then, [S−1 n QnS−1 n ]−1[ ˆS−1 n ˆQn ˆS−1 n ] p−→ IK by the continuous mapping theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' By Proposition 2, (ˆβ1 − β1)/[V (ˆβ)]1/2 11 d−→ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since ˆθ = ˆβ1, [V (ˆβ)]11 = V (ˆβ1) = V (ˆθ) = σ2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, (ˆθ − θ)/σn d−→ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' A further implication of Proposition 2 is that [ ˆV (ˆβ)]11/[V (ˆβ)]11 p−→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Using theorem 3 of Ding (2021), the Liang-Zeger estimators (Liang and Zeger, 1986) are numerically equivalent regardless of whether the long regression or the residualized regression were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Since the CGM estimator is a function of the Liang-Zeger estimators, ˆσ2 n = [ ˆV (ˆβ)]11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' Hence, ˆσ2 n/σ2 n p−→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' References Cameron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE2T4oBgHgl3EQfTwcd/content/2301.03805v1.pdf'} +page_content=', J.' metadata={'source': 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a/HtAzT4oBgHgl3EQfU_w0/content/tmp_files/2301.01275v1.pdf.txt b/HtAzT4oBgHgl3EQfU_w0/content/tmp_files/2301.01275v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2bd42c4d6f473b0cdc44eb9d8564934292376b69 --- /dev/null +++ b/HtAzT4oBgHgl3EQfU_w0/content/tmp_files/2301.01275v1.pdf.txt @@ -0,0 +1,3122 @@ +Fixed-time synchronization for quaternion-valued +memristor-based neural networks with mixed delays +Yanlin Zhanga, Liqiao Yanga, Kit Ian Koua,∗, Yang Liub +aDepartment of Mathematics, Faculty of Science and Technology, University of +Macau, Macau, 999078, China +b College of Mathematics and Computer Science, Zhejiang Normal +University, Jinhua, 321004, China +Abstract +In this paper, the fixed-time synchronization (FXTSYN) of unilateral coefficients +quaternion-valued memristor-based neural networks (UCQVMNNs) with mixed de- +lays is investigated. Instead of decomposition, a direct analytical method is proposed +to achieve FXTSYN of UCQVMNNs using one-norm smoothly. Then apply the set- +valued map and the differential inclusion theorem to handle discontinuity problems of +drive-response systems. The novel nonlinear controllers together with the Lyapunov +function are designed to achieve the control goal. Using the FXTSYN theory and +inequality techniques, some criteria of FXTSYN for UCQVMNNs are given. Further- +more, the estimated settling time is obtained explicitly. Finally, numerical simulations +are presented to demonstrate the correctness, effectiveness and practicability of the +obtained theoretical results. +Keywords: +Fixed-time synchronization, controllers, quaternion-valued, unilateral +coefficients, memristor-based neural networks +2010 MSC: 34D06, 37N35, 92B20, 93D05 +1. Introduction +Quaternion is a subset of Clifford algebra, which was invented by Hamilton in 1843 +[1] and is a natural extension of complex space. The state variables, input variables, +connection weights, and the activation functions of quaternion-valued neural networks +(QVNNs) all take values in the quaternion field. Moreover, QVNNs have been used in +∗Corresponding author +Email addresses: yanlnzhang@163.com (Yanlin Zhang), liqiaoyoung@163.com (Liqiao Yang), +kikou@umac.mo (Kit Ian Kou), liuyang@zjnu.edu.cn ( Yang Liu) +1 +arXiv:2301.01275v1 [eess.SY] 2 Jan 2023 + +a variety of practical applications, such as satellite TV, aerospace, 3-D wind process- +ing, color image processing, polarized waves, and space rotation [2, 3, 4, 5]. However, +in contrast to real- and complex-values, some arithmetic rules such as the commuta- +tivity of multiplication do not apply to quaternion. As a result, these researches on +NNs are primarily focused on the real-valued and complex-valued fields [6, 7, 8, 9], +while the related research on quaternion is relatively scarce in recent decades. +Due to the non-commutative of quaternion multiplication, there are a great variety +of polynomials in quaternion algebra than in the real and complex fields, such as +polynomials with left, right, and bilateral coefficients. +In QDEs, those equations +with bilateral coefficients are too difficult to solve, so there are not many results +[10, 11] about it. Therefore, in this paper, we first study the unilateral coefficients of +quaternion-valued neural networks (UCQVNNs). +As one of the basic circuit elements, the memristor (an abbreviation for the mem- +ory and the resistor) was first proposed by Chua [12] in 1971. The memristor has +the properties of memory and nanoscale, which can better and more realistically sim- +ulate the biological synapse. It describes the relationship between the charge and +the magnetic flux, so the memristor systems are more precise models of artificial +neural networks. +Furthermore, it has the potential to improve the application of +pattern recognition, combinatorial optimization, and data processing [13, 14, 15]. In +recent years, many scholars have studied many characteristics of memristor-based +neural networks (MNNs) due to the powerful functions in human brain computers +[16, 17, 18, 19, 20], such as dissipativity, stability, synchronization and so on. +It is widely known that synchronization is an expansion of stability, which is the +dynamic behavior of the drive-response system to achieve the same state at the same +time. Due to the powerful role of synchronization in non-linear systems [21], it is +widely used in areas such as information processing [22], security communication +[23], chemical reaction [24] and so on. Our research aims to achieve synchroniza- +tion as quickly as possible under the controller’s action. Fortunately, the fixed time +synchronization (FXTSYN) [25] was quickly proposed, in which the settling time +is completely irrelevant to initial values. And FXTSYN has a fast convergence time +and better interference inhibitory characteristics. Indeed, many practical phenomena, +such as information processing and biological systems, require rapid synchronization +to maintain normal order. Therefore, many scholars pay attention to the FXTSYN of +real-valued MNNs (RVMNNs) and complex-valued MNNs (CVMNNs), some promis- +ing applications can be found in [20, 26, 27, 28, 29, 30]. +However, as far as we know, compared to the fields of RVMNNs and CVMNNs, +there are quite rare results on FXTSNY of UCQVMNNs with mixed delays [31, 32, +33, 34, 35]. And the method used by most of them is the decomposition method, in +[31], the FXTSNY of a class of QVNNs with memristor was investigated via decompo- +sition methods. That is, the real and imaginary parts of a quaternion neural network +2 + +are re-expressed as four real-valued systems, which lose important information about +the original problem structure. Regrettably, it is generally accompanied by a more +complex derivation process, and it may also increase conservatism. Therefore, finding +an easier and no-decomposition method to research the FXTSNY of MNNs is neces- +sary and important. In [34], Peng et al. use a direct analytical method to study the +FNTSNY and FXTSNY problems of QVNNs by introducing an improved one-norm, +which is without decomposition techniques. Compared with the traditional decom- +posing method, even more specific FXTSNY is achieved by the one-norm method, +which shows the strong applicability and less conservative of this method. So, we +design two novel nonlinear controllers based on the improved one-norm method to +achieve the FXTSNY of UCQVMNNs with mixed delays in this paper. It is of great +significance and value for research. +Motivated by the above, it will be of great significance for some high-dimensional +dynamical systems with delays, such as secure communication [36] and image com- +pression systems [37] to effectively synchronize the complex dynamic behaviors of +these delayed systems in a fixed time. However, existing research results pay little +attention to it. +Therefore, this paper aims at is studied the FXTSNY of a class +of UCQVMNNs with mixed delays by the non-decomposition method. The main +contributions presented in this article can be described as follows. +• Most of the UCQVMNNs are decomposed into four RVMNNs by the decomposi- +tion method. The calculation and proof process has been carried out four times, +which is quite complicated and increases the difficulty of theoretical derivation. +Therefore, using the non-decomposition method of one-norm makes the proof +result easier to realize and the calculation process more simple. In addition, the +existence of memristor will cause the system to be discontinuous, few studies +have used this approach for FXTSNY analysis of UCQVNNs with memristors. +• A novel FXTSNY method is applied to UCQVMNNs for the first time, com- +bined with Lemma 2.2. Firstly, V (t) ≥ 1 or 0 < V (t) < 1 is judged and the +functioning part is intelligently chosen. Then, by using the improved one-norm, +a suitable Lyapunov function and controller are constructed, and a more robust +and accurate synchronization time estimation is obtained. +• To verify the effectiveness of the non-decomposition method of the UCQVMNNs, +several numerical simulations of the FXTSNY process are proposed. It not only +verifies the correctness and validity of the two theorems, but also shows the su- +periority of Theorem 2. Moreover, we can achieve the best synchronization ef- +fect by adjusting the values of different parameters. Interestingly, UCQVMNNs +have good applications in the rapid recovery of high-dimensional data, so it has +practical research significance. +3 + +The rest of this paper is organized as follows. Section 2 presents the description +of QVMNNs with bilateral coefficients and unilateral coefficients, respectively. And +several Definitions, Lemmas, and some new one-norm inequalities of the quaternion +are deduced. In Section 3, the FXTSNY scheme is proposed while the effectiveness +of the theoretical results is illustrated. The effectiveness of sufficient conditions is +checked by simulation examples in Section 4. Some conclusions are drawn in Section +5. +Notations: The sets of real and non-negative real numbers are represented by R and +R+, respectively. H represents the set of all quaternions and denoted by bold letter, +superscripts T and ∗ indicate transposition and conjugate transpose, respectively. +The set of all n-dimensional real numbers, complex numbers, and quaternions are +represented by Rn, Cn and Hn. Using ∥ · ∥1 and sgn(·) to denote the one-norm and +the sign function, respectively. Rn×m, Cn×m and Hn×m denote the n×m-dimensional +real, complex and quaternion matrix. Real and complex numbers are the special case +of quaternions (i.e. R ∈ C ∈ H). +2. Model Formulation and Preliminaries +2.1. Quaternion Algebra Fundamentals +A quaternion number x ∈ H is combined by a real part and three imagery parts, +and can be written as following form: +x = x(0) + x(1)i + x(2)j + x(3)k, +where x(ζ) ∈ R, ζ = 0, 1, 2, 3, and i, j, k are imaginary units, and the imaginary units +are defined by +� +i2 = j2 = k2 = ijk = −1, +ij = −ji = k, jk = −kj = i, ki = −ik = j. +The modulus of x are defined as |x| = √x¯x = +� +(x(0))2 + (x(1))2 + (x(2))2 + (x(3))2. +The transpose of vector x is denoted by xT, the conjugate and conjugate transpose of +x are denoted by ¯x = x(0)−x(1)i−x(2)j −x(3)k and x∗ = (x(0)−x(1)i−x(2)j −x(3)k)T, +respectively. For any two quaternion x and y = y(0)+iy(1)+jy(2)+ky(3), the addition +operation is defined as follows +x + y = (x(0) + y(0)) + (x(1) + y(1))i + (x(2) + y(2))j + (x(3) + y(3))k. +The multiplication operation is defined by +xy =(x(0)y(0) − x(1)y(1) − x(2)y(2) − x(3)y(3)) + (x(0)y(1) + x(1)y(0) + x(2)y(3) +− x(3)y(2))i + (x(0)y(2) − x(1)y(3) + x(2)y(0) + x(3)y(1))j + (x(0)y(3) ++ x(1)y(2) − x(2)y(1) + x(3)y(0))k. +4 + +It is important to note that the multiplication in the quaternion domain is not com- +mutative, i.e. xy ̸= yx. +The one-norm of vector v = (v1, v2, ..., vn) ∈ Rn and the quaternion x are writ- +ten as ∥v∥1 = �n +p=1 |vp| and ∥x∥1 = � +ζ=0,1,2,3 ∥x(ζ)∥1, respectively. +For e(t) = +(e1(t), e2(t), ..., en(t))T ∈ Hn, t ∈ R, the sign function and one-norm of vector +e(t) are denoted by sgn(e(t)) = (sgn(e1(t)), sgn(e2(t)), ..., sgn(en(t)))T, ∥e(t)∥1 = +�n +p=1 ∥ep(t)∥1 respectively, and [e(t)]r = ([e1(t)]r, [e2(t)]r, ..., [en(t)]r)T, moreover, +[ep(t)]r = sgn(ep(t))∥ep(t)∥r +1, p = 1, 2, ..., n and r > 0. +For any quaternion x = x(0) + x(1)i + x(2)j + x(3)k can be uniquely expressed +as x = x1 + x2j, where x1 = x(0) + x(1)i, and x2 = x(2) + x(3)i. Furthermore, this +expression can be used by quaternion matrix. +Definition 2.1 [38]. Given a quaternion matrix A ∈ Hm×n, its expression using the +Cayley-Dickson notation is A = Ap +Aqj, where Ap and Aq ∈ Cm×n. The quaternion +matrix can be denoted as an isomorphic complex matrix +A = +� Ap +Aq +− ¯ +Aq +¯ +Ap +� +, +where Ap = A0 + A1i ∈ Cm×n and Aq = A2 + A3i ∈ Cm×n. +2.2. QVMNNs with Bilateral Coefficients (BCQVMNNs) +In this section, we introduce a new kind of bilateral coefficients QVMNNs (BC- +QVMNNs), which has discrete and distributed time delays and with a form +d +dtxp(t) = −dpxp(t) + +n +� +q=1 +`apq(xp(t))fq(xq(t))´apq(xp(t)) ++ +n +� +q=1 +`bpq(xp(t))gq(xq(t − τ(t)))´bpq(xp(t)) ++ +n +� +q=1 +`cpq(xp(t))( +� t +t−π +hq(xq(s))ds)´cpq(xp(t)) + Ip, +(2.1) +for p = 1, 2, ..., n, where n corresponds to the number of neurons, xp(t) ∈ H stand +the state variable of the p-th neuron, dp > 0 is the real-valued self-feedback coef- +ficient, Ip ∈ H denotes the external input or bias, `apq(xp(t)), ´apq(xp(t)), `bpq(xp(t)), +´bpq(xp(t)), `cpq(xp(t)), and ´cpq(xp(t)) ∈ H stand for the memristive connection weights, +fq(xq(t)), gq(xq(t − τ(t))) and hq(xq(s)) ∈ H are the activation functions. The dis- +crete time delay and the distributed time delay are denoted by τ(t) and π, they +satisfies 0 ≤ τ(t) ≤ τ, τ = max{τ(t), π}. +5 + +For simpler expression, we can represent (2.1) as vector form +d +dtX(t) = −DX(t) + ` +Af(X(t)) ´ +A + `Bg(X(t − τ(t))) ´B + `C( +� t +t−π +h(X(s))ds) ´C + I, +(2.2) +where X(t) = (x1(t), x2(t), · · · , xn(t))T ∈ Hn, D = diag(d1, d2, · · · , dn) ∈ Rn×n, +f(X(t)) = (f1(x1(t)), f2(x2(t)), · · · , fn(xn(t)))T ∈ Hn, g(X(t − τ(t))) = (g1(x1(t − +τ(t))), g2(x2(t−τ(t))), · · · , gn(xn(t−τ(t))))T ∈ Hn, h(X(s)) = (h1(x1(s)), h2(x2(s)), +· · · , hn(xn(s)))T ∈ Hn, I = (I1, I2, · · · , In)T ∈ Hn and +` +A = +� +� +� +� +� +`a11(x1(t)) +`a12(x1(t)) +· · · +`a1n(x1(t)) +`a21(x2(t)) +`a22(x2(t)) +· · · +`a2n(x2(t)) +... +... +... +... +`an1(xn(t)) +`an2(xn(t)) +· · · +`ann(xn(t)) +� +� +� +� +� ∈ Hn×n, +´ +A = +� +� +� +� +� +´a11(x1(t)) +´a12(x1(t)) +· · · +´a1n(x1(t)) +´a21(x2(t)) +´a22(x2(t)) +· · · +´a2n(x2(t)) +... +... +... +... +´an1(xn(t)) +´an2(xn(t)) +· · · +´ann(xn(t)) +� +� +� +� +� ∈ Hn×n, +`B = +� +� +� +� +� +`b11(x1(t)) +`b12(x1(t)) +· · · +`b1n(x1(t)) +`b21(x2(t)) +`b22(x2(t)) +· · · +`b2n(x2(t)) +... +... +... +... +`bn1(xn(t)) +`bn2(xn(t)) +· · · +`bnn(xn(t)) +� +� +� +� +� ∈ Hn×n, +´B = +� +� +� +� +� +´b11(x1(t)) +´b12(x1(t)) +· · · +´b1n(x1(t)) +´b21(x2(t)) +´b22(x2(t)) +· · · +´b2n(x2(t)) +... +... +... +... +´bn1(xn(t)) +´bn2(xn(t)) +· · · +´bnn(xn(t)) +� +� +� +� +� ∈ Hn×n, +`C = +� +� +� +� +� +`c11(x1(t)) +`c12(x1(t)) +· · · +`c1n(x1(t)) +`c21(x2(t)) +`c22(x2(t)) +· · · +`c2n(x2(t)) +... +... +... +... +`cn1(xn(t)) +`cn2(xn(t)) +· · · +`cnn(xn(t)) +� +� +� +� +� ∈ Hn×n, +´C = +� +� +� +� +� +´c11(x1(t)) +´c12(x1(t)) +· · · +´c1n(x1(t)) +´c21(x2(t)) +´c22(x2(t)) +· · · +´c2n(x2(t)) +... +... +... +... +´cn1(xn(t)) +´cn2(xn(t)) +· · · +´cnn(xn(t)) +� +� +� +� +� ∈ Hn×n. +6 + +According to Definition 2.1, let X(t) = X1(t)+X2(t)j, ` +A = `A1 + `A2j, ´ +A = ´A1 + ´A2j, +`B = `B1 + `B2j, ´B = ´B1 + ´B2j, `C = `C1 + `C2j, ´C = ´C1 + ´C2j, and f(X(t)) = +f1(X(t))+f2(X(t))j, g(X(t−τ(t))) = g1(X(t−τ(t)))+g2(X(t−τ(t)))j, h(X(s)) = +h1(X(s))+h2(X(s))j, I = L1 +L2j, where X1(t), X2(t), L1, L2, f1(X(t)), f2(X(t)), +g1(X(t−τ(t))), g2(X(t−τ(t))), and h1(X(s)), h2(X(s)) ∈ Cn, `A1, `A2, ´A1, ´A2, `B1, `B2, +´B1, ´B2, `C1, `C2, and ´C1, ´C2 ∈ Cn×n. +Then system (2.2) can be converted to the following form +d +dt(X1(t) + X2(t)j) = −D(X1(t) + X2(t)j) + ( `A1 + `A2j)(f1(X(t)) + f2(X(t))j)( ´A1 + ´A2j) ++ ( `B1 + `B2j)(g1(X(t − τ(t))) + g2(X(t − τ(t)))j)( ´B1 + ´B2j) ++ ( `C1 + `C2j)( +� t +t−π +(h1(X(s)) + h2(X(s))j)ds)( ´C1 + ´C2j) ++ L1 + L2j. +Thus system (2.2) is equivalent to +d +dtX1(t) = −DX1(t) + `A1 ´A1f1(X(t)) − `A1 ¯´A2f2(X(t)) − `A2 ¯´A2 ¯f1(X(t)) − `A2 ´A1 ¯f2(X(t)) ++ `B1 ´B1g1(X(t − τ(t))) − `B1 ¯´B2g2(X(t − τ(t))) − `B2 ¯´B2 ¯g1(X(t − τ(t))) +− `B2 ´B1¯g2(X(t − τ(t))) + `C1 ´C1 +� t +t−π +(h1(X(s))ds − `C1 ¯´C2 +� t +t−π +h2(X(s))ds +− `C2 ¯´C2 +� t +t−π +(¯h1(X(s))ds − `C2 ´C1 +� t +t−π +¯h2(X(s)) + L1, +(2.3) +and +d +dtX2(t) = −DX2(t) + `A1 ´A2f1(X(t)) + `A1 ¯´A1f2(X(t)) + `A2 ¯´A1 ¯f1(X(t)) − `A2 ´A2 ¯f2(X(t)) ++ `B1 ´B2g1(X(t − τ(t))) + `B1 ¯´B1g2(X(t − τ(t))) + `B2 ¯´B1¯g1(X(t − τ(t))) +− `B2 ´B2¯g2(X(t − τ(t))) + `C1 ´C2 +� t +t−π +(h1(X(s))ds + `C1 ¯´C1 +� t +t−π +h2(X(s))ds ++ `C2 ¯´C1 +� t +t−π +(¯h1(X(s))ds − `C2 ´C2 +� t +t−π +¯h2(X(s)) + L2. +(2.4) +By the Cayley-Dickson transformation, BCQVMNNs (2.2) is transformed into +two CVMNNs (2.3) and (2.4). Due to the mixed time delay and integral term in +BCQVMNNs, it is difficult to directly study the FXTSYN of BCQVMNNs by non- +decomposition method. As a special case of BCQVMNNs, the following part of this +paper mainly illustrates the related properties of the bilateral systems by studying +the sufficient conditions for FXTSYN of UCQVMNNs. +7 + +2.3. QVMNNs with Unilateral Coefficients (UCQVMNNs) +The following UCQVMNNs with discrete and distributed time delays is considered +d +dtxp(t) = −dpxp(t) + +n +� +q=1 +apq(xp(t))fq(xq(t)) + +n +� +q=1 +bpq(xp(t))gq(xq(t − τ(t))) ++ +n +� +q=1 +cpq(xp(t)) +� t +t−π +hq(xq(s))ds + Ip, +(2.5) +for p = 1, 2, ..., n, xp(t) ∈ H is the state variable, apq(xp(t)), bpq(xp(t)), cpq(xp(t)) ∈ H +stand for the memristive connection weights. +Define system (2.5) as drive system, the response system is described as following +d +dtyp(t) = −dpyp(t) + +n +� +q=1 +apq(yp(t))fq(yq(t)) + +n +� +q=1 +bpq(yp(t))gq(yq(t − τ(t))) ++ +n +� +q=1 +cpq(yp(t)) +� t +t−π +hq(yq(s))ds + Ip + up(t), +(2.6) +where up(t) ∈ H is the designed controller. The initial conditions of systems (2.5) +and (2.6) are +xp(s) = φp(s), +yp(s) = ψp(s), s ∈ [−τ, 0]. +Based on the characteristics of the memristor and current–voltage, the memristive +connection weights in (2.5) and (2.6) are satisfy +apq(·) = +� +ˆapq, +| · | ≤ rp, +ˇapq, +| · | > rp, +bpq(·) = +�ˆbpq, +| · | ≤ rp, +ˇbpq, +| · | > rp, +cpq(·) = +� +ˆcpq, +| · | ≤ rp, +ˇcpq, +| · | > rp, +for p, q = 1, 2, ..., n, where ˆapq, ˇapq, ˆbpq, ˇbpq, ˆcpq, ˇcpq are known constants with respect +to the memristor, and the switching jumps rp > 0 is the threshold level. +Define +a+ +pq = max{|ˆapq|, |ˇapq|}, b+ +pq = max{|ˆbpq|, |ˇbpq|} and c+ +pq = max{|ˆcpq|, |ˇcpq|}. amax +pq += +max{ˆapq, ˇapq}, amin +pq += min{ˆapq, ˇapq}, bmax +pq += max{ˆbpq, ˇbpq}, bmin +pq += min{ˆbpq, ˇbpq}, +cmax +pq += max{ˆcpq, ˇcpq}, and cmin +pq += min{ˆcpq, ˇcpq}. +As shown above, due to the discontinuity of the memristive connection weights, +the system (2.5) and the system (2.6) are regarded as the discontinuous differential +equation of the right-hand side, which the existence and uniqueness of solutions are +not guaranteed. So the Filippov solutions are utilized to deal with the special case. +8 + +By the differential inclusions and the set-valued maps [39], the differential inclusion +of systems (2.5) and (2.6) are as follows +d +dtxp(t) ∈ −dpxp(t) + +n +� +q=1 +K(apq(xp(t)))fq(xq(t)) + +n +� +q=1 +K(bpq(xp(t))) +× gq(xq(t − τ(t))) + +n +� +q=1 +K(cpq(xp(t))) +� t +t−π +hq(xq(s))ds + Ip, +(2.7) +and +d +dtyp(t) ∈ −dpyp(t) + +n +� +q=1 +K(apq(yp(t)))fq(yq(t)) + +n +� +q=1 +K(bpq(yp(t))) +× gq(yq(t − τ(t))) + +n +� +q=1 +K(cpq(yp(t))) +� t +t−π +hq(yq(s))ds + Ip + up(t), +(2.8) +where +K(apq(·)) = +� +� +� +� +� +ˆapq, +| · | < rp, +co{ˆapq, ˇapq}, | · | = rp, +ˇapq, +| · | > rp, +K(bpq(·)) = +� +� +� +� +� +ˆbpq, +| · | < rp, +co{ˆbpq, ˇbpq}, | · | = rp, +ˇbpq, +| · | > rp, +K(cpq(·)) = +� +� +� +� +� +ˆcpq, +| · | < rp, +co{ˆcpq, ˇcpq}, | · | = rp, +ˇcpq, +| · | > rp, +in which co{a, b} is the closure of the convex hull, and co{ˆapq, ˇapq} = [amin +pq , amax +pq ], +co{ˆbpq, ˇbpq} = [bmin +pq , bmax +pq ], co{ˆcpq, ˇcpq} = [cmin +pq , cmax +pq ]. +By the measurable selection theorem [40], if (xp(t), yp(t)) is the solution of systems +(2.7)-(2.8), there exist bounded measurable functions ˜apq(t) ∈ K(apq(·)), ˜bpq(t) ∈ +K(bpq(·)) and ˜cpq(t) ∈ K(cpq(·)), which rely on xp(t) and yp(t) respectively, such +that +d +dtxp(t) = −dpxp(t) + +n +� +q=1 +˜apq(t)fq(xq(t)) + +n +� +q=1 +˜bpq(t) +× gq(xq(t − τ(t))) + +n +� +q=1 +˜cpq(t) +� t +t−π +hq(xq(s))ds + Ip, +(2.9) +9 + +and +d +dtyp(t) = −dpyp(t) + +n +� +q=1 +˜apq(t)fq(yq(t)) + +n +� +q=1 +˜bpq(t)gq(yq(t − τ(t))) ++ +n +� +q=1 +˜cpq(t) +� t +t−π +hq(yq(s))ds + Ip + up(t). +(2.10) +Based on the UCQVMNNs (2.9) and (2.10), the synchronization error can be defined +as ep(t) = yp(t) − xp(t), so we can get +d +dtep(t) = −dpep(t) + +n +� +q=1 +˜apq(t)(fq(yq(t) − fq(xq(t))) ++ +n +� +q=1 +˜bpq(t)(gq(yq(t − τ(t))) − gq(xq(t − τ(t)))) ++ +n +� +q=1 +˜cpq(t) +� t +t−π +(hq(yq(s)) − hq(xq(s)))ds + up(t), +(2.11) +with the initial condition ϕp(s) = ψp(s) − φp(s) for s ∈ [−τ, 0]. +To obtain the main results, the following Assumptions, Definitions and Lemmas +are necessary. +Assumption (A1). For any xq(t), yq(t) ∈ H, suppose the expression of activation +functions +fq(·) = fq(0)(·) + ifq(1)(·) + jfq(2)(·) + kfq(3)(·), +gq(·) = gq(0)(·) + igq(1)(·) + jgq(2)(·) + kgq(3)(·), +hq(·) = hq(0)(·) + ihq(1)(·) + jhq(2)(·) + kfq(3)(·). +where fq(0) = 0, gq(0) = 0, hq(0) = 0, fq(ζ)(·), gq(ζ)(·), hq(ζ)(·) ∈ R, ζ = 0, 1, 2, 3, then +there exist positive constant υq, ϱq and ιq, q = 1, 2, ..., n satisfying +∥fq(yq(t) − fq(xq(t))∥1 ≤ υq∥yq(t) − xq(t)∥1 = υq∥eq(t)∥1, +∥gq(yq(t) − gq(xq(t))∥1 ≤ ϱq∥yq(t) − xq(t)∥1 = ϱq∥eq(t)∥1, +∥hq(yq(t) − hq(xq(t))∥1 ≤ ιq∥yq(t) − xq(t)∥1 = ιq∥eq(t)∥1. +Definition 2.2 [17]. +The drive system (2.5) is said to be synchronized with the +response system (2.6) in finite time, if there exists a constant T(e0) > 0 and T(e0) +depends on the initial error e0 such that +lim +t→T(e0) ∥e(t)∥ = 0, +∥e(t)∥ = 0 +for ∀t ≥ T(e0), +10 + +where e(t) = (e1(t), e1(t), · · · en(t))T, and T(e0) is called the settling time. +Definition 2.3 [25]. The drive system (2.5) and the response system (2.6) are said +to reach the fixed-time synchronization, if the following two conditions hold +(1) The system (2.5) is said to be synchronized with the system (2.6) in finite +time, +(2) There exists a fixed constant Tmax, such that for any initial condition e0, the +corresponding settling time satisfies T(e0) ≤ Tmax. +Lemma 2.1 [25]. Assume that there exists a continuous radically unbounded function +V : Rn → R+ = [0, +∞) satisfying +(1) V (e(t)) = 0 if and only if e(t) = 0, +(2) +d +dtV (e(t)) ≤ −aV α(e(t)) − bV β(e(t)), +where a, b > 0, 0 < α < 1, and β > 1. +Then the origin of the error system (2.11) is fixed-time stable, and V (e(t)) = 0 +for t ≥ T(e0) where the settling time T(e0) is bounded by +T(e0) ≤ Tmax = 1 +a +�a +b +� 1−α +β−α � +1 +β − 1 + +1 +1 − α +� +for ∀ e0 ∈ Rn. +Remark 2.1. [25]. Under the same conditions in Lemma 2.1, the synchronization +time T(e0) can also be estimated by the following formula +T(e0) ≤ Tmax = +1 +a(1 − α) + +1 +b(β − 1) +for ∀e0 ∈ Rn. +Lemma 2.2 [41]. If there exists a continuous, positive definite, and radically un- +bounded function V (e(t)) : Rn → R+ such that any solution x(t) of system (2.11) +satisfies the inequality +d +dtV (e(t)) ≤ +� +aV (e(t)) − b1(V (e(t)))γ+sgn(V (e(t))−1), +V ⩾ 1, +aV (e(t)) − b2(V (e(t)))γ+sgn(V (e(t))−1), +0 ≤ V < 1, +(2.12) +in which a < min{b1, b2}, b1, b2 > 0, 1 ≤ γ < 2, then the origin of the error system +(2.11) is globally fixed-time stable, In addition, for any initial state e0 of system +(2.11), the settling time is described as +T = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +a(2 − γ) ln +b2 +b2 − a + +1 +γ(b1 − a), +a > 0, +1 +b2(2 − γ) + 1 +b1γ , +a = 0, +1 +a(2 − γ) ln +b2 +b2 − a + 1 +aγ ln +b1 +b1 − a, +a < 0, +(2.13) +11 + +Lemma 2.3 [42]. If x1, x2, · · · , xn ≥ 0, 0 < q1 ≤ 1 and p1 > 1, then +n +� +p=1 +xq1 +p ≥ +� n +� +p=1 +xp +�q1 +, +n +� +p=1 +xp1 +p ≥ n1−p1 +� n +� +p=1 +xp +�p1 +. +Definition 2.4 [32]. We define sign function of quaternion x = x(0) + x(1)i + x(2)j + +x(3)k as follows +sgn(x) = sgn(x(0)) + sgn(x(1))i + sgn(x(2))j + sgn(x(3))k, +and the conjugate transpose of sgn(x) are denoted by +sgn(x)∗ = +� +sgn(x(0)) − sgn(x(1))i − sgn(x(2))j − sgn(x(3))k +�T. +Definition 2.5. According to the characteristics of the sign function, it can be known +that |x| = sgn(x)x, x ∈ R, and the one-norm of quaternoin vector u = u(0) + u(1)i + +u(2)j + u(3)k ∈ Hn can be expressed as +∥u∥1 = ∥u(0)∥1 + ∥u(1)∥1 + ∥u(2)∥1 + ∥u(3)∥1 += (sgn(u(0)))Tu(0) + (sgn(u(1)))Tu(1) + (sgn(u(2)))Tu(2) + (sgn(u(3)))Tu(3) += 1 +2((sgn(u))∗u + sgn(u)u∗). +where u(ζ) ∈ Rn, ζ = 0, 1, 2, 3 represent the real and imaginary parts of the quaternoin +vector, respectively. +There are some Lemmas associated with this Definition is given below, so as to +facilitate the proof and calculation of the theorem later. +Lemma 2.4 [34]. Suppose that vector e(t) = (e1(t), e2(t), ..., en(t))T, l(t) = (l1(t), l2(t) +, ..., ln(t))T ∈ Hn, and e(t) = e(0)(t)+e(1)(t)i+e(2)(t)j+e(3)(t)k, l(t) = l(0)(t)+l(1)(t)i+ +l(2)(t)j + l(3)(t)k, where ep(t), lp(t) ∈ H, p = 1, 2, ..., n, and e(ζ)(t), l(ζ)(t) ∈ Rn, ζ = +0, 1, 2, 3. For any e(t), l(t) ∈ Hn, positive constant c > 0, if f(e(t)) is an integrable +function is defined on [t − τ, t], the following formulas hold +(1) sgn(e(t))∗e(t) + (e(t))∗sgn(e(t)) = 2∥e(t)∥1, +(2) sgn(e(t))∗l(t) + (l(t))∗sgn(e(t)) ≤ 2∥l(t)∥1, +(3) sgn(e(t))∗sgn(e(t)) = sgn(e(t))sgn(e(t))∗ = ∥sgn(e(t))∥1, +(4) sgn(e(t))∗(e(t))c + ((e(t))c)∗sgn(e(t)) = 2∥(e(t))c∥1 ≥ +� +2n1−c∥(e(t))∥c +1, c > 1, +2∥(e(t))∥c +1, +0 < c ≤ 1, +(5) ∥ +� t +t−π +f(e(s))ds∥1 ≤ +� t +t−π +∥f(e(s))∥1ds. +12 + +3. Main results +In this section, by designing effective controllers, some sufficient conditions are +established to achieve the synchronization of the drive system (2.5) and the response +system (2.6) in fixed time. The one norm of the quaternion is to demonstrate the +practicability of our method by achieving FXTSYN of UCQVMNNs. +To synchronize the drive system and the response system in a fixed time, a novel +controller up(t) is designed as follows +up(t) = λ1pep(t)−λ2pep(t)α −λ3pep(t)β +λ4pep(t−τ(t))+λ5p +� t +t−π +∥ep(s)∥1ds, (3.1) +where λ1p, λ2p, λ3p, λ4p and λ5p are real constants, and the numbers α, β ∈ R satisfy +0 < α < 1 and β > 1, respectively. +Theorem 3.1. Suppose that Assumptions (A1) holds. If parameters λ1p, λ2p, λ3p, λ4p +and λ5p in the controller (3.1) satisfy +dp − λ1p − +n +� +q=1 +υpa+ +qp ≥ 0, +n +� +q=1 +ϱpb+ +qp + λ4p ≤ 0, +n +� +q=1 +ιpc+ +qp + λ5p ≤ 0, +λ2p > 0, +λ3p > 0, +(3.2) +for p, q = 1, 2, ..., n, then the drive system (2.5) synchronizes to the response system +(2.6) in a fixed time. The settling time is estimated as follows +T1 = 1 +λ2 +� +λ2 +n2(1−β)λ3 +� 1−α +β−α� +1 +β − 1 + +1 +1 − α +� +. +(3.3) +where λ2 = min +1≤p≤n{λ2p}, λ3 = min +1≤p≤n{λ3p}. +Proof. Consider the Lyapunov function +V (t) = 1 +2 +n +� +p=1 +(sgn(ep(t))∗ep(t) + ep(t)∗sgn(ep(t))). +(3.4) +13 + +By applying Assumption (A1) to the derivative along the trajectory of the error +system (2.11), one can obtain +d +dtV (t) = 1 +2 +n +� +p=1 +(sgn(ep(t))∗ d +dtep(t) + ( d +dtep(t))∗sgn(ep(t))) += 1 +2 +n +� +p=1 +sgn(ep(t))∗� +− dpep(t) + +n +� +q=1 +˜apq(t) +� +fq(yq(t)) − fq(xq(t)) +� ++ +n +� +q=1 +˜bpq(t) +� +gq(yq(t − τ(t))) − gq(xq(t − τ(t))) +� ++ +n +� +q=1 +˜cpq(t) +� t +t−π +(hq(yq(s)) − hq(xq(s)))ds ++ λ1pep(t) − λ2p(ep(t))α − λ3p(ep(t))β + λ4pep(t − τ(t)) + λ5p +� t +t−π +∥ep(s)∥1ds +� ++ 1 +2 +n +� +p=1 +� +− dpep(t) + +n +� +q=1 +˜apq(t) +� +fq(yq(t)) − fq(xq(t)) +� ++ +n +� +q=1 +˜bpq(t) +� +gq(yq(t − τ(t))) − gq(xq(t − τ(t))) +� ++ +n +� +q=1 +˜cpq(t) +� t +t−π +(hq(yq(s)) − hq(xq(s)))ds ++ λ1pep(t) − λ2p(ep(t))α − λ3p(ep(t))β + λ4pep(t − τ(t)) ++ λ5p +� t +t−π +∥ep(s)∥1ds +�∗ +sgn(ep(t)) += 1 +2 +n +� +p=1 +� +sgn(ep(t))∗(−dp)ep(t) + (ep(t))∗(−dp)∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜apq(t)[fq(yq(t)) − fq(xq(t))] ++ [fq(yq(t)) − fq(xq(t))]∗(˜apq(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜bpq(t)[gq(yq(t − τ(t))) − gq(xq(t − τ(t)))] ++ [gq(yq(t − τ(t))) − gq(xq(t − τ(t)))]∗(˜bpq(t))∗sgn(ep(t)) +� +14 + ++ 1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜cpq(t) +� � t +t−π +(hq(yq(s)) − hq(xq(s)))ds +� ++ +� � t +t−π +(hq(yq(s)) − hq(xq(s)))ds +�∗(˜cpq(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +λ1p +� +sgn(ep(t))∗ep(t) + ep(t)∗sgn(ep(t)) +� +− 1 +2 +n +� +p=1 +λ2p +� +sgn(ep(t))∗(ep(t))α + ((ep(t))α)∗sgn(ep(t)) +� +− 1 +2 +n +� +p=1 +λ3p +� +sgn(ep(t))∗(ep(t))β + ((ep(t))β)∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +λ4p +� +sgn(ep(t))∗ep(t − τ(t)) + (eq(t − τ(t)))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +λ5p +� +sgn(ep(t))∗� � t +t−π +∥ep(s)∥1ds +� ++ +� � t +t−π +∥ep(s)∥1ds +�∗sgn(ep(t)) +� +≤ 1 +2 +n +� +p=1 +(−dp + λ1p) +� +sgn(ep(t))∗ep(t) + (ep(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜apq(t)υq∥eq(t)∥1 + [υq∥eq(t)∥1]∗(˜apq(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜bpq(t)ϱq∥eq(t − τ(t))∥1 ++ [ϱq∥eq(t − τ(t))∥1]∗(˜bpq(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜cpq(t) +� � t +t−π +(ιq∥eq(s)∥1)ds +� ++ +� � t +t−π +(ιq∥eq(s)∥1)ds +�∗(˜cpq(t))∗sgn(ep(t)) +� +− 1 +2 +n +� +p=1 +λ2p +� +sgn(ep(t))∗(ep(t))α + ((ep(t))α)∗sgn(ep(t)) +� +− 1 +2 +n +� +p=1 +λ3p +� +sgn(ep(t))∗(ep(t))β + ((ep(t))β)∗sgn(ep(t)) +� +15 + ++ 1 +2 +n +� +p=1 +λ4p +� +sgn(ep(t))∗ep(t − τ(t)) + (ep(t − τ(t)))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +λ5p +� +sgn(ep(t))∗� � t +t−π +∥ep(s)∥1ds +� ++ +� � t +t−π +∥ep(s)∥1ds +�∗sgn(ep(t)) +� +By Lemma 2.4 and Assumption (A1), we have +1 +2 +n +� +p=1 +(−dp + λ1p) +� +sgn(ep(t))∗ep(t) + (ep(t))∗sgn(ep(t)) +� += − +n +� +p=1 +(dp − λ1p)∥ep(t)∥1, +(3.5) +1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜apq(t)υq∥eq(t)∥1 + [υq∥eq(t)∥1]∗(˜apq(t))∗sgn(ep(t)) +� +≤ +n +� +p=1 +n +� +q=1 +υqa+ +pq∥eq(t)∥1 = +n +� +p=1 +n +� +q=1 +υpa+ +qp∥ep(t)∥1, +(3.6) +1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜bpq(t)ϱq∥eq(t − τ(t))∥1 + [ϱq∥eq(t − τ(t))∥1]∗(˜bpq(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +λ4p +� +sgn(ep(t))∗ep(t − τ(t)) + (ep(t − τ(t)))∗sgn(ep(t)) +� +≤ +n +� +p=1 +n +� +q=1 +ϱqb+ +pq∥eq(t − τ(t))∥1 + +n +� +p=1 +λ4p∥ep(t − τ(t))∥1 += +n +� +p=1 +� +n +� +q=1 +ϱpb+ +qp + λ4p +� +∥ep(t − τ(t))∥1. +(3.7) +And +1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜cpq(t) +� � t +t−π +(ιq∥eq(s)∥1)ds +� ++ +� � t +t−π +(ιq∥eq(s)∥1)ds +�∗(˜cpq(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +λ5p +� +sgn(ep(t))∗� � t +t−π +∥ep(s)∥1ds +� ++ +� � t +t−π +∥ep(s)∥1ds +�∗sgn(ep(t)) +� +16 + +≤ +n +� +p=1 +n +� +q=1 +ιqc+ +pq +� t +t−π +∥eq(s)∥1ds + +n +� +p=1 +λ5p +� t +t−π +∥ep(s)∥1ds += +n +� +p=1 +( +n +� +q=1 +ιpc+ +qp + λ5p) +� t +t−π +∥ep(s)∥1ds. +(3.8) +Analogously, using Lemma 2.3 and Lemma 2.4, it is not difficult to get the following +conclusion +− 1 +2 +n +� +p=1 +λ2p +� +sgn(ep(t))∗(ep(t))α + ((ep(t))α)∗sgn(ep(t)) +� +≤ − +n +� +p=1 +λ2p∥ep(t)∥α +1, +− 1 +2 +n +� +p=1 +λ3p +� +sgn(ep(t))∗(ep(t))β + ((ep(t))β)∗sgn(ep(t)) +� +≤ −n1−β +n +� +p=1 +λ3p∥ep(t)∥β +1. +(3.9) +Combined with the above formulas (3.5)-(3.9) and Lemma 2.3, one can obtain +d +dtV (t) ≤ − +n +� +p=1 +� +dp − λ1p − +n +� +q=1 +υpa+ +qp +� +∥ep(t)∥1 + +n +� +p=1 +� +n +� +q=1 +ϱpb+ +qp + λ4p +� +∥ep(t − τ(t))∥1 ++ +n +� +p=1 +( +n +� +q=1 +ιpc+ +qp + λ5p) +� t +t−π +∥ep(s)∥1ds − +n +� +p=1 +λ2p∥ep(t)∥α +1 +− n1−β +n +� +p=1 +λ3p∥ep(t)∥β +1 +≤ − +n +� +p=1 +λ2p∥ep(t)∥α +1 − n1−β +n +� +p=1 +λ3p∥ep(t)∥β +1 +≤ −λ2 +n +� +p=1 +∥ep(t)∥α +1 − n1−βλ3 +n +� +p=1 +∥ep(t)∥β +1 +≤ −λ2( +n +� +p=1 +∥ep(t)∥1)α − n2(1−β)λ3( +n +� +p=1 +∥ep(t)∥1)β += −λ2(V (t))α − n2(1−β)λ3(V (t))β, +where λ2 = min +1≤p≤n{λ2p}, λ3 = min +1≤p≤n{λ3p}. +Based on Lemma 2.1, it follows that the error system (2.11) is fixed-time stable, +that is the drive system (2.5) and the response system (2.6) achieve synchronization +17 + +in fixed-time with the controller (3.1). Furthermore, one can estimate the settling +time by the following equality +T1 = 1 +λ2 +� +λ2 +n2(1−β)λ3 +� 1−α +β−α� +1 +β − 1 + +1 +1 − α +� +. +(3.10) +The proof is completed. +□ +Corollary 3.1. By Remark 2.1, taking the same conditions and the controller as +in Theorem 3.1, one can obtain that system (2.5) and system (2.6) are synchronized +within a fixed time. Furthermore, the settling time can be estimated as +T2 = +1 +λ2(1 − α) + +1 +n2(1−β)λ3(β − 1). +(3.11) +It is well known that the error system can be stabilized within a fixed time, that +is, e(t) → 0 within time T, by developing an appropriate controller and Lyapunov +function. In general, it can be seen from the inequality +d +dtV (e(t)) ≤ −aV α(e(t)) − +bV β(e(t)) with 0 < α < 1, β > 1 in Lemma 2.1 that the right-hand side contains two +terms: the index of one term is bigger than 0 and smaller than 1, while the index +of the other one is larger than 1. In the controller (3.1), we can see that the first, +fourth, and fifth items are designed to allow the error system to achieve Lyapunov +stability. The second and third terms are designed to achieve synchronization of the +drive-response system in a fixed time. And then compute the settling time using the +parameters of these two items. However, in Lemma 2.1 and Remark 2.1, the settling +time estimate includes both errors from greater than 1 to 1 and then from 1 to 0. +This must be considered because the two terms exist in the inequality at the same +time, whether they play a role or not, which may cause the estimated settling time +to be inaccurate. +Li et al. [41] proposed a novel Lemma 2.2 to guarantee fixed-time synchronization +of discontinuous neural networks, where V (t) ≥ 1 or 0 < V (t) < 1 is first judged +and the functioning part is intelligently chosen to be more economical. It is clear +that Lemma 2.2 is an excellent alternative to existing techniques and can signifi- +cantly reduce energy consumption while achieving a more precise settling time than +most relevant research. For the first time, Lemma 2.2 is used to synchronize the +UCQVMNNs in a fixed time. +According to Lemma 2.2, in order to make the error system (2.11) stable in fixed +time, different from Theorem 3.1, a new nonlinear controller is designed as follows: +up(t) = −k1pep(t) + k2pep(t − τ(t)) − µ(ep(t))γ+sgn(∥e(t)∥1−1) + k3p +n +� +q=1 +� t +t−π +∥eq(s)∥1ds, +(3.12) +18 + +where e(t) = (e1(t), e2(t), · · · , en(t))T ∈ Hn, parameter γ ∈ R satisfies 1 ≤ γ < 2, +the feedback gains k1p, k2p and µ are real constants, and k1p ≥ 0, µ > 0 will be +determined later. +Theorem 3.2. Suppose that Assumption (A1) holds, and for constants k2P, k3P and +µ, the following inequalities are fulfilled +d < µ1, +n +� +q=1 +ϱpb+ +qp + k2p ≤ 0, +n +� +q=1 +ιqc+ +pq + k3p ≤ 0, +(3.13) +then, UCQVMNNs (2.5) and (2.6) can achieve FIXSYN with controller (3.12). Fur- +thermore, for any initial condition, the settling time can be estimated by +T4 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +d(2 − γ) ln +µ +µ − d + +1 +γ(µ1 − d), +d > 0, +1 +µ(2 − γ) + +1 +µ1γ , +d = 0, +1 +d(2 − γ) ln +µ +µ − d + 1 +dγ ln +µ1 +µ1 − d, +d < 0, +(3.14) +In particular, if the initial value is less than 1, the corresponding settling time is +estimated as follows +T3 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +d(2 − γ) ln +µ +µ − d + +1 +γ(µ − d), +d > 0, +1 +µ(2 − γ) + 1 +µγ , +d = 0, +1 +d(2 − γ) ln +µ +µ − d + 1 +dγ ln +µ +µ − d, +d < 0, +(3.15) +where d = max +1≤p,q≤n{−dp − k1p + �n +q=1 υpa+ +qp}, µ1 = µn−2γ. +Proof. Consider the Lyapunov function +V (t) = 1 +2 +n +� +p=1 +(sgn(ep(t))∗ep(t) + ep(t)∗sgn(ep(t))), +(3.16) +19 + +According to Assumption (A1), calculating the derivative along the trajectory of the +error system (2.11), one can get +d +dtV (t) = 1 +2 +n +� +p=1 +(sgn(ep(t))∗ d +dtep(t) + ( d +dtep(t))∗sgn(ep(t))) += 1 +2 +n +� +p=1 +sgn(ep(t))∗� +− dpep(t) + +n +� +q=1 +˜apq(t) +� +fq(yq(t)) − fq(xq(t)) +� ++ +n +� +q=1 +˜bpq(t) +� +gq(yq(t − τ(t))) − gq(xq(t − τ(t))) +� ++ +n +� +q=1 +˜cpq(t) +� t +t−π +(hq(yq(s)) − hq(xq(s)))ds +− k1pep(t) + k2pep(t − τ(t)) − µ(ep(t))γ+sgn(∥ep(t)∥1−1) + k3p +n +� +q=1 +� t +t−π +∥eq(s)∥1ds +� ++ 1 +2 +n +� +p=1 +� +− dpep(t) + +n +� +q=1 +˜apq(t) +� +fq(yq(t)) − fq(xq(t)) +� ++ +n +� +q=1 +˜bpq(t) +� +gq(yq(t − τ(t))) − gq(xq(t − τ(t))) +� ++ +n +� +q=1 +˜cpq(t) +� t +t−π +(hq(yq(s)) − hq(xq(s)))ds +− k1pep(t) + k2pep(t − τ(t)) − µ(ep(t))γ+sgn(∥ep(t)∥1−1) ++ k3p +n +� +q=1 +� t +t−π +∥eq(s)∥1ds +�∗ +sgn(ep(t)) += 1 +2 +n +� +p=1 +� +sgn(ep(t))∗(−dp)ep(t) + (ep(t))∗(−dp)∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜apq(t)[fq(yq(t)) − fq(xq(t))] ++ [fq(yq(t)) − fq(xq(t))]∗(˜apq(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜bpq(t)[gq(yq(t − τ(t))) − gq(xq(t − τ(t)))] +20 + ++ [gq(yq(t − τ(t))) − gq(xq(t − τ(t)))]∗(˜bpq(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜cpq(t) +� � t +t−π +(hq(yq(s)) − hq(xq(s)))ds +� ++ +� � t +t−π +(hq(yq(s)) − hq(xq(s)))ds +�∗(˜cpq(t))∗sgn(ep(t)) +� +− 1 +2 +n +� +p=1 +k1p +� +sgn(ep(t))∗ep(t) + ep(t)∗sgn(ep(t)) +� +− 1 +2 +n +� +p=1 +µ +� +sgn(ep(t))∗(ep(t))γ+sgn(∥e(t)∥1−1) + ((ep(t))γ+sgn(∥e(t)∥1−1))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +k2p +� +sgn(ep(t))∗ep(t − τ(t)) + (ep(t − τ(t)))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +k3p +� +sgn(ep(t))∗� � t +t−π +∥eq(s)∥1ds +� ++ +� � t +t−π +∥eq(s)∥1ds +�∗)sgn(ep(t)) +� +≤ 1 +2 +n +� +p=1 +� +(−dp − k1p) +� +sgn(ep(t))∗ep(t) + (ep(t))∗sgn(ep(t)) +�� ++ 1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜apq(t)υq∥eq(t)∥1 + [υq∥eq(t)∥1]∗(˜apq(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜bpq(t)ϱq∥eq(t − τ(t))∥1 ++ [ϱq∥eq(t − τ(t))∥1]∗(˜bpq(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +k2p +� +sgn(ep(t))∗ep(t − τ(t)) + (ep(t − τ(t)))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜cpq(t) +� � t +t−π +(ιq∥eq(s)∥1)ds +� ++ +� � t +t−π +(ιq∥eq(s)∥1)ds +�∗(˜cpq(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +k3p +� +sgn(ep(t))∗� � t +t−π +∥eq(s)∥1ds +� ++ +� � t +t−π +∥eq(s)∥1ds +�∗)sgn(ep(t)) +� +21 + +− 1 +2µ +n +� +p=1 +� +sgn(ep(t))∗(ep(t))γ+sgn(∥e(t)∥1−1) + ((ep(t))γ+sgn(∥e(t)∥1−1))∗sgn(ep(t)) +� +. +Using Lemma 2.4 can be get +1 +2 +n +� +p=1 +n +� +q=1 +� +sgn(ep(t))∗˜cpq(t) +� � t +t−π +(ιq∥eq(s)∥1)ds +� ++ +� � t +t−π +(ιq∥eq(s)∥1)ds +�∗(˜cpq(t))∗sgn(ep(t)) +� ++ 1 +2 +n +� +p=1 +n +� +q=1 +k3p +� +sgn(ep(t))∗� � t +t−π +∥eq(s)∥1ds +� ++ +� � t +t−π +∥eq(s)∥1ds +�∗)sgn(ep(t)) +� +≤ +n +� +p=1 +n +� +q=1 +(ιqc+ +pq + k3p) +� t +t−π +∥eq(s)∥1ds. +(3.17) +Next, in combination with Lemma 2.2, we will discuss two cases according to the +error value. +Case 1: when 0 < ∥e(t)∥1 < 1 (0 < V (t) < 1), so that sgn(∥e(t)∥1 − 1) = −1, +0 ≤ γ + sgn(∥e(t)∥1 − 1) < 1. According to the conclusion in Theorem 1 and Lemma +2.3 can be directly obtained following inequalities. +− 1 +2 +� +sgn(ep(t))∗(ep(t))γ+sgn(∥e(t)∥1−1) + ((ep(t))γ+sgn(∥e(t)∥1−1))∗sgn(ep(t)) +� +≤ −∥ep(t)∥γ+sgn(∥e(t)∥1−1) +1 +. +(3.18) +Therefore combined with the above formulas (3.5)-(3.7), (3.17) and (3.18), then +using Lemma 2.3 and Lemma 2.4, one can obtain +d +dtV (t) ≤ +n +� +p=1 +� +− dp − k1p + +n +� +q=1 +υpa+ +qp +� +∥ep(t)∥1 + +n +� +p=1 +� +n +� +q=1 +ϱpb+ +qp + k2p +� +∥ep(t − τ(t))∥1 ++ +n +� +p=1 +n +� +q=1 +(ιqc+ +pq + k3p) +� t +t−π +∥eq(s)∥1ds − µ +n +� +p=1 +∥ep(t)∥γ+sgn(∥e(t)∥1−1) +1 +≤ d +n +� +p=1 +∥ep(t)∥1 − µ +n +� +p=1 +∥ep(t)∥γ+sgn(∥e(t)∥1−1) +1 +≤ d +n +� +p=1 +∥ep(t)∥1 − µ( +n +� +p=1 +∥ep(t)∥1)γ+sgn(∥e(t)∥1−1) += dV (t) − µ(V (t))γ+sgn(V (t)−1), +(3.19) +22 + +where d = max +1≤p,q≤n{−dp − k1p + �n +q=1 υpa+ +qp}. +With conditions in (3.13) holding, based on Lemma 2.2, it follows that the error +system (2.11) is fixed-time stable, that is the QVMNNs (2.5) and the QVMNNs (2.6) +achieve synchronization in fixed-time with the controller (3.12). And the settling time +is estimated as +T3 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +d(2 − γ) ln +µ +µ − d + +1 +γ(µ − d), +d > 0, +1 +µ(2 − γ) + 1 +µγ , +d = 0, +1 +d(2 − γ) ln +µ +µ − d + 1 +dγ ln +µ +µ − d, +d < 0, +Case 2: +when ∥e(t)∥1 ≥ 1 (V (t) ≥ 1), so that sgn(∥e(t)∥1 − 1) ≥ 0, γ + +sgn(∥e(t)∥1 − 1) ≥ γ ≥ 1. +According to Lemma 2.3 and Lemma 2.4, it can ba +seen that +− 1 +2 +� +sgn(ep(t))∗(ep(t))γ+sgn(∥e(t)∥1−1) + ((ep(t))γ+sgn(∥e(t)∥1−1))∗sgn(ep(t)) +� +≤ −n1−γ−sgn(∥e(t)∥1−1)∥ep(t)∥γ+sgn(∥e(t)∥1−1) +1 +≤ −n−γ∥ep(t)∥γ+sgn(∥e(t)∥1−1) +1 +(3.20) +Therefore we have +d +dtV (t) ≤ +n +� +p=1 +� +− dp − k1p + +n +� +q=1 +υpa+ +qp +� +∥ep(t)∥1 + +n +� +p=1 +� +n +� +q=1 +ϱpb+ +qp + k2p +� +∥ep(t − τ(t))∥1 ++ +n +� +p=1 +n +� +q=1 +(ιpc+ +qp + k3p) +� t +t−π +∥eq(s)∥1ds − µ +n +� +p=1 +n−γ∥ep(t)∥γ+sgn(∥e(t)∥1−1) +1 +≤ d +n +� +p=1 +∥ep(t)∥1 − µ +n +� +p=1 +n−γ∥ep(t)∥γ+sgn(∥e(t)∥1−1) +1 +≤ d +n +� +p=1 +∥ep(t)∥1 − µn−γn1−γ−sgn(∥e(t)∥1−1)( +n +� +p=1 +∥ep(t)∥1)γ+sgn(∥e(t)∥1−1) +≤ d +n +� +p=1 +∥ep(t)∥1 − µn−2γ( +n +� +p=1 +∥ep(t)∥1)γ+sgn(∥e(t)∥1−1) += dV (t) − µ1(V (t))γ+sgn(V (t)−1), +(3.21) +where µ1 = µn−2γ. In order to guarantee the Lyapunov stability, an extra condition +d < µ1 must be satisfied. +23 + +Analogously, by Lemma 2.2, we can get that the UCQVMNNs (2.5) and (2.6) +achieve synchronization in fixed-time with the controller (3.12). +And the settling +time can be inferred as +T4 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +d(2 − γ) ln +µ +µ − d + +1 +γ(µ1 − d), +d > 0, +1 +µ(2 − γ) + +1 +µ1γ , +d = 0, +1 +d(2 − γ) ln +µ +µ − d + 1 +dγ ln +µ1 +µ1 − d, +d < 0, +The proof is completed. +□ +Remark 3.1. Different from Theorem 3.1, Theorem 3.2 does not require parameter +d < 0. Different d values yield different settling time estimates, resulting in greater +selectivity and robustness. If we take a proper k1p > 0 to make d < 0, any value of +positive µ can guarantee the existence of condition (3.13). If k1p = 0, for p = 1, 2, ..., n +in controller (3.12), maybe d > 0, so we should choose a large enough positive µ to +make inequality (3.13) hold. In this case, controller (3.12) only has three terms. As +a result, it is critical to choose an appropriate k1p for the given situation. +4. Numerical Simulations +In this section, three numerical instances are given to illustrate the effectiveness +of our theoretical results obtained in the previous section. +Example 1. Consider the following 2-dimensional UCQVMNNs with mixed delays +as the drive system +d +dtxp(t) = −dpxp(t) + +2 +� +q=1 +apq(xp(t))fq(xq(t)) + +2 +� +q=1 +bpq(xp(t))gq(xq(t − τ(t))) ++ +2 +� +q=1 +cpq(xp(t)) +� t +t−π +hq(xq(s))ds, +(4.1) +the response system is +d +dtyp(t) = −dpyp(t) + +2 +� +q=1 +apq(yp(t))fq(yq(t)) + +2 +� +q=1 +bpq(yp(t))gq(yq(t − τ(t))) ++ +2 +� +q=1 +cpq(yp(t)) +� t +t−π +hq(yq(s))ds + up(t), +(4.2) +24 + +where p, q = 1, 2, dp = 0.5, τ(t) = 0.3sin(t) + 0.4, π = 0.4, so let τ = 0.7, and the +memristive connection weights are +A = +� 1.8 − 1.6i − 2.3j − 1k +−1 − 1.5i − 1.7j + 1.3k +1.5 + 3.5i − 2j − 1.5k +−1.5 + 2i − 1.5j − 1.6k +� +B = +� −0.45 + 0.3i − 0.2j − 0.35k +−0.25 + 0.25i + 0.1j − 0.4k +0.2 − 0.5i + 0.35j − 0.25k +0.2 − 0.4i − 0.18j + 0.22k +� +C = +� +2.4 + 3i + 1j + 2k +−1.5 + 1.2i − 2j + 1.1k +−1 + 2.1i − 1.9j + 1.3k +2 + 2.3i + 1j − 3.2k +� +where |xp(t)| ≤ 1, p = 1, 2. +A = +� 1.8 − 1.6i − 3.5j − 2.3k +−0.6 + 1.5i − 1.7j + 1.3k +−1.9 − 1.5i − 1.7j + 1.3k +1.5 − 2i + 1.5j + 1.2k +� +B = +� −0.5 + 0.3i − 0.2j − 0.3k +−0.15 − 0.2i − 0.2j + 0.45k +0.3 − 0.65i − 0.15j + 0.1k +−0.44 + 0.1i + 0.16j + 0.3k +� +C = +� −2.4 − 2i − 1j + 1.6k +1.1 − 2i + 1.5j − 1.6k +1.4 − 1.7i + 2j − 1.4k +−2 − 2.3i + 1j − 1.8k +� +where |xp(t)| > 1, p = 1, 2. One can easily compute a+ +11 = 9.2, a+ +12 = 5.5, a+ +21 = +8.5, a+ +22 = 6.6, b+ +11 = 1.3, b+ +12 = 1.0, b+ +21 = 1.3, b+ +22 = 1.0 and c+ +11 = 8.4, c+ +12 = 6.2, c+ +21 = +6.5, c+ +22 = 7.1. And take the activation functions are +fq(xq(t)) = 2 tanh(xq(0)(t)) + 2 tanh(xq(1)(t))i + 2 tanh(xq(2)(t))j + 2 tanh(xq(3)(t))k, +gq(xq(t)) = 0.1 tanh(xq(0)(t))+0.1 tanh(xq(1)(t))i+0.1 tanh(xq(2)(t))j+0.1 tanh(xq(3)(t))k, +hq(xq(t)) = 0.7 tanh(xq(0)(t))+0.7 tanh(xq(1)(t))i+0.7 tanh(xq(2)(t))j+0.7 tanh(xq(3)(t))k. +According to Assumption (A1), a simple calculation yields that υ1 = υ2 = 2, ϱ1 = +ϱ2 = 0.1, ι1 = ι2 = 0.7. +The initial conditions of (4.1) and (4.2) are selected as +φ1(s) = 1.5 + 2i − 0.6j + 0.8k, φ2(s) = −1.2 − 1.5i + 1j − 0.5k, ψ1(s) = 2.5 − +2i − 1j + 1.2k, ψ2(s) = −3 + 1.6i + 0.8j − 2k, s ∈ [−0.7, 0]. +If there is no controller, i.e., u1(t) = u2(t) = 0, the trajectories of the error +system is simulated in Fig. 1, which implies that systems (4.1) and (4.2) cannot +obtain synchronization without control input. +Correspondingly, by (3.1), the controllers are designed as follows +u1(t) = λ11e1(t) − λ21eα +1(t) − λ31eβ +1(t) + λ41e1(t − τ(t)) + λ51 +� t +t−π +∥e1(s)∥1ds, (4.3) +25 + +u2(t) = λ12e2(t) − λ22eα +2(t) − λ32eβ +2(t) + λ42e2(t − τ(t)) + λ52 +� t +t−π +∥e2(s)∥1ds, (4.4) +Figure 1: The trajectories of error system with- +out controller. +Figure 2: The trajectories of error system with +controllers (4.3) and (4.4). +The values of the coefficients λ11 = −80, λ12 = −50, λ21 = λ22 = 1, λ31 = 30, λ32 = +35, λ41 = −0.26, λ42 = −0.2, λ51 = −10.45, λ52 = −9.35 can be calculated from (3.2), +α = 0.6, β = 1.6, the conditions in Theorem 3.1 hold. +(a) +(b) +(c) +(d) +Figure 3: Trajectories of the real and imaginary parts of drive-response system with controllers (4.3) +and (4.4) when p = 1. +Under the designed controllers (4.3) and (4.4), the trajectories of real and imag- +inary parts of system (4.1) and (4.2) are given in Fig. 3-4. These figures indicate +that once appending the controllers (4.3) and (4.4) to the system (4.2), it will be +synchronized with the system (4.1) in a fixed time. +26 + +150 +(t) +ek(t) +(t) +e,(t) +e,(t) +e,(t) +100 +50 +0 +-50 +-100 +-150 +0 +2 +3 +4 +5 +t5 +e,(t) +e(t) +ek(t) +4 +(t +3 +2 +0 +-2 +-3 +-4 +-5 +0 +0.2 +0.4 +0.6 +0.8 +t140 +120 +(t +100 +4.5 +80 +4 +60 +3.5 +40 +0 +0.020.04 +20 +0 +-20 +0 +2 +3 +4 +540 +3.5 +(a)x +30 + (t) +20 4.5 +-5 +10 +5.5 +0 +0 +0.05 +Λ0.1 +-10 +-20 +-30 +-40 +-50 +-60 +0 +2 +3 +4 +540 +0 +-0.5 +30 +-1 +-1.5 +20 +10 +0.05 +0.1 +0.15 +0 +-10 +-20 +-30 +-40 +0 +2 +3 +4 +525 +20 +t) +15 +10 +5 +0 +-5 +-10 +$.5 +-15 +3 +-20+5 +0.05 +0.1, +0.15 +0.2 +-25 +0 +1 +2 +3 +4 +5 +t(a) +(b) +(c) +(d) +Figure 4: Trajectories of the real and imaginary parts of drive-response system with controllers (4.3) +and (4.4) when p = 2. +Moreover, Fig. 2 shows the whole process of FXTSNY under one-norm. It can +be concluded that the error system reaches 0 in a fixed time, which once implies +that before T = 0.2 one can achieve FXTSNY of the drive-response system. We can +calculate the T1 ≈ 1.491 according to the settling time estimated in (3.3), which is +larger than the real synchronization time but is smaller than conventional estimation. +It shows that Theorem 3.1 is correct. In addition, according to the equation (3.11) +the settling time T2 ≈ 2.628 can be derived from Corollary 1. By comparison, the +estimated settling time derived from Lemma 2.1 is more accurate than Remark 2.1. +Example 2. The two-dimensional UCQVMNNs with mixed delays were given by +the drive system (4.1) and the response system (4.2), which consider the fixed-time +synchronization of Theorem 2. +The parameters take the same as in Example 1. +The initial conditions of (4.1) and (4.2) are chosen as φ1(s) = −3.8 + 4.8i − 1.5j + +1.6k, φ2(s) = −4 + 0.9i − 2.3j + 2k, ψ1(s) = 1.2 + 2.5i − 1.93j + 0.7k, ψ2(s) = +−2.3 + 2.8i − 1.4j − 2.3k, s ∈ [−0.7, 0]. +Correspondingly, using the 2-dimensional controller by (3.12) and we have adap- +tive rules for k1p, k2p, k3p(p = 1, 2) are define in (3.13) with coefficients k21 = +−0.26, k22 = −0.2, k31 = −10.25, k32 = −9.52. +The evolution of real and imaginary parts of system (4.1) and (4.2) are presented +in Fig. 5 under the effect of controller (3.12). And these two figures show that The +drive-response system can achieve FXTSYN in a very short time under the action of +the controller (3.12). Next, Matlab drawing verification is carried out mainly for the +initial error greater than 1. +27 + +100 +80 +K +60 +40 +20 +0 +-20 +-40 +4 +-60 +-6 +-8 +-80 +-10 +-12 +-100 +0.1 +0.2 +0 +1 +2 +3 +4 +560 +40 +6 +20 +0 +0 +0. 1 +-20 +-40 +-60 +-80 +0 +2 +3 +4 +570 +5 +60 +4.5 +50 +4 +3.5 +40 +0 +0,05 +0.1 +30 +20 +10 +0 +-10 +-20 +0 +2 +3 +4 +530 +20 +10 +-10 +-20 +-30 +2 +-40 +-50 +9- +-60 +0 +0.1 +0.2 +-70 +0 +1 +2 +3 +4 +5(a) +(b) +Figure 5: The drive-response system trajectories of p = 1, 2 with controller (3.13). +Firstly, let µ = 40, γ = 1.5, if k11 = 37, k12 = 130, we can get d = −5.1 < 0, µ1 = 5 +satisfy the condition (3.13) in Theorem 3.2. And the evolution trajectory of the error +system with controller (3.12) is shown in Fig. 6(a), we can clearly see that the error +system reaches stability in a very short time. According to Theorem 3.2, we can +calculate a relatively accurate estimate of the settling time, which is T4 ≈ 0.160 +(T3 ≈ 0.065). +(a) d < 0 +(b) d = 0 +(c) d > 0 +Figure 6: Evolution of error states between networks (4.1) and (4.2) under controller (3.13). +Secondly, let k11 = 50, k12 = 23.7, and µ = 56, γ = 1.5, it’s easy to can get +d = 0, µ1 = 7 satisfy the condition (3.13) in Theorem 3.2. Then the corresponding +evolution of the error state with controller (3.12) are depicted in Fig. 6(b). Therefore, +we can see from this picture that the synchronization of systems (4.1) and (4.2) can +be realized within t = 0.2. Furthermore, T4 ≈ 0.131 (T3 ≈ 0.048) according to the +settling time estimated in Theorem 3.2. +Finally, if k11 = 33, k12 = 130, we can get d = 1.9 > 0, µ1 = 4 satisfy the condition +d < µ1. Then the dynamics of the error system with controller (3.12) are depicted +in Fig. 6(c) with µ = 32, γ = 1.5. Obviously, the drive system (4.1) and response +28 + +40 +30 +20 +10 +0 +-10 +-20 +-30 +x(t) +yR(t) +y(t) +y,(t) +一yk(t) +-40 +0 +0.2 +0.4 +0.6 +0.8 +t40 +20 +0 +-20 +-40 +-60 +-80 +-100 +x,(t) +y2(t) +y,(t) +y,(t) +ye(t) +-120 +0 +0.2 +0.4 +0.6 +0.8 +t5 +e(t) +e(t) +4 +(t) +e;(t) e;(t) +(t) +3 +2 +0 +1 +-2 +-3 +-4 +-5 +0 +0.2 +0.4 +0.6 +0.8 +t3 +e,(t) +e(t) +e,(t) +-e,(t) +-e;(t) +ek(t) +0 +-1 +-2 +-3 +-4 +-5 +0 +0.2 +0.4 +0.6 +0.8 +t3 +e,(t) +e(t) +e,(t) +e;(t) e;(t) +2 +0 +-1 +-2 +-3 +-4 +-5 +0 +0.2 +0.4 +0.6 +0.8 +tsystem (4.2) achieve synchronization within the time t = 0.2. And, according to +the equation (3.15) in Theorem 3.2, we can calculate the settling time is T4 ≈ 0.382 +(T3 ≈ 0.087), which is larger than the real synchronization time, but is smaller than +the conventional estimation. +Table 1: Comparisons of the settling time between different k11. +k11 +31 +33 +34.9 +40 +45 +50 +d +3.9 +1.9 +0 +-5.1 +-10.1 +-15.1 +T3 +0.071 +0.069 +0.067 +0.063 +0.059 +0.057 +T4 +0.659 +0.266 +0.183 +0.139 +0.118 +0.104 +Table 2: Comparisons of the settling time between different µ. +µ +8 +16 +24 +32 +40 +48 +µ1 +1 +2 +3 +4 +5 +6 +T3 +0.216 +0.129 +0.093 +0.072 +0.059 +0.050 +T4 +0.321 +0.216 +0.167 +0.137 +0.118 +0.103 +Table 3: Comparisons of the settling time between different γ. +γ +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +T3 +0.091 +0.072 +0.059 +0.051 +0.045 +0.040 +T4 +0.158 +0.134 +0.118 +0.106 +0.096 +0.089 +From (3.14) and (3.15), we know that parameters k2p, k3p(p = 1, 2) have no impact +on the settling time. Tables 1-3 show the comparisons of the settling time for different +k11 (k12 = 130, µ = 40, γ = 1.5), µ (k12 = 130, k11 = 45, γ = 1.5), and γ (k12 = +130, k11 = 45, µ = 40). It is not hard to find that with controller parameters k11, µ +and γ increasing, the settling time T3 and T4 decrease. According to these, we can +choose more proper parameters as the requirements are met. In a nutshell, it is not +difficult to get through the comparison of the above two examples. The settling time +estimation value obtained through Theorem 3.2 is more accurate than Theorem 3.1. +Example 3. +Consider the 128 × 128 pixels color image pattern ”Baboon” that +is depicted in Fig. 7(a). Additionally, create UCQVMNNs that have the form of +the system (2.6) for associatively remembering the color image. +Considering the +computational complexity, we divided the image into 64 blocks for processing (Fig. +7(b)), each block with 16 × 16 pixels. Therefore, each block needs 256-dimensional +neurons to store it. So we need to design UCQVMNNs (2.6) composed of 256 neurons +that have a 256-dimensional equilibrium point storing the colors of the pattern. +29 + +(a) Baboon +(b) 8 × 8 blocks +Figure 7: The original image and its segmented image. +The original image (Fig. 7(a)) has an additional missing, which causes extremely +sparse initial values for the response system (2.6). The specific values are chosen as +φ1(t) = (0.8863i + 0.5373j + 0.4902k, 0, 0.8824i + 0.5059j + 0.4157k, · · · , 0.8235i + +0.3804j+0.3255k, 0) ∈ H256, φ2(t) = (0, · · · , 0.6549i+0.2275j+0.2029k, · · · , 0.8039i+ +0.3490j+0.3412k, · · · ) ∈ H256, · · · , φ64(t) = (0, · · · , 0.5021i+0.2196j+0.3294k, · · · ) ∈ +H256. The weight coefficient are C = (cpq) = diag(10, · · · , 10) ∈ H256×256, d = 0.01, +p, q = 1, 2, · · · , 256, and +apq(·) = +� +� +� +� +� +−0.2 + 0.2i − 0.5j + 0.4k, +q < p, +2 + 0.3i − 0.2j + 0.3k, +q = p, +−0.1 + 0.2i + 0.3j − 0.5k, +q > p, +bpq(·) = +� +0.04 + 0.04i − 0.03j + 0.05k, +q < p, +0.04 − 0.05i + 0.05j − 0.03k, +q ⩾ p, +Take the activation functions are fq(xq(t)) = 0.06(|xq(t)+2|−|xq(t)+1|), gq(xq(t)) = +0.05(|xq(t)+2|−|xq(t)+1|), hq(xq(t)) = 0.01(|xq(t)+2|−|xq(t)+1|), where xq(t) ∈ +H, q = 1, 2, ..., 256. The equilibrium points corresponding to the 64 small blocks color +image are x∗ +1 = (0.3608i+0.3216j+0.1490k, 0.4706i+0.4000j+0.1686k, · · · , 0.6157i+ +0.6431j+0.4275k) ∈ H256, · · · , x∗ +64 = (0.4941i+0.5333j+0.4235k, 0.4549i+0.4549j+ +0.3725k, · · · , 0.2471i + 0.2392j + 0.2235k) ∈ H256. Under the positive role of the +controller (3.12), the process of image restoration is the process of the system (2.6) +to reach the equilibrium state, and the time used for restoration is the time used for +the system to achieve equilibrium. +30 + +(a) Miss 80% +(b) 0.04s +(c) 0.06s +(d) 0.1s +(e) 0.3s +Figure 8: Color image completion results on Image (7). (a) is the missing image where the ratio of +missing pixels is 80%. (b) is the recovery image ( T=0.04s). (c) is the recovery image ( T=0.06s). +(d) is the recovery image ( T=0.1s). (e) is the recovery image ( T=0.3s). +(a) Noise 80% +(b) 0.04s +(c) 0.06s +(d) 0.1s +(e) 0.3s +Figure 9: Color image completion results on Image (7). (a) is the ”salt and pepper” noise image +where the noise density is 80%. (b) is the recovery image ( T=0.04s). (c) is the recovery image ( +T=0.06s). (d) is the recovery image ( T=0.1s). (e) is the recovery image ( T=0.3s). +It can be seen from Fig. 8(a), 8(b), 8(c), 8(d), and 8(e) that system (2.6) can reach +the equilibrium state in a short time under the effect of the controller (3.12). It also +shows that the color images can get rapid recovery when the ratio of missing pixels +is 80%. From Fig. 9(a), 9(b), 9(c), 9(d), and 9(e), we can see that the system can +reach the equilibrium point quickly under the controller (3.12), that is, the ”Baboon” +image under the premise of adding 80% density ”salt and pepper” noise can be quickly +recovered. +Therefore, as long as the suitable controller is designed, we can use UCQVMNNs +to quickly recover color images from any missing or noise state. Thus it shows us +the high efficiency of UCQVMNNs in dealing with highdimensional image restoration +problems, and the controller of the theorem has a significant practical application. +5. Conclusions +The FXTSNY is discussed in this paper for a class of UCQVMNNs with mixed +delays. Since the decomposition technology is generally accompanied by a more com- +31 + +plex derivation process about quaternion. As a result, the quaternion-valued state is +considered as a whole, with one-norm employed to achieve FXTSNY of UCQVMNNs +smoothly and directly. Then based on the Lyapunov stability theorem, set-valued +map, and differential inclusion theorem, we effectively deal with the system disconti- +nuity caused by the memristor’s weight coefficient in drive-response systems using the +measurable selection theorem. Furthermore, sufficient conditions for the FXTSNY of +delayed UCQVMNNs are proposed using the inequality technique and the Lyapunov +stability theorem. In addition, different estimation settling times are obtained based +on various FXTSNY criteria. It is simple to prove that Theorem 3.2’s estimation +value is more accurate than Theorem 3.1’s. Finally, three numerical examples are +provided to demonstrate the validity and effectiveness of theoretical results, as well +as the practical value in high-dimensional color image processing. +Preassigned-time synchronization is a more flexible synchronization method that +is used in conjunction with the non-commutativity of the quaternion. It should be +worthwhile to investigate new methods for studying the preassigned-time synchro- +nization of QVMNNs in future work. +6. Acknowledgement +This work was supported by University of Macau (MYRG2022-00108-FST), Sci- +ence and Technology Development Fund, Macao S.A.R (FDCT/0036/2021/AGJ). +References +[1] William Rowan Hamilton. 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Cambridge university press, 1952. +36 + diff --git a/HtAzT4oBgHgl3EQfU_w0/content/tmp_files/load_file.txt b/HtAzT4oBgHgl3EQfU_w0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..92220a3a2e48080684b763de0faa398d5b23b736 --- /dev/null +++ b/HtAzT4oBgHgl3EQfU_w0/content/tmp_files/load_file.txt @@ -0,0 +1,1820 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf,len=1819 +page_content='Fixed-time synchronization for quaternion-valued memristor-based neural networks with mixed delays Yanlin Zhanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Liqiao Yanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Kit Ian Koua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Yang Liub aDepartment of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Faculty of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' University of Macau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Macau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 999078,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' China b College of Mathematics and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Zhejiang Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Jinhua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 321004,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' China Abstract In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' the fixed-time synchronization (FXTSYN) of unilateral coefficients quaternion-valued memristor-based neural networks (UCQVMNNs) with mixed de- lays is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Instead of decomposition, a direct analytical method is proposed to achieve FXTSYN of UCQVMNNs using one-norm smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Then apply the set- valued map and the differential inclusion theorem to handle discontinuity problems of drive-response systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The novel nonlinear controllers together with the Lyapunov function are designed to achieve the control goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Using the FXTSYN theory and inequality techniques, some criteria of FXTSYN for UCQVMNNs are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Further- more, the estimated settling time is obtained explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Finally, numerical simulations are presented to demonstrate the correctness, effectiveness and practicability of the obtained theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Keywords: Fixed-time synchronization, controllers, quaternion-valued, unilateral coefficients, memristor-based neural networks 2010 MSC: 34D06, 37N35, 92B20, 93D05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Introduction Quaternion is a subset of Clifford algebra, which was invented by Hamilton in 1843 [1] and is a natural extension of complex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The state variables, input variables, connection weights, and the activation functions of quaternion-valued neural networks (QVNNs) all take values in the quaternion field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Moreover, QVNNs have been used in ∗Corresponding author Email addresses: yanlnzhang@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='com (Yanlin Zhang), liqiaoyoung@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='com (Liqiao Yang), kikou@umac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='mo (Kit Ian Kou), liuyang@zjnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='cn ( Yang Liu) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='01275v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='SY] 2 Jan 2023 a variety of practical applications, such as satellite TV, aerospace, 3-D wind process- ing, color image processing, polarized waves, and space rotation [2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' However, in contrast to real- and complex-values, some arithmetic rules such as the commuta- tivity of multiplication do not apply to quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' As a result, these researches on NNs are primarily focused on the real-valued and complex-valued fields [6, 7, 8, 9], while the related research on quaternion is relatively scarce in recent decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Due to the non-commutative of quaternion multiplication, there are a great variety of polynomials in quaternion algebra than in the real and complex fields, such as polynomials with left, right, and bilateral coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' In QDEs, those equations with bilateral coefficients are too difficult to solve, so there are not many results [10, 11] about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Therefore, in this paper, we first study the unilateral coefficients of quaternion-valued neural networks (UCQVNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' As one of the basic circuit elements, the memristor (an abbreviation for the mem- ory and the resistor) was first proposed by Chua [12] in 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The memristor has the properties of memory and nanoscale, which can better and more realistically sim- ulate the biological synapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' It describes the relationship between the charge and the magnetic flux, so the memristor systems are more precise models of artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Furthermore, it has the potential to improve the application of pattern recognition, combinatorial optimization, and data processing [13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' In recent years, many scholars have studied many characteristics of memristor-based neural networks (MNNs) due to the powerful functions in human brain computers [16, 17, 18, 19, 20], such as dissipativity, stability, synchronization and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' It is widely known that synchronization is an expansion of stability, which is the dynamic behavior of the drive-response system to achieve the same state at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Due to the powerful role of synchronization in non-linear systems [21], it is widely used in areas such as information processing [22], security communication [23], chemical reaction [24] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Our research aims to achieve synchroniza- tion as quickly as possible under the controller’s action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Fortunately, the fixed time synchronization (FXTSYN) [25] was quickly proposed, in which the settling time is completely irrelevant to initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' And FXTSYN has a fast convergence time and better interference inhibitory characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Indeed, many practical phenomena, such as information processing and biological systems, require rapid synchronization to maintain normal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Therefore, many scholars pay attention to the FXTSYN of real-valued MNNs (RVMNNs) and complex-valued MNNs (CVMNNs), some promis- ing applications can be found in [20, 26, 27, 28, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' However, as far as we know, compared to the fields of RVMNNs and CVMNNs, there are quite rare results on FXTSNY of UCQVMNNs with mixed delays [31, 32, 33, 34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' And the method used by most of them is the decomposition method, in [31], the FXTSNY of a class of QVNNs with memristor was investigated via decompo- sition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' That is, the real and imaginary parts of a quaternion neural network 2 are re-expressed as four real-valued systems, which lose important information about the original problem structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Regrettably, it is generally accompanied by a more complex derivation process, and it may also increase conservatism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Therefore, finding an easier and no-decomposition method to research the FXTSNY of MNNs is neces- sary and important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' In [34], Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' use a direct analytical method to study the FNTSNY and FXTSNY problems of QVNNs by introducing an improved one-norm, which is without decomposition techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Compared with the traditional decom- posing method, even more specific FXTSNY is achieved by the one-norm method, which shows the strong applicability and less conservative of this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' So, we design two novel nonlinear controllers based on the improved one-norm method to achieve the FXTSNY of UCQVMNNs with mixed delays in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' It is of great significance and value for research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Motivated by the above, it will be of great significance for some high-dimensional dynamical systems with delays, such as secure communication [36] and image com- pression systems [37] to effectively synchronize the complex dynamic behaviors of these delayed systems in a fixed time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' However, existing research results pay little attention to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Therefore, this paper aims at is studied the FXTSNY of a class of UCQVMNNs with mixed delays by the non-decomposition method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The main contributions presented in this article can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Most of the UCQVMNNs are decomposed into four RVMNNs by the decomposi- tion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The calculation and proof process has been carried out four times, which is quite complicated and increases the difficulty of theoretical derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Therefore, using the non-decomposition method of one-norm makes the proof result easier to realize and the calculation process more simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' In addition, the existence of memristor will cause the system to be discontinuous, few studies have used this approach for FXTSNY analysis of UCQVNNs with memristors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' A novel FXTSNY method is applied to UCQVMNNs for the first time, com- bined with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Firstly, V (t) ≥ 1 or 0 < V (t) < 1 is judged and the functioning part is intelligently chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Then, by using the improved one-norm, a suitable Lyapunov function and controller are constructed, and a more robust and accurate synchronization time estimation is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' To verify the effectiveness of the non-decomposition method of the UCQVMNNs, several numerical simulations of the FXTSNY process are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' It not only verifies the correctness and validity of the two theorems, but also shows the su- periority of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Moreover, we can achieve the best synchronization ef- fect by adjusting the values of different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Interestingly, UCQVMNNs have good applications in the rapid recovery of high-dimensional data, so it has practical research significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 3 The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Section 2 presents the description of QVMNNs with bilateral coefficients and unilateral coefficients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' And several Definitions, Lemmas, and some new one-norm inequalities of the quaternion are deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' In Section 3, the FXTSNY scheme is proposed while the effectiveness of the theoretical results is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The effectiveness of sufficient conditions is checked by simulation examples in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Some conclusions are drawn in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Notations: The sets of real and non-negative real numbers are represented by R and R+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' H represents the set of all quaternions and denoted by bold letter, superscripts T and ∗ indicate transposition and conjugate transpose, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The set of all n-dimensional real numbers, complex numbers, and quaternions are represented by Rn, Cn and Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Using ∥ · ∥1 and sgn(·) to denote the one-norm and the sign function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Rn×m, Cn×m and Hn×m denote the n×m-dimensional real, complex and quaternion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Real and complex numbers are the special case of quaternions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' R ∈ C ∈ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Model Formulation and Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Quaternion Algebra Fundamentals A quaternion number x ∈ H is combined by a real part and three imagery parts, and can be written as following form: x = x(0) + x(1)i + x(2)j + x(3)k, where x(ζ) ∈ R, ζ = 0, 1, 2, 3, and i, j, k are imaginary units, and the imaginary units are defined by � i2 = j2 = k2 = ijk = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The modulus of x are defined as |x| = √x¯x = � (x(0))2 + (x(1))2 + (x(2))2 + (x(3))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The transpose of vector x is denoted by xT, the conjugate and conjugate transpose of x are denoted by ¯x = x(0)−x(1)i−x(2)j −x(3)k and x∗ = (x(0)−x(1)i−x(2)j −x(3)k)T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' For any two quaternion x and y = y(0)+iy(1)+jy(2)+ky(3), the addition operation is defined as follows x + y = (x(0) + y(0)) + (x(1) + y(1))i + (x(2) + y(2))j + (x(3) + y(3))k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The multiplication operation is defined by xy =(x(0)y(0) − x(1)y(1) − x(2)y(2) − x(3)y(3)) + (x(0)y(1) + x(1)y(0) + x(2)y(3) − x(3)y(2))i + (x(0)y(2) − x(1)y(3) + x(2)y(0) + x(3)y(1))j + (x(0)y(3) + x(1)y(2) − x(2)y(1) + x(3)y(0))k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 4 It is important to note that the multiplication in the quaternion domain is not com- mutative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' xy ̸= yx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The one-norm of vector v = (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', vn) ∈ Rn and the quaternion x are writ- ten as ∥v∥1 = �n p=1 |vp| and ∥x∥1 = � ζ=0,1,2,3 ∥x(ζ)∥1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' For e(t) = (e1(t), e2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', en(t))T ∈ Hn, t ∈ R, the sign function and one-norm of vector e(t) are denoted by sgn(e(t)) = (sgn(e1(t)), sgn(e2(t)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', sgn(en(t)))T, ∥e(t)∥1 = �n p=1 ∥ep(t)∥1 respectively, and [e(t)]r = ([e1(t)]r, [e2(t)]r, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', [en(t)]r)T, moreover, [ep(t)]r = sgn(ep(t))∥ep(t)∥r 1, p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', n and r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' For any quaternion x = x(0) + x(1)i + x(2)j + x(3)k can be uniquely expressed as x = x1 + x2j, where x1 = x(0) + x(1)i, and x2 = x(2) + x(3)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Furthermore, this expression can be used by quaternion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Given a quaternion matrix A ∈ Hm×n, its expression using the Cayley-Dickson notation is A = Ap +Aqj, where Ap and Aq ∈ Cm×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The quaternion matrix can be denoted as an isomorphic complex matrix A = � Ap Aq − ¯ Aq ¯ Ap � , where Ap = A0 + A1i ∈ Cm×n and Aq = A2 + A3i ∈ Cm×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' QVMNNs with Bilateral Coefficients (BCQVMNNs) In this section, we introduce a new kind of bilateral coefficients QVMNNs (BC- QVMNNs), which has discrete and distributed time delays and with a form d dtxp(t) = −dpxp(t) + n � q=1 `apq(xp(t))fq(xq(t))´apq(xp(t)) + n � q=1 `bpq(xp(t))gq(xq(t − τ(t)))´bpq(xp(t)) + n � q=1 `cpq(xp(t))( � t t−π hq(xq(s))ds)´cpq(xp(t)) + Ip, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) for p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', n, where n corresponds to the number of neurons, xp(t) ∈ H stand the state variable of the p-th neuron, dp > 0 is the real-valued self-feedback coef- ficient, Ip ∈ H denotes the external input or bias, `apq(xp(t)), ´apq(xp(t)), `bpq(xp(t)), ´bpq(xp(t)), `cpq(xp(t)), and ´cpq(xp(t)) ∈ H stand for the memristive connection weights, fq(xq(t)), gq(xq(t − τ(t))) and hq(xq(s)) ∈ H are the activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The dis- crete time delay and the distributed time delay are denoted by τ(t) and π, they satisfies 0 ≤ τ(t) ≤ τ, τ = max{τ(t), π}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 5 For simpler expression, we can represent (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) as vector form d dtX(t) = −DX(t) + ` Af(X(t)) ´ A + `Bg(X(t − τ(t))) ´B + `C( � t t−π h(X(s))ds) ´C + I, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) where X(t) = (x1(t), x2(t), · · · , xn(t))T ∈ Hn, D = diag(d1, d2, · · · , dn) ∈ Rn×n, f(X(t)) = (f1(x1(t)), f2(x2(t)), · · · , fn(xn(t)))T ∈ Hn, g(X(t − τ(t))) = (g1(x1(t − τ(t))), g2(x2(t−τ(t))), · · · , gn(xn(t−τ(t))))T ∈ Hn, h(X(s)) = (h1(x1(s)), h2(x2(s)), · · , hn(xn(s)))T ∈ Hn, I = (I1, I2, · · · , In)T ∈ Hn and ` A = � � � � � `a11(x1(t)) `a12(x1(t)) · · `a1n(x1(t)) `a21(x2(t)) `a22(x2(t)) · · `a2n(x2(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' `an1(xn(t)) `an2(xn(t)) · · `ann(xn(t)) � � � � � ∈ Hn×n, ´ A = � � � � � ´a11(x1(t)) ´a12(x1(t)) · · ´a1n(x1(t)) ´a21(x2(t)) ´a22(x2(t)) · · ´a2n(x2(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' ´an1(xn(t)) ´an2(xn(t)) · · ´ann(xn(t)) � � � � � ∈ Hn×n, `B = � � � � � `b11(x1(t)) `b12(x1(t)) · · `b1n(x1(t)) `b21(x2(t)) `b22(x2(t)) · · `b2n(x2(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' `bn1(xn(t)) `bn2(xn(t)) · · `bnn(xn(t)) � � � � � ∈ Hn×n, ´B = � � � � � ´b11(x1(t)) ´b12(x1(t)) · · ´b1n(x1(t)) ´b21(x2(t)) ´b22(x2(t)) · · ´b2n(x2(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' ´bn1(xn(t)) ´bn2(xn(t)) · · ´bnn(xn(t)) � � � � � ∈ Hn×n, `C = � � � � � `c11(x1(t)) `c12(x1(t)) · · `c1n(x1(t)) `c21(x2(t)) `c22(x2(t)) · · `c2n(x2(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' `cn1(xn(t)) `cn2(xn(t)) · · `cnn(xn(t)) � � � � � ∈ Hn×n, ´C = � � � � � ´c11(x1(t)) ´c12(x1(t)) · · ´c1n(x1(t)) ´c21(x2(t)) ´c22(x2(t)) · · ´c2n(x2(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' ´cn1(xn(t)) ´cn2(xn(t)) · · ´cnn(xn(t)) � � � � � ∈ Hn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 6 According to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1, let X(t) = X1(t)+X2(t)j, ` A = `A1 + `A2j, ´ A = ´A1 + ´A2j, `B = `B1 + `B2j, ´B = ´B1 + ´B2j, `C = `C1 + `C2j, ´C = ´C1 + ´C2j, and f(X(t)) = f1(X(t))+f2(X(t))j, g(X(t−τ(t))) = g1(X(t−τ(t)))+g2(X(t−τ(t)))j, h(X(s)) = h1(X(s))+h2(X(s))j, I = L1 +L2j, where X1(t), X2(t), L1, L2, f1(X(t)), f2(X(t)), g1(X(t−τ(t))), g2(X(t−τ(t))), and h1(X(s)), h2(X(s)) ∈ Cn, `A1, `A2, ´A1, ´A2, `B1, `B2, ´B1, ´B2, `C1, `C2, and ´C1, ´C2 ∈ Cn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Then system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) can be converted to the following form d dt(X1(t) + X2(t)j) = −D(X1(t) + X2(t)j) + ( `A1 + `A2j)(f1(X(t)) + f2(X(t))j)( ´A1 + ´A2j) + ( `B1 + `B2j)(g1(X(t − τ(t))) + g2(X(t − τ(t)))j)( ´B1 + ´B2j) + ( `C1 + `C2j)( � t t−π (h1(X(s)) + h2(X(s))j)ds)( ´C1 + ´C2j) + L1 + L2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Thus system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) is equivalent to d dtX1(t) = −DX1(t) + `A1 ´A1f1(X(t)) − `A1 ¯´A2f2(X(t)) − `A2 ¯´A2 ¯f1(X(t)) − `A2 ´A1 ¯f2(X(t)) + `B1 ´B1g1(X(t − τ(t))) − `B1 ¯´B2g2(X(t − τ(t))) − `B2 ¯´B2 ¯g1(X(t − τ(t))) − `B2 ´B1¯g2(X(t − τ(t))) + `C1 ´C1 � t t−π (h1(X(s))ds − `C1 ¯´C2 � t t−π h2(X(s))ds − `C2 ¯´C2 � t t−π (¯h1(X(s))ds − `C2 ´C1 � t t−π ¯h2(X(s)) + L1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3) and d dtX2(t) = −DX2(t) + `A1 ´A2f1(X(t)) + `A1 ¯´A1f2(X(t)) + `A2 ¯´A1 ¯f1(X(t)) − `A2 ´A2 ¯f2(X(t)) + `B1 ´B2g1(X(t − τ(t))) + `B1 ¯´B1g2(X(t − τ(t))) + `B2 ¯´B1¯g1(X(t − τ(t))) − `B2 ´B2¯g2(X(t − τ(t))) + `C1 ´C2 � t t−π (h1(X(s))ds + `C1 ¯´C1 � t t−π h2(X(s))ds + `C2 ¯´C1 � t t−π (¯h1(X(s))ds − `C2 ´C2 � t t−π ¯h2(X(s)) + L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4) By the Cayley-Dickson transformation, BCQVMNNs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) is transformed into two CVMNNs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Due to the mixed time delay and integral term in BCQVMNNs, it is difficult to directly study the FXTSYN of BCQVMNNs by non- decomposition method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' As a special case of BCQVMNNs, the following part of this paper mainly illustrates the related properties of the bilateral systems by studying the sufficient conditions for FXTSYN of UCQVMNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' QVMNNs with Unilateral Coefficients (UCQVMNNs) The following UCQVMNNs with discrete and distributed time delays is considered d dtxp(t) = −dpxp(t) + n � q=1 apq(xp(t))fq(xq(t)) + n � q=1 bpq(xp(t))gq(xq(t − τ(t))) + n � q=1 cpq(xp(t)) � t t−π hq(xq(s))ds + Ip, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) for p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', n, xp(t) ∈ H is the state variable, apq(xp(t)), bpq(xp(t)), cpq(xp(t)) ∈ H stand for the memristive connection weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Define system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) as drive system, the response system is described as following d dtyp(t) = −dpyp(t) + n � q=1 apq(yp(t))fq(yq(t)) + n � q=1 bpq(yp(t))gq(yq(t − τ(t))) + n � q=1 cpq(yp(t)) � t t−π hq(yq(s))ds + Ip + up(t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) where up(t) ∈ H is the designed controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The initial conditions of systems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) are xp(s) = φp(s), yp(s) = ψp(s), s ∈ [−τ, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Based on the characteristics of the memristor and current–voltage, the memristive connection weights in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) are satisfy apq(·) = � ˆapq, | · | ≤ rp, ˇapq, | · | > rp, bpq(·) = �ˆbpq, | · | ≤ rp, ˇbpq, | · | > rp, cpq(·) = � ˆcpq, | · | ≤ rp, ˇcpq, | · | > rp, for p, q = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', n, where ˆapq, ˇapq, ˆbpq, ˇbpq, ˆcpq, ˇcpq are known constants with respect to the memristor, and the switching jumps rp > 0 is the threshold level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Define a+ pq = max{|ˆapq|, |ˇapq|}, b+ pq = max{|ˆbpq|, |ˇbpq|} and c+ pq = max{|ˆcpq|, |ˇcpq|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' amax pq = max{ˆapq, ˇapq}, amin pq = min{ˆapq, ˇapq}, bmax pq = max{ˆbpq, ˇbpq}, bmin pq = min{ˆbpq, ˇbpq}, cmax pq = max{ˆcpq, ˇcpq}, and cmin pq = min{ˆcpq, ˇcpq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' As shown above, due to the discontinuity of the memristive connection weights, the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) and the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) are regarded as the discontinuous differential equation of the right-hand side, which the existence and uniqueness of solutions are not guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' So the Filippov solutions are utilized to deal with the special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 8 By the differential inclusions and the set-valued maps [39], the differential inclusion of systems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) are as follows d dtxp(t) ∈ −dpxp(t) + n � q=1 K(apq(xp(t)))fq(xq(t)) + n � q=1 K(bpq(xp(t))) × gq(xq(t − τ(t))) + n � q=1 K(cpq(xp(t))) � t t−π hq(xq(s))ds + Ip, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7) and d dtyp(t) ∈ −dpyp(t) + n � q=1 K(apq(yp(t)))fq(yq(t)) + n � q=1 K(bpq(yp(t))) × gq(yq(t − τ(t))) + n � q=1 K(cpq(yp(t))) � t t−π hq(yq(s))ds + Ip + up(t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8) where K(apq(·)) = � � � � � ˆapq, | · | < rp, co{ˆapq, ˇapq}, | · | = rp, ˇapq, | · | > rp, K(bpq(·)) = � � � � � ˆbpq, | · | < rp, co{ˆbpq, ˇbpq}, | · | = rp, ˇbpq, | · | > rp, K(cpq(·)) = � � � � � ˆcpq, | · | < rp, co{ˆcpq, ˇcpq}, | · | = rp, ˇcpq, | · | > rp, in which co{a, b} is the closure of the convex hull, and co{ˆapq, ˇapq} = [amin pq , amax pq ], co{ˆbpq, ˇbpq} = [bmin pq , bmax pq ], co{ˆcpq, ˇcpq} = [cmin pq , cmax pq ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' By the measurable selection theorem [40], if (xp(t), yp(t)) is the solution of systems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8), there exist bounded measurable functions ˜apq(t) ∈ K(apq(·)), ˜bpq(t) ∈ K(bpq(·)) and ˜cpq(t) ∈ K(cpq(·)), which rely on xp(t) and yp(t) respectively, such that d dtxp(t) = −dpxp(t) + n � q=1 ˜apq(t)fq(xq(t)) + n � q=1 ˜bpq(t) × gq(xq(t − τ(t))) + n � q=1 ˜cpq(t) � t t−π hq(xq(s))ds + Ip, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='9) 9 and d dtyp(t) = −dpyp(t) + n � q=1 ˜apq(t)fq(yq(t)) + n � q=1 ˜bpq(t)gq(yq(t − τ(t))) + n � q=1 ˜cpq(t) � t t−π hq(yq(s))ds + Ip + up(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='10) Based on the UCQVMNNs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='10), the synchronization error can be defined as ep(t) = yp(t) − xp(t), so we can get d dtep(t) = −dpep(t) + n � q=1 ˜apq(t)(fq(yq(t) − fq(xq(t))) + n � q=1 ˜bpq(t)(gq(yq(t − τ(t))) − gq(xq(t − τ(t)))) + n � q=1 ˜cpq(t) � t t−π (hq(yq(s)) − hq(xq(s)))ds + up(t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='11) with the initial condition ϕp(s) = ψp(s) − φp(s) for s ∈ [−τ, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' To obtain the main results, the following Assumptions, Definitions and Lemmas are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Assumption (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' For any xq(t), yq(t) ∈ H, suppose the expression of activation functions fq(·) = fq(0)(·) + ifq(1)(·) + jfq(2)(·) + kfq(3)(·), gq(·) = gq(0)(·) + igq(1)(·) + jgq(2)(·) + kgq(3)(·), hq(·) = hq(0)(·) + ihq(1)(·) + jhq(2)(·) + kfq(3)(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' where fq(0) = 0, gq(0) = 0, hq(0) = 0, fq(ζ)(·), gq(ζ)(·), hq(ζ)(·) ∈ R, ζ = 0, 1, 2, 3, then there exist positive constant υq, ϱq and ιq, q = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', n satisfying ∥fq(yq(t) − fq(xq(t))∥1 ≤ υq∥yq(t) − xq(t)∥1 = υq∥eq(t)∥1, ∥gq(yq(t) − gq(xq(t))∥1 ≤ ϱq∥yq(t) − xq(t)∥1 = ϱq∥eq(t)∥1, ∥hq(yq(t) − hq(xq(t))∥1 ≤ ιq∥yq(t) − xq(t)∥1 = ιq∥eq(t)∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The drive system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) is said to be synchronized with the response system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) in finite time, if there exists a constant T(e0) > 0 and T(e0) depends on the initial error e0 such that lim t→T(e0) ∥e(t)∥ = 0, ∥e(t)∥ = 0 for ∀t ≥ T(e0), 10 where e(t) = (e1(t), e1(t), · · · en(t))T, and T(e0) is called the settling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The drive system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) and the response system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) are said to reach the fixed-time synchronization, if the following two conditions hold (1) The system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) is said to be synchronized with the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) in finite time, (2) There exists a fixed constant Tmax, such that for any initial condition e0, the corresponding settling time satisfies T(e0) ≤ Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Assume that there exists a continuous radically unbounded function V : Rn → R+ = [0, +∞) satisfying (1) V (e(t)) = 0 if and only if e(t) = 0, (2) d dtV (e(t)) ≤ −aV α(e(t)) − bV β(e(t)), where a, b > 0, 0 < α < 1, and β > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Then the origin of the error system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='11) is fixed-time stable, and V (e(t)) = 0 for t ≥ T(e0) where the settling time T(e0) is bounded by T(e0) ≤ Tmax = 1 a �a b � 1−α β−α � 1 β − 1 + 1 1 − α � for ∀ e0 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Under the same conditions in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1, the synchronization time T(e0) can also be estimated by the following formula T(e0) ≤ Tmax = 1 a(1 − α) + 1 b(β − 1) for ∀e0 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' If there exists a continuous, positive definite, and radically un- bounded function V (e(t)) : Rn → R+ such that any solution x(t) of system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='11) satisfies the inequality d dtV (e(t)) ≤ � aV (e(t)) − b1(V (e(t)))γ+sgn(V (e(t))−1), V ⩾ 1, aV (e(t)) − b2(V (e(t)))γ+sgn(V (e(t))−1), 0 ≤ V < 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12) in which a < min{b1, b2}, b1, b2 > 0, 1 ≤ γ < 2, then the origin of the error system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='11) is globally fixed-time stable, In addition, for any initial state e0 of system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='11), the settling time is described as T = � � � � � � � � � � � � � � � 1 a(2 − γ) ln b2 b2 − a + 1 γ(b1 − a), a > 0, 1 b2(2 − γ) + 1 b1γ , a = 0, 1 a(2 − γ) ln b2 b2 − a + 1 aγ ln b1 b1 − a, a < 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='13) 11 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' If x1, x2, · · · , xn ≥ 0, 0 < q1 ≤ 1 and p1 > 1, then n � p=1 xq1 p ≥ � n � p=1 xp �q1 , n � p=1 xp1 p ≥ n1−p1 � n � p=1 xp �p1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' We define sign function of quaternion x = x(0) + x(1)i + x(2)j + x(3)k as follows sgn(x) = sgn(x(0)) + sgn(x(1))i + sgn(x(2))j + sgn(x(3))k, and the conjugate transpose of sgn(x) are denoted by sgn(x)∗ = � sgn(x(0)) − sgn(x(1))i − sgn(x(2))j − sgn(x(3))k �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' According to the characteristics of the sign function, it can be known that |x| = sgn(x)x, x ∈ R, and the one-norm of quaternoin vector u = u(0) + u(1)i + u(2)j + u(3)k ∈ Hn can be expressed as ∥u∥1 = ∥u(0)∥1 + ∥u(1)∥1 + ∥u(2)∥1 + ∥u(3)∥1 = (sgn(u(0)))Tu(0) + (sgn(u(1)))Tu(1) + (sgn(u(2)))Tu(2) + (sgn(u(3)))Tu(3) = 1 2((sgn(u))∗u + sgn(u)u∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' where u(ζ) ∈ Rn, ζ = 0, 1, 2, 3 represent the real and imaginary parts of the quaternoin vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' There are some Lemmas associated with this Definition is given below, so as to facilitate the proof and calculation of the theorem later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Suppose that vector e(t) = (e1(t), e2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', en(t))T, l(t) = (l1(t), l2(t) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', ln(t))T ∈ Hn, and e(t) = e(0)(t)+e(1)(t)i+e(2)(t)j+e(3)(t)k, l(t) = l(0)(t)+l(1)(t)i+ l(2)(t)j + l(3)(t)k, where ep(t), lp(t) ∈ H, p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', n, and e(ζ)(t), l(ζ)(t) ∈ Rn, ζ = 0, 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' For any e(t), l(t) ∈ Hn, positive constant c > 0, if f(e(t)) is an integrable function is defined on [t − τ, t], the following formulas hold (1) sgn(e(t))∗e(t) + (e(t))∗sgn(e(t)) = 2∥e(t)∥1, (2) sgn(e(t))∗l(t) + (l(t))∗sgn(e(t)) ≤ 2∥l(t)∥1, (3) sgn(e(t))∗sgn(e(t)) = sgn(e(t))sgn(e(t))∗ = ∥sgn(e(t))∥1, (4) sgn(e(t))∗(e(t))c + ((e(t))c)∗sgn(e(t)) = 2∥(e(t))c∥1 ≥ � 2n1−c∥(e(t))∥c 1, c > 1, 2∥(e(t))∥c 1, 0 < c ≤ 1, (5) ∥ � t t−π f(e(s))ds∥1 ≤ � t t−π ∥f(e(s))∥1ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Main results In this section, by designing effective controllers, some sufficient conditions are established to achieve the synchronization of the drive system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) and the response system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) in fixed time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The one norm of the quaternion is to demonstrate the practicability of our method by achieving FXTSYN of UCQVMNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' To synchronize the drive system and the response system in a fixed time, a novel controller up(t) is designed as follows up(t) = λ1pep(t)−λ2pep(t)α −λ3pep(t)β +λ4pep(t−τ(t))+λ5p � t t−π ∥ep(s)∥1ds, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) where λ1p, λ2p, λ3p, λ4p and λ5p are real constants, and the numbers α, β ∈ R satisfy 0 < α < 1 and β > 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Suppose that Assumptions (A1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' If parameters λ1p, λ2p, λ3p, λ4p and λ5p in the controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) satisfy dp − λ1p − n � q=1 υpa+ qp ≥ 0, n � q=1 ϱpb+ qp + λ4p ≤ 0, n � q=1 ιpc+ qp + λ5p ≤ 0, λ2p > 0, λ3p > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) for p, q = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', n, then the drive system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) synchronizes to the response system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) in a fixed time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The settling time is estimated as follows T1 = 1 λ2 � λ2 n2(1−β)λ3 � 1−α β−α� 1 β − 1 + 1 1 − α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3) where λ2 = min 1≤p≤n{λ2p}, λ3 = min 1≤p≤n{λ3p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Consider the Lyapunov function V (t) = 1 2 n � p=1 (sgn(ep(t))∗ep(t) + ep(t)∗sgn(ep(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4) 13 By applying Assumption (A1) to the derivative along the trajectory of the error system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' one can obtain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='dtV (t) = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(sgn(ep(t))∗ d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='dtep(t) + ( d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='dtep(t))∗sgn(ep(t))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗� ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='gq(yq(t − τ(t))) − gq(xq(t − τ(t))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='˜cpq(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(hq(yq(s)) − hq(xq(s)))ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ λ1pep(t) − λ2p(ep(t))α − λ3p(ep(t))β + λ4pep(t − τ(t)) + λ5p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': 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τ(t))) − gq(xq(t − τ(t)))] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ [gq(yq(t − τ(t))) − gq(xq(t − τ(t)))]∗(˜bpq(t))∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='λ5p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗� � t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(s)∥1ds ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗˜bpq(t)ϱq∥eq(t − τ(t))∥1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ [ϱq∥eq(t − τ(t))∥1]∗(˜bpq(t))∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗˜cpq(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� � t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(ιq∥eq(s)∥1)ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� � t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(ιq∥eq(s)∥1)ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='�∗(˜cpq(t))∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='λ2p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗(ep(t))α + ((ep(t))α)∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='λ3p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗(ep(t))β + ((ep(t))β)∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='λ4p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗ep(t − τ(t)) + (ep(t − τ(t)))∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='λ5p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗� � t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(s)∥1ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� � t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(s)∥1ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='�∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 and Assumption (A1), we have 1 2 n � p=1 (−dp + λ1p) � sgn(ep(t))∗ep(t) + (ep(t))∗sgn(ep(t)) � = − n � p=1 (dp − λ1p)∥ep(t)∥1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) 1 2 n � p=1 n � q=1 � sgn(ep(t))∗˜apq(t)υq∥eq(t)∥1 + [υq∥eq(t)∥1]∗(˜apq(t))∗sgn(ep(t)) � ≤ n � p=1 n � q=1 υqa+ pq∥eq(t)∥1 = n � p=1 n � q=1 υpa+ qp∥ep(t)∥1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) 1 2 n � p=1 n � q=1 � sgn(ep(t))∗˜bpq(t)ϱq∥eq(t − τ(t))∥1 + [ϱq∥eq(t − τ(t))∥1]∗(˜bpq(t))∗sgn(ep(t)) � + 1 2 n � p=1 λ4p � sgn(ep(t))∗ep(t − τ(t)) + (ep(t − τ(t)))∗sgn(ep(t)) � ≤ n � p=1 n � q=1 ϱqb+ pq∥eq(t − τ(t))∥1 + n � p=1 λ4p∥ep(t − τ(t))∥1 = n � p=1 � n � q=1 ϱpb+ qp + λ4p � ∥ep(t − τ(t))∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7) And 1 2 n � p=1 n � q=1 � sgn(ep(t))∗˜cpq(t) � � t t−π (ιq∥eq(s)∥1)ds � + � � t t−π (ιq∥eq(s)∥1)ds �∗(˜cpq(t))∗sgn(ep(t)) � + 1 2 n � p=1 λ5p � sgn(ep(t))∗� � t t−π ∥ep(s)∥1ds � + � � t t−π ∥ep(s)∥1ds �∗sgn(ep(t)) � 16 ≤ n � p=1 n � q=1 ιqc+ pq � t t−π ∥eq(s)∥1ds + n � p=1 λ5p � t t−π ∥ep(s)∥1ds = n � p=1 ( n � q=1 ιpc+ qp + λ5p) � t t−π ∥ep(s)∥1ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8) Analogously, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4, it is not difficult to get the following conclusion − 1 2 n � p=1 λ2p � sgn(ep(t))∗(ep(t))α + ((ep(t))α)∗sgn(ep(t)) � ≤ − n � p=1 λ2p∥ep(t)∥α 1, − 1 2 n � p=1 λ3p � sgn(ep(t))∗(ep(t))β + ((ep(t))β)∗sgn(ep(t)) � ≤ −n1−β n � p=1 λ3p∥ep(t)∥β 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='9) Combined with the above formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='9) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' one can obtain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='dtV (t) ≤ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='dp − λ1p − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='υpa+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='qp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t)∥1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='ϱpb+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='qp + λ4p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t − τ(t))∥1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='ιpc+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='qp + λ5p) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(s)∥1ds − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='λ2p∥ep(t)∥α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='− n1−β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='λ3p∥ep(t)∥β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='≤ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='λ2p∥ep(t)∥α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 − n1−β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='λ3p∥ep(t)∥β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='≤ −λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t)∥α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 − n1−βλ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t)∥β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='≤ −λ2( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t)∥1)α − n2(1−β)λ3( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t)∥1)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='= −λ2(V (t))α − n2(1−β)λ3(V (t))β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' where λ2 = min 1≤p≤n{λ2p},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' λ3 = min 1≤p≤n{λ3p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Based on Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1, it follows that the error system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='11) is fixed-time stable, that is the drive system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) and the response system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) achieve synchronization 17 in fixed-time with the controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Furthermore, one can estimate the settling time by the following equality T1 = 1 λ2 � λ2 n2(1−β)λ3 � 1−α β−α� 1 β − 1 + 1 1 − α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='10) The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1, taking the same conditions and the controller as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1, one can obtain that system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) and system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) are synchronized within a fixed time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Furthermore, the settling time can be estimated as T2 = 1 λ2(1 − α) + 1 n2(1−β)λ3(β − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='11) It is well known that the error system can be stabilized within a fixed time, that is, e(t) → 0 within time T, by developing an appropriate controller and Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' In general, it can be seen from the inequality d dtV (e(t)) ≤ −aV α(e(t)) − bV β(e(t)) with 0 < α < 1, β > 1 in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 that the right-hand side contains two terms: the index of one term is bigger than 0 and smaller than 1, while the index of the other one is larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' In the controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1), we can see that the first, fourth, and fifth items are designed to allow the error system to achieve Lyapunov stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The second and third terms are designed to achieve synchronization of the drive-response system in a fixed time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' And then compute the settling time using the parameters of these two items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' However, in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1, the settling time estimate includes both errors from greater than 1 to 1 and then from 1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' This must be considered because the two terms exist in the inequality at the same time, whether they play a role or not, which may cause the estimated settling time to be inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' [41] proposed a novel Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 to guarantee fixed-time synchronization of discontinuous neural networks, where V (t) ≥ 1 or 0 < V (t) < 1 is first judged and the functioning part is intelligently chosen to be more economical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' It is clear that Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 is an excellent alternative to existing techniques and can signifi- cantly reduce energy consumption while achieving a more precise settling time than most relevant research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' For the first time, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 is used to synchronize the UCQVMNNs in a fixed time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2, in order to make the error system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='11) stable in fixed time, different from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1, a new nonlinear controller is designed as follows: up(t) = −k1pep(t) + k2pep(t − τ(t)) − µ(ep(t))γ+sgn(∥e(t)∥1−1) + k3p n � q=1 � t t−π ∥eq(s)∥1ds, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12) 18 where e(t) = (e1(t), e2(t), · · · , en(t))T ∈ Hn, parameter γ ∈ R satisfies 1 ≤ γ < 2, the feedback gains k1p, k2p and µ are real constants, and k1p ≥ 0, µ > 0 will be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Suppose that Assumption (A1) holds, and for constants k2P, k3P and µ, the following inequalities are fulfilled d < µ1, n � q=1 ϱpb+ qp + k2p ≤ 0, n � q=1 ιqc+ pq + k3p ≤ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='13) then, UCQVMNNs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) can achieve FIXSYN with controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Fur- thermore, for any initial condition, the settling time can be estimated by T4 = � � � � � � � � � � � � � � � 1 d(2 − γ) ln µ µ − d + 1 γ(µ1 − d), d > 0, 1 µ(2 − γ) + 1 µ1γ , d = 0, 1 d(2 − γ) ln µ µ − d + 1 dγ ln µ1 µ1 − d, d < 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='14) In particular, if the initial value is less than 1, the corresponding settling time is estimated as follows T3 = � � � � � � � � � � � � � � � 1 d(2 − γ) ln µ µ − d + 1 γ(µ − d), d > 0, 1 µ(2 − γ) + 1 µγ , d = 0, 1 d(2 − γ) ln µ µ − d + 1 dγ ln µ µ − d, d < 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='15) where d = max 1≤p,q≤n{−dp − k1p + �n q=1 υpa+ qp}, µ1 = µn−2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Consider the Lyapunov function V (t) = 1 2 n � p=1 (sgn(ep(t))∗ep(t) + ep(t)∗sgn(ep(t))), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='16) 19 According to Assumption (A1), calculating the derivative along the trajectory of the error system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' one can get ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='dtV (t) = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(sgn(ep(t))∗ d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='dtep(t) + ( d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='dtep(t))∗sgn(ep(t))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='− dpep(t) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='˜apq(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='fq(yq(t)) − fq(xq(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='˜bpq(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='gq(yq(t − τ(t))) − gq(xq(t − τ(t))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='˜cpq(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(hq(yq(s)) − hq(xq(s)))ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='− k1pep(t) + k2pep(t − τ(t)) − µ(ep(t))γ+sgn(∥ep(t)∥1−1) + k3p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥eq(s)∥1ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='− dpep(t) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='˜apq(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='fq(yq(t)) − fq(xq(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='˜bpq(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='gq(yq(t − τ(t))) − gq(xq(t − τ(t))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='˜cpq(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(hq(yq(s)) − hq(xq(s)))ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='− k1pep(t) + k2pep(t − τ(t)) − µ(ep(t))γ+sgn(∥ep(t)∥1−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ k3p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥eq(s)∥1ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='�∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(−dp − k1p) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗ep(t) + (ep(t))∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗˜apq(t)υq∥eq(t)∥1 + [υq∥eq(t)∥1]∗(˜apq(t))∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗˜bpq(t)ϱq∥eq(t − τ(t))∥1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ [ϱq∥eq(t − τ(t))∥1]∗(˜bpq(t))∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='k2p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗ep(t − τ(t)) + (ep(t − τ(t)))∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗˜cpq(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� � t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(ιq∥eq(s)∥1)ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� � t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(ιq∥eq(s)∥1)ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='�∗(˜cpq(t))∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='k3p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗� � t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥eq(s)∥1ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� � t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥eq(s)∥1ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='�∗)sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='sgn(ep(t))∗(ep(t))γ+sgn(∥e(t)∥1−1) + ((ep(t))γ+sgn(∥e(t)∥1−1))∗sgn(ep(t)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 can be get 1 2 n � p=1 n � q=1 � sgn(ep(t))∗˜cpq(t) � � t t−π (ιq∥eq(s)∥1)ds � + � � t t−π (ιq∥eq(s)∥1)ds �∗(˜cpq(t))∗sgn(ep(t)) � + 1 2 n � p=1 n � q=1 k3p � sgn(ep(t))∗� � t t−π ∥eq(s)∥1ds � + � � t t−π ∥eq(s)∥1ds �∗)sgn(ep(t)) � ≤ n � p=1 n � q=1 (ιqc+ pq + k3p) � t t−π ∥eq(s)∥1ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='17) Next, in combination with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2, we will discuss two cases according to the error value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Case 1: when 0 < ∥e(t)∥1 < 1 (0 < V (t) < 1), so that sgn(∥e(t)∥1 − 1) = −1, 0 ≤ γ + sgn(∥e(t)∥1 − 1) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' According to the conclusion in Theorem 1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3 can be directly obtained following inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' − 1 2 � sgn(ep(t))∗(ep(t))γ+sgn(∥e(t)∥1−1) + ((ep(t))γ+sgn(∥e(t)∥1−1))∗sgn(ep(t)) � ≤ −∥ep(t)∥γ+sgn(∥e(t)∥1−1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='18) Therefore combined with the above formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='18), then using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4, one can obtain d dtV (t) ≤ n � p=1 � − dp − k1p + n � q=1 υpa+ qp � ∥ep(t)∥1 + n � p=1 � n � q=1 ϱpb+ qp + k2p � ∥ep(t − τ(t))∥1 + n � p=1 n � q=1 (ιqc+ pq + k3p) � t t−π ∥eq(s)∥1ds − µ n � p=1 ∥ep(t)∥γ+sgn(∥e(t)∥1−1) 1 ≤ d n � p=1 ∥ep(t)∥1 − µ n � p=1 ∥ep(t)∥γ+sgn(∥e(t)∥1−1) 1 ≤ d n � p=1 ∥ep(t)∥1 − µ( n � p=1 ∥ep(t)∥1)γ+sgn(∥e(t)∥1−1) = dV (t) − µ(V (t))γ+sgn(V (t)−1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='19) 22 where d = max 1≤p,q≤n{−dp − k1p + �n q=1 υpa+ qp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' With conditions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='13) holding, based on Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2, it follows that the error system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='11) is fixed-time stable, that is the QVMNNs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) and the QVMNNs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) achieve synchronization in fixed-time with the controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' And the settling time is estimated as T3 = � � � � � � � � � � � � � � � 1 d(2 − γ) ln µ µ − d + 1 γ(µ − d), d > 0, 1 µ(2 − γ) + 1 µγ , d = 0, 1 d(2 − γ) ln µ µ − d + 1 dγ ln µ µ − d, d < 0, Case 2: when ∥e(t)∥1 ≥ 1 (V (t) ≥ 1), so that sgn(∥e(t)∥1 − 1) ≥ 0, γ + sgn(∥e(t)∥1 − 1) ≥ γ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4, it can ba seen that − 1 2 � sgn(ep(t))∗(ep(t))γ+sgn(∥e(t)∥1−1) + ((ep(t))γ+sgn(∥e(t)∥1−1))∗sgn(ep(t)) � ≤ −n1−γ−sgn(∥e(t)∥1−1)∥ep(t)∥γ+sgn(∥e(t)∥1−1) 1 ≤ −n−γ∥ep(t)∥γ+sgn(∥e(t)∥1−1) 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='Therefore we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='dtV (t) ≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='− dp − k1p + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='υpa+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='qp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t)∥1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='ϱpb+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='qp + k2p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t − τ(t))∥1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='q=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(ιpc+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='qp + k3p) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='t−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥eq(s)∥1ds − µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n−γ∥ep(t)∥γ+sgn(∥e(t)∥1−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='≤ d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t)∥1 − µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n−γ∥ep(t)∥γ+sgn(∥e(t)∥1−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='≤ d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t)∥1 − µn−γn1−γ−sgn(∥e(t)∥1−1)( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t)∥1)γ+sgn(∥e(t)∥1−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='≤ d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t)∥1 − µn−2γ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='p=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='∥ep(t)∥1)γ+sgn(∥e(t)∥1−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='= dV (t) − µ1(V (t))γ+sgn(V (t)−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='21) where µ1 = µn−2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' In order to guarantee the Lyapunov stability, an extra condition d < µ1 must be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 23 Analogously, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2, we can get that the UCQVMNNs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) achieve synchronization in fixed-time with the controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' And the settling time can be inferred as T4 = � � � � � � � � � � � � � � � 1 d(2 − γ) ln µ µ − d + 1 γ(µ1 − d), d > 0, 1 µ(2 − γ) + 1 µ1γ , d = 0, 1 d(2 − γ) ln µ µ − d + 1 dγ ln µ1 µ1 − d, d < 0, The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Different from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 does not require parameter d < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Different d values yield different settling time estimates, resulting in greater selectivity and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' If we take a proper k1p > 0 to make d < 0, any value of positive µ can guarantee the existence of condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' If k1p = 0, for p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', n in controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12), maybe d > 0, so we should choose a large enough positive µ to make inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='13) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' In this case, controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12) only has three terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' As a result, it is critical to choose an appropriate k1p for the given situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Numerical Simulations In this section, three numerical instances are given to illustrate the effectiveness of our theoretical results obtained in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Consider the following 2-dimensional UCQVMNNs with mixed delays as the drive system d dtxp(t) = −dpxp(t) + 2 � q=1 apq(xp(t))fq(xq(t)) + 2 � q=1 bpq(xp(t))gq(xq(t − τ(t))) + 2 � q=1 cpq(xp(t)) � t t−π hq(xq(s))ds, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) the response system is d dtyp(t) = −dpyp(t) + 2 � q=1 apq(yp(t))fq(yq(t)) + 2 � q=1 bpq(yp(t))gq(yq(t − τ(t))) + 2 � q=1 cpq(yp(t)) � t t−π hq(yq(s))ds + up(t), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) 24 where p, q = 1, 2, dp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5, τ(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3sin(t) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4, π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4, so let τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7, and the memristive connection weights are A = � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6i − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3j − 1k −1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5i − 2j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5k −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 + 2i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6k � B = � −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='45 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3i − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2j − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='35k −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='25 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='25i + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1j − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5i + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='35j − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='25k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4i − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='18j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='22k � C = � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 + 3i + 1j + 2k −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2i − 2j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1k −1 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='9j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3k 2 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3i + 1j − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2k � where |xp(t)| ≤ 1, p = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' A = � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6i − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5j − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3k −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3k −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='9 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 − 2i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2k � B = � −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3i − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2j − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3k −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='15 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2i − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='45k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='65i − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='15j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1k −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='44 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1i + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='16j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3k � C = � −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 − 2i − 1j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 − 2i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7i + 2j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4k −2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3i + 1j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8k � where |xp(t)| > 1, p = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' One can easily compute a+ 11 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2, a+ 12 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5, a+ 21 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5, a+ 22 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6, b+ 11 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3, b+ 12 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='0, b+ 21 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3, b+ 22 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='0 and c+ 11 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4, c+ 12 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2, c+ 21 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5, c+ 22 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' And take the activation functions are fq(xq(t)) = 2 tanh(xq(0)(t)) + 2 tanh(xq(1)(t))i + 2 tanh(xq(2)(t))j + 2 tanh(xq(3)(t))k, gq(xq(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 tanh(xq(0)(t))+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 tanh(xq(1)(t))i+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 tanh(xq(2)(t))j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 tanh(xq(3)(t))k, hq(xq(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7 tanh(xq(0)(t))+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7 tanh(xq(1)(t))i+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7 tanh(xq(2)(t))j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7 tanh(xq(3)(t))k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' According to Assumption (A1), a simple calculation yields that υ1 = υ2 = 2, ϱ1 = ϱ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1, ι1 = ι2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The initial conditions of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) are selected as φ1(s) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 + 2i − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8k, φ2(s) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5i + 1j − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5k, ψ1(s) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 − 2i − 1j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2k, ψ2(s) = −3 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6i + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8j − 2k, s ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' If there is no controller, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', u1(t) = u2(t) = 0, the trajectories of the error system is simulated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 1, which implies that systems (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) cannot obtain synchronization without control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Correspondingly, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1), the controllers are designed as follows u1(t) = λ11e1(t) − λ21eα 1(t) − λ31eβ 1(t) + λ41e1(t − τ(t)) + λ51 � t t−π ∥e1(s)∥1ds, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3) 25 u2(t) = λ12e2(t) − λ22eα 2(t) − λ32eβ 2(t) + λ42e2(t − τ(t)) + λ52 � t t−π ∥e2(s)∥1ds, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4) Figure 1: The trajectories of error system with- out controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Figure 2: The trajectories of error system with controllers (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The values of the coefficients λ11 = −80, λ12 = −50, λ21 = λ22 = 1, λ31 = 30, λ32 = 35, λ41 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='26, λ42 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2, λ51 = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='45, λ52 = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='35 can be calculated from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2), α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6, β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6, the conditions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 3: Trajectories of the real and imaginary parts of drive-response system with controllers (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4) when p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Under the designed controllers (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4), the trajectories of real and imag- inary parts of system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 3-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' These figures indicate that once appending the controllers (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4) to the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2), it will be synchronized with the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) in a fixed time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 26 150 (t) ek(t) (t) e,(t) e,(t) e,(t) 100 50 0 50 100 150 0 2 3 4 5 t5 e,(t) e(t) ek(t) 4 (t 3 2 0 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8 t140 120 (t 100 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 80 4 60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='04 20 0 20 0 2 3 4 540 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 (a)x 30 (t) 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 5 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='05 Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 10 20 30 40 50 60 0 2 3 4 540 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 30 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='15 0 10 20 30 40 0 2 3 4 525 20 t) 15 10 5 0 5 10 $.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 15 3 20+5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 25 0 1 2 3 4 5 t(a) (b) (c) (d) Figure 4: Trajectories of the real and imaginary parts of drive-response system with controllers (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4) when p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Moreover, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 2 shows the whole process of FXTSNY under one-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' It can be concluded that the error system reaches 0 in a fixed time, which once implies that before T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 one can achieve FXTSNY of the drive-response system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' We can calculate the T1 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='491 according to the settling time estimated in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3), which is larger than the real synchronization time but is smaller than conventional estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' It shows that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' In addition, according to the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='11) the settling time T2 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='628 can be derived from Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' By comparison, the estimated settling time derived from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 is more accurate than Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The two-dimensional UCQVMNNs with mixed delays were given by the drive system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) and the response system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2), which consider the fixed-time synchronization of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The parameters take the same as in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The initial conditions of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) are chosen as φ1(s) = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6k, φ2(s) = −4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='9i − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3j + 2k, ψ1(s) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='93j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7k, ψ2(s) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4j − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3k, s ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Correspondingly, using the 2-dimensional controller by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12) and we have adap- tive rules for k1p, k2p, k3p(p = 1, 2) are define in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='13) with coefficients k21 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='26, k22 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2, k31 = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='25, k32 = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The evolution of real and imaginary parts of system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 5 under the effect of controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' And these two figures show that The drive-response system can achieve FXTSYN in a very short time under the action of the controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Next, Matlab drawing verification is carried out mainly for the initial error greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 27 100 80 K 60 40 20 0 20 40 4 60 6 8 80 10 12 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 0 1 2 3 4 560 40 6 20 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 1 20 40 60 80 0 2 3 4 570 5 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 50 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 40 0 0,05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 30 20 10 0 10 20 0 2 3 4 530 20 10 10 20 30 2 40 50 9- 60 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 70 0 1 2 3 4 5(a) (b) Figure 5: The drive-response system trajectories of p = 1, 2 with controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Firstly, let µ = 40, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5, if k11 = 37, k12 = 130, we can get d = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 < 0, µ1 = 5 satisfy the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='13) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' And the evolution trajectory of the error system with controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 6(a), we can clearly see that the error system reaches stability in a very short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' According to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2, we can calculate a relatively accurate estimate of the settling time, which is T4 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='160 (T3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='065).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (a) d < 0 (b) d = 0 (c) d > 0 Figure 6: Evolution of error states between networks (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) under controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Secondly, let k11 = 50, k12 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7, and µ = 56, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5, it’s easy to can get d = 0, µ1 = 7 satisfy the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='13) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Then the corresponding evolution of the error state with controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12) are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Therefore, we can see from this picture that the synchronization of systems (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) can be realized within t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Furthermore, T4 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='131 (T3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='048) according to the settling time estimated in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Finally, if k11 = 33, k12 = 130, we can get d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='9 > 0, µ1 = 4 satisfy the condition d < µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Then the dynamics of the error system with controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12) are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 6(c) with µ = 32, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Obviously, the drive system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1) and response 28 40 30 20 10 0 10 20 30 x(t) yR(t) y(t) y,(t) 一yk(t) 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8 t40 20 0 20 40 60 80 100 x,(t) y2(t) y,(t) y,(t) ye(t) 120 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8 t5 e(t) e(t) 4 (t) e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(t) e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(t) (t) 3 2 0 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8 t3 e,(t) e(t) e,(t) e,(t) e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(t) ek(t) 0 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8 t3 e,(t) e(t) e,(t) e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(t) e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='(t) 2 0 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8 tsystem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2) achieve synchronization within the time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' And, according to the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='15) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2, we can calculate the settling time is T4 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='382 (T3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='087), which is larger than the real synchronization time, but is smaller than the conventional estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Table 1: Comparisons of the settling time between different k11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' k11 31 33 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='9 40 45 50 d 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='9 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 T3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='057 T4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='659 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='104 Table 2: Comparisons of the settling time between different µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' µ 8 16 24 32 40 48 µ1 1 2 3 4 5 6 T3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='050 T4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='103 Table 3: Comparisons of the settling time between different γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' γ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8 T3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='040 T4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='089 From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='15), we know that parameters k2p, k3p(p = 1, 2) have no impact on the settling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Tables 1-3 show the comparisons of the settling time for different k11 (k12 = 130, µ = 40, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5), µ (k12 = 130, k11 = 45, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5), and γ (k12 = 130, k11 = 45, µ = 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' It is not hard to find that with controller parameters k11, µ and γ increasing, the settling time T3 and T4 decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' According to these, we can choose more proper parameters as the requirements are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' In a nutshell, it is not difficult to get through the comparison of the above two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The settling time estimation value obtained through Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 is more accurate than Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Consider the 128 × 128 pixels color image pattern ”Baboon” that is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Additionally, create UCQVMNNs that have the form of the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) for associatively remembering the color image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Considering the computational complexity, we divided the image into 64 blocks for processing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 7(b)), each block with 16 × 16 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Therefore, each block needs 256-dimensional neurons to store it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' So we need to design UCQVMNNs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) composed of 256 neurons that have a 256-dimensional equilibrium point storing the colors of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 29 (a) Baboon (b) 8 × 8 blocks Figure 7: The original image and its segmented image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The original image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 7(a)) has an additional missing, which causes extremely sparse initial values for the response system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The specific values are chosen as φ1(t) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8863i + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5373j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4902k, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8824i + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5059j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4157k, · · · , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8235i + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3804j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3255k, 0) ∈ H256, φ2(t) = (0, · · · , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6549i+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2275j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2029k, · · · , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='8039i+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3490j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3412k, · · · ) ∈ H256, · · · , φ64(t) = (0, · · · , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5021i+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2196j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3294k, · · · ) ∈ H256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The weight coefficient are C = (cpq) = diag(10, · · · , 10) ∈ H256×256, d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='01, p, q = 1, 2, · · · , 256, and apq(·) = � � � � � −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2i − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4k, q < p, 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3i − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3k, q = p, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2i + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3j − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5k, q > p, bpq(·) = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='04 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='04i − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='03j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='05k, q < p, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='04 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='05i + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='05j − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='03k, q ⩾ p, Take the activation functions are fq(xq(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='06(|xq(t)+2|−|xq(t)+1|), gq(xq(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='05(|xq(t)+2|−|xq(t)+1|), hq(xq(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='01(|xq(t)+2|−|xq(t)+1|), where xq(t) ∈ H, q = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=', 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' The equilibrium points corresponding to the 64 small blocks color image are x∗ 1 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3608i+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3216j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1490k, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4706i+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4000j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1686k, · · · , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6157i+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6431j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4275k) ∈ H256, · · · , x∗ 64 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4941i+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='5333j+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4235k, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4549i+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='4549j+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3725k, · · · , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2471i + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2392j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2235k) ∈ H256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Under the positive role of the controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12), the process of image restoration is the process of the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) to reach the equilibrium state, and the time used for restoration is the time used for the system to achieve equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 30 (a) Miss 80% (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='04s (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='06s (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1s (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3s Figure 8: Color image completion results on Image (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (a) is the missing image where the ratio of missing pixels is 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (b) is the recovery image ( T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='04s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (c) is the recovery image ( T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='06s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (d) is the recovery image ( T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (e) is the recovery image ( T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (a) Noise 80% (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='04s (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='06s (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1s (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3s Figure 9: Color image completion results on Image (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (a) is the ”salt and pepper” noise image where the noise density is 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (b) is the recovery image ( T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='04s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (c) is the recovery image ( T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='06s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (d) is the recovery image ( T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' (e) is the recovery image ( T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='3s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 8(a), 8(b), 8(c), 8(d), and 8(e) that system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='6) can reach the equilibrium state in a short time under the effect of the controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' It also shows that the color images can get rapid recovery when the ratio of missing pixels is 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 9(a), 9(b), 9(c), 9(d), and 9(e), we can see that the system can reach the equilibrium point quickly under the controller (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='12), that is, the ”Baboon” image under the premise of adding 80% density ”salt and pepper” noise can be quickly recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Therefore, as long as the suitable controller is designed, we can use UCQVMNNs to quickly recover color images from any missing or noise state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Thus it shows us the high efficiency of UCQVMNNs in dealing with highdimensional image restoration problems, and the controller of the theorem has a significant practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Conclusions The FXTSNY is discussed in this paper for a class of UCQVMNNs with mixed delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Since the decomposition technology is generally accompanied by a more com- 31 plex derivation process about quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' As a result, the quaternion-valued state is considered as a whole, with one-norm employed to achieve FXTSNY of UCQVMNNs smoothly and directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Then based on the Lyapunov stability theorem, set-valued map, and differential inclusion theorem, we effectively deal with the system disconti- nuity caused by the memristor’s weight coefficient in drive-response systems using the measurable selection theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Furthermore, sufficient conditions for the FXTSNY of delayed UCQVMNNs are proposed using the inequality technique and the Lyapunov stability theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' In addition, different estimation settling times are obtained based on various FXTSNY criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' It is simple to prove that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='2’s estimation value is more accurate than Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content='1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Finally, three numerical examples are provided to demonstrate the validity and effectiveness of theoretical results, as well as the practical value in high-dimensional color image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' Preassigned-time synchronization is a more flexible synchronization method that is used in conjunction with the non-commutativity of the quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} +page_content=' It should be worthwhile to investigate new methods for studying the preassigned-time synchro- nization of QVMNNs in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfU_w0/content/2301.01275v1.pdf'} 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0000000000000000000000000000000000000000..accf27399edbcf95a15b09f70c9b7743e6190182 --- /dev/null +++ b/HtE4T4oBgHgl3EQfIAw0/content/tmp_files/2301.04908v1.pdf.txt @@ -0,0 +1,1353 @@ +Diagnostics for plasmon satellites and Hubbard bands in transition metal oxides +Steffen Backes1,2,3,∗ Hong Jiang4, and Silke Biermann3,5,6,7 +1Research Center for Advanced Science and Technology, +University of Tokyo, Komaba, Tokyo 153-8904, Japan +2Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan +3CPHT, CNRS, ´Ecole polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France +4College of Chemistry and Molecular Engineering, Peking University, China +5Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France +6European Theoretical Spectroscopy Facility, 91128 Palaiseau, France, Europe and +7Department of Physics, Division of Mathematical Physics, +Lund University, Professorsgatan 1, 22363 Lund, Sweden +(Dated: January 13, 2023) +Coulomb correlations between the electrons imprint characteristic signatures to the spectral prop- +erties of materials. Among others, they are at the origin of a rich phenomenology of satellite features, +either stemming from atomic-like multiplets or from interactions with particle-hole excitations or +plasmons. +While in many cases the latter lie at considerably higher energies than the former, +suggesting clear distinction criteria, this picture has recently become blurred by indications that +satellites of different types can coexist in the same energy range. It is now generally accepted that +the identification of the nature of spectral features is a highly non-trivial task. In this article we +propose a general procedure for tracing the origin of satellites of different types within modern ab +initio calculations. As an illustration, we analyze the ternary transition metal oxides SrVO3 and +SrMoO3, which are drosophila compounds for the coexistence of Hubbard and plasmonic satellites, +reconciling previous seemingly contradictory findings in an unexpected manner. +INTRODUCTION +Impressive progress in direct – and to a much lesser ex- +tent inverse – photoemission spectroscopy over the last +decades has resulted in a situation where the spectral +properties of electronic systems have become some of the +most commonly probed experimental properties of ma- +terials [1–3]. +The main quantity is the spectral func- +tion A(k, ω), which encodes information about the pos- +sible electron removal and addition processes, as probed +in direct and inverse photoemission. The knowledge of +A(k, ω) in turn is typically synonymous with a good first +understanding of the behaviour of the material under a +variety of probes, even those not directly encoded in A. +In normal metals, the low-energy behaviour is governed +by renormalized quasi-particle bands following the Lan- +dau Fermi liquid paradigm, while in insulators the spec- +trum is gapped around the Fermi level. Beyond these el- +ementary considerations, spectral functions can however +display a whole zoology of different features at interme- +diate or high energies (in typical transition metal oxides, +in energy ranges spanning a few tenths to a few tens of +eV). +Among the most prominent features in electronic sys- +tems with sizable Coulomb correlations are Hubbard +satellites, remnants of the atomic physics in the mate- +rial, corresponding to the atomic multiplets of an isolated +atom placed in the crystal field environment of its sur- +roundings but potentially acquiring some dispersion due +to the periodicity of the crystal. The energy scales of +these multiplet structures are given by the effective local +Coulomb interaction, often parametrized theoretically in +the form of a local Hubbard U (or more precisely a Hub- +bard U matrix including Hund’s exchange and orbital +structures) [4]. Such features have been studied in some +detail in the past with elaborate theoretical approaches, +starting from exact diagonalization[5, 6] and more re- +cently within Dynamical Mean-Field Theory (DMFT)[7– +9]. and are well-documented experimentally [6, 10]. +Another type of satellites appearing in the spectral +function of electronic materials are due to electrons cou- +pling to plasmons [11–18]. Plasmons have been experi- +mentally observed and theoretically investigated in mate- +rials ranging from elementary metals [19–21], bronzes[22, +23], oxides[18, 24–30], in particular ruthenates[31] and +cuprates[32–34] as well as in graphene[16, 35, 36]. Plas- +monic excitations are relevant and actively utilized in the +design of functional materials[37–39], such as in plasmon- +mediated photocatalysis[40, 41] and sensors[42]. +Plas- +mons are collective electronic excitations, which are in +general highly non-local in nature. They are encoded in +the dielectric function describing the dynamic response of +the electronic system as a whole to a perturbation. This +response can be mediated by particle-hole excitations +as well as by collective (plasmonic) excitations, which +both can give rise to shake-up satellites in the spectral +function. For simplicity, below, we will refer to any fea- +tures beyond a local atomic-like picture, i.e. originating +arXiv:2301.04908v1 [cond-mat.str-el] 12 Jan 2023 + +2 +from non-local collective excitations as plasmonic satel- +lites, and our aim will be to distinguish those from the +Hubbard-type satellites described above. This differenti- +ation becomes non-trivial when the energy scale of plas- +monic excitations is similar to that of the local Coulomb +interactions, which can lead to both Hubbard and plas- +mon satellites to appear at similar energies. Recently, +evidence has accumulated that this is the case in a large +number of transition metal oxides [43–51]. +In this letter we present a protocol for a quantitative +ab initio identification of Hubbard and plasmonic con- +tributions in low-energy satellites in real materials. Us- +ing this protocol, we reinvestigate two prototypical 3d +and 4d perovskite transition metal oxides, SrVO3 and +SrMoO3, and determine the Hubbard and plasmonic con- +tributions in the observed low-energy satellites. Contrary +to previous interpretations[48–51], we find both Hubbard +and plasmon satellites to be present, albeit with differ- +ent magnitude. On the other hand, our findings recon- +cile seemingly contradictory calculations within many- +body perturbation theory (within the GW approxima- +tion) and combined GW+Dynamical Mean Field Theory +(GW+DMFT) in a surprising manner. +SrVO3 is a 3d1 compound with metallic V t2g states +crossing the Fermi level, forming a typical 3-peak struc- +ture in the spectral function, both confirmed from +experiment[52–57] and theoretical calculations[53, 54, +57–60]. The proposed origin of the satellites though has +significantly evolved over the years. Early combined den- +sity functional theory and dynamical mean-field theory +(DFT+DMFT) calculations suggested that the satellites +arise from strong local V-t2g Coulomb interactions in the +form of Hubbard bands[53, 54, 58, 59, 61–63]. The ad- +vent of combined many-body perturbation theory and +dynamical mean field theory (”GW+DMFT”) [64], how- +ever, made it possible to include both, Hubbard bands +and plasmonic features, in the theoretical description, +and it was realized that in the low energy (< 5 eV) +range features of both types can coexist [60, 65]. More- +over, it was pointed out that the empty V-eg states that +are split off from the partially filled V-t2g states by the +octahedral crystal field lie in the same energy range as +the upper Hubbard band from the early DFT+DMFT +calculations. Interestingly, many-body perturbation the- +ory alone could also reproduce the observed satellite fea- +tures (albeit at slightly shifted energetic positions) [47], +a finding which seemed to be in contradiction with the +interpretation as Hubbard bands. Along this line, several +works [47–51] gave a purely plasmonic interpretation to +the lowest energy features both in the occupied and the +unoccupied part of the spectrum. A new twist appeared +when it was realized that oxygen vacancies contribute +spectral weight at the same energy as the satellite in the +occupied spectrum[57]. While the importance of oxygen +vacancies responsible for part of the spectral weight in +the energy range in question is now widely recognized, +no consensus has been reached so far concerning the ori- +gin of the remaining intrinsic part of the satellite. +We now turn to a brief discussion of the theoretical de- +scription of the creation of plasmonic features in the spec- +tral function, within DMFT-derived schemes. As is well- +known [13, 15, 48, 60, 64, 66–68] electronic screening is +a dynamical process, since the response of the electronic +density in a solid to a perturbation depends on the energy +scale of the perturbation. For a given set of orbitals of +interest, the charge redistribution and thus the screening +is energy dependent, and directly translates into the no- +tion of a frequency-dependent effective screened Coulomb +interaction U(ω) when higher energy degrees of freedom +are integrated out [66]. An approximate form of the ef- +fective U(ω) can be obtained for example within the con- +strained Random-Phase-Approximation (cRPA) [13, 66], +that considers only screening processes outside of a target +low-energy subspace. From U(ω) two important pieces +of physical information can be deduced: First, the value +of the static screened interaction U(ω = 0), determin- +ing in an atomic picture the energetic positions of the +atomic multiplets, which in a periodic crystal typically +result in non- or weakly dispersive broad satellites, the +Hubbard bands. Second, the crossover from the screened +to the bare Coulomb interaction at the plasma frequency +ω0 creates satellites from collective electron excitations +at multiples of ω0[15]. +In oxides with different manifolds of bands (e.g. cor- +responding to the d- or p- states), additional ”subplas- +mons” corresponding to collective excitations within spe- +cific subspaces of the full Hilbert space can occur. For +SrVO3, for example, besides the main plasmon (located +at ω0 ≈ 14.5 eV) multiple excitations are found in the +dielectric function, namely around 2.5 eV and 5 eV, +the former originating from charge-oscillations in the V +t2g manifold[47, 60, 65]. +This is precisely the energy +scale where Hubbard satellites have been reported in +SrVO3[53, 54, 58, 59], indicating that both plasmonic +and Hubbard satellite features may exist in this system, +and at comparable energies. +Different state-of-the-art methods usually obtain only +a partial picture of the satellites, as shown in Fig. 1. +Compared to a Density-Functional-Theory (DFT) cal- +culation, which neither can describe Hubbard or plas- +monic satellites, the consideration of dynamical screen- +ing processes within the GW approximation introduces +plasmonic satellites in the occupied and unoccupied part +of the spectrum. +Including the effects of correlations +originating from the low-energy part U(ω = 0) of the +Coulomb interaction but without dynamical screening +within DFT+DMFT, one also observes satellites at very +similar energies but now of Hubbard-type origin. This +hints at a possible coexistence of both features in the +final spectrum, but necessitates the use of a method +that treats both Hubbard and plasmon contributions on +equal footing, like the combination of GW and DMFT + +3 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 +-4 +-3 +-2 +-1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 +Spectral +function of +SrVO3 +Hubbard +plasmon +Hubbard +plasmon +Spectral function A(ω) +Energy ω [eV] +DFT +G0W0 +DFT+DMFT + 0 + 0.05 + 0.1 + 0.15 +-4 -2 0 2 4 6 +FIG. 1: The spectral function of SrVO3, calculated within +Density Functional Theory (DFT), the G0W0 approximation +and a low-energy model solved in DFT+Dynamical Mean- +Field Theory (DFT+DMFT). The G0W0 approximation in- +troduces corrections due to dynamical screening effects and +plasmon satellites, while DFT+DMFT describes low-energy +correlations and the emergence of Hubbard bands. The inset +shows the same data on a smaller scale. +(GW+EDMFT)[48, 49, 60, 64, 65]. In this method non- +local correlation and screening processes are accounted +for by the GW approximation, while the local part +is obtained from the DMFT solution of a local impu- +rity problem subject to the partially screened interac- +tion U(ω), which encodes all screening processes beyond +the low-energy subspace in its frequency dependence. +Since Hubbard satellites originate from the low-energy +part U(ω = 0), and plasmons from dynamical screen- +ing, i.e. they emergence in the local model via the fre- +quency dependence of U(ω), we can use this to disentan- +gle their contributions. Using GW+EDMFT in its causal +implementation[69], we propose the following protocol to +identify and separate out only the plasmonic contribu- +tions in the spectral function: The effective Coulomb in- +teraction U(ω) can be artificially reduced by a constant +shift such that the static effective Coulomb interaction +U(ω = 0) vanishes, but the full frequency dependence +is retained. This removes contributions from low-energy +correlations, i.e. +the Hubbard satellites, but fully re- +tains the plasmonic contribution. +(See appendix for a +one-orbital proof-of-principle example.) +The resulting spectral function for SrVO3 within this +scheme is shown in Fig. 2. +Without any artificial re- +duction of the interaction the result is very similar to +previous GW+EDMFT calculations[48, 49, 60], with a +renormalized quasi-particle peak and a main satellite in +the occupied and unoccupied part. Different from DMFT +but similar as in GW[47, 70] one observes an additional +plasmon satellite around −5 eV, originating from tran- + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 +-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 +SrVO3 +(GW+EDMFT) +100% Hubbard +75% Hubbard +100% Plasmon +Spectral function A(ω) +Energy ω [eV] +DFT +U(0)=3.4eV +U(0)=2.4eV +U(0)=1.4eV +U(0)=0.0eV +-6 -4 -2 0 2 4 6 +P +H +H+P +FIG. 2: +The spectral function of SrVO3 for different val- +ues of the screened static interaction U(0) as obtained +from GW+DMFT, including both Hubbard- and plasmonic +physics. The low-energy satellite in the occupied part of the +spectrum vanishes for U(0) = 0 eV, indicating it is purely +composed of a Hubbard satellite. +On the other hand, the +upper satellite is composed of ∼ 25/75% plasmonic/Hubbard +weight. +sitions outside the t2g space[47]. Reducing the static in- +teraction from the ab initio value U(0) = 3.4 eV to zero, +we observe, besides an expected increase in bandwidth, a +strong reduction of the two satellites closest to the Fermi +level, where the lower satellite completely vanishes for +U(0) = 0 eV. A small upper satellite remains with about +25% of the original weight. The satellite at −5 eV is not +affected. This indicates that the satellite around −2 eV +in SrVO3 is indeed purely a lower Hubbard band, albeit +with an intensity lower than reported in DFT+DMFT. +This in fact agrees with the experimental observation +that the lower intrinsic satellite is rather small and in +general contains significant contributions from oxygen +vacancies[57]. On the other hand, the remaining satel- +lites around ±5 eV correspond to the plasmon satellites +in SrVO3 originating from the 5 eV transition reported +in the energy loss function of SrVO3[47, 60, 65]. +The +upper satellite is thus composed of Hubbard (∼ 75%) +and plasmonic (∼ 25%) contributions at similar energies, +with the plasmon satellite effectively ’buried’ beneath the +dominant Hubbard satellite. +Eventually the GW+EDMFT spectral function and its +satellites are very similar to the G0W0 result, except for a +slight increase in renormalization (see appendix for a di- +rect comparison), in contrast to previous results[48, 49], +which found a reduction in correlation. This difference +stems from causality violations in the previous computa- +tional scheme, as discussed in Ref[69], whereas our cur- +rent scheme does not suffer from this issue. This agree- +ment between G0W0 and GW+EDMFT not only indi- +cates that the current level of self-consistency is suffi- + +4 + 0 + 0.02 + 0.04 + 0.06 + 0.08 + 0.1 +-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 +P +H +H +P +(GlocWloc) +Spectral function A(ω) +Energy ω [eV] +U(0)=3.4eV +U(0)=2.4eV +U(0)=1.4eV +U(0)=0.0eV + 0 + 5 + 10 + 15 +P +MP +H +-Im[W(ω)] +ω [eV] +FIG. 3: +The t2g spectral function of SrVO3 obtained from a +local G0W0 approximation for different values of the screened +interaction U(0) but retaining the full frequency dependence. +The inset shows the negative imaginary part of the resulting +fully screened interaction W(ω). The lower and upper Hub- +bard (H)-like peaks originate from a local charge oscillation in +the t2g orbitals, corresponding to the peak around 2.5 eV in +W(ω). As in Fig.2 these satellites vanish when the screened +static interaction U(0) becomes zero, and only the plasmon +contribution (P) remains. (MP) indicates the main plasmon +excitation. +cient, but also that SrVO3 is only moderately correlated +such that G0W0 is able to capture most of the relevant +physics. Therefore, the interpretation of the low-energy +satellites as Hubbard satellites in SrVO3 raises the ques- +tion about the true nature of the G0W0 low-energy satel- +lites. As they originate from charge excitations in the +vanadium t2g manifold[47, 60], we apply a similar lo- +cal G0W0 scheme to disentangle possible collective non- +local charge excitations (plasmons) from local Hubbard- +like physics. +In Fig. 3 we show the resulting spectral +function A(ω) and screened interaction W(ω) for SrVO3 +within a local low-energy G0W0 scheme. In this scheme +the local ’bare’ V t2g interaction U(ω) is screened by +only considering local transitions in the t2g space, and +the resulting W(ω) is convoluted with the local non- +interacting t2g Green’s function to obtain the effective +self-energy (i.e., the impurity model is solved within the +G0W0 approximation). +The resulting W(ω) and spec- +tral function almost perfectly reproduces the full G0W0 +calculation, besides an overestimation of the energetic +position of the t2g derived peak in W(ω) around 3 eV, +which leads to an overestimation of the satellite position. +As the calculation has been performed on the real fre- +quency axis, more pronounced structures are visible and +not smeared out by the analytic continuation procedure. +This result indicates that the low-energy peaks in G0W0 +can be explained by only considering local charge excita- +tions and a local Coulomb interaction. Similarly as for + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 +-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 +SrMoO3 +(GW+DMFT) +≈100% Hubbard +65% Hubbard +100% Plasmon +Spectral function A(ω) +Energy ω [eV] +DFT +U(0)=3.0eV +U(0)=2.0eV +U(0)=1.0eV +U(0)=0.0eV +-4 +-2 + 0 +PES +noncausal +causal +FIG. 4: The spectral function of SrMoO3 for different val- +ues of the screened static interaction U(0) as obtained from +GW+EDMFT. The low-energy satellite around −2.5 eV is +mostly composed of a Hubbard satellite, while the unoccupied +satellite is about 35% plasmonic and 65% Hubbard type ori- +gin. The inset shows the same spectral function at U = 3 eV, +compared to a noncausal implementation (taken from [49]) +and photoemission data (taken from [71]). +the GW+EDMFT result, the peaks vanish as the static +screened interaction is reduced, confirming their local +’Hubbard’-like nature. Even though G0W0 as a pertur- +bative approach cannot access strong electronic correla- +tions, the corresponding atomic multiplet excitations are +effectively encoded in W(ω) via the RPA approximation +and give rise to satellites representing the Hubbard satel- +lites obtained in non-perturbative methods. Thus, this +result confirms that the low-energy satellites in SrVO3 +do not originate from non-local collective excitations but +arise purely from local Hubbard-like charge excitations +given by the static local interaction U(ω = 0). Plasmon +satellites are only found at energies (and beyond) ±5 eV. +We apply the same method to the closely related 4d2 +material SrMoO3, which is isostructural to SrVO3. Due +to the more extended nature of the 4d orbitals, low- +energy electronic correlations are weaker and the experi- +mentally observed satellites have been proposed to be of +purely plasmonic origin[49, 71–73]. The resulting spec- +tral function for different values of the static interaction +is shown in Fig. 4. As in previous reports we obtain a +much broader quasiparticle peak than in SrVO3, with +a lower shoulder-like feature around 2.5 eV and an up- +per satellite. Upon reducing the static interaction, we +see a similar trend as in SrVO3, but less pronounced. +For U(0) = 0 the occupied shoulder seems to completely +merge with the quasiparticle peak, corresponding to a +Hubbard satellite, but we found the analytic continua- +tion procedure to be less reliable in that area. The ex- +istence of a small lower Hubbard satellite does not con- +tradict with a previous DFT+DMFT study[72], which + +5 +found no pronounced Hubbard satellite but still a signif- +icant amount of spectral weight shifted to lower energies +around −2.5 eV, very similar to our results. The unoc- +cupied satellite is significantly reduced, retaining about +35% of its weight as a plasmon satellite. Similar as in +SrVO3, we find that SrMoO3 shows pronounced plasmon +satellites around ±5 eV. Most noteworthy, the causal im- +plementation of GW+EDMFT used in this manuscript is +able to accurately reproduce the observed experimental +spectral function from photoemission experiments[71], +demonstrating that this approach is capable of capturing +the relevant physics in this material and thus strength- +ening our interpretation of the spectral features as Hub- +bard satellites. The noncausal formulation[49] shows a +significantly lower intensity for the Hubbard-like satel- +lite and increased bandwidth of the V t2g quasi-particle +dispersive states, indicating that the noncausal variant +underestimates the electronic correlation strength also in +SrMoO3, in line with previous observations[69, 74]. +In summary, we have revisited the decade-old prob- +lem of the nature of spectral satellites in ternary tran- +sition metal oxides, and proposed a theoretical ab initio +method to distinguish plasmonic satellites, which emerge +from collective electronic excitations, from Hubbard-type +satellites, resulting from a strong local Coulomb interac- +tion. For the prototypical transition metal oxide SrVO3 +we show that the occupied low-energy satellite is purely +composed of Hubbard-type incoherent weight, while in +the unoccupied satellite both Hubbard and plasmonic +contributions coexist at similar energies. In the weaker +correlated 4d2 SrMoO3 we observe a similar but less pro- +nounced picture of a Hubbard satellite in the occupied +part, and both plasmon and Hubbard satellites at sim- +ilar energies in the unoccupied part. These observation +call for a reinvestigation of similar and other correlated +materials and their satellite features, both theoretically +and experimentally, as they are in particular relevant for +plasmon-mediated applications in functional materials, +where precise knowledge of the intensity and energy of +plasmonic excitations is needed. +The scheme that we +have developed can be applied to a large class of mate- +rials, and can aid the development of such applications +by theoretically quantifying the plasmonic and Hubbard- +type contributions in the spectral satellites. +While this work was being prepared for publication, +a new joint experimental-theoretical work appeared on +SrVO3[75]. In that work, a purely plasmonic origin of the +satellites is advocated. 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B 105, +085102 (2022). +[75] C.-P. +Su, +K. +Ruotsalainen, +A. +Nico- +laou, +M. +Gatti, +and +A. +Gloter, +Advanced +Optical +Materials +n/a, +2202415 +(2023), +https://onlinelibrary.wiley.com/doi/pdf/10.1002/adom.202202415 +. + +Diagnostics for plasmon satellites and Hubbard bands in transition metal oxides - +Supplementary Material +Steffen Backes1,2,3,∗ Hong Jiang4, and Silke Biermann3,5,6,7 +1Research Center for Advanced Science and Technology, +University of Tokyo, Komaba, Tokyo 153-8904, Japan +2Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan +3CPHT, CNRS, ´Ecole polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France +4College of Chemistry and Molecular Engineering, Peking University, China +5Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France +6European Theoretical Spectroscopy Facility, 91128 Palaiseau, France, Europe and +7Department of Physics, Division of Mathematical Physics, +Lund University, Professorsgatan 1, 22363 Lund, Sweden +(Dated: January 13, 2023) +COMPUTATIONAL DETAILS +For the GW+EDMFT cycle we start with a well con- +verged DFT calculation from Wien2K [1], and perform a +constrained Random-Phase-Approximation and a G0W0 +calculation, as implemented in the FHI-gap Code[2], to +obtain the effective impurity interaction U(ω) and the +Selfenergy ΣGW (k, iωn), projected onto the t2g orbitals +of either SrVO3 or SrMoO3, using a maximally localized +Wannier basis. The cRPA and GW calculation were per- +formed on a 8 × 8 × 8 k-mesh, and the resulting U(ω) +and ΣGW (k, iωn) are then interpolated by cubic interpo- +lation onto a dense 30 × 30 × 30 k-mesh, which serves +as the input for the selfconsistent EDMFT calculation. +The impurity model is solved within the continuous-time +Quantum Monte-Carlo method in the hybridization ex- +pansion, as implemented in the ALPS package[3] at in- +verse temperature β = 40 1/eV, including the frequency +dependence of the monopole term F0(ω) of the effective +interaction. The analytical continuation from the imag- +inary to the real frequency axis is done using a combi- +nation of Pad´e approximants and the Maximum Entropy +code from Ref. [4], where we added plasmonic peaks in +the default model of exponentially decaying weight at +multiples of the plasma frequency to faciliate proper con- +tinuation. +The GW+EDMFT calculation was not done fully self- +consistently, i.e. the nonlocal GW self-energy remained +at the G0W0 level, and the effective interaction U(iω) was +not updated. As discussed in the main text the resulting +spectral function from this GW+EDMFT is very close to +the G0W0 result, as shown in Fig.1. This indicates that +further self-consistency will have only minor effects and +not qualitatively change the one-shot result. Therefore, +we only considered the ’one-shot’ GW+EDMFT results, +which are computationally less demanding. + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 +-3 +-2 +-1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 +Spectral +function of +SrVO3 +Spectral function A(ω) +Energy ω [eV] +DFT +DFT+DMFT +G0W0 +GW+EDMFT + 0 + 0.05 + 0.1 + 0.15 +-6-4-2 0 2 4 6 +FIG. 1: +The spectral function of SrVO3, comparing the De- +sity Functional Theory (DFT), the DFT+DMFT, the G0W0 +approximation and the GW+EDMFT result shown in the +main text. The G0W0 and the GW+EDMFT spectral func- +tions are almost identical, indicating that a fully selfconsis- +tent GW+EDMFT calculation will not significantly change +the result. +LOCAL G0W0 AND SATELLITES +The results shown in Fig. 3 in the main text were ob- +tained by employing a local G0W0 approximation: First, +the polarization was calculated from the projected lo- +cal DFT Green’s function for the t2g orbitals as Ploc = +GlocGloc. For the effective “bare” interaction the cRPA +derived impurity interaction U(ω) was used, which was +screened by the local polarization to obtain the screened +local interaction Wloc = U[1−PlocU]−1 (shown in the in- +set of Fig. 3 in the main text). Then the self-energy was +obtained by the convolution of the local Green’s function +and screened interaction as Σ = GlocWloc. +arXiv:2301.04908v1 [cond-mat.str-el] 12 Jan 2023 + +2 + 0 + 1 + 2 + 3 + 4 + 5 +-4 +-2 + 0 + 2 + 4 +U=2, V=0 +A(ω) +ω +G0W0 +exact + 0 + 1 + 2 + 3 + 4 + 5 +U=2, V=1 +A(ω) +G0W0 +exact + 0 + 1 + 2 + 3 + 4 + 5 +U=2, V=0.3 +A(ω) +G0W0 +exact +FIG. 2: +The ground state spectral function for a DMFT im- +purity model with one bath site solved within the G0W0 ap- +proximation for different values of the hybridization strength +V at fixed interaction U = 2. Except in the strong interac- +tion limit V → 0 G0W0 captures all qualitative features and +correlation satellites, albeit overestimating their energetic po- +sition. +To elucidate the appearance of the Hubbard correla- +tion satellites within G0W0 one can consider a simplified +’linearized’ DMFT impurity problem [5–7] with only one +bath site, which can be solved analytically +H = Un↑n↓ + V +� +σ +(c† +σfσ + f † +σcσ) − µ +� +σ +nσ, +(1) +with c, c†/f, f † the impurity/bath annihilation and cre- +ation operators, interaction U, hybridization strength V +and chemical potential µ. Solving this model exactly and +within the G0W0 approximation yields for the impurity +self-energy +Σexact(z) = U 2 +8 +� +1 +z − 3V + +1 +z + 3V +� +(2) +ΣG0W0(z) = U 2 +4a +� +1 +z − V (2a + 1) + +1 +z + V (2a + 1) +� +, +(3) +with a = +� +1 + U/(2V ). Both results qualitatively agree, +but G0W0 does not capture the correct position and +weight of the two peaks in Σ. We note, however, that +for U/V → 0 the peak position is correctly reproduced + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 +Satellite weight +U/V +G0W0 +exact + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 + 4.5 + 5 +Satellite position +G0W0 +exact +FIG. 3: +The satellite positions and weights for the DMFT +impurity model with one bath site as shown in Fig.2, as a +function of the interaction strength U for fixed hybridization +V = 1. +in G0W0, while the weight differs by a factor of 2. In +Fig.2 we show the resulting spectral function for different +values of the hybridization V for fixed interaction U = 2. +Except in the strong interaction limit V → 0 G0W0 cap- +tures all qualitative features and correlation satellites, +albeit overestimating their energetic position. This can +be seen explicitly in Fig.3, where we show the position +and weights of the two different satellites emerging in the +spectral function as a function of the interaction strength +U. G0W0 performs reasonably well for small values of +the interaction strength U/V ≲ 1.5, in particular for the +bonding/antibonding states and satellite weights. +The +emergence of the correlation satellites in G0W0, which in +this setup appear as Hubbard satellites, is in fact not sur- +prising, since a perturbative approach such as the G0W0 +approximation is expected to become more accurate in +the weakly correlated regime. +If Hubbard satellites in +this regime are present, it is expected that G0W0 is able +to qualitatively capture them, as shown above. +SEPARATION OF HUBBARD SATELLITES +Previously it had been suggested to use the ener- +getic separation of the unoccupied and occupied satel- +lites, and their dependence on the bandwidth to distin- +guish plasmon satellites from Hubbard satellites[8, 9]. In +the atomic limit Hubbard satellites are separated by the +static onsite interaction U(0), but for metallic systems +a finite bandwidth enhances the separation of the Hub- + +3 + 1 + 1.2 + 1.4 + 1.6 + 1.8 + 2 + 2.2 + 2.4 + 2.6 + 1 + 1.2 1.4 1.6 1.8 + 2 + 2.2 2.4 2.6 +DMFT +GW+EDMFT +Hubbard satellite separation +∆Hubbard/D +U/D +SrVO3 DMFT +FIG. 4: +The separation of the Hubbard satellites in SrVO3 +within a DMFT approach for different values of the inter- +action U, relative to the non-interacting bandwidth D. For +small interactions the separation is much larger that the value +of U, but becomes smaller for larger interactions. The non- +interacting bandwidth for GW+EDMFT is given by the spec- +tral function where the local self-energy has been removed +(≈ 3.25 eV). The error bars indicate the uncertainty from +analytic continuation. +bard satellites[10]. In systems far away from half-filling +such as SrVO3, the position of the Hubbard satellites is +further complicated by the breaking of particle-hole sym- +metry. Already at the DMFT level, the separation of the +Hubbard satellites ∆Hub ≈ 5 eV greatly surpasses the +value of the static interaction U ≈ 3.5 eV. +In Fig.4 we show the dependence of the Hubbard satel- +lite separation in SrVO3 on the interaction U, obtained +from a DMFT calculation. Up to moderate correlations +(for a filling of n = 1/6) the Hubbard satellite sep- +aration exceeds the value of U by almost a factor of +2. +On the other hand, for stronger interactions when +the quasiparticle peak is close to vanishing the situa- +tion reverses and the separation becomes smaller than +U. For the case of the GW+EDMFT result, which has +a larger ’non-interacting’ bandwidth D due to the re- +moval of the exchange-correlation potential and non-local +GW contributions[11], the Hubbard satellite separation +is larger than in standard DMFT, but is well within the +expected range, considering the value of U(0)/D (see la- +bel ’GW+EDMFT’ in Fig.4). Therefore, the Hubbard +satellite separation alone is not a good quantifier to dis- +tinguish plasmonic from Hubbard satellites, as in metallic +systems and systems away from half-filling, the Hubbard +satellite separation significantly differs from the usual +∆Hub ∼ U behavior. +ONE-ORBITAL MODEL FOR PLASMONIC AND +HUBBARD SATELLITES +In order to distinguish satellites of plasmonic origin +and that arising from strong electronic correlations, we +consider a simple model that describes the coupling of +electrons to a single bosonic degree of freedom, namely +the Hubbard-Holstein model +H = − +� +ij +tijc† +iσcjσ + Ubare +� +i +ni↑ni↓ ++ ω0 +� +i +b† +ibi + λ +� +i +ni +� +b† +i + bi +� +, +(4) +where tij is the hopping amplitude, Ubare is the local +instantaneous Coulomb repulsion, ω0 is the energy of +the bosonic mode (plasma frequency), generated by the +bosonic annihilation and creation operators b† +i, bi. The +coupling strength between the electronic charge ni and +the bosonic mode is given by λ. +Integrating out the bosonic degrees of freedom gives +rise to an effective dynamical interaction U(ω), where +the coupling to the bosonic mode is now encoded in the +frequency dependence +Re Ueff(ω) = Ubare − 2λ2 +ω0 +ω2 +0 − ω2 +(5) +Im Ueff(ω) = −λ2π (δ(ω − ω0) − δ(ω + ω0)) . +(6) +This model is an extension of the standard Hubbard +model, and it has been shown that this model exhibits +plasmonic replicas of the quasi-particle structure at mul- +tiples of the plasma frequency ω0, that originate from +plasmonic charge excitations[12, 13]. +We +use +extended +dynamical +mean-field +theory +(EDMFT) to solve the model at half-filling on the Bethe +lattice with bandwidth W = 4 eV, inverse temperature +β = 40 1/eV, and a single bosonic mode ω0 = 5 eV +and λ = 3.5 eV. The spectral function and Selfenergy for +different values of the static interaction U(ω) is shown +in Fig. 5. +We observe the Hubbard satellite to com- +pletely vanish when reducing the interaction by a con- +stant shift, where the quasiparticle peak recovers a renor- +malized semicircular form that corresponds to the origi- +nal non-interacting dispersion on the Bethe lattice, renor- +malized only by the transfer of spectral weight into plas- +mon satellites. +This behavior if even more evident in +the imaginary part of the Selfenergy in Fig. 5 b). The +low-energy peak responsible for the Hubbard satellite is +completely suppressed for vanishing U(0), while the ef- +fect on the plasmonic peaks is small. This confirms that +the reduction of the interaction by a constant shift is only +effecting the plasmon satellites to a minor degree, as they +originate from the frequency dependence in U(ω). On the +other hand, the Hubbard satellites completely vanish at +U(ω) = 0, allowing for a systematic classification of the +nature of the satellites. + +4 + 0 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 + 0.35 +A(ω) +λ=3.5eV const. +ω0=5eV +HUB +PLAS1 +PLAS2 +U0=3eV +U0=2eV +U0=1eV +U0=0eV + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 0 + 2 + 4 + 6 + 8 + 10 + 12 +-Im Σ(ω) +ω [eV] +HUB +PLAS1 +PLAS2 +U0=3eV +U0=2eV +U0=1eV +U0=0eV +FIG. 5: The spectral function and imaginary part of the Self- +energy for the half-filled Hubbard model with a one boson +screening mode. +We show only positive energies since the +spectrum is particle-hole symmetric. The frequency depen- +dence of the effective interaction U(ω) has been fixed with +U(∞) − U(0) = 5 eV, but a static shift has been applied +in order to reduce the strength of the interaction but keep +the transfer of spectral weight due to the bosonic coupling +constant. The bosonic energy is ω0 = 5 eV, giving rise to +plasmonic replica at multiples of ω0. We observe a disappear- +ance of the Hubbard satellite as U(0) approaches zero, while +the plasmonic satellites stay mostly unchanged. +∗ steffen-backes@g.ecc.u-tokyo.ac.jp +[1] B. P, S. M. G K H, K. D, and L. J, Karlheinz Schwarz, +Techn. Universit¨at Wien, Austria (2001). +[2] H. Jiang, R. I. G´omez-Abal, X. Li, C. Meisenbichler, +C. Ambrosch-Draxl, , and M. Scheffler, Computer Phys. +Commun.,184, 348 184, 348 (2012). +[3] A. Gaenko, A. Antipov, G. Carcassi, T. Chen, X. Chen, +Q. Dong, L. Gamper, J. Gukelberger, R. Igarashi, +S. Iskakov, M. K¨onz, J. LeBlanc, R. Levy, P. Ma, J. Paki, +H. Shinaoka, S. Todo, M. Troyer, and E. Gull, Computer +Physics Communications 213, 235 (2017). +[4] R. Levy, J. LeBlanc, and E. Gull, Computer Physics +Communications 215, 149 (2017). +[5] E. Lange, Modern Physics Letters B 12, 915 (1998). +[6] R. Bulla and M. Potthoff, The European Physical Jour- +nal B 13, 257 (2000). +[7] M. Potthoff, Phys. Rev. B 64, 165114 (2001). +[8] L. Boehnke, F. Nilsson, F. 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B +85, 035115 (2012). + diff --git a/HtE4T4oBgHgl3EQfIAw0/content/tmp_files/load_file.txt b/HtE4T4oBgHgl3EQfIAw0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..99f57a41458e35b500aea6bc52ae652d33333d65 --- /dev/null +++ b/HtE4T4oBgHgl3EQfIAw0/content/tmp_files/load_file.txt @@ -0,0 +1,1245 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf,len=1244 +page_content='Diagnostics for plasmon satellites and Hubbard bands in transition metal oxides Steffen Backes1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='∗ Hong Jiang4,' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Tokyo 153-8904,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Japan 2Center for Emergent Matter Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' RIKEN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Wako,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Saitama 351-0198,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Japan 3CPHT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' ´Ecole polytechnique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Institut Polytechnique de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 91120 Palaiseau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' France 4College of Chemistry and Molecular Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' China 5Coll`ege de France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 11 place Marcelin Berthelot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' France 6European Theoretical Spectroscopy Facility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 91128 Palaiseau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Europe and 7Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Division of Mathematical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Lund University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Professorsgatan 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 22363 Lund,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Sweden (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 2023) Coulomb correlations between the electrons imprint characteristic signatures to the spectral prop- erties of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Among others, they are at the origin of a rich phenomenology of satellite features, either stemming from atomic-like multiplets or from interactions with particle-hole excitations or plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' While in many cases the latter lie at considerably higher energies than the former, suggesting clear distinction criteria, this picture has recently become blurred by indications that satellites of different types can coexist in the same energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' It is now generally accepted that the identification of the nature of spectral features is a highly non-trivial task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In this article we propose a general procedure for tracing the origin of satellites of different types within modern ab initio calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' As an illustration, we analyze the ternary transition metal oxides SrVO3 and SrMoO3, which are drosophila compounds for the coexistence of Hubbard and plasmonic satellites, reconciling previous seemingly contradictory findings in an unexpected manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' INTRODUCTION Impressive progress in direct – and to a much lesser ex- tent inverse – photoemission spectroscopy over the last decades has resulted in a situation where the spectral properties of electronic systems have become some of the most commonly probed experimental properties of ma- terials [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The main quantity is the spectral func- tion A(k, ω), which encodes information about the pos- sible electron removal and addition processes, as probed in direct and inverse photoemission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The knowledge of A(k, ω) in turn is typically synonymous with a good first understanding of the behaviour of the material under a variety of probes, even those not directly encoded in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In normal metals, the low-energy behaviour is governed by renormalized quasi-particle bands following the Lan- dau Fermi liquid paradigm, while in insulators the spec- trum is gapped around the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Beyond these el- ementary considerations, spectral functions can however display a whole zoology of different features at interme- diate or high energies (in typical transition metal oxides, in energy ranges spanning a few tenths to a few tens of eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Among the most prominent features in electronic sys- tems with sizable Coulomb correlations are Hubbard satellites, remnants of the atomic physics in the mate- rial, corresponding to the atomic multiplets of an isolated atom placed in the crystal field environment of its sur- roundings but potentially acquiring some dispersion due to the periodicity of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The energy scales of these multiplet structures are given by the effective local Coulomb interaction, often parametrized theoretically in the form of a local Hubbard U (or more precisely a Hub- bard U matrix including Hund’s exchange and orbital structures) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Such features have been studied in some detail in the past with elaborate theoretical approaches, starting from exact diagonalization[5, 6] and more re- cently within Dynamical Mean-Field Theory (DMFT)[7– 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' and are well-documented experimentally [6, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Another type of satellites appearing in the spectral function of electronic materials are due to electrons cou- pling to plasmons [11–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Plasmons have been experi- mentally observed and theoretically investigated in mate- rials ranging from elementary metals [19–21], bronzes[22, 23], oxides[18, 24–30], in particular ruthenates[31] and cuprates[32–34] as well as in graphene[16, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Plas- monic excitations are relevant and actively utilized in the design of functional materials[37–39], such as in plasmon- mediated photocatalysis[40, 41] and sensors[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Plas- mons are collective electronic excitations, which are in general highly non-local in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' They are encoded in the dielectric function describing the dynamic response of the electronic system as a whole to a perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This response can be mediated by particle-hole excitations as well as by collective (plasmonic) excitations, which both can give rise to shake-up satellites in the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' For simplicity, below, we will refer to any fea- tures beyond a local atomic-like picture, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' originating arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='04908v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='str-el] 12 Jan 2023 2 from non-local collective excitations as plasmonic satel- lites, and our aim will be to distinguish those from the Hubbard-type satellites described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This differenti- ation becomes non-trivial when the energy scale of plas- monic excitations is similar to that of the local Coulomb interactions, which can lead to both Hubbard and plas- mon satellites to appear at similar energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Recently, evidence has accumulated that this is the case in a large number of transition metal oxides [43–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In this letter we present a protocol for a quantitative ab initio identification of Hubbard and plasmonic con- tributions in low-energy satellites in real materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Us- ing this protocol, we reinvestigate two prototypical 3d and 4d perovskite transition metal oxides, SrVO3 and SrMoO3, and determine the Hubbard and plasmonic con- tributions in the observed low-energy satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Contrary to previous interpretations[48–51], we find both Hubbard and plasmon satellites to be present, albeit with differ- ent magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' On the other hand, our findings recon- cile seemingly contradictory calculations within many- body perturbation theory (within the GW approxima- tion) and combined GW+Dynamical Mean Field Theory (GW+DMFT) in a surprising manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' SrVO3 is a 3d1 compound with metallic V t2g states crossing the Fermi level, forming a typical 3-peak struc- ture in the spectral function, both confirmed from experiment[52–57] and theoretical calculations[53, 54, 57–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The proposed origin of the satellites though has significantly evolved over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Early combined den- sity functional theory and dynamical mean-field theory (DFT+DMFT) calculations suggested that the satellites arise from strong local V-t2g Coulomb interactions in the form of Hubbard bands[53, 54, 58, 59, 61–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The ad- vent of combined many-body perturbation theory and dynamical mean field theory (”GW+DMFT”) [64], how- ever, made it possible to include both, Hubbard bands and plasmonic features, in the theoretical description, and it was realized that in the low energy (< 5 eV) range features of both types can coexist [60, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' More- over, it was pointed out that the empty V-eg states that are split off from the partially filled V-t2g states by the octahedral crystal field lie in the same energy range as the upper Hubbard band from the early DFT+DMFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Interestingly, many-body perturbation the- ory alone could also reproduce the observed satellite fea- tures (albeit at slightly shifted energetic positions) [47], a finding which seemed to be in contradiction with the interpretation as Hubbard bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Along this line, several works [47–51] gave a purely plasmonic interpretation to the lowest energy features both in the occupied and the unoccupied part of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' A new twist appeared when it was realized that oxygen vacancies contribute spectral weight at the same energy as the satellite in the occupied spectrum[57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' While the importance of oxygen vacancies responsible for part of the spectral weight in the energy range in question is now widely recognized, no consensus has been reached so far concerning the ori- gin of the remaining intrinsic part of the satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' We now turn to a brief discussion of the theoretical de- scription of the creation of plasmonic features in the spec- tral function, within DMFT-derived schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' As is well- known [13, 15, 48, 60, 64, 66–68] electronic screening is a dynamical process, since the response of the electronic density in a solid to a perturbation depends on the energy scale of the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' For a given set of orbitals of interest, the charge redistribution and thus the screening is energy dependent, and directly translates into the no- tion of a frequency-dependent effective screened Coulomb interaction U(ω) when higher energy degrees of freedom are integrated out [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' An approximate form of the ef- fective U(ω) can be obtained for example within the con- strained Random-Phase-Approximation (cRPA) [13, 66], that considers only screening processes outside of a target low-energy subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' From U(ω) two important pieces of physical information can be deduced: First, the value of the static screened interaction U(ω = 0), determin- ing in an atomic picture the energetic positions of the atomic multiplets, which in a periodic crystal typically result in non- or weakly dispersive broad satellites, the Hubbard bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Second, the crossover from the screened to the bare Coulomb interaction at the plasma frequency ω0 creates satellites from collective electron excitations at multiples of ω0[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In oxides with different manifolds of bands (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' cor- responding to the d- or p- states), additional ”subplas- mons” corresponding to collective excitations within spe- cific subspaces of the full Hilbert space can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' For SrVO3, for example, besides the main plasmon (located at ω0 ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 eV) multiple excitations are found in the dielectric function, namely around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 eV and 5 eV, the former originating from charge-oscillations in the V t2g manifold[47, 60, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This is precisely the energy scale where Hubbard satellites have been reported in SrVO3[53, 54, 58, 59], indicating that both plasmonic and Hubbard satellite features may exist in this system, and at comparable energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Different state-of-the-art methods usually obtain only a partial picture of the satellites, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Compared to a Density-Functional-Theory (DFT) cal- culation, which neither can describe Hubbard or plas- monic satellites, the consideration of dynamical screen- ing processes within the GW approximation introduces plasmonic satellites in the occupied and unoccupied part of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Including the effects of correlations originating from the low-energy part U(ω = 0) of the Coulomb interaction but without dynamical screening within DFT+DMFT, one also observes satellites at very similar energies but now of Hubbard-type origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This hints at a possible coexistence of both features in the final spectrum, but necessitates the use of a method that treats both Hubbard and plasmon contributions on equal footing, like the combination of GW and DMFT 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='8 1 4 3 2 1 0 1 2 3 4 5 6 Spectral function of SrVO3 Hubbard plasmon Hubbard plasmon Spectral function A(ω) Energy ω [eV] DFT G0W0 DFT+DMFT 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='15 4 -2 0 2 4 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 1: The spectral function of SrVO3, calculated within Density Functional Theory (DFT), the G0W0 approximation and a low-energy model solved in DFT+Dynamical Mean- Field Theory (DFT+DMFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The G0W0 approximation in- troduces corrections due to dynamical screening effects and plasmon satellites, while DFT+DMFT describes low-energy correlations and the emergence of Hubbard bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The inset shows the same data on a smaller scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' (GW+EDMFT)[48, 49, 60, 64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In this method non- local correlation and screening processes are accounted for by the GW approximation, while the local part is obtained from the DMFT solution of a local impu- rity problem subject to the partially screened interac- tion U(ω), which encodes all screening processes beyond the low-energy subspace in its frequency dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Since Hubbard satellites originate from the low-energy part U(ω = 0), and plasmons from dynamical screen- ing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' they emergence in the local model via the fre- quency dependence of U(ω), we can use this to disentan- gle their contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Using GW+EDMFT in its causal implementation[69], we propose the following protocol to identify and separate out only the plasmonic contribu- tions in the spectral function: The effective Coulomb in- teraction U(ω) can be artificially reduced by a constant shift such that the static effective Coulomb interaction U(ω = 0) vanishes, but the full frequency dependence is retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This removes contributions from low-energy correlations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' the Hubbard satellites, but fully re- tains the plasmonic contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' (See appendix for a one-orbital proof-of-principle example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=') The resulting spectral function for SrVO3 within this scheme is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Without any artificial re- duction of the interaction the result is very similar to previous GW+EDMFT calculations[48, 49, 60], with a renormalized quasi-particle peak and a main satellite in the occupied and unoccupied part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Different from DMFT but similar as in GW[47, 70] one observes an additional plasmon satellite around −5 eV, originating from tran- 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='7 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 SrVO3 (GW+EDMFT) 100% Hubbard 75% Hubbard 100% Plasmon Spectral function A(ω) Energy ω [eV] DFT U(0)=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4eV U(0)=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4eV U(0)=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4eV U(0)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='0eV 6 -4 -2 0 2 4 6 P H H+P FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 2: The spectral function of SrVO3 for different val- ues of the screened static interaction U(0) as obtained from GW+DMFT, including both Hubbard- and plasmonic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The low-energy satellite in the occupied part of the spectrum vanishes for U(0) = 0 eV, indicating it is purely composed of a Hubbard satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' On the other hand, the upper satellite is composed of ∼ 25/75% plasmonic/Hubbard weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' sitions outside the t2g space[47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Reducing the static in- teraction from the ab initio value U(0) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4 eV to zero, we observe, besides an expected increase in bandwidth, a strong reduction of the two satellites closest to the Fermi level, where the lower satellite completely vanishes for U(0) = 0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' A small upper satellite remains with about 25% of the original weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The satellite at −5 eV is not affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This indicates that the satellite around −2 eV in SrVO3 is indeed purely a lower Hubbard band, albeit with an intensity lower than reported in DFT+DMFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This in fact agrees with the experimental observation that the lower intrinsic satellite is rather small and in general contains significant contributions from oxygen vacancies[57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' On the other hand, the remaining satel- lites around ±5 eV correspond to the plasmon satellites in SrVO3 originating from the 5 eV transition reported in the energy loss function of SrVO3[47, 60, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The upper satellite is thus composed of Hubbard (∼ 75%) and plasmonic (∼ 25%) contributions at similar energies, with the plasmon satellite effectively ’buried’ beneath the dominant Hubbard satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Eventually the GW+EDMFT spectral function and its satellites are very similar to the G0W0 result, except for a slight increase in renormalization (see appendix for a di- rect comparison), in contrast to previous results[48, 49], which found a reduction in correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This difference stems from causality violations in the previous computa- tional scheme, as discussed in Ref[69], whereas our cur- rent scheme does not suffer from this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This agree- ment between G0W0 and GW+EDMFT not only indi- cates that the current level of self-consistency is suffi- 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='1 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 P H H P (GlocWloc) Spectral function A(ω) Energy ω [eV] U(0)=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4eV U(0)=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4eV U(0)=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4eV U(0)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='0eV 0 5 10 15 P MP H Im[W(ω)] ω [eV] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 3: The t2g spectral function of SrVO3 obtained from a local G0W0 approximation for different values of the screened interaction U(0) but retaining the full frequency dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The inset shows the negative imaginary part of the resulting fully screened interaction W(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The lower and upper Hub- bard (H)-like peaks originate from a local charge oscillation in the t2g orbitals, corresponding to the peak around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 eV in W(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2 these satellites vanish when the screened static interaction U(0) becomes zero, and only the plasmon contribution (P) remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' (MP) indicates the main plasmon excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' cient, but also that SrVO3 is only moderately correlated such that G0W0 is able to capture most of the relevant physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Therefore, the interpretation of the low-energy satellites as Hubbard satellites in SrVO3 raises the ques- tion about the true nature of the G0W0 low-energy satel- lites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' As they originate from charge excitations in the vanadium t2g manifold[47, 60], we apply a similar lo- cal G0W0 scheme to disentangle possible collective non- local charge excitations (plasmons) from local Hubbard- like physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 3 we show the resulting spectral function A(ω) and screened interaction W(ω) for SrVO3 within a local low-energy G0W0 scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In this scheme the local ’bare’ V t2g interaction U(ω) is screened by only considering local transitions in the t2g space, and the resulting W(ω) is convoluted with the local non- interacting t2g Green’s function to obtain the effective self-energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=', the impurity model is solved within the G0W0 approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The resulting W(ω) and spec- tral function almost perfectly reproduces the full G0W0 calculation, besides an overestimation of the energetic position of the t2g derived peak in W(ω) around 3 eV, which leads to an overestimation of the satellite position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' As the calculation has been performed on the real fre- quency axis, more pronounced structures are visible and not smeared out by the analytic continuation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This result indicates that the low-energy peaks in G0W0 can be explained by only considering local charge excita- tions and a local Coulomb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Similarly as for 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='7 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 SrMoO3 (GW+DMFT) ≈100% Hubbard 65% Hubbard 100% Plasmon Spectral function A(ω) Energy ω [eV] DFT U(0)=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='0eV U(0)=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='0eV U(0)=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='0eV U(0)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='0eV 4 2 0 PES noncausal causal FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 4: The spectral function of SrMoO3 for different val- ues of the screened static interaction U(0) as obtained from GW+EDMFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The low-energy satellite around −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 eV is mostly composed of a Hubbard satellite, while the unoccupied satellite is about 35% plasmonic and 65% Hubbard type ori- gin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The inset shows the same spectral function at U = 3 eV, compared to a noncausal implementation (taken from [49]) and photoemission data (taken from [71]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' the GW+EDMFT result, the peaks vanish as the static screened interaction is reduced, confirming their local ’Hubbard’-like nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Even though G0W0 as a pertur- bative approach cannot access strong electronic correla- tions, the corresponding atomic multiplet excitations are effectively encoded in W(ω) via the RPA approximation and give rise to satellites representing the Hubbard satel- lites obtained in non-perturbative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Thus, this result confirms that the low-energy satellites in SrVO3 do not originate from non-local collective excitations but arise purely from local Hubbard-like charge excitations given by the static local interaction U(ω = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Plasmon satellites are only found at energies (and beyond) ±5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' We apply the same method to the closely related 4d2 material SrMoO3, which is isostructural to SrVO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Due to the more extended nature of the 4d orbitals, low- energy electronic correlations are weaker and the experi- mentally observed satellites have been proposed to be of purely plasmonic origin[49, 71–73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The resulting spec- tral function for different values of the static interaction is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' As in previous reports we obtain a much broader quasiparticle peak than in SrVO3, with a lower shoulder-like feature around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 eV and an up- per satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Upon reducing the static interaction, we see a similar trend as in SrVO3, but less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' For U(0) = 0 the occupied shoulder seems to completely merge with the quasiparticle peak, corresponding to a Hubbard satellite, but we found the analytic continua- tion procedure to be less reliable in that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The ex- istence of a small lower Hubbard satellite does not con- tradict with a previous DFT+DMFT study[72], which 5 found no pronounced Hubbard satellite but still a signif- icant amount of spectral weight shifted to lower energies around −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 eV, very similar to our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The unoc- cupied satellite is significantly reduced, retaining about 35% of its weight as a plasmon satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Similar as in SrVO3, we find that SrMoO3 shows pronounced plasmon satellites around ±5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Most noteworthy, the causal im- plementation of GW+EDMFT used in this manuscript is able to accurately reproduce the observed experimental spectral function from photoemission experiments[71], demonstrating that this approach is capable of capturing the relevant physics in this material and thus strength- ening our interpretation of the spectral features as Hub- bard satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The noncausal formulation[49] shows a significantly lower intensity for the Hubbard-like satel- lite and increased bandwidth of the V t2g quasi-particle dispersive states, indicating that the noncausal variant underestimates the electronic correlation strength also in SrMoO3, in line with previous observations[69, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In summary, we have revisited the decade-old prob- lem of the nature of spectral satellites in ternary tran- sition metal oxides, and proposed a theoretical ab initio method to distinguish plasmonic satellites, which emerge from collective electronic excitations, from Hubbard-type satellites, resulting from a strong local Coulomb interac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' For the prototypical transition metal oxide SrVO3 we show that the occupied low-energy satellite is purely composed of Hubbard-type incoherent weight, while in the unoccupied satellite both Hubbard and plasmonic contributions coexist at similar energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In the weaker correlated 4d2 SrMoO3 we observe a similar but less pro- nounced picture of a Hubbard satellite in the occupied part, and both plasmon and Hubbard satellites at sim- ilar energies in the unoccupied part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' These observation call for a reinvestigation of similar and other correlated materials and their satellite features, both theoretically and experimentally, as they are in particular relevant for plasmon-mediated applications in functional materials, where precise knowledge of the intensity and energy of plasmonic excitations is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The scheme that we have developed can be applied to a large class of mate- rials, and can aid the development of such applications by theoretically quantifying the plasmonic and Hubbard- type contributions in the spectral satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' While this work was being prepared for publication, a new joint experimental-theoretical work appeared on SrVO3[75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In that work, a purely plasmonic origin of the satellites is advocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' However, we believe this apparent contradiction also to be resolved by our work, since we show that the corresponding features stem indeed from the dielectric function, but are nevertheless of multiplet- like origin rather than long-range collective excitations, see discussion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The authors gratefully acknowledge fruitful discussions with F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Aryasetiawan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Gatti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Lichtenstein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Petocchi, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Werner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' B 105, 085102 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' [75] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Su, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Ruotsalainen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Nico- laou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Gatti, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Gloter, Advanced Optical Materials n/a, 2202415 (2023), https://onlinelibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='com/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='1002/adom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='202202415 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Diagnostics for plasmon satellites and Hubbard bands in transition metal oxides - Supplementary Material Steffen Backes1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='∗ Hong Jiang4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' and Silke Biermann3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='7 1Research Center for Advanced Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Komaba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Tokyo 153-8904,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Japan 2Center for Emergent Matter Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' RIKEN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Wako,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Saitama 351-0198,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Japan 3CPHT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' ´Ecole polytechnique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Institut Polytechnique de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 91120 Palaiseau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' France 4College of Chemistry and Molecular Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' China 5Coll`ege de France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 11 place Marcelin Berthelot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' France 6European Theoretical Spectroscopy Facility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 91128 Palaiseau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Europe and 7Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Division of Mathematical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Lund University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Professorsgatan 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 22363 Lund,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Sweden (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 2023) COMPUTATIONAL DETAILS For the GW+EDMFT cycle we start with a well con- verged DFT calculation from Wien2K [1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' and perform a constrained Random-Phase-Approximation and a G0W0 calculation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' as implemented in the FHI-gap Code[2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' to obtain the effective impurity interaction U(ω) and the Selfenergy ΣGW (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' iωn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' projected onto the t2g orbitals of either SrVO3 or SrMoO3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' using a maximally localized Wannier basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The cRPA and GW calculation were per- formed on a 8 × 8 × 8 k-mesh, and the resulting U(ω) and ΣGW (k, iωn) are then interpolated by cubic interpo- lation onto a dense 30 × 30 × 30 k-mesh, which serves as the input for the selfconsistent EDMFT calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The impurity model is solved within the continuous-time Quantum Monte-Carlo method in the hybridization ex- pansion, as implemented in the ALPS package[3] at in- verse temperature β = 40 1/eV, including the frequency dependence of the monopole term F0(ω) of the effective interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The analytical continuation from the imag- inary to the real frequency axis is done using a combi- nation of Pad´e approximants and the Maximum Entropy code from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' [4], where we added plasmonic peaks in the default model of exponentially decaying weight at multiples of the plasma frequency to faciliate proper con- tinuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The GW+EDMFT calculation was not done fully self- consistently, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' the nonlocal GW self-energy remained at the G0W0 level, and the effective interaction U(iω) was not updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' As discussed in the main text the resulting spectral function from this GW+EDMFT is very close to the G0W0 result, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This indicates that further self-consistency will have only minor effects and not qualitatively change the one-shot result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Therefore, we only considered the ’one-shot’ GW+EDMFT results, which are computationally less demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='8 3 2 1 0 1 2 3 4 5 6 Spectral function of SrVO3 Spectral function A(ω) Energy ω [eV] DFT DFT+DMFT G0W0 GW+EDMFT 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='15 6-4-2 0 2 4 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 1: The spectral function of SrVO3, comparing the De- sity Functional Theory (DFT), the DFT+DMFT, the G0W0 approximation and the GW+EDMFT result shown in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The G0W0 and the GW+EDMFT spectral func- tions are almost identical, indicating that a fully selfconsis- tent GW+EDMFT calculation will not significantly change the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' LOCAL G0W0 AND SATELLITES The results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 3 in the main text were ob- tained by employing a local G0W0 approximation: First, the polarization was calculated from the projected lo- cal DFT Green’s function for the t2g orbitals as Ploc = GlocGloc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' For the effective “bare” interaction the cRPA derived impurity interaction U(ω) was used, which was screened by the local polarization to obtain the screened local interaction Wloc = U[1−PlocU]−1 (shown in the in- set of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 3 in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Then the self-energy was obtained by the convolution of the local Green’s function and screened interaction as Σ = GlocWloc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='04908v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='str-el] 12 Jan 2023 2 0 1 2 3 4 5 4 2 0 2 4 U=2, V=0 A(ω) ω G0W0 exact 0 1 2 3 4 5 U=2, V=1 A(ω) G0W0 exact 0 1 2 3 4 5 U=2, V=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='3 A(ω) G0W0 exact FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 2: The ground state spectral function for a DMFT im- purity model with one bath site solved within the G0W0 ap- proximation for different values of the hybridization strength V at fixed interaction U = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Except in the strong interac- tion limit V → 0 G0W0 captures all qualitative features and correlation satellites, albeit overestimating their energetic po- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' To elucidate the appearance of the Hubbard correla- tion satellites within G0W0 one can consider a simplified ’linearized’ DMFT impurity problem [5–7] with only one bath site, which can be solved analytically H = Un↑n↓ + V � σ (c† σfσ + f † σcσ) − µ � σ nσ, (1) with c, c†/f, f † the impurity/bath annihilation and cre- ation operators, interaction U, hybridization strength V and chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Solving this model exactly and within the G0W0 approximation yields for the impurity self-energy Σexact(z) = U 2 8 � 1 z − 3V + 1 z + 3V � (2) ΣG0W0(z) = U 2 4a � 1 z − V (2a + 1) + 1 z + V (2a + 1) � , (3) with a = � 1 + U/(2V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Both results qualitatively agree, but G0W0 does not capture the correct position and weight of the two peaks in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' We note, however, that for U/V → 0 the peak position is correctly reproduced 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 4 Satellite weight U/V G0W0 exact 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 5 Satellite position G0W0 exact FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 3: The satellite positions and weights for the DMFT impurity model with one bath site as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2, as a function of the interaction strength U for fixed hybridization V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' in G0W0, while the weight differs by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2 we show the resulting spectral function for different values of the hybridization V for fixed interaction U = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Except in the strong interaction limit V → 0 G0W0 cap- tures all qualitative features and correlation satellites, albeit overestimating their energetic position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This can be seen explicitly in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='3, where we show the position and weights of the two different satellites emerging in the spectral function as a function of the interaction strength U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' G0W0 performs reasonably well for small values of the interaction strength U/V ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5, in particular for the bonding/antibonding states and satellite weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The emergence of the correlation satellites in G0W0, which in this setup appear as Hubbard satellites, is in fact not sur- prising, since a perturbative approach such as the G0W0 approximation is expected to become more accurate in the weakly correlated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' If Hubbard satellites in this regime are present, it is expected that G0W0 is able to qualitatively capture them, as shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' SEPARATION OF HUBBARD SATELLITES Previously it had been suggested to use the ener- getic separation of the unoccupied and occupied satel- lites, and their dependence on the bandwidth to distin- guish plasmon satellites from Hubbard satellites[8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In the atomic limit Hubbard satellites are separated by the static onsite interaction U(0), but for metallic systems a finite bandwidth enhances the separation of the Hub- 3 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='6 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='6 DMFT GW+EDMFT Hubbard satellite separation ∆Hubbard/D U/D SrVO3 DMFT FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 4: The separation of the Hubbard satellites in SrVO3 within a DMFT approach for different values of the inter- action U, relative to the non-interacting bandwidth D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' For small interactions the separation is much larger that the value of U, but becomes smaller for larger interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The non- interacting bandwidth for GW+EDMFT is given by the spec- tral function where the local self-energy has been removed (≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='25 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The error bars indicate the uncertainty from analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' bard satellites[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In systems far away from half-filling such as SrVO3, the position of the Hubbard satellites is further complicated by the breaking of particle-hole sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Already at the DMFT level, the separation of the Hubbard satellites ∆Hub ≈ 5 eV greatly surpasses the value of the static interaction U ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4 we show the dependence of the Hubbard satel- lite separation in SrVO3 on the interaction U, obtained from a DMFT calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Up to moderate correlations (for a filling of n = 1/6) the Hubbard satellite sep- aration exceeds the value of U by almost a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' On the other hand, for stronger interactions when the quasiparticle peak is close to vanishing the situa- tion reverses and the separation becomes smaller than U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' For the case of the GW+EDMFT result, which has a larger ’non-interacting’ bandwidth D due to the re- moval of the exchange-correlation potential and non-local GW contributions[11], the Hubbard satellite separation is larger than in standard DMFT, but is well within the expected range, considering the value of U(0)/D (see la- bel ’GW+EDMFT’ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Therefore, the Hubbard satellite separation alone is not a good quantifier to dis- tinguish plasmonic from Hubbard satellites, as in metallic systems and systems away from half-filling, the Hubbard satellite separation significantly differs from the usual ∆Hub ∼ U behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' ONE-ORBITAL MODEL FOR PLASMONIC AND HUBBARD SATELLITES In order to distinguish satellites of plasmonic origin and that arising from strong electronic correlations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' we consider a simple model that describes the coupling of electrons to a single bosonic degree of freedom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' namely the Hubbard-Holstein model H = − � ij tijc† iσcjσ + Ubare � i ni↑ni↓ + ω0 � i b† ibi + λ � i ni � b† i + bi � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' (4) where tij is the hopping amplitude,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Ubare is the local instantaneous Coulomb repulsion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' ω0 is the energy of the bosonic mode (plasma frequency),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' generated by the bosonic annihilation and creation operators b† i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The coupling strength between the electronic charge ni and the bosonic mode is given by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' Integrating out the bosonic degrees of freedom gives rise to an effective dynamical interaction U(ω), where the coupling to the bosonic mode is now encoded in the frequency dependence Re Ueff(ω) = Ubare − 2λ2 ω0 ω2 0 − ω2 (5) Im Ueff(ω) = −λ2π (δ(ω − ω0) − δ(ω + ω0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' (6) This model is an extension of the standard Hubbard model, and it has been shown that this model exhibits plasmonic replicas of the quasi-particle structure at mul- tiples of the plasma frequency ω0, that originate from plasmonic charge excitations[12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' We use extended dynamical mean-field theory (EDMFT) to solve the model at half-filling on the Bethe lattice with bandwidth W = 4 eV, inverse temperature β = 40 1/eV, and a single bosonic mode ω0 = 5 eV and λ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The spectral function and Selfenergy for different values of the static interaction U(ω) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' We observe the Hubbard satellite to com- pletely vanish when reducing the interaction by a con- stant shift, where the quasiparticle peak recovers a renor- malized semicircular form that corresponds to the origi- nal non-interacting dispersion on the Bethe lattice, renor- malized only by the transfer of spectral weight into plas- mon satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This behavior if even more evident in the imaginary part of the Selfenergy in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 5 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The low-energy peak responsible for the Hubbard satellite is completely suppressed for vanishing U(0), while the ef- fect on the plasmonic peaks is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' This confirms that the reduction of the interaction by a constant shift is only effecting the plasmon satellites to a minor degree, as they originate from the frequency dependence in U(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' On the other hand, the Hubbard satellites completely vanish at U(ω) = 0, allowing for a systematic classification of the nature of the satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='35 A(ω) λ=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5eV const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' ω0=5eV HUB PLAS1 PLAS2 U0=3eV U0=2eV U0=1eV U0=0eV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='5 3 0 2 4 6 8 10 12 Im Σ(ω) ω [eV] HUB PLAS1 PLAS2 U0=3eV U0=2eV U0=1eV U0=0eV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' 5: The spectral function and imaginary part of the Self- energy for the half-filled Hubbard model with a one boson screening mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' We show only positive energies since the spectrum is particle-hole symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The frequency depen- dence of the effective interaction U(ω) has been fixed with U(∞) − U(0) = 5 eV, but a static shift has been applied in order to reduce the strength of the interaction but keep the transfer of spectral weight due to the bosonic coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' The bosonic energy is ω0 = 5 eV, giving rise to plasmonic replica at multiples of ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' We observe a disappear- ance of the Hubbard satellite as U(0) approaches zero, while the plasmonic satellites stay mostly unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content=' ∗ steffen-backes@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='ecc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfIAw0/content/2301.04908v1.pdf'} +page_content='u-tokyo.' metadata={'source': 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0000000000000000000000000000000000000000..8ba1486cc740ce82a25843b64de802708765bbc9 Binary files /dev/null and b/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf differ diff --git a/I9AzT4oBgHgl3EQfVPxY/content/tmp_files/2301.01280v1.pdf.txt b/I9AzT4oBgHgl3EQfVPxY/content/tmp_files/2301.01280v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2b2337371a8cd478e388578dce953ab4ebc7fc0f --- /dev/null +++ b/I9AzT4oBgHgl3EQfVPxY/content/tmp_files/2301.01280v1.pdf.txt @@ -0,0 +1,573 @@ +arXiv:2301.01280v1 [math.NA] 3 Jan 2023 +An asymptotic formula for Aldaz-Kounchev-Render +operators on the hypercube +Ana-Maria Acua, Ioan Ra¸sab +aLucian Blaga University of Sibiu, Department of Mathematics and Informatics, Romania, +e-mail: anamaria.acu@ulbsibiu.ro +bTechnical University of Cluj-Napoca, Faculty of Automation and Computer Science, +Department of Mathematics, Str. Memorandumului nr. 28, 400114 Cluj-Napoca, Romania +e-mail: ioan.rasa@math.utcluj.ro +Abstract +We prove a version of a conjecture concerning the asymptotic behavior of the +Aldaz-Kounchev-Render operators on the hypercube. +Keywords: +Aldaz-Kounchev-Render operators; Bernstein operator; +Voronovskaja-type formula; tensor product. +2010 MSC: 41A36 +1. Introduction +Let B[1] +n : C[0, 1] → C[0, 1] be the classical Bernstein operator defined as +B[1] +n f(x) = +n +� +i=0 +f +� i +n +� +pn,i(x), +where pn,i(x) = +�n +i +� +xi(1 − x)n−i, x ∈ [0, 1]. +For a fixed j ∈ N, j ≥ 2 and for n ≥ j, Aldaz, Kounchev and Render [2] +introduced a polynomial operator B[1] +n,j : C[0, 1] → C[0, 1] that fixes e0 and ej, +investigated its approximation properties and gave applications to CAGD. The +operator is explicitly given by +B[1] +n,jf(x) = +n +� +k=0 +f +� +tj +n,k +� +pn,k(x), +where +tj +n,k = +� k(k − 1) . . . (k − j + 1) +n(n − 1) . . . (n − j + 1) +�1/j +. +The Voronovskaja-type formula for the sequence (B[1] +n,j)n≥1 was conjectured in +[4] and proved in [3], [5]. +Preprint submitted to ... +January 4, 2023 + +For f ∈ C([0, 1]2), the tensor product B[1] +n ⊗ B[1] +n is given by +B[2] +n f(x, y) := (B[1] +n ⊗ B[1] +n )f(x, y) = +n +� +k=0 +n +� +l=0 +f +�k +n, l +n +� +pn,k(x)pn,l(y). +(1.1) +Let B[1] +n,j : C[0, 1] → C[0, 1] be the AKR operator and (x, y) ∈ [0, 1]2. Then, for +f ∈ C([0, 1]2), the tensor product B[1] +n,j ⊗ B[1] +n,j is given by +B[2] +n,jf(x, y) := (B[1] +n,j ⊗ B[1] +n,j)f(x, y) += +n +� +k=0 +n +� +l=0 +f +� +tj +n,k, tj +n,l +� +pn,k(x)pn,l(y), (x, y) ∈ [0, 1]2. +(1.2) +A conjecture concerning the Voronovskaja-type formula for the sequence +(B[2] +n,j) was formulated in [1]. The aim of this paper is to prove a version of this +conjecture. +2. Proof of Conjecture +For the sake of conciseness we consider only the case j = 2, but obviously +the proof can be extended to arbitrary j. +Let k and n be integers, n ≥ 2, 0 ≤ k ≤ n. Define +R(n, k) := k +n − +� +k(k − 1) +n(n − 1) − 1 +2n + +k +2n2 . +It is elementary to prove that +R(n, 0) = − 1 +2n, +(2.1) +R(n, k) ≥ 0, k = 1, 2, . . . , n, +(2.2) +0 ≤ k +n − +� +k(k − 1) +n(n − 1) ≤ 1 +n. +(2.3) +Lemma 2.1. If 0 < x ≤ 1, then +lim +n→∞ n +n +� +k=1 +pn,k(x)R(n, k) = 0. +(2.4) +Proof. Let x ∈ (0, 1] and f ∈ C2[0, 1]. It is known (see [3], [5]) that +lim +n→∞ n(B[1] +n,2f(x) − f(x)) = x(1 − x) +2 +f ′′(x) − 1 − x +2 +f ′(x). +It is also well known that +lim +n→∞ n(B[1] +n f(x) − f(x)) = x(1 − x) +2 +f ′′(x). +2 + +It follows that +lim +n→∞ n +� +B[1] +n,2f(x) − B[1] +n f(x) +� += −1 − x +2 +f ′(x). +In particular, for the function f(t) = t, we get +lim +n→∞ n +n +� +k=1 +pn,k(x) +�� +k(k − 1) +n(n − 1) − k +n +� += −1 − x +2 +. +This can be written as +lim +n→∞ n +n +� +k=1 +pn,k(x) +� 1 +2n +� +1 − k +n +� ++ R(n, k) +� += 1 − x +2 +, +i.e., +1 +2 lim +n→∞ +n +� +k=1 +pn,k(x) +� +1 − k +n +� ++ lim +n→∞ n +n +� +k=1 +pn,k(x)R(n, k) = 1 − x +2 +. +(2.5) +Let us remark that +1 +2 lim +n→∞ +n +� +k=1 +pn,k(x) +� +1 − k +n +� += 1 +2 lim +n→∞ +� +B[1] +n (1 − t; x) − (1 − x)n� += 1 +2(1 − x). +Combined with (2.5) this leads to (2.4), and the proof is finished. +Theorem 2.1. Let 0 < x ≤ 1, 0 < y ≤ 1, f ∈ C2([0, 1]2). Then +lim +n→∞ n +� +B[2] +n,2f(x, y) − f(x, y) +� += x(1 − x) +2 +f ′′ +x2(x, y) + y(1 − y) +2 +f ′′ +y2(x, y) − 1 − x +2 +f ′ +x(x, y) − 1 − y +2 +f ′ +y(x, y). +(2.6) +Proof. First we have +n +� +B[2] +n,2f(x, y) − B[2] +n f(x, y) +� += n +n +� +k=0 +n +� +l=0 +pn,k(x)pn,l(y) +� +f +�� +k(k − 1) +n(n − 1), +� +l(l − 1) +n(n − 1) +� +− f +�k +n, l +n +�� += Enf(x, y) + Fnf(x, y) + Gnf(x, y), +3 + +where +Enf(x, y) := n +n +� +k=0 +n +� +l=0 +pn,k(x)pn,l(y) +�� +k(k − 1) +n(n − 1) − k +n +� +f ′ +x +�k +n, l +n +� +, +Fnf(x, y) := n +n +� +k=0 +n +� +l=0 +pn,k(x)pn,l(y) +�� +l(l − 1) +n(n − 1) − l +n +� +f ′ +y +�k +n, l +n +� +, +Gnf(x, y) := n +2 +n +� +k=0 +n +� +l=0 +pn,k(x)pn,l(y) + + + +�� +k(k − 1) +n(n − 1) − k +n +�2 +f ′′ +x2(ξ, η) ++ 2 +�� +k(k − 1) +n(n − 1) − k +n +� �� +l(l − 1) +n(n − 1) − l +n +� +f ′′ +xy(ξ, η) ++ +�� +l(l − 1) +n(n − 1) − l +n +�2 +f ′′ +y2(ξ, η) + + + , +for suitable (ξ, η) furnished by Taylor’s formula. Using (2.3) we see that +lim +n→∞ Gnf(x, y) = 0. +(2.7) +Moreover, +lim +n→∞ Enf(x, y) += − lim +n→∞ n +n +� +k=0 +n +� +l=0 +pn,k(x)pn,l(y) +� 1 +2n +� +1 − k +n +� ++ R(n, k) +� +f ′ +x +�k +n, l +n +� += −1 +2 lim +n→∞ +n +� +k=0 +n +� +l=0 +pn,k(x)pn,l(y) +� +1 − k +n +� +f ′ +x +�k +n, l +n +� +− lim +n→∞ n +n +� +k=0 +n +� +l=0 +pn,k(x)pn,l(y)R(n, k)f ′ +x +�k +n, l +n +� += −1 +2 lim +n→∞ B[2] +n ((1 − s)f ′ +x(s, t); (x, y)) +− lim +n→∞ n +n +� +k=1 +n +� +l=0 +pn,k(x)pn,l(y)R(n, k)f ′ +x +�k +n, l +n +� ++ lim +n→∞ n +n +� +l=0 +(1 − x)npn,l(y) 1 +2nf ′ +x +� +0, l +n +� +. +The first term equals −1 +2(1 − x)f ′ +x(x, y). +4 + +Moreover, using (2.2) we have +�����n +n +� +k=1 +n +� +l=0 +pn,k(x)pn,l(y)R(n, k)f ′ +x +�k +n, l +n +������ +≤ +n +� +l=0 +� +n +n +� +k=1 +pn,k(x)R(n, k)∥f ′ +x∥∞ +� +pn,l(y) += n +n +� +k=1 +pn,k(x)R(n, k)∥f ′ +x∥∞, +and (2.4) shows that the second term is zero. The third one is also zero, and so +lim +n→∞ Enf(x, y) = −1 − x +2 +f ′ +x(x, y). +(2.8) +Similarly, +lim +n→∞ Fnf(x, y) = −1 − y +2 +f ′ +y(x, y). +(2.9) +Now (2.7), (2.8), (2.9) yield +lim +n→∞ n +� +B[2] +n,2f(x, y) − B[2] +n f(x, y) +� += −1 − x +2 +f ′ +x(x, y) − 1 − y +2 +f ′ +y(x, y). +(2.10) +On the other hand, it is well known that +lim +n→∞ n(B[2] +n f(x, y) − f(x, y)) = x(1 − x) +2 +f ′′ +x2(x, y) + y(1 − y) +2 +f ′′ +y2(x, y). (2.11) +From (2.10) and (2.11) we get (2.6) and the theorem is proved. +References +[1] A.M. Acu, S. De Marchi, I. Ra¸sa, Aldaz–Kounchev–Render Operators and +Their Approximation Properties. Results Math 78, 21 (2023). +[2] J.M. Aldaz, O. Kounchev, H. Render, Shape preserving properties of gener- +alized Bernstein operators on extended Chebyshev spaces, Numer. Math., +2009, 114(1), 1–25. +[3] M. Birou, A proof of a conjecture about the asymptotic formula of a Bern- +stein type operator, Results Math. 72 (2017), 1129–1138. +[4] D. C´ardenas-Morales, P. Garrancho, I. Rasa, Asymptotic Formulae via a +Korovkin-Type Result, Abstr. Appl. Anal. Volume 2012, Article ID 217464, +12 pages. +[5] I. Gavrea, M. Ivan, Complete asymptotic expansions related to conjecture +on a Voronovskaja-type theorem, J. Math. Anal. Appl. 458 (1) (2018), +452-463. +5 + diff --git a/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf b/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c9c578905380cf6e76eb11022fead8565bd0399d --- /dev/null +++ b/IdAyT4oBgHgl3EQf5vqw/content/2301.00811v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:39ff57b9bc8bc2411c5581dc91d3fa55536f54ddd5ef4671f60dda676cbfa707 +size 871382 diff --git a/IdAyT4oBgHgl3EQf5vqw/vector_store/index.pkl b/IdAyT4oBgHgl3EQf5vqw/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2f222e7f9dfe54faf920a7a846ae56712fd600d2 --- /dev/null +++ b/IdAyT4oBgHgl3EQf5vqw/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3df2d3185c70d4357487e5ecd1e1845f36a2fb8e2870656bf42afe7a4caaeb54 +size 166270 diff --git a/IdAyT4oBgHgl3EQfTPeX/content/tmp_files/2301.00102v1.pdf.txt b/IdAyT4oBgHgl3EQfTPeX/content/tmp_files/2301.00102v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd16ad1b7dd04de2a2cb4bf11ca8e2b2f9ca9ce6 --- /dev/null +++ b/IdAyT4oBgHgl3EQfTPeX/content/tmp_files/2301.00102v1.pdf.txt @@ -0,0 +1,2548 @@ +Giant excitonic magneto-optical Faraday rotation in single semimagnetic +CdTe/Cd1−xMnxTe quantum ring +Kalpana Panneerselvam1 and Bhaskaran Muralidharan1 +1Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India∗ +(Dated: January 3, 2023) +Magnetic tuning of the bound exciton states and corresponding giant Zeeman splitting (GZS) +between σ+ and σ− excitonic transitions in CdTe/Cd1−xMnxTe quantum ring has been investigated +in the Faraday configuration for various concentrations of Mn2+ ions at different temperatures, +using the variational technique in the effective mass approximation. The sp-d exchange interaction +between the localized magnetic impurity ions and the delocalized charge carriers has been accounted +via mean-field theory with the inclusion of modified Brillouin function. The enhancement of the +GZS, and in turn, the effective Landé g-factor with the application of an external magnetic field +is strikingly manifested in type-I – type-II transition in the band structure, which has been well +explained by computing the overlap integral between the electron and hole, and the in-plane exciton +radius. This highlights the extraordinary magneto-optical properties, including the giant Faraday +rotation and associated Verdet constant, which have been calculated using a single oscillator model. +The oscillator strength and exciton lifetime have been estimated and are found to be larger than in +the bulk diluted magnetic semiconductors and quantum wells, reflecting stronger confinement inside +the quantum ring. The results show that the DMS-based quantum ring exhibits more extensive +Zeeman splitting and Faraday rotation which are a few orders of magnitude larger than in the +existing quantum systems and magneto-optical materials. +I. +Introduction +Diluted magnetic semiconductors (DMS) are of special +interest for the past few decades because of its unique +combination of semiconducting and magnetic properties. +The sp-d exchange interaction between the localized +magnetic moments of the dopant magnetic ions and +the spins of the charge carriers (electrons/holes) in +DMS [1–6] significantly alters the energy spectra of +the carriers, which greatly enhances the spin dependent +effects. +Moreover, these effects can be widely tuned +by the external magnetic field, temperature and the +concentration of magnetic ions to induce fascinating +magneto-optical (MO) and magneto-electrical properties. +Among all the exciting signatures of such exchange +interaction, the striking consequences are the giant +Zeeman splitting (GZS) [7–9] and the giant Faraday +rotation (GFR) [10, 11]. The Zeeman splitting between +the band sates with different spin components generates +the spin – polarization of the conducting carriers which +is exploited as spin aligners in spintronic devices [12, 13], +an anomalous magnetoresistance at low temperature +[14], and vastly amplifies the conversion of spin current +into an electrical current [15]. +Hence, +the DMS +have been an active area of research as an alternative +to the ferromagnetic metal contacts for the efficient +spin-injection into non-magnetic semiconductors, spin +detection, +and the realization of the spin-polarized +transport +in +semiconductor +structures, +which +have +substantial +industrial +applications +in +the +field +of +magnetoelectronics, spintronics, and solid-state quantum +∗ bm@ee.iitb.ac.in +computing [16–22]. +The concept of the FR (a solid +rotation of the plane of polarization of light travels in +a magnetized medium along the applied magnetic field) +manifests itself in various MO devices such as optical +isolators, Faraday rotators, and optical circulators for +high-speed optical communication systems [11, 23–25], +for which DMS act as potential MO materials. +The carrier localization and its transport properties +have been examined using DMS materials in various +nanostructured systems like quantum wells, wires, and +dots [26–29]. +Considerable attention paid to quantum +ring (QR)-based infrared photodetectors and lasers +[30, 31] in recent times due to its doubly-connected +topological nature has engendered an interest in us +to study how the radial and axial confinement of the +individual carriers and the exciton in semimagnetic QR +impact the sp-d exchange interaction in the Faraday +configuration +(magnetic +field +is +applied +along +the +direction of observation (z) and parallel to the light wave +vector). The strong sp-d coupling makes magnetic ions +mediate the influence of the magnetic field on the band +gap engineering by enhancing the Zeeman splitting of +the energy levels, which is strikingly manifested in type-I +- type-II transition in the band offset [32–35]. +Hence, +DMS extends its potential applications to optoelectronics +due to the possible tuning of the band states, which in +turn tune the emission wavelength widely over Near- to +Far-IR, creating a giant optical response. +This article aims to delineate the magnetic tuning +of exciton energy states due to the GZS between +σ+ and σ− spin components, in turn the effective +Landé +g-factor +for +various +mole +fractions +of +(x) +magnetic dopants at different temperature baths in +CdTe/Cd1−xMnxTe QR since CdMnTe has well served +as a potential MO material for the past few decades +arXiv:2301.00102v1 [cond-mat.mes-hall] 31 Dec 2022 + +2 +towards +optoelectronic +applications. +Various +MO +parameters +have +been +evaluated +such +as +oscillator +strength, radiative lifetime, and radiative decay rate. +The occurrence of type-I - type-II transition in a +single semimagnetic QR has been well explained in +the present communication by computing the overlap +integral +between +the +electron +and +hole, +and +also +estimating the in-plane exciton radius. Though QRs are +more flexible for experimental developments [30, 36–39] +due to the advances in fabrication procedures, and DMS +addresses the fundamental challenges in the spintronic +devices in its unique way, the possible integration of DMS +into QR structures has not been yet developed to unveil +the hidden mystery. Few theoretical studies have been +proposed on single and concentric double QRs doped +with transition metal ions focusing on the magnetic and +thermal properties [40–42] but not on the MO properties +of excitons which is of novel interest in the present work. +We later show a theoretical evaluation of the Verdet +constant of a remarkable MO phenomenon, the GFR, +using single oscillator model. The source for the larger +values of the Verdet constant in DMS has been traced +down to the GZS of the energy band states near the +band gap resonance. Although most research has focused +on achieving a larger Verdet constant with various MO +materials, especially Cd1−xMnxTe, these studies have +been restricted only to bulk DMS [10, 43–46] and +epitaxial heterostructures in the form of wells [47–51] +and dots [11, 25, 52]. +Therefore, investigating the +impact of sp-d exchange interaction on the GFR in +DMS heterostructures with various topologies, like QR, +would generate unprecedented interest in developing +high-quality epitaxial structures for various technological +applications. +In the following, sec. +II A discusses the theoretical +formalism using the variational technique in the effective +mass approximation to solve for single-particle (electron +and hole) energy states in a QR doped with 10% Mn2+ +ions at liquid helium temperature. The mean-field theory +with the modified Brillouin function to account for sp-d +exchange interaction is also explained in detail. Section +II B discusses the solution for the excitonic case and +delineates the occurrence of the GZS in nanostructures. +Section III A presents the results of the binding energy +of single-particle energy states under the influence of +the external magnetic field for various dimensions of +the QR. The results of temperature-dependent variation +of interband transition energy, binding energy of σ± +magneto-exciton, and various MO properties, including +GZS, and GFR in QR doped with various concentrations +of Mn2+ ions (x = 0.5%, 1%, 5%, 10%, and 20% of Mn2+ +ions) are presented and discussed in sec. +III B-III F. +Section IV elucidates the significance of the experimental +validation in QR based on DMS by comparing the present +results with those already reported for the bulk and QWs. +II. +Theoretical Model +A. +Single-particle energy states +This +section +discusses +the +energy +states +of +a +single-particle +(electron/hole) +confined +in +a +CdTe/Cd0.9Mn0.1Te QR. The schematic diagram of +the single quantum ring (SQR) is displayed in Fig. +1(a). +The Schrodinger equation and corresponding +Hamiltonian +for +the +single-particle +energy +states +subjected to magnetic flux in DMS QR is written in a +dimensionless form, considering the effective Rydberg +(R∗) as a unit of energy and effective Bohr radius (a∗ +B) +as a unit of length, and is given by, +ˆHe,hΨe,h = Ee,hΨe,h ; ˆHe,h += ˆH0e,h + ˆHsp−d +(1) +ˆH0e,h = −∇2 − +2 +re,h ++ VB(ρe,h, ze,h) + +γ2 ρ2 +e,h +4 ++ γ Lze, h +2 +(2) +where, +e, +and h represent the electron and hole, +respectively, and re,h = +� +ρ2 +e,h + z2 +e,h gives the electron +(hole) location from the donor (acceptor) impurity. +The strength of the magnetic field is parametrized +by γ = ℏωc +2R∗ , ωc is the cyclotron frequency, and γ = 1 +corresponds to ≈ 30Tesla (885 Tesla) for the donor +(acceptor) impurity. +Lze, h is the z component of the +orbital angular momentum of electron and hole along the +’z’ directon. The sp-d exchange interaction between the +electron (hole) and the localized Mn2+ magnetic dopants +is denoted by ˆHsp−d, which causes the Zeeman splitting +of both the conduction and valence band edges in the +semimagntic (Cd1−xMnxTe) barrier, and is written as +[4, 53], +ˆHsp−d = − +� +i +J(re,h − Ri) ˆSi · ˆse,h +(3) +‘J’ is the coupling constant for the exchange interaction +between the electron (hole) of spin ˆse (ˆsh) located at re +(rh) and the spin ˆSi of the Mn2+ ions located at sites +Ri. VB(ρe,h, ze,h) in Eq. (2) is the confining potential of +the SQR and is modeled by an abrupt square potential: +VB(ρe,h, ze,h) = +� +� +� +� +� +0 +R1 < ρe,h ≤ R2, +−d/2 < ze,h ≤ +d/2 +V0e,h +otherwise +(4) +V0e = 70% ∆EB +g , andV0h = 30% ∆EB +g +represent +the +potential band offset formed in the conduction and +valence band, respectively. +Tuning of the potential +barrier height, V0e and V0h with the applied field, B, is +possible due to the Zeeman splitting of the band edges, +which is well established through the formulation used + +3 +Figure 1. +Schematics: (a) Profile of the CdTe/Cd1−xMnxTe SQR. (b) Giant Zeeman splitting of excitonic energy levels in +CdTe/Cd1−xMnxTe and corresponding optical transitions (σ+, σ−, π). (c) The concept of Faraday rotation in DMS SQR. +for bulk DMS and can be written as [32, 54], +V e +m(B) = ± αexc se xeff N0 +� +SMn +z +(B) +� +and, +V hh +m (B) = ± 1 +3 βexc shh xeff N0 +� +SMn +z +(B) +� +(5) +Here, +xeff +is +the +effective +concentration +of +Mn2+ +ions, +and +αexc +(βexc) +is +the +exchange +constant +for the conduction band (valence band), +which is +parametrized for Cd1−xMnxTe as αexcN0 = 220meV +(βexcN0 = −880meV) with the atomic concentration of +Cd to be N0 = 1.4701 × 1022cm−3. +� +SMn +z +(B) +� +is the +thermal average of the spin projection of Mn2+ ions with +spin SMn = 5/2 along the direction of B, and is given by +the modified Brillouin function, BS, as follows [4, 53]: +� +SMn +z +(B) +� += +� +Ψw +e(h)|S0(xin)BS(y1)|Ψw +e(h) +� ++ +� +Ψb +e(h)|S0(xout)BS(y2)|Ψb +e(h) +� +(6) +BS(yj) = 2S + 1 +2S +coth 2S + 1 +2S +yj − 1 +2S coth yj +2S +yj = gMn µB SMn B +kB (T + TAF ) +(7) +gMn = 2.01 is the g-factor of the Mn2+ ion, µB is the +Bohr Magneton, kB is the Boltzmann constant and T is +the lattice temperature. For the DMS of arbitrary ‘x’, +the antiferromagnetic interactions between the nearest +neighbouring Mn2+ ions are included in the calculation +through the phenomenological fitting parameters, S0 and +TAF, whose numerical values are obtained from the +available experimental results [1]. In order to investigate +the variation of potential barrier with the magnetic +field, K. Navaneethakrishnan et al [55] have suggested +a formula that satisfactorily fits the experimental data +available for the Mn2+ compositions x = 0.07, 0.24, +and 0.3 with a maximum error of 5%. +Hence, the +same formula is adopted here, and the fitting equation +and corresponding discussion are given in Appendix +A. The variational ansatz of the ground state donor +(acceptor) impurity in a SQR includes the envelope +Bessel function φ(ρe,h, ϕe,h) along the radial confinement +and the envelope function f(ze,h) along the z-direction +and is defined as, +Ψe,h = N1s,e,h φ(ρe,h, ϕe,h) f(ze,h) exp−λ re,h +(8) +φ(ρe,h, ϕe,h) = +� +� +� +� +� +� +� +� +� +C1,e,h I0 (βe,h, ρe,h) , +ρe,h < R1 +C2,e,h J0 (αe,h, ρe,h) + +C3,e,h Y0 (αe,h, ρe,h) , +R1 ≤ ρe,h ≤ R2 +C4,e,h K0 (βe,h, ρe,h) , +ρe,h > R2 +(9) +f(ze,h) = +� +� +� +� +� +Be,h exp[ke,h ze,h], +ze,h < −d/2 +cos(κe,h ze,h), +−d/2 < ze,h < +d/2 +Be,h exp[−ke,h ze,h], +ze,h > d/2 +(10) +where, βe,h = +m∗ +b(V0e,h−Eρe,h) +ℏ2 +; +αe,h = +m∗ +wEρe,h +ℏ2 +ke,h = +m∗ +b(V0e,h−Eze,h) +ℏ2 +; +κe,h = +m∗ +wEze,h +ℏ2 +Eρe,h, Eze,h are the subband energy levels formed due to + +(a) +(c) +Analyzer +R2 +R +Faraday Rotator +P +CdTe +(b) +Cd1-xMnxTe +CdTe +Cd1-xMnxTe +Polarizer ++3A ++1/2 +Input +-3A-.... +beam +DMS +4 + Noax(Sz) +言No βx(Sz) +元 +3B +-B +-1/2 +e- Spin +Mn2+ +B ++1/2 +3B ++3/24 +the radial and axial confinement of the QR. The binding +energy of the donor (acceptor) impurity is obtained by +the equation, +EBe,h = Eρe,h + Eze,h + γ − ⟨He,h⟩min +(11) +B. +Magnetic tuning of exciton energy levels +Within +the +effective +mass +approximation, +the +Schrodinger equation and corresponding Hamiltonian of +the ground state electron-hole pair confined in a SQR is +written as, +ˆHexΨex = EexΨex +(12) +ˆHex = − 1 +ρ2e +∂2 +∂ϕ2 − 1 +ρ2 +h +∂2 +∂ϕ2 − µ(T) +m∗e(T) +� +∇ρ2 +e + ∇z2 +e +� +− µ(T) +m∗ +h(T) +� +∇ρ2 +h + ∇z2 +h +� ++ V (ρe, ze) + V (ρh, zh) +− +e2 +ϵ(T)|⃗re − ⃗rh| + i γ m∗ +h − m∗ +e +m∗ +h + m∗e +∂ +∂ϕ + γ2ρ2 +4 +(13) +Since the electron and hole move freely along the annular +part of the ring, their motion no longer depends on ϕe and +ϕh separately, but on the relative angular displacement +ϕ = ϕe - ϕh, and it should be treated with the reduced +effective mass ‘µ’ of the exciton. Moreover, the material +parameters, effective mass, and spatial dielectric constant +are considered as temperature-dependent. +The most +appropriate trial wavefunction of a ground state exciton +is written in a non-separable form due to correlated +electron-hole pair as, +Ψex(re, rh) = N1s φ(ρe, ρh) f(ze, zh) Ω(ρe, ρh, ze, zh, ϕ) +(14) +where, +Ω(ρe, ρh, ze, zh, ϕ) = e−λ reh +describes +the +correlation +between +the +electron +and +hole +which +depends +mainly +on +the +distance, +reh = +� +|(ρe − ρh)|2 + |(ze − zh)|2 +between +the +two, +whereas, +|(ρe − ρh)|2 +denotes the projection of the +distance +between +the +electron +and +hole +on +the +plane of the QR and is given by, +|(ρe − ρh)|2 += +(ρ2 +e + ρ2 +h − 2ρeρh cos(ϕ))1/2. +Invoking the variational +technique, the binding energy (EBex) and the interband +transition energy (ETex) of the exciton is computed +using the form, +EBex = Eρe + Eρh + Eze + Ezh + γ − ⟨Hex⟩min +ETex = Eg(T) + ⟨Hex⟩min +(15) +In the Faraday geometry, the magnetic moments of the +ensemble of Mn2+ ions with spin angular momentum +SMn = 5/2 are subjected to the sp-d exchange interaction +with the conduction band electrons of spin s = 1/2 and +the heavy hole valence band with angular momentum +J = 3/2. +This causes the heavy hole exciton splits +into two components with angular momentum +1 and +-1 which is composed of sz = −1/2, Jz = +3/2 and +sz = 1/2, Jz = −3/2, respectively. The GZS between the +excitonic energy levels exhibited in the nanostructures +is as similar as in bulk DMS, but with a difference +in the potential barrier height experienced by the two +different spin states. +The schematic diagrams which +explain the Zeeman splitting of the energy levels in +DMS nanostructures and its resultant GFR are depicted +in Fig. +1(b) and Fig. +1(c). +The applied magnetic +field increases and decreases the potential barrier for the +spin up and spin down states, respectively, and thereby +the corresponding confinement of both the electron and +heavy hole with sz = +1/2, Jz = +3/2, and sz = −1/2, +Jz = −3/2 becomes stronger and weaker, respectively, +inside the QR. Therefore, by magnetically tuning the +potential barrier, the energy levels of the exciton inside +the QR could also be tuned, manifesting itself in two +different excitonic transitions, namely σ+ and σ−. σ+ +corresponds to the transition between Jz = −3/2 (heavy +hole) and sz = −1/2 (electron) states, and σ− transition +involves Jz = +3/2 and sz = +1/2 states. The splitting +of the energy level corresponding to two transitions is +expressed by [1], +E± = ±1 +2xeffN0 (βexc − αexc) +� +SMn +z +(B) +� +(16) +The Zeeman splitting energy between the two excitonic +transitions and its relation to the magnetization, M, is +given by [1, 43], +∆Esp−d +z += E+ − E− += βexc − αexc +gMn µB +M +(17) +The effective Landé g- factor, geff, corresponding to the +exciton Zeeman splitting is calculated by, +∆Esp−d +z += geff µB B +(18) +III. +Results and Discussion +A. +Binding energy of the donor and acceptor +impurities +The numerical calculations have been done for various +outer radii (R2) and heights (d) of the QR, keeping the +inner radius of the ring constant to R1 = 50Å. The results +for the single-particle energy states in the absence of +magnetic field have already been reported in our previous +work [40]. +In the present communication, a series of +calculations have been performed for various strengths +of the magnetic field by varying the ring width from 30 +to 350Å, keeping ‘d’ constant to d = 10, 20, 40, 150, +and 200Å, and the results are compared with the results + +5 +Figure 2. Impurity binding energy vs ring width at different strengths of the magnetic field for various heights of the ring. +Upper Panel: Donor Impurity (a) γ = 0, (b) γ = 0.05, and (c) γ = 0.1. +Lower Panel: Acceptor Impurity (a) γ = 0, (b) +γ = 0.001, and (c) γ = 0.002. +reported in Ref [40]. Figure 2(a)-2(c) shows the binding +energy of the donor impurity as a function of ring width +(R2 − R1 = R) for different values of ‘d’ by placing the +impurity at the centre of the radial (ρi = (R1 + R2)/2) +and the axial direction (zi = 0) of the QR. +For a given height of the ring, the binding energy +shows a non-monotonic behaviour irrespective of the +applied magnetic field, when the ring width increases +from narrow to a bulk limit which is a signature +of any low dimensional system. +EB increases until +it reaches a maximum at a particular value of ‘R’, +which falls in the quasi 1-D region, and thereafter it +decreases monotonically due to the delocalization of +the wavefunction. +However, the decrease of binding +energy below the quasi 1-D region is attributed to +the penetration of the carrier wavefunction through +CdMnTe barrier. In the absence of magnetic field, this +non-monotonicity as a function of ‘R’ is visible only +for d = 40Å, and the maximum is seen at R = 30Å, +which is a favourable ring width at which the carrier +is strongly bound to the impurity and is shifted to +higher R values when B starts to increase. The impurity +states are susceptible to the external magnetic field, as +could be seen from a rapid fall of binding energy as +B increases from γ = 0.05 (1.5Tesla) to γ = 0.1 (3Tesla). +This is because, the magnetic field tremendously reduces +the potential barrier height as a result of the interaction +between the Mn2+ impurity ions residing in the barrier +and the localized carriers in the well region, which in turn +reduces the confinement, and the binding energy peaks +at a ring width shifted to higher values. +The above discussion is also applicable for the heavy +hole bound to an acceptor impurity (Fig. +2(d)-2(f)) +confined inside such QR, except for the following: The +acceptor binding energy is larger than the donor’s +binding energy irrespective of the magnetic field. This +is mainly due to the larger effective mass of the +hole (m∗ +h = 0.67) than the electron’s effective mass +(m∗ +e = 0.090). +Moreover, no conspicuous turnover is +seen in the binding energy as a function of R2 for the +system with (γ = 0.001 (0.8Tesla), γ = 0.002 (1.5Tesla)), +and without (γ = 0) the application of magnetic field. +Most importantly, in the absence of B, the promising +height of the ring, which gives maximum confinement for +the hole is d ≈ 20Å rather than d ≈ 40Å as in the case +of donor. +The same can be noticed by comparing Fig. ??(a) with +Fig. ??(b) in Appendix B, which depicts the variation +of binding energy as a function of ’d’ for different outer +radii ’R2’ and magnetic field. It is seen from both these +figures that the binding energy increases as ’d’ decreases +and shows a maximum at a specific value of ’d’ and +then drops slowly for further reduction of the height of +the ring, which is again a tell-tale hallmark of any low +dimensional system. As the outer radius increases, the +peak value of the binding energy is suppressed, and there +is a shift in the peak when the system is subjected to the +external magnetic field. + +180 +100 +ifi +60 +ui +iii +(i) d = 10A +(a) = 0 +(b) = 0.05 +160 +90 +(c) = 0. +iv +(ii) d = 20A +140 +80 +50 +ii +(ii) d = 40A +120 +70 +iy +(iv) d = 150A +V +100li +60 +40 +i +(v) d = 200A +50 +80 +(meV) +iv +V +i +60 +40 +30 +40 +30 + Energy ( +20 +20 +20 +180 +140 +100 +ii +ii +iii +160 +(d) = 0 +(e) = 0.001 +(f) = 0.002) +90 +120 +Binding I +140 +1 +80 +i +100 +120 +70 +100 +iv +80 +iv +60 +iv +80 +50 +60 +60 +V +40 +40 +40 +30 +20 +20 +20 +50 +100 +150 +200 +250 +300 +350 +50 +100　150　200 +250 +300 +350 +50 +100 +150 +200 +250 +300 +350 +Ring Width (A)6 +Figure 3. +Interband transition energy as a function of +magnetic field for σ+ and σ− exciton at different temperatures +for various dopant concentrations. (a) x = 0.005, (b) x = 0.01, +(c) x = 0.05, (d) x = 0.1, and (e) x = 0.2. +B. +Magnetic-field induced excitonic interband +transition energy +Magnetic field dependence of the PL transition energy +(ET) for both σ+ and σ− polarization for various +concentrations of Mn2+ ions at different temperatures +is computed for ring dimensions R = 80Å, d = 20Å, +which is approximately equals to the effective Bohr radius +of the exciton, and the results are displayed in Fig. +3. The transition energy increases with the increasing +concentration of Mn2+ ion because the bandgap (Eg) +is directly proportional to the latter and decreases +with increasing temperature since Eg has a negative +temperature coefficient. The calculation using the above +theoretical model shows that at B = 0, the PL is +unpolarized, i.e., the energies of σ± magneto-exciton are +degenerate. However, the applied magnetic field breaks +the degeneracy and causes the PL to split into left (σ−) +and right (σ+) circularly polarized. This is indicated by +a monotonic shift of ET towards low and high energies +about zero field energy in Fig. 3, and the PL gets resolved +into two branches of exciton doublet corresponding to +σ+ and σ− polarization, respectively. +The reason for +this is attributed to the fact that the applied magnetic +field influences the potential barrier height of the two +different spin components in a unique way owing to the +sp-d exchange interaction, as discussed in sec. II B. +The variation of ET with B for the QR doped with +low Mn2+ concentration (x ≤ 0.01) is different from the +QR doped with high concentration. Instead of showing +a rapid fall with the magnetic field as seen for higher +concentration, the σ+ transition energy mimics the σ− +transition, as shown in Fig. +3(a) and 3(b), indicating +a change of the PL emission from right circular to left +circular polarization. +A vivid picture of this unusual +behaviour for low ’x’ has been well explained in a +DMS QD by Kai Chang et al [29], which is ascribed +to the tuning of the effective g-factor to zero with the +increasing field when the order of Zeeman splitting due +to sp-d exchange interaction is comparable to the order +of intrinsic Zeeman splitting. The sign of the former is +opposite to the latter. The presence of crossing between +σ+ and σ− transition energy in Ref [29] is missing +here for x = 0.01 because the data has been plotted +for the extended range of magnetic fields, including +the type-II region in Ref [29], whereas it is limited to +the type-I region in the present work. +Typically, the +order of intrinsic Zeeman splitting is much smaller than +the energy level splitting induced by the sp-d exchange +interaction; hence, the former is neglected in the present +calculation. +Figure 4. Zeeman shift related to zero field magneto-exciton +energy vs magnetic field for various dopant concentrations at +different temperatures. (a) x = 0.005, (b) x = 0.01, (c) x = +0.05, (d) x = 0.1, and (e) x = 0.2. +C. +Zeeman shift and Zeeman splitting of the +exciton energy levels +Figure +4 +plots +the +magnetic +field +dependence +of the exciton transition energy as Zeeman shifts +(Eex(B) − Eex(B = 0)) relative to the zero-field exciton + +1.60 +1.60 +4.2K +1.58 +1.58 +77K +.56 +(a) +(b) +1.56 +(e) +1.54 +x = 0.005 +x = 0.01 +1.54 + の+ +1.52 +1.52 + Transition +.50 + Transition +.50 +300K +8.0 +0.1 +0.2 +0.3 +0.4 +0.0 +0.2 +0.4 +0.6 +1.68 +.76 +(d) x = 0.1 +Interband +.72 +1.64 +Interband +1.68 +1.60 +.64 +1.56 +1.60 +.52 +1.56 +1.48 +x = 0.05 +c +1.52 +0.5 +1.5 +2.0 +0.2 +2.5 +0.4 +0.6 +0.8 +1.0 +1.2 +0.0 +1.0 +3.0 +0.0 +1.80 +1.75 +1.70 +1.65 +1.60 +1.55 += 0.2 +(e) +X +1.50 +2 +4 +6 +8 +10 +12 +14 +0 +Magnetic Field (Tesla)(a) x = 0.005 +(b) x = 0.01 +8 +6 +6 +4 +4.2K +4 +2 +77K +0 +(meV) +2 +300K +2 +:::::::::6 + Shift +4 +0.0 +0.1 +0.2 +0.3 +0.4 +0.0 +0.1 +0.2 +0.5 +0.6 +0.3 +0.4 +60 +(c) x = 0.05 +(d) x = 0.1 +40 +20 +20 +0 +0 +-20 +-20 +-40 +-40 +60 +1.5 +2.0 +2.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0.0 +0.5 +1.0 +3.0 +60 +x=0.2 +e +40 +20 +Shift +-20 +-40 +-60 +eman +-80 +-100 +-120 +140 +0 +2 +4 +6 +8 +10 +12 +14 +Magnetic Field (Tesla)7 +energy for both the transitions, which is also described by +Eq. (16). It is noted from figure that the shift increases +with increasing magnetic field for both the transition, +but it shows a positive and negative increment for σ− +and σ+ which corresponds to the blue and redshift in +the interband transition energy (Fig. +3), respectively. +Interestingly, one could observe the symmetric Zeeman +splitting about the zero-field energy for the QR doped +with 5% and 10% of Mn2+ ions, but for all other dopant +concentrations (0.5%, 1%, and 20%), the splitting seems +to be asymmetric. +On the quantitative footing, the Zeeman splitting +energy, ∆Esp−d +z +, plotted in Fig. +5(a) for x = 0.05 +is +described +as +the +energy +difference +between +the +two excitonic transitions under B and is determined +from the data plotted in Fig. +4 as given in Eq. +(17). +The numerical data for x = 0.005, 0.1, and +0.2 are depicted in Fig. +??(a)-??(c) in Appendix +C. Two opposing parameters, the magnetic field, and +temperature, interplay with each other to determine the +Zeeman splitting. +The magnetic field suppresses the +Mn2+ spin fluctuations by aligning the randomly oriented +Mn2+ spins along the field direction, indicating a state of +magnetic ordering, thereby increasing ⟨Sz⟩ and causing +the GZS. Contradicting this, the spin fluctuations are +large enough to keep the state with a maximum entropy +at elevated temperatures, and eventually, the Zeeman +splitting is abated by the thermal energy. It is interesting +Figure 5. (a) Zeeman splitting (∆Esp−d +z +) of the 1s exciton +at various temperatures for x = 0.05. +(b) Temperature +dependent magnetization (M) calculated using modified +Brillouin function for x = 0.05. (c) Magnetic susceptibility +(χ) data for various dopant concentrations as a function of +temperature. +to note from Fig. 5(a) and (??) that ∆Esp−d +z +increases +with the dopant concentration up to x = 0.05, and +thereafter it starts decreasing. +This is because the +Zeeman splitting is proportional to the effective dopant +concentration ‘xeff’ as given in Eq. (16), and the latter +increases with ‘x’ and shows a maximum at a particular +concentration. +Henceforth, it starts to move downhill +because of the antiferromagnetic interactions between +the nearest neighbouring magnetic ions, which cancels +the spins of the corresponding pairs and reduces the +effective contribution to the thermal average of the spin +polarization of Mn2+ ions, ⟨Sz⟩. +In the framework +of the theoretical model described in sec. +II B, the +effect of temperature on the absolute value of effective +Landé g-factor has been calculated, and the results are +tabulated in Table-I for various ‘x’. +The enhanced +g-factor in the applied magnetic field directly evidences +the strength of the Zeeman splitting, and the reduction +in g-factor for increased temperature is also anticipated +because of the depolarization of Mn2+ spins. +Table I. Effective Landé g-factor at T = 4.2K and +300K. +Effective Landé g-factor (geff) at B = 0.2Tesla +x +T = 4.2K +T = 300K +0.005 +102.375 +186.451 +0.01 +-383.885 +-313.412 +0.05 +-928.099 +-820.949 +0.1 +-681.671 +-601.917 +0.2 +-202.032 +-195.048 +Figure 5(b) shows the magnetization (M) vs magnetic +field curves at different temperatures for x = 0.05. +Magnetization increases with the magnetic field since it +enhances ⟨Sz⟩, showing a linear dependence on magnetic +field, which is an expected paramagnetic behaviour and +does not saturate even at higher magnetic fields, B>10T. +Similarly, +it decreases when temperature augments, +restricting all the magnetic moments from aligning along +the field direction by intensifying the spin fluctuations +so that the magnetic ordering is hampered. As already +discussed, when QR is populated with more magnetic +ions, the spin-spin interaction becomes more robust, +which results in a quenching of magnetization for high +‘x’ because of the lower value of ⟨Sz⟩ (Fig. ??(d)-??(f) +in Appendix C). To corroborate these results, +the +temperature dependent magnetic susceptibility, χ, data +for various concentrations keeping the magnetic field +constant to B = 0.2Tesla is illustrated in Fig. 5(c). +D. +Binding energy of σ± magneto-exciton +The variation of binding energy is plotted in Fig. 6(a) +for x = 0.05. +The trend of the binding energy for +both σ− and σ+ polarization concerning the magnetic + +10 +20 +(a) x = 0.05 +(b) x = 0.05 +0 +-10 +5 +-20 +-30 +10 +-40 +.50 +5 +-60 +0 +70 +1.2 +0.0 +0.2 +0.4 +0.6 +0.8 1.0 1.2 0.0 0.2 0.4 +0.6 +0.8 +1.0 +Magnetic Field (Tesla) +14 +(c) B = 0.2Tesla +12 +10 +8 +x=0.05 +-x=0.005 +6 +x=0.2 +x=0.1 +X +4 +(emu : +2 +0 +50 +100 +150 +200 +250 +300 +Temperature (K)8 +Figure 6. +(a) Binding energy of σ± magneto-exciton vs magnetic field for x = 0.05 at different temperatures. +In-plane +electron–hole distance corresponding to (b) σ+ and (c) σ− exciton vs magnetic field for various dopant concentrations at T = +4.2K. (d) Schematic explaining the overlap integral between the electron and hole under various strengths of magnetic field. +(i) B = 0, (ii) 0 < B < Bc, and (iii) B > Bc. +field is as same as the trend followed by the interband +transition energy, and this behaviour persists at different +temperatures and concentrations. +Nevertheless, for +σ+ polarization, there is a rapid decrease of binding +energy with the magnetic field as compared to the +steady increase for σ− polarization. This can be better +understood from the schematic in Fig. +6(b), which +explains how the applied magnetic field modifies the +electron-hole overlap inside a SQR. +At B = 0, the location of both the electron and +hole is in the same CdTe layer (Fig. 6(b-i)). Zeeman +splitting of the energy levels in the valence band is +highly sensitive to the applied field, which is not the +case with the conduction band. +This is because the +band offset formed in the conduction band is generally +larger than the valence band offset since 80% of the +bandgap difference falls in the former. +Moreover, +the absolute value of the exchange constant, which +represents the strength of the exchange interaction, is +larger for the heavy hole (|βexcN0 = 880meV|) than +for the electron (|αexcN0 = 220meV|). +Therefore the +electron with sz = −1/2 in the conduction band would +forever be confined in the non-magnetic CdTe layer +itself irrespective of the strength of the applied field +as its potential band offset is sufficiently larger than +the order of magnetic splitting (Fig. 6(b-ii)). However, +the potential barrier for the heavy hole with Jz = −3/2 +is tremendously reduced with the magnetic field, and +it encounters a flat band situation at critical field +value, beyond which the system undergoes a type-I - +type-II transition (Fig. +6(b-iii)). +As a result, the +electron remains in the CdTe layer, but the hole moves +towards the heterostructure interface and finally to the +CdMnTe layer. +Hence, the exciton will no longer be +spatially direct; rather, it becomes spatially indirect, +which reduces the overlap between the electron and hole, +whereby spin-down exciton states have reduced binding +energy. +To justify this discussion, the in-plane exciton radius, +Reh, the average distance between the electron and hole +in the plane of the QR, has been calculated and is plotted +in Fig. +6(c) and 6(d). +As anticipated, the monotonic +increase and decrease of Reh could be seen for σ+ and σ− +polarization, respectively, for all x. Moreover, the 3D plot +of the probability distribution of spin-down electrons and +holes (|Ψ|2) along ρ and z-directions of the QR, and the +density plot of the single-particle distribution depicted in +Fig. 7 helps to understand the effect of magnetic field on + +80 +B>B。 +x = 0.05 +(b) +B< B。 +(a) +75 +(ii) +(iii) +(i) +70 +4.2K +65 +60 +77K +electron +55 +300K +Type - I +Type - II +50 +45 +9+ +Heavy hol +40 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +80 (c) x= 0.005 +(d) +48 +T = 4.2K +T = 4.2K +45 ++ Polarization +70 +x= 0.01 + - Polarization +42 +39 +60 +36 +x = 0.005 +50 +33 +R +R +30 +40 +0.05 +x= 0.1 +27 +X= +=0.01 +X +30 +0.05 +24 +0.1 +X= 0.2 +21 +X02 +20 +1.2 +0.4 +0.8 +1.2 +1.6 +2.0 +2.4 +3.2 +0.0 +0.4 +0.8 +1.6 +2.0 +2.4 +2.8 +3.2 +0.0 +2.8 +Magnetic Field (Tesla)9 +Figure 7. Left panel: 3D-plot for the probability distribution of electrons and holes along axial, and radial direction. +Right panel: Density plot of the probability distribution of single-particle states along both radial and axial direction. The +data has been plotted for (a) B = 0, and (b) B = 1Tesla. +the carrier confinement inside the QR. Obviously, |Ψ|2 +is larger for zero magnetic field as one can compare the +order of magnitude between B = 0 and B = 1Tesla. +E. +Oscillator strength, radiative linewidth and +radiative lifetime of magneto-exciton +To gain further insight into the σ+ and σ− transition +and related radiative properties, the investigation of +oscillator strength (OS), radiative decay rate (RDR), and +radiative lifetime (RLT) have been performed, and the +results are delineated. +The expression for the exciton +oscillator strength follows [48, 56, 57], +f± = +EP +2 ET ± +����� +� +∞ +−∞ +I dρe dze +����� +2 +|Ω(0)|2 +(19) +where, +EP = 2.1eV for CdTe, +represents the Kane +energy. The OS mainly depends on the overlap integral +‘I’ between the electron and hole envelope wavefunctions: +I = | +� +∞ +−∞ N1s φe (ρe) φh(ρe) fe(ze) fh(ze) dρe dze|2, +and +Ω(0) denotes the probability of finding the electron and +hole at the same position. The oscillator strength per +unit area is proportional to the effective Bohr radius as, +F± = +1 +a∗ +B +2 f±. +Exciton radiative lifetime, ‘τ’ (radiative +decay rate, ‘Γ = 1/τ’) can be related to OS according to +[58, 59], +τ = 2πϵ0m0c3ℏ2 +ne2ET ±f± +(20) +Here, the fundamental physical constants have their +usual meaning and ‘n’ represents the refractive index of +the material CdTe. +The evolution of the oscillator strength as a function +of magnetic field solely depends on the spatial overlap +Figure 8. Temperature dependent variation of (a) overlap +integral, (b) oscillator strength, (c) radiative lifetime, and +(d) radiative decay rate with magnetic field for σ+ and σ− +transitions for x = 0.05. +between the electron and hole wave functions, which has +been depicted for x = 0.05 in Fig. 8. +The numerical +data pertaining to all other concentrations are depicted +in Fig. ?? and ?? in Appendix D. The applied magnetic +field increases the overlap between the electron and hole +ground states for σ− polarization, indicating larger OS +due to the increase of potential barrier height. +As +expected for the σ+ polarization, the OS sensitively +depends on B, which diminishes the excitonic effect by +spatially separating the electron and hole as explained in +sec. III D and thereby weakens the corresponding optical +transition. +The overlap integral is suppressed by the +thermal energy kBT, whereby the oscillator strength is +also abated. The overlap integral increases as the dopant +concentration increases due to the increased potential + +0.08 +0.06 +0.06 +0.04 +0.04 +70.50.03 +0.2 +/0.5 +- +0.04 +0.02 +2.0 +0.02 +2.0 +0.1 +0.02 +0.01 +0.0 +0.00 +1.5 +0.00 +/0.0 zh +0.00 +1.5 +0.0zh +0.5 +0.0 +0.5 +0.0 +0.5 +1.0ph +- +0.5 +1.0.ph +I +0.0 +1.0 +/0.5 +0.0 +1.0 +/0.5 +ze +pe +1.5 +- +ze +1.5 +-0.5 +(a) B = 0 +2.0 +0.5 +(b) B = 1 Tesla +0.5 +0.0 +I +0.5 +2.0 +0.0 +I +0.2 +0.2 +0.2 +0.2 +0.07 +0.04 +0.04 +0.06 +0.125 +0.1 +0.1 +0.1 +0.1 +0.05 +0.100 +0.03 +c0'0 +pe. +ph 0.0 +pe. +ph +0.0 +0.04 +0.075 +0.0 +0.0 +0.02 + 0.02 +0.03 +0.050 +0.1 +0.02 +-0.1 +0.1 +0.1 +0.025 +0.01 + 0.01 +0.01 +0.2 +0.2 +I +-0.2 +-0.2 +0.8 +1.0 +1.2 +1.4 +1.6 +0.8 +1.0 +1.2 +1.4 +1.6 +0.8 +1.0 +1.2 +1.4 +1.6 +0.8 +1.0 +1.2 +1.4 +1.6 +zh +ze +zh +ze0.9 +(a) +(b) +x= 0.05 +9 +0.8 +8 +7 +.6 +0. +６ +５ +0. +4 +0. +23 +T=4K +0.2 +2 +-T=77K +0.1 +1 +T=300K +.6 +C +(d) +3. +5 +3.0 +1.0 +0.8 +2.0 +0.6 +0.4 +1.0 +·····★ +0.2 +0.5 +0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 +Magnetic field (Tesla)10 +Figure 9. +Faraday rotation angle (ΘF) as a function of magnetic field for (a) x = 0.005, and (b) x = 0.05 at different +temperatures for a fixed photon energy of E=1.5eV. Verdet constant as a function of photon energy for various dopant +concentrations at (c) T = 4.2K, (d) T = 77K, and (e) T = 300K for a fixed strength of magnetic field, B = 0.2Tesla. +barrier height (Fig. ??). +Figure 8(c) and 8(d) shows the radiative lifetime and +radiative decay rate as a function of magnetic field +for 5% of Mn2+ concentration. +The RLT of exciton +increases with increasing B for σ+ polarization, which is +accompanied by a decrease in RDR. The exciton lifetime +is found to decrease from 5.04ns to 0.38ns when the +concentration of Mn2+ ion increases from x = 0.005 to x += 0.2 at low temperature and at B = 0, where radiative +recombination dominates (Fig. ??). The RDR, which +characterizes the decay of photon emitted by the exciton, +shows its maximum only for B = 0, which elucidates the +probability of finding an electron and hole at the same +position (re = rh) is more prominent in the absence of +magnetic field. +F. +GFR in semimagnetic SQR +The Faraday rotation (Fig. +1(c), also known as +an optical analogue of the Hall Effect, results from +a difference in refractive indices of the left and right +circularly polarized light after traveling through a +magnetized medium with a length ‘l’. +If there is a +difference in absorption of two circularly polarized light, +then the polarization vector will change its helicity from +linear to elliptical. However, in the present work, only +the changes in refractive indices are considered in the +numerical calculations and the latter is ignored. +The +phase difference in velocity between the two circularly +polarized components is expressed through the FR angle +as [43], +ΘF = ∆φ +2 += El +2ℏc (n− − n+) +(21) +Here, n− and n+ denote the refractive indices of the left +and right circular polarized light, and E is the incident +photon’s energy. +As aforementioned, the FR in DMS +alloys is a giant one due to the large Zeeman splitting of +the energy levels as a result of sp-d exchange interaction, +which has been computed using the single oscillator +model as preferred in the work of Bartholomew et al., +After performing a series of calculations, ΘF achieves the +final form as [43], +ΘF = +√F0l +2ℏc +�βexc − αexc +gMn µB +� +M 1 +E0 +y2 +(1 − y2)3/2 ; y = E +E0 +(22) +Here, F0 is a constant that involves the oscillator +strength, and E0 is the ground state interband transition +energy at the fundamental energy gap at zero magnetic +field. +The +angle +is +directly +proportional +to +the +GZS through the term ∆E = βexc−αexc +gMnµB M. +The Verdet +constant is written as the Faraday rotation per unit + +15.0 +(a) x = 0.005 +Eph=1.50eV +12.5 +-3 +(c) +10.0 +-6 +-9 +7.5 +-12 +5.0 +-15 +4.2K +-18 +T=300K +(/3 +2.5 +77K +-21F +B = 0.2 Tesla +0.0 +-24 +-3 +(degree) +→ 300K +T = 4.2K +-27 +-6 +-2.5 +(a) PA ++ x=0.005 +(a) PA +0.0 +0.1 +0.2 +0.3 +0.4 +-9 +-x=0.01 +25 +-12 ++ X=0.05 +(b) x = 0.05 +-15 +0 +-3 +(P) ++ x=0.1 +-18 +-25 +.6 +★ x=0.2 +-21 +-50 +-9 +-24 +-12 +-75 +1.2 +1.3 +1.6 +1.7 +-15 +Photon energy (eV) +-100 +-18 +-125 +-21 +-150 +-24 +T = 77K +-175 +-27 +0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +MagneticField (Tesla) +Photon energy (eV)11 +magnetic field per unit length, which is defined as [43], +Vd(E) = ΘF +Bl += +√F0 +2ℏc +�βexc − αexc +gMn µB +� ∂M +∂H +1 +E0 +y2 +(1 − y2)3/2 +(23) +Figure 9(a) and 9(b) depicts ΘF for the DMS QR in +the dilute regime (x=0.005) and doped with arbitrary +’x’ (x = 0.05) at a fixed photon energy of 1.5eV. It +is noted from figure that the rotation angle increases +with the increasing magnetic field since the applied field +enhances the Zeeman splitting and decreases with the +temperature since the magnetization is suppressed at +elevated temperatures. +The variation of the Verdet +constant with the incident photon energies at different +temperatures for a fixed magnetic field of B = 0.2Tesla +is plotted in Fig. 9(c)-9(e). The Verdet constant shows a +sharp increase whenever the band gap resonance occurs +(when the energy of the incident photon approaches +the absorption edge of the material), and the photon +energy, at which the Verdet constant shows a rapid +enhancement, shifted to higher energies for the heavily +doped QR because the absorption edge increases as the +concentration of Mn2+ ions increases. Though the single +oscillator model yields gratifying results, in which the +behaviour of E0 at Γ point has been crudely modelled as +constant at all temperatures, the success of using this +in QR could not be verified due to a lack of reliable +experimental data. +IV. +Concluding Remarks +Probing the single-particle and exciton energy states +in an applied magnetic field has been studied in +semimagnetic QR, and the theoretical investigation of +tuning related MO properties has been attempted. It is +found that the doubly-connected topological structures +like QR provide robust confinement for the carriers +compared to single-connected topological QDs [60]. The +difference in the behaviour of magneto-exciton energies +between the QR doped with low and high Mn2+ ion +concentrations has been explained in detail at various +temperatures. +The results show pronounced excitonic +Zeeman splitting for low ‘x and T’ than high ‘x and +T’, where the possibilities for the manganese ions to +form antiferromagnetic pairs in the latter case are +maximized. +Among all the concentrations discussed +here, x = 0.05 exhibits larger Zeeman splitting with the +absolute value of effective g-factor, geff = 928 (Table I, +and the corresponding Verdet constant is as large as +-15 degree/Tesla Å, and the latter is 104 − 106 orders of +magnitude larger than in bulk Cd1−xMnxTe [10, 43, 46], +thin films [61–63], and is 102 orders larger than in QWs +[48, 50], superlattices [49, 64] as reported in the previous +studies. This elucidates the importance of DMS-based +QR in MO devices operating at a wavelength shorter +than 1µm than already existing MO materials, such +as Yttrium Iron Garnet (YIG) and Terbium Aluminum +Garnet (TAG), organic molecules, conjugated polymers +[25, 43, 65]. +Moreover, +the low-temperature exciton lifetime is +715ps, +whereas it is ≈ 100ps in QWs doped with +25% Mn2+ +ion concentration [66]. +The study of +exciton lifetime in semimagnetic quantum systems is +impressive since it affects the optical properties and the +magnetization dynamics of the concerned systems to a +greater extent. The exciton lifetime in DMS determines +the formation of bound magnetic polaron (BMP) [67, 68] +or exciton magnetic polaron (EMP) [69], which causes +spontaneous ferromagnetic ordering even in the absence +of an external magnetic field due to the strong sp-d +exchange interaction. +Since the recombination limits +the exciton lifetime, it interrupts the EMP formation +before the polaron reaches its stable state. +If the +exciton does not decay during the process of EMP +formation, then the EMP would reach its equilibrium +state, which is accompanied by a decrease of exciton +energy and provides an additional localization for the +carriers. +The unique capability of manipulating the +MO properties at the nanoscale in external magnetic +fields and effective magnetic switching of the spins makes +DMS-based QR a judicious choice among promising +candidates for applications in future spintronic and +optoelectronic devices. +The reliability of the results +obtained using the single oscillator model could not be +verified due to the missing experimental data, but it is +believed to be improved using the multi oscillator model +as adopted in [44]. Since the low path length and the +modest magnetic field yields a high Verdet constant, +theoretical demonstration of generating larger FR and +higher Verdet constant in DMS QRs would incite interest +in preparing high-quality QR heterostructures based on +DMS. +With the unrivaled ability to modulate the magnetic +excitonic transitions and thereby the optical activity +of the materials at the nanoscale for a broader energy +spectrum with various mole fractions of Mn2+ ions, +the diluted magnetic semiconductors have potential +applications in spin-photonic and spin-electronic devices. + +12 +Appendix +A. +Fitting equation for magnetic field variation of +potential barrier height +The fitting equation to represent the changes in the +potential barrier height as a function of magnetic field is +given by [4, 53, 55], +∆EB +g = ∆E0 +g +ηe,h eζe,h γ − 1 +ηe,h − 1 +(A1) +∆EB +g +and +∆E0 +g +denotes +the +band +gap +difference +between +the +well +CdTe +layer +and +the +barrier +Cd1−xMnxTe +layer +in +the +presence +and +absence +of +applied +magnetic +field, +respectively. +The +composition +and +temperature +dependence +of +the +latter is written as: +∆E0 +g(x, T) = ∆Eg(x) + T C(x); +C(x) +is +known +as +temperature +coefficient +and +∆Eg(x) = Eg(Cd1−xMnxTe) − Eg(CdTe) = 1.587x. +Temperature +dependent +band +gap +of +the +CdTe +material +is +given +by, +Eg(CdTe) = Eg(0) − δT2 +T+ξ +; +Eg(0) = 1.606 eV, δ = 4.37 × 10−4 eV/K, ξ = 126.8K. +ηe,h = eζe,h γ0 +is +chosen +with +a +fitting +parameter +ζe(ζh) = 0.5(−0.5), and γ0 is a critical magnetic field +at which the barrier completely vanishes. +The critical +magnetic field γ0 in Tesla for different magnetic dopant +compositions is given for donor (acceptor) impurity as +γ0 = A enx with A = 0.734 and n = 19.082 (A = - 0.57 +and n = 16.706). +B. +Binding energy comparison between donor and +acceptor impurities +Figure B.1. Binding energy vs ’d’ at different strengths of +the magnetic field for various ring widths. (a) donor impurity, +and (b) acceptor impurity. +C. +Zeeman splitting and magnetization in dilute +and high ’x’ regimes +Figure C.1. Zeeman splitting of the excitonic energy levels +(∆Esp−d +z +), and temperature dependent magnetization (M) in +(a), (d) dilute and (b), (c), (e), (f) high x regimes. +D. +Magneto-optical data for dilute and high ’x’ +regimes +Temperature +dependent +variation +of +the +overlap +integral, OS, and RLT, RDR as a function of magnetic +field for 0.5%, 10%, and 20% dopant concentrations are +plotted in Fig. ?? and ??, respectively. +Figure D.1. Overlap integral (I) and oscillator strength (f±) +as a function of magnetic field for σ+ and σ− transition at +different temperatures in (a), (d) dilute and (b), (c), (e), (f) +high x regimes. +Figure D.2. Radiative lifetime (τ) and radiative decay rate +(Γ) as a function of magnetic field for σ+ and σ− transition +at different temperatures in (a), (d) dilute and (b), (c), (e), +(f) high x regimes. + +180 +180 +Acceptor +Donor +-=0 +----= 0.002 +-=0 +160 +160 +(meV) +(a) +(mel +(a) (d) R2 = 80A +140 +a +140 +(b) (e) R2 = 150A +120 +120 +(c) (f) R2 = 300A +100 +b +100 +80 +80 +60 +C +60 +40 +(f) +40 +(a) +b +20 +20 +60 +20 +80 100 120 140 160 180 200 +20　40 +40 +60 +80 100 120 140 160 180 200 +Height of the Ring (d) (A)8 +(meV) +(a) +x= 0.005 +0 +0 +-30 +6 +20 +→4.2K +60 +5 +-40 +←77K +90 +-60 +3 +-120 +300K +c) +(b) +-80 +S +-150 +1 +E +x = 0.2 +-180 +x= 0.1 +-100 +0.2 +8 10 12 14 +0.0 +0.1 +0.3 +0.4 +0.0 0.5 1.0 1.5 2.0 2.5 3.0 +6 +30 +50 +0.0 +(f) +(d) +(e) +ram) +25 +40 +-0.5 +20 +30 +60 +15 +-1.0 +(emu/: +20 +10 +-1.5 +10 +5 +M +0246 810121416 +8.0 +0.1 +0.2 +0.3( +0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 +Magnetic field (Tesla)1.4 +4 +0.12 +(a) x = 0.005 +(b) x = 0.1 +1.2 +C +0 +1.0 +1.0 +0.10 +T=4K +0.8 +0.8 +0.6 +T=77K +0.08 +0.6 +T=300K +0.4 +0.4 +0.2 +(c) x = 0.2 +0.06. +0.1 +0.0 +0.2 +0.3 +0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 +6 +10 +2 +4 +8 +12 +14 +1.4 +12 +10 +10 +1.2 +8 +8 +(1010, +6 +6 +4 +4 +2 +(e) +(e) +(d) +0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 +0 +2 +6 +8 +10121416 +0.0 +0.1 +0.2 +0.3 +Magnetic Field (Tesla)18 +(b) +(a) +(c) +6.5 +1.5 +15 += 0.005 +6.0 +x= 0.1 +x= 0.2 +X +12 +1.2 +(ns) +S +5.5 +T=4K +9 +n +0.9 +15.0 +·T=77K +6 +T4.5 +=300K +0.6 +T +3 +4.0 +0 +0.3 +0.0 +0.1 +0.2 +0.3 +0.4 +4 +6 +810 12 1416 +0.0 0.5 1.0 1.5 2.0 2.5 3.0 +0 +2 +0.18 +··达···齿 +1.8 +1.8 +(ueV) +1.5 +1.5 +0.16 +1.2 +1.2 +0.9 +0.9 + 0.12 +0.6 +0.6 +0.3 +(f) +(p)j +(e) +0.10 +0.3 +0.0 +0.0 +0.1 +0.2 +0.3 +0 +2 +0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 +4 +6 +8 +10 +1416 +Magnetic Field (Tesla)13 +[1] J. +Gaj, +R. +Planel, +and +G. +Fishman, +Solid +State +Communications 88, 927 (1993). +[2] V. 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Bayer, Physical Review B 95, 155303 +(2017). + diff --git a/IdAyT4oBgHgl3EQfTPeX/content/tmp_files/load_file.txt b/IdAyT4oBgHgl3EQfTPeX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e2eb2a6db674d14ff9acfacfcea6004ba4ea1478 --- /dev/null +++ b/IdAyT4oBgHgl3EQfTPeX/content/tmp_files/load_file.txt @@ -0,0 +1,1442 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf,len=1441 +page_content='Giant excitonic magneto-optical Faraday rotation in single semimagnetic CdTe/Cd1−xMnxTe quantum ring Kalpana Panneerselvam1 and Bhaskaran Muralidharan1 1Department of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Indian Institute of Technology Bombay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Powai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Mumbai-400076,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' India∗ (Dated: January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 2023) Magnetic tuning of the bound exciton states and corresponding giant Zeeman splitting (GZS) between σ+ and σ− excitonic transitions in CdTe/Cd1−xMnxTe quantum ring has been investigated in the Faraday configuration for various concentrations of Mn2+ ions at different temperatures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' using the variational technique in the effective mass approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The sp-d exchange interaction between the localized magnetic impurity ions and the delocalized charge carriers has been accounted via mean-field theory with the inclusion of modified Brillouin function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The enhancement of the GZS, and in turn, the effective Landé g-factor with the application of an external magnetic field is strikingly manifested in type-I – type-II transition in the band structure, which has been well explained by computing the overlap integral between the electron and hole, and the in-plane exciton radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' This highlights the extraordinary magneto-optical properties, including the giant Faraday rotation and associated Verdet constant, which have been calculated using a single oscillator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The oscillator strength and exciton lifetime have been estimated and are found to be larger than in the bulk diluted magnetic semiconductors and quantum wells, reflecting stronger confinement inside the quantum ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The results show that the DMS-based quantum ring exhibits more extensive Zeeman splitting and Faraday rotation which are a few orders of magnitude larger than in the existing quantum systems and magneto-optical materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Introduction Diluted magnetic semiconductors (DMS) are of special interest for the past few decades because of its unique combination of semiconducting and magnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The sp-d exchange interaction between the localized magnetic moments of the dopant magnetic ions and the spins of the charge carriers (electrons/holes) in DMS [1–6] significantly alters the energy spectra of the carriers, which greatly enhances the spin dependent effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Moreover, these effects can be widely tuned by the external magnetic field, temperature and the concentration of magnetic ions to induce fascinating magneto-optical (MO) and magneto-electrical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Among all the exciting signatures of such exchange interaction, the striking consequences are the giant Zeeman splitting (GZS) [7–9] and the giant Faraday rotation (GFR) [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The Zeeman splitting between the band sates with different spin components generates the spin – polarization of the conducting carriers which is exploited as spin aligners in spintronic devices [12, 13], an anomalous magnetoresistance at low temperature [14], and vastly amplifies the conversion of spin current into an electrical current [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Hence, the DMS have been an active area of research as an alternative to the ferromagnetic metal contacts for the efficient spin-injection into non-magnetic semiconductors, spin detection, and the realization of the spin-polarized transport in semiconductor structures, which have substantial industrial applications in the field of magnetoelectronics, spintronics, and solid-state quantum ∗ bm@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='in computing [16–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The concept of the FR (a solid rotation of the plane of polarization of light travels in a magnetized medium along the applied magnetic field) manifests itself in various MO devices such as optical isolators, Faraday rotators, and optical circulators for high-speed optical communication systems [11, 23–25], for which DMS act as potential MO materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The carrier localization and its transport properties have been examined using DMS materials in various nanostructured systems like quantum wells, wires, and dots [26–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Considerable attention paid to quantum ring (QR)-based infrared photodetectors and lasers [30, 31] in recent times due to its doubly-connected topological nature has engendered an interest in us to study how the radial and axial confinement of the individual carriers and the exciton in semimagnetic QR impact the sp-d exchange interaction in the Faraday configuration (magnetic field is applied along the direction of observation (z) and parallel to the light wave vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The strong sp-d coupling makes magnetic ions mediate the influence of the magnetic field on the band gap engineering by enhancing the Zeeman splitting of the energy levels, which is strikingly manifested in type-I type-II transition in the band offset [32–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Hence, DMS extends its potential applications to optoelectronics due to the possible tuning of the band states, which in turn tune the emission wavelength widely over Near- to Far-IR, creating a giant optical response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' This article aims to delineate the magnetic tuning of exciton energy states due to the GZS between σ+ and σ− spin components, in turn the effective Landé g-factor for various mole fractions of (x) magnetic dopants at different temperature baths in CdTe/Cd1−xMnxTe QR since CdMnTe has well served as a potential MO material for the past few decades arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='00102v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='mes-hall] 31 Dec 2022 2 towards optoelectronic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Various MO parameters have been evaluated such as oscillator strength, radiative lifetime, and radiative decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The occurrence of type-I - type-II transition in a single semimagnetic QR has been well explained in the present communication by computing the overlap integral between the electron and hole, and also estimating the in-plane exciton radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Though QRs are more flexible for experimental developments [30, 36–39] due to the advances in fabrication procedures, and DMS addresses the fundamental challenges in the spintronic devices in its unique way, the possible integration of DMS into QR structures has not been yet developed to unveil the hidden mystery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Few theoretical studies have been proposed on single and concentric double QRs doped with transition metal ions focusing on the magnetic and thermal properties [40–42] but not on the MO properties of excitons which is of novel interest in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' We later show a theoretical evaluation of the Verdet constant of a remarkable MO phenomenon, the GFR, using single oscillator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The source for the larger values of the Verdet constant in DMS has been traced down to the GZS of the energy band states near the band gap resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Although most research has focused on achieving a larger Verdet constant with various MO materials, especially Cd1−xMnxTe, these studies have been restricted only to bulk DMS [10, 43–46] and epitaxial heterostructures in the form of wells [47–51] and dots [11, 25, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Therefore, investigating the impact of sp-d exchange interaction on the GFR in DMS heterostructures with various topologies, like QR, would generate unprecedented interest in developing high-quality epitaxial structures for various technological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' In the following, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' II A discusses the theoretical formalism using the variational technique in the effective mass approximation to solve for single-particle (electron and hole) energy states in a QR doped with 10% Mn2+ ions at liquid helium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The mean-field theory with the modified Brillouin function to account for sp-d exchange interaction is also explained in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Section II B discusses the solution for the excitonic case and delineates the occurrence of the GZS in nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Section III A presents the results of the binding energy of single-particle energy states under the influence of the external magnetic field for various dimensions of the QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The results of temperature-dependent variation of interband transition energy, binding energy of σ± magneto-exciton, and various MO properties, including GZS, and GFR in QR doped with various concentrations of Mn2+ ions (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5%, 1%, 5%, 10%, and 20% of Mn2+ ions) are presented and discussed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' III B-III F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Section IV elucidates the significance of the experimental validation in QR based on DMS by comparing the present results with those already reported for the bulk and QWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Theoretical Model A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Single-particle energy states This section discusses the energy states of a single-particle (electron/hole) confined in a CdTe/Cd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='9Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1Te QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The schematic diagram of the single quantum ring (SQR) is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The Schrodinger equation and corresponding Hamiltonian for the single-particle energy states subjected to magnetic flux in DMS QR is written in a dimensionless form, considering the effective Rydberg (R∗) as a unit of energy and effective Bohr radius (a∗ B) as a unit of length, and is given by, ˆHe,hΨe,h = Ee,hΨe,h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ˆHe,h = ˆH0e,h + ˆHsp−d (1) ˆH0e,h = −∇2 − 2 re,h + VB(ρe,h, ze,h) + γ2 ρ2 e,h 4 + γ Lze, h 2 (2) where, e, and h represent the electron and hole, respectively, and re,h = � ρ2 e,h + z2 e,h gives the electron (hole) location from the donor (acceptor) impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The strength of the magnetic field is parametrized by γ = ℏωc 2R∗ , ωc is the cyclotron frequency, and γ = 1 corresponds to ≈ 30Tesla (885 Tesla) for the donor (acceptor) impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Lze, h is the z component of the orbital angular momentum of electron and hole along the ’z’ directon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The sp-d exchange interaction between the electron (hole) and the localized Mn2+ magnetic dopants is denoted by ˆHsp−d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' which causes the Zeeman splitting of both the conduction and valence band edges in the semimagntic (Cd1−xMnxTe) barrier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' and is written as [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 53],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ˆHsp−d = − � i J(re,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h − Ri) ˆSi · ˆse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h (3) ‘J’ is the coupling constant for the exchange interaction between the electron (hole) of spin ˆse (ˆsh) located at re (rh) and the spin ˆSi of the Mn2+ ions located at sites Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' VB(ρe,h, ze,h) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (2) is the confining potential of the SQR and is modeled by an abrupt square potential: VB(ρe,h, ze,h) = � � � � � 0 R1 < ρe,h ≤ R2, −d/2 < ze,h ≤ +d/2 V0e,h otherwise (4) V0e = 70% ∆EB g , andV0h = 30% ∆EB g represent the potential band offset formed in the conduction and valence band, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Tuning of the potential barrier height, V0e and V0h with the applied field, B, is possible due to the Zeeman splitting of the band edges, which is well established through the formulation used 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Schematics: (a) Profile of the CdTe/Cd1−xMnxTe SQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (b) Giant Zeeman splitting of excitonic energy levels in CdTe/Cd1−xMnxTe and corresponding optical transitions (σ+, σ−, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (c) The concept of Faraday rotation in DMS SQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' for bulk DMS and can be written as [32, 54], V e m(B) = ± αexc se xeff N0 � SMn z (B) � and, V hh m (B) = ± 1 3 βexc shh xeff N0 � SMn z (B) � (5) Here, xeff is the effective concentration of Mn2+ ions, and αexc (βexc) is the exchange constant for the conduction band (valence band), which is parametrized for Cd1−xMnxTe as αexcN0 = 220meV (βexcN0 = −880meV) with the atomic concentration of Cd to be N0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4701 × 1022cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' � SMn z (B) � is the thermal average of the spin projection of Mn2+ ions with spin SMn = 5/2 along the direction of B, and is given by the modified Brillouin function, BS, as follows [4, 53]: � SMn z (B) � = � Ψw e(h)|S0(xin)BS(y1)|Ψw e(h) � + � Ψb e(h)|S0(xout)BS(y2)|Ψb e(h) � (6) BS(yj) = 2S + 1 2S coth 2S + 1 2S yj − 1 2S coth yj 2S yj = gMn µB SMn B kB (T + TAF ) (7) gMn = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='01 is the g-factor of the Mn2+ ion, µB is the Bohr Magneton, kB is the Boltzmann constant and T is the lattice temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' For the DMS of arbitrary ‘x’, the antiferromagnetic interactions between the nearest neighbouring Mn2+ ions are included in the calculation through the phenomenological fitting parameters, S0 and TAF, whose numerical values are obtained from the available experimental results [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' In order to investigate the variation of potential barrier with the magnetic field, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Navaneethakrishnan et al [55] have suggested a formula that satisfactorily fits the experimental data available for the Mn2+ compositions x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='24, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='3 with a maximum error of 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Hence, the same formula is adopted here, and the fitting equation and corresponding discussion are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The variational ansatz of the ground state donor (acceptor) impurity in a SQR includes the envelope Bessel function φ(ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ϕe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h) along the radial confinement and the envelope function f(ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h) along the z-direction and is defined as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Ψe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h = N1s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h φ(ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ϕe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h) f(ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h) exp−λ re,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h (8) φ(ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ϕe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h) = � � � � � � � � � C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h I0 (βe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h < R1 C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h J0 (αe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h) + C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h Y0 (αe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' R1 ≤ ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h ≤ R2 C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h K0 (βe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h > R2 (9) f(ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h) = � � � � � Be,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h exp[ke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h < −d/2 cos(κe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' −d/2 < ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h < +d/2 Be,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h exp[−ke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h > d/2 (10) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' βe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h = m∗ b(V0e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h−Eρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='h) ℏ2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' αe,h = m∗ wEρe,h ℏ2 ke,h = m∗ b(V0e,h−Eze,h) ℏ2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' κe,h = m∗ wEze,h ℏ2 Eρe,h, Eze,h are the subband energy levels formed due to (a) (c) Analyzer R2 R Faraday Rotator P CdTe (b) Cd1-xMnxTe CdTe Cd1-xMnxTe Polarizer +3A +1/2 Input 3A-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='. beam DMS 4 Noax(Sz) 言No βx(Sz) 元 3B B 1/2 e- Spin Mn2+ B +1/2 3B +3/24 the radial and axial confinement of the QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The binding energy of the donor (acceptor) impurity is obtained by the equation, EBe,h = Eρe,h + Eze,h + γ − ⟨He,h⟩min (11) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Magnetic tuning of exciton energy levels Within the effective mass approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' the Schrodinger equation and corresponding Hamiltonian of the ground state electron-hole pair confined in a SQR is written as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ˆHexΨex = EexΨex (12) ˆHex = − 1 ρ2e ∂2 ∂ϕ2 − 1 ρ2 h ∂2 ∂ϕ2 − µ(T) m∗e(T) � ∇ρ2 e + ∇z2 e � − µ(T) m∗ h(T) � ∇ρ2 h + ∇z2 h � + V (ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ze) + V (ρh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' zh) − e2 ϵ(T)|⃗re − ⃗rh| + i γ m∗ h − m∗ e m∗ h + m∗e ∂ ∂ϕ + γ2ρ2 4 (13) Since the electron and hole move freely along the annular part of the ring,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' their motion no longer depends on ϕe and ϕh separately,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' but on the relative angular displacement ϕ = ϕe - ϕh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' and it should be treated with the reduced effective mass ‘µ’ of the exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Moreover, the material parameters, effective mass, and spatial dielectric constant are considered as temperature-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The most appropriate trial wavefunction of a ground state exciton is written in a non-separable form due to correlated electron-hole pair as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Ψex(re,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' rh) = N1s φ(ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ρh) f(ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' zh) Ω(ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ρh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' zh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ϕ) (14) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Ω(ρe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ρh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' zh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ϕ) = e−λ reh describes the correlation between the electron and hole which depends mainly on the distance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' reh = � |(ρe − ρh)|2 + |(ze − zh)|2 between the two,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' whereas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' |(ρe − ρh)|2 denotes the projection of the distance between the electron and hole on the plane of the QR and is given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' |(ρe − ρh)|2 = (ρ2 e + ρ2 h − 2ρeρh cos(ϕ))1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Invoking the variational technique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' the binding energy (EBex) and the interband transition energy (ETex) of the exciton is computed using the form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' EBex = Eρe + Eρh + Eze + Ezh + γ − ⟨Hex⟩min ETex = Eg(T) + ⟨Hex⟩min (15) In the Faraday geometry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' the magnetic moments of the ensemble of Mn2+ ions with spin angular momentum SMn = 5/2 are subjected to the sp-d exchange interaction with the conduction band electrons of spin s = 1/2 and the heavy hole valence band with angular momentum J = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' This causes the heavy hole exciton splits into two components with angular momentum +1 and 1 which is composed of sz = −1/2, Jz = +3/2 and sz = 1/2, Jz = −3/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The GZS between the excitonic energy levels exhibited in the nanostructures is as similar as in bulk DMS, but with a difference in the potential barrier height experienced by the two different spin states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The schematic diagrams which explain the Zeeman splitting of the energy levels in DMS nanostructures and its resultant GFR are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 1(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The applied magnetic field increases and decreases the potential barrier for the spin up and spin down states, respectively, and thereby the corresponding confinement of both the electron and heavy hole with sz = +1/2, Jz = +3/2, and sz = −1/2, Jz = −3/2 becomes stronger and weaker, respectively, inside the QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Therefore, by magnetically tuning the potential barrier, the energy levels of the exciton inside the QR could also be tuned, manifesting itself in two different excitonic transitions, namely σ+ and σ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' σ+ corresponds to the transition between Jz = −3/2 (heavy hole) and sz = −1/2 (electron) states, and σ− transition involves Jz = +3/2 and sz = +1/2 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The splitting of the energy level corresponding to two transitions is expressed by [1], E± = ±1 2xeffN0 (βexc − αexc) � SMn z (B) � (16) The Zeeman splitting energy between the two excitonic transitions and its relation to the magnetization, M, is given by [1, 43], ∆Esp−d z = E+ − E− = βexc − αexc gMn µB M (17) The effective Landé g- factor, geff, corresponding to the exciton Zeeman splitting is calculated by, ∆Esp−d z = geff µB B (18) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Results and Discussion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Binding energy of the donor and acceptor impurities The numerical calculations have been done for various outer radii (R2) and heights (d) of the QR, keeping the inner radius of the ring constant to R1 = 50Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The results for the single-particle energy states in the absence of magnetic field have already been reported in our previous work [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' In the present communication, a series of calculations have been performed for various strengths of the magnetic field by varying the ring width from 30 to 350Å, keeping ‘d’ constant to d = 10, 20, 40, 150, and 200Å, and the results are compared with the results 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Impurity binding energy vs ring width at different strengths of the magnetic field for various heights of the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Upper Panel: Donor Impurity (a) γ = 0, (b) γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05, and (c) γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Lower Panel: Acceptor Impurity (a) γ = 0, (b) γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='001, and (c) γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' reported in Ref [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Figure 2(a)-2(c) shows the binding energy of the donor impurity as a function of ring width (R2 − R1 = R) for different values of ‘d’ by placing the impurity at the centre of the radial (ρi = (R1 + R2)/2) and the axial direction (zi = 0) of the QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' For a given height of the ring, the binding energy shows a non-monotonic behaviour irrespective of the applied magnetic field, when the ring width increases from narrow to a bulk limit which is a signature of any low dimensional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' EB increases until it reaches a maximum at a particular value of ‘R’, which falls in the quasi 1-D region, and thereafter it decreases monotonically due to the delocalization of the wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' However, the decrease of binding energy below the quasi 1-D region is attributed to the penetration of the carrier wavefunction through CdMnTe barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' In the absence of magnetic field, this non-monotonicity as a function of ‘R’ is visible only for d = 40Å, and the maximum is seen at R = 30Å, which is a favourable ring width at which the carrier is strongly bound to the impurity and is shifted to higher R values when B starts to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The impurity states are susceptible to the external magnetic field, as could be seen from a rapid fall of binding energy as B increases from γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5Tesla) to γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 (3Tesla).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' This is because, the magnetic field tremendously reduces the potential barrier height as a result of the interaction between the Mn2+ impurity ions residing in the barrier and the localized carriers in the well region, which in turn reduces the confinement, and the binding energy peaks at a ring width shifted to higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The above discussion is also applicable for the heavy hole bound to an acceptor impurity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 2(d)-2(f)) confined inside such QR, except for the following: The acceptor binding energy is larger than the donor’s binding energy irrespective of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' This is mainly due to the larger effective mass of the hole (m∗ h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='67) than the electron’s effective mass (m∗ e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='090).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Moreover, no conspicuous turnover is seen in the binding energy as a function of R2 for the system with (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='8Tesla), γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='002 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5Tesla)), and without (γ = 0) the application of magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Most importantly, in the absence of B, the promising height of the ring, which gives maximum confinement for the hole is d ≈ 20Å rather than d ≈ 40Å as in the case of donor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The same can be noticed by comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (a) with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (b) in Appendix B, which depicts the variation of binding energy as a function of ’d’ for different outer radii ’R2’ and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' It is seen from both these figures that the binding energy increases as ’d’ decreases and shows a maximum at a specific value of ’d’ and then drops slowly for further reduction of the height of the ring, which is again a tell-tale hallmark of any low dimensional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' As the outer radius increases, the peak value of the binding energy is suppressed, and there is a shift in the peak when the system is subjected to the external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 180 100 ifi 60 ui iii (i) d = 10A (a) = 0 (b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 160 90 (c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' iv (ii) d = 20A 140 80 50 ii (ii) d = 40A 120 70 iy (iv) d = 150A V 100li 60 40 i (v) d = 200A 50 80 (meV) iv V i 60 40 30 40 30 Energy ( 20 20 20 180 140 100 ii ii iii 160 (d) = 0 (e) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='001 (f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='002) 90 120 Binding I 140 1 80 i 100 120 70 100 iv 80 iv 60 iv 80 50 60 60 V 40 40 40 30 20 20 20 50 100 150 200 250 300 350 50 100 150 200 250 300 350 50 100 150 200 250 300 350 Ring Width (A)6 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Interband transition energy as a function of magnetic field for σ+ and σ− exciton at different temperatures for various dopant concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (a) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005, (b) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='01, (c) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05, (d) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1, and (e) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Magnetic-field induced excitonic interband transition energy Magnetic field dependence of the PL transition energy (ET) for both σ+ and σ− polarization for various concentrations of Mn2+ ions at different temperatures is computed for ring dimensions R = 80Å, d = 20Å, which is approximately equals to the effective Bohr radius of the exciton, and the results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The transition energy increases with the increasing concentration of Mn2+ ion because the bandgap (Eg) is directly proportional to the latter and decreases with increasing temperature since Eg has a negative temperature coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The calculation using the above theoretical model shows that at B = 0, the PL is unpolarized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=', the energies of σ± magneto-exciton are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' However, the applied magnetic field breaks the degeneracy and causes the PL to split into left (σ−) and right (σ+) circularly polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' This is indicated by a monotonic shift of ET towards low and high energies about zero field energy in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 3, and the PL gets resolved into two branches of exciton doublet corresponding to σ+ and σ− polarization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The reason for this is attributed to the fact that the applied magnetic field influences the potential barrier height of the two different spin components in a unique way owing to the sp-d exchange interaction, as discussed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The variation of ET with B for the QR doped with low Mn2+ concentration (x ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='01) is different from the QR doped with high concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Instead of showing a rapid fall with the magnetic field as seen for higher concentration, the σ+ transition energy mimics the σ− transition, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 3(a) and 3(b), indicating a change of the PL emission from right circular to left circular polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' A vivid picture of this unusual behaviour for low ’x’ has been well explained in a DMS QD by Kai Chang et al [29], which is ascribed to the tuning of the effective g-factor to zero with the increasing field when the order of Zeeman splitting due to sp-d exchange interaction is comparable to the order of intrinsic Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The sign of the former is opposite to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The presence of crossing between σ+ and σ− transition energy in Ref [29] is missing here for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='01 because the data has been plotted for the extended range of magnetic fields, including the type-II region in Ref [29], whereas it is limited to the type-I region in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Typically, the order of intrinsic Zeeman splitting is much smaller than the energy level splitting induced by the sp-d exchange interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' hence, the former is neglected in the present calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Zeeman shift related to zero field magneto-exciton energy vs magnetic field for various dopant concentrations at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (a) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005, (b) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='01, (c) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05, (d) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1, and (e) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Zeeman shift and Zeeman splitting of the exciton energy levels Figure 4 plots the magnetic field dependence of the exciton transition energy as Zeeman shifts (Eex(B) − Eex(B = 0)) relative to the zero-field exciton 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='58 77K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='56 (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='56 (e) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='54 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='54 の+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='52 Transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='50 Transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='50 300K 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='68 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='76 (d) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 Interband .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='64 Interband 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='48 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='55 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 (e) X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='50 2 4 6 8 10 12 14 0 Magnetic Field (Tesla)(a) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005 (b) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='01 8 6 6 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2K 4 2 77K 0 (meV) 2 300K 2 :::::::::6 Shift 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 60 (c) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 (d) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 40 20 20 0 0 20 20 40 40 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 60 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 e 40 20 Shift 20 40 60 eman 80 100 120 140 0 2 4 6 8 10 12 14 Magnetic Field (Tesla)7 energy for both the transitions, which is also described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' It is noted from figure that the shift increases with increasing magnetic field for both the transition, but it shows a positive and negative increment for σ− and σ+ which corresponds to the blue and redshift in the interband transition energy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Interestingly, one could observe the symmetric Zeeman splitting about the zero-field energy for the QR doped with 5% and 10% of Mn2+ ions, but for all other dopant concentrations (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5%, 1%, and 20%), the splitting seems to be asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' On the quantitative footing, the Zeeman splitting energy, ∆Esp−d z , plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 5(a) for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 is described as the energy difference between the two excitonic transitions under B and is determined from the data plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 4 as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The numerical data for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='(a)-?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (c) in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Two opposing parameters, the magnetic field, and temperature, interplay with each other to determine the Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The magnetic field suppresses the Mn2+ spin fluctuations by aligning the randomly oriented Mn2+ spins along the field direction, indicating a state of magnetic ordering, thereby increasing ⟨Sz⟩ and causing the GZS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Contradicting this, the spin fluctuations are large enough to keep the state with a maximum entropy at elevated temperatures, and eventually, the Zeeman splitting is abated by the thermal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' It is interesting Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (a) Zeeman splitting (∆Esp−d z ) of the 1s exciton at various temperatures for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (b) Temperature dependent magnetization (M) calculated using modified Brillouin function for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (c) Magnetic susceptibility (χ) data for various dopant concentrations as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' to note from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 5(a) and (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=') that ∆Esp−d z increases with the dopant concentration up to x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05, and thereafter it starts decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' This is because the Zeeman splitting is proportional to the effective dopant concentration ‘xeff’ as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (16), and the latter increases with ‘x’ and shows a maximum at a particular concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Henceforth, it starts to move downhill because of the antiferromagnetic interactions between the nearest neighbouring magnetic ions, which cancels the spins of the corresponding pairs and reduces the effective contribution to the thermal average of the spin polarization of Mn2+ ions, ⟨Sz⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' In the framework of the theoretical model described in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' II B, the effect of temperature on the absolute value of effective Landé g-factor has been calculated, and the results are tabulated in Table-I for various ‘x’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The enhanced g-factor in the applied magnetic field directly evidences the strength of the Zeeman splitting, and the reduction in g-factor for increased temperature is also anticipated because of the depolarization of Mn2+ spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Effective Landé g-factor at T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2K and 300K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Effective Landé g-factor (geff) at B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2Tesla x T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2K T = 300K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='375 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='01 383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='885 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='099 820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='671 601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='032 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='048 Figure 5(b) shows the magnetization (M) vs magnetic field curves at different temperatures for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Magnetization increases with the magnetic field since it enhances ⟨Sz⟩, showing a linear dependence on magnetic field, which is an expected paramagnetic behaviour and does not saturate even at higher magnetic fields, B>10T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Similarly, it decreases when temperature augments, restricting all the magnetic moments from aligning along the field direction by intensifying the spin fluctuations so that the magnetic ordering is hampered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' As already discussed, when QR is populated with more magnetic ions, the spin-spin interaction becomes more robust, which results in a quenching of magnetization for high ‘x’ because of the lower value of ⟨Sz⟩ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='(d)-?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (f) in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' To corroborate these results, the temperature dependent magnetic susceptibility, χ, data for various concentrations keeping the magnetic field constant to B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2Tesla is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Binding energy of σ± magneto-exciton The variation of binding energy is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 6(a) for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The trend of the binding energy for both σ− and σ+ polarization concerning the magnetic 10 20 (a) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 (b) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 0 10 5 20 30 10 40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='50 5 60 0 70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 Magnetic Field (Tesla) 14 (c) B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2Tesla 12 10 8 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005 6 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 X 4 (emu : 2 0 50 100 150 200 250 300 Temperature (K)8 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (a) Binding energy of σ± magneto-exciton vs magnetic field for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' In-plane electron–hole distance corresponding to (b) σ+ and (c) σ− exciton vs magnetic field for various dopant concentrations at T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (d) Schematic explaining the overlap integral between the electron and hole under various strengths of magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (i) B = 0, (ii) 0 < B < Bc, and (iii) B > Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' field is as same as the trend followed by the interband transition energy, and this behaviour persists at different temperatures and concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Nevertheless, for σ+ polarization, there is a rapid decrease of binding energy with the magnetic field as compared to the steady increase for σ− polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' This can be better understood from the schematic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 6(b), which explains how the applied magnetic field modifies the electron-hole overlap inside a SQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' At B = 0, the location of both the electron and hole is in the same CdTe layer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 6(b-i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Zeeman splitting of the energy levels in the valence band is highly sensitive to the applied field, which is not the case with the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' This is because the band offset formed in the conduction band is generally larger than the valence band offset since 80% of the bandgap difference falls in the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Moreover, the absolute value of the exchange constant, which represents the strength of the exchange interaction, is larger for the heavy hole (|βexcN0 = 880meV|) than for the electron (|αexcN0 = 220meV|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Therefore the electron with sz = −1/2 in the conduction band would forever be confined in the non-magnetic CdTe layer itself irrespective of the strength of the applied field as its potential band offset is sufficiently larger than the order of magnetic splitting (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 6(b-ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' However, the potential barrier for the heavy hole with Jz = −3/2 is tremendously reduced with the magnetic field, and it encounters a flat band situation at critical field value, beyond which the system undergoes a type-I - type-II transition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 6(b-iii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' As a result, the electron remains in the CdTe layer, but the hole moves towards the heterostructure interface and finally to the CdMnTe layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Hence, the exciton will no longer be spatially direct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' rather, it becomes spatially indirect, which reduces the overlap between the electron and hole, whereby spin-down exciton states have reduced binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' To justify this discussion, the in-plane exciton radius, Reh, the average distance between the electron and hole in the plane of the QR, has been calculated and is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 6(c) and 6(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' As anticipated, the monotonic increase and decrease of Reh could be seen for σ+ and σ− polarization, respectively, for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Moreover, the 3D plot of the probability distribution of spin-down electrons and holes (|Ψ|2) along ρ and z-directions of the QR, and the density plot of the single-particle distribution depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 7 helps to understand the effect of magnetic field on 80 B>B。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 (b) B< B。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (a) 75 (ii) (iii) (i) 70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2K 65 60 77K electron 55 300K Type - I Type - II 50 45 9+ Heavy hol 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 80 (c) x= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005 (d) 48 T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2K T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2K 45 + Polarization 70 x= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='01 Polarization 42 39 60 36 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005 50 33 R R 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 x= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 27 X= =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='01 X 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 X= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 21 X02 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='8 Magnetic Field (Tesla)9 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Left panel: 3D-plot for the probability distribution of electrons and holes along axial, and radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Right panel: Density plot of the probability distribution of single-particle states along both radial and axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The data has been plotted for (a) B = 0, and (b) B = 1Tesla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' the carrier confinement inside the QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Obviously, |Ψ|2 is larger for zero magnetic field as one can compare the order of magnitude between B = 0 and B = 1Tesla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Oscillator strength, radiative linewidth and radiative lifetime of magneto-exciton To gain further insight into the σ+ and σ− transition and related radiative properties, the investigation of oscillator strength (OS), radiative decay rate (RDR), and radiative lifetime (RLT) have been performed, and the results are delineated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The expression for the exciton oscillator strength follows [48, 56, 57], f± = EP 2 ET ± ����� � +∞ −∞ I dρe dze ����� 2 |Ω(0)|2 (19) where, EP = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1eV for CdTe, represents the Kane energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The OS mainly depends on the overlap integral ‘I’ between the electron and hole envelope wavefunctions: I = | � +∞ −∞ N1s φe (ρe) φh(ρe) fe(ze) fh(ze) dρe dze|2, and Ω(0) denotes the probability of finding the electron and hole at the same position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The oscillator strength per unit area is proportional to the effective Bohr radius as, F± = 1 a∗ B 2 f±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Exciton radiative lifetime, ‘τ’ (radiative decay rate, ‘Γ = 1/τ’) can be related to OS according to [58, 59], τ = 2πϵ0m0c3ℏ2 ne2ET ±f± (20) Here, the fundamental physical constants have their usual meaning and ‘n’ represents the refractive index of the material CdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The evolution of the oscillator strength as a function of magnetic field solely depends on the spatial overlap Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Temperature dependent variation of (a) overlap integral, (b) oscillator strength, (c) radiative lifetime, and (d) radiative decay rate with magnetic field for σ+ and σ− transitions for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' between the electron and hole wave functions, which has been depicted for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The numerical data pertaining to all other concentrations are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' and ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The applied magnetic field increases the overlap between the electron and hole ground states for σ− polarization, indicating larger OS due to the increase of potential barrier height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' As expected for the σ+ polarization, the OS sensitively depends on B, which diminishes the excitonic effect by spatially separating the electron and hole as explained in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' III D and thereby weakens the corresponding optical transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The overlap integral is suppressed by the thermal energy kBT, whereby the oscillator strength is also abated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The overlap integral increases as the dopant concentration increases due to the increased potential 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='06 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 Magnetic field (Tesla)10 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Faraday rotation angle (ΘF) as a function of magnetic field for (a) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005, and (b) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 at different temperatures for a fixed photon energy of E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Verdet constant as a function of photon energy for various dopant concentrations at (c) T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2K, (d) T = 77K, and (e) T = 300K for a fixed strength of magnetic field, B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2Tesla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' barrier height (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Figure 8(c) and 8(d) shows the radiative lifetime and radiative decay rate as a function of magnetic field for 5% of Mn2+ concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The RLT of exciton increases with increasing B for σ+ polarization, which is accompanied by a decrease in RDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The exciton lifetime is found to decrease from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='04ns to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='38ns when the concentration of Mn2+ ion increases from x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005 to x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 at low temperature and at B = 0, where radiative recombination dominates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The RDR, which characterizes the decay of photon emitted by the exciton, shows its maximum only for B = 0, which elucidates the probability of finding an electron and hole at the same position (re = rh) is more prominent in the absence of magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' GFR in semimagnetic SQR The Faraday rotation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 1(c), also known as an optical analogue of the Hall Effect, results from a difference in refractive indices of the left and right circularly polarized light after traveling through a magnetized medium with a length ‘l’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' If there is a difference in absorption of two circularly polarized light, then the polarization vector will change its helicity from linear to elliptical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' However, in the present work, only the changes in refractive indices are considered in the numerical calculations and the latter is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The phase difference in velocity between the two circularly polarized components is expressed through the FR angle as [43], ΘF = ∆φ 2 = El 2ℏc (n− − n+) (21) Here, n− and n+ denote the refractive indices of the left and right circular polarized light, and E is the incident photon’s energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' As aforementioned, the FR in DMS alloys is a giant one due to the large Zeeman splitting of the energy levels as a result of sp-d exchange interaction, which has been computed using the single oscillator model as preferred in the work of Bartholomew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=', After performing a series of calculations, ΘF achieves the final form as [43], ΘF = √F0l 2ℏc �βexc − αexc gMn µB � M 1 E0 y2 (1 − y2)3/2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' y = E E0 (22) Here, F0 is a constant that involves the oscillator strength, and E0 is the ground state interband transition energy at the fundamental energy gap at zero magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The angle is directly proportional to the GZS through the term ∆E = βexc−αexc gMnµB M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The Verdet constant is written as the Faraday rotation per unit 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 (a) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005 Eph=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='50eV 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5 3 (c) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 6 9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2K 18 T=300K (/3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5 77K 21F B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 Tesla 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 24 3 (degree) → 300K T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2K 27 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5 (a) PA + x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005 (a) PA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 9 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='01 25 12 + X=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 (b) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 15 0 3 (P) + x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 18 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 ★ x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 21 50 9 24 12 75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='7 15 Photon energy (eV) 100 18 125 21 150 24 T = 77K 175 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='7 MagneticField (Tesla) Photon energy (eV)11 magnetic field per unit length, which is defined as [43], Vd(E) = ΘF Bl = √F0 2ℏc �βexc − αexc gMn µB � ∂M ∂H 1 E0 y2 (1 − y2)3/2 (23) Figure 9(a) and 9(b) depicts ΘF for the DMS QR in the dilute regime (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005) and doped with arbitrary ’x’ (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05) at a fixed photon energy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' It is noted from figure that the rotation angle increases with the increasing magnetic field since the applied field enhances the Zeeman splitting and decreases with the temperature since the magnetization is suppressed at elevated temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The variation of the Verdet constant with the incident photon energies at different temperatures for a fixed magnetic field of B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2Tesla is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 9(c)-9(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The Verdet constant shows a sharp increase whenever the band gap resonance occurs (when the energy of the incident photon approaches the absorption edge of the material), and the photon energy, at which the Verdet constant shows a rapid enhancement, shifted to higher energies for the heavily doped QR because the absorption edge increases as the concentration of Mn2+ ions increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Though the single oscillator model yields gratifying results, in which the behaviour of E0 at Γ point has been crudely modelled as constant at all temperatures, the success of using this in QR could not be verified due to a lack of reliable experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Concluding Remarks Probing the single-particle and exciton energy states in an applied magnetic field has been studied in semimagnetic QR, and the theoretical investigation of tuning related MO properties has been attempted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' It is found that the doubly-connected topological structures like QR provide robust confinement for the carriers compared to single-connected topological QDs [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The difference in the behaviour of magneto-exciton energies between the QR doped with low and high Mn2+ ion concentrations has been explained in detail at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The results show pronounced excitonic Zeeman splitting for low ‘x and T’ than high ‘x and T’, where the possibilities for the manganese ions to form antiferromagnetic pairs in the latter case are maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Among all the concentrations discussed here, x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='05 exhibits larger Zeeman splitting with the absolute value of effective g-factor, geff = 928 (Table I, and the corresponding Verdet constant is as large as 15 degree/Tesla Å, and the latter is 104 − 106 orders of magnitude larger than in bulk Cd1−xMnxTe [10, 43, 46], thin films [61–63], and is 102 orders larger than in QWs [48, 50], superlattices [49, 64] as reported in the previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' This elucidates the importance of DMS-based QR in MO devices operating at a wavelength shorter than 1µm than already existing MO materials, such as Yttrium Iron Garnet (YIG) and Terbium Aluminum Garnet (TAG), organic molecules, conjugated polymers [25, 43, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Moreover, the low-temperature exciton lifetime is 715ps, whereas it is ≈ 100ps in QWs doped with 25% Mn2+ ion concentration [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The study of exciton lifetime in semimagnetic quantum systems is impressive since it affects the optical properties and the magnetization dynamics of the concerned systems to a greater extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The exciton lifetime in DMS determines the formation of bound magnetic polaron (BMP) [67, 68] or exciton magnetic polaron (EMP) [69], which causes spontaneous ferromagnetic ordering even in the absence of an external magnetic field due to the strong sp-d exchange interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Since the recombination limits the exciton lifetime, it interrupts the EMP formation before the polaron reaches its stable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' If the exciton does not decay during the process of EMP formation, then the EMP would reach its equilibrium state, which is accompanied by a decrease of exciton energy and provides an additional localization for the carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The unique capability of manipulating the MO properties at the nanoscale in external magnetic fields and effective magnetic switching of the spins makes DMS-based QR a judicious choice among promising candidates for applications in future spintronic and optoelectronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The reliability of the results obtained using the single oscillator model could not be verified due to the missing experimental data, but it is believed to be improved using the multi oscillator model as adopted in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Since the low path length and the modest magnetic field yields a high Verdet constant, theoretical demonstration of generating larger FR and higher Verdet constant in DMS QRs would incite interest in preparing high-quality QR heterostructures based on DMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' With the unrivaled ability to modulate the magnetic excitonic transitions and thereby the optical activity of the materials at the nanoscale for a broader energy spectrum with various mole fractions of Mn2+ ions, the diluted magnetic semiconductors have potential applications in spin-photonic and spin-electronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 12 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Fitting equation for magnetic field variation of potential barrier height The fitting equation to represent the changes in the potential barrier height as a function of magnetic field is given by [4, 53, 55], ∆EB g = ∆E0 g ηe,h eζe,h γ − 1 ηe,h − 1 (A1) ∆EB g and ∆E0 g denotes the band gap difference between the well CdTe layer and the barrier Cd1−xMnxTe layer in the presence and absence of applied magnetic field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The composition and temperature dependence of the latter is written as: ∆E0 g(x, T) = ∆Eg(x) + T C(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' C(x) is known as temperature coefficient and ∆Eg(x) = Eg(Cd1−xMnxTe) − Eg(CdTe) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='587x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Temperature dependent band gap of the CdTe material is given by, Eg(CdTe) = Eg(0) − δT2 T+ξ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Eg(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='606 eV, δ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='37 × 10−4 eV/K, ξ = 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='8K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ηe,h = eζe,h γ0 is chosen with a fitting parameter ζe(ζh) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5), and γ0 is a critical magnetic field at which the barrier completely vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' The critical magnetic field γ0 in Tesla for different magnetic dopant compositions is given for donor (acceptor) impurity as γ0 = A enx with A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='734 and n = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='082 (A = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='57 and n = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='706).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Binding energy comparison between donor and acceptor impurities Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Binding energy vs ’d’ at different strengths of the magnetic field for various ring widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' (a) donor impurity, and (b) acceptor impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Zeeman splitting and magnetization in dilute and high ’x’ regimes Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Zeeman splitting of the excitonic energy levels (∆Esp−d z ), and temperature dependent magnetization (M) in (a), (d) dilute and (b), (c), (e), (f) high x regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Magneto-optical data for dilute and high ’x’ regimes Temperature dependent variation of the overlap integral, OS, and RLT, RDR as a function of magnetic field for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='5%, 10%, and 20% dopant concentrations are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' and ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=', respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Figure D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Overlap integral (I) and oscillator strength (f±) as a function of magnetic field for σ+ and σ− transition at different temperatures in (a), (d) dilute and (b), (c), (e), (f) high x regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Figure D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' Radiative lifetime (τ) and radiative decay rate (Γ) as a function of magnetic field for σ+ and σ− transition at different temperatures in (a), (d) dilute and (b), (c), (e), (f) high x regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content=' 180 180 Acceptor Donor =0 ----= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='002 =0 160 160 (meV) (a) (mel (a) (d) R2 = 80A 140 a 140 (b) (e) R2 = 150A 120 120 (c) (f) R2 = 300A 100 b 100 80 80 60 C 60 40 (f) 40 (a) b 20 20 60 20 80 100 120 140 160 180 200 20 40 40 60 80 100 120 140 160 180 200 Height of the Ring (d) (A)8 (meV) (a) x= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='005 0 0 30 6 20 →4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2K 60 5 40 ←77K 90 60 3 120 300K c) (b) 80 S 150 1 E x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 180 x= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='1 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfTPeX/content/2301.00102v1.pdf'} +page_content='2 8 10 12 14 0.' metadata={'source': 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Olivares S.1, Monika A. Mo´scibrodzka1, and Oliver Porth2 +1 Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands +2 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands +e-mail: holivares@science.ru.nl +January 31, 2023 +ABSTRACT +Context. Comparison of horizon-scale observations of Sgr A* and M87* with numerical simulations has provided considerable insight +in their interpretation. Most of these simulations are variations of the same physical scenario consisting of a rotation supported torus +seeded with a poloidal magnetic fields. However, this approach has several well known limitations such as secular decreasing trends +in mass accretion rate that render long term variability studies difficult, a lack of connection with the large-scale accretion flow which +is replaced by an artificial medium emulating vacuum, and important differences with respect to the predictions of models of accretion +onto Sgr A* fed by stellar winds. +Aims. We aim to study the flow patterns that arise at horizon scales in more general accretion scenarios, that have a clearer connection +with the large scale flow and are at the same time controlled by a reduced set of parameters. +Methods. As a first step in this direction, we perform three dimensional general relativistic hydrodynamic simulations of rotating tran- +sonic flows with velocity perturbations injected from a spherical boundary located far away from the central object (1000 gravitational +radii). We study the general properties of these flows with varying angular momentum and perturbation amplitudes. We analyze time +series of mass and angular momentum radial fluxes, angle- and time-averaged profiles, and synthetic Bremsstrahlung lightcurves, as +well as the three-dimensional structure of the flow, and quantify shock- and sonic transitions in the solutions. +Results. We observe a rich phenomenology in accretion patterns, that includes smooth Bondi-like flows, turbulent torus-like struc- +tures, shocks, filaments, and complex sonic structures. For sufficiently large perturbations and angular momentum, radial profiles +deviate from the constant entropy and constant angular momentum profiles used for initialization and resemble those of advection +dominated accretion flows, showing evidence of entropy generation and angular momentum redistribution not mediated by magnetic +fields. Time series do not show the secular decreasing trend and are suitable for long-term variability studies. We see that the fluctua- +tions are amplified and extend further in frequency than the injected spectrum, producing a red noise spectrum both for mass accretion +rate and the synthetic light curves. +Conclusions. We present a simulation setup that can produce a wide variety of flow patterns at horizon scales and incorporate +information from large scale accretion models. Future inclusion of magnetic fields and radiative cooling could make this type of +simulations a viable alternative for numerical modeling of general low-luminosity active galactic nuclei. +Key words. accretion, accretion disks - black hole physics - relativistic processes - methods: numerical +1. Introduction +Accretion onto compact objects such as black holes and neutron +stars powers some of the most spectacular phenomena in astro- +physics. While the focus of numerous studies in accretion theory +is on how matter and angular momentum are transported through +an accretion disk, much less studies put into focus the formation +of the accretion disk itself. In particular numerical simulations of +accretion disk formation are encumbered by the large scale sep- +aration between circularization radius of incoming matter and +the size of the accretor. There are certain applications however +where the accreted matter has comparatively low angular mo- +mentum, leading to circularization radii not much larger than the +accretor itself. Prime examples are the high-mass X-ray binaries +(HMXBs) and chaotic stellar wind-fed accretion in galactic nu- +clei such as our own Galactic center Sgr A*. +A central role in the interpretation of the event-horizon- +scale observations of Sgr A* and M87* by the EHT Collabo- +ration is played by general general relativistic magnetohydrody- +namic (GRMHD) simulations (Porth et al. 2019; Event Horizon +Telescope Collaboration et al. 2019, 2022). To date, all of the +models in the simulation library for Sgr A* and most of those +used for M87* follow variations of the same initial conditions of +a rotation-supported torus (Fishbone & Moncrief 1976) seeded +with a weak poloidal magnetic field. While a lot of physical in- +sights have already been gained by comparing observational data +against GRMHD simulations – leading to increasingly tight con- +straints of the parameters such as black hole mass, accretion rate, +inclination and black hole spin (Event Horizon Telescope Col- +laboration et al. 2019, 2021, 2022) – there are several limitations +intrinsic to the considered “black hole – torus (BHT)” simula- +tions. For example, since they are initialized with a finite amount +of matter contained in the torus, the matter content in the simu- +lation decreases over time, accompanied by a corresponding de- +crease in mass accretion rate. This secular trend renders the study +of long-term variability difficult. This systematic is particularly +important since the current set of GRMHD simulations produces +highly varying lightcurves which are tightly constrained by the +less variable data for Sgr A* (Event Horizon Telescope Collab- +oration et al. 2022; Wielgus et al. 2022). +Article number, page 1 of 17 +arXiv:2301.12020v1 [astro-ph.HE] 27 Jan 2023 + +A&A proofs: manuscript no. main +The most important limitation of the BHT simulations how- +ever concerns physical realism. It is now widely believed that our +galactic center black hole, Sgr A*, can be fed from the winds of +∼ 30 massive stars that orbit on the parsec scale (Quataert 2004; +Cuadra et al. 2008; Ressler et al. 2018). Whether an accretion +disk (torus) forms in this scenario depends not only on the ini- +tial wind parameters (Mo´scibrodzka et al. 2006; Shcherbakov & +Baganoff 2010) but also on the interactions of the unbound winds +which can give rise to shocks and hydrodynamic turbulence. +The flow patterns of realistic stellar wind accretion models for +low luminosity active galactic nuclei (AGNs) differ significantly +from the BHT scenario described above. In stellar wind accre- +tion, material forms clumpy structures and has a broad distribu- +tion of angular momentum without sufficient time to circularize, +and is not generally rotation supported. Instead, it is accreted +mainly due to an originally low angular momentum and remains +in large part unbound (Ressler et al. 2018). This latter property is +shared by different models of accretion from large scales such as +the constant-entropy solutions by Bondi (1952); Michel (1972); +Chakrabarti (1996) and models that include dissipation such as +the well-known advection dominated accretion flows (ADAFs) +(Narayan & Yi 1994). Recent magnetohydrodynamics (MHD) +simulations that focus on the large scale dynamics have revealed +further differences to the standard BHT scenario: while mag- +netic fields from stellar winds are initially weak and passively +advected, at horizon scales they accumulate and become dynam- +ically important and start to regulate accretion in a way similar +to Magnetically Arrested Disks (MADs) (Ressler et al. 2020a,b). +The lack of a predominant angular momentum or magnetic field +direction leads to erratic changes in the orientation of the accre- +tion disk (Ressler et al. 2021). Similar transient behavior can be +seen in the direction and power of the jet before the formation +of a steady jet (Lalakos et al. 2022). Simulations of accretion +from kpc scales onto the black hole (BH) event horizon have +also shown that the accretion flow in elliptic galaxies as M87 can +acquire a variety of patterns that range from rotation-supported +disks to chaotic streams (Guo et al. 2022). +In general, simulation-based studies of the horizon-scale +structure of the accretion flow resulting from large-scale feed- +ing present the computational challenge of having to simulate +length and time scales spanning ∼ 6 orders of magnitude, or +dealing with uncertain factors such as the details of stellar winds +astrophysics. It would be therefore desirable to gain more in- +sight on the properties of the accretion flow from the study of +transonic solutions connecting the event horizon to infinity, in +a similar manner as the theory of accretion disks has benefited +from the study of analytic solutions for fluids in circular mo- +tion around black holes. Depending on the specific angular mo- +mentum and energy, analytic studies of trans-sonic low angu- +lar momentum accretion flows (e.g. Fukue 1987; Chakrabarti +1989, 1996; Chakrabarti & Das 2004) have revealed different +regimes characterized by smooth Bondi-like flows, standing ac- +cretion shocks or the formation of circularized tori. Furthermore, +the solutions have been studied including the effects of viscosity +(Chakrabarti & Molteni 1995; Lanzafame et al. 1998), radiative +cooling (Molteni et al. 1996; Okuda et al. 2004) and magnetic +fields (Proga & Begelman 2003b; Okuda et al. 2019; Mitra et al. +2022), often with particular focus on the stability and dynam- +ics of the accretion shock. Numerical simulations of transonic +hydrodynamic solutions were presented more recently by Kim +et al. (2017, 2019) for the Schwarzschild and Kerr spacetimes, +showing that certain perturbations can trigger long-surviving +shocks at the location of predicted standing shocks (Chakrabarti +1996). +In fact, a realistic approach to the problem should consider +the destabilizing effect of inhomogeneities in the surrounding +medium. The stability of spherical Bondi accretion has been +studied analytically in a number of works; see for instance Mon- +crief (1980) and Kovalenko & Eremin (1998). In the latter work, +it is shown that this solution is unstable for non-radial perturba- +tions, although for the instability to manifest itself the size of the +accretor needs to be sufficiently small compared to the Bondi ra- +dius, precisely as it is the case for the nearest supermassive black +holes (SMBHs). +In this paper, we study a simulation setup that aims to ad- +dress the above described limitations of the BHT paradigm and +to facilitate the incorporation of information gained from larger +scale simulations. This model depends on a reduced set of pa- +rameters that can in principle be chosen to match the properties +inferred for known SMBHs such as Sgr A*, M87, and other tar- +gets of the Event Horizon Telescope (EHT) and the planned new +generation Event Horizon Telescope (ngEHT). By incorporat- +ing time-dependent properties of the surrounding medium in the +boundary conditions, the simulation domain can be of a mod- +est size comparable to that of existing GRMHD simulations in +the EHT library. The simulations presented here are run in pure +general relativistic hydrodynamics (GRHD), that is, with zero +magnetic field, as an intermediate step towards GRMHD simula- +tions that will be presented in a forthcoming work. We show that +the proposed setup produces steady time series that are in princi- +ple suitable for long-term variability studies, and exhibits a rich +phenomenology that can differ significantly both from typical +BHT simulations and from unperturbed Bondi-like accretion. +We describe this setup in Section 2 and provide a justification +for the physical parameters employed (Section 2.1). In Section +3, we report on the properties observed in the simulations, such +as time series of mass and angular momentum accretion rates +and radial profiles (Section 3.1), three-dimensional morphology, +including the presence of shocks and complex sonic structures +(Section 3.2) and variability properties 3.3. We summarize and +discuss our results in Section 4, and complement this work with +more information on the simulation setup in the Appendices. +2. Simulation setup +To explore the flow patterns arising from transonic accretion +of an inhomogeneous interstellar medium, we perform three- +dimensional GRHD simulations that continuously inject matter +from an outer boundary. We employ units such that G = c = 1, +so that the gravitational radius rg = GM/c2 and the gravitational +timescale tg = rg/c are rg = tg = M, where M is the mass of +the black hole. We adopt a Kerr spacetime with dimensionless +spin parameter a � J/M = 0.95 and the event horizon located at +rH/M = 1 + +√ +1 − a2. For all of our simulations, we set the sonic +radius to rs = 500 M and place the boundary at r = 1000 M. We +initialize the simulations with a smooth quasi-stationary back- +ground solution with a latitude-dependent angular momentum +profile. Following Proga & Begelman (2003b) we adopt an an- +gular momentum profile that peaks at the equator and vanishes +at the poles (see Appendix A for a detailed discussion of the +background flow): +ℓ(θ) = ℓ0(1 − | cos θ|). +(1) +The background flow is characterized by only two parameters, +the angular momentum at the equator ℓ0, and the sonic radius (or +alternatively fluid internal energy E = hut at the equator). +Article number, page 2 of 17 + +Héctor R. Olivares S. et al.: Perturbed Transonic Accretion +Table 1. Table of runs. +δ +ℓ0 = 0 +ℓ0 = 2.25 +ℓ0 = 3.25 +a = 0 +a = 0.95 +a = 0.95 +0.01 +l0p001 +l2p001 +l3p001 +0.1 +l0p01 +l2p01 +l3p01 +1 +l0p1 +l2p1 +l3p1 +10 +l0p10 +l2p10 +l3p10 +To model inhomogeneities in the interstellar medium, we in- +ject perturbations of varying amplitude to the (tangential-) veloc- +ity components at the outer boundary. Perturbations are modeled +as time-varying Gaussian random field with a white noise spec- +trum +S |δu|(k) ∼ constant , +(2) +in the wavelength range λk/M = 2π/kM ∈ [214, 2400], and in +the frequency range fk ∈ [3.7, 41] × 10−5M−1 (see Appendix B +for more details). The remaining fluid variables at the boundary +are set consistently with the initial condition, and are therefore +constantly injecting matter that should preserve the initial state in +absence of perturbations. The injected noise is controlled by the +parameter δ which specifies the ratio of the variance of velocity +perturbations to the radial component of the 4-velocity of the +unperturbed flow at the boundary, δ = ⟨δu2⟩1/2/ur. We adopt the +adiabatic index ˆγ = 4/3. +We perform several simulations varying ℓ0 and δ. In order to +isolate the effect of perturbations, we run a set of simulations in +a Bondi-Michel accretion scenario, that is, ℓ0 = 0 and a = 0. The +list of runs and parameters used is displayed in Table 2. +To run the simulations, we use the code BHAC (Porth et al. +2017; Olivares et al. 2019). We use a spherical polar grid in mod- +ified Kerr-Schild coordinates with logarithmic spacing in radius. +The base resolution is Nr × Nθ × Nφ = 96 × 48 × 48 and we em- +ploy 3 levels of Adaptive Mesh Refinement (AMR), obtaining +an effective resolution of 384 × 192 × 192. The inner bound- +ary is located inside of the central black hole event horizon, at +r = 1.19 M, in order to avoid boundary effects. We employ a +finite volume method with piecewise parabolic reconstruction +(PPM), a total variation diminishing Lax-Friedrichs (TVDLF) +approximate Riemann solver and a two-step method for time in- +tegration (see Porth et al. 2017, for more details on coordinates +and numerical methods). +To reduce the cost of simulations, we evolve a passive tracer +f that is initialized as f = 0 inside the domain and f = 1 for the +injected matter at the boundary, and evolve only those blocks of +8 × 8 × 8 cells for which f > 0.1 or which are surrounded by +blocks that satisfy this condition. For all of the simulations, this +tracer reaches the event horizon at t ≲ 30 000 M, after which +the simulation domain becomes active everywhere. We continue +the evolution up to t = 60 000 M, which corresponds to a total +simulation time of nearly 5 free-fall timescales from the sonic +radius, tff = π(rs/2)3/2 ≈ 12 418 M. +2.1. Physical parameters +Accretion onto an object at rest with respect to a spherically +symmetric, asymptotically uniform medium can be considered +to start at the Bondi radius, rB, the distance at which the asymp- +totic sound speed c∞ equals the escape velocity, that is, rB = +2GM/c2 +∞. Temperatures inferred from Chandra X-ray observa- +tions of the hot gas surrounding Sgr A* (Baganoff et al. 2003) +and the central black hole of M87 (Russell et al. 2015), combined +with the assumption of a monoatomic ideal gas with ˆγ = 5/3, +yield estimates for the Bondi radius of 6 × 105 M and 4 × 105 M, +respectively. The several orders of magnitude separation be- +tween the Bondi radius and the event horizon makes simulations +of accretion from the Bondi radius onto SMBHs prohibitive for +most numerical codes. In practice, due to temperature gradients, +the local sound speed cs does not coincide with escape veloc- +ity at the Bondi radius. This happens instead at the sonic radius +rs = 2GM/c2 +s, which marks the transition from subsonic to su- +personic flow. In Newtonian hydrodynamics, the case ˆγ = 5/3 is +degenerate and pushes the sonic radius to the origin. However, +by incorporating relativistic corrections and assuming c∞ ≪ c, +it takes a finite value that can be approximated as rs ≈ 3Mc/4c∞ +(see e.g. Rezzolla & Zanotti 2013). For the value of c∞ reported +above, this corresponds to rs ≈ 409M for Sgr A* and rs ≈ 335M +for M87. The value rs = 500M in our simulations is chosen ac- +cordingly within the same order of magnitude. +Following the same relativistic Bondi models, the dimen- +sionless temperature at the sonic radius can be estimated to be +Θ � kBT/mc2 = 7.3 × 10−4 – 8.8 × 10−4, where m is the ion +mass and kB is the Boltzmann constant. For monoatomic hydro- +gen, this corresponds to T ≈ 8×109K – 1010K (higher values cor- +respond to M87). The dimensionless temperatures attained at the +sonic radius for our simulations are similarly Θ ≈ 8 × 10−4. Ex- +pecting it to to increase by orders of magnitude when approach- +ing the black hole, we set ˆγ = 4/3. The effective adiabatic in- +dex in this regime is very dependent on uncertain factors such as +cooling and the ratio between ion and electron temperatures, and +a more self-consistent generation the background solution may +require the use of a relativistic equation of state as in Aguayo- +Ortiz et al. (2021). However, a fully accurate modeling of these +effects is beyond the scope of this project. +Turning to the second parameter, ℓ0, the specific angular mo- +mentum from stellar stellar wind accretion can be roughly esti- +mated by ℓ ≃ r2 +accΩ/4 (Frank et al. 2002). Here Ω is the orbital +angular velocity of the star and racc = 2GM/v2 +w is the accretion +radius for an assumed cold wind with velocity vw. Scaled to ge- +ometric units and for a star in Keplerian orbit with semi-major +axis a we have +ℓ ≃ 0.5 +� a +pc +�−3/2 � +vw +1000km s−1 +�−4 +. +(3) +Thus low angular momentum flows are indeed expected for these +fiducial values. Focusing on a particular source, it was argued in +Mo´scibrodzka et al. (2006) that the stellar complex known as +IRS 13 E3 (Maillard et al. 2004) exerts the the strongest ram- +pressure at the Galactic center which renders it the dominant +wind accretion source. Thus taking IRS 13 E3 with fiducial wind +velocity of 1000km s−1 as exemplary case and using the orbital +fits by Muži´c et al. (2008), we obtain ℓ in the range 0.1−16. This +large spread is caused by the large range of admitted semi-major +axes 0.1pc − 2.6pc reported in Muži´c et al. (2008). +As there are large uncertainties associated with the value of +ℓ in the Galactic center and to gain insight into the parameter de- +pendence, we here investigate three cases that correspond to the +different qualitative behaviors of the background solution: non- +rotating case (ℓ = 0), a rotating case where the solution is com- +plete, that is, it connects smoothly infinity and the event hori- +zon (ℓ = 2.25), and a rotating case with an incomplete solution +(ℓ = 3.25). Incomplete solutions of flows coming from infinity +are expected either to pass through a shock and transition to an- +other smooth solution that reaches horizon, or to represent flows +Article number, page 3 of 17 + +A&A proofs: manuscript no. main +that are unstable in absence of viscosity. For a sufficiently vis- +cous flow, some of these incomplete solutions can transition to a +torus (Chakrabarti 1996). It should be noted that a complete so- +lution can exist for ℓ even when there is a circularization radius +rcirc > rH at which ℓ is equal to the Keplerian angular momen- +tum, as it is the case for ℓ = 2.25 (rcirc ≈ 3.6 M). The reason +is that fluid elements have a nonzero radial velocity and depend- +ing on their energy (part of which is internal) their centrifugal +barrier is located further inside rcirc, and in some cases they can +even cross smoothly the event horizon. +Although Ressler et al. (2020b) showed that the orientation +of the flow angular momentum at horizon scales can vary wildly, +these variations occur on a scale of hundreds of years for Sgr +A*. The simulations presented here have a much shorter duration +– comparable to 14 days for the same source – which justifies +the assumption that the orientation of the angular momentum is +fixed. +Finally, the most uncertain parameter is the amplitude of in- +jected perturbations. To explore several possibilities, we have +considered a wide range with cases varying by orders of mag- +nitude with respect to the inflow velocity at the boundary. +3. Results +3.1. Global properties +In this section we briefly discuss and compare the salient global +features of the simulations. We start by computing time series of +the mass and angular momentum flux through the event horizon +˙M(t) � +� 2π +0 +� π +0 +ρur √−g dθ dφ , +(4) +˙L(t) � +� 2π +0 +� π +0 +T r +φ +√−g dθ dφ . +(5) +To set a typical scale that can be compared with real systems, we +normalize the accretion rate to the Bondi rate +˙MB = 4πλB(GM)2 ρ∞ +c3∞ +(6) +where +λB = 1 +4 +� +2 +5 − 3ˆγ +� 5−3ˆγ +2(ˆγ−1) +, +(7) +and ρ∞ and c∞ are the density and sound speed at infinity. In the +units employed here, ρ is normalized so that ρ = 1 at r = 6 M, +which leads to the numeric value ˙MB ≈ 246 code mass units per +gravitational timescale. +Figure 1 shows the time series in the interval t/M +∈ +[30 000, 60 000]. The first important feature shown in Figure 1 is +the long-term stability of the horizon penetrating fluxes over the +simulated timescales. While a quasi-stationary state is expected +due the constant mass supply at the inflow boundaries, it is reas- +suring that accretion rates are nearly constant after ∼ 2 freefall +timescales. +As it could be expected, there is a trend that relates a higher +angular momentum with a lower mass accretion rate. A larger +amplitude of perturbations also appears to result in smaller ac- +cretion rates, likely due to the extra angular momentum provided +by perturbations, which also contribute to centrifugal support. +For instance, the addition of δ = 10 perturbations for the ℓ = 0 +case reduces the accretion rate to a value of ∼ 0.75 ˙MB com- +parable to that obtained for the simulations for ℓ = 2.25 with +smaller perturbations. The simulation with largest perturbation +and angular momentum possesses the smallest accretion rate, at +≲ 0.25 ˙MB. +Inspecting the accretion of angular momentum, the solutions +show a surprising behavior: although it could be expected that a +flow with larger angular momentum would result in more an- +gular momentum accreted by the black hole, the simulations +with ℓ = 2.25 actually register slightly more angular momen- +tum accretion than those with ℓ = 3.25. Normalizing the angular +momentum accretion rate by the mass accretion rate, as it ap- +pears in the bottom panel of Figure 1, both cases show about the +same value of ˙L/ ˙M. The reason is likely that centrifugal support +prevents matter from accreting and carrying angular momentum +through the event horizon. In this respect, it is important to recall +the qualitative difference between the unperturbed flow configu- +rations corresponding to these two values: while ℓ = 2.25 allows +solutions that connect smoothly infinity with the event horizon, +ℓ = 3.25 produces an incomplete solution which for the viscous +case should connect to a rotation supported torus where the flow +is stalled (Chakrabarti 1996). +The time series in Figure 1 shows different variability prop- +erties for each simulation, with those having higher ℓ and larger +perturbations appearing more ‘noisy’. For the cases with ℓ = +3.25, this can be attributed again to the fact that the unperturbed +solution is incomplete, producing shocks and complex interac- +tions between the flow close to the centrifugal barrier even when +the injected perturbations are small. However, it is interesting to +see that the most variable time series corresponds to ℓ = 2.25, +for the simulation l2p10, where peaks in ˙M are sometimes even +larger than the Bondi accretion rate. We will diagnose the vari- +ability properties of the different solutions in more detail in Sec- +tion 3.3. +Simulations with δ ≤ 1 and ℓ ≤ 2.25 show transient oscilla- +tions near the time at which the innermost grids become active +(t ≈ 30 000 M), and decrease in amplitude and frequency as +the evolution proceeds. These are especially noticeable for the +cases ℓ = 0 for ˙M and ℓ = 2.25 for ˙L. For the other cases, the +oscillations are masked by the larger perturbations. +To quantify the departure of the perturbed solutions from the +initial background solution, in Figure 2 we show the angle- and +time-averaged radial profiles for all simulations. +These are computed as +⟨q⟩(r, t) � +� 2π +0 +� π +0 q(r, θ, φ, t) √−g dθ dφ +� 2π +0 +� π +0 +√−g dθ dφ +, +(8) +where q = ρ, p/ρ. We also show φ-averages of the rotation an- +gular velocity on the equatorial plane Ω = uφ/ut, +⟨Ω⟩(r, t) � 1 +2π +� 2π +0 +Ω(r, θ = π/2, φ, t) dθ dφ , +(9) +where axial symmetry has been used to eliminate the metric de- +terminant. All quantities are then time-averaged over the inter- +val t/M ∈ [50 000, 60 000]. We also plot the radial profiles ex- +pected for a self-similar ADAF model Narayan & Yi (1994) with +ˆγ = 4/3 and no radiative cooling, as well as those of the unper- +turbed Chakrabarti solutions with ℓ = 0, 2.25 and 3.25 which are +used as initial conditions and are exact on the equatorial plane. +For all simulations, the density profiles (left column of Fig- +ure 2) are well described an ADAF profile of ρ ∝ r−3/2 which +Article number, page 4 of 17 + +Héctor R. Olivares S. et al.: Perturbed Transonic Accretion +2.5 +3.0 +3.5 +4.0 +4.5 +t/tff +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +M/MB +l0p001 +l2p001 +l3p001 +l0p01 +l2p01 +l3p01 +l0p1 +l2p1 +l3p1 +l0p10 +l2p10 +l3p10 +30000 +35000 +40000 +45000 +50000 +55000 +60000 +t [M] +1.5 +1.0 +0.5 +0.0 +0.5 +L/M +Fig. 1. Mass and angular momentum flux through the event horizon for all of the simulations, starting from a time where perturbations have +reached the event horizon for all simulations. The upper horizontal scale measures time in units of the free-fall timescale from the sonic radius. +also holds approximately for the initial condition. In particular, +the profiles are inconsistent with the shallower convective so- +lution ρ−1/2 indicating that convection is not important in our +parameter regime (Narayan et al. 2000). +Although the averaged density profiles shown in Figure 2 +are smooth, some of the simulations exhibit density jumps at in- +dividual snapshots, in which case the profile possess the same +slope at either side of the jump. These are present in the simula- +tions with high angular momentum and large perturbations. As it +will be discussed in Section 3.2, they are related to shocks which +propagate outwards as they are smoothed away. Traces of these +jumps are visible in the profiles of simulations with ℓ = 3.25 +(especially of l3p10) at scales of r = 102 – 103 M. These are +expanding shocks produced by the fluid colliding with the cen- +trifugal barrier, as those studied for example by Suková et al. +(2017). The slow evolution timescales near the outer boundary +prevent them from being smoothed by the time average. +The averaged profiles of p/ρ ∝ Θ in the central column of +Figure 2 show a more interesting behavior. While initially they +coincide with those of the Chakrabarti solutions, those corre- +sponding to ℓ = 3.25 and δ = 10 gradually transition to the +profile of the ADAF solution. The clearest case is that of sim- +ulations l0p10 and l2p10, which transition form the constant- +entropy initial profile (∝ r−1/2 for the Bondi solution) to that of +the ADAF model ∝ r−1 at r ≈ 40 M. +The rise of the temperature profile close to the black hole is +likely a result of heating by shocks and turbulence, which trans- +form to thermal energy the kinetic energy injected through the +perturbations in the velocity. It can be noticed that the temper- +ature profiles of l0p1 and l2p1 start rising as well and deviate +from the initial profile at a shorter distance from the black hole. +This suggests that indeed the radius at which the transition to an +ADAF-like profile occurs is related to the amplitude of perturba- +tions in the medium. The fact that l0p10 acquires an ADAF-like +temperature profile close to the black hole is interesting. In fact, +this simulation differs from the scenario studied by Narayan & +Yi (1994) from which the self-similar solution is derived. Here +there is no coherent disk-like structure and the average of Ω is +close to zero (see rightmost panel of Figure 2). It is therefore +surprising that the heating provided by incoherent shocks and +turbulence results in a temperature profile similar to that of a +coherent viscous rotating flow. +The rightmost column of Figure 2 shows the angular veloc- +ity profile for all simulations. As expected, the profiles corre- +sponding to the cases with zero angular momentum in the unper- +turbed solution show negligible rotation velocity on the equato- +rial plane. The rest of profiles behave in a similar way as those +of dimensionless temperature: they follow the Chakrabarti con- +stant angular momentum profile at large radii (yielding a power +law slope of −2 far from the black hole) and transition to the +ADAF-like Keplerian profile ∝ r−3/2 closer to the black hole. +In general, it appears that at large distances the system is +well described by the adiabatic Bondi- and Chakrabarti-like so- +lutions, while once perturbations become enough amplified by +the geometry of the flow to produce shocks and turbulence, en- +tropy production starts and the system becomes better described +by ADAF-like profiles. +3.2. Three-dimensional morphology +Figures 3 and 4 show the density distribution for all the simula- +tions on the equatorial and the meridional plane, respectively, at +t = 60 000 M. The four simulations with lowest ℓ and pertur- +bation amplitude, l0p001, l2p001, l0p01, l2p01 are always +smooth and highly symmetric, and are practically indistinguish- +able from the initial conditions. Changes start becoming visible +for those with perturbations comparable to the incoming radial +velocity, l0p1 and l2p1, for which near-radial filaments can be +seen. +The simulations with higher angular momentum are qualita- +tively different, as it could be expected due to the incompleteness +of the Chakrabarti solution. In all of the ℓ = 3.25 runs it is possi- +ble to see the formation of a turbulent torus-like structure close +to the black hole. The larger the perturbations are, the more mis- +aligned this structure becomes with respect to the large-scale an- +gular momentum, which points into the +z direction (rightmost +panels of Figure 4). +Article number, page 5 of 17 + +A&A proofs: manuscript no. main +10 +3 +10 +2 +10 +1 +100 +101 +r +3/2 += 0 +10 +3 +10 +2 +p/ +r +1 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +( += /2) += 0.01 += 0.1 += 1 += 10 +r +3/2 +10 +3 +10 +2 +10 +1 +100 +101 +r +3/2 += 2.25 +10 +3 +10 +2 +p/ +r +1 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +( += /2) +r +3/2 +101 +102 +103 +r [M] +10 +3 +10 +2 +10 +1 +100 +101 +r +3/2 += 3.25 +101 +102 +103 +r [M] +10 +3 +10 +2 +p/ +r +1 +101 +102 +103 +r [M] +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +( += /2) +r +3/2 +Fig. 2. Radial profiles of density (left column), dimensionless temperature (middle column), and equatorial angular velocity (right column) averaged +over angle and time in the interval t/M ∈ [50 000, 60 000] for all simulations. From top to bottom, the columns correspond to ℓ0 = 0, ℓ0 = 2.25 +and ℓ0 = 3.25, respectively. The shaded regions indicate the standard deviation. The dot-dashed lines show the power laws expected for an ADAF +with γ = 4/3 and no radiative cooling, and the dashed lines are the profiles for the unperturbed configurations with constant angular momentum +used as initial condition. In most of the panels, the profiles for δ = 0.01 and δ = 0.1 overlap completely. +Despite the apparent similarity of these configurations with +those in BHT simulations, they exhibit important differences. +While for BHT simulations the toroidal structure is confined +mainly by the equilibrium between gravity and the centrifugal +force, for the simulations presented here it consists in large part +of unbound outflowing matter that is confined by its interaction +with inflowing matter. In addition, while for BHT simulations +most of the accretion flow occurs in the equatorial plane, here the +toroidal structure is an obstacle for the inflowing matter, causing +most of the accretion to occur through the poles. This behavior +is consistent with that observed for similar systems in absence +of magnetic fields, for example, by Proga & Begelman (2003a); +Moscibrodzka & Proga (2008); Suková et al. (2017). We expect +as well that the inclusion of magnetic fields will reverse the situ- +ation by producing accretion on the equatorial plane and a polar +outflow (Proga & Begelman 2003b). +Figures 5 and 6 show the relative pressure gradient as proxy +for the location of shocks. The simulations with low ℓ0 and δ, +(l0p001, l2p001, l0p01, l2p01) do not show important pres- +sure gradients. In contrast, those with ℓ0 = 3.25 and δ ≤ 1 +Article number, page 6 of 17 + +Héctor R. Olivares S. et al.: Perturbed Transonic Accretion +Fig. 3. Logarithmic density maps for all simulations at t = 60 000 M on the equatorial plane. Panels are organized in the same way as simulations +in Table 2, that is, increasing angular momentum from left to right, and amplitude of perturbations from top to bottom. Movies of simulations +l0p10, l2p10, and l3p10, are available at https://youtu.be/1TQV_aX13xE, https://youtu.be/oOh2reL9yK0, and https://youtu.be/ +VmCc3ZnDxEM, respectively. +(l3p001, l3p01, and l3p1) show clear spiral shocks This co- +herent large scale shock does not form in the strongly perturbed +case δ = 10. The colormap in Figures 5 and 6 also allow to see +sound waves traveling within the shocked regions. +It is also interesting to examine to what extent the causal +structure of the flow is preserved in presence of perturbations +and high angular momentum. The dashed lines in Figures 5 and +6 mark the surfaces for which the 4-velocity of an observer at +rest at infinity, ∂t = (1, 0, 0, 0), becomes null with respect to the +sonic metric (Moncrief 1980), +Gµν = ρ +hcs +� +gµν + (1 − c2 +s)uµuν +� +, +(10) +where ρ is the rest-mass density, h is the enthalpy, cs is the sound +speed and uµ is the four velocity of the fluid. This condition is +Article number, page 7 of 17 + +100 +0.0 +10p001 +12p001 +13p001 +50 +-0.5 +dot6ol +[W] +:0 +1.0 +-50 - +-1.5 +-100 +-2.0 +100 +10p01 +12p01 +13p01 +50 - +[W] +:0 +y +-50 - +-100 +100 +10p1 +12p1 +13p1 +50 +[W] +0 : +-50 - +-100 +100 +10p10 +12p10 +13p10 +50 - +[W] +0 +y +-50 - +-100 +-100 +0 +100-100 +0 +100-100 +0 +100 +x[M] +x [M] +x [M]A&A proofs: manuscript no. main +Fig. 4. Similar as Figure 3, for the meridional plane. +analogous to that defining static surfaces such as ergoregions and +event horizons for the spacetime metric gµν, and can be used to +characterize transitions between subsonic and supersonic flows +in an invariant way (Aguayo-Ortiz et al. 2021), especially in sit- +uations that lack symmetries such as the perturbed flows studied +here. +The sonic surface for the unperturbed solutions is a sphere +with radius rs = 500 M, centered at the black hole. Its structure +is practically unchanged for the four cases with lowest ℓ0 and +δ, (l0p001, l2p001, l0p01, l2p01). Cases with ℓ0 ≤ 2.25 and +δ = 1 (l0p1, l2p1) exhibit slight but noticeable changes in the +shape of the sonic surface, although it remains close to rs. This is +remarkable since perturbations already have an amplitude simi- +lar to the magnitude of the inflow radial velocity at the boundary. +Only when the perturbation amplitude is ten times the inflow ra- +dial velocity (l0p10, l2p10) we see large incursions of subsonic +matter inside rs, as well as islands of supersonic (subsonic) flow +within the former subsonic (supersonic) regions. +Models with ℓ0 = 3.25 show a more complex causal struc- +ture. The spiral shock produces an additional transition from su- +personic to subsonic flow, and it can erase the original sonic sur- +face as it propagates outwards. However, downstream the flow +Article number, page 8 of 17 + +100 +0.0 +10p001 +12p001 +13p001 +50 +-0.5 +dot6ol +0 +1.0 +N +50 - +-100 +-2.0 +100 +10p01 +12p01 +13p01 +50. +[W] +0. +N +-50 - +-100 +100 +10p1 +12p1 +13p1 +50 +[W] +0 +N +-50 +-100 +100 +10p10 +12p10 +13p10 +50 - +[W] +N +50 - +-100 +-100 +0 +100-100 +0 +100-100 +0 +100 +x [M] +x [M] +x[M]Héctor R. Olivares S. et al.: Perturbed Transonic Accretion +500 +0 +500 +y [M] +l0p001 +l2p001 +l3p001 +500 +0 +500 +y [M] +l0p01 +l2p01 +l3p01 +500 +0 +500 +y [M] +l0p1 +l2p1 +l3p1 +500 +0 +500 +x [M] +500 +0 +500 +y [M] +l0p10 +500 +0 +500 +x [M] +l2p10 +500 +0 +500 +x [M] +l3p10 +3 +2 +1 +0 +log10(| p|/p) +Fig. 5. Shocks and sonic surfaces for all simulations at t = 60 000 M. The color scale displays the relative pressure gradient, which is used as a +proxy for shock locations, and the dashed lines indicate the static limits of the sonic metric. +can become supersonic again. The spiral structure can then pro- +duce several sonic transitions between the distant regions, where +matter is injected subsonically, and the event horizon, that needs +to be crossed supersonically. For instance, in the panel of Fig- +ure 5 that corresponds to simulation l3p1, there can be even five +sonic transitions when approaching the black hole from certain +directions. For the case with δ = 10, l3p10, the original sonic +surface has disappeared completely, and a new one has formed +closer to the black hole. Also in this case, the spiral shock pro- +duces more than one sonic transition in some directions. +In addition to the entropy increase due to shocks, turbu- +lence could also play a role in heating the fluid and contribute to +the transition from a constant entropy temperature profile to an +ADAF-like profile, as shown in the middle panel of Figure 2. In +order to highlight the presence of turbulence, Figure 7 shows the +z-component of the vorticity vector on the equatorial plane for +simulations l3p001 and l0p10. The ℓ0 = 0 case shows vortic- +Article number, page 9 of 17 + +A&A proofs: manuscript no. main +500 +0 +500 +z [M] +l0p001 +l2p001 +l3p001 +500 +0 +500 +z [M] +l0p01 +l2p01 +l3p01 +500 +0 +500 +z [M] +l0p1 +l2p1 +l3p1 +500 +0 +500 +x [M] +500 +0 +500 +z [M] +l0p10 +500 +0 +500 +x [M] +l2p10 +500 +0 +500 +x [M] +l3p10 +3 +2 +1 +0 +log10(| p|/p) +Fig. 6. Similar as Figure 5, for the meridional plane. +ity sheets that can be associated with fluid streams approaching +the black hole at different speeds, and suggest the emergence of +smaller turbulent structures if simulated at a higher resolution. +For the incomplete ℓ0 = 3.25 analytical solution the fluid +is expected to form a torus at the circularization radius rcirc ≈ +8.8M, and no presence of fluid is expected at smaller radii. How- +ever, in all of our simulations we observe the flow occupying +this region without impediment, meaning that a means of angu- +lar momentum redistribution is operating. The same can also be +inferred from the rotation velocity profiles in the rightmost panel +of Figure 2, where several of them transition from constant to +Keplerian angular momentum profiles. This indicates that even +in the absence of magnetic fields, and thus MRI and large scale +Maxwell stresses, angular momentum redistribution occurs, and +it can be attributed to shocks and turbulence that could easily +appear in nature. +Article number, page 10 of 17 + +Héctor R. Olivares S. et al.: Perturbed Transonic Accretion +40 +20 +0 +20 +40 +x [M] +40 +20 +0 +20 +40 +y [M] +l3p001 +min: -11.58, max: 2.31 +40 +20 +0 +20 +40 +x [M] +l0p10 +min: -1.56, max: 0.90 +0.050 +0.025 +0.000 +0.025 +0.050 +z [M +1] +Fig. 7. Vertical component of the vorticity on the equatorial plane for +simulations l3p001 and l0p10. +3.3. Variability properties +To have a rough estimation of the observable properties of the +variability in our simulations, we have computed synthetic X- +ray light curves by integrating the total bremsstrahlung emissiv- +ity from free-free electron-ion collisions Rybicki & Lightman +(1986), +ϵBR = 5.54 × 10−9 Z2 +� mi +mp +�1/2 � +ni ρ +106 cm−3 +�2 � p +ρ +�1/2 +erg +cm3 s , (11) +where Z is the atomic number of ions, mi is the ion mass, and +mp is the proton mass, and ni is scaling factor that relates the +dimensionless code density ρ with the ion number density niρ. +The Gaunt factor has been assumed to be constant and equal to +1.2. The integration is performed as +LBR = 2.41×1038 +� +M +4.15 × 106 M⊙ +�3 � +ϵBR Γ √γ d3x erg +s , (12) +where Γ is the Lorentz factor, and the prefactor comes from +the conversion of volume in geometrized code units to physical +units. +The synthetic light curves within t/M ∈ [50 000, 60 000] for +each simulation are shown in Figure 8. The parameters have been +chosen for monoatomic hydrogen, with ni = 106 cm−3, and M as +the mass of Sgr A*, M = 4.15 × 106M⊙. This gives luminosities +that agree in order of magnitude with the ≈ 2.4 × 1033erg s−1 +estimated by Baganoff et al. (2003). For this source, the time in- +terval corresponds to ≈ 55.5 hours of observing time. We cal- +culated spectrograms in this interval using the Welch method +(Welch 1967) with time windows overlapping over 128 points +(= 128M). The power spectral densitys (PSDs) are shown in the +left panel of Figure 9. It can be seen that, as expected, the power +of fluctuations increases with the amplitude of the injected per- +turbations and with the angular momentum. The two simulations +with zero angular momentum and smallest perturbations show +small frequency peaks that are lost into the noise for the other +cases. +As perturbations and angular momentum increase, it is possi- +ble to observe a steepening in the slope of the PSD. Simulations +with ℓ0 = 3.25 or δ = 10 show a very similar spectrum with a +break from white noise to red noise around f ∼ 10−2M−1. Power +laws of red noise f −2 and f −4 are shown for comparison. Spec- +trograms are calculated over a frequency range higher than that +of the injected perturbations (see Secion 2), which therefore do +not appear in the PSD. +We Fourier-transformed these spectrograms in order to ob- +tain autocorrelation functions corr(LBR, LBR) for the synthetic +Table 2. Modulation index of the bremsstrahlung luminosity light curve +and the mass accretion rate through the event horizon (in parenthesis), +computed over the interval t/M = [50 000, 60 000]. +δuRMS/|ur| = +ℓ = 0 +ℓ = 2.25 +ℓ = 3.25 +0.01 +0.004 (0.003) +0.016 (0.002) +0.246 (0.066) +0.1 +0.003 (0.003) +0.028 (0.002) +0.319 (0.101) +1 +0.057 (0.001) +0.303 (0.012) +0.237 (0.075) +10 +0.105 (0.031) +0.225 (0.031) +0.269 (0.200) +lightcurve. For all of the simulations, positive correlations de- +cay below 1/e in about τ ∼ 5 – 15 M. However, autocorrelations +for noisier simulations (ℓ0 = 3.25 or δ = 10), are close to zero +for τ ∼ 30 M (≈ 10 minutes for Sgr A*), while simulations with +small angular momentum and perturbations still exhibit longer +term positive and negative correlations. +Overall, the correlation timescales are shorter than 40 M +for all simulations, which allow us to calculate modulation in- +dices σ/µ for statistically uncorrelated data by computing the +standard deviation σ and average µ over points separated by +50 M. Modulation indices are shown in Table 3.3. The modu- +lation index of the mass accretion rate through the event hori- +zon in the same time interval is shown in parenthesis. The fact +that the latter doesn’t show as much variations when the for- +mer varies by orders of magnitude indicates that an important +portion of Bremsstrahlung variability is not related to fluctu- +ations in the accretion rate close to the horizon. Instead the +Bremsstrahlung modulation index clearly increases for those +simulations in which shocks and turbulence are present. +In order to investigate the origin and properties of the fluctu- +ations observed in the mass accretion rate, we calculated PSDs +of ˙M(r, t) at several radial shells. These are shown for selected +radii and for all the simulations in Figure 10, for the same in- +terval used for the analysis of the synthetic Bremsstrahlung light +curve an using the same methodology as in Section 3.3. +It is evident that in general the spectrum at event-horizon +scales differs significantly from that at large distances. In most +of the panels it is possible to see higher frequencies increasingly +populated as one moves closer to the black hole. This can be in- +terpreted as the transfer of energy from longer lower frequency +modes to smaller and faster modes, and could be due to the +change in the characteristic scale of the system as the fluid moves +inwards, as well as to the development of turbulence. Cases pro- +ducing shocks (ℓ0 = 3.25 and δ > 1) show a larger power, and the +spectrum at the smaller radius acquires the form of a power law +∝ r−2 or steeper. The rest of cases with ℓ0 = 2.25 show a flatter +spectrum close to white noise. Similarly as for Bremsstrahlung +spectrograms, for cases with ℓ0 = 0 and δ = 0.01, 0.1 the +spectrum of +˙M has a very small power and is dominated by +oscillation peaks. These appear within r ≤ 15 M, indicating +that, for almost unperturbed Bondi-like accretion, fluctuations in +Bremsstrahlung emission do appear related to these fluctuations +in ˙M close to the horizon. +In contrast, our results suggest that for sufficiently large per- +turbations, as those injected manually or as those produced by +the movement of the spiral shock, a red noise spectrum will be +recovered, regardless of the injected perturbation spectrum. +4. Discussion and Conclusions +In this work we used 3D GRHD simulations to study the struc- +ture and variability patterns in perturbed transonic accretion +flows with low-angular momentum. Our aim was to explore an +Article number, page 11 of 17 + +A&A proofs: manuscript no. main +0 +10 +20 +30 +40 +50 +t [hr] +50000 +52000 +54000 +56000 +58000 +60000 +t [M] +1033 +1034 +LBR [erg s +1] +l0p001 +l2p001 +l3p001 +l0p01 +l2p01 +l3p01 +l0p1 +l2p1 +l3p1 +l0p10 +l2p10 +l3p10 +Fig. 8. Synthetic light curves of total Bremsstrahlung luminosity using the parameters of Sgr A*, for all simulations. The curve corresponding to +l0p001 overlaps completely to that of l0p01. The upper horizontal axis displays the time in hours. +10 +3 +10 +2 +f [Hz] +10 +2 +10 +1 +f [M +1] +10 +8 +10 +6 +10 +4 +10 +2 +100 +102 +104 +106 +PSD(LBR) +f +2 +f +4 +0 +5 +10 +15 +20 +[min] +0 +10 +20 +30 +40 +50 +60 +[M] +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +corr(LBR, LBR) +l0p001 +l2p001 +l3p001 +l0p01 +l2p01 +l3p01 +l0p1 +l2p1 +l3p1 +l0p10 +l2p10 +l3p10 +Fig. 9. Spectrograms (left) and correlation functions (right) of the +bremsstrahlung light curve proxy of all simulations. Dashed lines with +slopes of power laws are shown for comparison in the left panel, and +bound the region between ±1/e in the right panel. The upper horizontal +axes have been scaled for Sgr A*. +accretion scenario that generalizes torus simulations, is more +consistent with the properties of stellar wind-fed accretion, and +is controlled by a reduced set of parameters. Our simulation +setup also aims to overcome two of the known limitations of the +BHT simulation paradigm, namely, the secular decrease in torus +mass that complicates long-term variability studies and the artifi- +ciality of the medium beyond the close vicinity of the black hole. +We evaluated the general properties of this accretion scenario us- +ing several diagnostics, namely, (1) time series of the mass ac- +cretion rate and angular momentum flux through the event hori- +zon, (2) shell- and time-averaged profiles of several quantities +of interest, and (3) a synthetic Bremsstrahlung light curve used +to analyze its variability properties. We also investigated the 3- +dimensional morphology of the models, including the location +of shocks and sonic surfaces. +Our models, contrary to BHT simulations, have accretion +rates that do not decay exponentially, allowing for long-term +variability studies. We observe that ˙M decreases significantly for +models with larger angular momentum and perturbation ampli- +tude. This is consistent with the additional centrifugal support +provided by angular velocities and the additional pressure sup- +port due to heating caused by turbulence and shocks, that are +also more important for the same models. The reduction in mass +accretion rate for larger angular momentum models also results +in a smaller net angular momentum flux through the horizon, +suggesting that there is a finite value of ℓ0 that maximizes the +accretion of angular momentum. +The fact that these models are fed from solutions that ex- +tend to infinity allows to relate the mass accretion rate at horizon +scales to the Bondi accretion rate from the medium properties +at large scales. This in turn permits to obtain tighter constraints +on the models. For example, density scales need to match large +scale fluid properties in addition to electromagnetic flux con- +straints derived from radiative transfer calculations. +In addition to their significant variations in accretion rate, the +flows we studied possess a rich phenomenology and are in some +respects qualitatively different both from BHT simulations and +from unperturbed transonic flows. Some of the salient features +we observe include outflowing toroidal structures, turbulence, +shocks, filaments, and multiple sonic transitions. +Deviations from the transonic solutions used as initial con- +ditions are in some cases large enough to lead to different av- +eraged temperature and velocity profiles. In particular, models +with large perturbations and angular momentum deviate from +the isentropic temperature profiles and transition to profiles sim- +ilar to those of an ADAF. For the cases initialized from complete +solutions (ℓ0 = 0 and ℓ0 = 2.25), the radius at which the transi- +tion occurs seems related to the amplitude of perturbations. This +can be explained from the instability of supersonic spherical ac- +cretion to non-radial perturbations Kovalenko & Eremin (1998). +These perturbations grow without limit with smaller radii, pro- +ducing the ‘ADAF transition’ when they become sufficiently +large to produce shocks and generate entropy, which depends +on the injected perturbation amplitude. +For the cases initialized from the incomplete solutions with +small perturbation amplitudes, large scale spiral shocks are pro- +duced by the interaction between the inflowing fluid and the cen- +trifugal barrier. They likely play an important role in transporting +angular momentum outwards thus enabling accretion of matter +and the transition to a Keplerian rotation profile at small radii in +the absence of a magnetic field. +In the context of chaotic cold accretion (e.g. Gaspari et al. +2017; Prasad et al. 2017), it has been suggested that inelastic col- +lisions between clouds can cancel angular momentum, leading +to an increased accretion rate. In the simulations presented here, +Article number, page 12 of 17 + +Héctor R. Olivares S. et al.: Perturbed Transonic Accretion +10 +11 +10 +9 +10 +7 +10 +5 +10 +3 +10 +1 +101 +PSD +l0p001 +l2p001 +l3p001 +r = 750M +r = 500M +r = 300M +r = 150M +r = 50M +r = 15M +r = 5M +10 +11 +10 +9 +10 +7 +10 +5 +10 +3 +10 +1 +101 +PSD +l0p01 +l2p01 +l3p01 +10 +11 +10 +9 +10 +7 +10 +5 +10 +3 +10 +1 +101 +PSD +l0p1 +f +2 +l2p1 +l3p1 +10 +3 +10 +2 +f [M +1] +10 +11 +10 +9 +10 +7 +10 +5 +10 +3 +10 +1 +101 +PSD +l0p10 +10 +3 +10 +2 +f [M +1] +l2p10 +10 +3 +10 +2 +f [M +1] +l3p10 +Fig. 10. Power spectral density of the mass accretion rate measured at different radii for all of the simulations, during the interval t/M ∈ +[50 000, 60 000]. The dashed line represents a power law ∝ f −2. +Article number, page 13 of 17 + +A&A proofs: manuscript no. main +shocks could be expected to play a similar role; however, even +simulations where shocks are present show the same trend that +relates larger perturbations and angular momentum with smaller +accretion rates. The reason could be that shocked simulations are +precisely those with larger perturbations and angular momen- +tum, and this effect needs to compete with the additional support +provided by angular velocities and the pressure from gas heated +by shocks and turbulence. +As could be expected, the different qualitative behavior re- +sults in different variability properties of the simulations. How- +ever, we found that for sufficiently large angular momentum and +perturbations, a red noise spectrum is robustly recovered even +from a white noise injection perturbations. +Our simulations are complementary to similar hydrodynami- +cal simulations carried out by other authors. Ressler et al. (2018) +uses a conservative hydrodynamics code to study the formation +of the accretion flow onto Sgr A* in which matter is constantly +supplied by stars on orbits around the central black hole. This +setup leads to a somewhat chaotic accretion. Although their sim- +ulation domain is much larger in comparison to ours, their inner +boundary overlaps with our outer boundary. Ressler et al. (2018) +obtained solutions with density and temperature power-law pro- +files ∝ r−1 which is different compared to our results. Also, the +angular momentum in our simulations is significantly lower than +the value they obtain at comparable radii (In their Figure 14 +Ressler et al. (2018) reports that the ℓ ≈ 0.4 − 0.5ℓK at the inner +boundary). These differences in the profiles could be explained +by the differences in the physical scenario considered. In their +case, these include stronger rotation, the presence of important +outflows, as well as line and Bremsstrahlung cooling, which be- +comes important at the scales they consider. Similarly, very large +scale simulations performed by Guo et al. (2022) study the for- +mation of the accretion pattern at event horizon scales following +material from the Bondi radius scale in elliptical galaxies such +as M87. The information obtained from these works and future +large scale simulations can be incorporated to smaller scale sim- +ulation setups as those presented in this work, e.g. by specifying +initial density profiles and the spatial and temporal spectrum of +injected perturbations. +In a setting more similar to ours, Suková et al. (2017) uses +2D and 3D conservative GRHD simulations to study low angular +momentum flows on closer to horizon scales, but without man- +ually injecting perturbations. In the tests we performed while +implementing our initial condition we have recovered the gen- +eral behavior of some of their 2D models, although there were +some differences in implementation and parameter choices, that +we describe in Appendix A. +The inclusion of magnetic fields in the simulations will be +presented in a forthcoming publication; however, there are a few +expectations we can draw from our hydrodynamic models that +could be relevant for observations. For example, the fact that the +accretion pattern is almost isotropic for cases with low angular +momentum ℓ0 ≤ 2.25 may results in images that are also inde- +pendent from orientation, in particular contrast to Standard and +Normal Evolution (SANE) BHT models. In addition, the possi- +bility of producing synthetic synchrotron lightcurves that do not +suffer from secular torus depletion may contribute to some extent +to unravel the ongoing discussion on the suitability of GRMHD +models to met the tight variability constraints given by 230 GHz +observations of Sgr A* (Event Horizon Telescope Collaboration +et al. 2022). +Overall, we believe our simulations are an important middle- +step towards obtaining more realistic models of relativistic ac- +cretion flows in which matter is supplied by the turbulent inter- +stellar medium. +Acknowlegdements +We thank Jesse Vos, Aristomenis Yfantis, Alejandra Jimenez- +Rosales, Christiaan Brinkerink and other members of the EHT +group at Radboud University for discussions. We also thank +Jordy Davelaar and Agnieszka Janiuk for their comments. HROS +was supported part by a Virtual Institute of Accretion (VIA) +postdoctoral fellowship from the Netherlands Research School +for Astronomy (NOVA). We acknowledge that the results of this +research have been achieved using the DECI resource Snellius +based in the Netherlands at SURF with support from the PRACE +AISBL. This work made use of the following software libraries +not cited in the text: MATPLOTLIB (Hunter 2007) and NumPy +(Harris et al. 2020). This research has made use of NASA’s As- +trophysics Data System. +References +Aguayo-Ortiz, A., Tejeda, E., Sarbach, O., & López-Cámara, D. 2021, Monthly +Notices of the Royal Astronomical Society, 504, 5039 +Baganoff, F. K., Maeda, Y., Morris, M., et al. 2003, The Astrophysical Journal, +591, 891, aDS Bibcode: 2003ApJ...591..891B +Bondi, H. 1952, Monthly Notices of the Royal Astronomical Society, 112, 195 +Chakrabarti, S. K. 1989, The Astrophysical Journal, 347, 365 +Chakrabarti, S. K. 1996, Astrophys. J., 471, 237 +Chakrabarti, S. K. & Das, S. 2004, Monthly Notices of the Royal Astronomical +Society, 349, 649 +Chakrabarti, S. 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Audio & Electroacoust, 15, 70, aDS Bibcode: +1967ITAE...15...70W +Wielgus, M., Moscibrodzka, M., Vos, J., et al. 2022, Astronomy & Astrophysics, +665, L6, publisher: EDP Sciences +Article number, page 15 of 17 + +A&A proofs: manuscript no. main +Appendix A: Initial conditions +Our initial data is constructed from the semi-analytic rotating +transonic solutions by Chakrabarti (1996). These can be thought +of as a generalization of Michel accretion Michel (1972) for a +rotating flow and for the Kerr metric. +To solve for the fluid properties, one assumes that stream- +lines are radial when projected on the meridional plane (that is, +θ-components of the velocity are neglected). Mass flux √−g ρur, +entropy, internal energy E = hut and angular momentum L = +−huφ = ℓE are conserved along the streamline. +Similarly as for Michel and Bondi accretion, once L is spec- +ified, E can be chosen so that the sonic radius is at the desired +position. The flow configuration is then found by solving a pair +of coupled nonlinear algebraic equations for the sound speed and +the radial velocity in the co-rotating frame at every point (equa- +tions 30a,b of Chakrabarti 1996). The equations do not constrain +the density scale, which can be chosen later. In our case, we set +it so that ρ = 1 at r = 6 M. +Accretion solutions of these equations have a wide variety +of qualitative behaviors. A class of solutions connects smoothly +infinity and the event horizon similarly as the Michel solution. +Other solutions possess incomplete interior or exterior branches +that can be connected by a shock, and there exist also incom- +plete solutions that have a sonic point but do not extend super- +sonically to the event horizon (see Figure 2 and Section 4.1 of +Chakrabarti 1996, for a complete description). For simplicity, we +always solve only the exterior branch of the solution. +To initialize our simulations, we solve the system on a grid +covering the range θ ∈ [0, π/2] with 300 points. The supersonic +part of each streamline was solved with 300 points and the sub- +sonic part with 100 points, both logarithmically spaced in radius. +The calculated part of the subsonic region extends beyond the +simulation domain by 10% in order to be be used for the bound- +ary conditions. +The approximation of projected radial streamlines is, in gen- +eral, inconsistent with vertical equilibrium, however it holds on +the equatorial plane. For this reason, we choose an angular mo- +mentum profile with a sharp peak at the equator that decays to +ℓ = 0 at the poles, where again vertical equilibrium is fulfilled +(equation (1)). +In order to evaluate the adequacy of this approximation and +to test that the solution reproduced the expected qualitative be- +havior, we performed 2D simulations evolving only the initial +condition without injecting any perturbation. Figure A.1 shows +2D maps of rest-mass density and mass accretion rate per θ-angle +for some of the 2D simulations we performed for a = 0.95 and +different angular momenta. As expected, for those cases in which +the equatorial solution connects the event horizon and infinity +(ℓ = 1.75, 2.25), the artificial initial condition quickly relaxes +to a true solution in vertical equilibrium and remains stable un- +til the end of the simulation. Figure A.2 shows the evolution of +the radial 4-velocity profile of the unperturbed solution used for +simulations with ℓ = 2.25 at different latitudes. At ∼ 150 M, +the configuration has already relaxed to a stationary flow. This is +sufficiently adequate for our simulations, which have durations +more than 100 times longer. +For incomplete solutions (ℓ = 2.75, 3.25), material accret- +ing on the equatorial plane starts piling up due to the centrifugal +barrier, while accretion continues through the poles. The dense +toroidal structure that forms is different from the tori commonly +use in simulations in that it consists of ‘outflowing’ unbound +material, which produces a shock when interacting with the in- +coming accretion flow. Since there is no cooling, these struc- +tures grow until the end of the simulation (Molteni et al. 1996). +As described in Section 3.2, these shocks are present as well in +3D simulations; however, the lack of azimuthal symmetry intro- +duces important differences, such as the presence of turbulence +in the φ-direction and the change in shape of the shock from +spheroidal to spiral. +Our 2D unperturbed simulations show a behavior that is con- +sistent with that reported by Suková et al. (2017). A difference +with respect to that work is that their initial condition is a Bondi +flow to which rotation has been added, while in our case it is +a solution including rotation in a more self-consistent way (al- +beit exact only on the equatorial plane and the poles), for which +the assumption of low angular momentum is not necessary. An- +other difference is that Suková et al. (2017) is largely focused on +studying the parameter regime that produces oscillating shocks, +which we have not explored. +Appendix B: Boundary conditions +To emulate the turbulent flow entering from the boundary, we +filled the ghost zones with the same transonic solution used as +initial condition and added time-dependent noise in the form of +Gaussian random fields (GRFs). These fields are generated as +the superposition of plane waves with random phases. +The usual way of generating a time-dependent GRF in N +dimensions is by Fourier-transforming white noise in N + 1 di- +mensions, multiplying the Fourier transform by the desired PSD +and then transform back. One can then ‘play’ the time depen- +dent noise by successively applying slices of the resulting N +1- +dimensional array. +Although this way of generating GRFs is very fast due to the +elegance of the Fast-Fourier Transform (FFT) algorithm, it has +some disadvantages that lead us to follow a different procedure. +First, storing a large 4-dimensional array and communicating +it among different parallel processes to perform interpolations +can be a source of implementation and performance problems, +and second, we do not desire to apply the noise in a full three- +dimensional box, but only on the outer ghost cells, which make +an almost two-dimensional spherical shell embedded in three- +dimensional space. +For this reason, we build the GRF by directly evaluating a se- +ries of sine functions corresponding to plane waves with random +phases at the cells of interest. This sum has the form +GRF(t, xi) = +Nk +� +k1,k2,k3=−Nk +Ak sin +� 2π +λmax +(k · x − f(k) t − ϕk) +� +, +(B.1) +where k is a three-dimensional vector of integers, λmax is the +maximum wavelength, f(k) is a function determined by a user- +defined dispersion relation, and ϕk is the random phase corre- +sponding to k. We draw random phases from a uniform dis- +tribution over [0, 1) using a pseudo-random number generator +with a fixed seed, which allows to use the same random phases +without the need to store them between restarts. In order to en- +sure causality, we use the constant dispersion relation f(k) = cs, +where cs is the sound speed at the simulation boundary, although +more complicated relations are also possible. The coefficients Ak +are set according to the desired power spectral density S k, as +Ak ∝ (S k)1/2, and can be normalized to give the desired rms per- +turbation amplitude. We set Ak = 0 for k = (0, 0, 0) in order to +keep the average of the GRF to zero. +Article number, page 16 of 17 + +Héctor R. Olivares S. et al.: Perturbed Transonic Accretion +Fig. A.1. Rest-mass density and mass accretion rate per θ angle for 2D evolutions of unperturbed initial conditions with different angular momen- +tum. +Fig. A.2. Equatorial cuts of the radial velocity of background flow at +different times and latitudes for the case ℓ = 2.25. +In our simulation, we generate 3 GRFs to perturb the three +spatial components of the 4-velocity in Cartesian coordinates. +We then transform them to the code coordinates and add them +only to the angular components of the velocity given by the back- +ground transonic solution. +When giving up on the FFT, we pay the price of having to +evaluate a large number of transcendental functions, which can +slow down the code significantly. We find that Nk = 5 gives a +negligible slow down and the smallest-wavelength mode has a +size comparable to ∼ 1.6 cells of the outer boundary. +This leads to Nθ×Nφ×Nghost×Nfields×(2Nk+1)3 = 12 266 496 +evaluations of the sine function per time step, or 340 736 eval- +uations for each of the 8 × 8 × 8 AMR blocks at the boundary. +Here, Nghost = 4 is the number of ghost zones in the radial direc- +tion and Nfields = 3, since each of the spatial components of the +velocity is perturbed with a different GRF. +Finally, it is worth mentioning that, being a sum of peri- +odic functions, the noise is also periodic with the period of the +longest wavelength mode. Although in general this will not re- +sult in a periodic behavior of the simulation due to the changing +chaotic dynamics inside the domain, it may be desirable to pro- +duce non-periodic noise models. One possibility could be chang- +ing slightly the dispersion relation so that the period of some of +the modes is an irrational multiple of that of others. The search +for more appropriate non-periodic noise models is, however, out +of the scope of this work. +Article number, page 17 of 17 + +l= 1.75 +l = 2.25 +l = 2.75 +l=3.25 +75 +2 +t= 4000M +50 - +1 +25 - +0 +log1op +[W] +-0 +N +-2 +-25 - +-50 +-3 +-75 + +75 +20 +50 - +10 +25 - +ep/wp +[W] +0 +:0 +N +-25 - +-10 +-50 - +-75 +-20 +0 +50 +100 +150 +0 +50 +100 +150 +0 +50 +100 +150 +0 +50 +100 +150 +x[M] +x[M] +x[M] +x[M]-0.4 +-0.5 +-0.6 +-0.7 +t= 0, θ = π/3 +t= 150 M,0 = π/3 +- +t= 1000 M, θ = π/3 +-0.8 +t = 0 M, θ = π/2 +t = 150 M, 0 = π/2 +t = 1000 M, θ = π/2 +-0.9 +2 +3 +4 +5 +6 +7 +8 +9 \ No newline at end of file diff --git a/M9FLT4oBgHgl3EQfNi8W/content/tmp_files/load_file.txt b/M9FLT4oBgHgl3EQfNi8W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e697570fe4931132bb87da8f1ed06bd2c310dcc8 --- /dev/null +++ b/M9FLT4oBgHgl3EQfNi8W/content/tmp_files/load_file.txt @@ -0,0 +1,935 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf,len=934 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' main ©ESO 2023 January 31, 2023 General relativistic hydrodynamic simulations of perturbed transonic accretion Héctor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Olivares S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1, Monika A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Mo´scibrodzka1, and Oliver Porth2 1 Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Box 9010, 6500 GL Nijmegen, The Netherlands 2 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands e-mail: holivares@science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='nl January 31, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Comparison of horizon-scale observations of Sgr A* and M87* with numerical simulations has provided considerable insight in their interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Most of these simulations are variations of the same physical scenario consisting of a rotation supported torus seeded with a poloidal magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' However, this approach has several well known limitations such as secular decreasing trends in mass accretion rate that render long term variability studies difficult, a lack of connection with the large-scale accretion flow which is replaced by an artificial medium emulating vacuum, and important differences with respect to the predictions of models of accretion onto Sgr A* fed by stellar winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We aim to study the flow patterns that arise at horizon scales in more general accretion scenarios, that have a clearer connection with the large scale flow and are at the same time controlled by a reduced set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' As a first step in this direction, we perform three dimensional general relativistic hydrodynamic simulations of rotating tran- sonic flows with velocity perturbations injected from a spherical boundary located far away from the central object (1000 gravitational radii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We study the general properties of these flows with varying angular momentum and perturbation amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We analyze time series of mass and angular momentum radial fluxes, angle- and time-averaged profiles, and synthetic Bremsstrahlung lightcurves, as well as the three-dimensional structure of the flow, and quantify shock- and sonic transitions in the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We observe a rich phenomenology in accretion patterns, that includes smooth Bondi-like flows, turbulent torus-like struc- tures, shocks, filaments, and complex sonic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For sufficiently large perturbations and angular momentum, radial profiles deviate from the constant entropy and constant angular momentum profiles used for initialization and resemble those of advection dominated accretion flows, showing evidence of entropy generation and angular momentum redistribution not mediated by magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Time series do not show the secular decreasing trend and are suitable for long-term variability studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We see that the fluctua- tions are amplified and extend further in frequency than the injected spectrum, producing a red noise spectrum both for mass accretion rate and the synthetic light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We present a simulation setup that can produce a wide variety of flow patterns at horizon scales and incorporate information from large scale accretion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Future inclusion of magnetic fields and radiative cooling could make this type of simulations a viable alternative for numerical modeling of general low-luminosity active galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' accretion, accretion disks - black hole physics - relativistic processes - methods: numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Introduction Accretion onto compact objects such as black holes and neutron stars powers some of the most spectacular phenomena in astro- physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' While the focus of numerous studies in accretion theory is on how matter and angular momentum are transported through an accretion disk, much less studies put into focus the formation of the accretion disk itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In particular numerical simulations of accretion disk formation are encumbered by the large scale sep- aration between circularization radius of incoming matter and the size of the accretor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' There are certain applications however where the accreted matter has comparatively low angular mo- mentum, leading to circularization radii not much larger than the accretor itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Prime examples are the high-mass X-ray binaries (HMXBs) and chaotic stellar wind-fed accretion in galactic nu- clei such as our own Galactic center Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' A central role in the interpretation of the event-horizon- scale observations of Sgr A* and M87* by the EHT Collabo- ration is played by general general relativistic magnetohydrody- namic (GRMHD) simulations (Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2019, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' To date, all of the models in the simulation library for Sgr A* and most of those used for M87* follow variations of the same initial conditions of a rotation-supported torus (Fishbone & Moncrief 1976) seeded with a weak poloidal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' While a lot of physical in- sights have already been gained by comparing observational data against GRMHD simulations – leading to increasingly tight con- straints of the parameters such as black hole mass, accretion rate, inclination and black hole spin (Event Horizon Telescope Col- laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2019, 2021, 2022) – there are several limitations intrinsic to the considered “black hole – torus (BHT)” simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For example, since they are initialized with a finite amount of matter contained in the torus, the matter content in the simu- lation decreases over time, accompanied by a corresponding de- crease in mass accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This secular trend renders the study of long-term variability difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This systematic is particularly important since the current set of GRMHD simulations produces highly varying lightcurves which are tightly constrained by the less variable data for Sgr A* (Event Horizon Telescope Collab- oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Wielgus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Article number, page 1 of 17 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='12020v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='HE] 27 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' main The most important limitation of the BHT simulations how- ever concerns physical realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' It is now widely believed that our galactic center black hole, Sgr A*, can be fed from the winds of ∼ 30 massive stars that orbit on the parsec scale (Quataert 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Cuadra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Ressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Whether an accretion disk (torus) forms in this scenario depends not only on the ini- tial wind parameters (Mo´scibrodzka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Shcherbakov & Baganoff 2010) but also on the interactions of the unbound winds which can give rise to shocks and hydrodynamic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The flow patterns of realistic stellar wind accretion models for low luminosity active galactic nuclei (AGNs) differ significantly from the BHT scenario described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In stellar wind accre- tion, material forms clumpy structures and has a broad distribu- tion of angular momentum without sufficient time to circularize, and is not generally rotation supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Instead, it is accreted mainly due to an originally low angular momentum and remains in large part unbound (Ressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This latter property is shared by different models of accretion from large scales such as the constant-entropy solutions by Bondi (1952);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Michel (1972);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Chakrabarti (1996) and models that include dissipation such as the well-known advection dominated accretion flows (ADAFs) (Narayan & Yi 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Recent magnetohydrodynamics (MHD) simulations that focus on the large scale dynamics have revealed further differences to the standard BHT scenario: while mag- netic fields from stellar winds are initially weak and passively advected, at horizon scales they accumulate and become dynam- ically important and start to regulate accretion in a way similar to Magnetically Arrested Disks (MADs) (Ressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2020a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The lack of a predominant angular momentum or magnetic field direction leads to erratic changes in the orientation of the accre- tion disk (Ressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Similar transient behavior can be seen in the direction and power of the jet before the formation of a steady jet (Lalakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Simulations of accretion from kpc scales onto the black hole (BH) event horizon have also shown that the accretion flow in elliptic galaxies as M87 can acquire a variety of patterns that range from rotation-supported disks to chaotic streams (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In general, simulation-based studies of the horizon-scale structure of the accretion flow resulting from large-scale feed- ing present the computational challenge of having to simulate length and time scales spanning ∼ 6 orders of magnitude, or dealing with uncertain factors such as the details of stellar winds astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' It would be therefore desirable to gain more in- sight on the properties of the accretion flow from the study of transonic solutions connecting the event horizon to infinity, in a similar manner as the theory of accretion disks has benefited from the study of analytic solutions for fluids in circular mo- tion around black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Depending on the specific angular mo- mentum and energy, analytic studies of trans-sonic low angu- lar momentum accretion flows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Fukue 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Chakrabarti 1989, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Chakrabarti & Das 2004) have revealed different regimes characterized by smooth Bondi-like flows, standing ac- cretion shocks or the formation of circularized tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Furthermore, the solutions have been studied including the effects of viscosity (Chakrabarti & Molteni 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Lanzafame et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 1998), radiative cooling (Molteni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Okuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2004) and magnetic fields (Proga & Begelman 2003b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Okuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Mitra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2022), often with particular focus on the stability and dynam- ics of the accretion shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Numerical simulations of transonic hydrodynamic solutions were presented more recently by Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2017, 2019) for the Schwarzschild and Kerr spacetimes, showing that certain perturbations can trigger long-surviving shocks at the location of predicted standing shocks (Chakrabarti 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In fact, a realistic approach to the problem should consider the destabilizing effect of inhomogeneities in the surrounding medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The stability of spherical Bondi accretion has been studied analytically in a number of works;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' see for instance Mon- crief (1980) and Kovalenko & Eremin (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In the latter work, it is shown that this solution is unstable for non-radial perturba- tions, although for the instability to manifest itself the size of the accretor needs to be sufficiently small compared to the Bondi ra- dius, precisely as it is the case for the nearest supermassive black holes (SMBHs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In this paper, we study a simulation setup that aims to ad- dress the above described limitations of the BHT paradigm and to facilitate the incorporation of information gained from larger scale simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This model depends on a reduced set of pa- rameters that can in principle be chosen to match the properties inferred for known SMBHs such as Sgr A*, M87, and other tar- gets of the Event Horizon Telescope (EHT) and the planned new generation Event Horizon Telescope (ngEHT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' By incorporat- ing time-dependent properties of the surrounding medium in the boundary conditions, the simulation domain can be of a mod- est size comparable to that of existing GRMHD simulations in the EHT library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The simulations presented here are run in pure general relativistic hydrodynamics (GRHD), that is, with zero magnetic field, as an intermediate step towards GRMHD simula- tions that will be presented in a forthcoming work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We show that the proposed setup produces steady time series that are in princi- ple suitable for long-term variability studies, and exhibits a rich phenomenology that can differ significantly both from typical BHT simulations and from unperturbed Bondi-like accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We describe this setup in Section 2 and provide a justification for the physical parameters employed (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In Section 3, we report on the properties observed in the simulations, such as time series of mass and angular momentum accretion rates and radial profiles (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1), three-dimensional morphology, including the presence of shocks and complex sonic structures (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='2) and variability properties 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We summarize and discuss our results in Section 4, and complement this work with more information on the simulation setup in the Appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Simulation setup To explore the flow patterns arising from transonic accretion of an inhomogeneous interstellar medium, we perform three- dimensional GRHD simulations that continuously inject matter from an outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We employ units such that G = c = 1, so that the gravitational radius rg = GM/c2 and the gravitational timescale tg = rg/c are rg = tg = M, where M is the mass of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We adopt a Kerr spacetime with dimensionless spin parameter a � J/M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='95 and the event horizon located at rH/M = 1 + √ 1 − a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For all of our simulations, we set the sonic radius to rs = 500 M and place the boundary at r = 1000 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We initialize the simulations with a smooth quasi-stationary back- ground solution with a latitude-dependent angular momentum profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Following Proga & Begelman (2003b) we adopt an an- gular momentum profile that peaks at the equator and vanishes at the poles (see Appendix A for a detailed discussion of the background flow): ℓ(θ) = ℓ0(1 − | cos θ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (1) The background flow is characterized by only two parameters, the angular momentum at the equator ℓ0, and the sonic radius (or alternatively fluid internal energy E = hut at the equator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Article number, page 2 of 17 Héctor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Olivares S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' : Perturbed Transonic Accretion Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Table of runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' δ ℓ0 = 0 ℓ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 ℓ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 a = 0 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='95 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='01 l0p001 l2p001 l3p001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1 l0p01 l2p01 l3p01 1 l0p1 l2p1 l3p1 10 l0p10 l2p10 l3p10 To model inhomogeneities in the interstellar medium, we in- ject perturbations of varying amplitude to the (tangential-) veloc- ity components at the outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Perturbations are modeled as time-varying Gaussian random field with a white noise spec- trum S |δu|(k) ∼ constant , (2) in the wavelength range λk/M = 2π/kM ∈ [214, 2400], and in the frequency range fk ∈ [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='7, 41] × 10−5M−1 (see Appendix B for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The remaining fluid variables at the boundary are set consistently with the initial condition, and are therefore constantly injecting matter that should preserve the initial state in absence of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The injected noise is controlled by the parameter δ which specifies the ratio of the variance of velocity perturbations to the radial component of the 4-velocity of the unperturbed flow at the boundary, δ = ⟨δu2⟩1/2/ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We adopt the adiabatic index ˆγ = 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We perform several simulations varying ℓ0 and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In order to isolate the effect of perturbations, we run a set of simulations in a Bondi-Michel accretion scenario, that is, ℓ0 = 0 and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The list of runs and parameters used is displayed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' To run the simulations, we use the code BHAC (Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Olivares et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We use a spherical polar grid in mod- ified Kerr-Schild coordinates with logarithmic spacing in radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The base resolution is Nr × Nθ × Nφ = 96 × 48 × 48 and we em- ploy 3 levels of Adaptive Mesh Refinement (AMR), obtaining an effective resolution of 384 × 192 × 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The inner bound- ary is located inside of the central black hole event horizon, at r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='19 M, in order to avoid boundary effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We employ a finite volume method with piecewise parabolic reconstruction (PPM), a total variation diminishing Lax-Friedrichs (TVDLF) approximate Riemann solver and a two-step method for time in- tegration (see Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2017, for more details on coordinates and numerical methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' To reduce the cost of simulations, we evolve a passive tracer f that is initialized as f = 0 inside the domain and f = 1 for the injected matter at the boundary, and evolve only those blocks of 8 × 8 × 8 cells for which f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1 or which are surrounded by blocks that satisfy this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For all of the simulations, this tracer reaches the event horizon at t ≲ 30 000 M, after which the simulation domain becomes active everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We continue the evolution up to t = 60 000 M, which corresponds to a total simulation time of nearly 5 free-fall timescales from the sonic radius, tff = π(rs/2)3/2 ≈ 12 418 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Physical parameters Accretion onto an object at rest with respect to a spherically symmetric, asymptotically uniform medium can be considered to start at the Bondi radius, rB, the distance at which the asymp- totic sound speed c∞ equals the escape velocity, that is, rB = 2GM/c2 ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Temperatures inferred from Chandra X-ray observa- tions of the hot gas surrounding Sgr A* (Baganoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2003) and the central black hole of M87 (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2015), combined with the assumption of a monoatomic ideal gas with ˆγ = 5/3, yield estimates for the Bondi radius of 6 × 105 M and 4 × 105 M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The several orders of magnitude separation be- tween the Bondi radius and the event horizon makes simulations of accretion from the Bondi radius onto SMBHs prohibitive for most numerical codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In practice, due to temperature gradients, the local sound speed cs does not coincide with escape veloc- ity at the Bondi radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This happens instead at the sonic radius rs = 2GM/c2 s, which marks the transition from subsonic to su- personic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In Newtonian hydrodynamics, the case ˆγ = 5/3 is degenerate and pushes the sonic radius to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' However, by incorporating relativistic corrections and assuming c∞ ≪ c, it takes a finite value that can be approximated as rs ≈ 3Mc/4c∞ (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Rezzolla & Zanotti 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For the value of c∞ reported above, this corresponds to rs ≈ 409M for Sgr A* and rs ≈ 335M for M87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The value rs = 500M in our simulations is chosen ac- cordingly within the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Following the same relativistic Bondi models, the dimen- sionless temperature at the sonic radius can be estimated to be Θ � kBT/mc2 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='3 × 10−4 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='8 × 10−4, where m is the ion mass and kB is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For monoatomic hydro- gen, this corresponds to T ≈ 8×109K – 1010K (higher values cor- respond to M87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The dimensionless temperatures attained at the sonic radius for our simulations are similarly Θ ≈ 8 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Ex- pecting it to to increase by orders of magnitude when approach- ing the black hole, we set ˆγ = 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The effective adiabatic in- dex in this regime is very dependent on uncertain factors such as cooling and the ratio between ion and electron temperatures, and a more self-consistent generation the background solution may require the use of a relativistic equation of state as in Aguayo- Ortiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' However, a fully accurate modeling of these effects is beyond the scope of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Turning to the second parameter, ℓ0, the specific angular mo- mentum from stellar stellar wind accretion can be roughly esti- mated by ℓ ≃ r2 accΩ/4 (Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Here Ω is the orbital angular velocity of the star and racc = 2GM/v2 w is the accretion radius for an assumed cold wind with velocity vw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Scaled to ge- ometric units and for a star in Keplerian orbit with semi-major axis a we have ℓ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5 � a pc �−3/2 � vw 1000km s−1 �−4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (3) Thus low angular momentum flows are indeed expected for these fiducial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Focusing on a particular source, it was argued in Mo´scibrodzka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2006) that the stellar complex known as IRS 13 E3 (Maillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2004) exerts the the strongest ram- pressure at the Galactic center which renders it the dominant wind accretion source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Thus taking IRS 13 E3 with fiducial wind velocity of 1000km s−1 as exemplary case and using the orbital fits by Muži´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2008), we obtain ℓ in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1−16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This large spread is caused by the large range of admitted semi-major axes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1pc − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='6pc reported in Muži´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' As there are large uncertainties associated with the value of ℓ in the Galactic center and to gain insight into the parameter de- pendence, we here investigate three cases that correspond to the different qualitative behaviors of the background solution: non- rotating case (ℓ = 0), a rotating case where the solution is com- plete, that is, it connects smoothly infinity and the event hori- zon (ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25), and a rotating case with an incomplete solution (ℓ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Incomplete solutions of flows coming from infinity are expected either to pass through a shock and transition to an- other smooth solution that reaches horizon, or to represent flows Article number, page 3 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' main that are unstable in absence of viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For a sufficiently vis- cous flow, some of these incomplete solutions can transition to a torus (Chakrabarti 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' It should be noted that a complete so- lution can exist for ℓ even when there is a circularization radius rcirc > rH at which ℓ is equal to the Keplerian angular momen- tum, as it is the case for ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 (rcirc ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='6 M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The reason is that fluid elements have a nonzero radial velocity and depend- ing on their energy (part of which is internal) their centrifugal barrier is located further inside rcirc, and in some cases they can even cross smoothly the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Although Ressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2020b) showed that the orientation of the flow angular momentum at horizon scales can vary wildly, these variations occur on a scale of hundreds of years for Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The simulations presented here have a much shorter duration – comparable to 14 days for the same source – which justifies the assumption that the orientation of the angular momentum is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Finally, the most uncertain parameter is the amplitude of in- jected perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' To explore several possibilities, we have considered a wide range with cases varying by orders of mag- nitude with respect to the inflow velocity at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Global properties In this section we briefly discuss and compare the salient global features of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We start by computing time series of the mass and angular momentum flux through the event horizon ˙M(t) � � 2π 0 � π 0 ρur √−g dθ dφ , (4) ˙L(t) � � 2π 0 � π 0 T r φ √−g dθ dφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (5) To set a typical scale that can be compared with real systems, we normalize the accretion rate to the Bondi rate ˙MB = 4πλB(GM)2 ρ∞ c3∞ (6) where λB = 1 4 � 2 5 − 3ˆγ � 5−3ˆγ 2(ˆγ−1) , (7) and ρ∞ and c∞ are the density and sound speed at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In the units employed here, ρ is normalized so that ρ = 1 at r = 6 M, which leads to the numeric value ˙MB ≈ 246 code mass units per gravitational timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Figure 1 shows the time series in the interval t/M ∈ [30 000, 60 000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The first important feature shown in Figure 1 is the long-term stability of the horizon penetrating fluxes over the simulated timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' While a quasi-stationary state is expected due the constant mass supply at the inflow boundaries, it is reas- suring that accretion rates are nearly constant after ∼ 2 freefall timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' As it could be expected, there is a trend that relates a higher angular momentum with a lower mass accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' A larger amplitude of perturbations also appears to result in smaller ac- cretion rates, likely due to the extra angular momentum provided by perturbations, which also contribute to centrifugal support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For instance, the addition of δ = 10 perturbations for the ℓ = 0 case reduces the accretion rate to a value of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='75 ˙MB com- parable to that obtained for the simulations for ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 with smaller perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The simulation with largest perturbation and angular momentum possesses the smallest accretion rate, at ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 ˙MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Inspecting the accretion of angular momentum, the solutions show a surprising behavior: although it could be expected that a flow with larger angular momentum would result in more an- gular momentum accreted by the black hole, the simulations with ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 actually register slightly more angular momen- tum accretion than those with ℓ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Normalizing the angular momentum accretion rate by the mass accretion rate, as it ap- pears in the bottom panel of Figure 1, both cases show about the same value of ˙L/ ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The reason is likely that centrifugal support prevents matter from accreting and carrying angular momentum through the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In this respect, it is important to recall the qualitative difference between the unperturbed flow configu- rations corresponding to these two values: while ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 allows solutions that connect smoothly infinity with the event horizon, ℓ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 produces an incomplete solution which for the viscous case should connect to a rotation supported torus where the flow is stalled (Chakrabarti 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The time series in Figure 1 shows different variability prop- erties for each simulation, with those having higher ℓ and larger perturbations appearing more ‘noisy’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For the cases with ℓ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25, this can be attributed again to the fact that the unperturbed solution is incomplete, producing shocks and complex interac- tions between the flow close to the centrifugal barrier even when the injected perturbations are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' However, it is interesting to see that the most variable time series corresponds to ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25, for the simulation l2p10, where peaks in ˙M are sometimes even larger than the Bondi accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We will diagnose the vari- ability properties of the different solutions in more detail in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Simulations with δ ≤ 1 and ℓ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 show transient oscilla- tions near the time at which the innermost grids become active (t ≈ 30 000 M), and decrease in amplitude and frequency as the evolution proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' These are especially noticeable for the cases ℓ = 0 for ˙M and ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 for ˙L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For the other cases, the oscillations are masked by the larger perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' To quantify the departure of the perturbed solutions from the initial background solution, in Figure 2 we show the angle- and time-averaged radial profiles for all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' These are computed as ⟨q⟩(r, t) � � 2π 0 � π 0 q(r, θ, φ, t) √−g dθ dφ � 2π 0 � π 0 √−g dθ dφ , (8) where q = ρ, p/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We also show φ-averages of the rotation an- gular velocity on the equatorial plane Ω = uφ/ut, ⟨Ω⟩(r, t) � 1 2π � 2π 0 Ω(r, θ = π/2, φ, t) dθ dφ , (9) where axial symmetry has been used to eliminate the metric de- terminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' All quantities are then time-averaged over the inter- val t/M ∈ [50 000, 60 000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We also plot the radial profiles ex- pected for a self-similar ADAF model Narayan & Yi (1994) with ˆγ = 4/3 and no radiative cooling, as well as those of the unper- turbed Chakrabarti solutions with ℓ = 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 which are used as initial conditions and are exact on the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For all simulations, the density profiles (left column of Fig- ure 2) are well described an ADAF profile of ρ ∝ r−3/2 which Article number, page 4 of 17 Héctor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Olivares S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' : Perturbed Transonic Accretion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5 t/tff 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='50 M/MB l0p001 l2p001 l3p001 l0p01 l2p01 l3p01 l0p1 l2p1 l3p1 l0p10 l2p10 l3p10 30000 35000 40000 45000 50000 55000 60000 t [M] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5 L/M Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Mass and angular momentum flux through the event horizon for all of the simulations, starting from a time where perturbations have reached the event horizon for all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The upper horizontal scale measures time in units of the free-fall timescale from the sonic radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' also holds approximately for the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In particular, the profiles are inconsistent with the shallower convective so- lution ρ−1/2 indicating that convection is not important in our parameter regime (Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Although the averaged density profiles shown in Figure 2 are smooth, some of the simulations exhibit density jumps at in- dividual snapshots, in which case the profile possess the same slope at either side of the jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' These are present in the simula- tions with high angular momentum and large perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' As it will be discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='2, they are related to shocks which propagate outwards as they are smoothed away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Traces of these jumps are visible in the profiles of simulations with ℓ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 (especially of l3p10) at scales of r = 102 – 103 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' These are expanding shocks produced by the fluid colliding with the cen- trifugal barrier, as those studied for example by Suková et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The slow evolution timescales near the outer boundary prevent them from being smoothed by the time average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The averaged profiles of p/ρ ∝ Θ in the central column of Figure 2 show a more interesting behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' While initially they coincide with those of the Chakrabarti solutions, those corre- sponding to ℓ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 and δ = 10 gradually transition to the profile of the ADAF solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The clearest case is that of sim- ulations l0p10 and l2p10, which transition form the constant- entropy initial profile (∝ r−1/2 for the Bondi solution) to that of the ADAF model ∝ r−1 at r ≈ 40 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The rise of the temperature profile close to the black hole is likely a result of heating by shocks and turbulence, which trans- form to thermal energy the kinetic energy injected through the perturbations in the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' It can be noticed that the temper- ature profiles of l0p1 and l2p1 start rising as well and deviate from the initial profile at a shorter distance from the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This suggests that indeed the radius at which the transition to an ADAF-like profile occurs is related to the amplitude of perturba- tions in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The fact that l0p10 acquires an ADAF-like temperature profile close to the black hole is interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In fact, this simulation differs from the scenario studied by Narayan & Yi (1994) from which the self-similar solution is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Here there is no coherent disk-like structure and the average of Ω is close to zero (see rightmost panel of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' It is therefore surprising that the heating provided by incoherent shocks and turbulence results in a temperature profile similar to that of a coherent viscous rotating flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The rightmost column of Figure 2 shows the angular veloc- ity profile for all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' As expected, the profiles corre- sponding to the cases with zero angular momentum in the unper- turbed solution show negligible rotation velocity on the equato- rial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The rest of profiles behave in a similar way as those of dimensionless temperature: they follow the Chakrabarti con- stant angular momentum profile at large radii (yielding a power law slope of −2 far from the black hole) and transition to the ADAF-like Keplerian profile ∝ r−3/2 closer to the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In general, it appears that at large distances the system is well described by the adiabatic Bondi- and Chakrabarti-like so- lutions, while once perturbations become enough amplified by the geometry of the flow to produce shocks and turbulence, en- tropy production starts and the system becomes better described by ADAF-like profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Three-dimensional morphology Figures 3 and 4 show the density distribution for all the simula- tions on the equatorial and the meridional plane, respectively, at t = 60 000 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The four simulations with lowest ℓ and pertur- bation amplitude, l0p001, l2p001, l0p01, l2p01 are always smooth and highly symmetric, and are practically indistinguish- able from the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Changes start becoming visible for those with perturbations comparable to the incoming radial velocity, l0p1 and l2p1, for which near-radial filaments can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The simulations with higher angular momentum are qualita- tively different, as it could be expected due to the incompleteness of the Chakrabarti solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In all of the ℓ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 runs it is possi- ble to see the formation of a turbulent torus-like structure close to the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The larger the perturbations are, the more mis- aligned this structure becomes with respect to the large-scale an- gular momentum, which points into the +z direction (rightmost panels of Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Article number, page 5 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' main 10 3 10 2 10 1 100 101 r 3/2 = 0 10 3 10 2 p/ r 1 10 5 10 4 10 3 10 2 10 1 ( = /2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1 = 1 = 10 r 3/2 10 3 10 2 10 1 100 101 r 3/2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 10 3 10 2 p/ r 1 10 5 10 4 10 3 10 2 10 1 ( = /2) r 3/2 101 102 103 r [M] 10 3 10 2 10 1 100 101 r 3/2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 101 102 103 r [M] 10 3 10 2 p/ r 1 101 102 103 r [M] 10 5 10 4 10 3 10 2 10 1 ( = /2) r 3/2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Radial profiles of density (left column), dimensionless temperature (middle column), and equatorial angular velocity (right column) averaged over angle and time in the interval t/M ∈ [50 000, 60 000] for all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' From top to bottom, the columns correspond to ℓ0 = 0, ℓ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 and ℓ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The shaded regions indicate the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The dot-dashed lines show the power laws expected for an ADAF with γ = 4/3 and no radiative cooling, and the dashed lines are the profiles for the unperturbed configurations with constant angular momentum used as initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In most of the panels, the profiles for δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='01 and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1 overlap completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Despite the apparent similarity of these configurations with those in BHT simulations, they exhibit important differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' While for BHT simulations the toroidal structure is confined mainly by the equilibrium between gravity and the centrifugal force, for the simulations presented here it consists in large part of unbound outflowing matter that is confined by its interaction with inflowing matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In addition, while for BHT simulations most of the accretion flow occurs in the equatorial plane, here the toroidal structure is an obstacle for the inflowing matter, causing most of the accretion to occur through the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This behavior is consistent with that observed for similar systems in absence of magnetic fields, for example, by Proga & Begelman (2003a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Moscibrodzka & Proga (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Suková et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We expect as well that the inclusion of magnetic fields will reverse the situ- ation by producing accretion on the equatorial plane and a polar outflow (Proga & Begelman 2003b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Figures 5 and 6 show the relative pressure gradient as proxy for the location of shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The simulations with low ℓ0 and δ, (l0p001, l2p001, l0p01, l2p01) do not show important pres- sure gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In contrast, those with ℓ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 and δ ≤ 1 Article number, page 6 of 17 Héctor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Olivares S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' : Perturbed Transonic Accretion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Logarithmic density maps for all simulations at t = 60 000 M on the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Panels are organized in the same way as simulations in Table 2, that is, increasing angular momentum from left to right, and amplitude of perturbations from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Movies of simulations l0p10, l2p10, and l3p10, are available at https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='be/1TQV_aX13xE, https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='be/oOh2reL9yK0, and https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='be/ VmCc3ZnDxEM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (l3p001, l3p01, and l3p1) show clear spiral shocks This co- herent large scale shock does not form in the strongly perturbed case δ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The colormap in Figures 5 and 6 also allow to see sound waves traveling within the shocked regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' It is also interesting to examine to what extent the causal structure of the flow is preserved in presence of perturbations and high angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The dashed lines in Figures 5 and 6 mark the surfaces for which the 4-velocity of an observer at rest at infinity, ∂t = (1, 0, 0, 0), becomes null with respect to the sonic metric (Moncrief 1980), Gµν = ρ hcs � gµν + (1 − c2 s)uµuν � , (10) where ρ is the rest-mass density, h is the enthalpy, cs is the sound speed and uµ is the four velocity of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This condition is Article number, page 7 of 17 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='0 10p001 12p001 13p001 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5 dot6ol [W] :0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='0 50 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='0 100 10p01 12p01 13p01 50 - [W] :0 y 50 - 100 100 10p1 12p1 13p1 50 [W] 0 : 50 - 100 100 10p10 12p10 13p10 50 - [W] 0 y 50 - 100 100 0 100-100 0 100-100 0 100 x[M] x [M] x [M]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Similar as Figure 3, for the meridional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' analogous to that defining static surfaces such as ergoregions and event horizons for the spacetime metric gµν, and can be used to characterize transitions between subsonic and supersonic flows in an invariant way (Aguayo-Ortiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2021), especially in sit- uations that lack symmetries such as the perturbed flows studied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The sonic surface for the unperturbed solutions is a sphere with radius rs = 500 M, centered at the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Its structure is practically unchanged for the four cases with lowest ℓ0 and δ, (l0p001, l2p001, l0p01, l2p01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Cases with ℓ0 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 and δ = 1 (l0p1, l2p1) exhibit slight but noticeable changes in the shape of the sonic surface, although it remains close to rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This is remarkable since perturbations already have an amplitude simi- lar to the magnitude of the inflow radial velocity at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Only when the perturbation amplitude is ten times the inflow ra- dial velocity (l0p10, l2p10) we see large incursions of subsonic matter inside rs, as well as islands of supersonic (subsonic) flow within the former subsonic (supersonic) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Models with ℓ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 show a more complex causal struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The spiral shock produces an additional transition from su- personic to subsonic flow, and it can erase the original sonic sur- face as it propagates outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' However, downstream the flow Article number, page 8 of 17 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='0 10p001 12p001 13p001 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5 dot6ol 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='0 N 50 - 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='0 100 10p01 12p01 13p01 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' [W] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' N 50 - 100 100 10p1 12p1 13p1 50 [W] 0 N 50 100 100 10p10 12p10 13p10 50 - [W] N 50 - 100 100 0 100-100 0 100-100 0 100 x [M] x [M] x[M]Héctor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Olivares S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' : Perturbed Transonic Accretion 500 0 500 y [M] l0p001 l2p001 l3p001 500 0 500 y [M] l0p01 l2p01 l3p01 500 0 500 y [M] l0p1 l2p1 l3p1 500 0 500 x [M] 500 0 500 y [M] l0p10 500 0 500 x [M] l2p10 500 0 500 x [M] l3p10 3 2 1 0 log10(| p|/p) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Shocks and sonic surfaces for all simulations at t = 60 000 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The color scale displays the relative pressure gradient, which is used as a proxy for shock locations, and the dashed lines indicate the static limits of the sonic metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' can become supersonic again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The spiral structure can then pro- duce several sonic transitions between the distant regions, where matter is injected subsonically, and the event horizon, that needs to be crossed supersonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For instance, in the panel of Fig- ure 5 that corresponds to simulation l3p1, there can be even five sonic transitions when approaching the black hole from certain directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For the case with δ = 10, l3p10, the original sonic surface has disappeared completely, and a new one has formed closer to the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Also in this case, the spiral shock pro- duces more than one sonic transition in some directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In addition to the entropy increase due to shocks, turbu- lence could also play a role in heating the fluid and contribute to the transition from a constant entropy temperature profile to an ADAF-like profile, as shown in the middle panel of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In order to highlight the presence of turbulence, Figure 7 shows the z-component of the vorticity vector on the equatorial plane for simulations l3p001 and l0p10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The ℓ0 = 0 case shows vortic- Article number, page 9 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' main 500 0 500 z [M] l0p001 l2p001 l3p001 500 0 500 z [M] l0p01 l2p01 l3p01 500 0 500 z [M] l0p1 l2p1 l3p1 500 0 500 x [M] 500 0 500 z [M] l0p10 500 0 500 x [M] l2p10 500 0 500 x [M] l3p10 3 2 1 0 log10(| p|/p) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Similar as Figure 5, for the meridional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' ity sheets that can be associated with fluid streams approaching the black hole at different speeds, and suggest the emergence of smaller turbulent structures if simulated at a higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For the incomplete ℓ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 analytical solution the fluid is expected to form a torus at the circularization radius rcirc ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='8M, and no presence of fluid is expected at smaller radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' How- ever, in all of our simulations we observe the flow occupying this region without impediment, meaning that a means of angu- lar momentum redistribution is operating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The same can also be inferred from the rotation velocity profiles in the rightmost panel of Figure 2, where several of them transition from constant to Keplerian angular momentum profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This indicates that even in the absence of magnetic fields, and thus MRI and large scale Maxwell stresses, angular momentum redistribution occurs, and it can be attributed to shocks and turbulence that could easily appear in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Article number, page 10 of 17 Héctor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Olivares S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' : Perturbed Transonic Accretion 40 20 0 20 40 x [M] 40 20 0 20 40 y [M] l3p001 min: -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='58, max: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='31 40 20 0 20 40 x [M] l0p10 min: -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='56, max: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='050 z [M 1] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Vertical component of the vorticity on the equatorial plane for simulations l3p001 and l0p10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Variability properties To have a rough estimation of the observable properties of the variability in our simulations, we have computed synthetic X- ray light curves by integrating the total bremsstrahlung emissiv- ity from free-free electron-ion collisions Rybicki & Lightman (1986), ϵBR = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='54 × 10−9 Z2 � mi mp �1/2 � ni ρ 106 cm−3 �2 � p ρ �1/2 erg cm3 s , (11) where Z is the atomic number of ions, mi is the ion mass, and mp is the proton mass, and ni is scaling factor that relates the dimensionless code density ρ with the ion number density niρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The Gaunt factor has been assumed to be constant and equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The integration is performed as LBR = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='41×1038 � M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='15 × 106 M⊙ �3 � ϵBR Γ √γ d3x erg s , (12) where Γ is the Lorentz factor, and the prefactor comes from the conversion of volume in geometrized code units to physical units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The synthetic light curves within t/M ∈ [50 000, 60 000] for each simulation are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The parameters have been chosen for monoatomic hydrogen, with ni = 106 cm−3, and M as the mass of Sgr A*, M = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='15 × 106M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This gives luminosities that agree in order of magnitude with the ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='4 × 1033erg s−1 estimated by Baganoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For this source, the time in- terval corresponds to ≈ 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5 hours of observing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We cal- culated spectrograms in this interval using the Welch method (Welch 1967) with time windows overlapping over 128 points (= 128M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The power spectral densitys (PSDs) are shown in the left panel of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' It can be seen that, as expected, the power of fluctuations increases with the amplitude of the injected per- turbations and with the angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The two simulations with zero angular momentum and smallest perturbations show small frequency peaks that are lost into the noise for the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' As perturbations and angular momentum increase, it is possi- ble to observe a steepening in the slope of the PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Simulations with ℓ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 or δ = 10 show a very similar spectrum with a break from white noise to red noise around f ∼ 10−2M−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Power laws of red noise f −2 and f −4 are shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Spec- trograms are calculated over a frequency range higher than that of the injected perturbations (see Secion 2), which therefore do not appear in the PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We Fourier-transformed these spectrograms in order to ob- tain autocorrelation functions corr(LBR, LBR) for the synthetic Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Modulation index of the bremsstrahlung luminosity light curve and the mass accretion rate through the event horizon (in parenthesis), computed over the interval t/M = [50 000, 60 000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' δuRMS/|ur| = ℓ = 0 ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 ℓ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='004 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='003) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='016 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='002) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='246 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='066) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='003) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='028 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='002) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='319 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='101) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='057 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='001) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='303 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='012) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='237 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='075) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='105 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='031) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='225 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='031) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='269 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='200) lightcurve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For all of the simulations, positive correlations de- cay below 1/e in about τ ∼ 5 – 15 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' However, autocorrelations for noisier simulations (ℓ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 or δ = 10), are close to zero for τ ∼ 30 M (≈ 10 minutes for Sgr A*), while simulations with small angular momentum and perturbations still exhibit longer term positive and negative correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Overall, the correlation timescales are shorter than 40 M for all simulations, which allow us to calculate modulation in- dices σ/µ for statistically uncorrelated data by computing the standard deviation σ and average µ over points separated by 50 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Modulation indices are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The modu- lation index of the mass accretion rate through the event hori- zon in the same time interval is shown in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The fact that the latter doesn’t show as much variations when the for- mer varies by orders of magnitude indicates that an important portion of Bremsstrahlung variability is not related to fluctu- ations in the accretion rate close to the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Instead the Bremsstrahlung modulation index clearly increases for those simulations in which shocks and turbulence are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In order to investigate the origin and properties of the fluctu- ations observed in the mass accretion rate, we calculated PSDs of ˙M(r, t) at several radial shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' These are shown for selected radii and for all the simulations in Figure 10, for the same in- terval used for the analysis of the synthetic Bremsstrahlung light curve an using the same methodology as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' It is evident that in general the spectrum at event-horizon scales differs significantly from that at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In most of the panels it is possible to see higher frequencies increasingly populated as one moves closer to the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This can be in- terpreted as the transfer of energy from longer lower frequency modes to smaller and faster modes, and could be due to the change in the characteristic scale of the system as the fluid moves inwards, as well as to the development of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Cases pro- ducing shocks (ℓ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 and δ > 1) show a larger power, and the spectrum at the smaller radius acquires the form of a power law ∝ r−2 or steeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The rest of cases with ℓ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 show a flatter spectrum close to white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Similarly as for Bremsstrahlung spectrograms, for cases with ℓ0 = 0 and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1 the spectrum of ˙M has a very small power and is dominated by oscillation peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' These appear within r ≤ 15 M, indicating that, for almost unperturbed Bondi-like accretion, fluctuations in Bremsstrahlung emission do appear related to these fluctuations in ˙M close to the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In contrast, our results suggest that for sufficiently large per- turbations, as those injected manually or as those produced by the movement of the spiral shock, a red noise spectrum will be recovered, regardless of the injected perturbation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Discussion and Conclusions In this work we used 3D GRHD simulations to study the struc- ture and variability patterns in perturbed transonic accretion flows with low-angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Our aim was to explore an Article number, page 11 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' main 0 10 20 30 40 50 t [hr] 50000 52000 54000 56000 58000 60000 t [M] 1033 1034 LBR [erg s 1] l0p001 l2p001 l3p001 l0p01 l2p01 l3p01 l0p1 l2p1 l3p1 l0p10 l2p10 l3p10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Synthetic light curves of total Bremsstrahlung luminosity using the parameters of Sgr A*, for all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The curve corresponding to l0p001 overlaps completely to that of l0p01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The upper horizontal axis displays the time in hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 10 3 10 2 f [Hz] 10 2 10 1 f [M 1] 10 8 10 6 10 4 10 2 100 102 104 106 PSD(LBR) f 2 f 4 0 5 10 15 20 [min] 0 10 20 30 40 50 60 [M] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='0 corr(LBR, LBR) l0p001 l2p001 l3p001 l0p01 l2p01 l3p01 l0p1 l2p1 l3p1 l0p10 l2p10 l3p10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Spectrograms (left) and correlation functions (right) of the bremsstrahlung light curve proxy of all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Dashed lines with slopes of power laws are shown for comparison in the left panel, and bound the region between ±1/e in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The upper horizontal axes have been scaled for Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' accretion scenario that generalizes torus simulations, is more consistent with the properties of stellar wind-fed accretion, and is controlled by a reduced set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Our simulation setup also aims to overcome two of the known limitations of the BHT simulation paradigm, namely, the secular decrease in torus mass that complicates long-term variability studies and the artifi- ciality of the medium beyond the close vicinity of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We evaluated the general properties of this accretion scenario us- ing several diagnostics, namely, (1) time series of the mass ac- cretion rate and angular momentum flux through the event hori- zon, (2) shell- and time-averaged profiles of several quantities of interest, and (3) a synthetic Bremsstrahlung light curve used to analyze its variability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We also investigated the 3- dimensional morphology of the models, including the location of shocks and sonic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Our models, contrary to BHT simulations, have accretion rates that do not decay exponentially, allowing for long-term variability studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We observe that ˙M decreases significantly for models with larger angular momentum and perturbation ampli- tude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This is consistent with the additional centrifugal support provided by angular velocities and the additional pressure sup- port due to heating caused by turbulence and shocks, that are also more important for the same models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The reduction in mass accretion rate for larger angular momentum models also results in a smaller net angular momentum flux through the horizon, suggesting that there is a finite value of ℓ0 that maximizes the accretion of angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The fact that these models are fed from solutions that ex- tend to infinity allows to relate the mass accretion rate at horizon scales to the Bondi accretion rate from the medium properties at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This in turn permits to obtain tighter constraints on the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For example, density scales need to match large scale fluid properties in addition to electromagnetic flux con- straints derived from radiative transfer calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In addition to their significant variations in accretion rate, the flows we studied possess a rich phenomenology and are in some respects qualitatively different both from BHT simulations and from unperturbed transonic flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Some of the salient features we observe include outflowing toroidal structures, turbulence, shocks, filaments, and multiple sonic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Deviations from the transonic solutions used as initial con- ditions are in some cases large enough to lead to different av- eraged temperature and velocity profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In particular, models with large perturbations and angular momentum deviate from the isentropic temperature profiles and transition to profiles sim- ilar to those of an ADAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For the cases initialized from complete solutions (ℓ0 = 0 and ℓ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25), the radius at which the transi- tion occurs seems related to the amplitude of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This can be explained from the instability of supersonic spherical ac- cretion to non-radial perturbations Kovalenko & Eremin (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' These perturbations grow without limit with smaller radii, pro- ducing the ‘ADAF transition’ when they become sufficiently large to produce shocks and generate entropy, which depends on the injected perturbation amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For the cases initialized from the incomplete solutions with small perturbation amplitudes, large scale spiral shocks are pro- duced by the interaction between the inflowing fluid and the cen- trifugal barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' They likely play an important role in transporting angular momentum outwards thus enabling accretion of matter and the transition to a Keplerian rotation profile at small radii in the absence of a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In the context of chaotic cold accretion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Gaspari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2017), it has been suggested that inelastic col- lisions between clouds can cancel angular momentum, leading to an increased accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In the simulations presented here, Article number, page 12 of 17 Héctor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Olivares S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' : Perturbed Transonic Accretion 10 11 10 9 10 7 10 5 10 3 10 1 101 PSD l0p001 l2p001 l3p001 r = 750M r = 500M r = 300M r = 150M r = 50M r = 15M r = 5M 10 11 10 9 10 7 10 5 10 3 10 1 101 PSD l0p01 l2p01 l3p01 10 11 10 9 10 7 10 5 10 3 10 1 101 PSD l0p1 f 2 l2p1 l3p1 10 3 10 2 f [M 1] 10 11 10 9 10 7 10 5 10 3 10 1 101 PSD l0p10 10 3 10 2 f [M 1] l2p10 10 3 10 2 f [M 1] l3p10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Power spectral density of the mass accretion rate measured at different radii for all of the simulations, during the interval t/M ∈ [50 000, 60 000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The dashed line represents a power law ∝ f −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Article number, page 13 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' main shocks could be expected to play a similar role;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' however, even simulations where shocks are present show the same trend that relates larger perturbations and angular momentum with smaller accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The reason could be that shocked simulations are precisely those with larger perturbations and angular momen- tum, and this effect needs to compete with the additional support provided by angular velocities and the pressure from gas heated by shocks and turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' As could be expected, the different qualitative behavior re- sults in different variability properties of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' How- ever, we found that for sufficiently large angular momentum and perturbations, a red noise spectrum is robustly recovered even from a white noise injection perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Our simulations are complementary to similar hydrodynami- cal simulations carried out by other authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Ressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2018) uses a conservative hydrodynamics code to study the formation of the accretion flow onto Sgr A* in which matter is constantly supplied by stars on orbits around the central black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This setup leads to a somewhat chaotic accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Although their sim- ulation domain is much larger in comparison to ours, their inner boundary overlaps with our outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Ressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2018) obtained solutions with density and temperature power-law pro- files ∝ r−1 which is different compared to our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Also, the angular momentum in our simulations is significantly lower than the value they obtain at comparable radii (In their Figure 14 Ressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2018) reports that the ℓ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5ℓK at the inner boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' These differences in the profiles could be explained by the differences in the physical scenario considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In their case, these include stronger rotation, the presence of important outflows, as well as line and Bremsstrahlung cooling, which be- comes important at the scales they consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Similarly, very large scale simulations performed by Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2022) study the for- mation of the accretion pattern at event horizon scales following material from the Bondi radius scale in elliptical galaxies such as M87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The information obtained from these works and future large scale simulations can be incorporated to smaller scale sim- ulation setups as those presented in this work, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' by specifying initial density profiles and the spatial and temporal spectrum of injected perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In a setting more similar to ours, Suková et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2017) uses 2D and 3D conservative GRHD simulations to study low angular momentum flows on closer to horizon scales, but without man- ually injecting perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In the tests we performed while implementing our initial condition we have recovered the gen- eral behavior of some of their 2D models, although there were some differences in implementation and parameter choices, that we describe in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The inclusion of magnetic fields in the simulations will be presented in a forthcoming publication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' however, there are a few expectations we can draw from our hydrodynamic models that could be relevant for observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For example, the fact that the accretion pattern is almost isotropic for cases with low angular momentum ℓ0 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 may results in images that are also inde- pendent from orientation, in particular contrast to Standard and Normal Evolution (SANE) BHT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In addition, the possi- bility of producing synthetic synchrotron lightcurves that do not suffer from secular torus depletion may contribute to some extent to unravel the ongoing discussion on the suitability of GRMHD models to met the tight variability constraints given by 230 GHz observations of Sgr A* (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Overall, we believe our simulations are an important middle- step towards obtaining more realistic models of relativistic ac- cretion flows in which matter is supplied by the turbulent inter- stellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Acknowlegdements We thank Jesse Vos, Aristomenis Yfantis, Alejandra Jimenez- Rosales, Christiaan Brinkerink and other members of the EHT group at Radboud University for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We also thank Jordy Davelaar and Agnieszka Janiuk for their comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' HROS was supported part by a Virtual Institute of Accretion (VIA) postdoctoral fellowship from the Netherlands Research School for Astronomy (NOVA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We acknowledge that the results of this research have been achieved using the DECI resource Snellius based in the Netherlands at SURF with support from the PRACE AISBL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This work made use of the following software libraries not cited in the text: MATPLOTLIB (Hunter 2007) and NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This research has made use of NASA’s As- trophysics Data System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' References Aguayo-Ortiz, A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='70W Wielgus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=', Moscibrodzka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=', Vos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 2022, Astronomy & Astrophysics, 665, L6, publisher: EDP Sciences Article number, page 15 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' main Appendix A: Initial conditions Our initial data is constructed from the semi-analytic rotating transonic solutions by Chakrabarti (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' These can be thought of as a generalization of Michel accretion Michel (1972) for a rotating flow and for the Kerr metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' To solve for the fluid properties, one assumes that stream- lines are radial when projected on the meridional plane (that is, θ-components of the velocity are neglected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Mass flux √−g ρur, entropy, internal energy E = hut and angular momentum L = −huφ = ℓE are conserved along the streamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Similarly as for Michel and Bondi accretion, once L is spec- ified, E can be chosen so that the sonic radius is at the desired position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The flow configuration is then found by solving a pair of coupled nonlinear algebraic equations for the sound speed and the radial velocity in the co-rotating frame at every point (equa- tions 30a,b of Chakrabarti 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The equations do not constrain the density scale, which can be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In our case, we set it so that ρ = 1 at r = 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Accretion solutions of these equations have a wide variety of qualitative behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' A class of solutions connects smoothly infinity and the event horizon similarly as the Michel solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Other solutions possess incomplete interior or exterior branches that can be connected by a shock, and there exist also incom- plete solutions that have a sonic point but do not extend super- sonically to the event horizon (see Figure 2 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1 of Chakrabarti 1996, for a complete description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For simplicity, we always solve only the exterior branch of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' To initialize our simulations, we solve the system on a grid covering the range θ ∈ [0, π/2] with 300 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The supersonic part of each streamline was solved with 300 points and the sub- sonic part with 100 points, both logarithmically spaced in radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The calculated part of the subsonic region extends beyond the simulation domain by 10% in order to be be used for the bound- ary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The approximation of projected radial streamlines is, in gen- eral, inconsistent with vertical equilibrium, however it holds on the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For this reason, we choose an angular mo- mentum profile with a sharp peak at the equator that decays to ℓ = 0 at the poles, where again vertical equilibrium is fulfilled (equation (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In order to evaluate the adequacy of this approximation and to test that the solution reproduced the expected qualitative be- havior, we performed 2D simulations evolving only the initial condition without injecting any perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1 shows 2D maps of rest-mass density and mass accretion rate per θ-angle for some of the 2D simulations we performed for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='95 and different angular momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' As expected, for those cases in which the equatorial solution connects the event horizon and infinity (ℓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='75, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25), the artificial initial condition quickly relaxes to a true solution in vertical equilibrium and remains stable un- til the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='2 shows the evolution of the radial 4-velocity profile of the unperturbed solution used for simulations with ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 at different latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' At ∼ 150 M, the configuration has already relaxed to a stationary flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This is sufficiently adequate for our simulations, which have durations more than 100 times longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For incomplete solutions (ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='75, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25), material accret- ing on the equatorial plane starts piling up due to the centrifugal barrier, while accretion continues through the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The dense toroidal structure that forms is different from the tori commonly use in simulations in that it consists of ‘outflowing’ unbound material, which produces a shock when interacting with the in- coming accretion flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Since there is no cooling, these struc- tures grow until the end of the simulation (Molteni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' As described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='2, these shocks are present as well in 3D simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' however, the lack of azimuthal symmetry intro- duces important differences, such as the presence of turbulence in the φ-direction and the change in shape of the shock from spheroidal to spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Our 2D unperturbed simulations show a behavior that is con- sistent with that reported by Suková et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' A difference with respect to that work is that their initial condition is a Bondi flow to which rotation has been added, while in our case it is a solution including rotation in a more self-consistent way (al- beit exact only on the equatorial plane and the poles), for which the assumption of low angular momentum is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' An- other difference is that Suková et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' (2017) is largely focused on studying the parameter regime that produces oscillating shocks, which we have not explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Appendix B: Boundary conditions To emulate the turbulent flow entering from the boundary, we filled the ghost zones with the same transonic solution used as initial condition and added time-dependent noise in the form of Gaussian random fields (GRFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' These fields are generated as the superposition of plane waves with random phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The usual way of generating a time-dependent GRF in N dimensions is by Fourier-transforming white noise in N + 1 di- mensions, multiplying the Fourier transform by the desired PSD and then transform back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' One can then ‘play’ the time depen- dent noise by successively applying slices of the resulting N +1- dimensional array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Although this way of generating GRFs is very fast due to the elegance of the Fast-Fourier Transform (FFT) algorithm, it has some disadvantages that lead us to follow a different procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' First, storing a large 4-dimensional array and communicating it among different parallel processes to perform interpolations can be a source of implementation and performance problems, and second, we do not desire to apply the noise in a full three- dimensional box, but only on the outer ghost cells, which make an almost two-dimensional spherical shell embedded in three- dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' For this reason, we build the GRF by directly evaluating a se- ries of sine functions corresponding to plane waves with random phases at the cells of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This sum has the form GRF(t, xi) = Nk � k1,k2,k3=−Nk Ak sin � 2π λmax (k · x − f(k) t − ϕk) � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1) where k is a three-dimensional vector of integers, λmax is the maximum wavelength, f(k) is a function determined by a user- defined dispersion relation, and ϕk is the random phase corre- sponding to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We draw random phases from a uniform dis- tribution over [0, 1) using a pseudo-random number generator with a fixed seed, which allows to use the same random phases without the need to store them between restarts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In order to en- sure causality, we use the constant dispersion relation f(k) = cs, where cs is the sound speed at the simulation boundary, although more complicated relations are also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The coefficients Ak are set according to the desired power spectral density S k, as Ak ∝ (S k)1/2, and can be normalized to give the desired rms per- turbation amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We set Ak = 0 for k = (0, 0, 0) in order to keep the average of the GRF to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Article number, page 16 of 17 Héctor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Olivares S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' : Perturbed Transonic Accretion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Rest-mass density and mass accretion rate per θ angle for 2D evolutions of unperturbed initial conditions with different angular momen- tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Equatorial cuts of the radial velocity of background flow at different times and latitudes for the case ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' In our simulation, we generate 3 GRFs to perturb the three spatial components of the 4-velocity in Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We then transform them to the code coordinates and add them only to the angular components of the velocity given by the back- ground transonic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' When giving up on the FFT, we pay the price of having to evaluate a large number of transcendental functions, which can slow down the code significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' We find that Nk = 5 gives a negligible slow down and the smallest-wavelength mode has a size comparable to ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='6 cells of the outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' This leads to Nθ×Nφ×Nghost×Nfields×(2Nk+1)3 = 12 266 496 evaluations of the sine function per time step, or 340 736 eval- uations for each of the 8 × 8 × 8 AMR blocks at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Here, Nghost = 4 is the number of ghost zones in the radial direc- tion and Nfields = 3, since each of the spatial components of the velocity is perturbed with a different GRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Finally, it is worth mentioning that, being a sum of peri- odic functions, the noise is also periodic with the period of the longest wavelength mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Although in general this will not re- sult in a periodic behavior of the simulation due to the changing chaotic dynamics inside the domain, it may be desirable to pro- duce non-periodic noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' One possibility could be chang- ing slightly the dispersion relation so that the period of some of the modes is an irrational multiple of that of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' The search for more appropriate non-periodic noise models is, however, out of the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content=' Article number, page 17 of 17 l= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='75 l = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 l = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='75 l=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='25 75 2 t= 4000M 50 - 1 25 - 0 log1op [W] 0 N 2 25 - 50 3 75 + 75 20 50 - 10 25 - ep/wp [W] 0 :0 N 25 - 10 50 - 75 20 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 x[M] x[M] x[M] x[M]-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='7 t= 0, θ = π/3 t= 150 M,0 = π/3 t= 1000 M, θ = π/3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FLT4oBgHgl3EQfNi8W/content/2301.12020v1.pdf'} +page_content='8 t = 0 M, θ = π/2 t = 150 M, 0 = π/2 t = 1000 M, θ = π/2 0.' metadata={'source': 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Drew LaMar2 +1BioQUEST Curriculum Consortium, 5917 Alder St, Pittsburgh, +PA, 15232, USA +2William & Mary, Williamsburg, VA, 23187, USA +January 5, 2023 +Abstract +The QUBES platform was conceived as a “science education gateway” +and designed to accelerate innovation in undergraduate STEM education. +The technical infrastructure was purpose built to provide more equitable +access to professional resources, support learning that reflects authen- +tic science, and promote open education practices. +Four platform ser- +vices (OER Library Access; Professional Learning; Partner Support; and +Customizable Workspaces) support overlapping faculty user communities, +provide multiple points of entry, and enable manifold use case scenarios. +The integrated nature of the platform makes it possible to collect, curate, +and disseminate a diverse array of reform resources in a scalable and sus- +tainable manner. We believe that the QUBES platform has the capacity +to broaden participation in scholarship around teaching and learning and, +furthermore, that it can help to lower faculty barriers to the adoption of +reform practices. The role of cyberinfrastructure in undergraduate STEM +education is generally underappreciated and warrants further exploration. +INTRODUCTION +The impacts of science gateways on research outputs across diverse fields are +well documented [1]. Gateway infrastructures can be used to support emerging +practices that emphasize the interdisciplinary, collaborative, open, and compu- +tationally driven nature of science. However, the adoption of a science gateway +as a research platform can require significant adjustments to existing workflows +and challenge one’s assumptions about how to manage a research program. +Gateways that do not engage their user communities to build trust and support +adoption are less likely to establish a robust research community and therefore +1 +arXiv:2301.01760v1 [cs.CY] 4 Jan 2023 + +limit their potential scientific impacts [2]. The acceptance and use of scien- +tific gateways, like the diffusion of any innovation, depends in large part on the +perceived usefulness of the resource. Developing gateways collaboratively, with +input from user communities may accelerate the adoption of new disciplinary +practices afforded by the gateway platform. Centralized resources and online +collaboration are becoming more widely adopted in many contexts where we +believe the “gateway model” provides a context for addressing hard problems +and broadening access to key resources. +In this manuscript we describe an effort to build a Science Education Gate- +way to accelerate undergraduate STEM education reform. We define reform +very broadly to embrace the diversity and dynamic nature of the landscape +across which reform happens. Effective teaching must be supported in a wide +range of institutional settings, student populations, and delivery modalities. +There is broad recognition that teaching and learning strategies should evolve +to emphasize the adoption of evidence-based teaching methods, student en- +gagement with authentic scientific practices, and broaden participation among +traditionally underrepresented communities. These challenges are further com- +plicated by the need to continuously integrate new topics and skills to connect +classrooms to contemporary scientific practices and help prepare students to par- +ticipate in the technical workforce. Even when innovations are developed it is a +non-trivial undertaking to support the broad implementation of those strategies +so that the potential benefits reach as many learners as possible. Considered +at this scale, the acceleration of STEM education reform is a wicked problem +that will require the development of diverse overlapping strategies that can be +applied flexibly across the landscape. Furthermore, given the certainty that sci- +entific practices and computational resources will continue to evolve, education +reform should adopt a continuous quality improvement process in order to treat +reform as ongoing and context specific. +In 2014 NSF funded the project “Supporting Faculty in Quantitative Un- +dergraduate Biology Education and Synthesis (QUBES)” which was designed to +“address the Nation’s growing need to better prepare undergraduate biologists +with the quantitative and computational skills needed to be successful in the +workplace or in graduate school.” [3] Given the long history of quantitative +biology education reform efforts, the project was organized in part to highlight +the visibility of ongoing but isolated reform communities and coordinate faculty +access to a diverse collection of existing teaching and learning resources. We +adopted the HubZero platform and worked with the Science Gateways Commu- +nity Institute (SGCI) to design and deploy a gateway to support quantitative +biology education innovation and classroom implementation. Over time our mis- +sion has evolved to serve the STEM education reform community more broadly. +At the conclusion of the initial NSF funding the management of the QUBES +platform was moved into the BioQUEST Curriculum Consortium, a well es- +tablished 501(c)(3) nonprofit, where it is sustained as an open resource for the +reform community. +In this paper we describe the conceptualization and implementation of the +QUBES platform as a Science Education Gateway (SEG). After an overview of +2 + +the technical infrastructure (tools) we describe the ways that faculty use of the +gateway is facilitated using social infrastructure (practices). We end with a call +to action for the undergraduate STEM education reform community to explore +the potential use of science education gateways as a means to accelerate the +reform of teaching and learning. +THE QUBES PLATFORM AS A SCIENCE ED- +UCATION GATEWAY +Gateways refer to community-developed online environments that integrate ac- +cess to shared resources including software, data, collaboration tools, and high- +performance computing. As a Science Education Gateway QUBES is designed +to lower barriers to faculty participation in STEM education reform by making it +easier to engage in scholarship around teaching and learning. From finding new +teaching materials, to collaborating on projects, to accessing interdisciplinary +expertise the QUBES platform provides both technical and social support to +engage faculty. Our target audiences include faculty whose scholarship centers +on teaching and learning, with the platform designed to facilitate, document, +and disseminate faculty work as they participate in diverse professional activi- +ties. Following a high level overview of the technical infrastructure, we describe +a set of platform services that provide opportunities for faculty to engage with +reform resources and pursue professional opportunities (Figure 1). +Technical Infrastructure +The QUBES platform is a shared online space that can be used to publish and +disseminate Open Education Resources, host distributed meeting and workshop +activities, participate in professional learning, and support education reform +projects. QUBES is an instantiation of the open source content management +system HubZero, which initially was a branch of the Joomla! +content man- +agement system (CMS) but is now an independently developed platform for +scientific gateways. +The underlying technical infrastructure provides community building tools, +including communication, productivity, and collaboration functionality, through +membership controlled group spaces. These group spaces can be public or pri- +vate, with differing levels of openness to new members (e.g., completely open, +curated admission, invitation only). There can be multiple administrators as- +signed to a group who have additional abilities, such as changing group settings, +controlling membership, and creating or modifying group web pages. +Each +group space has optional community tools, such as discussion forums with email +digest capabilities, announcements, calendars, file sharing, blogs, Pinterest-style +file and image sharing (Collections), collaborative project spaces, a dedicated +OER library, and traditional usage and website metrics. Group spaces can be +fully customized to have their own web templates allowing for a customized +look-and-feel. +3 + +Figure 1: QUBES Science Education Gateway and its Four Platform +Services. The central element, labeled QUBES Science Education Gateway, +represents the underlying technical infrastructure that supports an integrated, +customizable online platform. The four platform services (Professional Learn- +ing; OER Library Access; Project Support; and Customizable Workspaces) sup- +port overlapping faculty user communities, provide multiple points of entry, and +enable manifold use case scenarios. Broadly, the gateway can be described as +both a platform for hosting diverse professional activities and as a highly cu- +rated repository of teaching and learning resources. +This dual functionality +supports a virtuous cycle where faculty engagement with STEM reform helps +them generate new products which are then captured and curated within the +gateway, making those resources accessible to the broader community. +4 + +RYR +Professional +Project +Learning +Support +QUBES +Science +Education +Gateway +OER Library +Customizable +Access +WorkspacesThe HubZero CMS also supports the publication and hosting of software +tools. QUBES has a dedicated execution host for providing cloud-based access +to any software that can be run on a linux machine using the OpenVZ mid- +dleware. Tools available on QUBES include data science environments (e.g., +RStudio Server, Jupyter Notebooks, Shiny apps), modeling tools (e.g., NetL- +ogo, Avida-Ed) and analysis tools (e.g., ImageJ, Mesquite). HubZero is cur- +rently working to replace OpenVZ with Docker containers, which would provide +additional functionality and better scalability. The QUBES platform currently +has the capacity to run 200 concurrent tool sessions which presents significant +scaling challenges when making software resources available to a national audi- +ence of STEM educators and students. +The QUBES platform has an open, self-publishing platform (QUBES OER +Library) that uses a git-like version control system for tracking versions, adap- +tations, attribution, and use metrics. For searchability, publications are catego- +rized via multiple standard OER library ontologies, such as activity length and +audience level. Additional optional ontologies are available to align resources +with evidence-based pedagogical frameworks, such as inclusive pedagogy, uni- +versal design for learning, and open science and education practices [4]–[7]. +Publications receive a digital object identifier (DOI) and several use metrics are +automatically tracked. Importantly, publications can be associated with not +just authors, but projects, organizations, or other QUBES hosted groups. This +makes it easy to extract and display subsets of the library within a project group +context, contributing to partner customization and autonomy. +While QUBES is utilizing the open-source HubZero CMS, we have pushed +for the development of additional functionality and UX improvements, both +independently and through the support of the SGCI Developer Program. For +example, in collaboration with HubZero developers we implemented an email +digests option for group discussion forums. +Similarly, in collaboration with +the SGCI Developer Program we implemented a forking/adaptation system for +open education resources that made it possible to support the full OER lifecy- +cle. Independently developed items include supporting instructor-only access to +particular files within a published resource, an OER commenting system, and +group webpage usage metrics. To support autonomy of hosted projects on the +QUBES platform, we increased group customization on the platform by allowing +overrides of components, plugins and modules, including fully autonomous OER +libraries (e.g., a dedicated search-and-browse for their community resources, and +a customized resource record view). Finally, we have developed the ability for +OER to have multiple aligned ontologies that can be easily used in a search- +and-browse interface with Solr search capabilities. +PLATFORM SERVICES +In order to raise faculty awareness and encourage adoption of the QUBES plat- +form we have developed a set of four platform services (Figure 1) tied to common +uses of the underlying technological infrastructure (tools). These platform ser- +5 + +vices promote a set of use scenarios, or practices, which help to contextualize +the ways that the gateway technology can be leveraged to meet faculty needs. +Platform services use a social infrastructure which provides training, templates, +activity structures, and support documents to lower barriers to participation +and promote successful engagement with the gateway. Developing and sharing +strategies for effectively using a gateway can address both technical challenges +associated with manipulating the gateway tools, and introduce new professional +practices that incorporate things like open licensing, distributed collaborations, +and online community building. +Open Education Resources Library Access +Too often innovative educational approaches developed by STEM faculty are not +widely shared, licensed to support reuse, or documented as professional scholar- +ship. Our limited ability to capture, curate, and disseminate the collective teach- +ing and learning knowledge of undergraduate STEM faculty severely limits the +development and implementation of reform practices. Collections of education +resources are often distributed across multiple web sites making them difficult +to find and manage, in addition to being more difficult to sustain over time. +The QUBES OER Library currently hosts over 2,100 resources that range +from conference posters to classroom activities. This centralized, well described +collection of teaching resources increases findability and can help faculty explore +new teaching and learning strategies. By hosting diverse resources across many +projects the QUBES OER Library promotes search both within and across +projects. Rich contextualization of the product reflecting authorship, project +association, group affiliation and other meta-data makes it possible for users +to naturally follow threads connecting seemingly disparate materials to support +the discovery of related content. This also makes it easy to extract and display +subsets of the library within a project group context, contributing to partner +customization and autonomy. +Access to the QUBES OER Library makes it easy for faculty to share their +own work. Because the library uses open licensing and the infrastructure con- +tains a “git-like” version management system it is possible for faculty to engage +in the full OER lifecycle. Adaptations of existing resources automatically con- +tain attribution information that link the resources, making it easy for users +to navigate between them. Use metrics address a range of data (e.g., views, +downloads, comments, versions, and adaptations) that help authors assess the +impacts of their products within the community. +The introduction of social infrastructure (practices) around open licensing +has been important to address community concerns about OER. Hosted projects +on QUBES are often very interested in publishing their work as part of their +dissemination and sustainability. +We also encourage projects to share non- +traditional products such as annual reports, conference presentations, protocols +and manuals to document outcomes and make them accessible to the broader +community. To support faculty use of the QUBES OER Library we structure +professional activities (e.g., project collaborations, professional learning, work- +6 + +shops) around existing published materials, or as a mechanism to collect newly +generated resources. This immediately helps to establish an authentic context +for participating in professional learning opportunities – run by BioQUEST or +hosted partner organizations – and generate products that are appropriate for +publication. We have adopted the language of a preprint server to help faculty +understand the benefits of sharing works in progress and recognise the utility of +self-publishing in the educational space. +The BioSkills Guide is an example of a resource that was shared early during +its development and then versioned as it was refined. To date it is in its fifth +version and in total it has been accessed more than 7,500 times and downloaded +more than 2,500 times [8]. When the manuscript describing the development of +the guide was published in a peer reviewed journal, the guide on QUBES was +referenced, with the journal citation added to the description of the guide on +QUBES, thereby pushing discoverability and traffic in both directions. +Project Support +Externally funded education reform projects play an important role in fostering +innovation and exploring effective teaching practices. However, funded projects +often face logistical challenges like coordinating project activities, document- +ing project impacts and sustaining their work beyond the funding period. It +is essential to the ongoing reform of STEM education that funded projects are +as successful as possible and that their findings are carefully documented and +shared to increase their impact over time. Effectively engaging user communi- +ties early in a project’s development cycle can play an important role in guiding +the work to be broadly useful and seed the effective dissemination of products. +The QUBES platform currently hosts over 80 partner projects. These projects +share a set of structural and technical needs that can burden the project team +and undermine the time available to pursue the innovative project agendas. +Hosting projects on the QUBES platform helps to avoid the inefficient and un- +sustainable practices of hosting work on a separate server. We have developed a +project support system that makes it easy to establish communication, collab- +oration, and dissemination mechanisms both within the project team and the +broader STEM education reform community. We provide a turnkey group space +that can be customized to address specific project needs. Our support services +are designed to address three common challenges groups face when hosting their +project on QUBES: understanding the operation of the platform; maintaining a +sense of ownership and branding; and adopting new collaboration and communi- +cation strategies to pursue their project. We also introduce planning resources +that help the project leadership coordinate phases of their projects with the +functionality they will require in their QUBES group space. These resources +are available within a partner support group on QUBES for onboarding of new +projects, which includes demonstrations of the effective use of the platform to +support the creation, maintenance, and sustainability of project workspaces. +The CCBioInsites project is an example of an NSF funded project that +hosted their activities on QUBES. The grant focused on helping community +7 + +college faculty participate in discipline based education research and improving +their teaching practices. The project recruited a distributed cohort of faculty +and used their QUBES site to coordinate their activities [9]. +Similarly the QB@CC project focused on teachers at 2-year schools who +have limited access to professional learning opportunities. They have brought +together both mathematics and biology faculty to collaboratively adapt existing +teaching modules so that they reflect effective mathematics, biology content, and +pedagogy. These products will persist and invite further customization in the +QUBES OER Library disseminating and sustaining their efforts well beyond +their active grant funding. +Professional Learning: Faculty Mentoring Networks +Professional learning refers to an intentionally designed collaborative environ- +ment where teachers work with one another to learn, develop and practice new +methods for educating students. In contrast to some professional development +models, professional learning promotes learning through engagement in reform +practices, deemphasizing the role of telling faculty what to do. Providing ef- +fective and efficient professional learning experiences is essential to faculty par- +ticipation in education reform. Equitable access to scholarly learning opportu- +nities requires that those opportunities are not exclusively tied to events like +conferences which can limit participation to those with available money and +time. In STEM specifically, both the disciplinary knowledge base and the sci- +entific tools are evolving rapidly, necessitating ongoing professional investment +in teaching and learning. Access to professional learning is a major contributor +to the implementation of evidence-based teaching practices discovered through +discipline-based education research. +Our primary professional learning model is called a Faculty Mentoring Net- +work (FMN). These involve geographically distributed groups of 10-15 faculty +working together with a facilitator over the course of a semester. They use a +QUBES platform group space to share resources, communicate asynchronously, +and publish their products. The schedule and activities are structured to lead +them through multiple stages: (1) a process of learning about a new teaching +resource; (2) customizing that resource for use in their teaching setting and with +their student audience; (3) implementing the module in their classroom; then +(4) refining and publishing their adaptation to the OER Library. We have tem- +plated the processes necessary to run an FMN and provide training for FMN +facilitators. Additionally, we have developed modules addressing common fac- +ulty interests such as inclusive teaching strategies, universal design for learning, +and overcoming math anxiety which can be integrated across diverse FMNs. +The basic FMN model has been implemented in over 90 professional learning +opportunities addressing diverse agendas. Groups have adapted our FMN model +to address different outcomes including learning how to use new computational +tools, designing new curricular activities, and conducting collaborative research. +One example of how an FMN has been used to engage faculty in a schol- +arly approach to teaching and learning involves the extension and adapta- +8 + +tion of a valuable teaching module. +“Investigating the footprint of climate +change on phenology and ecological interactions in north-central North Amer- +ica” was originally published as part of a NSF funded project coordinated +by the Ecological Society of America [10]. An FMN was hosted on QUBES +where faculty were mentored through customizing the original activity and +teaching this data-intensive lesson as part of their professional learning ex- +perience. This FMN led to 15 published adaptations of the original activity +where faculty produced customizations to incorporate different regional flora, +use different analysis tools, and fit within different course contexts (https: +//qubeshub.org/publications/267/forks/1). +Customizable Workspaces +There are a wide range of professional communities (both formal and informal) +that can play an important role in STEM education reform. In addition to the +grant funded projects described above, communities involving professional soci- +eties, education nonprofits, research collaborations, and special interest groups +can help engage faculty with professional opportunities. Programs such as con- +ferences, webinars, and workshops can also help expose faculty to new ideas +and potential collaborators. In order to be impactful these communities need to +establish and sustain faculty engagement. Everything from cross-institutional +course-based undergraduate research experiences (CUREs), to citizen science +projects, to journal clubs and special interest groups can play a role in helping +faculty pursue scholarship around teaching and learning. +The QUBES platform currently hosts over 450 online group workspaces con- +taining 1,200 project areas. We have a set of strategies for establishing, man- +aging, and disseminating resources from distributed communities of faculty as +they pursue new ideas for teaching and learning STEM. These involve integrat- +ing synchronous and asynchronous interactions, and using active facilitation +techniques to work toward a shared community goal. Through the creation of +customizable workspaces, the QUBES platform can be used to extend engage- +ment with traditional face-to-face conferences as well as support hybrid and +on-line only professional meetings. While participants can meet synchronously +during meetings and workshops, the infrastructure also supports asynchronous +engagement with material and other participants before and well after the syn- +chronous portion of the meeting has ended. +As an example, BioQUEST runs an annual online meeting called the BIOME +Institute which offers a unique opportunity to engage with a community of peers +to address an educational challenge with the ultimate goal of improving student +outcomes. A dedicated website and community space is created prior to the +BIOME Institute. Participants are invited to this space ahead of the meeting +to introduce themselves, bring attention to any educational reform projects +they are involved in, and to engage with introductory prompts addressing the +meeting focus and objective. During the meeting, talks and poster presentations +are given throughout, working groups are brainstormed and formed, with all the +material published as OER on the platform. Plans are made near the end of +9 + +the synchronous portion of the meeting to create online sub-communities for +working groups so that participants can continue their work asynchronously +over the Fall semester. Products from these asynchronous working groups are +then published as OER on the platform. Some of these working groups develop +into grant collaborations. You can see a list of the recent working groups spun off +from the BIOME meeting here (https://qubeshub.org/community/groups/ +summer2021). +ONGOING AND FUTURE WORK +In addition to the four platform services introduced above we continue to inte- +grate new functionality and use scenarios to support STEM education reform. +Since the inception of the QUBES platform in 2014, we have emphasized work- +ing closely with the user community to develop a shared vision for the ways +that a gateway can support faculty scholarship around teaching and learning +reflected in this quote. +If you want to build a ship, don’t drum up people to collect wood +and don’t assign them tasks and work, but rather teach them to long +for the endless immensity of the sea. – Antoine de Saint Exup´ery +Our ongoing and future development plans for the platform stem directly +from needs identified by hosted projects and users. Here we briefly introduce +four high priority areas including: scalable and robust hosting of software tools, +including tight integration of these tools with OER; design of teacher portfolios, +akin to a LinkedIn or ResearchGate for educators; a custom publishing platform +that supports peer-reviewed education journal tools, beyond the self-publishing +of OER already supported on the platform; and support for discipline-based +education research (DBER), which is constantly expanding our knowledge on +evidence-based teaching strategies (see Figure 2). +Software Tools +One of the key elements of a scientific gateway is simplified access to com- +putational tools for research and teaching. The QUBES platform utilizes the +open-source content management system HubZero in large part because of its +built-in support for hosting computational tools, in addition to its collaboration +and communication functionality. Addressing the scalability of access to these +tools for use in the classroom has been challenging. Education user communities +are much larger than research communities and require different types of server +support. This issue was exemplified most recently by the COVID pandemic. +During lockdown and social distancing, many biologists moved their research +and teaching out of the in-vitro/in-vivo and into the in-silica, leading to an +increased demand for simulations and software. +Data science has seen an explosion of demand and utility in the biological +sciences, with a specific increase in students and faculty learning and using +10 + +Figure 2: Example ongoing and future QUBES platform development +projects. +Each of the four panels represent ongoing or future development +to extend the functionality of the QUBES platform to further support faculty +engagement in scholarship around teaching and learning. (A) Software tools +involve implementing scalable and robust hosting of cloud based modeling and +analysis tools in a way that is tightly integrated with OER published in the +library. (B) Teacher portfolios refer to the design of an integrated system for +collecting, sharing, and contextualizing faculty teaching scholarship as a means +to document activities for consideration in hiring, promotion, and tenure. (C) +Custom publishing involves streamlining the deployment of manuscript manage- +ment workflows, and specialized publication collections. (D) Discipline based +education research support describes a suite of tools that would facilitate col- +laboration around classroom research and studies of reform practices. +11 + +A +B +Software +Teacher +Tools +Portfolios +C +D +Custom +DBER +Publishing +SupportR and Python in their courses and research. +There are many organizations +working to train faculty in modern data science skills and techniques, such as +The Carpentries, as well as offering professional development opportunities on +how to incorporate these skills into the classroom. There are many challenges, +however, in bringing these tools into the classroom. Posit Cloud, for example, +has free access for students to RStudio without the need to install R and RStudio +on their computers, but students can quickly run out of compute time, especially +as they are learning how to code. CyVerse offers many services around accessing +bioinformatics and computational tools in the cloud, but they mainly service the +research community. We are actively exploring ways to increase accessibility to +computational tools in classrooms, and fully integrate cloud based computing +within the context of OER published in the library. +Teaching Portfolios +As faculty participate in reform projects, publish OER materials, engage with +professional learning, and access materials from the OER library, they are build- +ing a record of their activity on the QUBES platform. Teaching scholarship is +often difficult to document because there are few publishing outlets for teaching +resources, with those that do get published tending to be cited less frequently as +their use happens in the informal context of the classroom. Our OER publish- +ing platform already provides DOIs, full citations, and some data about uptake +by the community. Furthermore, each version and adaptation of a publication +is tracked, showing iterative refinement, adoption, and use, with attribution +automatically assigned on adaptation. Our goal is to supplement these impact +metrics with additional ways for faculty to document their teaching scholar- +ship. We plan to reconfigure the existing dashboards available to QUBES users +to more effectively capture and share information about OER that has been +published, groups joined, and FMNs completed. We will also extend the user +profile module to capture more information related to faculty activities and +backgrounds. +We typically acknowledge successful completion of professional learning op- +portunities through FMNs and workshop/meeting experiences with certificates +and letters of participation. In the future we will establish a badge system to +document a wide range of activities that faculty complete. These badges can +be linked not just to learning opportunities completed, but also to products +produced via those opportunities. Essentially, our vision for teaching portfolios +involves providing more flexible opportunities to organize, present, and annotate +one’s contributions. We are already seeing many faculty, particularly those in +non-R1 settings, using information from their profiles and dashboards to capture +their scholarship in annual activity reports. +Custom Publishing +The vast majority of QUBES OER are submitted as a “QUBES Resource.” +This resource type contains a broad set of metadata for describing the resources, +12 + +their intended audience, and other features like their use of inclusive learning +practices, universal design principles followed, and racial equity strategies em- +ployed. QUBES currently provides custom publishing types for a small group +of projects including Math Modeling Hub (https://mmhub.qubeshub.org), +NIBLSE (https://niblse.qubeshub.org), CourseSource (https://coursesource. +qubeshub.org), and SIMIODE (https://simiode.qubeshub.org). These cus- +tom resource types have their own metadata schema and can be set to employ +a review process before public release. For example, the NIBLSE project devel- +oped and published a set of core competencies related to describing beneficial +outcomes of integrating bioinformatics techniques throughout the biology cur- +riculum. These competencies are represented in their metadata schema, thereby +raising awareness and improving findability across their collection of publica- +tions. CourceSource, a peer reviewed journal of biology and physics teaching +resources, transferred their OER library into QUBES, where we developed a cus- +tom metadata schema including alignment with learning outcomes and teaching +frameworks. In collaboration with CourseSource, we are currently building a +custom submission and editorial management pipeline that will allow authors, +editors, and reviewers to utilize QUBES throughout the submission, review, and +publication process. +The goal is to make the OER manuscript management process customizable +for easier and cost effective utilization by other communities. We see implica- +tions for ingesting existing education libraries looking for a sustainable home, +and support for distributed communities, such as CUREs and citizen science +projects. Custom designed metadata schemas and curation pipelines will sup- +port individualized, autonomous search-and-browse portals within their commu- +nities, with their resources also available across the entire QUBES OER Library +for broad dissemination and discoverability. +Discipline Based Education Research Support +There is growing awareness of and focus on discipline based education research +as an important component of faculty scholarship. We imagine QUBES as a +platform where curriculum specialists, education researchers, and teaching fac- +ulty could collaborate to scale up data collection and explore the impacts of +interventions in diverse teaching contexts and across student audiences. This +would likely involve some tools to facilitate data collection and management +(e.g., surveys and other online forms, activity tracking, and use logging). We +would also need to institute student account types that would appropriately han- +dle data de-identification to mitigate any personal risks. QUBES has already +been used to coordinate distributed research projects using faculty mentoring +networks to connect researchers with motivated teachers. We believe that the +gateway environment can play an important role facilitating research on fac- +ulty change, professional learning, project management, as well as documenting +emerging practices as communities adopt gateway infrastructures to evolve their +professional practices. +13 + +MOVING FORWARD - A CALL TO ACTION +The QUBES platform has been designed and implemented to help faculty pur- +sue scholarship around teaching and learning. Both the technical infrastructure +(tools) and social infrastructure (practices) were purpose built to advance inno- +vation in STEM education by supporting communities of practice, foreground- +ing equity, diversity and inclusion, and engaging faculty in the culture of open +education. Our four platform services (OER Library Access; Professional Learn- +ing; Partner Support; and Customizable Workspaces) provide multiple points +for faculty engagement and address key aspects of accelerating reform practices. +Hosting diverse activities on a single platform creates a synergistic effect and +has proven to be important to our success, allowing us to capture faculty work +and make it accessible to the broader community. We currently host a commu- +nity of over 20,000 registered users and we are beginning to see the impact of +both leveraging the platform to get work done and the ways that feeds forward +into the collection, curation, and documentation of those professional activities. +The integration and interoperability of the various gateway tools supports and +contextualizes a broad network of projects, organizations, and faculty. +We believe that the broad use of the QUBES platform by multiple stakehold- +ers demonstrates a significant need for a centralized infrastructure to support +reform practices. The National Science Foundation has identified QUBES, along +with other emerging science education gateways, as important resources for in- +creasing sustainability and broadening the impact of funded education projects +[11]. We encourage further exploration of the ways that science education gate- +ways might prove to be fruitful across STEM education, helping to develop other +purpose-built communities and platforms to accelerate innovation in teaching +and learning science. +ACKNOWLEDGMENTS +This material is based upon work supported by the National Science Foundation +under Grants 1346584, 1446258, 1446269, 1602989, and 1446284. The authors +would like to thank HubZero and the Science Gateways Community Institute +for their support. +REFERENCES +[1] M. Barker et al., “The global impact of science gateways, virtual research +environments and virtual laboratories,” Future Gener. Comput. Syst., vol. +95, pp. 240–248, 2019, doi:10.1016/j.future.2018.12.026. +[2] K. F. Kee, B. Le, and K. Jitkajornwanich, “If you build it, promote it, and +they trust you, then they will come: Diffusion strategies for science gateways +and cyberinfrastructure adoption to harness big data in the science, tech- +nology, engineering, and mathematics (STEM) community,” Concurrency +Computat. Pract. Exper., vol. 33, no. 19, 2021, doi:10.1002/cpe.6192. +14 + +[3] Sam Donovan et. al, “QUBES: a community focused on supporting teaching +and learning in quantitative biology,” Letters in Biomathematics, vol. 2, no. +1, 2015, doi:10.1080/23737867.2015.1049969. +[4] B. M. Dewsbury, “Deep Teaching in a College STEM Classroom,” Cultural +Studies of Science Education, vol. 15, pp. 169–191, 2020, doi:10.1007/s11422- +018-9891-z. +[5] B.M. Dewsbury and C. J. Brame, “Inclusive Teaching,” CBE—Life Sciences +Education, vol. 18, no. 2, 2019, doi:10.1187/cbe.19-01-0021. +[6] A. O. Hasley and H. Orndorf, “Introduction to the Universal Design for +Learning Guidelines,” Universal Design for Learning, QUBES Educational +Resources, 2021, doi:10.25334/3MJJ-4D08. +[7] K. Cangialosi and E. Bledsoe, +“Open Science and Education Prac- +tices Ontology,” B(ui)LDS: Biological, Universal, and Inclusive Learn- +ing in Data Science Community, QUBES Educational Resources, 2021, +doi:10.25334/9KKC-H752. +[8] A. Clemmons, J. Timbrook, J. Herron, and A. Crowe, “BioSkills Guide. Core +Competencies for Undergraduate Biology,” QUBES Educational Resources, +ver. 5.0, 2020. doi:10.25334/156H-T617 +[9] M. M. C. Musgrove, S. Nied, A. Cooley, J. N. Schinske, and L. A. Corwin, +“Engaging with CC Bio INSITES: Experiences of Barriers, Supports, and +Belonging in Community College Faculty Participating in Biology Education +Research,” CBE-Life Sci. Educ., vol. 21, no. 2, 2022. doi:10.1187/cbe.21-09- +0246. +[10] K. M. Calinger, “Investigating the footprint of climate change on phenol- +ogy and ecological interactions in north-central North America,” Teaching +Issues and Experiments in Ecology, vol. 10: Practice #1, 2014. [Online]. +Available: http://tiee.esa.org/vol/v10/issues/datasets/calinger/ +abstract.html (URL) +[11] NSF-DCL 21–026, “Dear Colleague Letter: Requesting proposals for online +biology education to the Research Coordination Networks for Undergradu- +ate Biology Education,” National Science Foundation, 03-Dec-2020. [On- +line]. Available: https://www.nsf.gov/pubs/2021/nsf21026/nsf21026. +jsp (URL) +15 + diff --git a/MNAzT4oBgHgl3EQfy_4H/content/tmp_files/load_file.txt b/MNAzT4oBgHgl3EQfy_4H/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d228fc1179348abd71540dc7b4638a3108f3e571 --- /dev/null +++ b/MNAzT4oBgHgl3EQfy_4H/content/tmp_files/load_file.txt @@ -0,0 +1,349 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf,len=348 +page_content='Using Science Education Gateways to improve undergraduate STEM education: The QUBES Platform as a case study Sam Donovan1 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Drew LaMar2 1BioQUEST Curriculum Consortium, 5917 Alder St, Pittsburgh, PA, 15232, USA 2William & Mary, Williamsburg, VA, 23187, USA January 5, 2023 Abstract The QUBES platform was conceived as a “science education gateway” and designed to accelerate innovation in undergraduate STEM education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The technical infrastructure was purpose built to provide more equitable access to professional resources, support learning that reflects authen- tic science, and promote open education practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Four platform ser- vices (OER Library Access;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Professional Learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Partner Support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' and Customizable Workspaces) support overlapping faculty user communities, provide multiple points of entry, and enable manifold use case scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The integrated nature of the platform makes it possible to collect, curate, and disseminate a diverse array of reform resources in a scalable and sus- tainable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We believe that the QUBES platform has the capacity to broaden participation in scholarship around teaching and learning and, furthermore, that it can help to lower faculty barriers to the adoption of reform practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The role of cyberinfrastructure in undergraduate STEM education is generally underappreciated and warrants further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' INTRODUCTION The impacts of science gateways on research outputs across diverse fields are well documented [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Gateway infrastructures can be used to support emerging practices that emphasize the interdisciplinary, collaborative, open, and compu- tationally driven nature of science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' However, the adoption of a science gateway as a research platform can require significant adjustments to existing workflows and challenge one’s assumptions about how to manage a research program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Gateways that do not engage their user communities to build trust and support adoption are less likely to establish a robust research community and therefore 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='01760v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='CY] 4 Jan 2023 limit their potential scientific impacts [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The acceptance and use of scien- tific gateways, like the diffusion of any innovation, depends in large part on the perceived usefulness of the resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Developing gateways collaboratively, with input from user communities may accelerate the adoption of new disciplinary practices afforded by the gateway platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Centralized resources and online collaboration are becoming more widely adopted in many contexts where we believe the “gateway model” provides a context for addressing hard problems and broadening access to key resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' In this manuscript we describe an effort to build a Science Education Gate- way to accelerate undergraduate STEM education reform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We define reform very broadly to embrace the diversity and dynamic nature of the landscape across which reform happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Effective teaching must be supported in a wide range of institutional settings, student populations, and delivery modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' There is broad recognition that teaching and learning strategies should evolve to emphasize the adoption of evidence-based teaching methods, student en- gagement with authentic scientific practices, and broaden participation among traditionally underrepresented communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' These challenges are further com- plicated by the need to continuously integrate new topics and skills to connect classrooms to contemporary scientific practices and help prepare students to par- ticipate in the technical workforce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Even when innovations are developed it is a non-trivial undertaking to support the broad implementation of those strategies so that the potential benefits reach as many learners as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Considered at this scale, the acceleration of STEM education reform is a wicked problem that will require the development of diverse overlapping strategies that can be applied flexibly across the landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Furthermore, given the certainty that sci- entific practices and computational resources will continue to evolve, education reform should adopt a continuous quality improvement process in order to treat reform as ongoing and context specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' In 2014 NSF funded the project “Supporting Faculty in Quantitative Un- dergraduate Biology Education and Synthesis (QUBES)” which was designed to “address the Nation’s growing need to better prepare undergraduate biologists with the quantitative and computational skills needed to be successful in the workplace or in graduate school.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' [3] Given the long history of quantitative biology education reform efforts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' the project was organized in part to highlight the visibility of ongoing but isolated reform communities and coordinate faculty access to a diverse collection of existing teaching and learning resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We adopted the HubZero platform and worked with the Science Gateways Commu- nity Institute (SGCI) to design and deploy a gateway to support quantitative biology education innovation and classroom implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Over time our mis- sion has evolved to serve the STEM education reform community more broadly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' At the conclusion of the initial NSF funding the management of the QUBES platform was moved into the BioQUEST Curriculum Consortium, a well es- tablished 501(c)(3) nonprofit, where it is sustained as an open resource for the reform community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' In this paper we describe the conceptualization and implementation of the QUBES platform as a Science Education Gateway (SEG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' After an overview of 2 the technical infrastructure (tools) we describe the ways that faculty use of the gateway is facilitated using social infrastructure (practices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We end with a call to action for the undergraduate STEM education reform community to explore the potential use of science education gateways as a means to accelerate the reform of teaching and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' THE QUBES PLATFORM AS A SCIENCE ED- UCATION GATEWAY Gateways refer to community-developed online environments that integrate ac- cess to shared resources including software, data, collaboration tools, and high- performance computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' As a Science Education Gateway QUBES is designed to lower barriers to faculty participation in STEM education reform by making it easier to engage in scholarship around teaching and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' From finding new teaching materials, to collaborating on projects, to accessing interdisciplinary expertise the QUBES platform provides both technical and social support to engage faculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Our target audiences include faculty whose scholarship centers on teaching and learning, with the platform designed to facilitate, document, and disseminate faculty work as they participate in diverse professional activi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Following a high level overview of the technical infrastructure, we describe a set of platform services that provide opportunities for faculty to engage with reform resources and pursue professional opportunities (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Technical Infrastructure The QUBES platform is a shared online space that can be used to publish and disseminate Open Education Resources, host distributed meeting and workshop activities, participate in professional learning, and support education reform projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' QUBES is an instantiation of the open source content management system HubZero, which initially was a branch of the Joomla!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' content man- agement system (CMS) but is now an independently developed platform for scientific gateways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The underlying technical infrastructure provides community building tools, including communication, productivity, and collaboration functionality, through membership controlled group spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' These group spaces can be public or pri- vate, with differing levels of openness to new members (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=', completely open, curated admission, invitation only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' There can be multiple administrators as- signed to a group who have additional abilities, such as changing group settings, controlling membership, and creating or modifying group web pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Each group space has optional community tools, such as discussion forums with email digest capabilities, announcements, calendars, file sharing, blogs, Pinterest-style file and image sharing (Collections), collaborative project spaces, a dedicated OER library, and traditional usage and website metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Group spaces can be fully customized to have their own web templates allowing for a customized look-and-feel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' 3 Figure 1: QUBES Science Education Gateway and its Four Platform Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The central element, labeled QUBES Science Education Gateway, represents the underlying technical infrastructure that supports an integrated, customizable online platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The four platform services (Professional Learn- ing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' OER Library Access;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Project Support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' and Customizable Workspaces) sup- port overlapping faculty user communities, provide multiple points of entry, and enable manifold use case scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Broadly, the gateway can be described as both a platform for hosting diverse professional activities and as a highly cu- rated repository of teaching and learning resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' This dual functionality supports a virtuous cycle where faculty engagement with STEM reform helps them generate new products which are then captured and curated within the gateway, making those resources accessible to the broader community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' 4 RYR Professional Project Learning Support QUBES Science Education Gateway OER Library Customizable Access WorkspacesThe HubZero CMS also supports the publication and hosting of software tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' QUBES has a dedicated execution host for providing cloud-based access to any software that can be run on a linux machine using the OpenVZ mid- dleware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Tools available on QUBES include data science environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=', RStudio Server, Jupyter Notebooks, Shiny apps), modeling tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=', NetL- ogo, Avida-Ed) and analysis tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=', ImageJ, Mesquite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' HubZero is cur- rently working to replace OpenVZ with Docker containers, which would provide additional functionality and better scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The QUBES platform currently has the capacity to run 200 concurrent tool sessions which presents significant scaling challenges when making software resources available to a national audi- ence of STEM educators and students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The QUBES platform has an open, self-publishing platform (QUBES OER Library) that uses a git-like version control system for tracking versions, adap- tations, attribution, and use metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' For searchability, publications are catego- rized via multiple standard OER library ontologies, such as activity length and audience level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Additional optional ontologies are available to align resources with evidence-based pedagogical frameworks, such as inclusive pedagogy, uni- versal design for learning, and open science and education practices [4]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Publications receive a digital object identifier (DOI) and several use metrics are automatically tracked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Importantly, publications can be associated with not just authors, but projects, organizations, or other QUBES hosted groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' This makes it easy to extract and display subsets of the library within a project group context, contributing to partner customization and autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' While QUBES is utilizing the open-source HubZero CMS, we have pushed for the development of additional functionality and UX improvements, both independently and through the support of the SGCI Developer Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' For example, in collaboration with HubZero developers we implemented an email digests option for group discussion forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Similarly, in collaboration with the SGCI Developer Program we implemented a forking/adaptation system for open education resources that made it possible to support the full OER lifecy- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Independently developed items include supporting instructor-only access to particular files within a published resource, an OER commenting system, and group webpage usage metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' To support autonomy of hosted projects on the QUBES platform, we increased group customization on the platform by allowing overrides of components, plugins and modules, including fully autonomous OER libraries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=', a dedicated search-and-browse for their community resources, and a customized resource record view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Finally, we have developed the ability for OER to have multiple aligned ontologies that can be easily used in a search- and-browse interface with Solr search capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' PLATFORM SERVICES In order to raise faculty awareness and encourage adoption of the QUBES plat- form we have developed a set of four platform services (Figure 1) tied to common uses of the underlying technological infrastructure (tools).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' These platform ser- 5 vices promote a set of use scenarios, or practices, which help to contextualize the ways that the gateway technology can be leveraged to meet faculty needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Platform services use a social infrastructure which provides training, templates, activity structures, and support documents to lower barriers to participation and promote successful engagement with the gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Developing and sharing strategies for effectively using a gateway can address both technical challenges associated with manipulating the gateway tools, and introduce new professional practices that incorporate things like open licensing, distributed collaborations, and online community building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Open Education Resources Library Access Too often innovative educational approaches developed by STEM faculty are not widely shared, licensed to support reuse, or documented as professional scholar- ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Our limited ability to capture, curate, and disseminate the collective teach- ing and learning knowledge of undergraduate STEM faculty severely limits the development and implementation of reform practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Collections of education resources are often distributed across multiple web sites making them difficult to find and manage, in addition to being more difficult to sustain over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The QUBES OER Library currently hosts over 2,100 resources that range from conference posters to classroom activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' This centralized, well described collection of teaching resources increases findability and can help faculty explore new teaching and learning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' By hosting diverse resources across many projects the QUBES OER Library promotes search both within and across projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Rich contextualization of the product reflecting authorship, project association, group affiliation and other meta-data makes it possible for users to naturally follow threads connecting seemingly disparate materials to support the discovery of related content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' This also makes it easy to extract and display subsets of the library within a project group context, contributing to partner customization and autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Access to the QUBES OER Library makes it easy for faculty to share their own work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Because the library uses open licensing and the infrastructure con- tains a “git-like” version management system it is possible for faculty to engage in the full OER lifecycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Adaptations of existing resources automatically con- tain attribution information that link the resources, making it easy for users to navigate between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Use metrics address a range of data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=', views, downloads, comments, versions, and adaptations) that help authors assess the impacts of their products within the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The introduction of social infrastructure (practices) around open licensing has been important to address community concerns about OER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Hosted projects on QUBES are often very interested in publishing their work as part of their dissemination and sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We also encourage projects to share non- traditional products such as annual reports, conference presentations, protocols and manuals to document outcomes and make them accessible to the broader community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' To support faculty use of the QUBES OER Library we structure professional activities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=', project collaborations, professional learning, work- 6 shops) around existing published materials, or as a mechanism to collect newly generated resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' This immediately helps to establish an authentic context for participating in professional learning opportunities – run by BioQUEST or hosted partner organizations – and generate products that are appropriate for publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We have adopted the language of a preprint server to help faculty understand the benefits of sharing works in progress and recognise the utility of self-publishing in the educational space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The BioSkills Guide is an example of a resource that was shared early during its development and then versioned as it was refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' To date it is in its fifth version and in total it has been accessed more than 7,500 times and downloaded more than 2,500 times [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' When the manuscript describing the development of the guide was published in a peer reviewed journal, the guide on QUBES was referenced, with the journal citation added to the description of the guide on QUBES, thereby pushing discoverability and traffic in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Project Support Externally funded education reform projects play an important role in fostering innovation and exploring effective teaching practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' However, funded projects often face logistical challenges like coordinating project activities, document- ing project impacts and sustaining their work beyond the funding period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' It is essential to the ongoing reform of STEM education that funded projects are as successful as possible and that their findings are carefully documented and shared to increase their impact over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Effectively engaging user communi- ties early in a project’s development cycle can play an important role in guiding the work to be broadly useful and seed the effective dissemination of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The QUBES platform currently hosts over 80 partner projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' These projects share a set of structural and technical needs that can burden the project team and undermine the time available to pursue the innovative project agendas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Hosting projects on the QUBES platform helps to avoid the inefficient and un- sustainable practices of hosting work on a separate server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We have developed a project support system that makes it easy to establish communication, collab- oration, and dissemination mechanisms both within the project team and the broader STEM education reform community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We provide a turnkey group space that can be customized to address specific project needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Our support services are designed to address three common challenges groups face when hosting their project on QUBES: understanding the operation of the platform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' maintaining a sense of ownership and branding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' and adopting new collaboration and communi- cation strategies to pursue their project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We also introduce planning resources that help the project leadership coordinate phases of their projects with the functionality they will require in their QUBES group space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' These resources are available within a partner support group on QUBES for onboarding of new projects, which includes demonstrations of the effective use of the platform to support the creation, maintenance, and sustainability of project workspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The CCBioInsites project is an example of an NSF funded project that hosted their activities on QUBES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The grant focused on helping community 7 college faculty participate in discipline based education research and improving their teaching practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The project recruited a distributed cohort of faculty and used their QUBES site to coordinate their activities [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Similarly the QB@CC project focused on teachers at 2-year schools who have limited access to professional learning opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' They have brought together both mathematics and biology faculty to collaboratively adapt existing teaching modules so that they reflect effective mathematics, biology content, and pedagogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' These products will persist and invite further customization in the QUBES OER Library disseminating and sustaining their efforts well beyond their active grant funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Professional Learning: Faculty Mentoring Networks Professional learning refers to an intentionally designed collaborative environ- ment where teachers work with one another to learn, develop and practice new methods for educating students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' In contrast to some professional development models, professional learning promotes learning through engagement in reform practices, deemphasizing the role of telling faculty what to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Providing ef- fective and efficient professional learning experiences is essential to faculty par- ticipation in education reform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Equitable access to scholarly learning opportu- nities requires that those opportunities are not exclusively tied to events like conferences which can limit participation to those with available money and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' In STEM specifically, both the disciplinary knowledge base and the sci- entific tools are evolving rapidly, necessitating ongoing professional investment in teaching and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Access to professional learning is a major contributor to the implementation of evidence-based teaching practices discovered through discipline-based education research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Our primary professional learning model is called a Faculty Mentoring Net- work (FMN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' These involve geographically distributed groups of 10-15 faculty working together with a facilitator over the course of a semester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' They use a QUBES platform group space to share resources, communicate asynchronously, and publish their products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The schedule and activities are structured to lead them through multiple stages: (1) a process of learning about a new teaching resource;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' (2) customizing that resource for use in their teaching setting and with their student audience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' (3) implementing the module in their classroom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' then (4) refining and publishing their adaptation to the OER Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We have tem- plated the processes necessary to run an FMN and provide training for FMN facilitators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Additionally, we have developed modules addressing common fac- ulty interests such as inclusive teaching strategies, universal design for learning, and overcoming math anxiety which can be integrated across diverse FMNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The basic FMN model has been implemented in over 90 professional learning opportunities addressing diverse agendas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Groups have adapted our FMN model to address different outcomes including learning how to use new computational tools, designing new curricular activities, and conducting collaborative research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' One example of how an FMN has been used to engage faculty in a schol- arly approach to teaching and learning involves the extension and adapta- 8 tion of a valuable teaching module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' “Investigating the footprint of climate change on phenology and ecological interactions in north-central North Amer- ica” was originally published as part of a NSF funded project coordinated by the Ecological Society of America [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' An FMN was hosted on QUBES where faculty were mentored through customizing the original activity and teaching this data-intensive lesson as part of their professional learning ex- perience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' This FMN led to 15 published adaptations of the original activity where faculty produced customizations to incorporate different regional flora, use different analysis tools, and fit within different course contexts (https: //qubeshub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='org/publications/267/forks/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Customizable Workspaces There are a wide range of professional communities (both formal and informal) that can play an important role in STEM education reform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' In addition to the grant funded projects described above, communities involving professional soci- eties, education nonprofits, research collaborations, and special interest groups can help engage faculty with professional opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Programs such as con- ferences, webinars, and workshops can also help expose faculty to new ideas and potential collaborators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' In order to be impactful these communities need to establish and sustain faculty engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Everything from cross-institutional course-based undergraduate research experiences (CUREs), to citizen science projects, to journal clubs and special interest groups can play a role in helping faculty pursue scholarship around teaching and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The QUBES platform currently hosts over 450 online group workspaces con- taining 1,200 project areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We have a set of strategies for establishing, man- aging, and disseminating resources from distributed communities of faculty as they pursue new ideas for teaching and learning STEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' These involve integrat- ing synchronous and asynchronous interactions, and using active facilitation techniques to work toward a shared community goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Through the creation of customizable workspaces, the QUBES platform can be used to extend engage- ment with traditional face-to-face conferences as well as support hybrid and on-line only professional meetings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' While participants can meet synchronously during meetings and workshops, the infrastructure also supports asynchronous engagement with material and other participants before and well after the syn- chronous portion of the meeting has ended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' As an example, BioQUEST runs an annual online meeting called the BIOME Institute which offers a unique opportunity to engage with a community of peers to address an educational challenge with the ultimate goal of improving student outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' A dedicated website and community space is created prior to the BIOME Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Participants are invited to this space ahead of the meeting to introduce themselves, bring attention to any educational reform projects they are involved in, and to engage with introductory prompts addressing the meeting focus and objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' During the meeting, talks and poster presentations are given throughout, working groups are brainstormed and formed, with all the material published as OER on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Plans are made near the end of 9 the synchronous portion of the meeting to create online sub-communities for working groups so that participants can continue their work asynchronously over the Fall semester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Products from these asynchronous working groups are then published as OER on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Some of these working groups develop into grant collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' You can see a list of the recent working groups spun off from the BIOME meeting here (https://qubeshub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='org/community/groups/ summer2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' ONGOING AND FUTURE WORK In addition to the four platform services introduced above we continue to inte- grate new functionality and use scenarios to support STEM education reform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Since the inception of the QUBES platform in 2014, we have emphasized work- ing closely with the user community to develop a shared vision for the ways that a gateway can support faculty scholarship around teaching and learning reflected in this quote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' If you want to build a ship, don’t drum up people to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' – Antoine de Saint Exup´ery Our ongoing and future development plans for the platform stem directly from needs identified by hosted projects and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Here we briefly introduce four high priority areas including: scalable and robust hosting of software tools, including tight integration of these tools with OER;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' design of teacher portfolios, akin to a LinkedIn or ResearchGate for educators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' a custom publishing platform that supports peer-reviewed education journal tools, beyond the self-publishing of OER already supported on the platform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' and support for discipline-based education research (DBER), which is constantly expanding our knowledge on evidence-based teaching strategies (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Software Tools One of the key elements of a scientific gateway is simplified access to com- putational tools for research and teaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The QUBES platform utilizes the open-source content management system HubZero in large part because of its built-in support for hosting computational tools, in addition to its collaboration and communication functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Addressing the scalability of access to these tools for use in the classroom has been challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Education user communities are much larger than research communities and require different types of server support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' This issue was exemplified most recently by the COVID pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' During lockdown and social distancing, many biologists moved their research and teaching out of the in-vitro/in-vivo and into the in-silica, leading to an increased demand for simulations and software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Data science has seen an explosion of demand and utility in the biological sciences, with a specific increase in students and faculty learning and using 10 Figure 2: Example ongoing and future QUBES platform development projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Each of the four panels represent ongoing or future development to extend the functionality of the QUBES platform to further support faculty engagement in scholarship around teaching and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' (A) Software tools involve implementing scalable and robust hosting of cloud based modeling and analysis tools in a way that is tightly integrated with OER published in the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' (B) Teacher portfolios refer to the design of an integrated system for collecting, sharing, and contextualizing faculty teaching scholarship as a means to document activities for consideration in hiring, promotion, and tenure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' (C) Custom publishing involves streamlining the deployment of manuscript manage- ment workflows, and specialized publication collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' (D) Discipline based education research support describes a suite of tools that would facilitate col- laboration around classroom research and studies of reform practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' 11 A B Software Teacher Tools Portfolios C D Custom DBER Publishing SupportR and Python in their courses and research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' There are many organizations working to train faculty in modern data science skills and techniques, such as The Carpentries, as well as offering professional development opportunities on how to incorporate these skills into the classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' There are many challenges, however, in bringing these tools into the classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Posit Cloud, for example, has free access for students to RStudio without the need to install R and RStudio on their computers, but students can quickly run out of compute time, especially as they are learning how to code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' CyVerse offers many services around accessing bioinformatics and computational tools in the cloud, but they mainly service the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We are actively exploring ways to increase accessibility to computational tools in classrooms, and fully integrate cloud based computing within the context of OER published in the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Teaching Portfolios As faculty participate in reform projects, publish OER materials, engage with professional learning, and access materials from the OER library, they are build- ing a record of their activity on the QUBES platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Teaching scholarship is often difficult to document because there are few publishing outlets for teaching resources, with those that do get published tending to be cited less frequently as their use happens in the informal context of the classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Our OER publish- ing platform already provides DOIs, full citations, and some data about uptake by the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Furthermore, each version and adaptation of a publication is tracked, showing iterative refinement, adoption, and use, with attribution automatically assigned on adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Our goal is to supplement these impact metrics with additional ways for faculty to document their teaching scholar- ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We plan to reconfigure the existing dashboards available to QUBES users to more effectively capture and share information about OER that has been published, groups joined, and FMNs completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We will also extend the user profile module to capture more information related to faculty activities and backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We typically acknowledge successful completion of professional learning op- portunities through FMNs and workshop/meeting experiences with certificates and letters of participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' In the future we will establish a badge system to document a wide range of activities that faculty complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' These badges can be linked not just to learning opportunities completed, but also to products produced via those opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Essentially, our vision for teaching portfolios involves providing more flexible opportunities to organize, present, and annotate one’s contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We are already seeing many faculty, particularly those in non-R1 settings, using information from their profiles and dashboards to capture their scholarship in annual activity reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Custom Publishing The vast majority of QUBES OER are submitted as a “QUBES Resource.” This resource type contains a broad set of metadata for describing the resources, 12 their intended audience, and other features like their use of inclusive learning practices, universal design principles followed, and racial equity strategies em- ployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' QUBES currently provides custom publishing types for a small group of projects including Math Modeling Hub (https://mmhub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='qubeshub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='org), NIBLSE (https://niblse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='qubeshub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='org), CourseSource (https://coursesource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' qubeshub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='org), and SIMIODE (https://simiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='qubeshub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' These cus- tom resource types have their own metadata schema and can be set to employ a review process before public release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' For example, the NIBLSE project devel- oped and published a set of core competencies related to describing beneficial outcomes of integrating bioinformatics techniques throughout the biology cur- riculum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' These competencies are represented in their metadata schema, thereby raising awareness and improving findability across their collection of publica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' CourceSource, a peer reviewed journal of biology and physics teaching resources, transferred their OER library into QUBES, where we developed a cus- tom metadata schema including alignment with learning outcomes and teaching frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' In collaboration with CourseSource, we are currently building a custom submission and editorial management pipeline that will allow authors, editors, and reviewers to utilize QUBES throughout the submission, review, and publication process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The goal is to make the OER manuscript management process customizable for easier and cost effective utilization by other communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We see implica- tions for ingesting existing education libraries looking for a sustainable home, and support for distributed communities, such as CUREs and citizen science projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Custom designed metadata schemas and curation pipelines will sup- port individualized, autonomous search-and-browse portals within their commu- nities, with their resources also available across the entire QUBES OER Library for broad dissemination and discoverability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Discipline Based Education Research Support There is growing awareness of and focus on discipline based education research as an important component of faculty scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We imagine QUBES as a platform where curriculum specialists, education researchers, and teaching fac- ulty could collaborate to scale up data collection and explore the impacts of interventions in diverse teaching contexts and across student audiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' This would likely involve some tools to facilitate data collection and management (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=', surveys and other online forms, activity tracking, and use logging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We would also need to institute student account types that would appropriately han- dle data de-identification to mitigate any personal risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' QUBES has already been used to coordinate distributed research projects using faculty mentoring networks to connect researchers with motivated teachers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We believe that the gateway environment can play an important role facilitating research on fac- ulty change, professional learning, project management, as well as documenting emerging practices as communities adopt gateway infrastructures to evolve their professional practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' 13 MOVING FORWARD - A CALL TO ACTION The QUBES platform has been designed and implemented to help faculty pur- sue scholarship around teaching and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Both the technical infrastructure (tools) and social infrastructure (practices) were purpose built to advance inno- vation in STEM education by supporting communities of practice, foreground- ing equity, diversity and inclusion, and engaging faculty in the culture of open education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Our four platform services (OER Library Access;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Professional Learn- ing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Partner Support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' and Customizable Workspaces) provide multiple points for faculty engagement and address key aspects of accelerating reform practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Hosting diverse activities on a single platform creates a synergistic effect and has proven to be important to our success, allowing us to capture faculty work and make it accessible to the broader community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We currently host a commu- nity of over 20,000 registered users and we are beginning to see the impact of both leveraging the platform to get work done and the ways that feeds forward into the collection, curation, and documentation of those professional activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The integration and interoperability of the various gateway tools supports and contextualizes a broad network of projects, organizations, and faculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We believe that the broad use of the QUBES platform by multiple stakehold- ers demonstrates a significant need for a centralized infrastructure to support reform practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' The National Science Foundation has identified QUBES, along with other emerging science education gateways, as important resources for in- creasing sustainability and broadening the impact of funded education projects [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' We encourage further exploration of the ways that science education gate- ways might prove to be fruitful across STEM education, helping to develop other purpose-built communities and platforms to accelerate innovation in teaching and learning science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='org/vol/v10/issues/datasets/calinger/ abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='html (URL) [11] NSF-DCL 21–026, “Dear Colleague Letter: Requesting proposals for online biology education to the Research Coordination Networks for Undergradu- ate Biology Education,” National Science Foundation, 03-Dec-2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' [On- line].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAzT4oBgHgl3EQfy_4H/content/2301.01760v1.pdf'} +page_content='nsf.' metadata={'source': 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Bliokh1, Ebrahim Karimi2, Miles J. Padgett3, Miguel A. Alonso4,5, +Mark R. Dennis6, Angela Dudley7, Andrew Forbes7, Sina Zahedpour8, Scott W. +Hancock8, Howard M. Milchberg8, Stefan Rotter9, Franco Nori1,10, Şahin K. +Özdemir11, Nicholas Bender12, Hui Cao12, Paul B. Corkum13, Carlos Hernández- +García14, Haoran Ren15, Yuri Kivshar16, Mário G. Silveirinha17, Nader Engheta18, +Arno Rauschenbeutel19, Philipp Schneeweiss19, Jürgen Volz19, Daniel Leykam20, +Daria A. Smirnova16, Kexiu Rong21, Bo Wang21,22, Erez Hasman21, Michela F. +Picardi23,24, Anatoly V. Zayats23, Francisco J. Rodríguez-Fortuño23, Chenwen +Yang25, Jie Ren25, Alexander B. Khanikaev26, Andrea Alù27, Etienne Brasselet28, +Michael Shats15, Jo Verbeeck29, Peter Schattschneider30, Dusan Sarenac31, +David G. Cory31, Dmitry Pushin31, Michael Birk32, Alexey Gorlach32, Ido +Kaminer32, Filippo Cardano33, Lorenzo Marrucci33, Mario Krenn34, and Florian +Marquardt34 +1Theoretical Quantum Physics Laboratory, RIKEN, Wako-shi, Saitama 351-0198, Japan +2Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada +3Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, Scotland +4Aix-Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille 13013, France +5The Institute of Optics, University of Rochester, Rochester, New York 14627, USA +6School of Physics and Astronomy, University of Birmingham, Birmingham, UK +7School of Physics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa +8Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, +Maryland 20742, USA +9Institute for Theoretical Physics, Vienna University of Technology (TU Wien), Vienna A-1040, Austria +10Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA +11Department of Engineering Science and Mechanics, and Materials Research Institute (MRI), The +Pennsylvania State University, University Park, Pennsylvania 16802 +12Department of Applied Physics, Yale University, New Haven, CT, USA +13Joint Attosecond Science Laboratory, National Research Council and University of Ottawa, Ottawa, +Ontario, Canada +14Grupo de Investigación en Aplicaciones del Láser y Fotónica, Departamento de Física Aplicada, +Universidad de Salamanca, E-37008 Salamanca, Spain +15MQ School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia +16Research School of Physics, Australian National University, Canberra, ACT 2601, Australia +17Instituto de Telecomunicações, Instituto Superior Técnico-University of Lisbon, Lisboa, Portugal +18Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA +19104, USA +19Department of Physics, Humboldt-Universität zu Berlin, 12489 Berlin, Germany +20Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, +Singapore, Singapore +21Russell Berrie Nanotechnology Institute, and Helen Diller Quantum Center, Technion–Israel Institute +of Technology, Haifa 3200003, Israel + +Journal of Optics (2022) #### +22Shanghai Jiao Tong University, 800 Dongchuan RD. Minhang District, Shanghai, China +23Department of Physics and London Centre for Nanotechnology, King’s College London, Strand, +London WC2R 2LS, United Kingdom +24ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, +Castelldefels (Barcelona) 08860, Spain +25Center for Phononics, School of Physics Science and Engineering, Tongji University, Shanghai 200092, +China +26Department of Electrical Engineering, City College of the City University of New York, 160 Convent +Avenue, New York, NY 10031, USA +27City University of New York, Photonics Initiative, Advanced Science Research Center, New York, USA +28Université de Bordeaux, CNRS, LOMA, UMR 5798, F-33400 Talence, France +29Electron Microscopy for Materials Science (EMAT), University of Antwerp, 2020 Antwerp, Belgium +30Vienna University of Technology (TU Wien), Vienna A-1040, Austria +31University of Waterloo, Waterloo, ON N2L3G1, Canada +32Department of Physics, Technion–Israel Institute of Technology, Haifa 3200003, Israel +33Dipartimento di Scienze Fisiche, Universita di Napoli ‘Federico II’, Complesso di Monte S Angelo, +80126 Napoli, Italy +34Max Planck Institute for the Science of Light, Erlangen, Germany + +Abstract +Structured waves are ubiquitous for all areas of wave physics, both classical and quantum, where the +wavefields are inhomogeneous and cannot be approximated by a single plane wave. Even the +interference of two plane waves, or a single inhomogeneous (evanescent) wave, provides a number +of nontrivial phenomena and additional functionalities as compared to a single plane wave. Complex +wavefields with inhomogeneities in the amplitude, phase, and polarization, including topological +structures and singularities, underpin modern nanooptics and photonics, yet they are equally +important, e.g., for quantum matter waves, acoustics, water waves, etc. Structured waves are crucial +in optical and electron microscopy, wave propagation and scattering, imaging, communications, +quantum optics, topological and non-Hermitian wave systems, quantum condensed-matter systems, +optomechanics, plasmonics and metamaterials, optical and acoustic manipulation, and so forth. This +Roadmap is written collectively by prominent researchers and aims to survey the role of structured +waves in various areas of wave physics. Providing background, current research, and anticipating +future developments, it will be of interest to a wide cross-disciplinary audience. + +Contents +1. +Introduction +2. +Phase matters (M. J. Padgett) +3. +The topology of 3D polarization (M. A. Alonso and M. R. Dennis) +4. +Shaping light (A. Dudley and A. Forbes) +5. +Spatiotemporal optical vortices and OAM (S. Zahedpour, S. W. Hancock, and H .M. Milchberg) +6. +Structured waves in Non-Hermitian systems (S. Rotter, F. Nori, and S. K. Ozdemir) +7. +Tailoring random light for imaging applications (N. Bender and H. Cao) +8. +Ultrafast structured beams and intense magnetic fields (P. B. Corkum and C. Hernández-García) +9. +Metaphotonics with structured Light (H. Ren and Y. S. Kivshar) +10. Structuring light with near-zero-index platforms (M. G. Silveirinha and N. Engheta) + +Journal of Optics (2022) #### +11. Strong coupling between atoms and guided light (A. Rauschenbeutel, P. Schneeweiss, and J. Volz) +12. Surface waves (D. Leykam and D. A. Smirnova) +13. Photonic spin-orbit interactions at metasurfaces: stochastic, Rashba, and quantum effects +(K. Rong, B. Wang, and E. Hasman) +14. Spin, momenta, and forces in evanescent waves - towards spatial and temporal structuring +(M. F. Picardi, A. V. Zayats, and F. J. Rodríguez-Fortuño) +15. Momentum and spin of electromagnetic, sound, and water waves (K. Y. Bliokh) +16. Acoustic spin (C. Yang and J. Ren) +17. Acoustic pseudospins for wave control and topological protection (A. B. Khanikaev and A. Alù) +18. Mechanical effects of structured sound waves (E. Brasselet) +19. Transport of surface matter in structured water waves (M. Shats) +20. Structured electron waves (J. Verbeeck and P. Schattschneider) +21. Structured neutron and atomic waves (D. Sarenac, D. G. Cory, and D. Pushin) +22. Structuring the quantum state of light (M. Birk, A. Gorlach, and I. Kaminer) +23. High-dimensional quantum communication (E. Karimi) +24. Simulating quantum systems with structured waves (F. Cardano and L. Marrucci) +25. Artificial intelligence for structured waves (M. Krenn and F. Marquardt) + + + + +Journal of Optics (2022) #### +1. Introduction +Konstantin Y. Bliokh1 and Ebrahim Karimi2 +1RIKEN +2University of Ottawa + +As it often happens with phenomena and discoveries in science, it is difficult to indicate a single +starting point for the study of structured waves. They have been in front of our eyes from the very +beginning of history: as waves on the sea surface, scattered sunlight and rainbows in the sky, etc. In +modern optics, seminal works by Nye and Berry [1–3], Baranova et al. [4], Soskin et al. [5, 6], and Allen +et al. [7, 8] stimulated the development of the fields of “singular optics” and “optical angular +momentum”. The development of these closely related areas was surveyed five years ago in the +“Roadmap on structured light” [9]. However, phase singularities (analysed by Nye and Berry for +ultrasonic pulses) previously appeared in the context of quantum matter waves in pioneering papers +by Fock [10], Dirac [11], Aharonov and Bohm [12], and Hirschfelder et al. [13, 14]. Furthermore, such +singularities can be found in the 19th century maps of tidal ocean waves [15]. Also, the orbital angular +momentum in localized wave states with vortices (described by Allen et al. for optical beams) appear +in the form of quantum electron states in atoms or in an external magnetic field as described in +textbooks on quantum mechanics [16]. This evidences that structured waves and their main +properties are universal phenomena across various wave systems, independently of their nature. +In this roadmap, we aim to review recent achievements related to structured waves in optics, +plasmonics, metamaterials, acoustics, electron and neutron optics, and quantum information. We +tried to avoid overlaps with the earlier “Roadmap on structured light” [9]. Therefore, the main focus +of this roadmap is shifted towards emerging directions which have been rapidly developing in the past +five years and to phenomena involving structured waves in non-optical fields. The areas addressed in +this roadmap include: non-Hermitian and topological wave systems; plasmonics, metasurfaces, and +near-zero index materials; quantum information and artificial intelligence; light-matter interactions +and waves in random media; electromagnetic, electron, neutron, acoustic and water waves. One can +notice the rapidly growing interest in such directions as: temporal structures, including time- +dependent media [17], space-time wavepackets [18], and spatiotemporal vortices [19–24]; novel +functionalities involving complex waves in non-Hermitian [384-386,124], topological [122,387], and +random [338,388] media; 3D topological polarization structures including polarization Mobius strips +[25–29], skyrmions [94,29–35,389], and structured polychromatic fields with commensurable +frequencies [36–40]; spin and polarization properties of sound [41–47], elastic [48–53], and water +waves [54, 55]; structured neutron and atomic waves [369,375,378,374]; simulating complex +quantum systems [56–58], high-dimensional communication [59–61], quantum cryptography [62–63], +and sensing [64–67]. Unfortunately, not all invited authors were able to contribute to this project, and +therefore, some areas were not covered. The most apparent omission is the absence of sections on +structured quantum condensed-matter waves, including BEC, cold atoms, exciton-polaritons, and +various quasiparticles in solids [68–75]. This topic deserves a special roadmap project. +This roadmap provides background, state-of-the-art, and perspectives for the interdisciplinary +physics of structured waves. We hope that it will illuminate the universality of structured-wave +phenomena across various areas of physics, highlight emergent directions involving structured waves, +and thus stimulate further development of this exciting field. + + +Journal of Optics (2022) #### +2. Phase structure matters +Miles J. Padgett +University of Glasgow + +Status +In a traditional sense, the importance of the spatial phase structure of light beams is inherent in any +image projection system. However, our eyes are insensitive to phase, perceiving only intensity and +hence for most user cases this image projection requires only the shaping of intensity. This shaping is +easily accomplished by transmission through an appropriate transparency, or with modern technology +in the form of a digital micromirror device normally associated with the projection of our slides in +scientific talks. In contrast to this intensity structuring, much of the current research in shaped light is +for those situations where the properties or application of the light arise from the phase structuring +of the beam. +One well known example of where it is the spatial phase structure of light that is the defining +feature is holography. In holography, the light scattered from the object is recorded by interfering this +light with a spatially coherent, plane-wave, reference beam. The constructive and destructive +interference with the reference captures not only the intensity but the phase of the scattered light +too. Traditionally, this interference pattern was captured using high resolution photographic film +which, once developed, could be illuminated by the same reference light to create the original light +beam as scattered by the object. This recreation of the scattered light creates a visual replica of the +object itself. It is interesting to reflect on the fact that despite the advances in the digital replacement +of film, these digital devices still lack the pixel count required to fully implement a realistic holographic +projector that can recreate the full 3D image of an everyday object. However, despite this technical +limitation, the modern advent of digital technologies acting as diffractive optical elements, i.e., digital +holography, has led to a multitude of related, albeit simpler, applications. +Beyond holography, the present-day interest in the phase structuring of light beam probably dates +to 1992, when the seminal paper published by Allen and co-workers reasoned that a light beam with +a helical phase structure, such as Laguerre-Gaussian laser modes, carried an “orbital angular +momentum” that was independent of, and additional to, light’s spin angular momentum [8]. This +postulate was rapidly experimentally verified by Rubinsztein-Dunlop and co-workers [76] along with +several other groups, transferring this angular momentum to microscopic particles held in optical +tweezers, causing the particles to spin. This early work was based on photographic film acting as +holograms, which when illuminated with a Gaussian reference beam, produced a diffracted beam with +the required helical phase [5]. While still focused on optical tweezers, Grier and co-workers replaced +the film with an interactive, pixelated liquid-crystal phase modulator for the holographic generation +of helically-phased and other beams [78]. The pioneering of these spatial light modulators and the +ease with which complex beams could be generated spawned work far beyond optical tweezers. In +the twenty years since [78], these spatial light modulators have driven an explosion in the study and +application of non-Gaussian laser beams and their use and application in areas ranging from the study +of optical phenomenology, imaging and sensing to quantum science. + +Current and Future Challenges +From a technical perspective, there are two key spatial light modulator technologies in widespread +use for the generation of complex phase and intensity structured beams: those based on liquid crystal + +Journal of Optics (2022) #### +and those based on digital micromirrors, both of which have video resolution. The liquid crystal +devices have phase-only modulation and, when used as a diffractive optical element, can reach well +over 50% conversion efficiency, albeit only at a video-frame rate switching. The micromirror devices +are intrinsically intensity modulators but can be used to create amplitude gratings and hence elements +with a low diffraction efficiency (< 5%), but at >10 kHz frame rate [79]. A clear challenge for the +development of new technologies is to simultaneously increase the diffraction efficiency while +maintaining or increasing further the frame rate. The combination of high efficiency and high speed +create new opportunities in real-time aberration correction and quantum information processing. +From a science perspective, most work to date on phase structured beams has focused on specific +beam types ranging from beams described by various polynomials—Laguerre, Bessel, Gegenbauer, +Airy—which often form complete orthonormal sets (i.e., a set of beams from which any other beam +can be synthesized). Typically this work has followed a logic of “what can beams of type X be used +for?”, whereas an alternative logic is “what design of beams will have the optimum performance in +application Y?” One example of these two logical approaches lies in single-pixel imaging systems +wherein a sequence of patterns is used to illuminate an object, and the backscattered light for each +projected pattern is measured. Summing the patterns, each weighted by the corresponding +backscattered single, reveals an image of the object. The majority of the work uses Hadamard or more +sophisticated patterns (e.g. [77]), but as an alternative, machine learning and related techniques can +be used to define a bespoke pattern set to enable a compressed sensing approach to emphasise +particular image properties or distinguish between specific object types [80], see Figure 1. + + +Rather than applying machine learning techniques to the optimum beam design, these machine +learning techniques can be applied to design a sequence of diffractive elements, creating an optical +implementation of a neural network [81], or the lossless transformation of one complicated modal set +into another simpler set [82]. All these transformations rely upon the phase structure of the beams. +As pioneered by Vellekoop and Mosk, another aspect of general complex beam design is in the +creation of light beams that deal with complex aberrations [83]. By precise shaping of the incident +light beam, it is possible to create a focused spot after transmission through highly scattering or +complex media such as a multimode optical fibre [84]. If the medium is characterised in terms of a +transmission matrix, then the inversion of this matrix defines the input beam required to produce any +spot at the output. If this spot is then raster scanned, the backscattered or similar signal reveals an +image from within or behind the media. + + +Figure 1. Within single-pixel imaging, it is possible to use deep learning to define an optimum measurement set based upon a library +of typical images. This approach to compressed sensing improves the quality of an image for a fixed number of under-sampled +measurements [80]. + +Previoustechnique:evolutionary +Deep-learningtechnique:deep- +Hadamardscanusing666patterns +learnedbasisof666patternsand +reconstructionJournal of Optics (2022) #### +Advances in Science and Technology to Meet Challenges +As discussed above, there are clear technical requirements and associated challenges for high-speed +spatial light modulators with high diffraction efficiency. Overdriving liquid crystal devices can perhaps +bring a speed which approaches that of a digital micromirror device, but progressing beyond a few +tens of kilohertz would seem to require a new, as yet unrealised, solid-state technology. If such +technology is to be developed it will possibly arise from image-based communication where the high- +dimensionality of the image state would bring a similar increase in communication bandwidth. +When it comes to the beams themselves, rather than those beams forming well-known complete +sets, future applications are likely to be based upon arbitrary beams optimised to a particular function. +Taking the example of aberration correction based upon the inversion of a transmission matrix, the +scale of the computation problem is clear. A system, or medium, of N modes is described by a N x N +matrix, the measurement time of which alone makes any real-time adaptation unlikely. A route to +solving these problems is possibly to identify a subspace so that the modified matrix can be inferred +from a relatively small set of additional measurements. The key is perhaps not to apply machine +learning to the recovery of the image data but to apply the compressed sensing to the measurement +of the transmission matrix itself. Whether a meaningful subspace exists will most likely depend on the +system type—a multi-mode fibre would seem a good place to start. + +Concluding Remarks +There are perhaps two main challenge and opportunity types ahead, both technical and conceptual. +A key technical challenge is for developing spatial light modulators with improved diffraction efficiency +and/or faster switching and/or much higher resolution. The last of these will undoubtedly be driven +by the consumer display market which might potentially enable full holographic projection of images, +whereas the first two are more likely to be addressed by specialist development. A key conceptual +challenge is to move away from specific beam types and their corresponding diffractive elements to +use machine learning and related techniques to create bespoke beams optimised for a particular need. +For example, the correction of the aberrations associated with complex media, creating the optimum +set of measurement patterns in imaging, or the creation of arbitrary diffractive elements in the optical +implementations of neural networks and optical mode transformation. This will be a fruitful ground +for the collaboration of optical engineers and computer scientists. + +Acknowledgements +This work is funded by the Royal Society and EPSRC under the grant number EP/M01326X/1. + + + +Journal of Optics (2022) #### +3. The topology of 3D polarisation +Miguel A. Alonso1,2 and Mark R. Dennis3 +1Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel +2University of Rochester +3University of Birmingham + +Status +Among the earliest applications of polarised light, going back to Brewster and Talbot in the 1800s, was +to explore structures of materials through crossed polarisers. Polarised light microscopy reveals the +optic axes of crystalline minerals, and Schlieren patterns reveal textures in liquid crystals, organised +by their topological defects [85]. In the modern study of structured light, where the state of +polarisation varies with position, light itself shows complex topological textures and structure [25, 86]. +The theory of polarisation for collimated light has a well-established formalism. When light travels +in a definite direction, only the two components of the electric field vector perpendicular to the +propagation direction are significant. For monochromatic light, the electric field vector at each point +traces an ellipse, whose eccentricity, handedness, and orientation constitute what we call +polarisation. The overall size (amplitude) and instantaneous position on the ellipse (phase) are usually +disregarded. The Poincaré sphere (Fig. 1(b)) is an elegant, abstract geometric representation, where +each polarised state corresponds to a point on the surface of a unit sphere. This point’s latitude and +longitude encode, respectively, the polarisation ellipse’s orientation and ellipticity. Its Cartesian +coordinates on the sphere, on the other hand, correspond to the Stokes parameters normalised by +the total intensity, which are easily measured. + + + +� +� +� +� +Figure 1. (a) 3D transverse polarisation texture with polarisation ellipses in one transverse plane. A C-line (white curve) of circular +polarisation passes through this plane several times, as does the L-surface on which the polarisation is linear. (b) Poincaré sphere of +transverse polarisations, indicating 2D polarisation states using a Runge colour scheme: polarisation azimuth by hue, and spin by +brightness, from black (left-handed circular) to white (right-handed circular). (c) Representation of mixed polarisation state where the E +vector traces a path in 3D (blue curve) that is not a simple ellipse: statistical E field described by the moment of inertia ellipsoid, and +average spin vector which must lie within the dual ellipsoid [89]. (d) Majorana sphere representation of polarisation as two points on the +sphere, projecting directly onto the polarisation ellipse as circles, or as the foci of the ellipse circumscribed by the sphere [90,91]. + +Journal of Optics (2022) #### + +The Poincaré sphere construction is more than just a map of two physical parameters onto a +surface. Amongst its physical properties, it represents: +• +the action of polarising and transforming optical elements as geodesic projections and +rotations over the sphere through the use of Jones calculus; +• +the geometric (Pancharatnam-Berry) phase due to cyclic changes in polarisation as half the +solid angle enclosed on the sphere by the path of polarisation transformations [87,88]; +• +partially polarised fields as points inside the sphere, with the radial coordinate giving the +degree of polarization. + +The standard polarisation formalism and the Poincaré sphere are not appropriate when light is +nonparaxial––i.e., the plane of the ellipse varies––and the normal to the polarisation ellipse is not tied +to the propagation direction. Various generalisations have been proposed which extend to the richer +structure of nonparaxial polarisation, albeit without the unifying simplicity of the Poincaré sphere. +Even for polarisation at a single point, the geometry and topology of the description for the +nonparaxial case are quite different. + +Current and Future Challenges +The topology textures in position-dependent polarisation fields strongly mirror topological textures in +liquid crystals; both can be described by headless vectors (directors), representing the major axes of +the polarisation ellipses or the axes of the rod-like liquid crystal molecules. Topological singularities +organise these patterns within a volume: C-lines of circular polarisation around which the director +rotates by ±180°, explored by Nye in the 1980s in Bristol where Frank had discovered disclinations— +the liquid crystal counterpart—30 years earlier. These singular filaments occur in transverse and non- +transverse polarisation fields. Their 3D structure in a volume can be controlled by holographic +manipulation, most often by liquid-crystal-based computer-controlled holograms (spatial light +modulators). This has allowed for the production and polarimetric measurement of beams with +knotted C-lines. +Determining the 3D state of polarisation directly, including the longitudinal component, is +nontrivial even in free space. Nowadays, the most common approach samples the field by scanning +with a nano-scatterer; this has allowed for the measurement of the fine structure around a C-line, +revealing Mobius band-like configurations [26]. +In addition to eccentricity and orientation in its plane, a nonparaxial polarisation ellipse has a +varying plane orientation (perpendicular to the direction of spin), requiring four parameters in total. +Based on Majorana’s spin formalism, Penrose proposed a representation in terms of not one but two +indistinguishable points over the surface of a unit sphere in 3D, with a simple geometric meaning: they +indicate the two directions for which the ellipse traced by the electric field projects onto a right- +handed circle (Fig. 1(d)) [90,91]. Mathematically, this corresponds to the symmetric product of two 2- +spheres, a topological representation of the complex projective plane CP2, corresponding to complex +3-vectors and disregarding an overall complex factor (corresponding to intensity and phase). This +representation allows for the optical geometric phase to be calculated when the plane of polarisation +varies, corresponding to a quantum spin-one geometric phase, as demonstrated, for instance, in the +Tomita-Chiao experiment for light in a twisted fibre [92]. +For partially polarised fields—that is, for fields that are polychromatic but whose frequency +components are uncorrelated—the electric field at a point does not trace an ellipse but a more + +Journal of Optics (2022) #### +complex, generally nonplanar and nonperiodic path. A generalisation of the Stokes parameters has +been proposed based on an expansion of the polarisation matrix using the 3x3 Gell-Mann matrices, +instead of the 2x2 Pauli matrices that yield the standard Stokes parameters [93]. This description +encodes polarisation as eight parameters, i.e., the number of real quantities needed to specify a trace- +normalized 3x3 Hermitian matrix. These parameters define an eight-dimensional hypervolume, +where—like for the Poincaré sphere—complete polarisation corresponds to points at unit distance +from the origin, and the remaining points correspond to partial polarisation states. However, without +a direct generalisation of optical elements or the Jones calculus to 3D, the power of this high- +dimensional description is yet to be realised. In the special case when the polychromatic field is +coherent and includes only a small discrete set of mutually rational frequencies—generated, for +example, by nonlinear harmonic generation—the electric field at each point traces a periodic path +that is not an ellipse but a Lissajous curve [36] that can be knotted in 3D [39]. +Recent experiments have revealed topological features in monochromatic polarisation +distributions not only along curves (e.g. Möbius strips) but over surfaces or volumes. For example, +Skyrmions are distributions that fully wrap around a parameter space corresponding to an n- +dimensional sphere. “Baby Skyrmions” for n=2 have been realised in different ways: +• +as full Poincaré beams, whose polarisations at any transverse plane cover the Poincaré +sphere’s surface [94] (Fig. 2(b)); +• +where, in the evanescent field over a planar metal surface, the linearly-polarised electric +field vector points in all 3D directions [30] (Fig. 2(a)); +• +where ellipses in a plane have spins (ellipse normal) in every 3D direction [31,83] (Fig. 2(c)). + + + + +A richer example of a 3D topological state (n=3) uses the fact that a normalised, transverse polarisation +state, including phase (Jones vector), corresponds to a point on a unit hypersphere in 4-dimensional +space. These 3-spheres admit the Hopf fibration into interlinking circles (on which the phase varies) +� +� +� +� +Figure 2. Different representations of Skrymion-like configurations in optical polarisation. (a) E-field points in all directions; (b) full Poincaré +beam (realising all points on Poincaré sphere); (c) circular polarised realising all directions of spin; (d) Skyrmionic Hopfion in 3D, realising +all polarisations and phases (on optical hypersphere). In all cases, the colour scheme is provided by the Runge sphere. + +O +0 +OJournal of Optics (2022) #### +parametrised by the Poincaré sphere. Such a structure—a Skyrmionic Hopfion (Fig. 2(d))—realising all +polarisations and phases in an entwined texture was recently designed, synthesised experimentally, +and measured [34]. The topological complexity of these polarisation distributions increases when +more degrees of freedom are considered, as is the case of polychromatic light [36,39]. +Concluding Remarks +Light polarisation at a point presents a simple yet elegant topological structure, which becomes more +complex when the paraxial limit is abandoned. The topological features are much richer when +considering the spatial variation of polarisation, echoing the structured topological states studied in +many forms of condensed matter. Because light can interact with matter, the structure of each can +shape that of the other; most prominently, light fields can be used to interrogate and shape liquid +crystals, and conversely. Modern microscopy techniques use light with varying polarisation to probe +the configurational 3D structure of biological matter (e.g. actin filaments). 3D topological polarisation +structures affect the orbital and spin dynamics of trapped particles. Topology and polarisation have +played a central role in the rapid development of the science of structured light over the past few +decades. However, the technological potential for many applications is still in its infancy. + +Acknowledgements +MAA acknowledges funding from the Excellence Initiative of Aix Marseille University – A∗MIDEX, a +French ‘Investissements d’Avenir’ programme, and from the Agence Nationale de Recherche (ANR) +through project ANR-21-CE24-0014-01. MRD acknowledges support from the EPSRC Centre for +Doctoral Training in Topological Design (EP/S02297X/1). + + + + +Journal of Optics (2022) #### +4. Shaping Light +Angela Dudley and Andrew Forbes +University of the Witwatersrand + +Status +Light can be shaped in all of its degrees of freedom (DoFs), in time and space, for so-called Structured +Light [96]. Traditionally, the spatial DoFs have been exploited—for example, amplitude, phase and +polarisation—first in 2D (the transverse plane) and later in 3D (all three components of the electric +field), while the time and frequency domains offer the potential for 4D control. Recently, there has +been a concerted effort to identify and control new DoFs, and to harness this control for emerging +applications, including communications, microscopy, imaging, optical trapping and tweezing, +quantum state engineering, and laser machining processes, to name but a few. The challenge is to +identify which DoFs can be controlled, to what extent, and with what toolkit [97]. For instance, +azimuthal phase control gives rise to orbital angular momentum (OAM) modes, which are easily +controlled both as scalar and vectorial superpositions (whose phase and polarization profiles are non- +separable). In this sense, OAM can be treated as an easily controllable DoF. In contrast, amplitude and +phase shaping to create radially structured modes (the p indexed modes in the Laguerre-Gaussian +basis, for example) is common-place [82], but this DoF is not easily controlled with our existing toolkit, +limiting its applicability. Path has long been associated with quantum states of light, but less so in +classical light, while ray-wave duality in classical fields (see Figure 4) is yet to be fully exploited. Given +the many DoFs of light, how can we realise and control them? +Optical cycles are much too fast to allow direct temporal light shaping, and so such light shaping is +done spatially. For example, to shape light temporally, the frequency components are usually path +separated by a dispersive element, mapping frequency to space, subsequently shaped in amplitude +and phase before a return mapping to reconstruct the desired temporal pulse [98]. In the spatial +domain, we may control the amplitude and phase of each polarisation component, the latter by +propagation and geometric phase, both of which can be made polarisation specific. The ubiquitous +lens is a simple example of a beam-shaping optic, adjusting the propagation phase by material +thickness and refractive index to shape light by refraction. Recent developments in free-form optics +has given new impetuses to refractive shaping of light, with unprecedented control possible [99]. In +the early 1990s, there was an explosion of activity in diffractive optical elements (DOEs), where a +computer-generated hologram was etched into a material to form a holographic plate of negligible +thickness (i.e., no refraction). Here, the underlying concept stems from Denis Gabor’s Nobel award- +winning development of holographic imaging, allowing for the recording of light’s amplitude and +phase information. Geometric phase has been exploited for complex beam shaping [100]—which by +definition is polarisation sensitive—allowing for the creation of vectorial light fields. A more recent +move to sub-wavelength structures in the visible has allowed for polarisation dependent propagation +phase control using metasurfaces [101], paving the way for all phases to be exploited. +The aforementioned approaches are all static, which greatly limits their versatility. The +introduction of liquid crystal spatial light modulators (SLMs) [102] ignited a plethora of investigations +into novel light shaping techniques and their corresponding applications. These devices are void of +customer-specific production requirements, span the visible and near-infrared wavelength ranges, +and offer instantaneous and rewritable amplitude, phase, and polarization control. Intriguingly, recent +times has seen a return to amplitude-only devices in the form of digital micromirror devices (DMDs), + +Journal of Optics (2022) #### +which are fast, cheap, and versatile. Direct light shaping at the source, within the laser, allows for high- +efficiency and high-purity modes as the output, and can be achieved through a variety of intra-cavity +and cavity geometry approaches [103], with complex light possible from very simple laser cavities +[104]. For example, a combination of internal and external path control has resulted in eight- +dimensional structured light across multiple DoFs (see Figure 4 and Ref. [104]). + + +Figure 1. A simple laser cavity can shape light that appears ray-like but with wave-like properties, across multiple DoFs. Eight-dimensional +structured light has been shown when internal and external light shaping are combined, producing the classical analogue to the famous GHZ +states of quantum light. Reproduced from [97] with permission. + +Challenges and opportunities +The algorithmic approaches to shaping light (the recipes) are very well established [105], dating back +to the early work on pattern recognition, and improved recently with the aid of machine learning. The +challenge lies mostly in the hardware. Although huge advancements have been made in the light- +shaping toolkit and its employment in a diverse range of applications, there remain some challenges. +Foremost amongst these are the need for the miniaturization of the modulation technology, increased +power thresholds, and a broader range in wavelength control, particularly at shorter ultra-violet (UV) +or extreme UV wavelengths. For example, in order to integrate and interface existing (miniature) +electronic components with fast photonic technologies that exploit the many photonic DoFs, light- +shaping approaches have to be miniaturised. The development of miniature, integrated on-chip +devices will bolster the optical computing, imaging, and communications industries. Miniaturising the +light-shaping toolbox has been restrained by the material of the beam-shaping device. Liquid crystals +typically allow for micro-meter-sized pixels, while refractive elements are limited to feature sizes +much greater than the operating wavelength. Recent developments in metamaterials and +nanostructured materials are offering promising avenues to engineer structured light with +subwavelength thick optics. Gradient metasurfaces are of particular interest in that they possess a +spatially varying phase response allowing for arbitrary wavefront control with subwavelength +resolution. These novel and minute components allow for integration into existing photonic +technologies, such as photonic circuits and optical fibres. Another promising opportunity is based on +two-photon polymerization (2PP) which is a direct laser writing technique fabricating complex 3D +micro-optic structures. Here wafer-thin optics with feature sizes of approximately 160 nm can be +fabricated [106]. +Another major challenge is to extend the current light-shaping techniques to high-power levels, +where levels exceeding kilowatts are needed for laser machining and laser-enabled manufacturing. + +Bellstate +Greenberger-Horne-Zeilingerstate +4(0-+(H(+ +(≥(z/(-/+(αl(i1(+) +z/(-0-+(a(t/(+(0+ ++2m)/+0)]+)[1)[R)+|m)/e) /)[2)[2) +_asercavityJournal of Optics (2022) #### +Commercially available single and multimode fibre lasers used for welding, cutting, and additive +manufacturing operate at the multi-kilowatt level, but to date, very few beam-shaping technologies +can tolerate such extreme power levels, although efforts to remedy this are underway. Optics used in +high-power kilowatt systems are large and bulky due to the finite absorption limitation—the complete +opposite of miniature-sized components. More recently, SLMs are being supplied with thermal control +units, where water-controlled heat sinks allow for a 10-fold increase in incident power. There is +promise in revisiting well-known and well-established adaptive optics solutions such as deformable +mirrors. These mirrors can tolerate and efficiently reflect powers on the order of kilowatts. However, +efforts will need to be made in order to enhance their response rate and stroke to achieve fast, high- +purity spatial mode creation while reducing their cost. In attempting to reduce the size of high-power +beam-shaping devices, the previously discussed metasurfaces and nanostructures could provide +promise; however, their power-handling capabilities have yet to be tested. +The aforementioned approaches are all based on linear optics. An exciting avenue that is very +much in its infancy is to shape light by nonlinear approaches. One possible solution is to shape the +light at low power and amplify it post-shaping, while another is to transfer shapes from one +wavelength to another through parametric processes. These approaches may overcome both the +power and wavelength challenges, while on-chip nonlinear optics has a long history, holding promise +for compact solutions too. Traditionally, amplification and nonlinear processes have focused on “how +much light do I have?”, and less so on “what does the light look like?” To explore these possible +solution pathways will require a paradigm shift in our thinking on such topics. +Other challenges lie in spatially controlling short wavelengths, as the light shaping toolbox for these +wavelengths are rare (neither SLMs nor DMDs work), while hardcoded DOEs and metasurfaces +likewise struggle with material choice and feature size. All of the previously mentioned tools are only +valid and applicable to coherent beams, but how can we achieve and similarly control the DoFs of +incoherent sources? Already, the advancement of “Li-Fi” may benefit from further DoF control if such +beam-shaping approaches are feasible for incoherent light. +Concluding Remarks +Techniques to achieve light shaping have advanced tremendously during recent years, offering a +diverse set of tools, and opening an abundance of exciting applications. Although a broad scope in +light shaping functionality has been achieved, photonic control in optical circuits is still primitive in +comparison to electronic control, with practical and commercial realisation still far off. Further +advances in efficiency, compactness, broader wavelength control, and power-handling capabilities are +sorely needed for shaping light to advance from science to application. + + + +Journal of Optics (2022) #### +5. Spatiotemporal optical vortices and OAM +Sina Zahedpour, Scott W. Hancock, and Howard M. Milchberg +University of Maryland + +Status +It is well established that monochromatic beams of light can support vortices where +electromagnetic energy density circulates around a local axis. Examples of such +orbital angular momentum (OAM)-carrying beams are the Laguerre-Gauss (LG) or +Bessel-Gauss (BG) modes in free space, where the OAM axis coincides with the +direction of propagation [8], or spatial optical solitons with vortex rings [107]. In all +cases, local vortical axes are fixed in space and energy density flow is described in +purely spatial coordinates. However, because electromagnetic vortices and OAM are +fundamentally associated with energy density circulation, there is no prohibition, in +principle, for a local vortical axis to +deviate +from +the +propagation +direction while embedded in a +propagating +pulse. +This +would +require a polychromatic beam [19, +21]. +Such +polychromatic +vortices +embedded in spacetime were first +measured as a naturally emergent +and +universal +structure +in +the +collapse arrest dynamics and self- +guiding of intense laser pulses in +nonlinear media [21]. Were it not for +‘collapse arrest’—a response to high +intensity such as ionization—the +beam would collapse to a singularity. +Collapse arrest enables a quasi- +stable accumulation of phase shear– +a very sharp phase gradient in the +transverse direction--between the +inner and outer parts of the beam, +triggering a phase defect (and field null) that wraps around the propagation axis and +spawns toroidal spacetime vortices (Fig. 1(a)). These toroidal spatiotemporal optical +vortices were observed for the first time in [21] and dubbed STOVs; they direct energy +density flow in a self-guided pulse, and are robust and topologically protected. +The realization that STOVs were generated by phase shear in spacetime led to a +method to generate them linearly and controllably, using a pulse shaper to apply +shear in the spatiospectral domain [22,23] and then return the pulse to the +spatiotemporal domain. The STOV pulse thus generated resembles an ‘edge first +flying donut’ (see Fig. 1(b) and Fig. 2), with phase circulation in spacetime and OAM +orthogonal to the propagation direction. Experiments [22] and theory [24,110] have + + +Figure 1. (a) Propagation simulation showing birth and evolution of +toroidal STOVs wrapping around a nonlinearly self-guided laser pulse. +The STOVs are generated by spatio-temporal phase shear [21]. (b) +Example [22] of 4𝑓 pulse shaper for linearly generating STOVs by +applying spatio-spectral phase shear. Here, SHG is also shown [109]. + + + + + + + + + + + + + + + + + + + +Figure 1 – We allow at most two figures that +are roughly the size of this box. +phase plate +cylindrical lens +BBO +-step +(b) +input grating +output grating + +phase +500um +50fs +intensityJournal of Optics (2022) #### +shown STOV propagation is governed by diffraction in both space and time. Later +simulations have shown that STOV-carrying pulses can be formed by compact +nanostructures that impose a specified field null axis in (𝜔/𝑐, 𝐤) space, leading to +arbitrarily oriented STOV axes in spacetime (𝑐𝑡, 𝐫) [111]. That spatiotemporal OAM is +a property of single photons was recently confirmed in experiments showing OAM +conservation under second harmonic generation (SHG) with STOV pulses [109]. +Recent work has investigated extreme ultraviolet STOV photons produced by high +harmonic generation [112]. + +Current and Future Challenges +Merely measuring STOVs presents new challenges. Unlike spatial optical vortex- +carrying beams, whose intensity and phase profiles can be measured by CW beam +imaging and interferometry, more complex methods are needed to visualize STOVs. +Narrow bandwidth, low resolution multi-shot pump-probe cross-correlation +measurements [23] can be used; these are limited to long duration STOV pulses with +high pulse-to-pulse stability. For examining ultrafast STOV evolution in nonlinear +propagation experiments, where sensitivity to small fluctuations is expected, a +broadband single-shot technique has +been +demonstrated +that +measures +pulses with spacetime phase defects +[22]. As STOV-structured light increases +in complexity, such as with short +wavelength attosecond pulses [112], +there will be a need for much higher +space and time resolution diagnostics +extending across the spectrum. +As the study of STOVs is in its +infancy, they could be viewed as a +solution in search of a problem. +Because they are integral to energy +density flow in both nonlinear self- +guided propagation [21] and in linear +propagation +[22,110], +an +unusual +robustness may apply to these beams +owing to conservation of topological +charge and angular momentum. As with +space-defined OAM, the excitation and probing of STOV-OAM states in materials and +structures will open completely new research directions. Any system involving +spacetime phase shear, such as transient current densities in nanostructures, could +be interrogated by STOV pulses with a prepared OAM content. Robust methods for +filtering or sorting the OAM components of STOV pulses will therefore be needed. In +analogy with super-resolution microscopy using monochromatic OAM beams, +STOVs may even provide a super-resolution capability in spacetime. + +Figure 2. Spatiotemporal intensity and phase profiles of a +𝑙 = 1 STOV propagating through its beam waist (𝑧 = 0), as +predicted by the modal theory (left) and measured (right) +[110]. Within each frame, pulse propagation is right-to-left. +Symbols: 𝑧! = 𝜋𝑤"# +$ +𝜆 +⁄ , 𝑤"# and 𝑤"% are space-like and +time-like modal scale lengths [110]. + + + + + + + + + + + + +2 +0 +0. +2 +0 +2-2 +0 +2-2 +0 +2-2Journal of Optics (2022) #### +At a fundamental level, only very recently has there been theoretical work on +angular momentum of STOV pulses. One approach calculates spin angular +momentum and OAM of STOV pulses in vacuum [24] and the other derives the mode +structure and OAM for paraxial STOV pulses in dispersive media [110]. These papers +differ in their choice of OAM operator, with differing results for the OAM per photon +ℒ in a STOV-carrying pulse of topological charge 𝑙: ℒ = 𝑙 ℏ(𝛼 + 𝛼!") 2 +⁄ [24] vs. ℒ = +𝑙 ℏ(𝛼 − 𝛽#𝛼!") 2 +⁄ [110], where 𝛼 = 𝑤$% 𝑤$& +⁄ + is the ratio of the STOV’s time-like to +space-like spot sizes and 𝛽# is the medium’s normalized group velocity dispersion. +The latter operator is conserved, and predicts half-integer OAM in vacuum (owing to lack +of energy density flux in the local time domain) and the existence of “STOV polaritons” +in dispersive media. Just as the pioneering work connecting LG modes to photon +OAM [8] was followed by quantum field theories, there are likely to be similar follow-ups +to [24,110]. + +Concluding Remarks +Experimental realization of electromagnetic pulses with spatiotemporal OAM has +opened a promising new avenue for studying OAM and its applications. + +Acknowledgements +The authors thank Konstantin Bliokh for fruitful discussions, and acknowledge the +support of the Air Force Office of Scientific Research, the Office of Naval Research, +and the National Science Foundation. + + + +Journal of Optics (2022) #### +6. Structured waves in non-Hermitian systems +Stefan Rotter1, Franco Nori2,3, and Şahin K. Özdemir4 +1Vienna University of Technology (TU Wien) +2RIKEN +3The University of Michigan +4The Pennsylvania State University + +Status +The structuring of waves typically involves the propagation of an incoming wave field through a device +that shapes the transmission in a desired fashion. Engineered structures like gratings and waveplates +can equivalently be operated in reflection mode. In both cases, the desired wavefront of the outgoing +wave is achieved through appropriate interferences induced by the wave-shaping device that tunes +the wave’s spatial and spectral (or temporal) degrees of freedom. In a very recent line of research, +one tries to extend the possibilities to structure waves of different kinds (electromagnetic, acoustic, +etc.) by working with non-Hermitian wave-shaping tools. The term “non-Hermitian” refers here to the +fact that the time evolution of the wave passing through the wave shaper is governed by a non- +Hermitian Hamiltonian. +While, formally, a simple wall that absorbs all of the incoming waves would already constitute such +a non-Hermitian system, one typically refers to non-Hermitian systems only when they contain a non- +trivial combination of gain (amplification) and loss (dissipation). Typical examples include non- +Hermitian meta-surfaces for steering transmitted or reflected waves in desired ways [113], and +waveguides that are designed such as to guide incoming waves around non-Hermitian degeneracies +known as exceptional points (EPs) [114]. Such EP-encircling protocols build on the remarkable +property that the output state on either side of the waveguide depends only on the direction in which +the EP is encircled—a property determined solely by the input port through which the wave is injected +into the waveguide. +Going beyond such asymmetric mode-switching protocols, spatially tailored gain-loss landscapes +can also be used to guide incoming waves [115–117], even through disordered scattering regions [118, +119]. When the patterning of gain and loss is done right, waves can not only be perfectly transmitted, +but can also maintain a well-behaved intensity profile (without any interference fringes) even inside +strongly disordered media (see Fig. 1). Under certain circumstances, the details of the system that is +patterned with a certain gain-loss distribution do not even have to be known to apply a non-Hermitian +structuring. In cases such as random lasers that have an unknown and typically inaccessible internal +structure, the spatial pattern of the pump beam that delivers the optical gain to the laser can be +optimised through appropriate algorithms to engineer the laser’s spectral and spatial emission +properties [120, 121]. In this way, single-mode operation and a directional far-field pattern of a +random laser can be achieved. +Recently, concepts from topology have also been employed to create robust unidirectional +propagation of light and to design lasers that operate on topologically protected edge modes (see +[122] for a recent review of this emerging field of research) or on modes that encircle an EP [123]. + +Current and Future Challenges +On the conceptual level, the major lines of research currently concentrate on the question: which new +functionalities, advantages, or unconventional features may the engineering of a system’s non- +Hermitian degrees of freedom bring along? Is it possible to use non-Hermitian engineering to, for + +Journal of Optics (2022) #### +example, overcome the stringent material requirements for meta-materials used for optical cloaking? +Can one exploit topological concepts to build photonic structures that are robust against fabrication +imperfections? Can such progress be achieved without having to work with very exotic materials or +cumbersome setups? + + + +On the technical level, one of the major challenges is the accurate positioning and control of the +non-Hermitian (gain and loss) components. While lossy media are ubiquitous and comparatively easy +to pattern, media with gain are more challenging to handle. This is because gain necessarily requires +an external source of energy to provide the amplification to the wave—think here of an active medium +such as a laser cavity that requires an external pump (optical or electrical) to operate the laser. For +optical devices in one or two dimensions, one typically works with optically active materials that are +pumped through an external beam with a spatial pattern that is controlled by a suitable mask or a +spatial light modulator (the size and costs of the latter, of course, also impose limitations). A three- +dimensional structuring of the pump profile is naturally more challenging to implement. Moreover, +non-linear effects, such as those induced by gain saturation and spatial hole burning, further +complicate the situation. For other types of waves, like sound or matter waves, suitable gain +mechanisms may not even be available. We emphasise that both the spatial and the spectral tunability +of a gain medium have certain restrictions: while the spatial pattern of the pump cannot be arbitrarily +fine in its structure, the spectral profile of the gain is determined by the fundamental constituents of +the active medium as well as by other restrictions like the Kramers-Kronig relations that follow from +the principle of causality. +A viable strategy to overcome these limitations is to consider discrete instead of continuous +systems, where the discreteness of the former may refer to their spatial structure or to their time- +evolution. A corresponding mapping onto arrays of waveguides or coupled fibre loops has the +advantage that each spatial or temporal element involved in such a discrete system can be individually +controlled. Recently, this advantage has been used in various circumstances to implement theoretical +concepts on non-Hermitian wave engineering—from the creation of constant-pressure sound waves +to the realisation of system designs based on the concept of non-Hermitian topology (for a review see +[124]). +Figure 1. (a) Two-dimensional distribution of a real refractive index nR (top panel). A Gaussian beam entering this disordered scattering +landscape and becomes severely distorted (bottom panel). (b) Adding a tailored distribution of gain and loss through an imaginary +refractive index nI (top panel), the Gaussian beam propagates as through a homogeneous medium (bottom panel). Image adapted from +[119]. + +(a) +nR +(b) +ni +10 +0.3 +0 +-10 +0.3 +10 +30 +50 +x/a +10 +30 +50 +x/2 +4 +IE|2 +60 +60 +T +[E]2 +0 +30 +0 +30 +15 +x/2 +15 +x/2 +0 +y/a +15 +0 +y/a +-15 +0Journal of Optics (2022) #### + +Advances in Science and Technology to Meet Challenges +On the theoretical level, there are still many questions left open for exploration. When considering +smaller and smaller constituents of non-Hermitian media, where quantum effects start playing a role, +the influence of the noise induced by both gain and loss has to be taken into account. As it turns out, +such noise processes are not just an annoying side effect, but they may constitute the major +bottleneck for the performance of a non-Hermitian device, such as for sensors that operate at an EP. +Dealing with such noise processes remains an outstanding theoretical challenge. +Also in the domain of non-Hermitian topology, numerous puzzles are waiting to be resolved. Going +beyond the question of how to correctly transfer concepts from Hermitian topology to the domain of +non-Hermitian physics, entirely new topics emerge; consider here, for example, nonlinear effects that +lead to strongly correlated states of light and interesting collective processes. It also still needs to be +clarified which features non-Hermitian materials can provide that were so far only associated with +other types of materials, such as anisotropic media and those with a vanishing or negative index of +refraction. Specifically, it will be interesting to clarify if and in which way non-Hermitian media can be +used for cloaking an object [125], for breaking Lorentz-reciprocity (in combination with nonlinearity +or spatio-temporal modulation), and for imaging beyond the diffraction-limit. +For the experimental implementation of flexible non-Hermitian media, progress on many +different sides will be beneficial. Advances in the speed, precision, and cost of spatial light modulators +will help to accurately control spatially tailored pump beams. Advances in the development of active +materials will help to address the challenging requirements in the gain values required for +implementing certain theoretical concepts. Non-Hermitian metasurfaces with well-controlled +distribution of discrete gain and loss ingredients may exhibit interference effects originating from +gain- and loss-induced phase responses, leading to phase singularities and vortices which can help to +control and shape the light transport [126]. + +Concluding Remarks +To conclude, the research on structured waves in non-Hermitian media is still in its infancy. While a +whole range of interesting theoretical concepts and experimental platforms for their implementation +have recently emerged, the field is still growing in several directions such as towards the inclusion of +topological, quantum, and nonlinear effects. Another appealing prospect is to apply non-Hermitian +design concepts not just on externally generated light fields, but to integrate them directly into the +design of the laser light source itself. Important challenges are the flexible and precise control of the +non-Hermitian gain and loss components in both their spatial and spectral degrees of freedom. With +rapid technological progress in this direction, we expect non-Hermitian tailoring of light fields to +become a standard tool in wave engineering. Ultimately, non-Hermitian elements may achieve +comparable relevance to spatial light modulators and conventional diffractive metasurfaces made +from dielectric or plasmonic structures without gain inclusion, paving the way to a new era of +wavefront shaping. + +Acknowledgements +S.R. acknowledges support by the Austrian Science Fund (FWF, grant P32300 WAVELAND) and by the +European Commission (grant MSCA-RISE 691209 NHQWAVE). F.N. is supported in part by NTT +Research, and Ş.K.Ö. by the Air Force Office of Scientific Research (AFOSR) Multidisciplinary University +Research Initiative (MURI) Award No. FA9550-21-1-0202. + + +Journal of Optics (2022) #### +7. Tailoring Random Light for Imaging Applications +Nicholas Bender and Hui Cao +Yale University + +Status +Spatially random light has the hallmark appearance of an irregular mosaic of diffraction-limited +speckle grains. Speckle formation is a phenomenon inherent to both classical and quantum waves, +occurring when a coherent wave undergoes a disorder-inducing scattering process. The speckle +patterns are described by a statistically stationary and ergodic random process. Stationarity requires +the statistical properties of an ensemble of speckle patterns to be the same as those of an individual +speckle pattern within the ensemble. Ergodicity requires the statistical properties of two spatial +positions—separated by more than one speckle grain size—to be independent and identical to those +of the ensemble. The speckle patterns are categorized by the joint probability-density function (PDF) +of their complex-valued field. Rayleigh speckles—the most common family of speckles—obey a +circular-Gaussian field PDF which results in a negative exponential intensity PDF [127,128]. The phase +PDF is independent of the amplitude PDF, and constant over a 2π range. The circular invariance of the +field-PDF results in a “fully developed” speckle pattern. Typically, non-Rayleigh speckles are classified +as either under-developed (the sum of a small number of scattered waves, or the sum of not fully +randomized waves) or partially coherent (the sum of incoherent partial waves). Furthermore, fully +developed speckles typically possess only short-ranged spatial intensity correlations which are +determined by the average speckle grain shape, which is dictated by the diffraction limit. +Because of their pervasiveness, speckle patterns have been adapted for use in a wide range of +optical applications ranging from imaging [129, 130] to optical manipulation [131]. While Rayleigh +speckles are the most common family of speckled light, their statistical properties and spatial +correlations are not necessarily ideal in different applications. There has been a plethora of interest +in creating speckle patterns with tailored statistics and spatial correlations [132-136], due to their +potential applications in structured-illumination imaging. Specific examples include dynamic speckle +illumination microscopy, super-resolution imaging, and pseudo-thermal light sources for high-order +ghost imaging. Furthermore, a general method for customizing both the statistics and topology of +laser speckle patterns would be a valuable tool for synthesizing optical potentials for cold atoms, +microparticles, and active media. + +Current and Future Challenges +Because the conditions required to generate Rayleigh speckles are general, creating fully developed +non-Rayleigh speckles is challenging. Specifically, the difficulty lies in altering the intensity PDF without +changing other statistical properties, e.g., phase PDF, stationarity, ergodicity. Recently, a simple +method for creating non-Rayleigh speckle patterns with a phase-only spatial light modulator (SLM) +was developed [132]. High-order correlations were encoded into the field reflected by the SLM, +resulting in a redistribution of the light intensity among the speckle grains in the far-field (Fourier- +plane). The resulting speckle pattern possesses an intensity PDF with a tail decaying either slower or +faster than a negative-exponential function. Subsequently, a general method for tailoring the intensity +statistics of speckle patterns was developed based on the same principle of modulating the phase +front of a laser beam with a SLM [133]. Experimentally, speckle patterns governed by arbitrary + +Journal of Optics (2022) #### +intensity PDFs were created. The speckle patterns exhibit distinct topologies from Rayleigh speckles, +without introducing spatial correlations beyond the diffraction-limited speckle grain size. + +A foundational principle of statistical physics is the Siegert equation. Specifically for the case of +spatial correlations in random light, the Siegert equation proportionally relates the intensity +correlation function with the squared magnitude of the field correlation function. As such, the spatial +intensity correlation function of a typical speckle pattern does not possess additional structure beyond +what is present in the field correlation function, which is determined by the diffraction-limited average +speckle-grain shape. In [132,133], the spatial intensity correlations of the customized speckle patterns +adhered to the Siegert equation. It was experimentally shown in [134] that a speckle pattern can +dramatically break the Siegert relation when non-local correlations are controllably encoded into a +speckle field by a SLM, specifically by tailoring the 4th order correlations in the Fourier plane. + + + + +While the techniques presented in [133, 134] independently modify different properties of +speckle patterns, [135] combined these methods to arbitrarily tailor the PDF and spatial intensity- +correlations of speckles. Figure 1 presents two examples of fully developed speckle patterns (a), (b) +with different tailored spatial correlations (c), (d) and customized statistics (e), (f). + +Figure 1. Two example speckle patterns (a), (b) from stationary and ergodic ensembles of 100 tailored speckle patterns, with +customised spatial intensity-corelations (c), (d) and intensity PDFs (e), (f). The example speckle pattern in (a) possesses non-local +intensity-correlations (c), and a unimodal intensity PDF (e). The speckle pattern represented by (b) is tailored to have ring-shaped, long- +range intensity-correlations (d), and adhere to a bimodal intensity PDF (f). + +aDesignedSpeckles +DesignedSpeckles +max +Ci(Ar) +i(△r) +ma +66um +66μm +Cmin +e +IntensityPDF +IntensityPDF +1 +/1)d +2Journal of Optics (2022) #### +Advances in Science and Technology to Meet Challenges +The scientific advances made in speckle customization have begun to translate into speckle-based +applications. For example, a proof of principle demonstration [136] has shown that 2D customized +speckles can significantly out-perform Rayleigh speckles in nonlinear pattern illumination imaging +techniques [130]. Figure 1(a) presents a specially tailored speckle pattern to photoconvert a uniform +fluorescent sample. Within the central square, the speckle pattern was designed to consist of a +random array of circular vortices embedded in an approximately constant-intensity background; +outside the square are Rayleigh speckles. Figure 1(b) is the fluorescence image of the sample after +being photoconverted by the speckle pattern. Outside the central square, the fluorescent pattern +consists of a sprawling anisotropic web, which reflects the topology of the low-intensity regions +surrounding the optical vortices in Rayleigh speckles. In stark contrast, the fluorescence pattern within +the central square c features isolated isotropic fluorescent spots. The isotropy exhibited by the +fluorescent spots originates from the high degree of rotational symmetry of the vortices in the +customized speckles. Apart from these spots, the fluorescent intensity is uniformly low due to the +homogeneity of the customized speckles’ intensity away from optical vortices. Qualitative comparison +between the two distinct fluorescent patterns illustrates the degree to which customizing the speckle +intensity statistics can enhance the performance of speckled illumination. Quantitatively, the +customized speckles were able to create fluorescent spots three times smaller than the diffraction +limit set by the illumination optics. + + + + +Nevertheless, significant scientific challenges remain to be solved in order to further speckle- +based applications: namely, developing a technique to customize speckled light inside a random +scattering medium, tailoring 3D volumetric speckles, and creating vector speckle fields with +customized statistics. Technical advances in wavefront shaping devices will facilitate addressing these +challenges. For example, increasing the effective number of independent phase modulating pixels on +a SLM can provide more degrees of freedom for creating customised 3D speckle patterns, or vector +light field for high-NA imaging applications. Future improvements to the SLM operation speed will +facilitate tailoring random light inside dynamic scattering media, potentially even allowing random +light tailoring inside live biological systems. + +Figure 2. A speckle pattern (a) is designed to photoconvert a uniform protein sample. Within the yellow square in (a), optical vortices +are randomly embedded in a bright background, and outside are Rayleigh speckles. Fluorescence image of unconverted protein inside +the square (b) shows isometric and isotropic spots produced by the vortices in the tailored speckles (c). Unconverted protein outside the +square in (b) features large, irregular, and interconnected fluorescent grains. + +a DesignedSpeckles +FluorescentSignal +C.-Target Region +xew +max +100umJournal of Optics (2022) #### +Concluding Remarks +Recent developments in customizing laser speckles [132-135] have resulted in simple, yet versatile, +techniques for creating and controlling random light, which can easily be adapted for use in a diverse +range of optical experiments and applications. For example, the ability to arbitrarily control the non- +local correlations and intensity PDFs of speckle patterns can be used to create complex optical- +potentials for studies on the transport of cold atoms, active media, and microparticles [131]. +Potentially, it can also enhance many structured-illumination applications like speckle illumination +microscopy [345], super-resolution imaging [129], and high-order ghost imaging [346]. A proof-of- +principle experimental demonstration [136] has demonstrated that intelligently tailored speckles can +significantly outperform commonly used Rayleigh speckles in nonlinear pattern illumination +microscopy. In the future, new techniques for customizing speckle patterns can potentially provide +drastic improvements to the myriad of speckle-based applications currently in existence, and +potentially lead to the development of new applications. + +Acknowledgements +The authors thank their co-workers Yaron Bromberg, Hasan Yılmaz, and collaborators Joerg +Bewersdorf and Mengyuan Sun for their contributions to the works presented here. They also +acknowledge financial support from the Office of Naval Research (N00014-20-1-2197) and the +National Science Foundation (DMR-1905465). + + +Journal of Optics (2022) #### +8. Ultrafast structured beams and intense magnetic fields +P. B. Corkum1 and Carlos Hernández-García2 +1University of Ottawa and the National Research Council of Canada +2Universidad de Salamanca + +Status +With spatial light modulators, Q- or S-plates, it is possible to impose any retardation on any spatial +element of an optical beam in the image plane of the retardation plate, enabling the generation of +vector or vortex beams. This ability is limited in space by pixel size and the resolution of the image +system, in intensity by damage to the phase plates, and in time by the spectral bandwidth of the spatial +element that forms the waveplate. The latter restricts the generation of ultrafast structured beams in +the femtosecond or even attosecond regimes, where high frequencies and broad spectral bandwidths +are required. At the femtosecond timescales, this restriction can be relaxed through femtosecond +pulse compression (with fiber compression or in thin dielectric windows), reaching the few-cycle limit +of light [137,138]. + +Current and Future Challenges +Femtosecond cylindrical vector beams have been demonstrated recently to allow for coherent control +over currents, transferring the light’s topology to a material in an ultrafast fashion [139,140]. Electrons +created by structured light beams can re-radiate, thereby transferring the original properties of light +to a new frequency—higher frequency for generating high harmonic radiation, lower frequency for +THz magnetic fields. Information about ultrafast dynamics in the medium is encoded in the generated +radiation. + +High-harmonic generation is one of the most robust mechanisms to transfer structured light to +the ultrafast, high-frequency regime. In gas-phase high-harmonic generation, the phase-matched +emission of all dipole emitters, together with angular momentum conservation rules, allow for the +vectorial [141,144] and orbital angular momentum [141,142,145,146] properties from the infrared to +be transferred to the extreme-ultraviolet (XUV). Thus, highly nonlinear up-conversion can produce +high-harmonic radiation with almost any orbital angular momentum and polarization, and more +generally, with any vectorial structure. Interestingly, high-harmonic generation allows for the +topological properties of harmonic light to be controlled at the attosecond time scale. If several +harmonics with different orbital angular momenta are composed, ultrafast light is arranged like a coil +spring—a “light spring” [141,142]. In addition, high-harmonic generation allows for the orbital angular +momentum of light beams to be varied in the sub-femtosecond scale, thus creating light beams with + +Figure 1. (a) Attosecond light spring [129], [130] and (b) XUV beam with self-torque or time-dependent OAM [131]. + +a +Attosecond light spring +b +Lightbeamwithself-torque +x(um) +50 +35 +50 +0 +50 +31 +29 +Intensity +(wrl) +17 +0 +27 +(arb. units) +52207 +0.5 +-50 +2 +0 +2=47nm +Intensity +Self-torque: +E=1.32fs +(a.u.) +10 +15 +20 +25 +30 +Time (fs)Journal of Optics (2022) #### +time-dependent orbital angular momentum or self-torque [143]. Such ultrafast structured beams, +“light springs”, or self-torque beams are not found in other spectral regimes. +In gases, where the electron moves primarily in the vacuum, the fundamental field is almost solely +responsible for the electron’s motion and for the high harmonics that the gas emits. Phase matching +forces the conservation of orbital angular momentum [141,142,145,146]. In solids, the electron’s +interaction with the solid cannot be ignored, but still, angular momentum is conserved. Turning this +around, intense infrared pulses can serve as a flexible probe of solids wherein the electron’s motion +is partially controlled by the infrared light and partially by the electron’s interaction with the material, +but the high harmonics that emerge still must satisfy phase matching. For example, as high harmonics +are developed in metals, metals will produce vector beams and be probed by vector beams [147]. +A different way to modify a light pulse is to use a structured material. For example, when a metal +containing a hole interacts with azimuthally polarized light [148,149], a large ring current can oscillate +at the laser frequency. This current generates ultrafast, intense longitudinal magnetic fields isolated +from the electric field (see Fig. 2(a)). +An alternative method is to drive ring currents using quantum control. A quantum-controlled +current is not limited by the angular momentum of the incident beam because electrons and holes +are created together, and each gain equal but opposite momentum and angular momentum. In +semiconductors, quantum control allows for any current structure to be engineered on any pixel [150]. +In the 1990s, there was a lot of work on coherent control of semiconductor currents, using linear +or circular polarization. For this work, a detector was developed in cold-grown GaAs. That detector +has been adapted to measure current driven by azimuthally polarized fundamental and second +harmonic light. Fig 2(b) shows the ring currents that are needed to launch a THz “flying torus” [151, +152]. Neither linear nor nonlinear spectroscopy with flying tori have been studied. +Although quantum control is feasible whenever there are interfering pathways to a final state (in +this case, the direction of the current), the work in GaAs is in the perturbative limit. In contrast, +breakdown in gases or dielectrics will allow high intensity control and high magnetic fields. In gases, +the combination of fundamental and second harmonic light transforms a gas into a plasma that has +initial conditions imposed by the transformation process. + +Figure 2. (a) Enhanced magnetic field (blue) at a metal sample irradiated by an infrared azimuthally polarized beam [136], [137]. (b) Ring +current measured with a 25x25 μm2 detector (left) and the magnetic field (right) calculated [127], [128] using the Biot-Savart law. + +a +SchemetoinduceintenseultrafastBfields +E +Ix +5 +B +B.(t) +2.5 +Current +oops +Aperture +B +-2.5 +2/2 +..-Incident +Atsample +-5 +Sample +0 +10 +20 +Time [fs] +Bz +FWHM=248μmJournal of Optics (2022) #### +If we were to ask for the highest magnetic fields that humans can controllably generate, these +fields are created within intensely irradiated plasmas. However, magnetic fields within plasmas are +difficult to use since they are internal to the plasma. High-intensity quantum control will allow us to +generate large, controllable, magnetic fields that are as isolated as possible from the plasma, making +these fields useful. + +Concluding Remarks +Producing and applying XUV/soft X-ray structured beams and ultrafast intense magnetic field pulses +will become important forefronts of research with structured light. The study of magnetic helicoidal +dichroism or magnetization switching with structured laser pulses [153, 154] are examples of +applications of ultrafast structured beams. While it is early days for soft X-ray structured beams, there +are two frontiers. First, high-harmonic generation is a robust source of structured beams in the XUV +that can be used to observe electrons and, through them, the response of matter to extreme radiation. +This matter can be any type of new material—chiral, magnetic, or topological materials. XUV or soft +X-ray radiation can also be integrated with attosecond science, allowing for unusual pulses such as +“light springs” or light beams with self-torque. +Further advances in the technology of midinfrared structured driving beams may enable the +generation of attosecond structured beams deeper in the soft X-rays. In addition, the use of solid +targets may hinder new scenarios for structured high-harmonic generation. Secondly, the +development of the quantum optics of soft X-ray beams is required for the application of such +structured beams in fields such as long-distance space communications which can benefit from their +very low beam divergence and high photon energy. + + + +Journal of Optics (2022) #### +9. Metaphotonics with Structured Light +Haoran Ren1 and Yuri Kivshar2 +1Monash University +2Australian National University + +Status +The wave-particle nature of light leads to multiple degrees of freedom such as wavelength, amplitude, +phase, polarisation, and angular momentum, which can be controlled in spatial, temporal, and spatial- +temporal domains. Structured light patterns were first observed in the double-slit experiment of +Thomas Young, where the amplitude and phase interference created bright and dark fringes. Today +we understand that a light beam can be structured into millions of transverse modes (e.g., Hermite– +Gaussian, Laguerre–Gaussian, etc.) in a square millimetre [96], an extraordinary resource for boosting +information capacity. In structured light, singular photonics exhibits topological properties possessing +dark singularity centres in a phase vortex with the orbital angular momentum (OAM) of 𝑙ℏ per photon +(𝑙 can take any integer value in [-∞,∞], and ℏ is the reduced Planck constant) (Fig. 1(a)); a polarisation +vortex manifested by a tensor product of the polarisation and OAM degrees of freedom (defined as +|𝜓⟩ = cos + +, +-, |𝑙⟩|𝑅⟩ + e./sin + +, +-, |−𝑙⟩|𝐿⟩, where 𝜃 and α denote the weighted contribution of and +relative phase between left- (𝐿) and right-handed (𝑅) circular polarisations) (Fig. 1(b)); and a plasmonic +vortex carrying the total angular momentum resulting from spin-orbit coupling (Fig. 1(c)). +Metaphotonics has recently transformed the photonic design for the control of multi-dimensional +photonic vortices. To implement phase vortices in real space, different dielectric and plasmonic +metasurfaces were designed. A high-index dielectric nanopillar with strong mode confinement was +developed as a truncated waveguide with an effective mode index and phase response (Fig. 1(d)). The +high-index nanopillar can be designed as a subwavelength waveplate with strong birefringence, +exhibiting different phase accumulations for the polarisation along the long and short axe (Fig. 1(e)). +Geometric metasurfaces based on the Pancharatnam–Berry phase have been used to create a phase +vortex through the in-plane rotation of asymmetric nanopillars [155]. Meanwhile, each anisotropic +nanopillar can function as a subwavelength waveplate for implementing polarisation vortices in real +space [156]. +Huygens’ metasurfaces offer an alternative platform to realize phase vortices through spectrally +overlapping electric and magnetic resonances (Fig. 1(f)) [157]. Ultrathin plasmonic metasurfaces +based on the near-field mode hybridization have been used to create phase vortices (Fig. 1(g)) [158]. +Additionally, metal–insulator–metal meta-atoms that support the gap plasmon resonance in a +magnetic field could enable highly efficient generation of phase vortices in reflection (Fig. 1(h)). +Photonic crystal slabs possess an inherent polarisation vortex in momentum space around bound +states in the continuum (BIC) of the periodic structures, featuring the in-plane winding of a vector field +and thereby a polarisation vortex (Fig. 1(i)) [159]. In the nonparaxial limit, the space and polarisation +degrees of freedom are non-separable, giving rise to the total angular momentum, a measurable +quantity through spin-orbit coupling. Surface plasmon polaritons (SPPs)—a tightly confined surface +wave beyond the diffraction limit—open the possibility of producing subwavelength plasmonic +vortices in the near-field region. For instance, Fig. 1(j) presents a plasmonic nanoring aperture used +for the multiplexing generation and detection of different plasmonic vortices [160]. Fig. 1(k) shows +the use of plasmonic grooves for the excitation and ultrafast imaging of optical skyrmions in SPPs +[161]. + +Journal of Optics (2022) #### + + + +Current and Future Challenges +We have provided a review summary on singular metaphotonics and highlighted some nanophotonic +structures designed based on different physics for the control of phase, polarisation, and plasmonic +vortices. For the phase vortex generation in real space, high efficiency metasurfaces can be designed +in both reflection and transmission. For reflection metasurfaces, plasmonic materials featuring a +metal-insulator-metal configuration could offer superior efficiency. For transmission metasurfaces, +low-loss and high-index dielectric metasurfaces featuring an ultrathin thickness (<λ, where λ is the +wavelength of incident light) or a relatively large thickness (~λ) can be designed for Huygens’ and +waveguide metasurfaces, respectively. However, to achieve the most accurate phase digitalization +when considering the fabrication error, geometric metasurfaces exploiting the rotation angle- +controlled phase response are perhaps more desired. Meanwhile, to achieve high efficiency +polarisation vortex generation in real space, anisotropic meta-atoms based on the metal-insulator- +metal configuration and all-dielectrics can be designed for the subwavelength polarisation control in +reflection and transmission, respectively. Even though metasurface generation of polarisation vortices +faces the same challenges as for the phase vortices, it is generally more difficult to use a metasurface +device to distinguish and sort structured polarisation singularities. +In addition to the fabrication challenges, metaphotonics devices are most usually passive and +unlikely to be able to dynamically switch vortex modes. Besides, structured optical fields are typically +limited to a 2D transverse plane without the wavefront control in the propagation direction. The ability +to tailor light beyond 2D structured light, towards 3D control (in all spatial coordinates and field +components), and even 4D control with spatiotemporal control of structured light, is of fundamental +and practical interest for future research. On the other hand, even though BIC-induced polarisation +vortices in momentum space feature robustness, alleviation of coaxial beam alignment, and +Figure 1. Illustration of singular metaphotonics based on the manipulation of phase vortex (a), polarisation vortex (b), and plasmonic +vortex (c). (d-h) Design principles of different meta-atoms used for the generation of phase and polarisation vortices, including (d) a high- +index dielectric waveguide, (e) a nanopillar waveguide, (f) an ultrathin dielectric cylinder in a Huygens’ metasurface, (g) plasmonic +nanoantennas with hybridized plasmon modes, (h) a metal-insulator-metal structure supporting gap plasmon resonance. (i) A photonic +crystal slab designed for creating polarisation vortex lasing modes in momentum space around the BIC resonance. (j) A plasmonic nanoring +aperture used for on-chip OAM multiplexing through the total angular momentum mode-sorting nanoring slits. (k) Plasmonic grooves +used for ultrafast imaging of optical skyrmions carried by propagating SPPs. (i-k) Reprinted by permission from American Association for +the Advancement of Science in References [159–161]. + +(d) +(f)Huygensmetasurface +(g) +Waveguidemetasurface +Plasmonicnanoantennas +ED- +MD +(e) +(h)Gap plasmon metasurface +Nanopillarmetasurface +neffi +a) +IHI +neff2 +Phasevortex +(b) +(c) +Polarizationvortex +Plasmonicvortex +() +Nanoring aperture +(i) +Photoniccrystalslab +BIC +(k)Plasmonicgrooves +m +Kx- ++ky +SingularmetaphotonicsJournal of Optics (2022) #### +unrestricted choice of materials, symmetry-protected photonic crystal slabs designed near the BIC at +the Γ point are extremely sensitive to the change of refractive index, incident wavelength, and incident +angle that may easily break the system symmetry. To create plasmonic vortices, nanogrooves +engraved in a metal film were generally employed, but they usually have a low coupling efficiency of +SPPs. Traditional noble metals such as gold and silver also suffer from high metal losses due to +interband transitions in the ultraviolet and visible frequency ranges. More critically, strong dissipation +of the highly localized plasmonic vortex fields in the near-field region hinders the SPPs applications for +on-chip vortex transmission and processing. + +Advances in Science and Technology to Meet Challenges +Nowadays, nanofabrication techniques are available to create functional metaphotonics with the +resolution down to a few nanometres. Systems such as EBL, focused-ion beam (FIB), and 3LN can +enable ultrahigh resolution (EBL, FIB) or 3D manufacturing, but are limited by a low throughput due +to a sequential writing process. Masked techniques, such as photolithography, soft lithography, NIL, +and colloidal lithography enable high throughput by replicating the entire mask simultaneously but +are often limited in resolution or flexibility. The NIL technique is scalable, and large-area roll-to-roll or +roll-to-plate techniques have already been developed to enable high-throughput metasurface +production towards industrial applications. Alternatively, self-assembly techniques have emerged and +could circumvent the need for clean-room facilities and expensive equipment, though this technique +lacks the flexibility to create different-shaped nanostructures. +Recently, digital vectorial holography has enabled the generation of advanced vortex beams, in +which the phase and polarisation singularity centres can spatially vary either in 3D polarisations [163] +or along the propagation direction [164]. Moreover, a 3D wave packet that carries a spatiotemporal +optical vortex with a controllable purely transverse OAM has been realized [23]. Multi-dimensional +structured light could offer extra degrees of freedom for versatile light-matter interactions, quantum +entanglement, optical trappings, harmonic generation, and optical sensing, holding great potential for +novel applications that may not be possible otherwise. Even though strong dissipation of the highly +localized plasmonic vortex fields hinders on-chip plasmonic vortex transmission and processing, +superior transmission efficiency can be offered by low-loss semiconductor nanowires sustaining highly +confined optical modes. Recently, an OAM-controlled hybrid nanowire plasmonic circuit was +introduced, demonstrating OAM-controlled optical logic operations including AND and OR gates [165]. +OAM beams with different topological charges exhibit selective excitation of single-crystalline +cadmium sulfide nanowires through coupling OAM-distinct plasmonic fields into nanowire +waveguides for long-distance transportation on-a-chip. + +Conclusion +We believe that metaphotonics provides a great playground for structured light manipulation, and it +will lead to a diverse range of ultracompact, ultrahigh-capacity, and ultrahigh-speed devices +harnessing multi-dimensional structured light. We believe it is of paramount importance to integrate +developed metaphotonics devices with established optical systems for advanced optical imaging, +holographic displays, optical and quantum communications, nonlinear and ultrafast light shaping, and +turbulence- and scattering-resilient communications and imaging. + +Acknowledgements +H.R. acknowledges a support from the Australian Research Council DECRA Fellowship DE220101085. +Y. K. acknowledges a support from the Australian Research Council (grant DP210101292). + + +Journal of Optics (2022) #### +10. Structuring Light with Near-Zero-Index Platforms +Mário G. Silveirinha1 and Nader Engheta2 +1University of Lisbon and Instituto de Telecomunicações +2University of Pennsylvania + +Status +Materials provide the means to structure light. Judiciously engineered material platforms, known as +metamaterials and metasurfaces, have provided scientists and engineers with versatile tools to +control, manipulate, and sculpt electromagnetic waves and fields. In particular, materials whose real +part of relative permittivity and/or permeability attain near-zero values at given operating frequencies +offer specially interesting platforms for structuring electromagnetic and optical waves [166-168]. At +such frequencies, these materials exhibit (a) refractive index near zero, and consequently (b) the wave +phase velocity attains very high values (theoretically infinite values) which leads to (c) a “stretched” +wavelength and (d) uniform phase distributions. When both relative permittivity and permeability +are zero, the electric and magnetic phenomena are effectively decoupled in such materials [166] +yielding static-like spatial distributions of electric and magnetic fields, while at the same time they are +dynamically time varying. This special feature makes epsilon-near-zero (ENZ), mu-near-zero (MNZ), +and epsilon-and-mu-near-zero (EMNZ), which form the general class of near-zero-index (NZI) +materials, particularly interesting for wave manipulations, beam shaping and lensing [167-171], for +example the wavefronts emerging from an ENZ material block typically inherit the shape of the ENZ- +material surface [170]. +Since the wavelength of waves in such media can be long even for high frequencies, one can +effectively think of this effect as “loosening” the connection between the frequency and the +wavelength. Moreover, a block of material in which the wavelength is very long can be viewed +electromagnetically as a “point”, even though it can be large compared to the free-space wavelength. +This phenomenon has enabled numerous exciting features in wave interaction with such NZI media. +The “supercoupling” effect [167-169] is an interesting example of such features; when two metallic +waveguides are joined together, the connecting segment (i.e., transition region) between the two +waveguides can be of any shape and size, if that region is filled with an NZI medium [167,168]. If an +ENZ (or MNZ) medium fills the transition region, then the connecting segment needs to be narrow (or +wide), while it can be bent and can have arbitrary shapes [167-169] (see Fig. 1(a) and 1(b)). This +“electromagnetically point-like” region may provide an unusual coupling between two emitters, +effectively causing “near-field coupling” even when the emitters are far apart in space [172] (see Fig. +2(a) and 2(b)). This is an interesting way to structure light in dipole-dipole coupling among quantum +emitters. +Another intriguing by-product of such NZI-enabled stretched wavelength can be considered in +“sampling and squeezing” waves through narrow channels [167,168], in which one can effectively +transfer an “image” through a subwavelength opening (see Fig. 1(c) and 1(d)). The electromagnetic +ENZ phenomena have also inspired efforts on other physical, non-electromagnetic, platforms in which +other parameters can attain near-zero values. For example, we theoretically studied how one can +conceive electronic metamaterials in which effective mass of electrons can be engineered to be near +zero, exploring the topic of “transformation electronics” [173]. + + +Journal of Optics (2022) #### + + + + +Current and Future Challenges +Passive NZI materials must inherently be dispersive, and therefore have a finite bandwidth of interest. +However, depending on the feature of interest and the frequency domain of interest, the bandwidth +may be sufficient to achieve a desired functionality. For example, consider the phenomenon of +“photonic doping” [169], in which a dielectric rod is inserted in an ENZ medium with an arbitrary cross- +sectional geometry. As viewed by an outside observer, according to the effective medium theory, for +two-dimensional scenarios, this “single-inclusion metamaterial” can be treated as an effective +medium such that the effective permittivity is still zero (even though the dielectric rod is inserted in +the ENZ host) but the effective permeability can be different from unity (notwithstanding both the +dielectric rod and the ENZ host are non-magnetic with unity relative permeability). Such engineered +effective permeability exhibits a resonant dispersion as a function of frequency, but this resonance is +mainly due to the size and the permittivity of the dielectric rod, which exhibits narrower bandwidth +than the bandwidth over which the ENZ host behaves as a material with near-zero permittivity. So in +this example, the bandwidth of the ENZ host can be sufficient for the resonant behaviour of the +relative permeability. +The idea of photonic doping combined with the fact that in the ENZ media the wavelength is “long” +has led to the idea of ENZ-based cavity resonators in which the resonance frequency does not depend +on the external shape and geometry of the cavity, but instead it is pinned to the ENZ frequency of the +materials [174]. Such geometry-independent cavity resonators have several salient features: (a) they +can be the basis for the notion of “flexible photonics”, a paradigm in which changing, bending and +morphing the external shape of cavities would not affect its resonance frequency; and (b) while +changing the external shape and size does not change the resonance frequency, it does affect the +quality factor (Q) of the cavity, if a small amount of loss is considered [174]. So here is another unusual +Figure 1. Illustration of the ENZ supercoupling effect. (a) Two parallel-plate waveguides are connected through a narrow ENZ channel. As +shown in b) for a 180º bend with a narrow channel, the reflection level at the ENZ frequency (w=wp) can approach zero in the limit of +vanishing material loss and is typically much weaker than for an empty channel. (c) The supercoupling effect may enable the transmission +of a complex image through a narrow aperture in a metallic screen embedded in an ENZ material. The image is sampled by an array of +metallic wires, which are then “squeezed” through the narrow hole to the other side of the opaque screen. (d) Image transported by the +array of metallic wires at the ENZ frequency. Adapted from Refs. [167,168] with permission, copyright (c) American Physical Society. + +(a) +(b) +6,=15 +6,=~1.6 +8,=0 +1.0 +6~0.47 +Region +F/o,=0 +Amplitudeofp +0.8r/e=0.05 +PECwalls +9:0 +x=0 +0.4 +a: +Region 2 +0.2 +Empty +x-0 +0.0 +channel +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Normalizedfrequency,/o, +(c) +(p) +2-10.31Am +-1.86 -0.73 -0.35 -0.140 0.07 0.22 0.51 1.86Journal of Optics (2022) #### +situation for structured light in which the cavity’s resonance frequency and Q are effectively +decoupled from each other, whereas in conventional cavities they are intimately connected. +Furthermore, remarkably, in the limit of vanishing material loss, core-shell ENZ resonators can +support embedded eigenstates in the continuum, i.e., non-radiative bound states which despite being +coupled to the radiation continuum do not decay in time [175,176]. Geometry-independent ENZ +nanoresonators can be exploited in the field of quantum optics [108] where the coupling between an +excited atom with a resonant cavity is usually considered. In conventional situations of atom-cavity +coupling the resonance frequency of the cavity should match the transition frequency of the excited +atoms. This is a delicate balance because a slight change in the shape of the cavity can shift its +resonance frequency (since such cavities are usually high-Q cavities) and therefore the cavity would +be detuned. Moreover, the vacuum Rabi oscillation depends on the cavity Q, which can also be +affected by the slight change in the cavity shape. The ENZ-based cavities can provide an interesting +solution in this scenario in that one can change the Q of the cavity (and thus engineer the vacuum +Rabi oscillation) while the cavity resonance frequency stays tuned [108]. (See Fig. 2c) + + + +Advances in Science and Technology to Meet Challenges +Material loss in some ENZ platforms can be a limiting factor. Since one of the important features of +NZI media is the wavelength stretching, it is important to note how material loss can affect this +phenomenon. As the refractive index is given by +, it follows that at the ENZ frequency +r +i +e +e += ++ +n +i +Figure 2. (a), (b) An ideal epsilon-and-mu-near-zero (EMNZ) 2D material block with arbitrary cross-sectional shape does not influence the +field distribution in the unfilled 2D parallel-plate waveguide sections, and thus behaves electromagnetically as a “single point”. In +particular, the interaction between two quantum emitters placed in the unfilled waveguide sections is the same as for a straight waveguide +regardless of the relative orientation of the unfilled waveguide sections connected to the EMNZ block. (c) Illustration of the decay of an +excited quantum emitter enclosed in an ENZ cavity. The cavity resonance is independent of the shell thickness, and thereby the transition +frequency of the emitter is always matched to the cavity. However, the coupling strength is sensitive to the shell radius, and hence the +Rabi frequency also is. i) Time evolution of the probability of the excited state for different radii of an ideally lossless ENZ shell. ii) +Normalized spectral density for a slightly lossy ENZ shell and different shell radii. iii) Time evolution of the excited state in the presence of +the lossy ENZ shells. Adapted from Refs. [172,108] with permission. Panels (a) and (b), Copyright is Open access from OPTICA, panels (ci), +(cii) and (ciii) Copyright by National Academy of Sciences of the United States. + +PEC +S (a) +EMNZ +2 +(b) +(ci) +P(t) +(cii) +rg=5入0 +0.8 +5.0 +0.0 +10 +g(w) +0.752 +r2=1入0 +0.2 +0.999 +26660 +1.0005 +1.001 +(cii) +r2=0.75 +P(t) +Rt/(2#) +2mt/(2m)Journal of Optics (2022) #### +, resulting in +. Thus the wavelength stretching is approximately limited +by +. Therefore, the lower the +, the longer the wavelength stretched in ENZ media, +and the more uniform the phase distribution in the structure. As a result, for any NZI-based application +of interest that depends on the phase uniformity, one needs to ask the following questions: How large +is the structure and how much phase variation can be tolerated with the effect still observable? The +answer to this question determines how large + can be. For example, some of the transparent +conducting oxides (TCO), such as indium tin oxide (ITO), can exhibit ENZ behavior with + below unity +in the near-IR regime, while silicon carbide (SiC) behaves as ENZ with smaller + in mid IR wavelength +[193]. So each of them can be suitable for a different set of applications, as they have different levels +of material loss. It is important to note that one can also engineer metastructures that mimic some +of the NZI properties, while the loss can reduced. For example, metallic rectangular waveguides +operating at the TE10 cut-off frequency possess wave properties resembling some of the ENZ features, +so they can be suitable for microwave frequencies. Furthermore, photonic crystals with the Dirac +dispersion with accidental degeneracy exhibit effective refractive index near zero, thus providing a +platform for NZI properties at optical frequencies [219]. + +Concluding Remarks +Near-zero-index (NZI) photonics is an exciting field of optics and electromagnetics. It encompasses +unconventional ways of structuring light due to wavelength stretching, with the material bulk +behaving as an electromagnetic point, exhibiting unique features in light-matter interaction, and +offering exciting potential applications. Numerous phenomena resulting from the ENZ, MNZ, and NZI +features, such as flexible photonics, supercoupling, photonic doping, directive thermal radiation, +engineering vacuum fluctuations, ENZ-based quantum optics, super-radiance, emitter-emitter long- +range coupling, ENZ electric-based levitation, giant nonlinearity, embedded eigenstates, and more +have been explored. Moreover, electromagnetic NZI concepts have also inspired wave phenomena in +other physical domains such as acoustics, electronics, amongst others. We are hopeful that this field +continues to grow, expand, and reveal other exciting wave phenomena. + +Acknowledgements +M. G. S. acknowledges partial support from Simons Foundation/Collaboration on Extreme Wave +Phenomena Based on Symmetries, from the Institution of Engineering and Technology (IET) under the +A F Harvey Research Prize 2018, and from Instituto de Telecomunicações under project +UIDB/50008/2020. N.E. acknowledges partial support from Simons Foundation/Collaboration on +Extreme Wave Phenomena Based on Symmetries, and from the US Air Force Office of Scientific +Research (AFOSR) Multidisciplinary University Research Initiative (MURI) grant number FA9550-21-1- +0312. + + +r +0 +e = +( +) +i +i 1 +2 +e +e += += ++ +n +i +i +ENZ +0 +i / 2 +l +l +e += +ie +ie +ie +ie + +Journal of Optics (2022) #### +11. Strong Coupling Between Atoms and Guided Light +Arno Rauschenbeutel, Philipp Schneeweiss, and Jürgen Volz +Humboldt-Universität zu Berlin + +Status +The past decade has seen remarkable advances in the field of quantum nonlinear optics, where a +strong interaction between individual photons is mediated by quantum emitters [177]. Such strong +photon-photon interactions are of both fundamental and technological interest: they are the +prerequisite for implementing deterministic quantum logic gate operations for processing optical +quantum information [178]. Moreover, photons that strongly interact via a quantum nonlinear +medium exhibit complex out-of-equilibrium dynamics that, e.g., enable one to tailor and control the +photon statistics of light [179]. +Using free-space light fields, photon-photon interactions have been successfully demonstrated in +a number of experimental settings. The most established method is to couple atoms with photons +that are confined inside a high-finesse optical resonator [180], see Figure 1(b). This allows one to +increase the coupling of such a so-called resonator-enhanced atom to the input and output mode of +the resonator. In this way, the inherently nonlinear response of the atom mediates strong photon- +photon interactions. Alternatively, strong photon-photon interactions have also been demonstrated +using the collectively enhanced coupling between propagating light fields and ensembles of strongly +interacting Rydberg atoms, so-called Rydberg superatoms [171], see Figure 1(c). Also here the aim is +to enhance the coupling of the effective atoms with the input and output light mode to the point +where coupling to other modes becomes negligible. In these scenarios, a key figure of merit is given +by the so-called 𝛽-factor, +β = +0!"#! +0!$! , +(1) +which is the ratio of the emission rate of the initially excited (effective) atom into the target mode, +Γ1231, and the total emission rate into all possible modes, Γ141, see Figure 1(a). +With regards to employing strong photon-photon interactions in future research and technology, +it is, however, essential to couple quantum emitters to guided fields in integrated optical platforms. +This so-called waveguide quantum electrodynamics (QED) setting has been realised with a variety of +emitters and waveguide types [182]. They include semiconductor quantum dots coupled to +nanophotonic waveguides, silicon vacancies coupled to diamond waveguides, organic dye molecules +coupled to waveguides consisting of an organic crystal-filled glass capillary or to sub-wavelength- +diameter silica fibres, so-called optical nanofibres, as well as cold, laser-trapped atoms coupled to +nanofibres or one-dimensional photonic crystal waveguides [183, 184]. Finally, resonator-enhanced +atoms, realised by single atoms trapped in the evanescent field of whispering-gallery-mode (WGM) +microresonators, have been coupled to nanofibres [185], see Figure 2(a). +Waveguide QED systems lend themselves to distributing and processing optical quantum +information, to deterministically preparing non-classical states of light, and to realising an almost ideal +model-system for strongly correlated, open many-body quantum physics. However, in addition to the +nonlinear response at the single-photon level, many of the corresponding experimental protocols +require high 𝛽-factors, which should ideally reach 100%. A 𝛽-factor that falls short of this value will, +at best, lead to a reduced success probability, e.g., in the case of photon-photon quantum gates [178]. +In the worst case, a too small 𝛽-value impedes the implementation of the protocol altogether, e.g., in +the case of a photon number-dependent delay line based on quantum nonlinearities [186]. + +Journal of Optics (2022) #### + + + +Current and Future Challenges +It is important to note that, for the above-mentioned applications, Γ1231 in Eq. (1) refers to the +emission into a single spatiotemporal target mode, such that all emitted photons exhibit the same +lifetime-limited spectrum. Moreover, for many quantum applications, the photons have to be +indistinguishable when they are emitted at different times or by different waveguide-coupled +emitters. In this context, cold atoms stand out in terms of their superior coherence properties and +their negligible spread of resonance frequencies, or inhomogeneous broadening. As a consequence, +large ensembles of waveguide-coupled atoms can interact collectively with the guided light. This +makes them a prime candidate for scaling up waveguide QED systems. Nonetheless, perfect coupling, +with 𝛽 ∼ 100%, of a single optical mode to a large number of identical and fully coherent quantum +emitters remains an important challenge for existing implementations. For example, when coupling +laser-cooled atoms to the evanescent field surrounding optical nanofibres, 𝛽-factors in the few- +percent regime are expected and have been realised experimentally. +For a single atom that is coupled to the evanescent field of a nanophotonic waveguide, maximizing +the 𝛽-factor for a given decay rate into the free space modes amounts to maximizing the single photon +Rabi frequency, +Ω5 = 𝑑⃗ ⋅ 𝐸C⃗5(𝑟⃗)/ℏ , +(2) +where Ω5 is chosen to be real and positive, 𝑑⃗ is the dipole moment of the atomic transition, and 𝐸C⃗5(𝑟⃗) +the field per guided photon at the position of the atom, 𝑟⃗. Now, |𝐸C⃗5(𝑟⃗)| increases approximately +exponentially when approaching the waveguide. From this perspective, it is thus advantageous to trap + + + + + + + + + + + + + + + + + + + + +Figure 1 – We allow at most two figures +that are roughly the size of this box. +Figure 1. (a) Quantum emitter (yellow sphere) coupled to a target mode (grey). The 𝛽-factor is defined according to Eq. (1) with 𝛤&'& = +𝛤(')) + 𝛤&*+&. (b) Resonator-enhanced atom with optical input and output modes (arrows). (c) Rydberg superatom composed of a cloud of +laser-cooled atoms in the ground state (yellow) and one optically excited Rydberg atom (red). The presence of the latter prevents other +atoms inside the so-called blockade radius to be excited. + +M +trgtJournal of Optics (2022) #### +the atoms at the smallest possible distance from the waveguide surface. However, for distances +smaller than ∼ 200 nm, the van der Waals force becomes so large that it can no longer be +straightforwardly counteracted by optical dipole forces. Furthermore, we have |𝐸C⃗5(𝑟⃗)| ∝ 1/√𝐴, +where A is the cross-sectional area of the guided mode, which can in principle be decreased by +increasing the refractive index of the waveguide material and concomitantly reducing the transverse +waveguide dimensions. However, for the refractive indices accessible with low-loss dielectric +materials, even for an atom placed inside a (possibly slotted) waveguide, near-unity 𝛽-values are still +out of reach. + +Advances in Science and Technology to Meet Challenges +We now elaborate on three possible strategies that lend themselves to meeting the grand challenge +of reaching near-unity 𝛽-factors for single (effective) atoms. First, 𝛽-factors of about 47%, that have +been experimentally demonstrated for the resonator-enhanced atom, can in principle be further +increased by improving the Purcell factor of the WGM resonator, 𝜂 ∝ 𝑄/𝑉. Here, 𝑄 is the quality +factor of the resonator, and 𝑉 is the effective resonator mode volume. Currently, for fused silica-based +WGM resonators, 𝜂 is limited by scattering-induced losses due to surface roughness and pollution. +Here, we thus expect a major step forward by employing advanced resonator production and post- +processing techniques. +Along the same lines, the single-atom 𝛽 can be increased by reducing the group index, 𝑛32 = +𝑐5/𝑣32, of the guided light using photonic crystal waveguides, see Figure 2(b), where 𝑐5 is the speed +of light in the unstructured waveguide, and 𝑣32 is the group velocity. The 𝛽-factor of the photonic +crystal waveguide is then given by, +𝛽:; = +𝑛32𝛽5 +𝑛32𝛽5 + (1 − 𝛽5) , +(3) +where 𝛽5 is the 𝛽-factor of a single atom coupled to the unstructured waveguide. Here, the major +advancement with respect to previous work will be to realise stable trapping of atoms in a region of +the waveguide where 𝛽5 is large while realising a photonic crystal waveguide with small group velocity +and small propagation losses. While the latter is mostly a technical challenge, atom trapping in high- +𝛽5 regions of a photonic crystal waveguide is still subject to current research. +Finally, the coupling of the input and output mode to an ensemble of atoms can be collectively +enhanced. Indeed, the collective 𝛽-factor scales with the atom number, 𝑁<1, as, +𝛽;4== = +𝑁<1𝛽5 +𝑁<1𝛽5 + (1 − 𝛽5) . +(4) +Thus, using only a few hundred atoms, 𝛽;4== ∼ 1 is reached. However, the response of an ensemble +of independent atoms differs from that of a single quantum emitter when the ensemble interacts with +more than one photon. Specifically, the inherent nonlinearity featured by each of the atoms is +“diluted” because two consecutive photons can interact with different atoms. Thus, in order to profit +from a large collective 𝛽-factor for implementing quantum nonlinearities, one needs to introduce +atom-atom interactions, e.g., in the form of a dipole blockade in Rydberg superatoms, see Figure 2(c). +Here, the major advancement will be to control the detrimental influence of the nearby waveguide +on the coherence properties of the atomic Rydberg levels [187], e.g., by working with particularly thin +nanofibres that feature super-extended evanescent fields [188]. + + +Journal of Optics (2022) #### + + +Concluding Remarks +All three approaches towards reaching near-unity 𝛽-factors laid out above, resonator-enhanced +atoms, large group index, and waveguide-coupled Rydberg superatoms, see Figure 2(a)–(c), come with +considerable technical and conceptual challenges. Meeting these challenges will mark important +advances in science and technology in their own right, ranging from next-generation ultra-high +𝑄 factor WGM microresonators to slow-light waveguides with record-low propagation loss to +passivation and charge control of dielectrics at the level of single elementary charges. Most of all, the +corresponding research and development effort is justified by the exciting applications that are +enabled by atomic waveguide QED in the high-𝛽 range. In particular, it opens the route towards the +implementation of near-ideal fibre-coupled nonlinear quantum devices, which will mark a major +breakthrough in quantum optics and constitute a key resource in quantum sensing, quantum +metrology, quantum communication, as well as quantum simulations. + +Acknowledgements +We acknowledge funding by the Alexander von Humboldt Foundation in the framework of the +Alexander von Humboldt Professorship endowed by the Federal Ministry of Education and Research. +Moreover, financial support from the European Union’s Horizon 2020 research and innovation +program under grant agreement No. 899275 (DAALI) is gratefully acknowledged. + + + + + + + + + + + + + + + + +Figure 1 – We allow at most two figures that are roughly the size +of this box. +Figure 2. (a) Resonator-enhanced atom realised by an atom coupled to the evanescent field of a WGM resonator, interfaced using an +optical nanofibre. The latter is realised as the subwavelength-diameter waist of a tapered optical fibre (TOF). (b) Atom coupled to a +photonic-crystal-based waveguide with large group index. (c) Rydberg superatom coupled to an optical nanofibre. + +WGMJournal of Optics (2022) #### +12. Surface Waves +Daniel Leykam1 and Daria A. Smirnova2 +1National University of Singapore +2Australian National University + +Status +Surface physics is messy; surface waves are no exception. Analytical solutions are scarce, numerical +calculations are resource-intensive, and there is a myriad of possible interface configurations to +consider. Emerging from these challenges are elegant theories and numerous applications. +Surface waves are important because they exhibit remarkable properties unattainable using +isolated bulk wave media. Long-studied examples include gravity waves at liquid surfaces, elastic +waves at the surfaces of solids, subgap Tamm or Shockley electronic states at terminated +semiconductors, and electromagnetic plasmon-polaritons at metal-dielectric interfaces, illustrated in +Figure 1. +Initial interest in surface waves stemmed from their ability to guide and strongly confine energy, +observed most strikingly in the destructive power of seismic Rayleigh waves predicted in 1885. At +smaller scales, electromagnetic surface waves guide and localise light below the diffraction limit [189]. +More recently, the non-trivial spatial structure of surface waves such as their transverse spin enables +chiral coupling between localised sources and guided modes. +A long-standing challenge has been ab-initio prediction of the existence and properties of surface +waves. Even in the simplest case of homogeneous media described by a few material parameters, +novel solutions continue to be discovered, such as Dyakonov surface waves of anisotropic +electromagnetic media [190]. +Since the 2000s, studies of surface waves have been reinvigorated thanks to the development of +topological band theory [191]. Topological band theory enables prediction of novel types of surface +waves of periodic media such as photonic crystals, i.e. systems with wavelength-scale variations in +their material parameters. + + + +Figure 1. Magnetic field distribution (the out-of-plane component Hz) in the TM-polarised surface plasmon-polariton wave, showing different +transverse localization scales in the two media (air and metal) brought into contact. + + +Journal of Optics (2022) #### +Topological band theory shows promise as a systematic approach for designing surface waves and +optimising their properties for applications including precision sensing, compact waveguides, and +signal processing. At the same time, techniques from the structured light community are being +fruitfully applied to study topological bands [192]. +Ongoing research aims to better understand connections between surface waves emerging for +different classes of waves. This will not only allow us to design and optimise surface waves in a variety +of wave systems, but also observe novel physics. For example, Weyl semimetals are topological +materials supporting coexisting surface and bulk modes exhibiting Weyl quasiparticles originally +hypothesised in 1929. In 2015, Weyl semimetals were observed for the first time using an electronic +system (TaAs) and an analogous microwave photonic crystal [122]. + +Current and Future Challenges +While there has been great success in emulating edge modes of 1D and 2D condensed matter systems, +studies of protected surface waves of 3D systems remain in their infancy [194]. Such surface waves +exhibit linear Dirac-like dispersion, with locking between their spin and momentum, illustrated in +Figure 2 and elaborated on further in Section 14. +One challenge is that many models of topological surface waves were originally formulated for +electronic condensed matter systems with fermionic spin-orbit coupling. To implement similar surface +waves for classical wave systems requires other effects such as bi-anisotropy, use of orbital angular +momentum modes as a spin-like degree of freedom, or additional crystalline symmetries. Limits to +the strength of these effects may lead to non-ideal dispersion relations, such as surface waves co- +existing with bulk bands, resulting in bulk scattering losses. + + +Figure 2. Massless Dirac-like dispersion of topological surface waves with spin–momentum locking within the bulk gap in momentum space. +The (pseudo-)spin texture is illustrated by black arrows. + + +Bulk modes +Frequency +Surface waves +Bulk modes +ky +kzJournal of Optics (2022) #### +Not all classes of surface waves may be accessible in a given material platform. For example, +sound-based phononic crystals are based on manipulating scalar waves and can therefore be well- +described by simple tight binding models [195]. By contrast, 3D photonic crystals typically have band +structures complicated by orbital and polarisation degrees of freedom, making space group +arguments insufficient to guarantee the existence of protected surface waves. +The holy grail is the ability to identify the best possible surface wave for a given application subject +to material or design constraints. At first glance, this seems like a hopeless task given the explosion in +degrees of freedom compared to uniform media. Topology allows us to make concrete statements +about some properties of surface waves, such as the difference between the number of forward and +backward propagating waves, independent of details such as precise material parameters or the +interface shape. However, continuous parameters such as the wave speed or degree of localization +are not topologically protected and must be optimised using conventional methods. +Another challenge is to understand the robustness of surface waves against effects including +scattering losses, fabrication imperfections, imperfect symmetries, and incomplete band gaps. +Topological band theory as originally developed for electronic condensed matter materials was limited +to lossless, non-interacting wave systems described by the Schrödinger equation, analogous to the +paraxial wave equation. Generalising beyond these constraints will allow us to identify robust surface +states for new kinds of wave media. + +Advances in Science and Technology to Meet Challenges +Advances in fabrication technologies will expand the platforms available for implementing surface +waves and give us new tools to tune their properties. Areas of active research include heterogeneous +multilayer metamaterials, novel two-dimensional materials such as graphene, hexagonal boron nitride +and twisted monolayers [196], nanostructured metasurfaces, hybrid polariton systems, surface +magnetoplasmons in gyrotropic materials, and even electronic topological materials [197]. +3D printing is maturing as a fast and flexible approach towards prototyping topological surface +waves, with many recent high-profile works in acoustics [195] covered in Section 17. For +electromagnetic waves, 3D printing is practical for microwave photonic crystals, but scaling up to +optical frequencies remains challenging due to the need for 3D nanofabrication. State-of-the-art +methods such as direct laser writing have been used to realise topological edge and surface waves in +the terahertz and near-infrared [122]. Advances in 3D printing including finer resolution and the ability +to incorporate more combinations of materials will open up new possibilities for surface waves in the +visible frequency range. +Methods from the field of structured light may offer an easier route towards generating and +finding useful applications of topological surface waves. Specifically, one can use internal degrees of +freedom such as orbital angular momentum as synthetic dimensions, with hopping along the synthetic +dimension mediated by periodic spatial or temporal modulation [198]. Surface waves that mix spatial +and internal degrees of freedom show promise for applications such as robust, high-efficiency mode +conversion and non-reciprocity in planar integrated photonic circuits. Realising this goal will, however, +require the integration of high-efficiency optical modulators. +Generalisations of topological band theory to broader classes of wave media are being actively +pursued. These include active or lossy (non-Hermitian) systems (see also Section 6), effective medium +theories describing metamaterials [199], non-periodic, dispersive, and nonlinear media [200]. +Advances in these directions will allow us to identify novel combinations of materials supporting +surface waves and better understand their robustness to losses and other perturbations. For example, + +Journal of Optics (2022) #### +for high power device applications, it is essential to understand the conditions under which nonlinear +surface waves remain stable. +There is growing interest in applying machine learning techniques to physics problems [201]. +Potential applications to surface waves include discovery of interface designs with superior surface +wave properties via generative modelling, identification of new classes of topological wave media +supporting robust surface modes, and mesh-free neural network-based beam propagation methods +for numerical simulation of complex interface geometries [202]. Applications of machine learning to +the broader field of structured waves are discussed further in Section 25.. + +Concluding Remarks +Topological band theory has led to a resurgence of interest in surface waves in quantum and classical +systems in a similar vein to how analogies with bulk electronic band structures gave rise to the fields +of photonic and phononic crystals. While researchers are still attempting to understand all the +subtleties of topological phases, there is no doubt that these discoveries will require band structure +textbooks to be rewritten. +So far, the flow of ideas has largely been unidirectional, from electronics to photonics and +acoustics. Given the greater appreciation of the universality of bulk and surface waves occurring in +various fields, there is great potential for recent advances in structured light to be applied to electronic +and acoustic systems [44]. +Engineered interfaces and surface waves will serve as a flexible testbed for probing relativistic +physics (fundamental science, quasiparticles) and strong light-matter interactions (due to the field +confinement). Advances in their basic science will lead to technological breakthroughs in more applied +areas, such as ultra-thin, resilient, and flexible surface wave-based devices. We have only scratched +the surface of potential applications. + +Acknowledgements +D. A. S. acknowledges support from the Australian Research Council (DE190100430). + + + + +Journal of Optics (2022) #### +13. Photonic spin-orbit interactions at metasurfaces: stochastic, Rashba and +quantum effects +Kexiu Rong1, Bo Wang2,1 and Erez Hasman1 +1Technion – Israel Institute of Technology +2Shanghai Jiao Tong University + +Status +Light possesses both spin and orbital angular momentum (OAM); the former is associated with circular +polarisation states, and the latter arises from azimuthal phase gradients of the light field. A coupling +between spin and OAM (or linear momentum) occurs when light interacts with anisotropic or +inhomogeneous structures, giving rise to optical phenomena in which the spin of light affects and +controls the spatial degrees of freedom of light, such as the vectorial field distribution and propagation +path [203,204]. These spin-orbit interactions (SOIs) bring forth novel spin-optical effects (e.g., +photonic spin Hall effect (PSHE) and photonic Aharonov-Bohm effect) and enable efficient spin- +dependent light manipulations [205-208]. +Metamaterials are artificial structures assembled from multiple elements smaller in scale than the +wavelength of external stimuli, endowing a medium with unique electromagnetic responses and +functionalities. Metasurfaces [209-215], metamaterials of reduced dimensionality, are phased arrays +composed of resonant optical nanoantennas, which facilitate substantial control of local light +scattering properties. Controlling the electromagnetic response of metasurfaces can be achieved by a +geometric phase (e.g., Pancharatnam-Berry phase) mechanism [88,209], enabling an excellent +platform to investigate new types of SOI effects from the classical to quantum regime. In this roadmap, +we would like to introduce novel types of SOIs utilising geometric phase metasurfaces (GPMs): (i) +Stochastic PSHE [211], (ii) Photonic Rashba effect [212], (iii) Quantum entanglement between the spin +and the OAM of photons [213]. +(i) The study of SOIs in disordered systems offers a wealth of interesting effects and numerous +potential applications, such as suppressing undesired optical scatterings and achieving ultra-sensitive +optical metrologies. An optical metrology that can detect extremely weak disorders in a deep- +subwavelength resolution is critical for nanotechnology. Recently, we reported on a stochastic PSHE +arising from space-variant Berry-Zak phases, which are generated by disordered magneto-optical +effects. This effect is observed from a spatially bounded lattice of ferromagnetic meta-atoms +displaying nanoscale disorders. Our approach may be used for sensing deep-subwavelength disorders +by actively breaking the photonic spin symmetry and may enable investigations of fluctuation effects +in magnetic nano-systems. +(ii) Heterostructures combining a thin layer of quantum emitters and planar nanostructures +enable custom-tailored photoluminescence in an integrated fashion. Recently, there has been a surge +of interest in selectively manipulating quantum emitters—that is, valley excitons—in transition metal +dichalcogenide (TMD) monolayers due to their potential as an alternative information carrier in +valleytronics. GPMs constructed of anisotropic nanoantennas with space-variant orientations allow +the manipulation of light by the spin degree of freedom [209,210]. This is enabled by the polarisation +evolution of light on the Poincaré sphere, thus generating spin-dependent Pancharatnam-Berry +phases for the spin-flipped components. Hence, the GPMs represent an attractive candidate to +perform the desired valley separation required by valleytronics, inspired by spin-dependent +phenomena such as the PSHE and photonic Rashba effect underpinned by SOIs [216-218]. + +Journal of Optics (2022) #### +(iii) Quantum information provides a route to solve problems in reduced time and complexity by +exploiting fundamental quantum principles such as superposition and entanglement. Moreover, due +to a relatively easy manipulation and long quantum coherence time, single photons encoded with +quantum states are an appealing candidate to implement quantum algorithms. Hence, generating and +manipulating entangled photon states using SOIs mediated by metamaterials is at the heart of the +field of photonic quantum information. + + +Current and Future Challenges +A challenge of dealing with weakly disordered nanostructures is the limited opportunities to extract +the information from subtle light-matter interactions. The disorder and stochastic nature hinder light +from detecting any useful information other than that from a homogeneous medium. The previous +strategies to overcome this limitation involve the implementation of special optical conditions +including critical angle, symmetry broken, and resonant enhancement. Specifically, the emerging +photonic spin-dependent effects due to symmetry broken in arrays of anisotropic nanoantennas +provide a SOI mechanism to achieve a sensitive optical metrology [214]. However, the spin-dependent +effects in these architectures naturally disappear when the anisotropic nanoantennas are replaced by +isotropic ones. More importantly, how to accurately quantify the structure fluctuations as a function +of the measurable spin-split effects remains largely unexplored. +In the pursuit of new spin-optical devices possessing large information capacity and high +processing speed, a long-thought goal is to interface spinoptics and spintronics for an interchange of +spin information between photons and electrons. This requires the miniaturisation of spin-polarised +sources down to a nanometric scale and beyond. Currently, a family of atomic-thin materials has + + + + + + + + + + + + + + + + + + +Figure 1 – We allow at most two figures that +are roughly the size of this box. +Figure 1. Optical metrology with stochastic PSHE. (a) Sketched PSHE from a magnetized disorder metasurface. A polarised incident beam +(polarisations indicated by the cyan arrows) is reflected and split into spin-up (σ+) and spin-down (σ–) components with a subdiffraction- +limited angle δ, due to disordered magneto-optical Kerr rotations (the disordered cyan arrows). R indicates the radius of a circular +nanoantenna, D is the beam’s diameter, and B is the magnetic field. The bottom panel exemplifies a radius distribution of disordered +nanoantennas, with ΔR being the fluctuation. (b) Probability distribution of stochastic PSHEs P(δ), with Δδ being the standard deviation +of the Gaussian distribution. λ is the wavelength of light. (c) Experimental and calculated Δδ vs. radius fluctuation ΔR. M is magnetization. +Insets: scanning electron microscopy images. Scale bar, 1 μm. Reprinted with permission from [211], copyright 2020 The Authors. + + +(a) +B +(b) +(c) +×10-5 +MAR +D +(a.u.) +5 +(art)ev +248 +4 +P(a) +0.5 +N +0 +-15 +-10 +-5 +5 +10 +15 +Noise leve +8×105 (/D) +0 +5 +10 +15 +20 +25 +30 +△R (nm)Journal of Optics (2022) #### +triggered intense research due to their exotic electrical, optical, and thermal properties. Particularly, +direct bandgap TMD monolayers show opposite electronic spins at ±K valleys. Consequently, the valley +information can be selectively encoded and retrieved by the photonic spin according to the valley- +dependent optical selection rules. Although great efforts have been devoted into this field, previous +strategies using metallic structures inherited intrinsic losses and limited functionalities. Alternatively, +versatile GPMs represent an attractive candidate to perform such a task, inspired by, for example, the +photonic Rashba effect that describes a momentum-space spin-split dispersion from inversion- +asymmetric structures [215]. However, conventional GPMs are generally designed for plane waves, +preventing an efficient interaction between nanoantennas and integrated valley excitons behaving as +in-plane circular dipole emitters. A similar issue also arises when single quantum emitters are +integrated with GPMs to investigate SOIs in the quantum regime, where a further requirement of long +quantum coherence times (or low losses) should be fulfilled. + +Advances in Science and Technology to Meet Challenges +(i) Stochastic PSHE: In a recent work [211], we approached the metrology challenge in weakly +disordered systems by exploiting magnetised disordered metasurfaces (Figure 1). Nanoscale size +fluctuations were revealed by the probability distribution of a stochastic PSHE, which was induced by +disordered magneto-optical Kerr rotations. Here, the metasurfaces are consisted of circular nickel +nanoantennas with radii randomly fluctuated in several nanometers. The random variations in sizes +of nanoantennas give rise to disordered geometric phases from magneto-optical Kerr rotations. This +leads to a spin-dependent beam shift being several orders of magnitude smaller than the diffraction +limit of light, i.e., a PSHE. By evaluating the PSHEs via weak measurements from many disordered +metasurfaces with different randomisations, we observe a Gaussian probability distribution for the +spin shifts. Notably, the standard deviation of the Gaussian distribution is proportional to the size +fluctuation of the nanoantennas. This result enabled us to detect a five-nanometer size fluctuation of +nanoantennas. +(ii) Photonic Rashba effect from quantum emitters: On the other hand, we tackled the weak +interaction between integrated valley excitons and nanoantennas by exploiting a novel platform of +Berry phase defective photonic crystals (BP-PhCs) [212,213]. The BP-PhCs are composed of a PhC slab +with isotropic nanopillars and a GPM with space-variant anisotropic nanoantennas that serve as +defects (Figure 2). By utilising the bandgap of the PhC slab, the insertion of the GPM into the PhC slab +gives rise to a near-field geometric phase defect mode, which couples the defects for an effective +interaction with the integrated valley excitons, resulting in site-controlled excitation, +photoluminescence enhancement, and spin-dependent manipulation of individual valley excitons. +Consequently, a spin-split dispersion from valley excitons is observed in momentum space, +manifesting as the photonic Rashba effect. Particularly, the spin-up and -down branches correspond +to emission from ±K valley excitons, respectively, indicating a valley separation in momentum space +at room temperature. Moreover, this basic interaction mechanism between circular dipole emitters +and nanostructures can be generalised to quantum emitters with arbitrary in-plane polarisations and +PhC structures with distinct symmetries. +(iii) Quantum photonic metasurfaces: In the preparation of entangled photons [213], we used +lossless dielectric metasurfaces, the near unity efficiency of which enables the manipulation of single +photons under a sufficient long quantum coherence time. To entangle single photons’ spin and OAM, +a GPM embedded with a spin-dependent helical phase was designed. Depending on the sign of the +photon spin, the GPM performs a unitary transformation that adds or subtracts one quanta of OAM. + +Journal of Optics (2022) #### +In our case, the spatial wavefunction of the photon is paraxial; therefore, the spin and the OAM are +independent and have Hilbert spaces of different dimensions. Consequently, single photons in +entangled spin and OAM states, and photon pair with nonlocal correlations between the spin of one +photon and the OAM of another photon are generated. + + +Concluding Remarks +Optical SOIs are ubiquitous in nano- and atomic-scale systems. An in-depth understanding of them +contributes to both fundamental physics and advanced applications. Our results suggest that +Pancharatnam-Berry phase optical elements are suitable for discovering new types of SOIs, which +show promising applications in novel photon transport controls, such as entangled photons, spin- +polarised light sources, and ultra-sensitive optical metrologies utilising splits of non-degenerated spin +modes distinguished by quantum weak measurements. To investigate nanophotonics under +unprecedented extreme conditions, the spin-controlled generation, manipulation, and detection of +atomic-scale light sources of various statistical properties—e.g., spontaneous emission (super- +Poissonian), stimulated emission (Poissonian), and quantum emission (sub-Poissonian)—are +promising fields, which we foresee many possibilities in the coming future. In general, introducing +spin-orbit coupling of electromagnetic waves into contemporary photonics and atomic-scale optics +may result in the development of a new area of research, that is, atomic-scale spinoptics. + +Acknowledgements +The authors gratefully acknowledge financial support from the Israel Science Foundation (ISF), the U.S. +Air Force Office of Scientific Research (FA9550-18-1-0208) through their program on Photonic +Metamaterials, the Israel Ministry of Science, Technology and Space. The fabrication was performed +at the Micro-Nano Fabrication & Printing Unit (MNF&PU), Technion. + + + + + + + + + + + + + + + + +Figure 1 – We allow at most two figures that +are roughly the size of this box. +Figure 2. Photonic Rashba effect from valley excitons. (a) Illustration of a heterostructure combining a BP-PhC and a WSe2 monolayer. +The PhC slab composed of isotropic nanopillars is arranged in one lattice, and the GPM composed of anisotropic nanoantennas is arranged +in another lattice, serving as defects to the PhC slab. By exploiting the emerging Berry-phase defect mode, valley excitons effectively +interact with the defects for coherent geometric phase pickups, leading to a photonic Rashba effect in momentum space. Inset: Schematics +of valley-dependent optical selection rules for ±K valley excitons in WSe2 monolayers. (b) Simulated defect band of a BP-PhC. (c) Measured +spin-split dispersion in momentum space (i.e., photonic Rashba effect) from valley excitons. Reprinted with permission from [212], +copyright 2020 The Authors. + + +(a) +WSe, +Si +(b) +(c) +0.7 +defectband +0.60 ++ exp. +00Z +theory +90 +Wavelength (nm) +0.58 +760 +Wavelength (nm) +006 +wa/(2nc) +/em) +0.56 +0.4 +M +1300 +0.3 +0.54 +820 +-0.2 +-0.1 +0 +0.1 +0.2 +k, (2r/a)Journal of Optics (2022) #### +14. Spin, momenta, and forces in evanescent waves – towards spatial and +temporal structuring. +Michela F. Picardi1,2, Anatoly V. Zayats1 and Francisco J. Rodríguez-Fortuño1 +1King’s College London and London Centre for Nanotechnology +2ICFO – Institut de Ciencies Fotoniques + +Status +Coupling between spin and orbital angular momenta is a fundamental property of electromagnetic +(EM) waves, but it is especially pronounced in the evanescent fields. When modes are spatially +confined along at least one dimension, as is the case of dielectric or plasmonic waveguides or fibres, +their wavevector becomes complex. The wavevector component along the direction of confinement +at the interface between two media is imaginary, resulting in the field exponentially decaying away +from the interface. Many interesting phenomena originate from the topological features of +evanescent waves related to spin-momentum locking [207]. The transversality condition (𝐤 ⋅ 𝐄 = 0) +requires the evanescent fields to be elliptically polarised with a field component parallel to the +propagation direction, in sharp contrast to free-space fields. This locks the handedness of the +waveguided fields with their propagation direction [207], meaning that when either of the two is +reversed, the other must be reversed too. This behaviour arises from basic laws of electromagnetism +and exists even in unpolarised light [220]. Such topological protection of the propagation direction of +guided waves is valid until the spin flips, e.g., due to scattering. +Because of spin-momentum locking, elliptically polarised EM sources excite guided modes +unidirectionally via evanescent coupling [221]. Selective mode excitation was also achieved +considering the reactive power of evanescent waves Im(𝐄∗ × 𝐇) . Being perpendicular to the +interface, the reactive power is locked with the direction of evanescent decay so that multipolar EM +sources may excite a guided mode, without any directionality, only if their reactive power is parallel +(not antiparallel) to that of the evanescent wave, as in the case of Janus dipoles [222]. Selectivity and +directionality of the guided mode launching may also be understood as near-field interference, with +the same evanescent modes being excited and interfering with their different symmetries inherited +from different EM source multipoles [221,359]. +Reciprocal effects in controlling the polarisation of far-field directional scattering with the +direction of guiding modes were also demonstrated [223]. The spinning fields of guided modes were +widely exploited to achieve polarisation-dependent optical forces on achiral objects [360]. All these +effects stem from topological properties of evanescent fields of the guided modes and have been +observed in an extremely broad spectral range, from optical to radio frequencies, with various types +of waveguides and EM sources, such as multipolar emitters and scatterers, atoms and quantum dots, +and predicted for other kinds of evanescent waves, such as acoustic and gravitational ones. + +Current and Future Challenges +The phenomena described above follow from classical photonics but can also be applied in the +quantum realm. The coupling between quantum dots and complex light, such as evanescent waves, +leads to chiral directional single photon routing [361]. Single-photon evanescent waves and their +entanglement open up new possibilities in the development of light-matter interfaces and quantum +technology. + + +Journal of Optics (2022) #### + + +Structured evanescent waves are obtained by relaxing the condition of a single-wavevector, such +as in surface plasmon vortices [225,362]. Structured evanescent waves are already proving a fertile +ground for physical phenomena such as dynamic field-skyrmionic lattices. Exploiting spin to orbital +angular momentum coupling, spin-skyrmions in the evanescent field have been demonstrated (Fig. +1(e)) [31]. Structured evanescent waves are at their infancy, and inspiration can be taken from the +myriad of structured beams considered in free space to find and exploit analogues in the near-field. +As well as structuring in space, time-domain structuring can be introduced. For propagating +waves, time-dependence leads to remarkable polarization patterns in multicoloured light [38]. It is +therefore to be expected that the routing and metrological capabilities of single-coloured evanescent +waves—based on their unique polarization patterns—will acquire additional features and versatility +when the time-harmonic assumption is dropped, in favour of multi-coloured and pulsed evanescent +waves. +The combination of both spatial structuring and time-dependence, e.g., in spatio-temporal +vortices [19-24], will greatly extend the evanescent wave playground, providing time-dependent +photonic topologies in the presence of a complex wavevector. Guided spatio-temporally structured +waves could be used for encoding classical or quantum information into topological invariants such as +vortex winding numbers as a novel means of guided wave information transfer. +Optical forces near surfaces is another vast field enabled by the properties of guided waves. +Counter-intuitive effects such as levitation, polarisation-controlled forces, or chiral sorting for +particles and molecules is an active field of study [363-365] based on the peculiar spin and momentum +properties of simple non-structured time-harmonic surface evanescent waves. The use of structured +Figure 1. (a) Magnetic field amplitude of a surface plasmon-polariton unidirectionally excited by a circularly polarised electric dipole, +adapted from reference [221]. (b) Reciprocal control of far-field scattering polarization via unidirectional near field excitation, adapted +from reference [223]. (c) SEM image of the nanosphere placed on the waveguide and light intensity at the two ends of a fibre measured +by varying the polarisation of the light incident on the nanosphere. The nanosphere acts as a point dipole, and when it is circularly +polarised, it will couple unidirectionally to the mode guided by the fibre, adapted from reference [224]. (d) Structured surface plasmon- +polariton vortex excited via a plasmonic vortex lens, reprinted from [225]. (e) Skyrmion-like structure of the spin angular momentum +vector (top) and the intensity distribution (bottom) of a surface wave vortex, adapted from reference [31]. + +a +b +LCP +air +metal +C +1.0 +0.8 +e +? +2 +Max +JEPJournal of Optics (2022) #### +and multi-coloured evanescent wave illumination can lead to precisely engineered optical force fields +for optically tuneable manipulation of nanoparticles, atoms, and molecules near surfaces. +Even more layers of complexity will be introduced if the assumption of polarised and coherent +light is dropped. Transverse spin of the evanescent field persists even when light is fully unpolarised +[220]. Incoherent light (with a fluctuating phase) will show similar behaviour, opening a vast +playground of statistical optics with evanescent waves. This could enable new applications, for +example, chiral optical force separation of enantiomers using non-laser light sources, and applications +using thermal emission or even sunlight. + + + + +Advances in Science and Technology to Meet Challenges +Despite significant progress in describing spin-orbit properties of light fields, further advances in the +understanding of dynamical properties of near-fields are needed. Light properties are typically defined +via their effect on material probes. For instance, the spin angular momentum density of an evanescent +wave is proportional to the mechanical torque exerted by the wavefield on a dipolar particle. Novel +dynamical properties are uncovered when considering more complex particles. The optical torque on +a quadrupolar particle is not proportional to the spin, but instead to another quantity: the spin of the +field gradient [227]. Given potential applications of optical spin effects in evanescent fields, such as +transverse spin and skyrmions, higher-order field properties—such as the above-described field +gradient spin and other related properties that arise in the evanescent field interactions with higher- +order multipoles—call out for an in-depth analysis and deeper understanding of light properties in the +near field, especially in structured, dynamic, and multi-colour settings. +More immediate important technological challenges are related to reducing the effects of losses +if surface plasmons are chosen as a platform for evanescent waves, and to improving the coupling +Figure 2. Schematic representation of the main interesting phenomena observed in non-structured time-harmonic surface waves (top), +and the possible direction of expansion into new domains which can be accessed via space and time structuring of surface waves (bottom). + +Evanescentwaves +Single wavevector +Time harmonic +Quantumspin-Halleffect +Near-field directionality +Near-fieldopticalforces +Structured evanescentwaves +Structured +Time +Unpolarised/ +wavevector spectrum +dependent +Incoherent +Surface wave +Polycromatic +Transverse +vortices +pulsed +spinof +Photonic +unpolarised +evanescent +light +skyrmions +waves +Spatiotemporal vortex pulsed +surfacewaves +FutureadvancesJournal of Optics (2022) #### +efficiencies from emitters and scatterers to evanescent wave platforms such as rib and slot +nanophotonic waveguides and nanofibres. The temporal structuring of surface waves will need +advances in attosecond physics and the control of the intra-pulse polarization evolution. +Advances in material science, nanophotonic platforms, and optical force instrumentation will all +be needed to exploit the full potential of spatially and temporally structured surface evanescent waves +and their effects. The miniaturisation of EM emitters and improvement in their coupling efficiencies +as well as the successful integration of quantum dots, molecules, and atoms into nanophotonic +devices are requirements for many of the proposed and potential applications: from nanoscale +quantum technologies, requiring single-photon spin routing based on evanescent wave spin- +momentum locking, which could form part of optical quantum computing or secure quantum +communication platforms, to the use of evanescent fields for on-chip chiral optical forces that might +enable important applications like all-optical on-chip separation of enantiomers. + +Concluding Remarks +The applications in quantum technologies, nanophotonics, sensing, metrology and nano-opto- +mechanics drive requirements on making use and manipulation of all degrees of freedom of guided +light. In addition to polarisation-controlled routing and coupling of electromagnetic waves, which have +already resulted in applications in position sensing with nanometric displacement sensitivity, +integrated miniaturised polarimeters and coherent-optical receivers as well as unusual lateral and +repulsive optical forces, the use of structured evanescent waves with new dynamical and topological +properties above and beyond spin-orbit interaction offer innovative solutions towards +miniaturisation, energy efficiency, and ultrafast operation. These future advances will be made +possible in close interaction of theory, developing new understanding of angular momenta properties +in structured and dynamic evanescent fields, nanoscience, and photonic instrumentation for realising +and measuring these properties, and development of photonic and nanophotonic platforms tailored +for specific applications. + +Acknowledgements +This work was supported by the European Research Council projects iCOMM (789340) and Starting +Grant ERC-2016-STG-714151-PSINFONI. + + + +Journal of Optics (2022) #### +15. Momentum and spin of electromagnetic, sound, and water waves +Konstantin Y. Bliokh +RIKEN + +Status +Wave momentum has a long (and probably never-ending) history starting from 18th century studies +by Euler, with the landmarks of the electromagnetic momentum by Poynting and quantum- +mechanical momentum by de Broglie. It includes numerous controversies: the momentum of sound +waves [228], the Abraham-Minkowski dilemma for the momentum of light in a medium [229], the +Belinfante-Rosenfeld problem for the energy-momentum tensor in field theories [230], etc. Angular +momentum (AM) is closely related to the momentum, and is also involved in these problems. Recent +great interest in structured waves, materials, and wave-matter interactions prompted thorough +revision of local momentum and AM properties of generic inhomogeneous (although often restricted +by monochromaticity) wave fields. +Starting with the general field-theory approach, there are two types of momentum and AM +densities: (i) the kinetic momentum density 𝚷 (e.g., the Poynting momentum in electromagnetism) +and kinetic AM density 𝐌 = 𝐫 × 𝚷; (ii) the canonical momentum density 𝐏 and the canonical AM +density 𝐉 = 𝐫 × 𝐏 + 𝐒, where 𝐫 × 𝐏 = 𝐋 and 𝐒 are the orbital and spin (intrinsic AM) densities. The +kinetic and canonical quantities are related by the Belinfante-Rosenfeld equation [230]: + +𝚷 = 𝐏 + +? +- 𝛁 × 𝐒 . +(1) +One can regard the canonical momentum and spin densities as two fundamental independent +quantities; other momentum/AM characteristics are their derivatives. Both the kinetic and canonical +quantities are important and have their own advantages. For example, in relativistic field theory, the +kinetic quantities are expressed via fields and are explicitly gauge-invariant, while the canonical ones +follow directly from Noether’s theorem, i.e., provide generators of translations and rotations in the +quantum-mechanical formalism. Notably, the canonical momentum and spin densities allow +meaningful gauge-invariant expressions only for monochromatic fields, but these are directly +observable via forces and torques exerted by monochromatic fields on dipole particles [231]. +For monochromatic electromagnetic fields in free space, the canonical momentum and spin +densities read [231–234], + +𝐏 = +? +@A Im[𝜀 𝐄∗ ⋅ (𝛁)𝐄 + 𝜇 𝐇∗ ⋅ (𝛁)𝐇] , +(2) + +𝐒 = +? +@A Im(𝜀 𝐄∗ × 𝐄 + 𝜇 𝐇∗ × 𝐇) . +(3) +Here, 𝐄(𝐫) and 𝐇(𝐫) are the complex electric and magnetic field amplitudes, whereas 𝜀 and 𝜇 are +the vacuum permittivity and permeability. Substituting Eqs. (2) and (3) into Eq. (1) and using Maxwell +equations yields the kinetic Poynting momentum 𝚷 = +? +-;% Re(𝐄∗ × 𝐇), where 𝑐 = 1/√𝜀𝜇 is the +speed of light. +Importantly, there is a freedom in defining the canonical momentum and spin densities (2) and +(3) even when the Poynting momentum is fixed. While Eqs. (2) and (3) represent symmetric ‘arithmetic +mean’ of the electric and magnetic contributions, one can use either purely electric or purely magnetic +quantities instead. The reason for choosing the dual (electric-magnetic) symmetric expressions is + +Journal of Optics (2022) #### +mostly aesthetic: to preserve the dual symmetry inherent in free-space Maxwell equations [231–234]. +However, the presence of charges breaks this symmetry because only electric charges exist in nature. +In addition, electric-dipole and magnetic-dipole particles effectively interact with the electric and +magnetic parts of the canonical densities (2) and (3) so that their electric/magnetic/symmetric form +is not fixed fundamentally but rather chosen in each particular problem using auxiliary arguments. +Sound waves in a fluid or gas have similar momentum and spin properties. Although sound waves +are often regarded as ‘scalar’ or ‘spinless’, these are actually vector waves; the local displacement (or +velocity) of the medium particles provides the vector wavefield. Akin to electromagnetic waves, one +can describe monochromatic sound waves via two complex fields: velocity 𝐯(𝐫) and pressure 𝑝(𝐫). +The canonical momentum and spin densities of monochromatic sound waves are [41–44]: + +𝐏 = +B +-A Im[𝐯∗ ⋅ (𝛁)𝐯] , +(4) + +𝐒 = +B +-A Im(𝐯∗ × 𝐯) , +(5) +where 𝜌 is the mass density of the medium. Substituting Eqs. (4) and (5) into Eq. (1), supplied with the +acoustic wave equations for 𝐯 and 𝑝, yields the acoustic analogue of the Poynting vector: 𝚷 = +? +-;&% Re(𝑝∗𝐯), where 𝑐𝑠 = 1/>𝜌𝛽 is the speed of sound with 𝛽 being the compressibility of the +medium. +From the field-theory viewpoint, the canonical quantities (4) and (5) also allow different forms +compatible with the same kinetic momentum. Namely, instead of the velocity-related quantities (4) +and (5), one can use the pressure-related quantities (𝐏 = +D +-A Im(𝑝∗𝛁𝑝) = 𝚷 and the spin density +vanishes in this case), or an ‘arithmetic mean’ of the velocity-related and pressure-related +contributions [43,44]. Furthermore, monopole and dipole acoustic particles are effectively coupled to +the pressure-related and velocity-related forms of the canonical quantities [43]. Then why do we +prefer the ‘asymmetric’ velocity-related definitions (4) and (5)? This is because, in contrast to +electromagnetism, sound waves exist only in a medium, and one can associate their dynamical +properties with microscopic mechanical properties of the medium particles. Indeed, local rotation of +the medium particles in a generic sound-wave field with an elliptical polarisation (polarisation of the +velocity field 𝒗 corresponds of the microscopic real-space trajectory of the particle) generates exactly +the canonical AM density (5) [41,42]. Furthermore, the medium particles slowly drift in the sound- +wave field due to the second-order difference between the Euler and Lagrange coordinates. This is +the Stokes drift with the velocity 𝐮 exactly corresponding to the canonical momentum density (4): +𝐏 = 𝜌𝐮 [55,235]. Thus, microscopic mechanical properties of the medium allow one to +unambiguously determine the canonical momentum and spin densities in sound waves. Remarkably, +Eqs. (4) and (5) are quite general and are also valid for elastic waves in isotropic solids [49,50,53] or +Langmuir plasma waves [236]. +The above features make the canonical momentum and spin in acoustic waves directly +observable, at least in principle, via microscopic motion of the medium particles. In practice, such +observation is challenging with typical sound waves. Larger-scale waves, such as water-surface waves, +can serve as a perfect platform for the observation of microscopic medium properties in structured +wave fields. Considering monochromatic deep-water gravity waves with the dispersion 𝜔2 = 𝑔𝑘 (𝑔 +is the gravitational acceleration) as a quasi-2D wave system, we recently derived the canonical + +Journal of Optics (2022) #### +momentum density in the unperturbed surface (𝑥, 𝑦)-plane and the corresponding spin density in +the vertical 𝑧-direction [55]: + +𝐏 = +B +-A Im[𝐕∗ ⋅ (𝛁-)𝐕 + 𝑊∗𝛁-𝑊] , +(6) + +𝐒 = +B +-A Im(𝐕∗ × 𝐕) . +(7) +Here, 𝛁2 = (𝜕𝑥, 𝜕𝑦), whereas 𝐕 = (𝑣&, 𝑣*) and 𝑊 = 𝑣+ are the in-plane and vertical velocity +components of water particles. Substituting Eqs. (6) and (7) into Eq. (1), and using the equations of +motion for gravity waves, yields the kinetic momentum 𝚷 = +BF +A Im(𝑊∗𝐕), which is consistent with +Ref. [237]. Akin to sound waves, Eqs. (6) and (7) correspond to the mechanical momentum (due to the +Stokes drift) and the microscopic mechanical angular momentum (due to the local elliptical +trajectories) of water particles, as shown in Figure 1 [55]. + +Thus, the momentum and angular momentum properties of electromagnetic, acoustic, and water +waves have profound similarities related to the presence of spin and the fundamental Belinfante- +Rosenfeld relation (1). There is also a principal difference in that the free-space electromagnetic +quantities are not associated with any medium (‘ether’) and cannot be derived from microscopic +mechanical considerations. However, electromagnetic waves in a medium represent mixed light- +matter waves and do involve microscopic mechanical properties of the medium. The kinetic +momentum of an electromagnetic wave in an isotropic dispersive medium with permittivity 𝜀 = +𝜀(𝜔) and permeability 𝜇 = 𝜇(𝜔) is still given by the Poynting (or Abraham) momentum 𝚷 = +Figure 1. Theoretically calculated surface distributions of the canonical momentum 𝐏 (black arrows) and spin 𝑆, (blue-red) densities, +Eqs. (6) and (7), in the interference of two plane gravity (water-surface) waves with orthogonal wavevectors 𝐤-,$. Numerical and +experimental plots show trajectories of microscopic water particles for three wave periods 6π/ω. The Stokes drift of the particles +and their elliptical motion correspond to the canonical momentum and spin, respectively. The normalized surface coordinates are +𝑋 = √2𝑘𝑥, 𝑌 = √2𝑘𝑦. Adapted from Ref. [55]. + +theory +numerics +experiment +V +canonical momentum +(0) +(O) +(O) +(0) +0) +m +元 +00 +O) +(0)) +(O) +(O) +0 +(0) +(O) +) +0 +元 +spin +(C) +(0) +(0) +(O) +000 +Q00 +2元 +wavevectors +Stokesdrift +spin +X +0 +(a.u.)Journal of Optics (2022) #### +? +-;% Re(𝐄∗ × 𝐇), while the canonical momentum and spin densities can be described by Eqs. (2) and +(3) with the substitution [236,238], + +𝜀 → 𝜀̃ = 𝜀 + 𝜔 +PQ +PA , 𝜇 → 𝜇I = 𝜇 + 𝜔 +PR +PA . +(8) +In this case, microscopic contributions from the motion of the medium particles (e.g., electrons or +atoms) play a crucial role [235,236,238,239], and the canonical expressions can be regarded as the +Minkowski-type momentum and spin densities [238]. Equations (2), (3), and (8) are found within the +dual-symmetric formalism, but the electric-field-biased approach is also possible and can be relevant +because the medium is usually electric-biased on the microscopic level [236] (i.e., consisting of electric +charges and dipoles). + +Current and Future Challenges +Despite the great progress in the description of the momentum and angular momentum of +structured waves, there are still many unsolved questions. To name a few: extension of the +above approach (if possible) to anisotropic media, polychromatic fields, and fields with +complex frequencies (e.g., Mie quasimodes). + +Advances in Science and Technology to Meet Challenges +Rapid development of nanophotonics, including metamaterials and plasmonics, provides a +perfect platform for theoretical, numerical, and experimental studies of fundamental +dynamical properties of complex fields and their interactions with matter. + +Concluding Remarks +We have briefly described the canonical and kinetic momentum and AM properties of +monochromatic electromagnetic, acoustic, and water waves. Despite enormous differences +in scales and their nature, the momentum and AM of these waves share profound similarities. +This reflects the universality of these concepts, as well as the remarkable role of the spin and +field-theory relations even in ‘spinless’ classical waves. + + + + +Journal of Optics (2022) #### +16. Acoustic spin +Chenwen Yang and Jie Ren +Tonji University + +Status +The acoustic waves in fluids were traditionally mis-regarded as spinless fields because of their curl- +free nature. Recently, several works about elastic and acoustic waves show that even pure longitudinal +waves that can be fully described with a scalar field still have the ability to carry spin angular +momentum (SAM). The existence of SAM does not require an extra spatial degree of freedom but +needs a locally temporal rotation of the vector field, like displacement or velocity. A clear definition +of SAM in acoustic wave systems will pave a new way to understand and realise various structured +acoustic wave systems, included but not limited to the spin-momentum locking in acoustic wave +systems and the symmetry selective excitation. +The expression of spin angular momentum (SAM) in acoustic/elastic waves could be derived from +the definition of angular momentum in acoustic/elastic waves, which is [42,50,53,240,355]: + +𝐒 = 𝜌𝜔 +2 Im[𝐮∗ × 𝐮] = 𝜌 +2𝜔 Im[𝐯∗ × 𝐯], +(1) + +where 𝐮∗ represents the conjugate of the displacement vector 𝐮, 𝐯 is the particle velocity which is the +time derivative of 𝐮, 𝜔 is the angular frequency of wave, and 𝜌 is the density of media. Under linear +configuration of small wave amplitude, 𝜌 is regarded as a constant. It is worth noting that the spin +angular momentum of phonons is also studied as early as the 1960s [48,241], which are quasiparticles +more suitable for describing quantised lattice vibration in the quantum scene. The spin angular +momentum of phonons shares a similar expression as Eq. (1), the SAM of acoustic/elastic waves, +reflecting the fact of wave-particle duality. However, initial works about phonon spin ignored the spin +of curl-free longitudinal mode [355]. The SAM of elastic/acoustic waves describes the locally temporal +rotation of displacement/velocity vector, not the spatial curl of the vector field. In other words, if a +displacement/velocity field has both longitudinal and transverse components (or could be non-zero +decomposed into two directions), it could have the SAM, and this is irrelevant with the curl of the +vector field. As such, even longitudinal waves (curl-free wave), e.g., acoustic waves, possess the ability +to carry SAM. Next, we will give a brief explanation through several derivations. +The displacement field 𝐮 could be written as the combination of the gradient of a scalar potential +and curl of a vector potential [50,51]: +𝐮 = 𝛁𝜙 + 𝛁 × 𝝍. +(2) + +In longitudinal wave fields, 𝛁 × 𝐮 = 0 and 𝝍 = 𝟎. Note that in the field of acoustics, people usually +describe the acoustic wave with the media particle velocity 𝐯 and acoustic pressure 𝑃 instead of +displacement 𝐮 and the scalar potential 𝜙, which gives [35,242]: + +−𝛁𝑃 = 𝜕(𝜌𝐯) +𝜕𝑡 += −𝑖𝜌𝜔𝐯, + 𝐯 = 𝜕𝐮 +𝜕𝑡 = −𝑖𝜔𝐮. +(3) + + + +Journal of Optics (2022) #### +Under this configuration, the acoustic SAM could be express as 𝐒 = (𝜌/2𝜔) Im[𝐯∗ × 𝐯] as shown in +Eq. (1) [42,43,45,240,243]. These configuration differences do not affect our discussion below. + +To clarify the existence of SAM in longitudinal waves, we could start with a common form of the +scalar potential in 𝑥-𝑦 plane: +𝜙 = 𝜙5𝑒SF'T𝑒SF(U𝑒VSA1, +(4) + +Where 𝑘U (𝑘T) represents the wave number along 𝑥 (𝑦) axis. According to Eq. (2), the displacement +field of a longitudinal wave can be written as: +𝑢U = 𝜕𝜙 +𝜕𝑥 = 𝑖𝑘U𝜙, +𝑢T = 𝜕𝜙 +𝜕𝑦 = 𝑖𝑘T𝜙. +(5) + + +Clearly, the displacement vector 𝑢T, or the transverse component perpendicular to the transport +direction, is non-zero in this evanescent wave. Although the material which only supports longitudinal +waves has no shear modulus, the transverse component 𝑢U still exists. As such, the spin angular +momentum is: +𝜌𝜔 +2 Im[𝐮∗ × 𝐮] = 𝐳r 𝜌𝜔 +2 Ims𝑢U∗𝑢T − 𝑢T∗𝑢Ut += 𝐳r 𝜌𝜔 +2 Imsu𝑘U∗𝑘T − 𝑘T∗𝑘Uv𝜙5 +-t, +(6) + + +where 𝐳r is the unit vector along the 𝑧-axis. If both 𝑘U and 𝑘T are real constants, e.g., a plane wave +transported in the 𝑥-𝑦 plane, nothing interesting happens. As 𝑘U∗𝑘T − 𝑘T∗𝑘U = 0, the acoustic SAM is +zero. However, things change when we consider a more complex situation, e.g., the evanescent +acoustic wave transported along the 𝑥-axis and decaying along +𝑦-axis from 𝑦 = 0. As such, 𝑘T could +Figure 1. Acoustic spin as a rotating particle velocity field (8). (a) A rotating particle velocity (black arrow) can be decomposed into +two components 𝑣# (blue arrow) and 𝑣/ (red arrow) along the 𝑥 and 𝑦 directions. (b) Acoustic spin in the interference of two acoustic +beams. Two beams with equal amplitudes propagating along the 𝑥 and 𝑦 directions contribute 𝑣# and 𝑣/ components of the particle +velocity field, respectively. + + + +(a) +-Ae（+) +Y, = Aei(+$+90) +(b) +Acousticbeam1 +'360° +315° +-270° +225° +180° +1350 +90° +45° +0° +Phasedifference between v,andy +Acousticbeam2 +PolarizationJournal of Optics (2022) #### +be rewritten as 𝑖𝜏, where 𝜏 is a real component. Also, the acoustic SAM will be a non-zero quantity: +𝐳r BA +- Im[(𝑘U(𝑖𝜏) − (−𝑖𝜏)𝑘U)𝜙5 +-] = 𝐳r𝜌𝜔𝜏𝑘U𝜙5 +-. We may call this acoustic SAM as the transverse spin +because it is perpendicular to the wave direction, similar with the transverse spin of an optical wave +[231]. Note that the sign of 𝑘U determines the sign of SAM in this evanescent acoustic wave +configuration. This indicates the spin-momentum locking effect in evanescent acoustic waves +[42,43,45], which is similar with the electromagnetic evanescent waves in optics [207]. +Nevertheless, similar things happen to the phase difference of strain components 𝜖UU and 𝜖UT. For +example, a surface longitudinal wave in fluid, like air or water, can have transverse shear strain, which +could be expressed as: +𝜖UU = 𝜕𝑢U +𝜕𝑥 = −𝑘U-𝜙, +𝜖UT = 1 +2 {𝜕𝑢U +𝜕𝑦 + 𝜕𝑢T +𝜕𝑥 | = −𝑘U𝑘T𝜙. +(7) + +Although 𝛁 ∙ 𝐮 = 0, the transverse shear strain 𝜖UT ≠ 0, and +X(( +X(' = +F( +F'. If we consider a plane wave in +both the x and y direction, +F( +F' is always a real number, and there is no phase difference between 𝜖UU +and 𝜖UT . However, if we consider a surface wave, +X(( +X(' = +F +Y 𝑒S) +% , this gives a 𝜋/2 phase difference +between 𝜖UU and 𝜖UT. Also, one can say that the phase difference between the normal and shear +strains shows the existence of acoustic SAM in this scene. It is worth noting that the phase difference +between 𝜖UU and 𝜖UT is also important in SAW-induced nonreciprocal ferromagnetic resonance [244]. +According to the analysis above, the acoustic SAM may also excite the ferromagnetic resonance. +Actually, the existence of non-zero acoustic SAM relies on the local phase difference between 𝑢U +and 𝑢T (or 𝑣U and 𝑣T), as indicated by Eq. (1) and (6). Non-zero SAM exists when the phase is different +between two different directions [42,240,356], e.g., the acoustic field generated with two acoustic +beams with different phases, as shown in Figure 1. + +Current and Future Challenges +Acoustic spin induced torque. The acoustic wave with non-zero SAM will introduce a torque to an +absorption particle. This spin-matter interaction can be characterised by the rates of the angular +momentum transfer between the acoustic field and the particle [43]. Because the monopole vibration +mode is isotropic and does not provide the circularly polarised local states, only the dipole momentum +can introduce the torque. With the help of a meta-atom which supports dipole resonance, one can +measure the torque induced by the acoustic SAM [42], as shown in Figure 2(a). These acoustic spin +and torque also exist in the topological meta structure [245]. Although the present work shows a clear +correlation between the pseudo-spin state and SAM in this kind of quantum spin hall effect acoustic +metamaterial, due to the challenge in directly measuring the SAM-induced torque in complex acoustic +system, the experimental evidence is still missing, and the physical origin of this correlation still needs +further clarifying. +The structured acoustic wave with SAM. The acoustic SAM also improves the fundamental +understanding of the inherent near-field symmetry and directional coupling in acoustics. Along with +the time average energy flow and reactive power, acoustic SAM density can be used to characterise +the time and space symmetry of the evanescent acoustic wave [45], as shown in Figure 2(b). These + +Journal of Optics (2022) #### +evanescent wave modes could be selectively excited by the acoustic source with particular +symmetries, which provides a feasible approach for designing functional acoustic devices. +Besides the near-field acoustics, the spin-dependent transportation could also be realised in wave +guides with symmetry-breaking boundary conditions [243], as shown in Figure 2(c). The momentum +of the acoustic wave in such waveguides are tightly coupled with the acoustic SAM. This spin- +momentum locking effect in wave guides will raise the SAM dependent selective transportation and +enhance the backscattering suppression. +As a kind of special structure, skyrmions are also proposed in acoustic systems [35,29]. Under the +help of a well-designed hexagonal acoustic metasurface, the acoustic velocity fields can raise clear +skyrmion lattice patterns, which unveil a fundamental property of acoustic fields and may inspire +future research in structured acoustic waves. However, the excitation and controlling method of +skyrmion patterns of acoustic SAM is still not presented yet. In the future, acoustic skyrmions may +pave a new way for the research in structured acoustic waves and functional acoustic devices. + + + +The SAM interaction between elastic and magnetic/optical systems. As the acoustic spin we talk +about here can carry real SAM, it is natural to assume that angular momentum could transfer between +acoustic systems and magnetic/optical systems. +In the field of research about magnetic materials, the interaction of spin waves and ultrasonic +waves in ferromagnetic crystals was theoretically demonstrated with the phonon-magnon interaction +as early as 1958 [246]. The experimental demonstrations about the interaction between surface +acoustic waves (SAW) and magnon systems also attract many attentions, including but not limited to +the SAW spin-pumping [247], SAW-driven ferromagnetic resonance [248] and SAW-controlled +magnetisation [237]. Meanwhile, the angular momentum interaction transfer between elastic and + + + + + + + + + + + + + + + + + + + + + + +Figure 1 – We allow at most two figures that +Figure 2. (a) Experimental set-up for the measurement of acoustic spin and the measured SAM-induced torque of acoustic waves +which possess positive (red line) and negative (blue line) SAM [42]. (b) The near-field selective excitation of spin source [45]. (c) The +spin-dependent propagations in waveguide with symmetry-breaking boundary conditions [243]. The excited waveguide modes have +different propagation directions and carry opposite SAM texture. + + +a) +0.4 +0.2 +0.2 +.0.40 +50 +100 +150 +200 +250 +Input amplitude (mV) +b) +Po +Media +Media +Air +Air +Media +iD +Media +DX+iD +(C)Journal of Optics (2022) #### +electromagnetic systems is also proposed with the help of optical fibres [250] or piezoelectric +materials [251]. These works may offer a new way to control dynamic states of magnetic, optical, and +elastic systems. However, existing research mainly describes this spin transfer with the elastic strains +or phonons rather than acoustic/elastic SAM. To give a simpler and more practical theory of the SAM +coupling between various physical systems in the micro scene, a definition of the transition between +acoustic SAM and magnetic/optical SAM is still needed. + +Concluding Remarks +Spin angular momentum helps people to give a fundamental explanation of structured waves and +practical implications for wave devices. The acoustic SAM proves that the ability of wave to carry SAM +is more related with the polarised vibration other than the number of intrinsic spatial degree of +freedom. We expect that these concepts of acoustic SAM will assist the development of structured +waves in optical, acoustic, and elastic systems, as well as the SAM transfer between different physical +systems. + + + +Journal of Optics (2022) #### +17. Acoustic Pseudospins for Wave Control and Topological Protection +Alexander B. Khanikaev1 and Andrea Alù2 +1The City College of New York +2Photonics Initiative, CUNY Advanced Science Research Center + +Status +In recent years, synthetic degrees of freedom deliberately introduced into the design of metamaterials +via symmetry engineering have significantly expanded the landscape of classical wave phenomena. In +particular, synthetic pseudo-spins spanning Hilbert spaces of desirable dimensions and leveraging +effective Hamiltonians with nearly any form and structure have been enabling a wide range of +eigenstate and spectral engineering in metamaterial responses. Advances in material engineering and +manufacturing on an unprecedented deeply subwavelength scale have been enabling a precise +control over the structure of such effective Hamiltonians, leading to the emulation of numerous +fascinating physical systems, from field theory with synthetic gauge fields to relativistic and +topological phenomena. +In this context, acoustic metamaterials arguably represent the most straightforward and easy-to- +work platform for emulating complex wave phenomena. In addition to the relative simplicity of +acoustic experiments, recent advances in additive manufacturing have made acoustic systems very +appealing. On the other hand, unlike other vector wave-fields, such as electromagnetic and +mechanical platforms, which offer natural intrinsic degrees of freedom that can be leveraged to +construct effective Hamiltonians, the acoustic pressure field is scalar in nature. Therefore, synthetic +degrees of freedom represent an absolute necessity for the emulation of effective Hamiltonians with +acoustic pressure-wave systems [252]. Figure 1(a) illustrates how synthetic pseudospin can be +produced in an acoustic Kagome lattice of 3D printed trimers and used for directional (valley) +excitation of bulk Bloch waves. For mechanical waves of vector nature, engineering the additional +degrees of freedom represents a powerful tool to further expand the Hilbert space and entangle +synthetic and natural degrees of freedom. +Based on this approach, a broad range of topological phenomena, including higher-order 2D and +3D topological phases, and emulation of Dirac and Weyl physics, have been realised in acoustic +metamaterials [253-256]. These advances have led to numerous demonstrations of unprecedented +control over spectral features, propagation, and scattering of sound waves. Spectral pinning of +resonant higher-order topological states via lattice symmetries, pseudo-spin polarised topological +edge transport, and topological resilience represent just some examples of recent demonstrations. +Beyond these demonstrations, artificial acoustic media with synthetic pseudospins are posed to +bring even more fascinating advances, both in the demonstration of fundamentally new wave +phenomena and in practical applications. These advances will likely stem from new physics enabled +by sound-matter interactions and more exotic responses of structured acoustic media, which can +radically expand the range of attainable effective Hamiltonians. Indeed, recent demonstrations of +multiphysics-enabled phenomena in classical-wave polaritonic systems have shown how structured +nature of waves in one physical subsystem can be transferred onto the second one via interactions, +to yield a new type of topological excitations, as illustrated by Fig. 1(b) [257]. From the technological +point of view, synthetic pseudospins have not been exploited yet, the context of controlling scattering +and radiative properties of acoustic metamaterials, which is another promising direction to pursue, +e.g., for achieving pseudo-spin-controlled directional emission of sound waves. + +Journal of Optics (2022) #### + + + +Current and Future Challenges +Emergent topological phenomena involving nonlinear and active regimes, where new pseudo-spin- +dependent and topological nonlinear responses and nonlinear modes could be observed, are +therefore of great interest [258-261]. The traditional approach to enable this challenging vision has +been to realise acoustic metamaterials loaded with active elements by incorporating electric circuits +with nonlinear elements and amplifiers, which may provide desirable feedback via microphones and +transducers. While this approach does enable testing of both nonlinear and non-Hermitian regimes, +it is hardly scalable, and therefore, it is not suitable beyond proof-of-principle demonstrations. +Moreover, acoustic excitations carry no charge, spin or magnetic momentum, and thus they do +not easily interact with other types of excitations. This in turn makes it difficult to actively control +acoustic waves, exploit nonlinearities that can be inherited from such excitations, and enable multi- +physics phenomena. Indeed, by mixing and hybridizing acoustic modes with excitations of different a +nature, such as mechanical waves, electromagnetic and optical waves, magnons and spin-waves, etc., +as illustrated in Fig. 2, one could promote nonlinear effects, drain their energy to induce acoustic gain, +or expand the number of available degrees of freedom to explore richer physics, e.g., by engineering +synthetic Hamiltonians of a more complex structure and enlarged dimensionality. It is, therefore, a +major current challenge in the field of topological acoustics to find material systems that could +efficiently and actively interact with acoustic waves and, at the same time, can be suitable for +integration into acoustic metamaterials [258-260]. +In addition to these materials’ related challenges, there is still a great need in better +understanding how synthetic pseudospins and spatially varying gauge fields can affect the radiative +properties of open acoustic metamaterials. While photonic pseudospins have been shown to enable +unprecedented control over radiation, from basic control over the radiative lifetime of topological +modes to the generation of vortex beams within topological cavities, these successes have not been +translated into the acoustic domain yet. Besides the control over far-field radiative profiles, the +possibility to control near-fields with synthetic pseudospins could enable a plethora of applications +unique to acoustics, including trapping and moving objects along pathways defined by synthetic gauge +fields, and transfer of angular momentum from acoustic fields to trapped objects for controllable +rotations. + + + + + + + + + + + + + + + + + +Figure 1 – We allow at most two figures that are roughly the size of this box. +a +b +Figure 1. (a) Acoustic pseudo-spin in 3D printed kagome lattice of acoustic resonators (left) and control over the propagation directions +(K or K’) of the spin-full bulk modes via circularly polarized excitation (right). (b) Illustration of multiphysics with synthetic degrees of +freedom—polaritonic systems where structured optical field endowed with pseudo-spins induces chiral vibrational modes in van der +Waals materials [257]. + +Journal of Optics (2022) #### +Advances in Science and Technology to Meet Challenges +While basic models—such as tight-binding model and coupled mode theory, which are widely used to +describe acoustic metamaterials with synthetic degrees of freedom—can sometimes be sufficient for +understanding basic properties of metamaterials, they also tend to oversimplify the physics by largely +ignoring the structure of pressure, displacement and velocity fields, and long-range coupling within +metamaterials. On the other end, first-principles methods, such as finite element methods which +allow to directly solve wave-equations, provide little insight into the fundamental features of +topological metamaterials. While these two approaches typically agree well and meet the needs of +systems based on discrete resonances, such as discrete lattices of printed or machined acoustic +cavities, they diverge when considering systems where the continuous nature of wavefields should be +properly treated. One example is long-range interactions that naturally exist in non-discrete systems +and can give rise to significant corrections to spatial dispersion of the modes, and even to new types +of topological excitations [262]. Similarly, in open systems where acoustic fields may have an +evanescent nature, discrete models would not capture the complex structure of the near-fields +carrying nonzero angular momentum [242,263]. It is therefore crucial to develop analytical and semi- +analytical techniques that can be simple enough to provide an effective Hamiltonian description, yet +capable of accounting for the structure of the wavefield. One of the candidates for such description is +mode matching, which has been widely used in various contexts, but not for the description of +pseudospins and gauge fields in acoustic metamaterials. +A comprehensive theoretical description of the acoustic field structure is also crucial for +understanding and modelling the interaction of sound with active matter. For example, overlap of +chiral hotspots of the near-field in topological materials can be used for the generation of pseudospin- +dependent synthetic gauge fields and spin-selective control of sound waves. Active matter could be +mechanically, electrically, or optically driven, and contain internal degrees of freedom which could be +engineered to interact selectively with synthetic acoustic pseudospins. In this case, interactions can +be leveraged to induce a desirable form of non-Hermitian and nonlinear synthetic gauge fields. + + + +Concluding Remarks +Acoustic metamaterials with symmetry-engineered pseudospins have already become an exciting +platform for control of propagation of sound waves via synthetic gauge fields acting in the expanded +Hilbert space span by pseudospins. Coupling such metamaterials with active materials interacting with +sound waves opens even broader opportunities via realisation of pseudospin-dependent tuneable + + + + + + + + + + + + + + + + + + +Figure 2. Concept of nascent acoustic metamaterials with synthetic pseudo-spins coupled to active and nonlinear materials for unmatched +control over propagation and radiation of acoustic waves. Judicious design of coupling of structured sound waves with active, nonlinear, +and time-modulated materials envisions emulation of non-Hermitian and interacting topological phases in real and synthetic dimensions. + +() +Dimensionts) +Synthetic +2D ++Journal of Optics (2022) #### +gauge potentials, including the realisation of phenomena that have so far evaded the field of acoustics, +such as novel non-Hermitian and nonlinear topological phases, including real and synthetic +dimensions. Hybrid acoustic/active-matter systems also offer a broad range of novel applications, +from active control of acoustic radiation as well as structure of the near- and far-fields which can +enable new approaches for acoustic communications, imaging and for mechanical trapping—acoustic +tweezers, all enhanced with additional degrees of control via synthetic degrees of freedom. + + +Acknowledgements +Our work in this area has been funded by the National Science Foundation, the Office of Naval +Research, and the Simons Foundation. + + + + +Journal of Optics (2022) #### +18. Mechanical effects of structured sound waves +Etienne Brasselet +University of Bordeaux, CNRS + +Status +Acoustic waves are mechanical in nature because their existence requires a medium whose vibratory +characteristics allow us to classify the waves into two families: longitudinal (compression) and +transverse (shear). Considering the idealized situation of a plane wave propagating in a homogeneous +medium, the former refers to a 1D back-and-forth motion along the direction of propagation, while +the latter refers to a motion, usually 2D, in a plane orthogonal to the propagation direction. +More than a century ago, physicists discovered that the propagation of acoustic waves is +accompanied by mechanical effects on the media in which they propagate. This is illustrated by several +pioneering experimental works on the spatial manipulation of microscopic [264] or macroscopic [265] +solid objects, and on the deformation of fluid interfaces [266]. These phenomena involve a rich set of +physical effects such as heating, fluid dynamics, and radiation stresses. The relative simplicity of the +experimental implementation contrasts with the difficulty of a quantitative description which must +account for the transfer of energy, linear momentum, and angular momentum between the wave and +the material. +Figure 1 illustrates dissipative and non-dissipative wave-matter interactions, which occur jointly +at different length scales depending on the nature of the material inhomogeneities and the properties +of the media involved. In homogeneous media, the attenuation of waves, which corresponds to the +thermo-viscous dissipation inherent to the setting in motion of matter at the microscopic scale, leads +to a transfer of energy and momentum. Energy transfer produces a heating while a force per unit +volume exerted on the matter results from momentum transfer, which generates flows in usual fluids, +deformations in viscoelastic media, and mechanical stresses in solids. The transfer of momentum can +also occur at the interface between two homogeneous media where the discontinuous change in +material properties results in a force per unit area exerted on the interface, which can then be +deformed, even—and perhaps somewhat counterintuitively—if the interface is acoustically +transparent [347]. +All these phenomena have led to the emergence of numerous applications such as quantitative +imaging of inert or living media, metrology of wave or material properties, non-contact manipulation +and processing of fluids and objects. Knowing that any real-world field is a superposition of plane +waves, and that any system is finite, structured acoustics meets structured matter is the norm, for +which ongoing conceptual and technical developments aim to fully exploit the advantages of +acoustomechanics based on translational and rotational degrees of freedom. + +Current and Future Challenges +When considering the mechanical effects of (compression) sound waves, it is striking how often +theoretical and experimental developments are carried out in the context of their electromagnetic +counterparts. This highlights the power of analogies in wave physics as well as the fundamental +distinctions and opportunities associated with fields of a distinct nature. To name but a few, we are +dealing with: longitudinal (sound) versus transverse (light) waves; coupled scalar and vectorial fields +(pressure and velocity) versus coupled vectorial fields (electric and magnetic); a series of material +parameters involved in the constitutive relations describing how the fields evolve, which offer more + +Journal of Optics (2022) #### +practical flexibility in acoustics (density and compressibility) than in optics (dielectric and magnetic +permittivity); and generic wave scattering problems involving multipole expansions in the treatment +of the wave-matter interaction. + + + +Analytical toolkits are emerging which provide a better fundamental understanding of the generic +and specific aspects of the physics involved, e.g. [43,267]. Furthermore, they equip experimentalists +with rational design and fabrication strategies of soft metamaterials whose properties go beyond +those of their constituents, mediated by spatially extended acoustic force landscapes. Exploiting +acoustomechanics opens the way to programmable functionalities [268] provided that the structured +sound fields can be adapted to demand. Usually, the soughtafter "meta-atom" architectures +correspond to periodic networks, for which standing waves are well suited. However, this prevents +the development of meta-devices endowed with spatially distributed functionalities. Non-periodic +radiation force networks therefore appear as a natural and time-dependent next step as a way to +actively control the local interactions of many-body systems and thus their collective behaviour [269], +seeding the development of active matter fuelled by sound. +Although rotational mechanical effects date back to the beginnings of acoustomechanics, as +recalled by the implementation of the sound mill by Dvorak and Mayer in the 1870s [265], it is only +recently that angular momentum characteristics of structured sound are experimentally exploited. On +the one hand, pressure fields endowed with phase singularities make it possible to exert acoustic +radiation couples on matter either by dissipative [270] or non-dissipative [271] orbital angular +momentum transfer processes. On the other hand, a spin contribution to the angular momentum of +sound, which is intimately linked to inhomogeneous fields (two plane waves are sufficient) and is +associated with the local elliptical vibration of the medium, has also been demonstrated +experimentally recently by using a dissipative and polarisable subwavelength object [42]. While +a +!"# +$ +$ +$ +% +b +c +d +% +% +Figure 1. Illustration of the main mechanisms driving mechanical effects mediated by wave-matter transfer of energy and momentum. +(a) Wave attenuation during propagation, which is associated with the characteristic attenuation length 𝛼0-, where 𝛼 is the field +attenuation coefficient. (b) Geometric reflection and refraction of acoustic beams when the characteristic length 𝑎 associated with +inhomogeneities of the material properties typically satisfies 𝑎/𝜆 ≫ 1 , where 𝜆 is the wavelength. (c) Wave diffraction when +inhomogeneities are of the order of the wavelength, hence typically 𝑎/𝜆~1. (d) Scattering for point-like inhomogeneities, hence in the +regime 𝑎/𝜆 ≪ 1. Note that there is no formal boundaries between the latter regimes; still, the parameter 𝑘𝑎 is a key parameter to define +the relevant framework when describing the wave-matter interaction. + +Journal of Optics (2022) #### +taming the different facets of spin and orbital angular momentum of sound is still in its infancy, the +interest it has generated suggests that it will not remain a mere scientific curiosity for long. + +Advances in Science and Technology to Meet Challenges +Now that conceptual frameworks, structural design approaches, and fabrication tools are available for +waves and materials, the study and exploitation of the mechanical effects of structured sound has a +bright future, which depends in part on the ability of often distinct research communities to open up +to each other. +The contactless manipulation of matter highlights how fundamental and technological advances +can come together in a simple way. As an example, we mention the prototypical situation of a focused +vortex beam interacting with a spherical particle, which refers to the recently introduced single-beam +acoustic tweezers [272] and inherently involves mechanical translational and rotational degrees of +freedom. This represents the acoustic analogue of their famous optical counterpart celebrated by the +2018 Nobel Prize in Physics, but endowed with force and torque resources that are—Watt per Watt— +several orders of magnitude larger. Indeed, the linear and angular acoustic momenta respectively +scale as 1/𝑐 and 1/𝜔, where 𝑐 is the speed of sound and 𝜔 is the angular frequency of sound. +To date, the spin, orbital, and viscous contributions to the mechanical actions exerted on a particle +trapped in vortex tweezers remain little explored. This invites one to address the +acoustohydrodynamic problem as a whole towards developing applications such as sound-driven +micro-machines from physical, chemical, and biological perspectives. In particular, it appears +necessary to go beyond the simple, yet instructive, case study of an isotropic spherical particle +immersed in an isotropic standard fluid. Conversely, technologies relying on the mechanical effects of +"unstructured" sound, such as radiation-force based ultrasound imaging techniques, are likely to +benefit from the advantages deriving from structured waves since other mechanical degrees of +freedom are involved, such as acoustic vortex elastography [348]. +From a general point of view, the growing interest in acoustic spin and orbital angular momentum +opens up questions such as: How can the polarization state of sound be used to manipulate +anisotropic media as is done in electromagnetism for many decades with optical spin angular +momentum? How can the singularities and topological textures of inhomogeneously polarised +acoustic fields be used to shape matter in non-trivial ways? How can spin-orbit interaction mediated +by anisotropic or inhomogeneous media be used to enrich the toolbox of acoustomechanics? In this +context, topological properties of artificially structured materials and elasticity are some of the new +key players at play. + +Concluding Remarks +Although speaking of the mechanical effects of sound is formally a pleonasm—sound being itself a +mechanical movement—their surprisingly rich consequences across length scales, both for waves and +for matter, make them still attractive. Recalling that, more than a century ago, the first steps of +acoustomechanics dealt with the use of spatially textured sound fields owing to wave interferences +[264], the exploration of linear and angular mechanical effects exerted on objects [265] and the use +of deformable materials allowing nonlinear feedback phenomena [266], it is remarkable how easy it +is to find echoes of them in current research themes. Now that acoustics has entered into our +everyday life as sensors, transducers, and imaging systems, among other things, the mechanical +effects of sound remain a source of inspiration for improving knowledge as well as for designing and +implementing new technologies. We may hear about structured sound for a long time to come. + + +Journal of Optics (2022) #### +19. Transport of surface matter in structured water waves +Michael Shats +The Australian National University + +Status +Water surface waves share many similarities with their optical and acoustic counterparts, except for +a very different dispersion relation, 𝜔2 = 𝑔𝑘 + 𝜎𝑘3/𝜌, where 𝜔 and 𝑘 are the wave frequency and +wave number, 𝑔 is the gravity acceleration, 𝜎 is the surface tension coefficient, and 𝜌 is the density of +the liquid. The restoring force for the surface perturbation at long wavelengths (longer than about 20 +mm in water) is the gravity force and 𝜔~√𝑘. For shorter waves (less than 10 mm), the restoring force +is capillary and 𝜔~𝑘a/-. Though water waves have been studied for centuries, there is no universal +theory which would describe the motion of fluid particles even in relatively simple structured waves. +On the other hand, recent progress in laboratory studies of the particle motion on the water surface +perturbed by waves revealed rich phenomenology related to the generation of horizontal vortices and +vortex lattices [273], direct and reversed jets [274], and to developed two-dimensional turbulence +[275]. Some of these phenomena, e.g., vortex lattices, are related to the wave momentum and spin +[55], while others, for example, turbulent motion of fluid particles driven by steep nonlinear waves, +require new theoretical approaches. Horizontal motion of fluid particles at the surface is coupled to +the wave pattern and to the width of the wave spectra [277]. Chaotic fluid motion at the surface leads +to the increased disorder in the wave field as manifested in the broadening of the wave spectrum. +This suggests new theoretical approaches which would allow to predict rms velocities of the fluid +particles from the wave spectrum width and vice versa. It is interesting to note that, to generate 2D +turbulence at the liquid-air interface, it is not necessary to drive turbulence in the wave field; a slightly +broadened wave spectrum results in a broad spectrum of the horizontal fluid velocities matching +classical Kolmogorov-Kraichnan spectrum 𝐸𝑘 ∝ 𝑘−5/3. +The mass transport driven by surface water waves is well recognised in natural applications, for +example, in oceanology. However, there is also growing interest in controlled manipulation of +particles on a liquid surface for engineering applications, such as mixing, particle sorting and +clustering, as well as for controlling properties of tunable ‘metafluids’ [278]. Better understanding of +the wave-driven transport of particles opens opportunities for the development of new biomaterials +in liquid media by applying waves to the growing culture [279]. Waves can also be used to promote or +discourage the formation of biofilms on solid substrates [279]. + +Current and Future Challenges +Recent progress in experimental research advanced our understanding of the mass transport driven +by the surface waves. Experiments revealed a different nature of the particle motion in small- +amplitude, or weakly nonlinear waves, and in parametrically excited, strongly nonlinear waves, also +known as the Faraday waves. Traditionally, the mass transport by propagating waves was described +within the framework of the Stokes drift in 2D wave fields. Such drift along the y-axis (z-axis is in the +vertical direction) produces vertically polarised trochoids, or the motion with horizontal spin. +Recently, it was shown that, in 3D waves, fluid particles have both horizontal and vertical spin and +corresponding generalized Stokes drift [55]. An example of the particle trajectory is illustrated in Fig. 1. +Good agreement between theory and experiments gives hope that the field theory approach can be + +Journal of Optics (2022) #### +productive in developing theories and models capable of predicting the mass transport for the weakly +nonlinear waves. + + +The existence of the vertical and horizontal spin in 3D waves is also important for the motion of +larger inertial particles possessing internal spin. The motion of such particles is governed by the +interaction between the wave spin and particle spin, and it opens the opportunity to manipulate and +sort spinning particles using structured surface waves [280]. +The existence of the vertical spin in some wave configurations offers a new conceptual base for +the development of the surface wave spintronics [281], where the spin of passive particles can be +controlled by imposed surface waves [55]. The challenge here is to account for the return flows in +continuous medium as a reaction to the wave-driven drifts. +The situation is more complex for steep nonlinear Faraday waves. Fast motion of the surface fluid +particles differs qualitatively from the slow drift in linear waves; particle trajectories in such waves +have no resemblance with classical Stokes drift [275], as seen in Fig. 2. The development of theory +considering particle inertia is a challenging problem. The importance of the inertia of fluid parcels in +Faraday waves is manifested in the extended inertial interval in the spectra of horizontal fluid +velocities [275]. Though Faraday waves generate chaotic particle motion, the mass transport is +statistically predictable and allows fine control over particle dispersion at the surface [282]. The main +challenge is the development of theory based on the Lagrangian description of fluid motion. The +progress can be made through the development of theoretical models verified and fine-tuned in +experiments. +x +y +z +b +a +x +y +z +Figure 1. (a) Measured surface elevation produced by two orthogonal standing waves. (b) Measured 3D trajectory (red) of a surface +particle drifting within a unit cell (dashed line in (a)) and its projection on the horizontal plane (green). + + + +Journal of Optics (2022) #### + + + +Advances in Science and Technology to Meet Challenges +Recent demonstration of importance of the wave-generated spin of fluid parcels on the surface +perturbed by structured waves is an important step towards developing new applications relevant for +particle manipulation and sorting. These would require overcoming several problems related to (a) +incorporation into theoretical models of the transverse angular momentum–induced transport in +continuous media, or (b) balancing in experiments of the return flows caused by the spatially varying +Stokes drift. These problems are related to the configuration driven by two orthogonal propagating +waves [55], while they are not important in the field produced by two orthogonal standing waves, as +in Fig. 1(b), where the Stokes-drift flow closes on itself and thus remains stationary. In this example, +waves generate large-scale vortices which form a vortex lattice. +The generation of a vortex lattice is also a feature of the nonlinear Faraday waves [275]. However, +in that case, due to the larger fluid velocities, vortices strongly interact with each other causing 2D +turbulent motion. Experiments demonstrated that turbulence is dominated by the randomly moving +coherent bundles of particles, or meandering ‘rivers’, whose width is about half the Faraday +wavelength [282]. The existence of such ‘rivers’ is an important feature of the wave-driven turbulence. +In particular, the ‘rivers’ can be guided by solid boundaries within the flow which allows the +rectification of the turbulent (mean-zero) velocity fluctuations. It has been shown that this effect can +be used to create unidirectionally propagating floaters which tap turbulence energy (self-propelled +floating devices) [283]. Similarly, the turbulence-driven rotors powered by turbulence have been +demonstrated in laboratory experiments [284]. The latter ability can be used for efficient utilization +of the wave energy. This direction requires better theoretical basis and models capable of deriving +flow parameters relevant to engineering applications from the properties of the disordered but +structured wave fields. +Periodic and quasi-periodic wave-driven flows have a potential to be used in biological flows, such +as bacterial suspensions. It has been demonstrated that waves can shape the patterns of the bacterial +biofilms developing in the wave fields [55]. This should also be investigated in the wave-driven +turbulence. A potentially important theme is the formation of the structure of the bacterial cellulose. +This important biomaterial grows near the media-air interface, and it is affected by the surface waves. +This would require joint efforts by physicists and microbiologists to understand the formation of +extracellular polymeric matrices in moving fluid environment. + +Figure 2. Fluid particle motion in Faraday waves. Pink and blue wave fields correspond to two consecutive phase extrema of the waves +separated in time by a half-wave period. Green: three-dimensional particle trajectory followed for 100 Faraday wave periods. + + +(a) +4mmJournal of Optics (2022) #### +Concluding Remarks +The structured water wave field is an emerging tool to control mass transport at the gas-liquid +interface. Weakly nonlinear surface waves share many similarities with optical and acoustic waves +[55] and can drive deterministic transport of the surface matter. They promise new directions, such +as liquid interface spintronics based on the transverse angular momentum-induced transport, +resembling the spin Hall effect. Due to the modest wave steepness, the prediction of the mass +transport in such waves is applicable to oceanic waves, or they can be generated in various industrial +flows. +Steep nonlinear waves (Faraday waves) generate intense turbulent motion on the surface due to +the strong interaction between wave-driven horizontal eddies. Such turbulence shows statistical +properties of 2D turbulence since it is based on the inverse energy cascade, a process of the spectral +energy transfer from intermediate to large scales. In a bounded domain, such a transfer can lead to +the accumulation of spectral energy at the domain scale, a.k.a. spectral condensation, which is a form +of turbulence self-organization into a coherent vortex. Wall-guided self-organization of turbulence can +be used to rectify turbulent energy for the development of self-propelled surface vehicles and +unidirectional rotors for the wave energy conversion. + +Acknowledgements +This work was supported by the Australian Research Council Discovery Project DP190100406. + + + + +Journal of Optics (2022) #### +20. Structured electron waves +J. Verbeeck1 and P. Schattschneider2 +1EMAT, University of Antwerp +2TU Wien + +Status +Electron beams are used in a wide variety of applications ranging from vacuum tubes that formed the +basis of electronics more than half a century ago with specific high power and high frequency tubes +still being indispensable today, to electron microscopes providing atomic resolution images of +materials, and free electron lasers providing intense X-ray beams, radiotherapy and surface +treatment, e- beam lithography and chip inspection tools, displays, portable X-ray sources and many +more. +Most of these applications rely on a classical picture of a ray of accelerated electrons, providing a +current through vacuum. But as with light, the wave nature of electrons limits the spatial resolution +to the de Broglie wavelength reaching picometer levels for energies greater than a keV, at least five +orders of magnitude smaller than the wavelength of visible light. This unique property of electron +beams is the essence of its use in electron microscopy and e-beam lithography, providing spatial +resolution that reveals the atomic structure of materials on a routine basis. In order to approach the +ultimate resolution limit, wave aberrations induced by magnetic round lenses of electron microscopes +had to be corrected. The tremendous evolution of such correctors in the last two decades, based on +phase modifying magnetic multipoles, has generated ideas to apply phase control of the electron for +even more sophisticated applications. +Quantum mechanically, electrons are described by the Schrödinger/Dirac equation. The relevant +paraxial solutions are highly similar to solutions of the Helmholtz equation used to describe wave +phenomena in optics and acoustics. This indicates that all wave shaping that is investigated in these +areas (see e.g. Sections 7 and 18) can, at least in principle, be carried over to electron beams. +Wave shaping of electrons can be obtained by interaction with electric or magnetic fields provided +either by (macroscopic) sources, as realized in aberration correctors, by phase plates, or by the +microscopic electric or magnetic potential of matter (Fig. 1). +In this roadmap, we want to look beyond resolution revolution, exploiting the quantum nature of +the electron, in analogy to the revolution of adaptive light optics or phased arrays in +radiocommunication/radar and acoustics. + + + + + + + + + + + + + + +V, A +electron +source +lens +plane wave +structured wave +propagation direction +EM field +Figure 1. Sketch of the interaction of a paraxial electron beam with purposely designed scalar (V) and vector (A) electromagnetic +potentials creating a structured electron wave. + +Journal of Optics (2022) #### + +Current and Future Challenges +So far, electron vortex beams (EVB) have received the most attention with a wide range of +experiments showing ways to create these topological electron waves, carrying quantized orbital +angular momentum (OAM), similar to its optical counterpart. The charge of the electron results in a +magnetic moment mµB parallel to the propagation axis, with m a quantum number defining the OAM +and µB the Bohr magneton -—a unique property of charged matter waves. So far, EVBs have been +created, by and large, using forked amplitude gratings [285], magnetic monopole-like fields, and thin +refractive elements with spiral height profiles. These methods are static and absorb a significant +fraction of the beam intensity [286,226]. A phase plate in combination with cylinder lenses (magnetic +quadrupoles) conserves the intensity and can be tuned so as to achieve EVBs with + or – helicity. It +can as well be operated in reverse [287]. Tuneable electrostatic phaseplates have been demonstrated +as well [276]. Figure 2 shows some vortex beams and an overview of wave shaping methods. Besides +creating EVBs, significant progress has been made in EVB filters to decompose arbitrary electron +beams in its OAM components [349]. + + + + +A +C +D +Figure 2. A) a series of electron vortex beams carrying topological charge -2 to 2, with their wavefront structure. B) A series of holographic +gratings producing a variety of structured waves. C) Mode converter with cylindrical lenses. D) Spiral phase plate. E) 2x2 electrostatic +programmable phase plate for electrons. Reprinted from ref 2 (A, B, D), ref 3 (C), ref 7 (E) with permission from Elsevier. + + +1stcylinderlens +2nd cylinder lens +HG input beam +free space propagation +LG outputbeamx4 +x +OAM +current +(=1 +phasefronts +L +O +XAholograms +far-field images +((=3)+((=-3) +(6=3)+((=-3)Journal of Optics (2022) #### +EVBs can reveal chirality in crystals [288] or apply torque on nanoparticles and atoms with +predicted very high rotational speeds. Interaction with optical excitations in materials could reveal +local information on optically chiral nano-objects [289]. Structured electron waves carry information +equivalent to polarized light, but at far higher resolution [289], a fact upon which EMCD, the electron +counterpart of XMCD, is based [290]. +Besides EVBs, other non-diffracting beam classes have been studied, like Airy beams seemingly +following a parabolic trajectory in field-free space (like a curve ball), or Bessel beams that stay focused +over long distances, providing attractive prospects to study thicker samples [350-353]. +Challenges in arbitrary wavefront shaping relate to pushing experimental trials to technical +maturity, such as scaling up an array of electrostatic Einzel lenses [291], or using the ponderomotive +effect when electrons travel through regions of intense light, shifting the challenge towards making a +programmable light field with extreme intensities [292]. Other attempts use light fields in combination +with an electron transparent thin film, increasing the electron-photon coupling at the expense of +inserting material in the beam [293]. Yet other experiments have achieved EVBs by pulsed laser +illuminating of round apertures with circularly polarised light via interaction with surface plasmon +polaritons [294]. + +Concluding Remarks +The emerging ability to freely shape electron wavefronts in practical instruments opens interesting +avenues for future research and application. +One is the ability to encode quantum information in electron beams e. g. in a vortex basis. This +provides a quantum communication channel, which could open new avenues for quantum computing. +The short wavelength combined with strong interaction with electromagnetic fields and the +robustness against decoherence, the topological protection and ease of single particle detection +provides attractive features. +Another emerging opportunity is the ability to implement adaptive optics in electron beam +instruments, where the instrument can optimise its beam conditions in a feedback loop to avoid +tedious alignment procedures and to provide the highest possible contrast for specific features of +interest [354]. This is especially attractive for life science as contrast is notoriously poor, demanding a +high electron dose, often damaging the material before images can be obtained. Dynamic structuring +of the wavefronts could maximize the information per dose. +Other applications include imaging through thick objects, coded aperture approaches to directly +reveal the phase, dynamic phase scrambling to reduce effects of dynamic scattering in both imaging +and diffraction and many more. + +Acknowledgements +JV acknowledges funding from the eBEAM project supported by the European Union’s Horizon 2020 +research and innovation programme under grant agreement No 101017720 (FET-Proactive EBEAM), +FWO project G042820N 'Exploring adaptive optics in transmission electron microscopy' and European +Union's Horizon 2020 Research Infrastructure - Integrating Activities for Advanced Communities grant +agreement No 823717 – ESTEEM3. PS acknowledges the support of the Austrian Science Fund under +project nr. P29687-N36. + + + +Journal of Optics (2022) #### +21. Structured neutron and atomic waves +Dusan Sarenac, David G. Cory, and Dmitry Pushin +University of Waterloo + +Status +The successful extension of the structured waves toolbox to neutron and atomic beams promises an +array of exciting applications in fundamental physics and material characterization techniques. For +example, neutrons offer a complimentary probe of nature and materials when compared to photons +and electrons, as they possess unique penetrating capabilities and interaction strengths due to the +strong force and electroneutrality. However, given the wave-particle duality, the methods first +developed for generating and characterizing optical structured waves are typically the backbone of +the methods to create structured neutron and atomic waves [366]. Although the methods are +theoretically and conceptually analogous, the practical realizations are complicated by the technical +challenges associated with controlling and manipulating these de Broglie waves. For example, the +extension of holography techniques to neutron waves was accomplished via perfect-crystal silicon +interferometer which through Bragg diffraction provided a coherent superposition of an angled +reference beam and a beam that had passed through a macroscopic object [367]. This is a direct +adaptation of the two-beam wedge optics technique introduced by Leith and Upatnieks [368]. The +first experiments with neutron orbital angular momentum (OAM) focused on manipulating the OAM +of incoming neutrons, though given that the input beam had a small transverse coherence length +(ranging from ≈nm to µm) relative to the beam diameter (≈cm) the value of the OAM was not well +defined [369]. Several theoretical studies of incorporating spin correlations to OAM [370-372] led to +an experiment that prepared and characterized neutron lattices of spin coupled OAM beams [373]. +Shown in Fig. 1 is the experimental setup of Ref. [373] that relied on a sequences of magnetic field +gradients produced by spatially oriented triangular coils to achieve programmable spin topologies. + + + +In regards to inducing azimuthal phase shifts over the wavepacket coherence lengths, the first +experimental achievement of atomic and molecular beams carrying quantized OAM came in 2021 +[374]. The demonstration was done with helium atoms and metastable helium dimers. This work +Figure 1. The setup used in Ref [8] to prepare and characterize neutron lattices of spin coupled orbital angular momentum. +Sequences of triangular magnetic fields are used in conjunction with 3He cells. Here the magnetically polarized 3He cells act +as neutron spin polarizers due to their neutron absorption cross section being highly dependent on the neutron spin +direction relative to the helium polarization state. Shown are the simulated spin dependant intensity profiles at each stage of +the setup. Reprinted with permission from Ref. [373]. + +KJ,>2 +Phase Structure +KI,I>2 +of N-2 lattice +K1,>2 +Neutron +Camera +Slit +Analyzeralong ++zdirection +Neutron +Beam +PermalloyTube +LOVPrismPair +Polarizeralong +Guide +z direction +CoilJournal of Optics (2022) #### +opens new avenues in using the OAM to probe particle collisions between atoms and/or molecules. +Soon after, the first demonstration for quantized neutron OAM (depicted in Fig. 2) was achieved in +2022 [375]. The convenient integration of this method with material characterization studies at Small +Angle Neutron Scattering facilities promises to extend neutrons as probes of topological material’s +bulk properties [376,377], which cannot be directly probed via photons or electrons. + +Current and Future Challenges +The grand scientific challenges revolve around incorporating the additional degrees of freedom +brought forth by structured waves, such as OAM, into the existing scattering theory and material +characterization methods. Spin textures and spin topologies, such as skyrmions and merons, possess +a non-trivial coupling between spin and other dynamical degrees of freedom which manifest a rich +variety of emergent dynamics and phases of matter. While conventional probes provide indirect +transport measurements of topological excitations, neutrons with specific spin-orbit couplings are +strongly desired because they may act as direct probes of the target’s topology. + + + +The technical challenges that deter the progression of the field revolve around the difficulties in +preparing, manipulating, and detecting neutron and atomic beams. It is expected that as those +capabilities evolve, the field follows the progression of optical structured waves. First to note is that +there is no device equivalent to a laser which outputs coherent light. Whereas coherent and well- +defined Gaussian states are the common inputs to optical experiments with optical structured waves, +experiments with neutrons and atoms are limited to working with beams whose transverse coherence +lengths are much smaller than the size of the actual beam. Typical methods of beam collimation rely +on circular slit pairs to define the beam divergence and thus the transverse coherence. Here we can +also note that the low neutron flux ensures that only one neutron at a time is present in the entire +setup, and therefore all the experiments in essence are done with a post-selection on there being a +neutron. In relation to this, a notable challenge is the access and availability of high intensity neutron +sources. Likewise, it is worth mentioning that many other electromagnetic components which are +taken for granted in optics setups are also not yet practical. For example, the neutron index of +refraction for most materials is around n≈1-10-5, which makes the production of a neutron lens +impractical. Lastly the position sensitive neutron detectors possess much poorer spatial resolution +Figure 2. Grating arrays have been the enabler of experimental demonstrations of neutron, atom, and molecular helical +waves. a) The SEM images of the grating arrays used in the neutron demonstration. b) The conceptual illustration where +each grating of the array coherently acts on individual neutrons. c) Example of the observed intensity in the far field where +the OAM signature profiles are observed in the diffraction orders. Reprinted with permission from Ref. [375]. + +a) +b) +Arrayofforkdislocation +TopView +Incoming +phase-gratings with q=3 +MEN +neutron +ma-1 +2oum +OAM=-3 +0000 +TopView. +m=0 +OAM=0 +Incoming +m=1 +0AM=3 +neutron +m=-1 +OAM=-3 +SideView450tilted +00000 +m=0 +OAM=0 +Q.(A") +m=] +OAM=3Journal of Optics (2022) #### +when compared to optical cameras. For example, the OAM demonstration with neutrons of Ref. [375] +used a neutron camera with a pixel size of around 5mm by 5mm. + +Advances in Science and Technology to Meet Challenges +The holy grail technological advance that would push the field of neutron and atom structured waves +is a component analogous to a spatial light modulator (SLM). The SLM’s ability to provide arbitrary +wavefront shaping has revolutionized the field and enabled the widespread use of optical structured +waves. However, a practical active device that enables either phase or intensity modulation is +currently out of reach for neutron and atomic beams. For the time being, the main enablers have been +the advances in programming and control the full range of neutron and atomic degrees of freedom, +for example the nanofabrication methods of passive devices. It is interesting to note that both OAM +works of Ref. [374] with helium atoms and molecules as well as Ref. [375] with neutrons relied on +nanofabricated arrays of gratings. The former relied on absorption gratings while the latter on phase- +gratings. Due to the small transverse coherence length in both cases the gratings need to have +periodicities on the nanometre scale in order to induce coherent diffraction. The arrays in Ref. [375] +consisting of 6,250,000 individual fork dislocation phase gratings and they are depicted on Fig. 2. The +motivation for the large arrays was to increase the observed signal which otherwise would have been +too small to detect. Even with such an increase in signal every intensity image took around an hour of +acquisition time. In the case of absorption gratings, the thickness of the gratings needs to be sufficient +to spatially remove parts of the beam. Whereas in the case of phase-gratings, they need a high aspect +ratio to induce an appreciable phase-shift. Given the suitable material options, the available/possible +fabrication methods determine whether phase or intensity gratings are more practical. + +Concluding Remarks +The power of neutrons was most striking in the hallmark fundamental physics experiments such as +the first demonstration of gravity on a quantum particle and the observation of the 4π symmetry of +spinor rotation, and the impactful industry applications with the neutron imaging of fuel cells and +lithium batteries. This was enabled by the control of the three easily accessible degrees of freedom: +spin, path, and energy. The addition of the structured waves toolbox and the OAM degree of freedom +is expected to set forth the next generation of fundamental experiments [378] and material +characterization techniques [379,380]. The neutron’s penetrating abilities are well suited for bulk +studies of materials with spin textures and spin topologies such as skyrmions. + +Acknowledgements +The authors would like to thank their many collaborators including Wangchun Chen, Charles W. Clark, +Lisa DeBeer-Schmitt, Huseyin Ekinci, Melissa Henderson, Michael Huber, Connor Kapahi, Ivar +Taminiau, and Kirill Zhernenkov. The authors would also like to acknowledge their funding sources: +the Canadian Excellence Research Chairs (CERC) program, the Natural Sciences and Engineering +Research Council of Canada (NSERC), the Canada First Research Excellence Fund (CFREF). + + + +Journal of Optics (2022) #### +22. Structuring the Quantum State of Light +Michael Birk, Alexey Gorlach, and Ido Kaminer +Technion–Israel Institute of Technology + +Status +The shaping of various degrees of freedom of light has been a key factor in the progress of optical +sciences and engineering; many degrees of freedom have been brought under control. Spatial shaping +has been used to create nondiffracting beams such as Bessel [298] and Airy [295-297] beams, and to +imbue light with orbital angular momentum (OAM) [8], while frequency and temporal shaping lie at +the core of optical communications and ultrafast optical sciences. Advances that combine both spatial +and temporal shaping now push the boundaries of this field even further [299,300]. The richness of +capabilities for structuring light made clear that the more degrees of freedom of light are under +control, the more applications become possible. +However, it is pertinent to remember that all degrees of freedom addressed so far in the context +of structured light are related to classical light. From a quantum perspective, these degrees form only +a small subspace that the photonic state can inhabit. The quantum degrees of freedom can be +described as a function of generalized phase-space coordinates. This function is known as a Wigner +quasiprobability distribution, describing the quantum state of light for a single or multiple optical +modes. Our purpose in this roadmap is to propose the concept of structured quantum light. The +impact made by the field of structured light shows the prospects that can emerge from developing +capabilities for shaping the quantum properties of light. Below, we discuss some of the many open +challenges in quantum optics that remain to be resolved if one wishes to acquire control over the +quantum shape of light—structuring of the photonic Wigner function. +Current efforts and proposals in this direction can be divided into deterministic schemes [301- +303] or schemes based on post-selection [304,305]. Most of these efforts have been focused on +creating single- or few-photon quantum states. Even though few-photon quantum states have +applications in quantum technologies, the ability to create many-photon quantum light states has +seen a rising need for a wider range of applications, both classical and quantum. Such states are +needed for ghost imaging, precision measurements, quantum communication protocols, and photonic +quantum computation based on the so called “bosonic codes” or continuous-variables quantum +information [307]. A prime example is the quest toward the generation of the Gottesman–Kitaev– +Preskill (GKP) states [308], which are the much-needed resource for scalable fault-tolerant photonic +quantum computation [304, 307], and yet they have never been generated in the optical range. From +the perspective of basic science, robust control over the quantum light state can be used to +significantly enhance even the most fundamental nonlinear optical processes [301]. +The applications of structured quantum light will certainly be greatly expanded when tools for +many-photon Wigner function shaping will be developed—as demonstrated in the past by the +richness of discoveries brought forth by the invention and commercialization of spatial light +modulators (SLMs) for spatial shaping of light. + +Current and Future Challenges +The basic theoretical toolset for shaping the light Wigner function in the optical domain is provided +by the operations of rotations, displacement, squeezing, and amplitude dispersion, which can be +performed by phase delays, beam splitters, amplifiers, and nonlinear optical effects such as + +Journal of Optics (2022) #### +parametric down conversion and the Kerr nonlinearity. However, what is possible in theory turns out +to be difficult in practice. While it has been predicted that Kerr nonlinearity can generate Schrödinger +cat states [309], experimental efforts to generate macroscopic quantum states in free space have +been thwarted by low interaction strengths and dissipation. More recent attempts to overcome these +challenges use optical nonlinearities in microcavities, optomechanical cavities, and various integrated +photonic platforms. Such approaches have been successful in generating few-photon quantum states +but have not yet led to the generation of many-photon states with useful quantum properties. + + + +The most advanced demonstrations of many-photon squeezed light states to date are using four- +wave mixing and Kerr nonlinearities in fibres, on-chip nonlinearities in waveguides and cavities or +parametric down-conversion in free-space optical setups. Such experiments enabled manipulation of +photon statistics, leading to enhancement of nonlinear processes and generation of exotic indefinite- +mean photonic states [302]. These efforts are mostly limited to generation of a limited set of photonic +states (called Gaussian states), insufficient for quantum computation [307]. +Other methods of quantum state generation involve post-selection methods [306,381] rather +than relying on optical nonlinearities. These methods have a different bottleneck—the probability to +measure the desired outcome of a heralding observable, and the ultimate sensitivity of the detector. +Such schemes are inherently probabilistic, mandating either low generation rates (hence low +throughput), or the use of resource-intensive architectures such as parallelizing a large number of +generators to raise throughput [304]. +Classical properties of light +Quantum properties of light +spatial/ +angular +temporal/ +spectral +shaping +methods +Phase masks +Holograms +SLM +Fourier +transform +optical pulse +shaping +apps +statistics/ +entanglement +shaping +methods +Nonlinear +optical elements +Post-selection +GKP states +Phys. Rev. A 64, 012310 (2001) +Cat states +Phys. Rev. A 103, 013710 (2021) +Entangled photons +Rev. Mod. Phys. 71, S288 (1999) +Spatially shaped beams +J. Opt 19, 013001 (2017) +Space-time pulses +Nature Photonics 4, 103 (2010) +apps +BS +Wavefront shaping +Nature Photonics 6, 283 (2012) +SLM +incident +wave +scattering sample +transmitted intensity +Optical AWG +Nature Photonics 1, 463 (2007) +WDM switching +J. Light. Technol. 5, 904 (1999) +Figure 1. Structured light across all degrees of freedom. Current efforts focus on shaping the classical properties of light (top), including +spatial/angular/temporal/spectral/polarization degrees of freedom. We envision developments in shaping the quantum properties of +light (bottom), including the photon statistics and high-order coherences. While shaping classical properties relies on established +technology such as spatial light modulators (SLM), there are currently no standard methods for generic control of quantum properties, +but they are expected to rely on nonlinear optics and on post-selection using photon number resolving detectors (PNRD). Techniques for +structuring classical properties of light enabled a vast range of capabilities such as imaging and focusing in complex media, optical arbitrary +wave generation (AWG), and wavelength division multiplexing (WDM) for applications in optical communication. Spatial and temporal +shaping can be combined to gain full 3+1D control over the light field. However, the quantum nature of light contains many more degrees +of freedom, such as squeezing and photon entanglement. Finding ways to structure the quantum degrees of freedom could lead to +breakthroughs in many areas of quantum technologies. For example, many-photon superpositions of coherent states (cat states) and +Gottesman-Kitaev-Preskill (GKP) states are the sought-after building blocks for fault-tolerant optical quantum computation. + +....Journal of Optics (2022) #### +A conceptually different approach for the shaping and entanglement of light is utilizing the +interaction of light with free electrons [305, 310]. The key to this approach is the nonlinear nature of +electron-light interactions and the ability to pre-shape [311] the electron wavepacket before the +interaction. A recent experiment demonstrated the effect of quantum photon statistics on the +interaction [312], and recent predictions proposed novel methods for utilizing the interaction to +create the desired many-photon quantum states [305, 310], including a GKP state [313]. Such schemes +may also involve post-selection on the electron energy, since it becomes entangled with the photonic +state. The advantages of electron-based approaches are the use of mature techniques for electron +wavepacket shaping [311], and a post-selection process that avoids the dependence on photon +number resolving detectors, which are a bottleneck in conventional post-selection schemes. The +obstacle facing these approaches is the complexity of high-quality electron sources, which are +currently mostly studied in expensive state-of-the-art electron microscopes [312]. + +Advances in Science and Technology to Meet Challenges +Considering the described challenges, we highlight a few selected paths toward the full quantum +structuring of light: +1) The use of photon number resolving detectors is critical for the current post-selection schemes +enabling the creation of low-number quantum light states [304]. Efforts for development of better +detectors capable of resolving higher photon numbers will propel the frontiers of the field toward +the goal of generating many-photon quantum states of light. +Figure 2. Exemplary quantum states of light along with selected generation methods (inner circle) and examples of applications +(outer circle). The plots along the outer circle present example single-mode Wigner functions of the corresponding light states. The text +in the inner circle describes several known and theorized methods of generation of the quantum light states. The text in the outer circle +provides examples of current and potential applications of the states. The special case of classical light – the (Glauber) coherent state – +is nested in the green segment. All other segments contain quantum states that cannot be described classically, with example +applications in quantum metrology, computing, and communications. +Coherent states +Squeezed states +Cat states +GKP states +Fock states +Quantum +structuring +of light +Lasers +Parametric down +conversion +Four-wave mixing +Photon subtraction +Kerr nonlinearity +Atoms and +artificial atoms +Heralded +parametric down +conversion +Fock laser +Post-processing +on cat states +Post-selection on +squeezed states +Sub-shot-noise +metrology +Nature Photonics 7, +613 (2013) +Nonlinear yield +enhancement +Phys. Rev. Lett. 119, +223603 (2020) +Quantum +computing +Phys. Rev. A 87, 042315 +(2013); +Quantum +cryptography +Rev. Mod. Phys. 74, 145 +(2022) +Restoring +information and +error correction +Phys. Rev. Lett. 111, +120501 (2013) +Dual-rail qubit +Phys. Rev. Lett. 76, 4281 +(1996) +Fault-tolerant +photonic quantum +computing +Phys. Rev. A 101, 012316 +(2020); +Nature 584, 368 (2020) +WDM switching +J. Light. Technol. 5, 904 +(1999) +Optical AWG +Nature Photonics 1, +463 (2007) +Spatial/temporal shaping +Nature Photonics 4, 103 (2010) +Free-electron +lasers + +Journal of Optics (2022) #### +2) The ability to generate quantum light using optical nonlinearities depends on the ratio between +the nonlinearity coupling efficiency and losses. This ratio is being gradually improved using better +quality nonlinear microcavities, nonlinear photonic crystal fibres and integrated-photonic +waveguides [382]. An utterly different method is to utilize mechanisms of extreme nonlinear +optics, such as the ones behind high-harmonic generation, which may lead to strong emission of +quantum light, bypassing the limitations in efficiency and losses [383]. +3) Light-electron interaction schemes for quantum light shaping can become more practical by +miniaturizing high-quality electron sources and integrating them with existing methods for +shaping electron wavepackets [313]. Additional advances in nanophotonic-based electron–light +couplers are necessary to increase the intrinsic interaction strength, an important requirement +for generating many-photon states. + +Concluding Remarks +The development of generic methods for shaping the quantum state of light, akin to those available +for the spatial and temporal degrees of freedom of light, could spawn a great variety of applications +in quantum optics and the wider fields of quantum technologies. While such methods may not be +immediately in sight, the many advances in quantum optics over the past decade leave us optimistic +about the prospects for structured quantum light in the coming decade. + + + +Journal of Optics (2022) #### +23. High-dimensional quantum communication +Ebrahim Karimi +University of Ottawa + +Status +Electromagnetic (EM) waves are widely used in our global communication network to transmit +information through free-space, underwater, and fibre networks. EM fields—photons in the quantum +regime—are the main ‘resource’ for classical and quantum communication infrastructures because +they do not possess a net electric charge and rest mass. Generation, manipulation, transmission, and +detection of the EM field’s internal degrees of freedom (DOF) play curtail roles in both classical and +quantum communications. For instance, the classical communication bandwidth is directly +proportional to the dimension in which the information is encoded and logarithmically depends on +the detection signal-to-noise ratio (𝑆/𝑁), respectively—according to the Shannon–Hartley theorem, +the channel capacity is given by 𝑘 Log-(1 + 𝑆/𝑁), where 𝑘 is the communication bandwidth [314]. +This clearly indicates the importance of multiplexing in order to increase the communication +bandwidth. +EM fields possess several DOF, including polarisation, frequency, amplitude, phase, and +spatiotemporal modes, which can be used to share information. An EM field that is a coherent or +incoherent superposition of all of these DOFs is referred to as structured light (or structured photons +in the quantum regime) [96]. Polarisation is inherently bi-dimensional, and thus can only be used to +share one bit (‘0’ or ‘1’) of information. Meanwhile, frequency, amplitude, phase, and spatiotemporal +modes—although completely different in nature—are unbounded, and thus can be used to increase +information beyond ‘0’ and ‘1’. Frequency (wavelength), amplitude, and phase are currently used in +telecommunication multiplexing, allowing for the transmission of data at several terabit rates. +Moreover, during the past decades with the progress in the generation and detection of orthogonal +spatial modes, e.g., Laguerre-Gauss and Hermite-Gauss, the communication channel capacity has +been increased by a couple of orders of magnitude, allowing information to be transmitted much +faster using spatial mode multiplexing [315]. +The security of the current classical communication network is guaranteed by rigorous +mathematical algorithms, e.g., Rivest–Shamir–Adleman (RSA), which is mainly based on the difficulties +of finding prime factors of integer numbers using classical computers. However, algorithms—such as +Shor's algorithm—implemented on a quantum computer would help to break these classical +encryption techniques, and thus threatens our current classical encryption techniques. Quantum +communication, e.g., quantum cryptography, employing laws of quantum physics provides +approaches to monitoring a communication link and verifying the security threats. These methods are +based on two main laws of quantum physics: the superposition principle, wherein a quantum entity +can be in a superposition of two or more quantum states simultaneously; and the uncertainty +principle, in which the measurement of conjugate quantities with arbitrarily high precision is not +allowed [316]. The former resulted in a ‘no-go’ theorem, referred to as the no-cloning theorem, which +directly indicates that a quantum state cannot be copied perfectly without introducing noise to both +(multiple) copies [317]. Therefore, whenever a “quantum” message/key is shared, the attacker’s +(namely Eve’s) presence introduces an impurity (noise) to the message/key, and by monitoring the +noise, one can verify the security threat. + + +Journal of Optics (2022) #### +Current and Future Challenges +There have been significant advances in quantum communication, both at discrete and continuous +variable regimes since the seminal quantum key distribution (QKD) proposal of Bennet and Brassard. +Different QKD protocols based on a single photon, entangled photon pairs, attenuated coherent +beams, and squeezed light are developed with proper security analysis [316]. Without considering the +network architecture challenges, e.g., quantum repeaters and quantum memory, discrete and +continuous variables QKD have their own advantages, difficulties, and challenges. Discrete variable +QKD allows for long-range distance but limited key rates, while continuous variables have a limited +range but provide higher key rates. Employing high-dimensional encryption in discrete variable QKD +can potentially improve the key rate, and thus has received much attention over the last few decades, +e.g., 𝑑 = 2d-dimensional encryption provides 𝑛-bits of information per sifted photon [62,318]. It has +been shown that such high-dimensional encryption is more noise-tolerant and more sensitive to Eve’s +presence—see Figure 1. +For instance, the best quantum cloning machine introduces + +? +- − +? +Pe?, noise to the cloned wave +functions; the introduced error is 0.17 for qubit (𝑑 = 2-dimensional) encryption, but increases to 0.25 +for qutrit (𝑑 = 3-dimensional) encryption [319]. The classical methods using different DOF to perform +multiplexing to increase classical communications would provide qudit states in the quantum regime. +This includes complex photonics states with well-engineered spatial and temporal modes, +polarisation, and frequency. Photonic polarisation (polarisation qubit) and time-bin (time-bin qudit) +can be generated by means of electro-optic and/or optical devices at very high speeds (GHz to THz), +which made polarisation and time-bin QKD a promising venue for technological advances. However, +implementing both temporal and spatial modes in a practical QKD setup/network, due to the technical +difficulties in the fast generation and detection, has hitherto remained a venue to explore and +consider. Thus, developing novel linear and nonlinear approaches to generate, manipulate, and +determine spatiotemporal modes at a high-speed rate will be highly rewarding but remains a +challenging task for communication. + + + + +Another promising venue, not only applicable to quantum communication but for photonics +quantum computing, is to explore the generation of multi-photon high-dimensional entangled states +QBER +Secret Key Rate +AB6nicbVDLTgJBEOzF+IL9ehlIjHxRH +YJUY9ELx4xyiOBlcwOvTBhdnYzM2tCJ/gxYPGePWLvPk3DrAHBSvpFLV +ne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsG +G4EthOFNAoEtoLRzcxvPaHSPJYPZpygH9GB5CFn1FjpvLo9Yolt+zOQV +aJl5ESZKj3il/dfszSCKVhgmrd8dzE+BOqDGcCp4VuqjGhbEQH2LFU0gi1 +P5mfOiVnVumTMFa2pCFz9fEhEZaj6PAdkbUDPWyNxP/8zqpCa/8CZdJa +lCyxaIwFcTEZPY36XOFzIixJZQpbm8lbEgVZcamU7AheMsvr5JmpexdlK +t31VLtOosjDydwCufgwSXU4Bbq0AGA3iGV3hzhPivDsfi9ack80cwx8 +4nz+iZo1i +21 +AB6nicbVDLTgJBEOzF+IL9ehlIjHxRH +YJUY9ELx4xyiOBlcwOszBhdnYz02tCJ/gxYPGePWLvPk3DrAHBSvpFLV +ne6uIJHCoOt+O7m19Y3Nrfx2YWd3b/+geHjUNHGqGW+wWMa6HVDpVC8g +QIlbyea0yiQvBWMbmZ+64lrI2L1gOE+xEdKBEKRtFK95XHSq9YcsvuHG +SVeBkpQYZ6r/jV7csjbhCJqkxHc9N0J9QjYJPi10U8MTykZ0wDuWKhpx +40/mp07JmVX6JIy1LYVkrv6emNDImHEU2M6I4tAsezPxP6+TYnjlT4RKU +uSKLRaFqSQYk9nfpC80ZyjHlCmhb2VsCHVlKFNp2BD8JZfXiXNStm7KF +fvqXadRZHk7gFM7Bg0uowS3UoQEMBvAMr/DmSOfFeXc+Fq05J5s5hj9 +wPn8Ao+qNYw= +22 +AB6nicbVDLTgJBEOz1ifhCPXqZSEw8kV +0k6pHoxSNGeSwktlhFibMzm5mek0I4RO8eNAYr36RN/GAfagYCWdVKq6 +090VJFIYdN1vZ2V1bX1jM7eV397Z3dsvHBw2TJxqxuslrFuBdRwKRSvo +0DJW4nmNAokbwbDm6nfOLaiFg94CjhfkT7SoSCUbTSfnxvFsouiV3Br +JMvIwUIUOtW/jq9GKWRlwhk9SYtucm6I+pRsEkn+Q7qeEJZUPa521LFY24 +8cezUyfk1Co9EsbalkIyU39PjGlkzCgKbGdEcWAWvan4n9dOMbzyx0IlK +XLF5ovCVBKMyfRv0hOaM5QjSyjTwt5K2IBqytCmk7cheIsvL5NGueRdlC +p3lWL1OosjB8dwAmfgwSVU4RZqUAcGfXiGV3hzpPivDsf89YVJ5s5gj9 +wPn8ApW6NZA= +23 +AB6nicbVDLTgJBEOzF+IL9ehlIjHxRH +YJUY9ELx4xyiOBlcwOszBhdnYz02tCJ/gxYPGePWLvPk3DrAHBSvpFLV +ne6uIJHCoOt+O7m19Y3Nrfx2YWd3b/+geHjUNHGqGW+wWMa6HVDpVC8g +QIlbyea0yiQvBWMbmZ+64lrI2L1gOE+xEdKBEKRtFK95XHaq9YcsvuHG +SVeBkpQYZ6r/jV7csjbhCJqkxHc9N0J9QjYJPi10U8MTykZ0wDuWKhpx +40/mp07JmVX6JIy1LYVkrv6emNDImHEU2M6I4tAsezPxP6+TYnjlT4RKU +uSKLRaFqSQYk9nfpC80ZyjHlCmhb2VsCHVlKFNp2BD8JZfXiXNStm7KF +fvqXadRZHk7gFM7Bg0uowS3UoQEMBvAMr/DmSOfFeXc+Fq05J5s5hj9 +wPn8ApvKNZQ= +24 +AB6nicbVDLTgJBEOz1ifhCPXqZSEw8kV +2CjyPRi0eM8khgJbNDL0yYnd3MzJoQwid48aAxXv0ib/6NA+xBwUo6qVR1 +p7srSATXxnW/nZXVtfWNzdxWfntnd2+/cHDY0HGqGNZLGLVCqhGwSXWD +TcCW4lCGgUCm8HwZuo3n1BpHsHM0rQj2hf8pAzaqx0X3487xaKbsmdgS +wTLyNFyFDrFr46vZilEUrDBNW67bmJ8cdUGc4ETvKdVGNC2ZD2sW2pBFq +fzw7dUJOrdIjYaxsSUNm6u+JMY20HkWB7YyoGehFbyr+57VTE175Yy6T1 +KBk80VhKoiJyfRv0uMKmREjSyhT3N5K2IAqyoxNJ29D8BZfXiaNcsm7KF +XuKsXqdRZHDo7hBM7Ag0uowi3UoA4M+vAMr/DmCOfFeXc+5q0rTjZzBH/ +gfP4AqHaNZg= +25 +AB6nicbVDLTgJBEOz1ifhCPXqZSEw8kV +1C0CPRi0eM8khgJbNDL0yYnd3MzJoQwid48aAxXv0ib/6NA+xBwUo6qVR1 +p7srSATXxnW/nbX1jc2t7dxOfndv/+CwcHTc1HGqGDZYLGLVDqhGwSU2D +DcC24lCGgUCW8HoZua3nlBpHsHM07Qj+hA8pAzaqx0X36s9gpFt+TOQV +aJl5EiZKj3Cl/dfszSCKVhgmrd8dzE+BOqDGcCp/luqjGhbEQH2LFU0gi1 +P5mfOiXnVumTMFa2pCFz9fEhEZaj6PAdkbUDPWyNxP/8zqpCa/8CZdJa +lCyxaIwFcTEZPY36XOFzIixJZQpbm8lbEgVZcamk7cheMsvr5JmueRVS5 +W7SrF2ncWRg1M4gwvw4BJqcAt1aACDATzDK7w5wnlx3p2PReuak82cwB8 +4nz+p+o1n +26 +AB6nicbVDLTgJBEOz1ifhCPXqZSEw8kV +1CxCPRi0eM8khgJbNDL0yYnd3MzJoQwid48aAxXv0ib/6NA+xBwUo6qVR1 +p7srSATXxnW/nbX1jc2t7dxOfndv/+CwcHTc1HGqGDZYLGLVDqhGwSU2D +DcC24lCGgUCW8HoZua3nlBpHsHM07Qj+hA8pAzaqx0X36s9gpFt+TOQV +aJl5EiZKj3Cl/dfszSCKVhgmrd8dzE+BOqDGcCp/luqjGhbEQH2LFU0gi1 +P5mfOiXnVumTMFa2pCFz9fEhEZaj6PAdkbUDPWyNxP/8zqpCa/8CZdJa +lCyxaIwFcTEZPY36XOFzIixJZQpbm8lbEgVZcamk7cheMsvr5JmueRdli +p3lWLtOosjB6dwBhfgQRVqcAt1aACDATzDK7w5wnlx3p2PReuak82cwB8 +4nz+rfo1o +27 +AB6nicbVDLTgJBEOz1ifhCPXqZSEw8kV +1ClCPRi0eM8khgJbPDLEyYnd3M9JoQwid48aAxXv0ib/6NA+xBwUo6qVR1 +p7srSKQw6Lrfztr6xubWdm4nv7u3f3BYODpumjVjDdYLGPdDqjhUijeQ +IGStxPNaRI3gpGNzO/9cS1EbF6wHC/YgOlAgFo2il+/JjtVcouiV3Dr +JKvIwUIUO9V/jq9mOWRlwhk9SYjucm6E+oRsEkn+a7qeEJZSM64B1LFY24 +8SfzU6fk3Cp9EsbalkIyV39PTGhkzDgKbGdEcWiWvZn4n9dJMaz6E6GSF +Lli0VhKgnGZPY36QvNGcqxJZRpYW8lbEg1ZWjTydsQvOWXV0mzXPIuS5 +W7SrF2ncWRg1M4gwvw4ApqcAt1aACDATzDK7w50nlx3p2PReuak82cwB8 +4nz+tAo1p +28 +Figure 1. Secrete key rate as a function of Quantum Bit Error Rate (QBER) for high-dimensional BB84 protocols. When the QBER is zero, +each sifted photon transmits 𝑛 = log 𝑑 = 21 bits of information, where 𝑛 = {1, … ,8}. However, the key rate decreases exponentially with +the noise, and no positive key is shared when the secrete key rate is zero. The QBER for qubit (𝑑 = 2-), ququart (𝑑 = 2$) and quoct (𝑑 = +22) is about 11%, 19%, and 25%, respectively, and asymptotically reaches 50% for 𝑑 → +∞. This indicates that high-dimensional QKD is +robust against noise while providing higher key rates in the QKD protocols. + +LO +4 +2 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4Journal of Optics (2022) #### +[320]. In QKD, access to multi-photon high-dimensional entangled states grants certain protocols, e.g., +device-independent QKD, enhanced security of the communication channel. Furthermore, high- +dimensional quantum communication channels—fibre [321], free-space [59-60], underwater [61], +and satellite [322] links—either need to be developed or fully characterized. For instance, one may +need to extend ‘vortex’- or ‘twisted’-like fibres to support higher spatiotemporal quantum states +without introducing significant distortion or crosstalk among the modes or develop dynamic real-time +mode-distortion analysis in the current fibre network to recover the coherency among the transmitted +qudit states. Of course, transmitting the photonics system in more complex media, such as turbulent +and scattering channels or non-flat curved spacetime geometry [323], needs to be properly +investigated theoretically and experimentally. This channel analysis will help us to use the desired +quantum communication protocols, allowing one to handle certain noise thresholds, which without +knowing the channel process matrix the communication key rate would decrease significantly. + +Concluding Remarks +Engineering photonic quantum states provides access to high-dimensional vector spaces which can +enhance the communication key rate. In addition, in the quantum regime, employing qudit QKD allows +for the implementation of non-trivial and novel protocols that would enhance the security further or +are more noise tolerant. There are three main obstacles or studies, in general, to be considered for +the future of high-dimensional quantum communication: (1) linear or nonlinear methods to quickly +manipulate/generate single photon, multi-photon entangled states, or in general, non-classical light; +(2) explore different (QKD) protocols to optimise the key rate and noise tolerance after sifting as well +as optimising the communication physical range, and (3) develop new fibres or methods to use current +fibre infrastructure to support more spatiotemporal modes. Of course, the former includes different +communication channels, e.g., ground-to-satellite, submersible-to-satellite, and intra-satellites, +where the physics is less explored, and thus extensive experimental and theoretical studies must be +conducted. + +Acknowledgements +EK acknowledges the support of Canada Research Chairs, Ontario’s Early Research Award, and NRC- +uOttawa Joint Centre for Extreme Quantum Photonics (JCEP) via the High Throughput and Secure +Networks Challenge Program at the National Research Council of Canada. + + + +Journal of Optics (2022) #### +24. Simulating quantum systems with structured waves +Filippo Cardano and Lorenzo Marrucci +Università di Napoli Federico II + +Status +Large quantum systems cannot be generally simulated through classical computers, as the number of +required resources increases exponentially with the system size. As first suggested by Richard +Feynman, a quantum system could be simulated more efficiently by another controllable quantum +system. Nearly 50 years after this proposal, many quantum-simulating machines have been +developed, ranging from proof-of-principle devices to rather sophisticated hardware that can handle +tens (even hundreds) of qubits or bosonic modes and perform complex tasks. Besides the original +motivation to overcome the computational complexity of large quantum systems, these simulators +keep providing us with an extraordinary machinery to investigate quantum phenomena in controllable +and accessible architectures [324]. +Here we will briefly discuss how the accurate control of structured waves at the quantum level +has contributed to this field, focusing on open issues and challenges, and trying to identify key +technological steps that could boost the capabilities of current machines, or pave the way to novel +approaches. To this end, let us consider the simulation of a single particle whose wavefunction |𝜓⟩ is +defined in a discrete Hilbert space of dimension 𝑁, i.e. |𝜓⟩ = ∑ +𝑐S|𝜓S⟩ +t +Su? +, where |𝜓S⟩ form a complete +set of states and 𝑐S(𝑡) are complex time-dependent coefficients evolving according to an Hamiltonian +operator 𝐻. As an example, these states may correspond to the positions that an electron can take in +a crystalline lattice (see Fig. 1(a)). In the simulation, the basis kets |𝜓S⟩ can in turn be associated with +distinct wave states, for example localized at different spatial positions (e.g. photons propagating in +distinct waveguides within an integrated network). A viable alternative is provided by structured +waves, corresponding to modes (or superposition of modes) having envelopes that are spatially +overlapping, yet they feature a distinct internal mode structure that makes them orthogonal. Relevant +examples of the latter approach are optical or atomic modes carrying quantized values of orbital +angular momentum (OAM) or of transverse linear momentum, whose superposition leads to complex +spatial patterns due to multi-mode interference. +Our ability to tailor the evolution of such structured waves, e.g. for the simulation of tight-binding +Hamiltonians by coupling structured modes of atoms and photons via controlled momentum kicks +(see Fig. 1(b,c)), have enabled the direct observation of many physical phenomena. Remarkable +examples are optical analogues of the anomalous Quantum Hall effect [58], the formation of complex +patterns in interacting atomic condensates [325], or uncharted effects at the boundary between +quantum mechanics and general relativity [326]. Other relevant experiments will be mentioned in the +following, yet an exhaustive review of relevant literature is out of the scope of this article. For the +same reason, we will not discuss the case of temporally-structured waves, as for instance trains of +optical pulses recently used to achieve demonstrate a quantum advantage in a photonic simulation +[357]. + +Current and Future Challenges +Quantum simulators based on structured waves have not reached yet the level of complexity that +would undermine their classical simulation. A first issue is that most platforms effectively simulate +single-particle properties, often by using classical coherent fields. To increase the system complexity, + +Journal of Optics (2022) #### +and to observe genuine quantum features, it would be crucial to implement quantum states of many +indistinguishable particles. +If one critical issue is associated with the quantum nature of the structured wave, another +important aspect is related to the possibility of engineering complex Hamiltonians, with particular +attention to the case of interacting systems. Strongly correlated many-body states can be nowadays +realized in diverse simulators, thanks to the natural interactions that take place between particles in +the simulator, such as for instance trapped atoms or ions. These powerful setups are currently much +more expensive than photonic counterparts, yet the latter have only permitted simulation of mean- +field approximations of many-body systems thus far, thanks to light propagation through structured +nonlinear media [327]. + + + +As discussed above, the interest in these simulators does not only depend on the complexity of +their classical simulation, but it is also related to our ability to use them to study novel states of matter, +possibly designed artificially to address a specific task. While structured waves have already proved +key to the demonstration of a variety of topological effects, it would be crucial to remove barriers that +nowadays force us to implement only a certain class of Hamiltonians. First, these are often translation- +invariant, which prevents the simulation of finite systems with open boundaries, or random unitary +evolutions that are at the basis of sampling experiments. Next, the simulators have some degree of +tunability, but they are not completely reconfigurable, which is a crucial ingredient in view of the +realization of universal simulators. Finally, when simulating out-of-equilibrium dynamics or when +simply simulating a particle that explores large lattices, it is important to be able to control long +temporal evolutions, preserving the coherence of the quantum state against decoherence +Figure 1. (a) Typical honeycomb lattice reproducing the position of carbon atoms in a graphene layer. (b) Liquid-crystal g-plates are +arranged to give photons transverse momentum kicks to simulate 2D quantum walks [58]. In the focal plane of a lens light is distributed +in spots forming a square lattice. (c) Atoms in a Bose-Einstein condensate are given linear momentum kicks by exploiting suitable optical +pulses, according to the coupling scheme shown in the central part. The interference of these multiple waves results in a complex +pattern, as shown to the right [325]. Figures sources: (a) from https://en.wikipedia.org/wiki/Graphene (CC BY-SA 3.0 license), (b) from +Ref. [57], (c) from Ref. [325]. + +a +b +120 μm +N +k=k +B(t) +0um +20 +60 +100 +Density, n (um"*)Journal of Optics (2022) #### +mechanisms and avoiding losses in photonic experiments, or coherence decay between different +atomic states. + +Advances in Science and Technology to Meet Challenges +Generic simulators can be conceptually divided into three functional blocks: encoding, manipulating, +retrieving. +Encoding – Photonic simulations of complex quantum systems require multi-particle states made +of indistinguishable photons. Important progress has been achieved in recent years, with state-of-the- +art quantum-dot sources that can provide Fock states with ∼10 photons. While most implementations +have focused on single-particle physics, which keeps providing new exotic and uncharted properties, +we anticipate a growing use of multi-particle states. Recently, the use of squeezed quantum states +with many photons has attracted broad attention, e.g. with the first demonstration of quantum +supremacy in an optical processor, and promise exciting possibilities [328]. In general, setups based +on structured waves also require being able to manipulate simultaneously many modes associated +with diverse degrees of freedom (position/momentum, time/energy, polarization, atomic internal +structure, etc.) to increase the number of synthetic dimensions [329]. + + + +Manipulating – In the framework of non-interacting Hamiltonians, the possibility to explore novel +dynamics will rely on our ability to controllably mix spatial modes of structured waves [330] with +reconfigurable platforms (see Fig. 2(a)), e.g. as recently shown in Ref. [331] for simulating tunable +spin-orbit Hamiltonians. Control of the spatial structure of materials embedded in optical +microcavities (see Fig. 2(b)) is also enabling the simulation of topological Hamiltonians [332]. It will +also be crucial to reduce optical losses for the simulation of multi-particle systems and long temporal +Figure 2. (a) Multiple phase modulations allow one to mix suitably a set of modes carrying orbital angular momentum. (b) Perovskite +crystals in a planar microcavity are arranged so that polaritons dynamics in the sample obey 2D topological Hamiltonians, with a +carefully designed Berry curvature. (c) Simulation of a 2D topological insulator within a 1D array of optical waveguides. The second +spatial dimension is provided by internal spatial modes, shown on the right, that are coupled thanks to the bending of the waveguides. +Figure sources: (a) from Ref. [330], (b) from Ref. [332], (c) from Ref. [329]. + + +a +SLM-B +SLM-C +1 +P +M +IV +SLM-A +M +Detector +Source +b +C +Mode +Coupling +N +PbJournal of Optics (2022) #### +evolutions, for instance by implementing the entire evolution with a reduced number of suitably +engineered 3D devices, instead of a long sequence of 2D elements. The same concept could apply to +atomic systems, by replacing pulses implementing single momentum kicks with more complex long- +range excitations equivalent to a superposition of multiple kicks. When coming to the simulation of +interacting systems, new approaches to engineering interaction in synthetic atomic dimensions will +enable access to unexplored many-body physics [333]. On the optical side, it is well known that +photons cannot easily mimic interacting systems. In this sense, developments in the engineering of +photon-matter couplings will be crucial here, as recently shown in [334]. +Detection – In a simulator hosting multi-particle quantum states, retrieving at any time the +associated wavefunction would be an important feature. In photonic systems, it is often sufficient to +count photons that are simultaneously in different modes, yet this is not equivalent to retrieving the +whole quantum state. Challenges are both associated with the discrimination of multiple photons per +mode and with managing systems made of many modes. Ideally, it would be desirable to have +“quantum cameras” capable of discriminating the number of photons impinging on each pixel, +determining both their spatial position and arrival-time, to allow a full retrieval of correlation maps. + +Concluding Remarks +The search of novel approaches to structuring wave propagation and confinement in atomic, photonic +or hybrid systems will surely provide us with the possibility of simulating novel quantum states and +engineer complex Hamiltonians. We conclude our analysis by focusing on recent hybrid approaches +that mix localized modes (as, for instance, those confined in optical cavities) with the synthetic +dimensions provided by structured waves. As an example, a proof-of-principle simulation of a 2D +topological insulator has been recently performed in a 1D array of optical waveguides, by using +different guided modes within the same waveguide as an additional synthetic dimension [329] (see +Fig. 2(c), where the guided-mode internal structure is revealed). In this direction, all the techniques +that have been developed to control structured waves are now ready to be embedded in simulators +based on localized excitations, paving the way to experiments involving a much larger number of +modes and simulating quantum systems in high spatial dimensions. + +Acknowledgments +We thank Pietro Massignan and Alexandre Dauphin for useful discussions and suggestions. + + + +Journal of Optics (2022) #### +25. Artificial Intelligence for Structured Waves +Mario Krenn and Florian Marquardt +Max Planck Institute for the Science of Light + +Status +The additional resources provided by the complexity of structured waves offer enormous potential +for technological applications and new scientific research. However, this potential comes with a price. +The increased complexity is challenging for human intuition and classical algorithmic approaches +when dealing with the design or control of new systems or large-scale data analysis. The community +recently started developing and employing methods from the domain of machine learning, or more +broadly, artificial intelligence (AI), to overcome these new complications. These methods have seen +revolutionary progress over the last decade, solving practical real-world challenges. Here, we will offer +a big picture of the current applications to structured waves, followed by an outline of exciting +opportunities that might lie ahead of us. +AI for Data Analysis: The experimental classification of structured waves can be challenging. An +example is long-distance communication, where spatial structures are used to increase the +information density of light. Those structures are strongly influenced by atmospheric turbulence, +making it challenging to decode the information. Here, machine learning can help classify the modes +even though they are distorted, as shown in a 143-kilometre turbulent link [335]. The combination of +polarisation with spatial modes leads to highly complex vector vortex beams. Their classification with +classical methods is expensive and can be enhanced by neural networks to classify the complex +polarisation patterns with high quality [336]. Other works exploited neural networks to characterise +spectral structures of beams and identify spectral instabilities upon propagation through optical fibres +or improve the high-quality recognition of vortices in beams, for example by correcting aberration +introduced by turbulence [358]. +Rather than characterising the beams themselves, machine learning has also significantly +advanced the spatial-waves-based analysis of objects. For example, the field of deep microscopy aims +to advance the resolution of classical microscopy images solely by computation. A series of +experiments showed how neural networks could improve single-pixel cameras by autonomously +learning the ideal illumination patterns of the objects [337]. A very active field of research deals with +wavefront shaping and imaging using complex media. Deep learning has become a powerful tool to +replace classical, deterministic algorithms and directly learn from and act according to measured data +[338]. +AI for Design: Due to their complexity, the design of photonic or band-gap structures or quantum +experiments can become infeasible for human researchers. Therefore, various AI approaches have +been applied to this challenge in recent years. One example is experimental designs for high- +dimensional multi-particle entangled systems [339]. This has led to numerous new experimental +results involving the first observation of a multilevel GHZ state and the discovery of new entanglement +and interference principles. An advanced and highly active field of research deals with designing new +photonic nanostructures and metamaterials with AI. This field has recently been pioneered by +gradient-based and deep learning methodologies [340]. In the search for new topological phenomena, +AI systems have massively sped up the computation of the resulting properties directly from the +geometric structure of photonic crystals [341], which can be used for rapid exploration and + +Journal of Optics (2022) #### +optimization. In a similar spirit, neural networks helped to predict ultrafast nonlinear dynamics in +fibres [342]. + +Current and Future Challenges +For applying AI technologies in science in general, and for structured-wave research in particular, five +critical points need to be considered: +1. Data: Large, high-quality labelled datasets are necessary for many ML approaches (supervised +learning). Usually, even for simple problems, 100s or more examples are necessary. The dataset also +needs to avoid any biases as neural networks are extremely good at exploiting those weaknesses. For +example, the selection of the training examples should be random, from an appropriate distribution +(and it is not always clear what the right distribution is). Furthermore, the data needs to be labelled. +Hand-annotating 1000s of examples is infeasible in most situations. An alternative is to generate +simulated or experimental data where labels, for example, correspond to experimental outcomes. +2. Data from Simulations: Experiments often are too expensive for generating a lot of training +data; thus, simulators of the physical system are powerful. It is crucial that the simulators reflect the +physical situations well, as neural networks will learn the biases of the simulator. Another issue might +be that the simulator is very slow, making it difficult to generate a large amount of training data. +3. Data from Experiments: If experimental data can be collected for training data, biases need to +be avoided as well. For example, the environment or parameters of the setup might change over time, +which needs to be taken into account. +4. False positives and negatives: If neural networks are used in the experimental data analysis, +they might make wrong predictions. For example, in NN-enhanced microscopy, the machine could +predict an interesting effect in the image where there is none (“false positive”). This can be corrected +by human intervention. More dramatic are situations where the machine overlooks an interesting +effect in a microscopy image. This question is specifically important when the machine is used in a +system for which there is no training data (out-of-distribution). + + +Figure 1. Three different applications of AI in the realm of structured waves. A) For data analysation. Here, the illumination patterns of a 3d +single-pixel camera are constructed via deep neural networks and lead to significantly better results than classical methods. [337] B) For +Design questions. Here, periodic nanostructures with interesting topological band structures are designed using deep neural networks [341]. +C) For adaptive measurement. In the learning phase, the algorithm outputs a hypothesis of the state from randomly drawn measurements. +Afterwards, the goal is to predict the outcome of a randomly chosen measurement. The "probably approximately correct model" method +can speed up experimental quantum state estimation [343]. + + +A)Alfordataanalysation +C)Alforadaptivemeasurements +Acq.Time0.5s +1s +2s +4s +10s +20s +40s +60s +100s +Ei +Tr(Ep) +128×128 +DL +32×32 +DL +Hadamard +64x64 +Learningphase +Hadamard +128×128 +Hadamard +Tr(Ep) +B)Alfor Design +DE +Tr(Eo) +Geometry +Tight-binding model +Bandstructure +(and symmetries) +Predictionphase +(symmetryenhanced)Journal of Optics (2022) #### +Advances in Science and Technology to Meet Challenges +Advances in technology can solve several of these challenges. The requirement of large datasets can +be overcome by transfer learning. The system learns in simpler situations, and the pre-trained system +requires only a small number of examples to adapt to the more challenging situation. +One promising direction is the application of AI for choosing the optimal next measurement to +gain maximum information. Pioneering work on adaptive measurements for structured waves [343] +considered the quantum state of a complicated 2-particle vector-vortex beam. Conventionally, the +number of measurements scales exponentially with the qubit number for quantum state tomography. +However, the authors applied an adaptive measurement procedure that allowed a “probably +approximately correct” estimate of the quantum state in just a linear number of measurements. +In many situations, the decisions need to be fast (e.g. light through turbulence), which requires +additional considerations. The requirement for rapid neural networks (for instance, in active control +of experimental systems) can be achieved by hardwiring the neural network directly in FPGAs. One +exciting alternative is the use of optics and photonics itself as a speed-up for AI, either as a co- +processor or an end-to-end computing machine [344]. +Many novel developments in photonic technologies, such as new modalities of sensors, could +potentially benefit from the advanced capability of AI for data analysis. Similarly, we believe that a +large degree of freedom in the automated control of experimental components (such as variable +detection schemes or automated alignment via piezo-controlled mirrors) might enable fully AI- +powered alignment and control systems of experimental setups. Even the experiments’ design could +be aided by AI [339-342]. +Artificial intelligence is a culmination of very diverse, advanced computational algorithms that can +be applied in a wide variety of fields—exemplified by the concrete examples above. Those techniques +have been proven to be enormously powerful by computer scientists. The question is which new +scientific and technological problems can we address with AI? In some way, we have a mighty hammer +and need to identify suitable nails. + +Concluding Remarks +With the emergence of AI, we have a powerful tool at our disposal with the potential to advance open +questions in the study of complex structured waves. While numerous and diverse questions have +already been addressed with AI, we believe that a vast amount of intriguing applications are yet to be +explored. Finding new nails for this powerful hammer is a challenge that a community effort might +best address. A great opportunity to do so is the biannual community conference ICOAM. A half-day +online workshop with focused discussion groups on predefined topics could bring together domain +experts on the theory and experiments of structured waves and researchers with expertise and a +broad overview of AI technologies and their applications in science. AI approaches work well when +large amounts of data are available, where huge search spaces hinder systematic scanning or where +computationally very expensive functions are involved in the data analysis. Given the complexity of +structured waves, we strongly expect exciting applications of AI to emerge from such collaborations. + + + + +Journal of Optics (2022) #### +References +[1] +Nye J F and Berry M V 1974 Dislocations in wave trains Proc. R. Soc. Lond. 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' King’s College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Strand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' London WC2R 2LS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' United Kingdom 24ICFO – Institut de Ciencies Fotoniques,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The Barcelona Institute of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Castelldefels (Barcelona) 08860,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Spain 25Center for Phononics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' School of Physics Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Tongji University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Shanghai 200092,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' China 26Department of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' City College of the City University of New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 160 Convent Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' NY 10031,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' USA 27City University of New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Photonics Initiative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advanced Science Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' USA 28Université de Bordeaux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' LOMA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' UMR 5798,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' F-33400 Talence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' France 29Electron Microscopy for Materials Science (EMAT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' University of Antwerp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2020 Antwerp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Belgium 30Vienna University of Technology (TU Wien),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Vienna A-1040,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Austria 31University of Waterloo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Waterloo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' ON N2L3G1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Canada 32Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Technion–Israel Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Haifa 3200003,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Israel 33Dipartimento di Scienze Fisiche,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Universita di Napoli ‘Federico II’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Complesso di Monte S Angelo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 80126 Napoli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Italy 34Max Planck Institute for the Science of Light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Erlangen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Germany Abstract Structured waves are ubiquitous for all areas of wave physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' both classical and quantum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' where the wavefields are inhomogeneous and cannot be approximated by a single plane wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Even the interference of two plane waves, or a single inhomogeneous (evanescent) wave, provides a number of nontrivial phenomena and additional functionalities as compared to a single plane wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Complex wavefields with inhomogeneities in the amplitude, phase, and polarization, including topological structures and singularities, underpin modern nanooptics and photonics, yet they are equally important, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', for quantum matter waves, acoustics, water waves, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structured waves are crucial in optical and electron microscopy, wave propagation and scattering, imaging, communications, quantum optics, topological and non-Hermitian wave systems, quantum condensed-matter systems, optomechanics, plasmonics and metamaterials, optical and acoustic manipulation, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This Roadmap is written collectively by prominent researchers and aims to survey the role of structured waves in various areas of wave physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Providing background, current research, and anticipating future developments, it will be of interest to a wide cross-disciplinary audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Introduction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Phase matters (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Padgett) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The topology of 3D polarization (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Alonso and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Dennis) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Shaping light (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Dudley and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Forbes) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Spatiotemporal optical vortices and OAM (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Zahedpour, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Hancock, and H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Milchberg) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structured waves in Non-Hermitian systems (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rotter, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Nori, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Ozdemir) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Tailoring random light for imaging applications (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Bender and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Cao) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Ultrafast structured beams and intense magnetic fields (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Corkum and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Hernández-García) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Metaphotonics with structured Light (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Ren and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Kivshar) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structuring light with near-zero-index platforms (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Silveirinha and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Engheta) Journal of Optics (2022) #### 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Strong coupling between atoms and guided light (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rauschenbeutel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Schneeweiss, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Volz) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Surface waves (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Leykam and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Smirnova) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Photonic spin-orbit interactions at metasurfaces: stochastic, Rashba, and quantum effects (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rong, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Wang, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Hasman) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Spin, momenta, and forces in evanescent waves - towards spatial and temporal structuring (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Picardi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Zayats, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rodríguez-Fortuño) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Momentum and spin of electromagnetic, sound, and water waves (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Bliokh) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acoustic spin (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Yang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Ren) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acoustic pseudospins for wave control and topological protection (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Khanikaev and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Alù) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Mechanical effects of structured sound waves (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Brasselet) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Transport of surface matter in structured water waves (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Shats) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structured electron waves (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Verbeeck and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Schattschneider) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structured neutron and atomic waves (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Sarenac, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Cory, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Pushin) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structuring the quantum state of light (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Birk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Gorlach, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Kaminer) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' High-dimensional quantum communication (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Karimi) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Simulating quantum systems with structured waves (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Cardano and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Marrucci) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Artificial intelligence for structured waves (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Krenn and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Marquardt) Journal of Optics (2022) #### 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Introduction Konstantin Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Bliokh1 and Ebrahim Karimi2 1RIKEN 2University of Ottawa As it often happens with phenomena and discoveries in science, it is difficult to indicate a single starting point for the study of structured waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' They have been in front of our eyes from the very beginning of history: as waves on the sea surface, scattered sunlight and rainbows in the sky, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In modern optics, seminal works by Nye and Berry [1–3], Baranova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [4], Soskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [5, 6], and Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [7, 8] stimulated the development of the fields of “singular optics” and “optical angular momentum”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The development of these closely related areas was surveyed five years ago in the “Roadmap on structured light” [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, phase singularities (analysed by Nye and Berry for ultrasonic pulses) previously appeared in the context of quantum matter waves in pioneering papers by Fock [10], Dirac [11], Aharonov and Bohm [12], and Hirschfelder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Furthermore, such singularities can be found in the 19th century maps of tidal ocean waves [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Also, the orbital angular momentum in localized wave states with vortices (described by Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' for optical beams) appear in the form of quantum electron states in atoms or in an external magnetic field as described in textbooks on quantum mechanics [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This evidences that structured waves and their main properties are universal phenomena across various wave systems, independently of their nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In this roadmap, we aim to review recent achievements related to structured waves in optics, plasmonics, metamaterials, acoustics, electron and neutron optics, and quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' We tried to avoid overlaps with the earlier “Roadmap on structured light” [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Therefore, the main focus of this roadmap is shifted towards emerging directions which have been rapidly developing in the past five years and to phenomena involving structured waves in non-optical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The areas addressed in this roadmap include: non-Hermitian and topological wave systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' plasmonics, metasurfaces, and near-zero index materials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' quantum information and artificial intelligence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' light-matter interactions and waves in random media;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' electromagnetic, electron, neutron, acoustic and water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' One can notice the rapidly growing interest in such directions as: temporal structures, including time- dependent media [17], space-time wavepackets [18], and spatiotemporal vortices [19–24];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' novel functionalities involving complex waves in non-Hermitian [384-386,124], topological [122,387], and random [338,388] media;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 3D topological polarization structures including polarization Mobius strips [25–29], skyrmions [94,29–35,389], and structured polychromatic fields with commensurable frequencies [36–40];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' spin and polarization properties of sound [41–47], elastic [48–53], and water waves [54, 55];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' structured neutron and atomic waves [369,375,378,374];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' simulating complex quantum systems [56–58], high-dimensional communication [59–61], quantum cryptography [62–63], and sensing [64–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Unfortunately, not all invited authors were able to contribute to this project, and therefore, some areas were not covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The most apparent omission is the absence of sections on structured quantum condensed-matter waves, including BEC, cold atoms, exciton-polaritons, and various quasiparticles in solids [68–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This topic deserves a special roadmap project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This roadmap provides background, state-of-the-art, and perspectives for the interdisciplinary physics of structured waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' We hope that it will illuminate the universality of structured-wave phenomena across various areas of physics, highlight emergent directions involving structured waves, and thus stimulate further development of this exciting field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Phase structure matters Miles J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Padgett University of Glasgow Status In a traditional sense, the importance of the spatial phase structure of light beams is inherent in any image projection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, our eyes are insensitive to phase, perceiving only intensity and hence for most user cases this image projection requires only the shaping of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This shaping is easily accomplished by transmission through an appropriate transparency, or with modern technology in the form of a digital micromirror device normally associated with the projection of our slides in scientific talks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In contrast to this intensity structuring, much of the current research in shaped light is for those situations where the properties or application of the light arise from the phase structuring of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' One well known example of where it is the spatial phase structure of light that is the defining feature is holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In holography, the light scattered from the object is recorded by interfering this light with a spatially coherent, plane-wave, reference beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The constructive and destructive interference with the reference captures not only the intensity but the phase of the scattered light too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Traditionally, this interference pattern was captured using high resolution photographic film which, once developed, could be illuminated by the same reference light to create the original light beam as scattered by the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This recreation of the scattered light creates a visual replica of the object itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It is interesting to reflect on the fact that despite the advances in the digital replacement of film, these digital devices still lack the pixel count required to fully implement a realistic holographic projector that can recreate the full 3D image of an everyday object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, despite this technical limitation, the modern advent of digital technologies acting as diffractive optical elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', digital holography, has led to a multitude of related, albeit simpler, applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Beyond holography, the present-day interest in the phase structuring of light beam probably dates to 1992, when the seminal paper published by Allen and co-workers reasoned that a light beam with a helical phase structure, such as Laguerre-Gaussian laser modes, carried an “orbital angular momentum” that was independent of, and additional to, light’s spin angular momentum [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This postulate was rapidly experimentally verified by Rubinsztein-Dunlop and co-workers [76] along with several other groups, transferring this angular momentum to microscopic particles held in optical tweezers, causing the particles to spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This early work was based on photographic film acting as holograms, which when illuminated with a Gaussian reference beam, produced a diffracted beam with the required helical phase [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While still focused on optical tweezers, Grier and co-workers replaced the film with an interactive, pixelated liquid-crystal phase modulator for the holographic generation of helically-phased and other beams [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The pioneering of these spatial light modulators and the ease with which complex beams could be generated spawned work far beyond optical tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the twenty years since [78], these spatial light modulators have driven an explosion in the study and application of non-Gaussian laser beams and their use and application in areas ranging from the study of optical phenomenology, imaging and sensing to quantum science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges From a technical perspective, there are two key spatial light modulator technologies in widespread use for the generation of complex phase and intensity structured beams: those based on liquid crystal Journal of Optics (2022) #### and those based on digital micromirrors, both of which have video resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The liquid crystal devices have phase-only modulation and, when used as a diffractive optical element, can reach well over 50% conversion efficiency, albeit only at a video-frame rate switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The micromirror devices are intrinsically intensity modulators but can be used to create amplitude gratings and hence elements with a low diffraction efficiency (< 5%), but at >10 kHz frame rate [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A clear challenge for the development of new technologies is to simultaneously increase the diffraction efficiency while maintaining or increasing further the frame rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The combination of high efficiency and high speed create new opportunities in real-time aberration correction and quantum information processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' From a science perspective, most work to date on phase structured beams has focused on specific beam types ranging from beams described by various polynomials—Laguerre, Bessel, Gegenbauer, Airy—which often form complete orthonormal sets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', a set of beams from which any other beam can be synthesized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Typically this work has followed a logic of “what can beams of type X be used for?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', whereas an alternative logic is “what design of beams will have the optimum performance in application Y?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' One example of these two logical approaches lies in single-pixel imaging systems wherein a sequence of patterns is used to illuminate an object, and the backscattered light for each projected pattern is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Summing the patterns, each weighted by the corresponding backscattered single, reveals an image of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The majority of the work uses Hadamard or more sophisticated patterns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [77]), but as an alternative, machine learning and related techniques can be used to define a bespoke pattern set to enable a compressed sensing approach to emphasise particular image properties or distinguish between specific object types [80], see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rather than applying machine learning techniques to the optimum beam design, these machine learning techniques can be applied to design a sequence of diffractive elements, creating an optical implementation of a neural network [81], or the lossless transformation of one complicated modal set into another simpler set [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' All these transformations rely upon the phase structure of the beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As pioneered by Vellekoop and Mosk, another aspect of general complex beam design is in the creation of light beams that deal with complex aberrations [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' By precise shaping of the incident light beam, it is possible to create a focused spot after transmission through highly scattering or complex media such as a multimode optical fibre [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' If the medium is characterised in terms of a transmission matrix, then the inversion of this matrix defines the input beam required to produce any spot at the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' If this spot is then raster scanned, the backscattered or similar signal reveals an image from within or behind the media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Within single-pixel imaging, it is possible to use deep learning to define an optimum measurement set based upon a library of typical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This approach to compressed sensing improves the quality of an image for a fixed number of under-sampled measurements [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Previoustechnique:evolutionary Deep-learningtechnique:deep- Hadamardscanusing666patterns learnedbasisof666patternsand reconstructionJournal of Optics (2022) #### Advances in Science and Technology to Meet Challenges As discussed above, there are clear technical requirements and associated challenges for high-speed spatial light modulators with high diffraction efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Overdriving liquid crystal devices can perhaps bring a speed which approaches that of a digital micromirror device, but progressing beyond a few tens of kilohertz would seem to require a new, as yet unrealised, solid-state technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' If such technology is to be developed it will possibly arise from image-based communication where the high- dimensionality of the image state would bring a similar increase in communication bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' When it comes to the beams themselves, rather than those beams forming well-known complete sets, future applications are likely to be based upon arbitrary beams optimised to a particular function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Taking the example of aberration correction based upon the inversion of a transmission matrix, the scale of the computation problem is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A system, or medium, of N modes is described by a N x N matrix, the measurement time of which alone makes any real-time adaptation unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A route to solving these problems is possibly to identify a subspace so that the modified matrix can be inferred from a relatively small set of additional measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The key is perhaps not to apply machine learning to the recovery of the image data but to apply the compressed sensing to the measurement of the transmission matrix itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Whether a meaningful subspace exists will most likely depend on the system type—a multi-mode fibre would seem a good place to start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks There are perhaps two main challenge and opportunity types ahead, both technical and conceptual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A key technical challenge is for developing spatial light modulators with improved diffraction efficiency and/or faster switching and/or much higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The last of these will undoubtedly be driven by the consumer display market which might potentially enable full holographic projection of images, whereas the first two are more likely to be addressed by specialist development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A key conceptual challenge is to move away from specific beam types and their corresponding diffractive elements to use machine learning and related techniques to create bespoke beams optimised for a particular need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, the correction of the aberrations associated with complex media, creating the optimum set of measurement patterns in imaging, or the creation of arbitrary diffractive elements in the optical implementations of neural networks and optical mode transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This will be a fruitful ground for the collaboration of optical engineers and computer scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements This work is funded by the Royal Society and EPSRC under the grant number EP/M01326X/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The topology of 3D polarisation Miguel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Alonso1,2 and Mark R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Dennis3 1Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel 2University of Rochester 3University of Birmingham Status Among the earliest applications of polarised light, going back to Brewster and Talbot in the 1800s, was to explore structures of materials through crossed polarisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Polarised light microscopy reveals the optic axes of crystalline minerals, and Schlieren patterns reveal textures in liquid crystals, organised by their topological defects [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the modern study of structured light, where the state of polarisation varies with position, light itself shows complex topological textures and structure [25, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The theory of polarisation for collimated light has a well-established formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' When light travels in a definite direction, only the two components of the electric field vector perpendicular to the propagation direction are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For monochromatic light, the electric field vector at each point traces an ellipse, whose eccentricity, handedness, and orientation constitute what we call polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The overall size (amplitude) and instantaneous position on the ellipse (phase) are usually disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The Poincaré sphere (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(b)) is an elegant, abstract geometric representation, where each polarised state corresponds to a point on the surface of a unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This point’s latitude and longitude encode, respectively, the polarisation ellipse’s orientation and ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Its Cartesian coordinates on the sphere, on the other hand, correspond to the Stokes parameters normalised by the total intensity, which are easily measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' � � � � Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) 3D transverse polarisation texture with polarisation ellipses in one transverse plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A C-line (white curve) of circular polarisation passes through this plane several times, as does the L-surface on which the polarisation is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Poincaré sphere of transverse polarisations, indicating 2D polarisation states using a Runge colour scheme: polarisation azimuth by hue, and spin by brightness, from black (left-handed circular) to white (right-handed circular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (c) Representation of mixed polarisation state where the E vector traces a path in 3D (blue curve) that is not a simple ellipse: statistical E field described by the moment of inertia ellipsoid, and average spin vector which must lie within the dual ellipsoid [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (d) Majorana sphere representation of polarisation as two points on the sphere, projecting directly onto the polarisation ellipse as circles, or as the foci of the ellipse circumscribed by the sphere [90,91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### The Poincaré sphere construction is more than just a map of two physical parameters onto a surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Amongst its physical properties, it represents: • the action of polarising and transforming optical elements as geodesic projections and rotations over the sphere through the use of Jones calculus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' • the geometric (Pancharatnam-Berry) phase due to cyclic changes in polarisation as half the solid angle enclosed on the sphere by the path of polarisation transformations [87,88];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' • partially polarised fields as points inside the sphere, with the radial coordinate giving the degree of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The standard polarisation formalism and the Poincaré sphere are not appropriate when light is nonparaxial––i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', the plane of the ellipse varies––and the normal to the polarisation ellipse is not tied to the propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Various generalisations have been proposed which extend to the richer structure of nonparaxial polarisation, albeit without the unifying simplicity of the Poincaré sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Even for polarisation at a single point, the geometry and topology of the description for the nonparaxial case are quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges The topology textures in position-dependent polarisation fields strongly mirror topological textures in liquid crystals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' both can be described by headless vectors (directors), representing the major axes of the polarisation ellipses or the axes of the rod-like liquid crystal molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Topological singularities organise these patterns within a volume: C-lines of circular polarisation around which the director rotates by ±180°, explored by Nye in the 1980s in Bristol where Frank had discovered disclinations— the liquid crystal counterpart—30 years earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These singular filaments occur in transverse and non- transverse polarisation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Their 3D structure in a volume can be controlled by holographic manipulation, most often by liquid-crystal-based computer-controlled holograms (spatial light modulators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This has allowed for the production and polarimetric measurement of beams with knotted C-lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Determining the 3D state of polarisation directly, including the longitudinal component, is nontrivial even in free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Nowadays, the most common approach samples the field by scanning with a nano-scatterer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' this has allowed for the measurement of the fine structure around a C-line, revealing Mobius band-like configurations [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In addition to eccentricity and orientation in its plane, a nonparaxial polarisation ellipse has a varying plane orientation (perpendicular to the direction of spin), requiring four parameters in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Based on Majorana’s spin formalism, Penrose proposed a representation in terms of not one but two indistinguishable points over the surface of a unit sphere in 3D, with a simple geometric meaning: they indicate the two directions for which the ellipse traced by the electric field projects onto a right- handed circle (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(d)) [90,91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Mathematically, this corresponds to the symmetric product of two 2- spheres, a topological representation of the complex projective plane CP2, corresponding to complex 3-vectors and disregarding an overall complex factor (corresponding to intensity and phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This representation allows for the optical geometric phase to be calculated when the plane of polarisation varies, corresponding to a quantum spin-one geometric phase, as demonstrated, for instance, in the Tomita-Chiao experiment for light in a twisted fibre [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For partially polarised fields—that is, for fields that are polychromatic but whose frequency components are uncorrelated—the electric field at a point does not trace an ellipse but a more Journal of Optics (2022) #### complex, generally nonplanar and nonperiodic path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A generalisation of the Stokes parameters has been proposed based on an expansion of the polarisation matrix using the 3x3 Gell-Mann matrices, instead of the 2x2 Pauli matrices that yield the standard Stokes parameters [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This description encodes polarisation as eight parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', the number of real quantities needed to specify a trace- normalized 3x3 Hermitian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These parameters define an eight-dimensional hypervolume, where—like for the Poincaré sphere—complete polarisation corresponds to points at unit distance from the origin, and the remaining points correspond to partial polarisation states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, without a direct generalisation of optical elements or the Jones calculus to 3D, the power of this high- dimensional description is yet to be realised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the special case when the polychromatic field is coherent and includes only a small discrete set of mutually rational frequencies—generated, for example, by nonlinear harmonic generation—the electric field at each point traces a periodic path that is not an ellipse but a Lissajous curve [36] that can be knotted in 3D [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recent experiments have revealed topological features in monochromatic polarisation distributions not only along curves (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Möbius strips) but over surfaces or volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, Skyrmions are distributions that fully wrap around a parameter space corresponding to an n- dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' “Baby Skyrmions” for n=2 have been realised in different ways: • as full Poincaré beams, whose polarisations at any transverse plane cover the Poincaré sphere’s surface [94] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2(b));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' • where, in the evanescent field over a planar metal surface, the linearly-polarised electric field vector points in all 3D directions [30] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2(a));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' • where ellipses in a plane have spins (ellipse normal) in every 3D direction [31,83] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A richer example of a 3D topological state (n=3) uses the fact that a normalised, transverse polarisation state, including phase (Jones vector), corresponds to a point on a unit hypersphere in 4-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These 3-spheres admit the Hopf fibration into interlinking circles (on which the phase varies) � � � � Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Different representations of Skrymion-like configurations in optical polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) E-field points in all directions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) full Poincaré beam (realising all points on Poincaré sphere);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (c) circular polarised realising all directions of spin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (d) Skyrmionic Hopfion in 3D, realising all polarisations and phases (on optical hypersphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In all cases, the colour scheme is provided by the Runge sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' O 0 OJournal of Optics (2022) #### parametrised by the Poincaré sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such a structure—a Skyrmionic Hopfion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2(d))—realising all polarisations and phases in an entwined texture was recently designed, synthesised experimentally, and measured [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The topological complexity of these polarisation distributions increases when more degrees of freedom are considered, as is the case of polychromatic light [36,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks Light polarisation at a point presents a simple yet elegant topological structure, which becomes more complex when the paraxial limit is abandoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The topological features are much richer when considering the spatial variation of polarisation, echoing the structured topological states studied in many forms of condensed matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Because light can interact with matter, the structure of each can shape that of the other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' most prominently, light fields can be used to interrogate and shape liquid crystals, and conversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Modern microscopy techniques use light with varying polarisation to probe the configurational 3D structure of biological matter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' actin filaments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 3D topological polarisation structures affect the orbital and spin dynamics of trapped particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Topology and polarisation have played a central role in the rapid development of the science of structured light over the past few decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, the technological potential for many applications is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements MAA acknowledges funding from the Excellence Initiative of Aix Marseille University – A∗MIDEX, a French ‘Investissements d’Avenir’ programme, and from the Agence Nationale de Recherche (ANR) through project ANR-21-CE24-0014-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' MRD acknowledges support from the EPSRC Centre for Doctoral Training in Topological Design (EP/S02297X/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Shaping Light Angela Dudley and Andrew Forbes University of the Witwatersrand Status Light can be shaped in all of its degrees of freedom (DoFs), in time and space, for so-called Structured Light [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Traditionally, the spatial DoFs have been exploited—for example, amplitude, phase and polarisation—first in 2D (the transverse plane) and later in 3D (all three components of the electric field), while the time and frequency domains offer the potential for 4D control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recently, there has been a concerted effort to identify and control new DoFs, and to harness this control for emerging applications, including communications, microscopy, imaging, optical trapping and tweezing, quantum state engineering, and laser machining processes, to name but a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The challenge is to identify which DoFs can be controlled, to what extent, and with what toolkit [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For instance, azimuthal phase control gives rise to orbital angular momentum (OAM) modes, which are easily controlled both as scalar and vectorial superpositions (whose phase and polarization profiles are non- separable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In this sense, OAM can be treated as an easily controllable DoF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In contrast, amplitude and phase shaping to create radially structured modes (the p indexed modes in the Laguerre-Gaussian basis, for example) is common-place [82], but this DoF is not easily controlled with our existing toolkit, limiting its applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Path has long been associated with quantum states of light, but less so in classical light, while ray-wave duality in classical fields (see Figure 4) is yet to be fully exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Given the many DoFs of light, how can we realise and control them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Optical cycles are much too fast to allow direct temporal light shaping, and so such light shaping is done spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, to shape light temporally, the frequency components are usually path separated by a dispersive element, mapping frequency to space, subsequently shaped in amplitude and phase before a return mapping to reconstruct the desired temporal pulse [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the spatial domain, we may control the amplitude and phase of each polarisation component, the latter by propagation and geometric phase, both of which can be made polarisation specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The ubiquitous lens is a simple example of a beam-shaping optic, adjusting the propagation phase by material thickness and refractive index to shape light by refraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recent developments in free-form optics has given new impetuses to refractive shaping of light, with unprecedented control possible [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the early 1990s, there was an explosion of activity in diffractive optical elements (DOEs), where a computer-generated hologram was etched into a material to form a holographic plate of negligible thickness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', no refraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Here, the underlying concept stems from Denis Gabor’s Nobel award- winning development of holographic imaging, allowing for the recording of light’s amplitude and phase information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Geometric phase has been exploited for complex beam shaping [100]—which by definition is polarisation sensitive—allowing for the creation of vectorial light fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A more recent move to sub-wavelength structures in the visible has allowed for polarisation dependent propagation phase control using metasurfaces [101], paving the way for all phases to be exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The aforementioned approaches are all static, which greatly limits their versatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The introduction of liquid crystal spatial light modulators (SLMs) [102] ignited a plethora of investigations into novel light shaping techniques and their corresponding applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These devices are void of customer-specific production requirements, span the visible and near-infrared wavelength ranges, and offer instantaneous and rewritable amplitude, phase, and polarization control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Intriguingly, recent times has seen a return to amplitude-only devices in the form of digital micromirror devices (DMDs), Journal of Optics (2022) #### which are fast, cheap, and versatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Direct light shaping at the source, within the laser, allows for high- efficiency and high-purity modes as the output, and can be achieved through a variety of intra-cavity and cavity geometry approaches [103], with complex light possible from very simple laser cavities [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, a combination of internal and external path control has resulted in eight- dimensional structured light across multiple DoFs (see Figure 4 and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [104]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A simple laser cavity can shape light that appears ray-like but with wave-like properties, across multiple DoFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Eight-dimensional structured light has been shown when internal and external light shaping are combined, producing the classical analogue to the famous GHZ states of quantum light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Reproduced from [97] with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Challenges and opportunities The algorithmic approaches to shaping light (the recipes) are very well established [105], dating back to the early work on pattern recognition, and improved recently with the aid of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The challenge lies mostly in the hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Although huge advancements have been made in the light- shaping toolkit and its employment in a diverse range of applications, there remain some challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Foremost amongst these are the need for the miniaturization of the modulation technology, increased power thresholds, and a broader range in wavelength control, particularly at shorter ultra-violet (UV) or extreme UV wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, in order to integrate and interface existing (miniature) electronic components with fast photonic technologies that exploit the many photonic DoFs, light- shaping approaches have to be miniaturised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The development of miniature, integrated on-chip devices will bolster the optical computing, imaging, and communications industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Miniaturising the light-shaping toolbox has been restrained by the material of the beam-shaping device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Liquid crystals typically allow for micro-meter-sized pixels, while refractive elements are limited to feature sizes much greater than the operating wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recent developments in metamaterials and nanostructured materials are offering promising avenues to engineer structured light with subwavelength thick optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Gradient metasurfaces are of particular interest in that they possess a spatially varying phase response allowing for arbitrary wavefront control with subwavelength resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These novel and minute components allow for integration into existing photonic technologies, such as photonic circuits and optical fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Another promising opportunity is based on two-photon polymerization (2PP) which is a direct laser writing technique fabricating complex 3D micro-optic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Here wafer-thin optics with feature sizes of approximately 160 nm can be fabricated [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Another major challenge is to extend the current light-shaping techniques to high-power levels, where levels exceeding kilowatts are needed for laser machining and laser-enabled manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Bellstate Greenberger-Horne-Zeilingerstate 4(0-+(H(+ (≥(z/(-/+(αl(i1(+) z/(-0-+(a(t/(+(0+ +2m)/+0)]+)[1)[R)+|m)/e) /)[2)[2) _asercavityJournal of Optics (2022) #### Commercially available single and multimode fibre lasers used for welding, cutting, and additive manufacturing operate at the multi-kilowatt level, but to date, very few beam-shaping technologies can tolerate such extreme power levels, although efforts to remedy this are underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Optics used in high-power kilowatt systems are large and bulky due to the finite absorption limitation—the complete opposite of miniature-sized components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' More recently, SLMs are being supplied with thermal control units, where water-controlled heat sinks allow for a 10-fold increase in incident power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' There is promise in revisiting well-known and well-established adaptive optics solutions such as deformable mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These mirrors can tolerate and efficiently reflect powers on the order of kilowatts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, efforts will need to be made in order to enhance their response rate and stroke to achieve fast, high- purity spatial mode creation while reducing their cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In attempting to reduce the size of high-power beam-shaping devices, the previously discussed metasurfaces and nanostructures could provide promise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' however, their power-handling capabilities have yet to be tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The aforementioned approaches are all based on linear optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' An exciting avenue that is very much in its infancy is to shape light by nonlinear approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' One possible solution is to shape the light at low power and amplify it post-shaping, while another is to transfer shapes from one wavelength to another through parametric processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These approaches may overcome both the power and wavelength challenges, while on-chip nonlinear optics has a long history, holding promise for compact solutions too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Traditionally, amplification and nonlinear processes have focused on “how much light do I have?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', and less so on “what does the light look like?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' To explore these possible solution pathways will require a paradigm shift in our thinking on such topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Other challenges lie in spatially controlling short wavelengths, as the light shaping toolbox for these wavelengths are rare (neither SLMs nor DMDs work), while hardcoded DOEs and metasurfaces likewise struggle with material choice and feature size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' All of the previously mentioned tools are only valid and applicable to coherent beams, but how can we achieve and similarly control the DoFs of incoherent sources?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Already, the advancement of “Li-Fi” may benefit from further DoF control if such beam-shaping approaches are feasible for incoherent light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks Techniques to achieve light shaping have advanced tremendously during recent years, offering a diverse set of tools, and opening an abundance of exciting applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Although a broad scope in light shaping functionality has been achieved, photonic control in optical circuits is still primitive in comparison to electronic control, with practical and commercial realisation still far off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Further advances in efficiency, compactness, broader wavelength control, and power-handling capabilities are sorely needed for shaping light to advance from science to application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Spatiotemporal optical vortices and OAM Sina Zahedpour, Scott W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Hancock, and Howard M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Milchberg University of Maryland Status It is well established that monochromatic beams of light can support vortices where electromagnetic energy density circulates around a local axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Examples of such orbital angular momentum (OAM)-carrying beams are the Laguerre-Gauss (LG) or Bessel-Gauss (BG) modes in free space, where the OAM axis coincides with the direction of propagation [8], or spatial optical solitons with vortex rings [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In all cases, local vortical axes are fixed in space and energy density flow is described in purely spatial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, because electromagnetic vortices and OAM are fundamentally associated with energy density circulation, there is no prohibition, in principle, for a local vortical axis to deviate from the propagation direction while embedded in a propagating pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This would require a polychromatic beam [19, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such polychromatic vortices embedded in spacetime were first measured as a naturally emergent and universal structure in the collapse arrest dynamics and self- guiding of intense laser pulses in nonlinear media [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Were it not for ‘collapse arrest’—a response to high intensity such as ionization—the beam would collapse to a singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Collapse arrest enables a quasi- stable accumulation of phase shear– a very sharp phase gradient in the transverse direction--between the inner and outer parts of the beam, triggering a phase defect (and field null) that wraps around the propagation axis and spawns toroidal spacetime vortices (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These toroidal spatiotemporal optical vortices were observed for the first time in [21] and dubbed STOVs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' they direct energy density flow in a self-guided pulse, and are robust and topologically protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The realization that STOVs were generated by phase shear in spacetime led to a method to generate them linearly and controllably, using a pulse shaper to apply shear in the spatiospectral domain [22,23] and then return the pulse to the spatiotemporal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The STOV pulse thus generated resembles an ‘edge first flying donut’ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2), with phase circulation in spacetime and OAM orthogonal to the propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Experiments [22] and theory [24,110] have Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Propagation simulation showing birth and evolution of toroidal STOVs wrapping around a nonlinearly self-guided laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The STOVs are generated by spatio-temporal phase shear [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Example [22] of 4𝑓 pulse shaper for linearly generating STOVs by applying spatio-spectral phase shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Here, SHG is also shown [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1 – We allow at most two figures that are roughly the size of this box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' phase plate cylindrical lens BBO -step (b) input grating output grating phase 500um 50fs intensityJournal of Optics (2022) #### shown STOV propagation is governed by diffraction in both space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Later simulations have shown that STOV-carrying pulses can be formed by compact nanostructures that impose a specified field null axis in (𝜔/𝑐, 𝐤) space, leading to arbitrarily oriented STOV axes in spacetime (𝑐𝑡, 𝐫) [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' That spatiotemporal OAM is a property of single photons was recently confirmed in experiments showing OAM conservation under second harmonic generation (SHG) with STOV pulses [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recent work has investigated extreme ultraviolet STOV photons produced by high harmonic generation [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges Merely measuring STOVs presents new challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Unlike spatial optical vortex- carrying beams, whose intensity and phase profiles can be measured by CW beam imaging and interferometry, more complex methods are needed to visualize STOVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Narrow bandwidth, low resolution multi-shot pump-probe cross-correlation measurements [23] can be used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' these are limited to long duration STOV pulses with high pulse-to-pulse stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For examining ultrafast STOV evolution in nonlinear propagation experiments, where sensitivity to small fluctuations is expected, a broadband single-shot technique has been demonstrated that measures pulses with spacetime phase defects [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As STOV-structured light increases in complexity, such as with short wavelength attosecond pulses [112], there will be a need for much higher space and time resolution diagnostics extending across the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As the study of STOVs is in its infancy, they could be viewed as a solution in search of a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Because they are integral to energy density flow in both nonlinear self- guided propagation [21] and in linear propagation [22,110], an unusual robustness may apply to these beams owing to conservation of topological charge and angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As with space-defined OAM, the excitation and probing of STOV-OAM states in materials and structures will open completely new research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Any system involving spacetime phase shear, such as transient current densities in nanostructures, could be interrogated by STOV pulses with a prepared OAM content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Robust methods for filtering or sorting the OAM components of STOV pulses will therefore be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In analogy with super-resolution microscopy using monochromatic OAM beams, STOVs may even provide a super-resolution capability in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Spatiotemporal intensity and phase profiles of a 𝑙 = 1 STOV propagating through its beam waist (𝑧 = 0), as predicted by the modal theory (left) and measured (right) [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Within each frame, pulse propagation is right-to-left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Symbols: 𝑧!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' = 𝜋𝑤"# $ 𝜆 ⁄ , 𝑤"# and 𝑤"% are space-like and time-like modal scale lengths [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2 0 2-2 0 2-2 0 2-2Journal of Optics (2022) #### At a fundamental level, only very recently has there been theoretical work on angular momentum of STOV pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' One approach calculates spin angular momentum and OAM of STOV pulses in vacuum [24] and the other derives the mode structure and OAM for paraxial STOV pulses in dispersive media [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These papers differ in their choice of OAM operator, with differing results for the OAM per photon ℒ in a STOV-carrying pulse of topological charge 𝑙: ℒ = 𝑙 ℏ(𝛼 + 𝛼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='") 2 ⁄ [24] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' ℒ = 𝑙 ℏ(𝛼 − 𝛽#𝛼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='") 2 ⁄ [110], where 𝛼 = 𝑤$% 𝑤$& ⁄ is the ratio of the STOV’s time-like to space-like spot sizes and 𝛽# is the medium’s normalized group velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The latter operator is conserved, and predicts half-integer OAM in vacuum (owing to lack of energy density flux in the local time domain) and the existence of “STOV polaritons” in dispersive media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Just as the pioneering work connecting LG modes to photon OAM [8] was followed by quantum field theories, there are likely to be similar follow-ups to [24,110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks Experimental realization of electromagnetic pulses with spatiotemporal OAM has opened a promising new avenue for studying OAM and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements The authors thank Konstantin Bliokh for fruitful discussions, and acknowledge the support of the Air Force Office of Scientific Research, the Office of Naval Research, and the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structured waves in non-Hermitian systems Stefan Rotter1, Franco Nori2,3, and Şahin K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Özdemir4 1Vienna University of Technology (TU Wien) 2RIKEN 3The University of Michigan 4The Pennsylvania State University Status The structuring of waves typically involves the propagation of an incoming wave field through a device that shapes the transmission in a desired fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Engineered structures like gratings and waveplates can equivalently be operated in reflection mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In both cases, the desired wavefront of the outgoing wave is achieved through appropriate interferences induced by the wave-shaping device that tunes the wave’s spatial and spectral (or temporal) degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In a very recent line of research, one tries to extend the possibilities to structure waves of different kinds (electromagnetic, acoustic, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=') by working with non-Hermitian wave-shaping tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The term “non-Hermitian” refers here to the fact that the time evolution of the wave passing through the wave shaper is governed by a non- Hermitian Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While, formally, a simple wall that absorbs all of the incoming waves would already constitute such a non-Hermitian system, one typically refers to non-Hermitian systems only when they contain a non- trivial combination of gain (amplification) and loss (dissipation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Typical examples include non- Hermitian meta-surfaces for steering transmitted or reflected waves in desired ways [113], and waveguides that are designed such as to guide incoming waves around non-Hermitian degeneracies known as exceptional points (EPs) [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such EP-encircling protocols build on the remarkable property that the output state on either side of the waveguide depends only on the direction in which the EP is encircled—a property determined solely by the input port through which the wave is injected into the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Going beyond such asymmetric mode-switching protocols, spatially tailored gain-loss landscapes can also be used to guide incoming waves [115–117], even through disordered scattering regions [118, 119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' When the patterning of gain and loss is done right, waves can not only be perfectly transmitted, but can also maintain a well-behaved intensity profile (without any interference fringes) even inside strongly disordered media (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Under certain circumstances, the details of the system that is patterned with a certain gain-loss distribution do not even have to be known to apply a non-Hermitian structuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In cases such as random lasers that have an unknown and typically inaccessible internal structure, the spatial pattern of the pump beam that delivers the optical gain to the laser can be optimised through appropriate algorithms to engineer the laser’s spectral and spatial emission properties [120, 121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In this way, single-mode operation and a directional far-field pattern of a random laser can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recently, concepts from topology have also been employed to create robust unidirectional propagation of light and to design lasers that operate on topologically protected edge modes (see [122] for a recent review of this emerging field of research) or on modes that encircle an EP [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges On the conceptual level, the major lines of research currently concentrate on the question: which new functionalities, advantages, or unconventional features may the engineering of a system’s non- Hermitian degrees of freedom bring along?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Is it possible to use non-Hermitian engineering to, for Journal of Optics (2022) #### example, overcome the stringent material requirements for meta-materials used for optical cloaking?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Can one exploit topological concepts to build photonic structures that are robust against fabrication imperfections?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Can such progress be achieved without having to work with very exotic materials or cumbersome setups?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' On the technical level, one of the major challenges is the accurate positioning and control of the non-Hermitian (gain and loss) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While lossy media are ubiquitous and comparatively easy to pattern, media with gain are more challenging to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This is because gain necessarily requires an external source of energy to provide the amplification to the wave—think here of an active medium such as a laser cavity that requires an external pump (optical or electrical) to operate the laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For optical devices in one or two dimensions, one typically works with optically active materials that are pumped through an external beam with a spatial pattern that is controlled by a suitable mask or a spatial light modulator (the size and costs of the latter, of course, also impose limitations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A three- dimensional structuring of the pump profile is naturally more challenging to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Moreover, non-linear effects, such as those induced by gain saturation and spatial hole burning, further complicate the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For other types of waves, like sound or matter waves, suitable gain mechanisms may not even be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' We emphasise that both the spatial and the spectral tunability of a gain medium have certain restrictions: while the spatial pattern of the pump cannot be arbitrarily fine in its structure, the spectral profile of the gain is determined by the fundamental constituents of the active medium as well as by other restrictions like the Kramers-Kronig relations that follow from the principle of causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A viable strategy to overcome these limitations is to consider discrete instead of continuous systems, where the discreteness of the former may refer to their spatial structure or to their time- evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A corresponding mapping onto arrays of waveguides or coupled fibre loops has the advantage that each spatial or temporal element involved in such a discrete system can be individually controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recently, this advantage has been used in various circumstances to implement theoretical concepts on non-Hermitian wave engineering—from the creation of constant-pressure sound waves to the realisation of system designs based on the concept of non-Hermitian topology (for a review see [124]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Two-dimensional distribution of a real refractive index nR (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A Gaussian beam entering this disordered scattering landscape and becomes severely distorted (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Adding a tailored distribution of gain and loss through an imaginary refractive index nI (top panel), the Gaussian beam propagates as through a homogeneous medium (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Image adapted from [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) nR (b) ni 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='3 0 -10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='3 10 30 50 x/a 10 30 50 x/2 4 IE|2 60 60 T [E]2 0 30 0 30 15 x/2 15 x/2 0 y/a 15 0 y/a -15 0Journal of Optics (2022) #### Advances in Science and Technology to Meet Challenges On the theoretical level, there are still many questions left open for exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' When considering smaller and smaller constituents of non-Hermitian media, where quantum effects start playing a role, the influence of the noise induced by both gain and loss has to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As it turns out, such noise processes are not just an annoying side effect, but they may constitute the major bottleneck for the performance of a non-Hermitian device, such as for sensors that operate at an EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Dealing with such noise processes remains an outstanding theoretical challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Also in the domain of non-Hermitian topology, numerous puzzles are waiting to be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Going beyond the question of how to correctly transfer concepts from Hermitian topology to the domain of non-Hermitian physics, entirely new topics emerge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' consider here, for example, nonlinear effects that lead to strongly correlated states of light and interesting collective processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It also still needs to be clarified which features non-Hermitian materials can provide that were so far only associated with other types of materials, such as anisotropic media and those with a vanishing or negative index of refraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Specifically, it will be interesting to clarify if and in which way non-Hermitian media can be used for cloaking an object [125], for breaking Lorentz-reciprocity (in combination with nonlinearity or spatio-temporal modulation), and for imaging beyond the diffraction-limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For the experimental implementation of flexible non-Hermitian media, progress on many different sides will be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in the speed, precision, and cost of spatial light modulators will help to accurately control spatially tailored pump beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in the development of active materials will help to address the challenging requirements in the gain values required for implementing certain theoretical concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Non-Hermitian metasurfaces with well-controlled distribution of discrete gain and loss ingredients may exhibit interference effects originating from gain- and loss-induced phase responses, leading to phase singularities and vortices which can help to control and shape the light transport [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks To conclude, the research on structured waves in non-Hermitian media is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While a whole range of interesting theoretical concepts and experimental platforms for their implementation have recently emerged, the field is still growing in several directions such as towards the inclusion of topological, quantum, and nonlinear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Another appealing prospect is to apply non-Hermitian design concepts not just on externally generated light fields, but to integrate them directly into the design of the laser light source itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Important challenges are the flexible and precise control of the non-Hermitian gain and loss components in both their spatial and spectral degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' With rapid technological progress in this direction, we expect non-Hermitian tailoring of light fields to become a standard tool in wave engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Ultimately, non-Hermitian elements may achieve comparable relevance to spatial light modulators and conventional diffractive metasurfaces made from dielectric or plasmonic structures without gain inclusion, paving the way to a new era of wavefront shaping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' acknowledges support by the Austrian Science Fund (FWF, grant P32300 WAVELAND) and by the European Commission (grant MSCA-RISE 691209 NHQWAVE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' is supported in part by NTT Research, and Ş.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='Ö.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' by the Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' FA9550-21-1-0202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Tailoring Random Light for Imaging Applications Nicholas Bender and Hui Cao Yale University Status Spatially random light has the hallmark appearance of an irregular mosaic of diffraction-limited speckle grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Speckle formation is a phenomenon inherent to both classical and quantum waves, occurring when a coherent wave undergoes a disorder-inducing scattering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The speckle patterns are described by a statistically stationary and ergodic random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Stationarity requires the statistical properties of an ensemble of speckle patterns to be the same as those of an individual speckle pattern within the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Ergodicity requires the statistical properties of two spatial positions—separated by more than one speckle grain size—to be independent and identical to those of the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The speckle patterns are categorized by the joint probability-density function (PDF) of their complex-valued field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rayleigh speckles—the most common family of speckles—obey a circular-Gaussian field PDF which results in a negative exponential intensity PDF [127,128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The phase PDF is independent of the amplitude PDF, and constant over a 2π range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The circular invariance of the field-PDF results in a “fully developed” speckle pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Typically, non-Rayleigh speckles are classified as either under-developed (the sum of a small number of scattered waves, or the sum of not fully randomized waves) or partially coherent (the sum of incoherent partial waves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Furthermore, fully developed speckles typically possess only short-ranged spatial intensity correlations which are determined by the average speckle grain shape, which is dictated by the diffraction limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Because of their pervasiveness, speckle patterns have been adapted for use in a wide range of optical applications ranging from imaging [129, 130] to optical manipulation [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While Rayleigh speckles are the most common family of speckled light, their statistical properties and spatial correlations are not necessarily ideal in different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' There has been a plethora of interest in creating speckle patterns with tailored statistics and spatial correlations [132-136], due to their potential applications in structured-illumination imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Specific examples include dynamic speckle illumination microscopy, super-resolution imaging, and pseudo-thermal light sources for high-order ghost imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Furthermore, a general method for customizing both the statistics and topology of laser speckle patterns would be a valuable tool for synthesizing optical potentials for cold atoms, microparticles, and active media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges Because the conditions required to generate Rayleigh speckles are general, creating fully developed non-Rayleigh speckles is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Specifically, the difficulty lies in altering the intensity PDF without changing other statistical properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', phase PDF, stationarity, ergodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recently, a simple method for creating non-Rayleigh speckle patterns with a phase-only spatial light modulator (SLM) was developed [132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' High-order correlations were encoded into the field reflected by the SLM, resulting in a redistribution of the light intensity among the speckle grains in the far-field (Fourier- plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The resulting speckle pattern possesses an intensity PDF with a tail decaying either slower or faster than a negative-exponential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Subsequently, a general method for tailoring the intensity statistics of speckle patterns was developed based on the same principle of modulating the phase front of a laser beam with a SLM [133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Experimentally, speckle patterns governed by arbitrary Journal of Optics (2022) #### intensity PDFs were created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The speckle patterns exhibit distinct topologies from Rayleigh speckles, without introducing spatial correlations beyond the diffraction-limited speckle grain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A foundational principle of statistical physics is the Siegert equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Specifically for the case of spatial correlations in random light, the Siegert equation proportionally relates the intensity correlation function with the squared magnitude of the field correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As such, the spatial intensity correlation function of a typical speckle pattern does not possess additional structure beyond what is present in the field correlation function, which is determined by the diffraction-limited average speckle-grain shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In [132,133], the spatial intensity correlations of the customized speckle patterns adhered to the Siegert equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It was experimentally shown in [134] that a speckle pattern can dramatically break the Siegert relation when non-local correlations are controllably encoded into a speckle field by a SLM, specifically by tailoring the 4th order correlations in the Fourier plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While the techniques presented in [133, 134] independently modify different properties of speckle patterns, [135] combined these methods to arbitrarily tailor the PDF and spatial intensity- correlations of speckles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1 presents two examples of fully developed speckle patterns (a), (b) with different tailored spatial correlations (c), (d) and customized statistics (e), (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Two example speckle patterns (a), (b) from stationary and ergodic ensembles of 100 tailored speckle patterns, with customised spatial intensity-corelations (c), (d) and intensity PDFs (e), (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The example speckle pattern in (a) possesses non-local intensity-correlations (c), and a unimodal intensity PDF (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The speckle pattern represented by (b) is tailored to have ring-shaped, long- range intensity-correlations (d), and adhere to a bimodal intensity PDF (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' aDesignedSpeckles DesignedSpeckles max Ci(Ar) i(△r) ma 66um 66μm Cmin e IntensityPDF IntensityPDF 1 /1)d 2Journal of Optics (2022) #### Advances in Science and Technology to Meet Challenges The scientific advances made in speckle customization have begun to translate into speckle-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, a proof of principle demonstration [136] has shown that 2D customized speckles can significantly out-perform Rayleigh speckles in nonlinear pattern illumination imaging techniques [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1(a) presents a specially tailored speckle pattern to photoconvert a uniform fluorescent sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Within the central square, the speckle pattern was designed to consist of a random array of circular vortices embedded in an approximately constant-intensity background;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' outside the square are Rayleigh speckles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1(b) is the fluorescence image of the sample after being photoconverted by the speckle pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Outside the central square, the fluorescent pattern consists of a sprawling anisotropic web, which reflects the topology of the low-intensity regions surrounding the optical vortices in Rayleigh speckles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In stark contrast, the fluorescence pattern within the central square c features isolated isotropic fluorescent spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The isotropy exhibited by the fluorescent spots originates from the high degree of rotational symmetry of the vortices in the customized speckles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Apart from these spots, the fluorescent intensity is uniformly low due to the homogeneity of the customized speckles’ intensity away from optical vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Qualitative comparison between the two distinct fluorescent patterns illustrates the degree to which customizing the speckle intensity statistics can enhance the performance of speckled illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Quantitatively, the customized speckles were able to create fluorescent spots three times smaller than the diffraction limit set by the illumination optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Nevertheless, significant scientific challenges remain to be solved in order to further speckle- based applications: namely, developing a technique to customize speckled light inside a random scattering medium, tailoring 3D volumetric speckles, and creating vector speckle fields with customized statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Technical advances in wavefront shaping devices will facilitate addressing these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, increasing the effective number of independent phase modulating pixels on a SLM can provide more degrees of freedom for creating customised 3D speckle patterns, or vector light field for high-NA imaging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Future improvements to the SLM operation speed will facilitate tailoring random light inside dynamic scattering media, potentially even allowing random light tailoring inside live biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A speckle pattern (a) is designed to photoconvert a uniform protein sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Within the yellow square in (a), optical vortices are randomly embedded in a bright background, and outside are Rayleigh speckles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Fluorescence image of unconverted protein inside the square (b) shows isometric and isotropic spots produced by the vortices in the tailored speckles (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Unconverted protein outside the square in (b) features large, irregular, and interconnected fluorescent grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' a DesignedSpeckles FluorescentSignal C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='-Target Region xew max 100umJournal of Optics (2022) #### Concluding Remarks Recent developments in customizing laser speckles [132-135] have resulted in simple, yet versatile, techniques for creating and controlling random light, which can easily be adapted for use in a diverse range of optical experiments and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, the ability to arbitrarily control the non- local correlations and intensity PDFs of speckle patterns can be used to create complex optical- potentials for studies on the transport of cold atoms, active media, and microparticles [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Potentially, it can also enhance many structured-illumination applications like speckle illumination microscopy [345], super-resolution imaging [129], and high-order ghost imaging [346].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A proof-of- principle experimental demonstration [136] has demonstrated that intelligently tailored speckles can significantly outperform commonly used Rayleigh speckles in nonlinear pattern illumination microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the future, new techniques for customizing speckle patterns can potentially provide drastic improvements to the myriad of speckle-based applications currently in existence, and potentially lead to the development of new applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements The authors thank their co-workers Yaron Bromberg, Hasan Yılmaz, and collaborators Joerg Bewersdorf and Mengyuan Sun for their contributions to the works presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' They also acknowledge financial support from the Office of Naval Research (N00014-20-1-2197) and the National Science Foundation (DMR-1905465).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Ultrafast structured beams and intense magnetic fields P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Corkum1 and Carlos Hernández-García2 1University of Ottawa and the National Research Council of Canada 2Universidad de Salamanca Status With spatial light modulators, Q- or S-plates, it is possible to impose any retardation on any spatial element of an optical beam in the image plane of the retardation plate, enabling the generation of vector or vortex beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This ability is limited in space by pixel size and the resolution of the image system, in intensity by damage to the phase plates, and in time by the spectral bandwidth of the spatial element that forms the waveplate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The latter restricts the generation of ultrafast structured beams in the femtosecond or even attosecond regimes, where high frequencies and broad spectral bandwidths are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' At the femtosecond timescales, this restriction can be relaxed through femtosecond pulse compression (with fiber compression or in thin dielectric windows), reaching the few-cycle limit of light [137,138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges Femtosecond cylindrical vector beams have been demonstrated recently to allow for coherent control over currents, transferring the light’s topology to a material in an ultrafast fashion [139,140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Electrons created by structured light beams can re-radiate, thereby transferring the original properties of light to a new frequency—higher frequency for generating high harmonic radiation, lower frequency for THz magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Information about ultrafast dynamics in the medium is encoded in the generated radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' High-harmonic generation is one of the most robust mechanisms to transfer structured light to the ultrafast, high-frequency regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In gas-phase high-harmonic generation, the phase-matched emission of all dipole emitters, together with angular momentum conservation rules, allow for the vectorial [141,144] and orbital angular momentum [141,142,145,146] properties from the infrared to be transferred to the extreme-ultraviolet (XUV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Thus, highly nonlinear up-conversion can produce high-harmonic radiation with almost any orbital angular momentum and polarization, and more generally, with any vectorial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Interestingly, high-harmonic generation allows for the topological properties of harmonic light to be controlled at the attosecond time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' If several harmonics with different orbital angular momenta are composed, ultrafast light is arranged like a coil spring—a “light spring” [141,142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In addition, high-harmonic generation allows for the orbital angular momentum of light beams to be varied in the sub-femtosecond scale, thus creating light beams with Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Attosecond light spring [129], [130] and (b) XUV beam with self-torque or time-dependent OAM [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' a Attosecond light spring b Lightbeamwithself-torque x(um) 50 35 50 0 50 31 29 Intensity (wrl) 17 0 27 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' units) 52207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='5 -50 2 0 2=47nm Intensity Self-torque: E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='32fs (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=') 10 15 20 25 30 Time (fs)Journal of Optics (2022) #### time-dependent orbital angular momentum or self-torque [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such ultrafast structured beams, “light springs”, or self-torque beams are not found in other spectral regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In gases, where the electron moves primarily in the vacuum, the fundamental field is almost solely responsible for the electron’s motion and for the high harmonics that the gas emits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Phase matching forces the conservation of orbital angular momentum [141,142,145,146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In solids, the electron’s interaction with the solid cannot be ignored, but still, angular momentum is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Turning this around, intense infrared pulses can serve as a flexible probe of solids wherein the electron’s motion is partially controlled by the infrared light and partially by the electron’s interaction with the material, but the high harmonics that emerge still must satisfy phase matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, as high harmonics are developed in metals, metals will produce vector beams and be probed by vector beams [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A different way to modify a light pulse is to use a structured material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, when a metal containing a hole interacts with azimuthally polarized light [148,149], a large ring current can oscillate at the laser frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This current generates ultrafast, intense longitudinal magnetic fields isolated from the electric field (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' An alternative method is to drive ring currents using quantum control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A quantum-controlled current is not limited by the angular momentum of the incident beam because electrons and holes are created together, and each gain equal but opposite momentum and angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In semiconductors, quantum control allows for any current structure to be engineered on any pixel [150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the 1990s, there was a lot of work on coherent control of semiconductor currents, using linear or circular polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For this work, a detector was developed in cold-grown GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' That detector has been adapted to measure current driven by azimuthally polarized fundamental and second harmonic light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Fig 2(b) shows the ring currents that are needed to launch a THz “flying torus” [151, 152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Neither linear nor nonlinear spectroscopy with flying tori have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Although quantum control is feasible whenever there are interfering pathways to a final state (in this case, the direction of the current), the work in GaAs is in the perturbative limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In contrast, breakdown in gases or dielectrics will allow high intensity control and high magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In gases, the combination of fundamental and second harmonic light transforms a gas into a plasma that has initial conditions imposed by the transformation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Enhanced magnetic field (blue) at a metal sample irradiated by an infrared azimuthally polarized beam [136], [137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Ring current measured with a 25x25 μm2 detector (left) and the magnetic field (right) calculated [127], [128] using the Biot-Savart law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' a SchemetoinduceintenseultrafastBfields E Ix 5 B B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (t) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='5 Current oops Aperture B -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='5 2/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='.-Incident Atsample -5 Sample 0 10 20 Time [fs] Bz FWHM=248μmJournal of Optics (2022) #### If we were to ask for the highest magnetic fields that humans can controllably generate, these fields are created within intensely irradiated plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, magnetic fields within plasmas are difficult to use since they are internal to the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' High-intensity quantum control will allow us to generate large, controllable, magnetic fields that are as isolated as possible from the plasma, making these fields useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks Producing and applying XUV/soft X-ray structured beams and ultrafast intense magnetic field pulses will become important forefronts of research with structured light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The study of magnetic helicoidal dichroism or magnetization switching with structured laser pulses [153, 154] are examples of applications of ultrafast structured beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While it is early days for soft X-ray structured beams, there are two frontiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' First, high-harmonic generation is a robust source of structured beams in the XUV that can be used to observe electrons and, through them, the response of matter to extreme radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This matter can be any type of new material—chiral, magnetic, or topological materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' XUV or soft X-ray radiation can also be integrated with attosecond science, allowing for unusual pulses such as “light springs” or light beams with self-torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Further advances in the technology of midinfrared structured driving beams may enable the generation of attosecond structured beams deeper in the soft X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In addition, the use of solid targets may hinder new scenarios for structured high-harmonic generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Secondly, the development of the quantum optics of soft X-ray beams is required for the application of such structured beams in fields such as long-distance space communications which can benefit from their very low beam divergence and high photon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Metaphotonics with Structured Light Haoran Ren1 and Yuri Kivshar2 1Monash University 2Australian National University Status The wave-particle nature of light leads to multiple degrees of freedom such as wavelength, amplitude, phase, polarisation, and angular momentum, which can be controlled in spatial, temporal, and spatial- temporal domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structured light patterns were first observed in the double-slit experiment of Thomas Young, where the amplitude and phase interference created bright and dark fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Today we understand that a light beam can be structured into millions of transverse modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', Hermite– Gaussian, Laguerre–Gaussian, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=') in a square millimetre [96], an extraordinary resource for boosting information capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In structured light, singular photonics exhibits topological properties possessing dark singularity centres in a phase vortex with the orbital angular momentum (OAM) of 𝑙ℏ per photon (𝑙 can take any integer value in [-∞,∞], and ℏ is the reduced Planck constant) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(a));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' a polarisation vortex manifested by a tensor product of the polarisation and OAM degrees of freedom (defined as |𝜓⟩ = cos + , -, |𝑙⟩|𝑅⟩ + e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='/sin + , -, |−𝑙⟩|𝐿⟩, where 𝜃 and α denote the weighted contribution of and relative phase between left- (𝐿) and right-handed (𝑅) circular polarisations) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(b));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' and a plasmonic vortex carrying the total angular momentum resulting from spin-orbit coupling (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Metaphotonics has recently transformed the photonic design for the control of multi-dimensional photonic vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' To implement phase vortices in real space, different dielectric and plasmonic metasurfaces were designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A high-index dielectric nanopillar with strong mode confinement was developed as a truncated waveguide with an effective mode index and phase response (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The high-index nanopillar can be designed as a subwavelength waveplate with strong birefringence, exhibiting different phase accumulations for the polarisation along the long and short axe (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Geometric metasurfaces based on the Pancharatnam–Berry phase have been used to create a phase vortex through the in-plane rotation of asymmetric nanopillars [155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Meanwhile, each anisotropic nanopillar can function as a subwavelength waveplate for implementing polarisation vortices in real space [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Huygens’ metasurfaces offer an alternative platform to realize phase vortices through spectrally overlapping electric and magnetic resonances (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(f)) [157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Ultrathin plasmonic metasurfaces based on the near-field mode hybridization have been used to create phase vortices (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(g)) [158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Additionally, metal–insulator–metal meta-atoms that support the gap plasmon resonance in a magnetic field could enable highly efficient generation of phase vortices in reflection (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Photonic crystal slabs possess an inherent polarisation vortex in momentum space around bound states in the continuum (BIC) of the periodic structures, featuring the in-plane winding of a vector field and thereby a polarisation vortex (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(i)) [159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the nonparaxial limit, the space and polarisation degrees of freedom are non-separable, giving rise to the total angular momentum, a measurable quantity through spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Surface plasmon polaritons (SPPs)—a tightly confined surface wave beyond the diffraction limit—open the possibility of producing subwavelength plasmonic vortices in the near-field region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For instance, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(j) presents a plasmonic nanoring aperture used for the multiplexing generation and detection of different plasmonic vortices [160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(k) shows the use of plasmonic grooves for the excitation and ultrafast imaging of optical skyrmions in SPPs [161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### Current and Future Challenges We have provided a review summary on singular metaphotonics and highlighted some nanophotonic structures designed based on different physics for the control of phase, polarisation, and plasmonic vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For the phase vortex generation in real space, high efficiency metasurfaces can be designed in both reflection and transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For reflection metasurfaces, plasmonic materials featuring a metal-insulator-metal configuration could offer superior efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For transmission metasurfaces, low-loss and high-index dielectric metasurfaces featuring an ultrathin thickness (<λ, where λ is the wavelength of incident light) or a relatively large thickness (~λ) can be designed for Huygens’ and waveguide metasurfaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, to achieve the most accurate phase digitalization when considering the fabrication error, geometric metasurfaces exploiting the rotation angle- controlled phase response are perhaps more desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Meanwhile, to achieve high efficiency polarisation vortex generation in real space, anisotropic meta-atoms based on the metal-insulator- metal configuration and all-dielectrics can be designed for the subwavelength polarisation control in reflection and transmission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Even though metasurface generation of polarisation vortices faces the same challenges as for the phase vortices, it is generally more difficult to use a metasurface device to distinguish and sort structured polarisation singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In addition to the fabrication challenges, metaphotonics devices are most usually passive and unlikely to be able to dynamically switch vortex modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Besides, structured optical fields are typically limited to a 2D transverse plane without the wavefront control in the propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The ability to tailor light beyond 2D structured light, towards 3D control (in all spatial coordinates and field components), and even 4D control with spatiotemporal control of structured light, is of fundamental and practical interest for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' On the other hand, even though BIC-induced polarisation vortices in momentum space feature robustness, alleviation of coaxial beam alignment, and Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Illustration of singular metaphotonics based on the manipulation of phase vortex (a), polarisation vortex (b), and plasmonic vortex (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (d-h) Design principles of different meta-atoms used for the generation of phase and polarisation vortices, including (d) a high- index dielectric waveguide, (e) a nanopillar waveguide, (f) an ultrathin dielectric cylinder in a Huygens’ metasurface, (g) plasmonic nanoantennas with hybridized plasmon modes, (h) a metal-insulator-metal structure supporting gap plasmon resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (i) A photonic crystal slab designed for creating polarisation vortex lasing modes in momentum space around the BIC resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (j) A plasmonic nanoring aperture used for on-chip OAM multiplexing through the total angular momentum mode-sorting nanoring slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (k) Plasmonic grooves used for ultrafast imaging of optical skyrmions carried by propagating SPPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (i-k) Reprinted by permission from American Association for the Advancement of Science in References [159–161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (d) (f)Huygensmetasurface (g) Waveguidemetasurface Plasmonicnanoantennas ED- MD (e) (h)Gap plasmon metasurface Nanopillarmetasurface neffi a) IHI neff2 Phasevortex (b) (c) Polarizationvortex Plasmonicvortex () Nanoring aperture (i) Photoniccrystalslab BIC (k)Plasmonicgrooves m Kx- +ky SingularmetaphotonicsJournal of Optics (2022) #### unrestricted choice of materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' symmetry-protected photonic crystal slabs designed near the BIC at the Γ point are extremely sensitive to the change of refractive index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' incident wavelength,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' and incident angle that may easily break the system symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' To create plasmonic vortices, nanogrooves engraved in a metal film were generally employed, but they usually have a low coupling efficiency of SPPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Traditional noble metals such as gold and silver also suffer from high metal losses due to interband transitions in the ultraviolet and visible frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' More critically, strong dissipation of the highly localized plasmonic vortex fields in the near-field region hinders the SPPs applications for on-chip vortex transmission and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in Science and Technology to Meet Challenges Nowadays, nanofabrication techniques are available to create functional metaphotonics with the resolution down to a few nanometres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Systems such as EBL, focused-ion beam (FIB), and 3LN can enable ultrahigh resolution (EBL, FIB) or 3D manufacturing, but are limited by a low throughput due to a sequential writing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Masked techniques, such as photolithography, soft lithography, NIL, and colloidal lithography enable high throughput by replicating the entire mask simultaneously but are often limited in resolution or flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The NIL technique is scalable, and large-area roll-to-roll or roll-to-plate techniques have already been developed to enable high-throughput metasurface production towards industrial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Alternatively, self-assembly techniques have emerged and could circumvent the need for clean-room facilities and expensive equipment, though this technique lacks the flexibility to create different-shaped nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recently, digital vectorial holography has enabled the generation of advanced vortex beams, in which the phase and polarisation singularity centres can spatially vary either in 3D polarisations [163] or along the propagation direction [164].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Moreover, a 3D wave packet that carries a spatiotemporal optical vortex with a controllable purely transverse OAM has been realized [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Multi-dimensional structured light could offer extra degrees of freedom for versatile light-matter interactions, quantum entanglement, optical trappings, harmonic generation, and optical sensing, holding great potential for novel applications that may not be possible otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Even though strong dissipation of the highly localized plasmonic vortex fields hinders on-chip plasmonic vortex transmission and processing, superior transmission efficiency can be offered by low-loss semiconductor nanowires sustaining highly confined optical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recently, an OAM-controlled hybrid nanowire plasmonic circuit was introduced, demonstrating OAM-controlled optical logic operations including AND and OR gates [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' OAM beams with different topological charges exhibit selective excitation of single-crystalline cadmium sulfide nanowires through coupling OAM-distinct plasmonic fields into nanowire waveguides for long-distance transportation on-a-chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Conclusion We believe that metaphotonics provides a great playground for structured light manipulation, and it will lead to a diverse range of ultracompact, ultrahigh-capacity, and ultrahigh-speed devices harnessing multi-dimensional structured light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' We believe it is of paramount importance to integrate developed metaphotonics devices with established optical systems for advanced optical imaging, holographic displays, optical and quantum communications, nonlinear and ultrafast light shaping, and turbulence- and scattering-resilient communications and imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' acknowledges a support from the Australian Research Council DECRA Fellowship DE220101085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' acknowledges a support from the Australian Research Council (grant DP210101292).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structuring Light with Near-Zero-Index Platforms Mário G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Silveirinha1 and Nader Engheta2 1University of Lisbon and Instituto de Telecomunicações 2University of Pennsylvania Status Materials provide the means to structure light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Judiciously engineered material platforms, known as metamaterials and metasurfaces, have provided scientists and engineers with versatile tools to control, manipulate, and sculpt electromagnetic waves and fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In particular, materials whose real part of relative permittivity and/or permeability attain near-zero values at given operating frequencies offer specially interesting platforms for structuring electromagnetic and optical waves [166-168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' At such frequencies, these materials exhibit (a) refractive index near zero, and consequently (b) the wave phase velocity attains very high values (theoretically infinite values) which leads to (c) a “stretched” wavelength and (d) uniform phase distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' When both relative permittivity and permeability are zero, the electric and magnetic phenomena are effectively decoupled in such materials [166] yielding static-like spatial distributions of electric and magnetic fields, while at the same time they are dynamically time varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This special feature makes epsilon-near-zero (ENZ), mu-near-zero (MNZ), and epsilon-and-mu-near-zero (EMNZ), which form the general class of near-zero-index (NZI) materials, particularly interesting for wave manipulations, beam shaping and lensing [167-171], for example the wavefronts emerging from an ENZ material block typically inherit the shape of the ENZ- material surface [170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Since the wavelength of waves in such media can be long even for high frequencies, one can effectively think of this effect as “loosening” the connection between the frequency and the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Moreover, a block of material in which the wavelength is very long can be viewed electromagnetically as a “point”, even though it can be large compared to the free-space wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This phenomenon has enabled numerous exciting features in wave interaction with such NZI media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The “supercoupling” effect [167-169] is an interesting example of such features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' when two metallic waveguides are joined together, the connecting segment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', transition region) between the two waveguides can be of any shape and size, if that region is filled with an NZI medium [167,168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' If an ENZ (or MNZ) medium fills the transition region, then the connecting segment needs to be narrow (or wide), while it can be bent and can have arbitrary shapes [167-169] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(a) and 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This “electromagnetically point-like” region may provide an unusual coupling between two emitters, effectively causing “near-field coupling” even when the emitters are far apart in space [172] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2(a) and 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This is an interesting way to structure light in dipole-dipole coupling among quantum emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Another intriguing by-product of such NZI-enabled stretched wavelength can be considered in “sampling and squeezing” waves through narrow channels [167,168], in which one can effectively transfer an “image” through a subwavelength opening (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(c) and 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The electromagnetic ENZ phenomena have also inspired efforts on other physical, non-electromagnetic, platforms in which other parameters can attain near-zero values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, we theoretically studied how one can conceive electronic metamaterials in which effective mass of electrons can be engineered to be near zero, exploring the topic of “transformation electronics” [173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### Current and Future Challenges Passive NZI materials must inherently be dispersive, and therefore have a finite bandwidth of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, depending on the feature of interest and the frequency domain of interest, the bandwidth may be sufficient to achieve a desired functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, consider the phenomenon of “photonic doping” [169], in which a dielectric rod is inserted in an ENZ medium with an arbitrary cross- sectional geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As viewed by an outside observer, according to the effective medium theory, for two-dimensional scenarios, this “single-inclusion metamaterial” can be treated as an effective medium such that the effective permittivity is still zero (even though the dielectric rod is inserted in the ENZ host) but the effective permeability can be different from unity (notwithstanding both the dielectric rod and the ENZ host are non-magnetic with unity relative permeability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such engineered effective permeability exhibits a resonant dispersion as a function of frequency, but this resonance is mainly due to the size and the permittivity of the dielectric rod, which exhibits narrower bandwidth than the bandwidth over which the ENZ host behaves as a material with near-zero permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' So in this example, the bandwidth of the ENZ host can be sufficient for the resonant behaviour of the relative permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The idea of photonic doping combined with the fact that in the ENZ media the wavelength is “long” has led to the idea of ENZ-based cavity resonators in which the resonance frequency does not depend on the external shape and geometry of the cavity, but instead it is pinned to the ENZ frequency of the materials [174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such geometry-independent cavity resonators have several salient features: (a) they can be the basis for the notion of “flexible photonics”, a paradigm in which changing, bending and morphing the external shape of cavities would not affect its resonance frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' and (b) while changing the external shape and size does not change the resonance frequency, it does affect the quality factor (Q) of the cavity, if a small amount of loss is considered [174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' So here is another unusual Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Illustration of the ENZ supercoupling effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Two parallel-plate waveguides are connected through a narrow ENZ channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As shown in b) for a 180º bend with a narrow channel, the reflection level at the ENZ frequency (w=wp) can approach zero in the limit of vanishing material loss and is typically much weaker than for an empty channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (c) The supercoupling effect may enable the transmission of a complex image through a narrow aperture in a metallic screen embedded in an ENZ material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The image is sampled by an array of metallic wires, which are then “squeezed” through the narrow hole to the other side of the opaque screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (d) Image transported by the array of metallic wires at the ENZ frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Adapted from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [167,168] with permission, copyright (c) American Physical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) (b) 6,=15 6,=~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='6 8,=0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='0 6~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='47 Region F/o,=0 Amplitudeofp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='8r/e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='05 PECwalls 9:0 x=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='4 a: Region 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='2 Empty x-0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='0 channel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='4 Normalizedfrequency,/o, (c) (p) 2-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='31Am -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='86 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='73 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='35 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='86Journal of Optics (2022) #### situation for structured light in which the cavity’s resonance frequency and Q are effectively decoupled from each other, whereas in conventional cavities they are intimately connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Furthermore, remarkably, in the limit of vanishing material loss, core-shell ENZ resonators can support embedded eigenstates in the continuum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', non-radiative bound states which despite being coupled to the radiation continuum do not decay in time [175,176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Geometry-independent ENZ nanoresonators can be exploited in the field of quantum optics [108] where the coupling between an excited atom with a resonant cavity is usually considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In conventional situations of atom-cavity coupling the resonance frequency of the cavity should match the transition frequency of the excited atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This is a delicate balance because a slight change in the shape of the cavity can shift its resonance frequency (since such cavities are usually high-Q cavities) and therefore the cavity would be detuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Moreover, the vacuum Rabi oscillation depends on the cavity Q, which can also be affected by the slight change in the cavity shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The ENZ-based cavities can provide an interesting solution in this scenario in that one can change the Q of the cavity (and thus engineer the vacuum Rabi oscillation) while the cavity resonance frequency stays tuned [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2c) Advances in Science and Technology to Meet Challenges Material loss in some ENZ platforms can be a limiting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Since one of the important features of NZI media is the wavelength stretching, it is important to note how material loss can affect this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As the refractive index is given by , it follows that at the ENZ frequency r i e e = + n i Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a), (b) An ideal epsilon-and-mu-near-zero (EMNZ) 2D material block with arbitrary cross-sectional shape does not influence the field distribution in the unfilled 2D parallel-plate waveguide sections, and thus behaves electromagnetically as a “single point”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In particular, the interaction between two quantum emitters placed in the unfilled waveguide sections is the same as for a straight waveguide regardless of the relative orientation of the unfilled waveguide sections connected to the EMNZ block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (c) Illustration of the decay of an excited quantum emitter enclosed in an ENZ cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The cavity resonance is independent of the shell thickness, and thereby the transition frequency of the emitter is always matched to the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, the coupling strength is sensitive to the shell radius, and hence the Rabi frequency also is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' i) Time evolution of the probability of the excited state for different radii of an ideally lossless ENZ shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' ii) Normalized spectral density for a slightly lossy ENZ shell and different shell radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' iii) Time evolution of the excited state in the presence of the lossy ENZ shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Adapted from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [172,108] with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Panels (a) and (b), Copyright is Open access from OPTICA, panels (ci), (cii) and (ciii) Copyright by National Academy of Sciences of the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' PEC S (a) EMNZ 2 (b) (ci) P(t) (cii) rg=5入0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='0 10 g(w) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='752 r2=1入0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='999 26660 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='0005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='001 (cii) r2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='75 P(t) Rt/(2#) 2mt/(2m)Journal of Optics (2022) #### , resulting in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Thus the wavelength stretching is approximately limited by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Therefore, the lower the , the longer the wavelength stretched in ENZ media, and the more uniform the phase distribution in the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As a result, for any NZI-based application of interest that depends on the phase uniformity, one needs to ask the following questions: How large is the structure and how much phase variation can be tolerated with the effect still observable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The answer to this question determines how large can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, some of the transparent conducting oxides (TCO), such as indium tin oxide (ITO), can exhibit ENZ behavior with below unity in the near-IR regime, while silicon carbide (SiC) behaves as ENZ with smaller in mid IR wavelength [193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' So each of them can be suitable for a different set of applications, as they have different levels of material loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It is important to note that one can also engineer metastructures that mimic some of the NZI properties, while the loss can reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, metallic rectangular waveguides operating at the TE10 cut-off frequency possess wave properties resembling some of the ENZ features, so they can be suitable for microwave frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Furthermore, photonic crystals with the Dirac dispersion with accidental degeneracy exhibit effective refractive index near zero, thus providing a platform for NZI properties at optical frequencies [219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks Near-zero-index (NZI) photonics is an exciting field of optics and electromagnetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It encompasses unconventional ways of structuring light due to wavelength stretching, with the material bulk behaving as an electromagnetic point, exhibiting unique features in light-matter interaction, and offering exciting potential applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Numerous phenomena resulting from the ENZ, MNZ, and NZI features, such as flexible photonics, supercoupling, photonic doping, directive thermal radiation, engineering vacuum fluctuations, ENZ-based quantum optics, super-radiance, emitter-emitter long- range coupling, ENZ electric-based levitation, giant nonlinearity, embedded eigenstates, and more have been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Moreover, electromagnetic NZI concepts have also inspired wave phenomena in other physical domains such as acoustics, electronics, amongst others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' We are hopeful that this field continues to grow, expand, and reveal other exciting wave phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' acknowledges partial support from Simons Foundation/Collaboration on Extreme Wave Phenomena Based on Symmetries, from the Institution of Engineering and Technology (IET) under the A F Harvey Research Prize 2018, and from Instituto de Telecomunicações under project UIDB/50008/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' acknowledges partial support from Simons Foundation/Collaboration on Extreme Wave Phenomena Based on Symmetries, and from the US Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) grant number FA9550-21-1- 0312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' r 0 e = ( ) i i 1 2 e e = = + n i i ENZ 0 i / 2 l l e = ie ie ie ie Journal of Optics (2022) #### 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Strong Coupling Between Atoms and Guided Light Arno Rauschenbeutel, Philipp Schneeweiss, and Jürgen Volz Humboldt-Universität zu Berlin Status The past decade has seen remarkable advances in the field of quantum nonlinear optics, where a strong interaction between individual photons is mediated by quantum emitters [177].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such strong photon-photon interactions are of both fundamental and technological interest: they are the prerequisite for implementing deterministic quantum logic gate operations for processing optical quantum information [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Moreover, photons that strongly interact via a quantum nonlinear medium exhibit complex out-of-equilibrium dynamics that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', enable one to tailor and control the photon statistics of light [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Using free-space light fields, photon-photon interactions have been successfully demonstrated in a number of experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The most established method is to couple atoms with photons that are confined inside a high-finesse optical resonator [180], see Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This allows one to increase the coupling of such a so-called resonator-enhanced atom to the input and output mode of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In this way, the inherently nonlinear response of the atom mediates strong photon- photon interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Alternatively, strong photon-photon interactions have also been demonstrated using the collectively enhanced coupling between propagating light fields and ensembles of strongly interacting Rydberg atoms, so-called Rydberg superatoms [171], see Figure 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Also here the aim is to enhance the coupling of the effective atoms with the input and output light mode to the point where coupling to other modes becomes negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In these scenarios, a key figure of merit is given by the so-called 𝛽-factor, β = 0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='"#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' , (1) which is the ratio of the emission rate of the initially excited (effective) atom into the target mode, Γ1231, and the total emission rate into all possible modes, Γ141, see Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' With regards to employing strong photon-photon interactions in future research and technology, it is, however, essential to couple quantum emitters to guided fields in integrated optical platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This so-called waveguide quantum electrodynamics (QED) setting has been realised with a variety of emitters and waveguide types [182].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' They include semiconductor quantum dots coupled to nanophotonic waveguides, silicon vacancies coupled to diamond waveguides, organic dye molecules coupled to waveguides consisting of an organic crystal-filled glass capillary or to sub-wavelength- diameter silica fibres, so-called optical nanofibres, as well as cold, laser-trapped atoms coupled to nanofibres or one-dimensional photonic crystal waveguides [183, 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Finally, resonator-enhanced atoms, realised by single atoms trapped in the evanescent field of whispering-gallery-mode (WGM) microresonators, have been coupled to nanofibres [185], see Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Waveguide QED systems lend themselves to distributing and processing optical quantum information, to deterministically preparing non-classical states of light, and to realising an almost ideal model-system for strongly correlated, open many-body quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, in addition to the nonlinear response at the single-photon level, many of the corresponding experimental protocols require high 𝛽-factors, which should ideally reach 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A 𝛽-factor that falls short of this value will, at best, lead to a reduced success probability, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', in the case of photon-photon quantum gates [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the worst case, a too small 𝛽-value impedes the implementation of the protocol altogether, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', in the case of a photon number-dependent delay line based on quantum nonlinearities [186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### Current and Future Challenges It is important to note that, for the above-mentioned applications, Γ1231 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (1) refers to the emission into a single spatiotemporal target mode, such that all emitted photons exhibit the same lifetime-limited spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Moreover, for many quantum applications, the photons have to be indistinguishable when they are emitted at different times or by different waveguide-coupled emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In this context, cold atoms stand out in terms of their superior coherence properties and their negligible spread of resonance frequencies, or inhomogeneous broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As a consequence, large ensembles of waveguide-coupled atoms can interact collectively with the guided light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This makes them a prime candidate for scaling up waveguide QED systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Nonetheless, perfect coupling, with 𝛽 ∼ 100%, of a single optical mode to a large number of identical and fully coherent quantum emitters remains an important challenge for existing implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, when coupling laser-cooled atoms to the evanescent field surrounding optical nanofibres, 𝛽-factors in the few- percent regime are expected and have been realised experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For a single atom that is coupled to the evanescent field of a nanophotonic waveguide, maximizing the 𝛽-factor for a given decay rate into the free space modes amounts to maximizing the single photon Rabi frequency, Ω5 = 𝑑⃗ ⋅ 𝐸C⃗5(𝑟⃗)/ℏ , (2) where Ω5 is chosen to be real and positive, 𝑑⃗ is the dipole moment of the atomic transition, and 𝐸C⃗5(𝑟⃗) the field per guided photon at the position of the atom, 𝑟⃗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Now, |𝐸C⃗5(𝑟⃗)| increases approximately exponentially when approaching the waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' From this perspective, it is thus advantageous to trap Figure 1 – We allow at most two figures that are roughly the size of this box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Quantum emitter (yellow sphere) coupled to a target mode (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The 𝛽-factor is defined according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=" (1) with 𝛤&'& = 𝛤(')) + 𝛤&*+&." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Resonator-enhanced atom with optical input and output modes (arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (c) Rydberg superatom composed of a cloud of laser-cooled atoms in the ground state (yellow) and one optically excited Rydberg atom (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The presence of the latter prevents other atoms inside the so-called blockade radius to be excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' M trgtJournal of Optics (2022) #### the atoms at the smallest possible distance from the waveguide surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, for distances smaller than ∼ 200 nm, the van der Waals force becomes so large that it can no longer be straightforwardly counteracted by optical dipole forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Furthermore, we have |𝐸C⃗5(𝑟⃗)| ∝ 1/√𝐴, where A is the cross-sectional area of the guided mode, which can in principle be decreased by increasing the refractive index of the waveguide material and concomitantly reducing the transverse waveguide dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, for the refractive indices accessible with low-loss dielectric materials, even for an atom placed inside a (possibly slotted) waveguide, near-unity 𝛽-values are still out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in Science and Technology to Meet Challenges We now elaborate on three possible strategies that lend themselves to meeting the grand challenge of reaching near-unity 𝛽-factors for single (effective) atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' First, 𝛽-factors of about 47%, that have been experimentally demonstrated for the resonator-enhanced atom, can in principle be further increased by improving the Purcell factor of the WGM resonator, 𝜂 ∝ 𝑄/𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Here, 𝑄 is the quality factor of the resonator, and 𝑉 is the effective resonator mode volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Currently, for fused silica-based WGM resonators, 𝜂 is limited by scattering-induced losses due to surface roughness and pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Here, we thus expect a major step forward by employing advanced resonator production and post- processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Along the same lines, the single-atom 𝛽 can be increased by reducing the group index, 𝑛32 = 𝑐5/𝑣32, of the guided light using photonic crystal waveguides, see Figure 2(b), where 𝑐5 is the speed of light in the unstructured waveguide, and 𝑣32 is the group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The 𝛽-factor of the photonic crystal waveguide is then given by, 𝛽:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' = 𝑛32𝛽5 𝑛32𝛽5 + (1 − 𝛽5) , (3) where 𝛽5 is the 𝛽-factor of a single atom coupled to the unstructured waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Here, the major advancement with respect to previous work will be to realise stable trapping of atoms in a region of the waveguide where 𝛽5 is large while realising a photonic crystal waveguide with small group velocity and small propagation losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While the latter is mostly a technical challenge, atom trapping in high- 𝛽5 regions of a photonic crystal waveguide is still subject to current research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Finally, the coupling of the input and output mode to an ensemble of atoms can be collectively enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Indeed, the collective 𝛽-factor scales with the atom number, 𝑁<1, as, 𝛽;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='4== = 𝑁<1𝛽5 𝑁<1𝛽5 + (1 − 𝛽5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (4) Thus, using only a few hundred atoms, 𝛽;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='4== ∼ 1 is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, the response of an ensemble of independent atoms differs from that of a single quantum emitter when the ensemble interacts with more than one photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Specifically, the inherent nonlinearity featured by each of the atoms is “diluted” because two consecutive photons can interact with different atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Thus, in order to profit from a large collective 𝛽-factor for implementing quantum nonlinearities, one needs to introduce atom-atom interactions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', in the form of a dipole blockade in Rydberg superatoms, see Figure 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Here, the major advancement will be to control the detrimental influence of the nearby waveguide on the coherence properties of the atomic Rydberg levels [187], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', by working with particularly thin nanofibres that feature super-extended evanescent fields [188].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### Concluding Remarks All three approaches towards reaching near-unity 𝛽-factors laid out above, resonator-enhanced atoms, large group index, and waveguide-coupled Rydberg superatoms, see Figure 2(a)–(c), come with considerable technical and conceptual challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Meeting these challenges will mark important advances in science and technology in their own right, ranging from next-generation ultra-high 𝑄 factor WGM microresonators to slow-light waveguides with record-low propagation loss to passivation and charge control of dielectrics at the level of single elementary charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Most of all, the corresponding research and development effort is justified by the exciting applications that are enabled by atomic waveguide QED in the high-𝛽 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In particular, it opens the route towards the implementation of near-ideal fibre-coupled nonlinear quantum devices, which will mark a major breakthrough in quantum optics and constitute a key resource in quantum sensing, quantum metrology, quantum communication, as well as quantum simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements We acknowledge funding by the Alexander von Humboldt Foundation in the framework of the Alexander von Humboldt Professorship endowed by the Federal Ministry of Education and Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Moreover, financial support from the European Union’s Horizon 2020 research and innovation program under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 899275 (DAALI) is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1 – We allow at most two figures that are roughly the size of this box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Resonator-enhanced atom realised by an atom coupled to the evanescent field of a WGM resonator, interfaced using an optical nanofibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The latter is realised as the subwavelength-diameter waist of a tapered optical fibre (TOF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Atom coupled to a photonic-crystal-based waveguide with large group index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (c) Rydberg superatom coupled to an optical nanofibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' WGMJournal of Optics (2022) #### 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Surface Waves Daniel Leykam1 and Daria A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Smirnova2 1National University of Singapore 2Australian National University Status Surface physics is messy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' surface waves are no exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Analytical solutions are scarce, numerical calculations are resource-intensive, and there is a myriad of possible interface configurations to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Emerging from these challenges are elegant theories and numerous applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Surface waves are important because they exhibit remarkable properties unattainable using isolated bulk wave media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Long-studied examples include gravity waves at liquid surfaces, elastic waves at the surfaces of solids, subgap Tamm or Shockley electronic states at terminated semiconductors, and electromagnetic plasmon-polaritons at metal-dielectric interfaces, illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Initial interest in surface waves stemmed from their ability to guide and strongly confine energy, observed most strikingly in the destructive power of seismic Rayleigh waves predicted in 1885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' At smaller scales, electromagnetic surface waves guide and localise light below the diffraction limit [189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' More recently, the non-trivial spatial structure of surface waves such as their transverse spin enables chiral coupling between localised sources and guided modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A long-standing challenge has been ab-initio prediction of the existence and properties of surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Even in the simplest case of homogeneous media described by a few material parameters, novel solutions continue to be discovered, such as Dyakonov surface waves of anisotropic electromagnetic media [190].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Since the 2000s, studies of surface waves have been reinvigorated thanks to the development of topological band theory [191].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Topological band theory enables prediction of novel types of surface waves of periodic media such as photonic crystals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' systems with wavelength-scale variations in their material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Magnetic field distribution (the out-of-plane component Hz) in the TM-polarised surface plasmon-polariton wave, showing different transverse localization scales in the two media (air and metal) brought into contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### Topological band theory shows promise as a systematic approach for designing surface waves and optimising their properties for applications including precision sensing, compact waveguides, and signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' At the same time, techniques from the structured light community are being fruitfully applied to study topological bands [192].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Ongoing research aims to better understand connections between surface waves emerging for different classes of waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This will not only allow us to design and optimise surface waves in a variety of wave systems, but also observe novel physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, Weyl semimetals are topological materials supporting coexisting surface and bulk modes exhibiting Weyl quasiparticles originally hypothesised in 1929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In 2015, Weyl semimetals were observed for the first time using an electronic system (TaAs) and an analogous microwave photonic crystal [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges While there has been great success in emulating edge modes of 1D and 2D condensed matter systems, studies of protected surface waves of 3D systems remain in their infancy [194].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such surface waves exhibit linear Dirac-like dispersion, with locking between their spin and momentum, illustrated in Figure 2 and elaborated on further in Section 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' One challenge is that many models of topological surface waves were originally formulated for electronic condensed matter systems with fermionic spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' To implement similar surface waves for classical wave systems requires other effects such as bi-anisotropy, use of orbital angular momentum modes as a spin-like degree of freedom, or additional crystalline symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Limits to the strength of these effects may lead to non-ideal dispersion relations, such as surface waves co- existing with bulk bands, resulting in bulk scattering losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Massless Dirac-like dispersion of topological surface waves with spin–momentum locking within the bulk gap in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The (pseudo-)spin texture is illustrated by black arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Bulk modes Frequency Surface waves Bulk modes ky kzJournal of Optics (2022) #### Not all classes of surface waves may be accessible in a given material platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, sound-based phononic crystals are based on manipulating scalar waves and can therefore be well- described by simple tight binding models [195].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' By contrast, 3D photonic crystals typically have band structures complicated by orbital and polarisation degrees of freedom, making space group arguments insufficient to guarantee the existence of protected surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The holy grail is the ability to identify the best possible surface wave for a given application subject to material or design constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' At first glance, this seems like a hopeless task given the explosion in degrees of freedom compared to uniform media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Topology allows us to make concrete statements about some properties of surface waves, such as the difference between the number of forward and backward propagating waves, independent of details such as precise material parameters or the interface shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, continuous parameters such as the wave speed or degree of localization are not topologically protected and must be optimised using conventional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Another challenge is to understand the robustness of surface waves against effects including scattering losses, fabrication imperfections, imperfect symmetries, and incomplete band gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Topological band theory as originally developed for electronic condensed matter materials was limited to lossless, non-interacting wave systems described by the Schrödinger equation, analogous to the paraxial wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Generalising beyond these constraints will allow us to identify robust surface states for new kinds of wave media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in Science and Technology to Meet Challenges Advances in fabrication technologies will expand the platforms available for implementing surface waves and give us new tools to tune their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Areas of active research include heterogeneous multilayer metamaterials, novel two-dimensional materials such as graphene, hexagonal boron nitride and twisted monolayers [196], nanostructured metasurfaces, hybrid polariton systems, surface magnetoplasmons in gyrotropic materials, and even electronic topological materials [197].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 3D printing is maturing as a fast and flexible approach towards prototyping topological surface waves, with many recent high-profile works in acoustics [195] covered in Section 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For electromagnetic waves, 3D printing is practical for microwave photonic crystals, but scaling up to optical frequencies remains challenging due to the need for 3D nanofabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' State-of-the-art methods such as direct laser writing have been used to realise topological edge and surface waves in the terahertz and near-infrared [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in 3D printing including finer resolution and the ability to incorporate more combinations of materials will open up new possibilities for surface waves in the visible frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Methods from the field of structured light may offer an easier route towards generating and finding useful applications of topological surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Specifically, one can use internal degrees of freedom such as orbital angular momentum as synthetic dimensions, with hopping along the synthetic dimension mediated by periodic spatial or temporal modulation [198].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Surface waves that mix spatial and internal degrees of freedom show promise for applications such as robust, high-efficiency mode conversion and non-reciprocity in planar integrated photonic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Realising this goal will, however, require the integration of high-efficiency optical modulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Generalisations of topological band theory to broader classes of wave media are being actively pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These include active or lossy (non-Hermitian) systems (see also Section 6), effective medium theories describing metamaterials [199], non-periodic, dispersive, and nonlinear media [200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in these directions will allow us to identify novel combinations of materials supporting surface waves and better understand their robustness to losses and other perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, Journal of Optics (2022) #### for high power device applications, it is essential to understand the conditions under which nonlinear surface waves remain stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' There is growing interest in applying machine learning techniques to physics problems [201].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Potential applications to surface waves include discovery of interface designs with superior surface wave properties via generative modelling, identification of new classes of topological wave media supporting robust surface modes, and mesh-free neural network-based beam propagation methods for numerical simulation of complex interface geometries [202].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Applications of machine learning to the broader field of structured waves are discussed further in Section 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='. Concluding Remarks Topological band theory has led to a resurgence of interest in surface waves in quantum and classical systems in a similar vein to how analogies with bulk electronic band structures gave rise to the fields of photonic and phononic crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While researchers are still attempting to understand all the subtleties of topological phases, there is no doubt that these discoveries will require band structure textbooks to be rewritten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' So far, the flow of ideas has largely been unidirectional, from electronics to photonics and acoustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Given the greater appreciation of the universality of bulk and surface waves occurring in various fields, there is great potential for recent advances in structured light to be applied to electronic and acoustic systems [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Engineered interfaces and surface waves will serve as a flexible testbed for probing relativistic physics (fundamental science, quasiparticles) and strong light-matter interactions (due to the field confinement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in their basic science will lead to technological breakthroughs in more applied areas, such as ultra-thin, resilient, and flexible surface wave-based devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' We have only scratched the surface of potential applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' acknowledges support from the Australian Research Council (DE190100430).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Photonic spin-orbit interactions at metasurfaces: stochastic, Rashba and quantum effects Kexiu Rong1, Bo Wang2,1 and Erez Hasman1 1Technion – Israel Institute of Technology 2Shanghai Jiao Tong University Status Light possesses both spin and orbital angular momentum (OAM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' the former is associated with circular polarisation states, and the latter arises from azimuthal phase gradients of the light field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A coupling between spin and OAM (or linear momentum) occurs when light interacts with anisotropic or inhomogeneous structures, giving rise to optical phenomena in which the spin of light affects and controls the spatial degrees of freedom of light, such as the vectorial field distribution and propagation path [203,204].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These spin-orbit interactions (SOIs) bring forth novel spin-optical effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', photonic spin Hall effect (PSHE) and photonic Aharonov-Bohm effect) and enable efficient spin- dependent light manipulations [205-208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Metamaterials are artificial structures assembled from multiple elements smaller in scale than the wavelength of external stimuli, endowing a medium with unique electromagnetic responses and functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Metasurfaces [209-215], metamaterials of reduced dimensionality, are phased arrays composed of resonant optical nanoantennas, which facilitate substantial control of local light scattering properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Controlling the electromagnetic response of metasurfaces can be achieved by a geometric phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', Pancharatnam-Berry phase) mechanism [88,209], enabling an excellent platform to investigate new types of SOI effects from the classical to quantum regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In this roadmap, we would like to introduce novel types of SOIs utilising geometric phase metasurfaces (GPMs): (i) Stochastic PSHE [211], (ii) Photonic Rashba effect [212], (iii) Quantum entanglement between the spin and the OAM of photons [213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (i) The study of SOIs in disordered systems offers a wealth of interesting effects and numerous potential applications, such as suppressing undesired optical scatterings and achieving ultra-sensitive optical metrologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' An optical metrology that can detect extremely weak disorders in a deep- subwavelength resolution is critical for nanotechnology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recently, we reported on a stochastic PSHE arising from space-variant Berry-Zak phases, which are generated by disordered magneto-optical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This effect is observed from a spatially bounded lattice of ferromagnetic meta-atoms displaying nanoscale disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Our approach may be used for sensing deep-subwavelength disorders by actively breaking the photonic spin symmetry and may enable investigations of fluctuation effects in magnetic nano-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (ii) Heterostructures combining a thin layer of quantum emitters and planar nanostructures enable custom-tailored photoluminescence in an integrated fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recently, there has been a surge of interest in selectively manipulating quantum emitters—that is, valley excitons—in transition metal dichalcogenide (TMD) monolayers due to their potential as an alternative information carrier in valleytronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' GPMs constructed of anisotropic nanoantennas with space-variant orientations allow the manipulation of light by the spin degree of freedom [209,210].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This is enabled by the polarisation evolution of light on the Poincaré sphere, thus generating spin-dependent Pancharatnam-Berry phases for the spin-flipped components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Hence, the GPMs represent an attractive candidate to perform the desired valley separation required by valleytronics, inspired by spin-dependent phenomena such as the PSHE and photonic Rashba effect underpinned by SOIs [216-218].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### (iii) Quantum information provides a route to solve problems in reduced time and complexity by exploiting fundamental quantum principles such as superposition and entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Moreover, due to a relatively easy manipulation and long quantum coherence time, single photons encoded with quantum states are an appealing candidate to implement quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Hence, generating and manipulating entangled photon states using SOIs mediated by metamaterials is at the heart of the field of photonic quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges A challenge of dealing with weakly disordered nanostructures is the limited opportunities to extract the information from subtle light-matter interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The disorder and stochastic nature hinder light from detecting any useful information other than that from a homogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The previous strategies to overcome this limitation involve the implementation of special optical conditions including critical angle, symmetry broken, and resonant enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Specifically, the emerging photonic spin-dependent effects due to symmetry broken in arrays of anisotropic nanoantennas provide a SOI mechanism to achieve a sensitive optical metrology [214].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, the spin-dependent effects in these architectures naturally disappear when the anisotropic nanoantennas are replaced by isotropic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' More importantly, how to accurately quantify the structure fluctuations as a function of the measurable spin-split effects remains largely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the pursuit of new spin-optical devices possessing large information capacity and high processing speed, a long-thought goal is to interface spinoptics and spintronics for an interchange of spin information between photons and electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This requires the miniaturisation of spin-polarised sources down to a nanometric scale and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Currently, a family of atomic-thin materials has Figure 1 – We allow at most two figures that are roughly the size of this box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Optical metrology with stochastic PSHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Sketched PSHE from a magnetized disorder metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A polarised incident beam (polarisations indicated by the cyan arrows) is reflected and split into spin-up (σ+) and spin-down (σ–) components with a subdiffraction- limited angle δ, due to disordered magneto-optical Kerr rotations (the disordered cyan arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' R indicates the radius of a circular nanoantenna, D is the beam’s diameter, and B is the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The bottom panel exemplifies a radius distribution of disordered nanoantennas, with ΔR being the fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Probability distribution of stochastic PSHEs P(δ), with Δδ being the standard deviation of the Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' λ is the wavelength of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (c) Experimental and calculated Δδ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' radius fluctuation ΔR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' M is magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Insets: scanning electron microscopy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Scale bar, 1 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Reprinted with permission from [211], copyright 2020 The Authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) B (b) (c) ×10-5 MAR D (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=') 5 (art)ev 248 4 P(a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='5 N 0 -15 -10 -5 5 10 15 Noise leve 8×105 (/D) 0 5 10 15 20 25 30 △R (nm)Journal of Optics (2022) #### triggered intense research due to their exotic electrical, optical, and thermal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Particularly, direct bandgap TMD monolayers show opposite electronic spins at ±K valleys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Consequently, the valley information can be selectively encoded and retrieved by the photonic spin according to the valley- dependent optical selection rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Although great efforts have been devoted into this field, previous strategies using metallic structures inherited intrinsic losses and limited functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Alternatively, versatile GPMs represent an attractive candidate to perform such a task, inspired by, for example, the photonic Rashba effect that describes a momentum-space spin-split dispersion from inversion- asymmetric structures [215].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, conventional GPMs are generally designed for plane waves, preventing an efficient interaction between nanoantennas and integrated valley excitons behaving as in-plane circular dipole emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A similar issue also arises when single quantum emitters are integrated with GPMs to investigate SOIs in the quantum regime, where a further requirement of long quantum coherence times (or low losses) should be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in Science and Technology to Meet Challenges (i) Stochastic PSHE: In a recent work [211], we approached the metrology challenge in weakly disordered systems by exploiting magnetised disordered metasurfaces (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Nanoscale size fluctuations were revealed by the probability distribution of a stochastic PSHE, which was induced by disordered magneto-optical Kerr rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Here, the metasurfaces are consisted of circular nickel nanoantennas with radii randomly fluctuated in several nanometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The random variations in sizes of nanoantennas give rise to disordered geometric phases from magneto-optical Kerr rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This leads to a spin-dependent beam shift being several orders of magnitude smaller than the diffraction limit of light, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', a PSHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' By evaluating the PSHEs via weak measurements from many disordered metasurfaces with different randomisations, we observe a Gaussian probability distribution for the spin shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Notably, the standard deviation of the Gaussian distribution is proportional to the size fluctuation of the nanoantennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This result enabled us to detect a five-nanometer size fluctuation of nanoantennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (ii) Photonic Rashba effect from quantum emitters: On the other hand, we tackled the weak interaction between integrated valley excitons and nanoantennas by exploiting a novel platform of Berry phase defective photonic crystals (BP-PhCs) [212,213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The BP-PhCs are composed of a PhC slab with isotropic nanopillars and a GPM with space-variant anisotropic nanoantennas that serve as defects (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' By utilising the bandgap of the PhC slab, the insertion of the GPM into the PhC slab gives rise to a near-field geometric phase defect mode, which couples the defects for an effective interaction with the integrated valley excitons, resulting in site-controlled excitation, photoluminescence enhancement, and spin-dependent manipulation of individual valley excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Consequently, a spin-split dispersion from valley excitons is observed in momentum space, manifesting as the photonic Rashba effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Particularly, the spin-up and -down branches correspond to emission from ±K valley excitons, respectively, indicating a valley separation in momentum space at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Moreover, this basic interaction mechanism between circular dipole emitters and nanostructures can be generalised to quantum emitters with arbitrary in-plane polarisations and PhC structures with distinct symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (iii) Quantum photonic metasurfaces: In the preparation of entangled photons [213], we used lossless dielectric metasurfaces, the near unity efficiency of which enables the manipulation of single photons under a sufficient long quantum coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' To entangle single photons’ spin and OAM, a GPM embedded with a spin-dependent helical phase was designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Depending on the sign of the photon spin, the GPM performs a unitary transformation that adds or subtracts one quanta of OAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### In our case, the spatial wavefunction of the photon is paraxial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' therefore, the spin and the OAM are independent and have Hilbert spaces of different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Consequently, single photons in entangled spin and OAM states, and photon pair with nonlocal correlations between the spin of one photon and the OAM of another photon are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks Optical SOIs are ubiquitous in nano- and atomic-scale systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' An in-depth understanding of them contributes to both fundamental physics and advanced applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Our results suggest that Pancharatnam-Berry phase optical elements are suitable for discovering new types of SOIs, which show promising applications in novel photon transport controls, such as entangled photons, spin- polarised light sources, and ultra-sensitive optical metrologies utilising splits of non-degenerated spin modes distinguished by quantum weak measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' To investigate nanophotonics under unprecedented extreme conditions, the spin-controlled generation, manipulation, and detection of atomic-scale light sources of various statistical properties—e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', spontaneous emission (super- Poissonian), stimulated emission (Poissonian), and quantum emission (sub-Poissonian)—are promising fields, which we foresee many possibilities in the coming future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In general, introducing spin-orbit coupling of electromagnetic waves into contemporary photonics and atomic-scale optics may result in the development of a new area of research, that is, atomic-scale spinoptics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements The authors gratefully acknowledge financial support from the Israel Science Foundation (ISF), the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Air Force Office of Scientific Research (FA9550-18-1-0208) through their program on Photonic Metamaterials, the Israel Ministry of Science, Technology and Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The fabrication was performed at the Micro-Nano Fabrication & Printing Unit (MNF&PU), Technion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1 – We allow at most two figures that are roughly the size of this box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Photonic Rashba effect from valley excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Illustration of a heterostructure combining a BP-PhC and a WSe2 monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The PhC slab composed of isotropic nanopillars is arranged in one lattice, and the GPM composed of anisotropic nanoantennas is arranged in another lattice, serving as defects to the PhC slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' By exploiting the emerging Berry-phase defect mode, valley excitons effectively interact with the defects for coherent geometric phase pickups, leading to a photonic Rashba effect in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Inset: Schematics of valley-dependent optical selection rules for ±K valley excitons in WSe2 monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Simulated defect band of a BP-PhC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (c) Measured spin-split dispersion in momentum space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', photonic Rashba effect) from valley excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Reprinted with permission from [212], copyright 2020 The Authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) WSe, Si (b) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='7 defectband 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='60 + exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 00Z theory 90 Wavelength (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='58 760 Wavelength (nm) 006 wa/(2nc) /em) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='4 M 1300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='54 820 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='2 k, (2r/a)Journal of Optics (2022) #### 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Spin, momenta, and forces in evanescent waves – towards spatial and temporal structuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Michela F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Picardi1,2, Anatoly V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Zayats1 and Francisco J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rodríguez-Fortuño1 1King’s College London and London Centre for Nanotechnology 2ICFO – Institut de Ciencies Fotoniques Status Coupling between spin and orbital angular momenta is a fundamental property of electromagnetic (EM) waves, but it is especially pronounced in the evanescent fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' When modes are spatially confined along at least one dimension, as is the case of dielectric or plasmonic waveguides or fibres, their wavevector becomes complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The wavevector component along the direction of confinement at the interface between two media is imaginary, resulting in the field exponentially decaying away from the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Many interesting phenomena originate from the topological features of evanescent waves related to spin-momentum locking [207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The transversality condition (𝐤 ⋅ 𝐄 = 0) requires the evanescent fields to be elliptically polarised with a field component parallel to the propagation direction, in sharp contrast to free-space fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This locks the handedness of the waveguided fields with their propagation direction [207], meaning that when either of the two is reversed, the other must be reversed too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This behaviour arises from basic laws of electromagnetism and exists even in unpolarised light [220].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such topological protection of the propagation direction of guided waves is valid until the spin flips, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', due to scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Because of spin-momentum locking, elliptically polarised EM sources excite guided modes unidirectionally via evanescent coupling [221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Selective mode excitation was also achieved considering the reactive power of evanescent waves Im(𝐄∗ × 𝐇) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Being perpendicular to the interface, the reactive power is locked with the direction of evanescent decay so that multipolar EM sources may excite a guided mode, without any directionality, only if their reactive power is parallel (not antiparallel) to that of the evanescent wave, as in the case of Janus dipoles [222].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Selectivity and directionality of the guided mode launching may also be understood as near-field interference, with the same evanescent modes being excited and interfering with their different symmetries inherited from different EM source multipoles [221,359].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Reciprocal effects in controlling the polarisation of far-field directional scattering with the direction of guiding modes were also demonstrated [223].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The spinning fields of guided modes were widely exploited to achieve polarisation-dependent optical forces on achiral objects [360].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' All these effects stem from topological properties of evanescent fields of the guided modes and have been observed in an extremely broad spectral range, from optical to radio frequencies, with various types of waveguides and EM sources, such as multipolar emitters and scatterers, atoms and quantum dots, and predicted for other kinds of evanescent waves, such as acoustic and gravitational ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges The phenomena described above follow from classical photonics but can also be applied in the quantum realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The coupling between quantum dots and complex light, such as evanescent waves, leads to chiral directional single photon routing [361].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Single-photon evanescent waves and their entanglement open up new possibilities in the development of light-matter interfaces and quantum technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### Structured evanescent waves are obtained by relaxing the condition of a single-wavevector, such as in surface plasmon vortices [225,362].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structured evanescent waves are already proving a fertile ground for physical phenomena such as dynamic field-skyrmionic lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Exploiting spin to orbital angular momentum coupling, spin-skyrmions in the evanescent field have been demonstrated (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(e)) [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structured evanescent waves are at their infancy, and inspiration can be taken from the myriad of structured beams considered in free space to find and exploit analogues in the near-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As well as structuring in space, time-domain structuring can be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For propagating waves, time-dependence leads to remarkable polarization patterns in multicoloured light [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It is therefore to be expected that the routing and metrological capabilities of single-coloured evanescent waves—based on their unique polarization patterns—will acquire additional features and versatility when the time-harmonic assumption is dropped, in favour of multi-coloured and pulsed evanescent waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The combination of both spatial structuring and time-dependence, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', in spatio-temporal vortices [19-24], will greatly extend the evanescent wave playground, providing time-dependent photonic topologies in the presence of a complex wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Guided spatio-temporally structured waves could be used for encoding classical or quantum information into topological invariants such as vortex winding numbers as a novel means of guided wave information transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Optical forces near surfaces is another vast field enabled by the properties of guided waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Counter-intuitive effects such as levitation, polarisation-controlled forces, or chiral sorting for particles and molecules is an active field of study [363-365] based on the peculiar spin and momentum properties of simple non-structured time-harmonic surface evanescent waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The use of structured Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Magnetic field amplitude of a surface plasmon-polariton unidirectionally excited by a circularly polarised electric dipole, adapted from reference [221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Reciprocal control of far-field scattering polarization via unidirectional near field excitation, adapted from reference [223].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (c) SEM image of the nanosphere placed on the waveguide and light intensity at the two ends of a fibre measured by varying the polarisation of the light incident on the nanosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The nanosphere acts as a point dipole, and when it is circularly polarised, it will couple unidirectionally to the mode guided by the fibre, adapted from reference [224].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (d) Structured surface plasmon- polariton vortex excited via a plasmonic vortex lens, reprinted from [225].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (e) Skyrmion-like structure of the spin angular momentum vector (top) and the intensity distribution (bottom) of a surface wave vortex, adapted from reference [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' a b LCP air metal C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='8 e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2 Max JEPJournal of Optics (2022) #### and multi-coloured evanescent wave illumination can lead to precisely engineered optical force fields for optically tuneable manipulation of nanoparticles, atoms, and molecules near surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Even more layers of complexity will be introduced if the assumption of polarised and coherent light is dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Transverse spin of the evanescent field persists even when light is fully unpolarised [220].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Incoherent light (with a fluctuating phase) will show similar behaviour, opening a vast playground of statistical optics with evanescent waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This could enable new applications, for example, chiral optical force separation of enantiomers using non-laser light sources, and applications using thermal emission or even sunlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in Science and Technology to Meet Challenges Despite significant progress in describing spin-orbit properties of light fields, further advances in the understanding of dynamical properties of near-fields are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Light properties are typically defined via their effect on material probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For instance, the spin angular momentum density of an evanescent wave is proportional to the mechanical torque exerted by the wavefield on a dipolar particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Novel dynamical properties are uncovered when considering more complex particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The optical torque on a quadrupolar particle is not proportional to the spin, but instead to another quantity: the spin of the field gradient [227].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Given potential applications of optical spin effects in evanescent fields, such as transverse spin and skyrmions, higher-order field properties—such as the above-described field gradient spin and other related properties that arise in the evanescent field interactions with higher- order multipoles—call out for an in-depth analysis and deeper understanding of light properties in the near field, especially in structured, dynamic, and multi-colour settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' More immediate important technological challenges are related to reducing the effects of losses if surface plasmons are chosen as a platform for evanescent waves, and to improving the coupling Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Schematic representation of the main interesting phenomena observed in non-structured time-harmonic surface waves (top), and the possible direction of expansion into new domains which can be accessed via space and time structuring of surface waves (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='efficiencies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='emitters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='scatterers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='evanescent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='wave ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='platforms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='such ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='rib ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='slot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='nanophotonic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='waveguides ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='nanofibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The temporal structuring of surface waves will need advances in attosecond physics and the control of the intra-pulse polarization evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in material science, nanophotonic platforms, and optical force instrumentation will all be needed to exploit the full potential of spatially and temporally structured surface evanescent waves and their effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The miniaturisation of EM emitters and improvement in their coupling efficiencies as well as the successful integration of quantum dots,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' molecules,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' and atoms into nanophotonic devices are requirements for many of the proposed and potential applications: from nanoscale quantum technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' requiring single-photon spin routing based on evanescent wave spin- momentum locking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' which could form part of optical quantum computing or secure quantum communication platforms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' to the use of evanescent fields for on-chip chiral optical forces that might enable important applications like all-optical on-chip separation of enantiomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks The applications in quantum technologies, nanophotonics, sensing, metrology and nano-opto- mechanics drive requirements on making use and manipulation of all degrees of freedom of guided light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In addition to polarisation-controlled routing and coupling of electromagnetic waves,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' which have already resulted in applications in position sensing with nanometric displacement sensitivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' integrated miniaturised polarimeters and coherent-optical receivers as well as unusual lateral and repulsive optical forces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' the use of structured evanescent waves with new dynamical and topological properties above and beyond spin-orbit interaction offer innovative solutions towards miniaturisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' energy efficiency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' and ultrafast operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These future advances will be made possible in close interaction of theory, developing new understanding of angular momenta properties in structured and dynamic evanescent fields, nanoscience, and photonic instrumentation for realising and measuring these properties, and development of photonic and nanophotonic platforms tailored for specific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements This work was supported by the European Research Council projects iCOMM (789340) and Starting Grant ERC-2016-STG-714151-PSINFONI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Momentum and spin of electromagnetic, sound, and water waves Konstantin Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Bliokh RIKEN Status Wave momentum has a long (and probably never-ending) history starting from 18th century studies by Euler, with the landmarks of the electromagnetic momentum by Poynting and quantum- mechanical momentum by de Broglie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It includes numerous controversies: the momentum of sound waves [228], the Abraham-Minkowski dilemma for the momentum of light in a medium [229], the Belinfante-Rosenfeld problem for the energy-momentum tensor in field theories [230], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Angular momentum (AM) is closely related to the momentum, and is also involved in these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recent great interest in structured waves, materials, and wave-matter interactions prompted thorough revision of local momentum and AM properties of generic inhomogeneous (although often restricted by monochromaticity) wave fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Starting with the general field-theory approach, there are two types of momentum and AM densities: (i) the kinetic momentum density 𝚷 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', the Poynting momentum in electromagnetism) and kinetic AM density 𝐌 = 𝐫 × 𝚷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (ii) the canonical momentum density 𝐏 and the canonical AM density 𝐉 = 𝐫 × 𝐏 + 𝐒, where 𝐫 × 𝐏 = 𝐋 and 𝐒 are the orbital and spin (intrinsic AM) densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The kinetic and canonical quantities are related by the Belinfante-Rosenfeld equation [230]: 𝚷 = 𝐏 + ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' - 𝛁 × 𝐒 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (1) One can regard the canonical momentum and spin densities as two fundamental independent quantities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' other momentum/AM characteristics are their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Both the kinetic and canonical quantities are important and have their own advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, in relativistic field theory, the kinetic quantities are expressed via fields and are explicitly gauge-invariant, while the canonical ones follow directly from Noether’s theorem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', provide generators of translations and rotations in the quantum-mechanical formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Notably, the canonical momentum and spin densities allow meaningful gauge-invariant expressions only for monochromatic fields, but these are directly observable via forces and torques exerted by monochromatic fields on dipole particles [231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For monochromatic electromagnetic fields in free space, the canonical momentum and spin densities read [231–234], 𝐏 = ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' @A Im[𝜀 𝐄∗ ⋅ (𝛁)𝐄 + 𝜇 𝐇∗ ⋅ (𝛁)𝐇] , (2) 𝐒 = ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' @A Im(𝜀 𝐄∗ × 𝐄 + 𝜇 𝐇∗ × 𝐇) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (3) Here, 𝐄(𝐫) and 𝐇(𝐫) are the complex electric and magnetic field amplitudes, whereas 𝜀 and 𝜇 are the vacuum permittivity and permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (2) and (3) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (1) and using Maxwell equations yields the kinetic Poynting momentum 𝚷 = ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' -;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='% Re(𝐄∗ × 𝐇), where 𝑐 = 1/√𝜀𝜇 is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Importantly, there is a freedom in defining the canonical momentum and spin densities (2) and (3) even when the Poynting momentum is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (2) and (3) represent symmetric ‘arithmetic mean’ of the electric and magnetic contributions, one can use either purely electric or purely magnetic quantities instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The reason for choosing the dual (electric-magnetic) symmetric expressions is Journal of Optics (2022) #### mostly aesthetic: to preserve the dual symmetry inherent in free-space Maxwell equations [231–234].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, the presence of charges breaks this symmetry because only electric charges exist in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In addition, electric-dipole and magnetic-dipole particles effectively interact with the electric and magnetic parts of the canonical densities (2) and (3) so that their electric/magnetic/symmetric form is not fixed fundamentally but rather chosen in each particular problem using auxiliary arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Sound waves in a fluid or gas have similar momentum and spin properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Although sound waves are often regarded as ‘scalar’ or ‘spinless’, these are actually vector waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' the local displacement (or velocity) of the medium particles provides the vector wavefield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Akin to electromagnetic waves, one can describe monochromatic sound waves via two complex fields: velocity 𝐯(𝐫) and pressure 𝑝(𝐫).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The canonical momentum and spin densities of monochromatic sound waves are [41–44]: 𝐏 = B -A Im[𝐯∗ ⋅ (𝛁)𝐯] , (4) 𝐒 = B -A Im(𝐯∗ × 𝐯) , (5) where 𝜌 is the mass density of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (4) and (5) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (1), supplied with the acoustic wave equations for 𝐯 and 𝑝, yields the acoustic analogue of the Poynting vector: 𝚷 = ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' -;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='&% Re(𝑝∗𝐯), where 𝑐𝑠 = 1/>𝜌𝛽 is the speed of sound with 𝛽 being the compressibility of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' From the field-theory viewpoint, the canonical quantities (4) and (5) also allow different forms compatible with the same kinetic momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Namely, instead of the velocity-related quantities (4) and (5), one can use the pressure-related quantities (𝐏 = D -A Im(𝑝∗𝛁𝑝) = 𝚷 and the spin density vanishes in this case), or an ‘arithmetic mean’ of the velocity-related and pressure-related contributions [43,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Furthermore, monopole and dipole acoustic particles are effectively coupled to the pressure-related and velocity-related forms of the canonical quantities [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Then why do we prefer the ‘asymmetric’ velocity-related definitions (4) and (5)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This is because, in contrast to electromagnetism, sound waves exist only in a medium, and one can associate their dynamical properties with microscopic mechanical properties of the medium particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Indeed, local rotation of the medium particles in a generic sound-wave field with an elliptical polarisation (polarisation of the velocity field 𝒗 corresponds of the microscopic real-space trajectory of the particle) generates exactly the canonical AM density (5) [41,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Furthermore, the medium particles slowly drift in the sound- wave field due to the second-order difference between the Euler and Lagrange coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This is the Stokes drift with the velocity 𝐮 exactly corresponding to the canonical momentum density (4): 𝐏 = 𝜌𝐮 [55,235].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Thus, microscopic mechanical properties of the medium allow one to unambiguously determine the canonical momentum and spin densities in sound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Remarkably, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (4) and (5) are quite general and are also valid for elastic waves in isotropic solids [49,50,53] or Langmuir plasma waves [236].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The above features make the canonical momentum and spin in acoustic waves directly observable, at least in principle, via microscopic motion of the medium particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In practice, such observation is challenging with typical sound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Larger-scale waves, such as water-surface waves, can serve as a perfect platform for the observation of microscopic medium properties in structured wave fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Considering monochromatic deep-water gravity waves with the dispersion 𝜔2 = 𝑔𝑘 (𝑔 is the gravitational acceleration) as a quasi-2D wave system, we recently derived the canonical Journal of Optics (2022) #### momentum density in the unperturbed surface (𝑥, 𝑦)-plane and the corresponding spin density in the vertical 𝑧-direction [55]: 𝐏 = B A Im[𝐕∗ ⋅ (𝛁 )𝐕 + 𝑊∗𝛁 𝑊] , (6) 𝐒 = B -A Im(𝐕∗ × 𝐕) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (7) Here, 𝛁2 = (𝜕𝑥, 𝜕𝑦), whereas 𝐕 = (𝑣&, 𝑣*) and 𝑊 = 𝑣+ are the in-plane and vertical velocity components of water particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (6) and (7) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (1), and using the equations of motion for gravity waves, yields the kinetic momentum 𝚷 = BF A Im(𝑊∗𝐕), which is consistent with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [237].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Akin to sound waves, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (6) and (7) correspond to the mechanical momentum (due to the Stokes drift) and the microscopic mechanical angular momentum (due to the local elliptical trajectories) of water particles, as shown in Figure 1 [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Thus, the momentum and angular momentum properties of electromagnetic, acoustic, and water waves have profound similarities related to the presence of spin and the fundamental Belinfante- Rosenfeld relation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' There is also a principal difference in that the free-space electromagnetic quantities are not associated with any medium (‘ether’) and cannot be derived from microscopic mechanical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, electromagnetic waves in a medium represent mixed light- matter waves and do involve microscopic mechanical properties of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The kinetic momentum of an electromagnetic wave in an isotropic dispersive medium with permittivity 𝜀 = 𝜀(𝜔) and permeability 𝜇 = 𝜇(𝜔) is still given by the Poynting (or Abraham) momentum 𝚷 = Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Theoretically calculated surface distributions of the canonical momentum 𝐏 (black arrows) and spin 𝑆, (blue-red) densities, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (6) and (7), in the interference of two plane gravity (water-surface) waves with orthogonal wavevectors 𝐤-,$.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Numerical and experimental plots show trajectories of microscopic water particles for three wave periods 6π/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The Stokes drift of the particles and their elliptical motion correspond to the canonical momentum and spin, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The normalized surface coordinates are 𝑋 = √2𝑘𝑥, 𝑌 = √2𝑘𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Adapted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' theory numerics experiment V canonical momentum (0) (O) (O) (0) 0) m 元 00 O) (0)) (O) (O) 0 (0) (O) ) 0 元 spin (C) (0) (0) (O) 000 Q00 2元 wavevectors Stokesdrift spin X 0 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' )Journal of Optics (2022) #### ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' -;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='% Re(𝐄∗ × 𝐇), while the canonical momentum and spin densities can be described by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (2) and (3) with the substitution [236,238], 𝜀 → 𝜀̃ = 𝜀 + 𝜔 PQ PA , 𝜇 → 𝜇I = 𝜇 + 𝜔 PR PA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (8) In this case, microscopic contributions from the motion of the medium particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', electrons or atoms) play a crucial role [235,236,238,239], and the canonical expressions can be regarded as the Minkowski-type momentum and spin densities [238].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Equations (2), (3), and (8) are found within the dual-symmetric formalism, but the electric-field-biased approach is also possible and can be relevant because the medium is usually electric-biased on the microscopic level [236] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', consisting of electric charges and dipoles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges Despite the great progress in the description of the momentum and angular momentum of structured waves, there are still many unsolved questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' To name a few: extension of the above approach (if possible) to anisotropic media, polychromatic fields, and fields with complex frequencies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', Mie quasimodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in Science and Technology to Meet Challenges Rapid development of nanophotonics, including metamaterials and plasmonics, provides a perfect platform for theoretical, numerical, and experimental studies of fundamental dynamical properties of complex fields and their interactions with matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks We have briefly described the canonical and kinetic momentum and AM properties of monochromatic electromagnetic, acoustic, and water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Despite enormous differences in scales and their nature, the momentum and AM of these waves share profound similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This reflects the universality of these concepts, as well as the remarkable role of the spin and field-theory relations even in ‘spinless’ classical waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acoustic spin Chenwen Yang and Jie Ren Tonji University Status The acoustic waves in fluids were traditionally mis-regarded as spinless fields because of their curl- free nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recently, several works about elastic and acoustic waves show that even pure longitudinal waves that can be fully described with a scalar field still have the ability to carry spin angular momentum (SAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The existence of SAM does not require an extra spatial degree of freedom but needs a locally temporal rotation of the vector field, like displacement or velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A clear definition of SAM in acoustic wave systems will pave a new way to understand and realise various structured acoustic wave systems, included but not limited to the spin-momentum locking in acoustic wave systems and the symmetry selective excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The expression of spin angular momentum (SAM) in acoustic/elastic waves could be derived from the definition of angular momentum in acoustic/elastic waves, which is [42,50,53,240,355]: 𝐒 = 𝜌𝜔 2 Im[𝐮∗ × 𝐮] = 𝜌 2𝜔 Im[𝐯∗ × 𝐯], (1) where 𝐮∗ represents the conjugate of the displacement vector 𝐮, 𝐯 is the particle velocity which is the time derivative of 𝐮, 𝜔 is the angular frequency of wave, and 𝜌 is the density of media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Under linear configuration of small wave amplitude, 𝜌 is regarded as a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It is worth noting that the spin angular momentum of phonons is also studied as early as the 1960s [48,241], which are quasiparticles more suitable for describing quantised lattice vibration in the quantum scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The spin angular momentum of phonons shares a similar expression as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (1), the SAM of acoustic/elastic waves, reflecting the fact of wave-particle duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, initial works about phonon spin ignored the spin of curl-free longitudinal mode [355].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The SAM of elastic/acoustic waves describes the locally temporal rotation of displacement/velocity vector, not the spatial curl of the vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In other words, if a displacement/velocity field has both longitudinal and transverse components (or could be non-zero decomposed into two directions), it could have the SAM, and this is irrelevant with the curl of the vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As such, even longitudinal waves (curl-free wave), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', acoustic waves, possess the ability to carry SAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Next, we will give a brief explanation through several derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The displacement field 𝐮 could be written as the combination of the gradient of a scalar potential and curl of a vector potential [50,51]: 𝐮 = 𝛁𝜙 + 𝛁 × 𝝍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (2) In longitudinal wave fields, 𝛁 × 𝐮 = 0 and 𝝍 = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Note that in the field of acoustics, people usually describe the acoustic wave with the media particle velocity 𝐯 and acoustic pressure 𝑃 instead of displacement 𝐮 and the scalar potential 𝜙, which gives [35,242]: −𝛁𝑃 = 𝜕(𝜌𝐯) 𝜕𝑡 = −𝑖𝜌𝜔𝐯, 𝐯 = 𝜕𝐮 𝜕𝑡 = −𝑖𝜔𝐮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (3) Journal of Optics (2022) #### Under this configuration, the acoustic SAM could be express as 𝐒 = (𝜌/2𝜔) Im[𝐯∗ × 𝐯] as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (1) [42,43,45,240,243].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These configuration differences do not affect our discussion below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=" To clarify the existence of SAM in longitudinal waves, we could start with a common form of the scalar potential in 𝑥-𝑦 plane: 𝜙 = 𝜙5𝑒SF'T𝑒SF(U𝑒VSA1, (4) Where 𝑘U (𝑘T) represents the wave number along 𝑥 (𝑦) axis." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (2), the displacement field of a longitudinal wave can be written as: 𝑢U = 𝜕𝜙 𝜕𝑥 = 𝑖𝑘U𝜙, 𝑢T = 𝜕𝜙 𝜕𝑦 = 𝑖𝑘T𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (5) Clearly, the displacement vector 𝑢T, or the transverse component perpendicular to the transport direction, is non-zero in this evanescent wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Although the material which only supports longitudinal waves has no shear modulus, the transverse component 𝑢U still exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As such, the spin angular momentum is: 𝜌𝜔 2 Im[𝐮∗ × 𝐮] = 𝐳r 𝜌𝜔 2 Ims𝑢U∗𝑢T − 𝑢T∗𝑢Ut = 𝐳r 𝜌𝜔 2 Imsu𝑘U∗𝑘T − 𝑘T∗𝑘Uv𝜙5 -t, (6) where 𝐳r is the unit vector along the 𝑧-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' If both 𝑘U and 𝑘T are real constants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', a plane wave transported in the 𝑥-𝑦 plane, nothing interesting happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As 𝑘U∗𝑘T − 𝑘T∗𝑘U = 0, the acoustic SAM is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, things change when we consider a more complex situation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', the evanescent acoustic wave transported along the 𝑥-axis and decaying along +𝑦-axis from 𝑦 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As such, 𝑘T could Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acoustic spin as a rotating particle velocity field (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) A rotating particle velocity (black arrow) can be decomposed into two components 𝑣# (blue arrow) and 𝑣/ (red arrow) along the 𝑥 and 𝑦 directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Acoustic spin in the interference of two acoustic beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Two beams with equal amplitudes propagating along the 𝑥 and 𝑦 directions contribute 𝑣# and 𝑣/ components of the particle velocity field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=" (a) -Ae（+) Y, = Aei(+$+90) (b) Acousticbeam1 '360° 315° -270° 225° 180° 1350 90° 45° 0° Phasedifference between v,andy Acousticbeam2 PolarizationJournal of Optics (2022) #### be rewritten as 𝑖𝜏, where 𝜏 is a real component." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Also, the acoustic SAM will be a non-zero quantity: 𝐳r BA - Im[(𝑘U(𝑖𝜏) − (−𝑖𝜏)𝑘U)𝜙5 -] = 𝐳r𝜌𝜔𝜏𝑘U𝜙5 -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' We may call this acoustic SAM as the transverse spin because it is perpendicular to the wave direction, similar with the transverse spin of an optical wave [231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Note that the sign of 𝑘U determines the sign of SAM in this evanescent acoustic wave configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This indicates the spin-momentum locking effect in evanescent acoustic waves [42,43,45], which is similar with the electromagnetic evanescent waves in optics [207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Nevertheless, similar things happen to the phase difference of strain components 𝜖UU and 𝜖UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, a surface longitudinal wave in fluid, like air or water, can have transverse shear strain, which could be expressed as: 𝜖UU = 𝜕𝑢U 𝜕𝑥 = −𝑘U-𝜙, 𝜖UT = 1 2 {𝜕𝑢U 𝜕𝑦 + 𝜕𝑢T 𝜕𝑥 | = −𝑘U𝑘T𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=" (7) Although 𝛁 ∙ 𝐮 = 0, the transverse shear strain 𝜖UT ≠ 0, and X(( X(' = F( F'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=" If we consider a plane wave in both the x and y direction, F( F' is always a real number, and there is no phase difference between 𝜖UU and 𝜖UT ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=" However, if we consider a surface wave, X(( X(' = F Y 𝑒S) % , this gives a 𝜋/2 phase difference between 𝜖UU and 𝜖UT." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Also, one can say that the phase difference between the normal and shear strains shows the existence of acoustic SAM in this scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It is worth noting that the phase difference between 𝜖UU and 𝜖UT is also important in SAW-induced nonreciprocal ferromagnetic resonance [244].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' According to the analysis above, the acoustic SAM may also excite the ferromagnetic resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Actually, the existence of non-zero acoustic SAM relies on the local phase difference between 𝑢U and 𝑢T (or 𝑣U and 𝑣T), as indicated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (1) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Non-zero SAM exists when the phase is different between two different directions [42,240,356], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', the acoustic field generated with two acoustic beams with different phases, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges Acoustic spin induced torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The acoustic wave with non-zero SAM will introduce a torque to an absorption particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This spin-matter interaction can be characterised by the rates of the angular momentum transfer between the acoustic field and the particle [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Because the monopole vibration mode is isotropic and does not provide the circularly polarised local states, only the dipole momentum can introduce the torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' With the help of a meta-atom which supports dipole resonance, one can measure the torque induced by the acoustic SAM [42], as shown in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These acoustic spin and torque also exist in the topological meta structure [245].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Although the present work shows a clear correlation between the pseudo-spin state and SAM in this kind of quantum spin hall effect acoustic metamaterial, due to the challenge in directly measuring the SAM-induced torque in complex acoustic system, the experimental evidence is still missing, and the physical origin of this correlation still needs further clarifying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The structured acoustic wave with SAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The acoustic SAM also improves the fundamental understanding of the inherent near-field symmetry and directional coupling in acoustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Along with the time average energy flow and reactive power, acoustic SAM density can be used to characterise the time and space symmetry of the evanescent acoustic wave [45], as shown in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These Journal of Optics (2022) #### evanescent wave modes could be selectively excited by the acoustic source with particular symmetries, which provides a feasible approach for designing functional acoustic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Besides the near-field acoustics, the spin-dependent transportation could also be realised in wave guides with symmetry-breaking boundary conditions [243], as shown in Figure 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The momentum of the acoustic wave in such waveguides are tightly coupled with the acoustic SAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This spin- momentum locking effect in wave guides will raise the SAM dependent selective transportation and enhance the backscattering suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As a kind of special structure, skyrmions are also proposed in acoustic systems [35,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Under the help of a well-designed hexagonal acoustic metasurface, the acoustic velocity fields can raise clear skyrmion lattice patterns, which unveil a fundamental property of acoustic fields and may inspire future research in structured acoustic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, the excitation and controlling method of skyrmion patterns of acoustic SAM is still not presented yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the future, acoustic skyrmions may pave a new way for the research in structured acoustic waves and functional acoustic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The SAM interaction between elastic and magnetic/optical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As the acoustic spin we talk about here can carry real SAM, it is natural to assume that angular momentum could transfer between acoustic systems and magnetic/optical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the field of research about magnetic materials, the interaction of spin waves and ultrasonic waves in ferromagnetic crystals was theoretically demonstrated with the phonon-magnon interaction as early as 1958 [246].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The experimental demonstrations about the interaction between surface acoustic waves (SAW) and magnon systems also attract many attentions, including but not limited to the SAW spin-pumping [247], SAW-driven ferromagnetic resonance [248] and SAW-controlled magnetisation [237].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Meanwhile, the angular momentum interaction transfer between elastic and Figure 1 – We allow at most two figures that Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Experimental set-up for the measurement of acoustic spin and the measured SAM-induced torque of acoustic waves which possess positive (red line) and negative (blue line) SAM [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) The near-field selective excitation of spin source [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (c) The spin-dependent propagations in waveguide with symmetry-breaking boundary conditions [243].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The excited waveguide modes have different propagation directions and carry opposite SAM texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='40 50 100 150 200 250 Input amplitude (mV) b) Po Media Media Air Air Media iD Media DX+iD (C)Journal of Optics (2022) #### electromagnetic systems is also proposed with the help of optical fibres [250] or piezoelectric materials [251].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These works may offer a new way to control dynamic states of magnetic, optical, and elastic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, existing research mainly describes this spin transfer with the elastic strains or phonons rather than acoustic/elastic SAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' To give a simpler and more practical theory of the SAM coupling between various physical systems in the micro scene, a definition of the transition between acoustic SAM and magnetic/optical SAM is still needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks Spin angular momentum helps people to give a fundamental explanation of structured waves and practical implications for wave devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The acoustic SAM proves that the ability of wave to carry SAM is more related with the polarised vibration other than the number of intrinsic spatial degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' We expect that these concepts of acoustic SAM will assist the development of structured waves in optical, acoustic, and elastic systems, as well as the SAM transfer between different physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acoustic Pseudospins for Wave Control and Topological Protection Alexander B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Khanikaev1 and Andrea Alù2 1The City College of New York 2Photonics Initiative, CUNY Advanced Science Research Center Status In recent years, synthetic degrees of freedom deliberately introduced into the design of metamaterials via symmetry engineering have significantly expanded the landscape of classical wave phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In particular, synthetic pseudo-spins spanning Hilbert spaces of desirable dimensions and leveraging effective Hamiltonians with nearly any form and structure have been enabling a wide range of eigenstate and spectral engineering in metamaterial responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in material engineering and manufacturing on an unprecedented deeply subwavelength scale have been enabling a precise control over the structure of such effective Hamiltonians, leading to the emulation of numerous fascinating physical systems, from field theory with synthetic gauge fields to relativistic and topological phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In this context, acoustic metamaterials arguably represent the most straightforward and easy-to- work platform for emulating complex wave phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In addition to the relative simplicity of acoustic experiments, recent advances in additive manufacturing have made acoustic systems very appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' On the other hand, unlike other vector wave-fields, such as electromagnetic and mechanical platforms, which offer natural intrinsic degrees of freedom that can be leveraged to construct effective Hamiltonians, the acoustic pressure field is scalar in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Therefore, synthetic degrees of freedom represent an absolute necessity for the emulation of effective Hamiltonians with acoustic pressure-wave systems [252].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1(a) illustrates how synthetic pseudospin can be produced in an acoustic Kagome lattice of 3D printed trimers and used for directional (valley) excitation of bulk Bloch waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For mechanical waves of vector nature, engineering the additional degrees of freedom represents a powerful tool to further expand the Hilbert space and entangle synthetic and natural degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Based on this approach, a broad range of topological phenomena, including higher-order 2D and 3D topological phases, and emulation of Dirac and Weyl physics, have been realised in acoustic metamaterials [253-256].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These advances have led to numerous demonstrations of unprecedented control over spectral features, propagation, and scattering of sound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Spectral pinning of resonant higher-order topological states via lattice symmetries, pseudo-spin polarised topological edge transport, and topological resilience represent just some examples of recent demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Beyond these demonstrations, artificial acoustic media with synthetic pseudospins are posed to bring even more fascinating advances, both in the demonstration of fundamentally new wave phenomena and in practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These advances will likely stem from new physics enabled by sound-matter interactions and more exotic responses of structured acoustic media, which can radically expand the range of attainable effective Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Indeed, recent demonstrations of multiphysics-enabled phenomena in classical-wave polaritonic systems have shown how structured nature of waves in one physical subsystem can be transferred onto the second one via interactions, to yield a new type of topological excitations, as illustrated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(b) [257].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' From the technological point of view, synthetic pseudospins have not been exploited yet, the context of controlling scattering and radiative properties of acoustic metamaterials, which is another promising direction to pursue, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', for achieving pseudo-spin-controlled directional emission of sound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### Current and Future Challenges Emergent topological phenomena involving nonlinear and active regimes, where new pseudo-spin- dependent and topological nonlinear responses and nonlinear modes could be observed, are therefore of great interest [258-261].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The traditional approach to enable this challenging vision has been to realise acoustic metamaterials loaded with active elements by incorporating electric circuits with nonlinear elements and amplifiers, which may provide desirable feedback via microphones and transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While this approach does enable testing of both nonlinear and non-Hermitian regimes, it is hardly scalable, and therefore, it is not suitable beyond proof-of-principle demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Moreover, acoustic excitations carry no charge, spin or magnetic momentum, and thus they do not easily interact with other types of excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This in turn makes it difficult to actively control acoustic waves, exploit nonlinearities that can be inherited from such excitations, and enable multi- physics phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Indeed, by mixing and hybridizing acoustic modes with excitations of different a nature, such as mechanical waves, electromagnetic and optical waves, magnons and spin-waves, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2, one could promote nonlinear effects, drain their energy to induce acoustic gain, or expand the number of available degrees of freedom to explore richer physics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', by engineering synthetic Hamiltonians of a more complex structure and enlarged dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It is, therefore, a major current challenge in the field of topological acoustics to find material systems that could efficiently and actively interact with acoustic waves and, at the same time, can be suitable for integration into acoustic metamaterials [258-260].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In addition to these materials’ related challenges, there is still a great need in better understanding how synthetic pseudospins and spatially varying gauge fields can affect the radiative properties of open acoustic metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While photonic pseudospins have been shown to enable unprecedented control over radiation, from basic control over the radiative lifetime of topological modes to the generation of vortex beams within topological cavities, these successes have not been translated into the acoustic domain yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Besides the control over far-field radiative profiles, the possibility to control near-fields with synthetic pseudospins could enable a plethora of applications unique to acoustics, including trapping and moving objects along pathways defined by synthetic gauge fields, and transfer of angular momentum from acoustic fields to trapped objects for controllable rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1 – We allow at most two figures that are roughly the size of this box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' a b Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Acoustic pseudo-spin in 3D printed kagome lattice of acoustic resonators (left) and control over the propagation directions (K or K’) of the spin-full bulk modes via circularly polarized excitation (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Illustration of multiphysics with synthetic degrees of freedom—polaritonic systems where structured optical field endowed with pseudo-spins induces chiral vibrational modes in van der Waals materials [257].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### Advances in Science and Technology to Meet Challenges While basic models—such as tight-binding model and coupled mode theory, which are widely used to describe acoustic metamaterials with synthetic degrees of freedom—can sometimes be sufficient for understanding basic properties of metamaterials, they also tend to oversimplify the physics by largely ignoring the structure of pressure, displacement and velocity fields, and long-range coupling within metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' On the other end, first-principles methods, such as finite element methods which allow to directly solve wave-equations, provide little insight into the fundamental features of topological metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While these two approaches typically agree well and meet the needs of systems based on discrete resonances, such as discrete lattices of printed or machined acoustic cavities, they diverge when considering systems where the continuous nature of wavefields should be properly treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' One example is long-range interactions that naturally exist in non-discrete systems and can give rise to significant corrections to spatial dispersion of the modes, and even to new types of topological excitations [262].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Similarly, in open systems where acoustic fields may have an evanescent nature, discrete models would not capture the complex structure of the near-fields carrying nonzero angular momentum [242,263].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It is therefore crucial to develop analytical and semi- analytical techniques that can be simple enough to provide an effective Hamiltonian description, yet capable of accounting for the structure of the wavefield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' One of the candidates for such description is mode matching, which has been widely used in various contexts, but not for the description of pseudospins and gauge fields in acoustic metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A comprehensive theoretical description of the acoustic field structure is also crucial for understanding and modelling the interaction of sound with active matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, overlap of chiral hotspots of the near-field in topological materials can be used for the generation of pseudospin- dependent synthetic gauge fields and spin-selective control of sound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Active matter could be mechanically, electrically, or optically driven, and contain internal degrees of freedom which could be engineered to interact selectively with synthetic acoustic pseudospins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In this case, interactions can be leveraged to induce a desirable form of non-Hermitian and nonlinear synthetic gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks Acoustic metamaterials with symmetry-engineered pseudospins have already become an exciting platform for control of propagation of sound waves via synthetic gauge fields acting in the expanded Hilbert space span by pseudospins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Coupling such metamaterials with active materials interacting with sound waves opens even broader opportunities via realisation of pseudospin-dependent tuneable Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concept of nascent acoustic metamaterials with synthetic pseudo-spins coupled to active and nonlinear materials for unmatched control over propagation and radiation of acoustic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Judicious design of coupling of structured sound waves with active, nonlinear, and time-modulated materials envisions emulation of non-Hermitian and interacting topological phases in real and synthetic dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' () Dimensionts) Synthetic 2D +Journal of Optics (2022) #### gauge potentials, including the realisation of phenomena that have so far evaded the field of acoustics, such as novel non-Hermitian and nonlinear topological phases, including real and synthetic dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Hybrid acoustic/active-matter systems also offer a broad range of novel applications, from active control of acoustic radiation as well as structure of the near- and far-fields which can enable new approaches for acoustic communications, imaging and for mechanical trapping—acoustic tweezers, all enhanced with additional degrees of control via synthetic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements Our work in this area has been funded by the National Science Foundation, the Office of Naval Research, and the Simons Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Mechanical effects of structured sound waves Etienne Brasselet University of Bordeaux, CNRS Status Acoustic waves are mechanical in nature because their existence requires a medium whose vibratory characteristics allow us to classify the waves into two families: longitudinal (compression) and transverse (shear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Considering the idealized situation of a plane wave propagating in a homogeneous medium, the former refers to a 1D back-and-forth motion along the direction of propagation, while the latter refers to a motion, usually 2D, in a plane orthogonal to the propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' More than a century ago, physicists discovered that the propagation of acoustic waves is accompanied by mechanical effects on the media in which they propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This is illustrated by several pioneering experimental works on the spatial manipulation of microscopic [264] or macroscopic [265] solid objects, and on the deformation of fluid interfaces [266].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These phenomena involve a rich set of physical effects such as heating, fluid dynamics, and radiation stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The relative simplicity of the experimental implementation contrasts with the difficulty of a quantitative description which must account for the transfer of energy, linear momentum, and angular momentum between the wave and the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 1 illustrates dissipative and non-dissipative wave-matter interactions, which occur jointly at different length scales depending on the nature of the material inhomogeneities and the properties of the media involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In homogeneous media, the attenuation of waves, which corresponds to the thermo-viscous dissipation inherent to the setting in motion of matter at the microscopic scale, leads to a transfer of energy and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Energy transfer produces a heating while a force per unit volume exerted on the matter results from momentum transfer, which generates flows in usual fluids, deformations in viscoelastic media, and mechanical stresses in solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The transfer of momentum can also occur at the interface between two homogeneous media where the discontinuous change in material properties results in a force per unit area exerted on the interface, which can then be deformed, even—and perhaps somewhat counterintuitively—if the interface is acoustically transparent [347].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' All these phenomena have led to the emergence of numerous applications such as quantitative imaging of inert or living media, metrology of wave or material properties, non-contact manipulation and processing of fluids and objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Knowing that any real-world field is a superposition of plane waves, and that any system is finite, structured acoustics meets structured matter is the norm, for which ongoing conceptual and technical developments aim to fully exploit the advantages of acoustomechanics based on translational and rotational degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges When considering the mechanical effects of (compression) sound waves, it is striking how often theoretical and experimental developments are carried out in the context of their electromagnetic counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This highlights the power of analogies in wave physics as well as the fundamental distinctions and opportunities associated with fields of a distinct nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' To name but a few, we are dealing with: longitudinal (sound) versus transverse (light) waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' coupled scalar and vectorial fields (pressure and velocity) versus coupled vectorial fields (electric and magnetic);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' a series of material parameters involved in the constitutive relations describing how the fields evolve, which offer more Journal of Optics (2022) #### practical flexibility in acoustics (density and compressibility) than in optics (dielectric and magnetic permittivity);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' and generic wave scattering problems involving multipole expansions in the treatment of the wave-matter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Analytical toolkits are emerging which provide a better fundamental understanding of the generic and specific aspects of the physics involved, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [43,267].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Furthermore, they equip experimentalists with rational design and fabrication strategies of soft metamaterials whose properties go beyond those of their constituents, mediated by spatially extended acoustic force landscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Exploiting acoustomechanics opens the way to programmable functionalities [268] provided that the structured sound fields can be adapted to demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Usually, the soughtafter "meta-atom" architectures correspond to periodic networks, for which standing waves are well suited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, this prevents the development of meta-devices endowed with spatially distributed functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Non-periodic radiation force networks therefore appear as a natural and time-dependent next step as a way to actively control the local interactions of many-body systems and thus their collective behaviour [269], seeding the development of active matter fuelled by sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Although rotational mechanical effects date back to the beginnings of acoustomechanics, as recalled by the implementation of the sound mill by Dvorak and Mayer in the 1870s [265], it is only recently that angular momentum characteristics of structured sound are experimentally exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' On the one hand, pressure fields endowed with phase singularities make it possible to exert acoustic radiation couples on matter either by dissipative [270] or non-dissipative [271] orbital angular momentum transfer processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' On the other hand, a spin contribution to the angular momentum of sound, which is intimately linked to inhomogeneous fields (two plane waves are sufficient) and is associated with the local elliptical vibration of the medium, has also been demonstrated experimentally recently by using a dissipative and polarisable subwavelength object [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While a !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' "# $ $ $ % b c d % % Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Illustration of the main mechanisms driving mechanical effects mediated by wave-matter transfer of energy and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Wave attenuation during propagation, which is associated with the characteristic attenuation length 𝛼0-, where 𝛼 is the field attenuation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Geometric reflection and refraction of acoustic beams when the characteristic length 𝑎 associated with inhomogeneities of the material properties typically satisfies 𝑎/𝜆 ≫ 1 , where 𝜆 is the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (c) Wave diffraction when inhomogeneities are of the order of the wavelength, hence typically 𝑎/𝜆~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (d) Scattering for point-like inhomogeneities, hence in the regime 𝑎/𝜆 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Note that there is no formal boundaries between the latter regimes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' still, the parameter 𝑘𝑎 is a key parameter to define the relevant framework when describing the wave-matter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### taming the different facets of spin and orbital angular momentum of sound is still in its infancy, the interest it has generated suggests that it will not remain a mere scientific curiosity for long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in Science and Technology to Meet Challenges Now that conceptual frameworks, structural design approaches, and fabrication tools are available for waves and materials, the study and exploitation of the mechanical effects of structured sound has a bright future, which depends in part on the ability of often distinct research communities to open up to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The contactless manipulation of matter highlights how fundamental and technological advances can come together in a simple way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' As an example, we mention the prototypical situation of a focused vortex beam interacting with a spherical particle, which refers to the recently introduced single-beam acoustic tweezers [272] and inherently involves mechanical translational and rotational degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This represents the acoustic analogue of their famous optical counterpart celebrated by the 2018 Nobel Prize in Physics, but endowed with force and torque resources that are—Watt per Watt— several orders of magnitude larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Indeed, the linear and angular acoustic momenta respectively scale as 1/𝑐 and 1/𝜔, where 𝑐 is the speed of sound and 𝜔 is the angular frequency of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' To date, the spin, orbital, and viscous contributions to the mechanical actions exerted on a particle trapped in vortex tweezers remain little explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This invites one to address the acoustohydrodynamic problem as a whole towards developing applications such as sound-driven micro-machines from physical, chemical, and biological perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In particular, it appears necessary to go beyond the simple, yet instructive, case study of an isotropic spherical particle immersed in an isotropic standard fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Conversely, technologies relying on the mechanical effects of "unstructured" sound, such as radiation-force based ultrasound imaging techniques, are likely to benefit from the advantages deriving from structured waves since other mechanical degrees of freedom are involved, such as acoustic vortex elastography [348].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' From a general point of view, the growing interest in acoustic spin and orbital angular momentum opens up questions such as: How can the polarization state of sound be used to manipulate anisotropic media as is done in electromagnetism for many decades with optical spin angular momentum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' How can the singularities and topological textures of inhomogeneously polarised acoustic fields be used to shape matter in non-trivial ways?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' How can spin-orbit interaction mediated by anisotropic or inhomogeneous media be used to enrich the toolbox of acoustomechanics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In this context, topological properties of artificially structured materials and elasticity are some of the new key players at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks Although speaking of the mechanical effects of sound is formally a pleonasm—sound being itself a mechanical movement—their surprisingly rich consequences across length scales, both for waves and for matter, make them still attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recalling that, more than a century ago, the first steps of acoustomechanics dealt with the use of spatially textured sound fields owing to wave interferences [264], the exploration of linear and angular mechanical effects exerted on objects [265] and the use of deformable materials allowing nonlinear feedback phenomena [266], it is remarkable how easy it is to find echoes of them in current research themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Now that acoustics has entered into our everyday life as sensors, transducers, and imaging systems, among other things, the mechanical effects of sound remain a source of inspiration for improving knowledge as well as for designing and implementing new technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' We may hear about structured sound for a long time to come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Transport of surface matter in structured water waves Michael Shats The Australian National University Status Water surface waves share many similarities with their optical and acoustic counterparts, except for a very different dispersion relation, 𝜔2 = 𝑔𝑘 + 𝜎𝑘3/𝜌, where 𝜔 and 𝑘 are the wave frequency and wave number, 𝑔 is the gravity acceleration, 𝜎 is the surface tension coefficient, and 𝜌 is the density of the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The restoring force for the surface perturbation at long wavelengths (longer than about 20 mm in water) is the gravity force and 𝜔~√𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For shorter waves (less than 10 mm), the restoring force is capillary and 𝜔~𝑘a/-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Though water waves have been studied for centuries, there is no universal theory which would describe the motion of fluid particles even in relatively simple structured waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' On the other hand, recent progress in laboratory studies of the particle motion on the water surface perturbed by waves revealed rich phenomenology related to the generation of horizontal vortices and vortex lattices [273], direct and reversed jets [274], and to developed two-dimensional turbulence [275].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Some of these phenomena, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', vortex lattices, are related to the wave momentum and spin [55], while others, for example, turbulent motion of fluid particles driven by steep nonlinear waves, require new theoretical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Horizontal motion of fluid particles at the surface is coupled to the wave pattern and to the width of the wave spectra [277].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Chaotic fluid motion at the surface leads to the increased disorder in the wave field as manifested in the broadening of the wave spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This suggests new theoretical approaches which would allow to predict rms velocities of the fluid particles from the wave spectrum width and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It is interesting to note that, to generate 2D turbulence at the liquid-air interface, it is not necessary to drive turbulence in the wave field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' a slightly broadened wave spectrum results in a broad spectrum of the horizontal fluid velocities matching classical Kolmogorov-Kraichnan spectrum 𝐸𝑘 ∝ 𝑘−5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The mass transport driven by surface water waves is well recognised in natural applications, for example, in oceanology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, there is also growing interest in controlled manipulation of particles on a liquid surface for engineering applications, such as mixing, particle sorting and clustering, as well as for controlling properties of tunable ‘metafluids’ [278].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Better understanding of the wave-driven transport of particles opens opportunities for the development of new biomaterials in liquid media by applying waves to the growing culture [279].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Waves can also be used to promote or discourage the formation of biofilms on solid substrates [279].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges Recent progress in experimental research advanced our understanding of the mass transport driven by the surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Experiments revealed a different nature of the particle motion in small- amplitude, or weakly nonlinear waves, and in parametrically excited, strongly nonlinear waves, also known as the Faraday waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Traditionally, the mass transport by propagating waves was described within the framework of the Stokes drift in 2D wave fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such drift along the y-axis (z-axis is in the vertical direction) produces vertically polarised trochoids, or the motion with horizontal spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Recently, it was shown that, in 3D waves, fluid particles have both horizontal and vertical spin and corresponding generalized Stokes drift [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' An example of the particle trajectory is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Good agreement between theory and experiments gives hope that the field theory approach can be Journal of Optics (2022) #### productive in developing theories and models capable of predicting the mass transport for the weakly nonlinear waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The existence of the vertical and horizontal spin in 3D waves is also important for the motion of larger inertial particles possessing internal spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The motion of such particles is governed by the interaction between the wave spin and particle spin, and it opens the opportunity to manipulate and sort spinning particles using structured surface waves [280].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The existence of the vertical spin in some wave configurations offers a new conceptual base for the development of the surface wave spintronics [281], where the spin of passive particles can be controlled by imposed surface waves [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The challenge here is to account for the return flows in continuous medium as a reaction to the wave-driven drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The situation is more complex for steep nonlinear Faraday waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Fast motion of the surface fluid particles differs qualitatively from the slow drift in linear waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' particle trajectories in such waves have no resemblance with classical Stokes drift [275], as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The development of theory considering particle inertia is a challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The importance of the inertia of fluid parcels in Faraday waves is manifested in the extended inertial interval in the spectra of horizontal fluid velocities [275].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Though Faraday waves generate chaotic particle motion, the mass transport is statistically predictable and allows fine control over particle dispersion at the surface [282].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The main challenge is the development of theory based on the Lagrangian description of fluid motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The progress can be made through the development of theoretical models verified and fine-tuned in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' x y z b a x y z Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) Measured surface elevation produced by two orthogonal standing waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (b) Measured 3D trajectory (red) of a surface particle drifting within a unit cell (dashed line in (a)) and its projection on the horizontal plane (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### Advances in Science and Technology to Meet Challenges Recent demonstration of importance of the wave-generated spin of fluid parcels on the surface perturbed by structured waves is an important step towards developing new applications relevant for particle manipulation and sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These would require overcoming several problems related to (a) incorporation into theoretical models of the transverse angular momentum–induced transport in continuous media, or (b) balancing in experiments of the return flows caused by the spatially varying Stokes drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These problems are related to the configuration driven by two orthogonal propagating waves [55], while they are not important in the field produced by two orthogonal standing waves, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1(b), where the Stokes-drift flow closes on itself and thus remains stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In this example, waves generate large-scale vortices which form a vortex lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The generation of a vortex lattice is also a feature of the nonlinear Faraday waves [275].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, in that case, due to the larger fluid velocities, vortices strongly interact with each other causing 2D turbulent motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Experiments demonstrated that turbulence is dominated by the randomly moving coherent bundles of particles, or meandering ‘rivers’, whose width is about half the Faraday wavelength [282].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The existence of such ‘rivers’ is an important feature of the wave-driven turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In particular, the ‘rivers’ can be guided by solid boundaries within the flow which allows the rectification of the turbulent (mean-zero) velocity fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It has been shown that this effect can be used to create unidirectionally propagating floaters which tap turbulence energy (self-propelled floating devices) [283].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Similarly, the turbulence-driven rotors powered by turbulence have been demonstrated in laboratory experiments [284].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The latter ability can be used for efficient utilization of the wave energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This direction requires better theoretical basis and models capable of deriving flow parameters relevant to engineering applications from the properties of the disordered but structured wave fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Periodic and quasi-periodic wave-driven flows have a potential to be used in biological flows, such as bacterial suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It has been demonstrated that waves can shape the patterns of the bacterial biofilms developing in the wave fields [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This should also be investigated in the wave-driven turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A potentially important theme is the formation of the structure of the bacterial cellulose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This important biomaterial grows near the media-air interface, and it is affected by the surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This would require joint efforts by physicists and microbiologists to understand the formation of extracellular polymeric matrices in moving fluid environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Fluid particle motion in Faraday waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Pink and blue wave fields correspond to two consecutive phase extrema of the waves separated in time by a half-wave period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Green: three-dimensional particle trajectory followed for 100 Faraday wave periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (a) 4mmJournal of Optics (2022) #### Concluding Remarks The structured water wave field is an emerging tool to control mass transport at the gas-liquid interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Weakly nonlinear surface waves share many similarities with optical and acoustic waves [55] and can drive deterministic transport of the surface matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' They promise new directions, such as liquid interface spintronics based on the transverse angular momentum-induced transport, resembling the spin Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Due to the modest wave steepness, the prediction of the mass transport in such waves is applicable to oceanic waves, or they can be generated in various industrial flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Steep nonlinear waves (Faraday waves) generate intense turbulent motion on the surface due to the strong interaction between wave-driven horizontal eddies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such turbulence shows statistical properties of 2D turbulence since it is based on the inverse energy cascade, a process of the spectral energy transfer from intermediate to large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In a bounded domain, such a transfer can lead to the accumulation of spectral energy at the domain scale, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' spectral condensation, which is a form of turbulence self-organization into a coherent vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Wall-guided self-organization of turbulence can be used to rectify turbulent energy for the development of self-propelled surface vehicles and unidirectional rotors for the wave energy conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements This work was supported by the Australian Research Council Discovery Project DP190100406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structured electron waves J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Verbeeck1 and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Schattschneider2 1EMAT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' University of Antwerp 2TU Wien Status Electron beams are used in a wide variety of applications ranging from vacuum tubes that formed the basis of electronics more than half a century ago with specific high power and high frequency tubes still being indispensable today,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' to electron microscopes providing atomic resolution images of materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' and free electron lasers providing intense X-ray beams,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' radiotherapy and surface treatment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' e- beam lithography and chip inspection tools,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' displays,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' portable X-ray sources and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Most of these applications rely on a classical picture of a ray of accelerated electrons, providing a current through vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' But as with light, the wave nature of electrons limits the spatial resolution to the de Broglie wavelength reaching picometer levels for energies greater than a keV, at least five orders of magnitude smaller than the wavelength of visible light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This unique property of electron beams is the essence of its use in electron microscopy and e-beam lithography, providing spatial resolution that reveals the atomic structure of materials on a routine basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In order to approach the ultimate resolution limit, wave aberrations induced by magnetic round lenses of electron microscopes had to be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The tremendous evolution of such correctors in the last two decades, based on phase modifying magnetic multipoles, has generated ideas to apply phase control of the electron for even more sophisticated applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Quantum mechanically, electrons are described by the Schrödinger/Dirac equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The relevant paraxial solutions are highly similar to solutions of the Helmholtz equation used to describe wave phenomena in optics and acoustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This indicates that all wave shaping that is investigated in these areas (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Sections 7 and 18) can, at least in principle, be carried over to electron beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Wave shaping of electrons can be obtained by interaction with electric or magnetic fields provided either by (macroscopic) sources, as realized in aberration correctors, by phase plates, or by the microscopic electric or magnetic potential of matter (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In this roadmap, we want to look beyond resolution revolution, exploiting the quantum nature of the electron, in analogy to the revolution of adaptive light optics or phased arrays in radiocommunication/radar and acoustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' V, A electron source lens plane wave structured wave propagation direction EM field Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Sketch of the interaction of a paraxial electron beam with purposely designed scalar (V) and vector (A) electromagnetic potentials creating a structured electron wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### Current and Future Challenges So far, electron vortex beams (EVB) have received the most attention with a wide range of experiments showing ways to create these topological electron waves, carrying quantized orbital angular momentum (OAM), similar to its optical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The charge of the electron results in a magnetic moment mµB parallel to the propagation axis, with m a quantum number defining the OAM and µB the Bohr magneton -—a unique property of charged matter waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' So far, EVBs have been created, by and large, using forked amplitude gratings [285], magnetic monopole-like fields, and thin refractive elements with spiral height profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These methods are static and absorb a significant fraction of the beam intensity [286,226].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A phase plate in combination with cylinder lenses (magnetic quadrupoles) conserves the intensity and can be tuned so as to achieve EVBs with + or – helicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It can as well be operated in reverse [287].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Tuneable electrostatic phaseplates have been demonstrated as well [276].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 2 shows some vortex beams and an overview of wave shaping methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Besides creating EVBs, significant progress has been made in EVB filters to decompose arbitrary electron beams in its OAM components [349].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A C D Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A) a series of electron vortex beams carrying topological charge -2 to 2, with their wavefront structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' B) A series of holographic gratings producing a variety of structured waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' C) Mode converter with cylindrical lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' D) Spiral phase plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' E) 2x2 electrostatic programmable phase plate for electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Reprinted from ref 2 (A, B, D), ref 3 (C), ref 7 (E) with permission from Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1stcylinderlens 2nd cylinder lens HG input beam free space propagation LG outputbeamx4 x OAM current (=1 phasefronts L O XAholograms far-field images ((=3)+((=-3) (6=3)+((=-3)Journal of Optics (2022) #### EVBs can reveal chirality in crystals [288] or apply torque on nanoparticles and atoms with predicted very high rotational speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Interaction with optical excitations in materials could reveal local information on optically chiral nano-objects [289].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structured electron waves carry information equivalent to polarized light, but at far higher resolution [289], a fact upon which EMCD, the electron counterpart of XMCD, is based [290].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Besides EVBs, other non-diffracting beam classes have been studied, like Airy beams seemingly following a parabolic trajectory in field-free space (like a curve ball), or Bessel beams that stay focused over long distances, providing attractive prospects to study thicker samples [350-353].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Challenges in arbitrary wavefront shaping relate to pushing experimental trials to technical maturity, such as scaling up an array of electrostatic Einzel lenses [291], or using the ponderomotive effect when electrons travel through regions of intense light, shifting the challenge towards making a programmable light field with extreme intensities [292].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Other attempts use light fields in combination with an electron transparent thin film, increasing the electron-photon coupling at the expense of inserting material in the beam [293].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Yet other experiments have achieved EVBs by pulsed laser illuminating of round apertures with circularly polarised light via interaction with surface plasmon polaritons [294].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks The emerging ability to freely shape electron wavefronts in practical instruments opens interesting avenues for future research and application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' One is the ability to encode quantum information in electron beams e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' in a vortex basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This provides a quantum communication channel, which could open new avenues for quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The short wavelength combined with strong interaction with electromagnetic fields and the robustness against decoherence, the topological protection and ease of single particle detection provides attractive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Another emerging opportunity is the ability to implement adaptive optics in electron beam instruments, where the instrument can optimise its beam conditions in a feedback loop to avoid tedious alignment procedures and to provide the highest possible contrast for specific features of interest [354].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This is especially attractive for life science as contrast is notoriously poor, demanding a high electron dose, often damaging the material before images can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Dynamic structuring of the wavefronts could maximize the information per dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Other applications include imaging through thick objects, coded aperture approaches to directly reveal the phase, dynamic phase scrambling to reduce effects of dynamic scattering in both imaging and diffraction and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=" Acknowledgements JV acknowledges funding from the eBEAM project supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 101017720 (FET-Proactive EBEAM), FWO project G042820N 'Exploring adaptive optics in transmission electron microscopy' and European Union's Horizon 2020 Research Infrastructure - Integrating Activities for Advanced Communities grant agreement No 823717 – ESTEEM3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' PS acknowledges the support of the Austrian Science Fund under project nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' P29687-N36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structured neutron and atomic waves Dusan Sarenac, David G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Cory, and Dmitry Pushin University of Waterloo Status The successful extension of the structured waves toolbox to neutron and atomic beams promises an array of exciting applications in fundamental physics and material characterization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, neutrons offer a complimentary probe of nature and materials when compared to photons and electrons, as they possess unique penetrating capabilities and interaction strengths due to the strong force and electroneutrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, given the wave-particle duality, the methods first developed for generating and characterizing optical structured waves are typically the backbone of the methods to create structured neutron and atomic waves [366].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Although the methods are theoretically and conceptually analogous, the practical realizations are complicated by the technical challenges associated with controlling and manipulating these de Broglie waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, the extension of holography techniques to neutron waves was accomplished via perfect-crystal silicon interferometer which through Bragg diffraction provided a coherent superposition of an angled reference beam and a beam that had passed through a macroscopic object [367].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This is a direct adaptation of the two-beam wedge optics technique introduced by Leith and Upatnieks [368].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The first experiments with neutron orbital angular momentum (OAM) focused on manipulating the OAM of incoming neutrons, though given that the input beam had a small transverse coherence length (ranging from ≈nm to µm) relative to the beam diameter (≈cm) the value of the OAM was not well defined [369].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Several theoretical studies of incorporating spin correlations to OAM [370-372] led to an experiment that prepared and characterized neutron lattices of spin coupled OAM beams [373].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 1 is the experimental setup of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [373] that relied on a sequences of magnetic field gradients produced by spatially oriented triangular coils to achieve programmable spin topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In regards to inducing azimuthal phase shifts over the wavepacket coherence lengths, the first experimental achievement of atomic and molecular beams carrying quantized OAM came in 2021 [374].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The demonstration was done with helium atoms and metastable helium dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This work Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The setup used in Ref [8] to prepare and characterize neutron lattices of spin coupled orbital angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Sequences of triangular magnetic fields are used in conjunction with 3He cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Here the magnetically polarized 3He cells act as neutron spin polarizers due to their neutron absorption cross section being highly dependent on the neutron spin direction relative to the helium polarization state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Shown are the simulated spin dependant intensity profiles at each stage of the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Reprinted with permission from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [373].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' KJ,>2 Phase Structure KI,I>2 of N-2 lattice K1,>2 Neutron Camera Slit Analyzeralong +zdirection Neutron Beam PermalloyTube LOVPrismPair Polarizeralong Guide z direction CoilJournal of Optics (2022) #### opens new avenues in using the OAM to probe particle collisions between atoms and/or molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Soon after, the first demonstration for quantized neutron OAM (depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2) was achieved in 2022 [375].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The convenient integration of this method with material characterization studies at Small Angle Neutron Scattering facilities promises to extend neutrons as probes of topological material’s bulk properties [376,377], which cannot be directly probed via photons or electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges The grand scientific challenges revolve around incorporating the additional degrees of freedom brought forth by structured waves, such as OAM, into the existing scattering theory and material characterization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Spin textures and spin topologies, such as skyrmions and merons, possess a non-trivial coupling between spin and other dynamical degrees of freedom which manifest a rich variety of emergent dynamics and phases of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While conventional probes provide indirect transport measurements of topological excitations, neutrons with specific spin-orbit couplings are strongly desired because they may act as direct probes of the target’s topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The technical challenges that deter the progression of the field revolve around the difficulties in preparing, manipulating, and detecting neutron and atomic beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It is expected that as those capabilities evolve, the field follows the progression of optical structured waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' First to note is that there is no device equivalent to a laser which outputs coherent light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Whereas coherent and well- defined Gaussian states are the common inputs to optical experiments with optical structured waves, experiments with neutrons and atoms are limited to working with beams whose transverse coherence lengths are much smaller than the size of the actual beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Typical methods of beam collimation rely on circular slit pairs to define the beam divergence and thus the transverse coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Here we can also note that the low neutron flux ensures that only one neutron at a time is present in the entire setup, and therefore all the experiments in essence are done with a post-selection on there being a neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In relation to this, a notable challenge is the access and availability of high intensity neutron sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Likewise, it is worth mentioning that many other electromagnetic components which are taken for granted in optics setups are also not yet practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, the neutron index of refraction for most materials is around n≈1-10-5, which makes the production of a neutron lens impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Lastly the position sensitive neutron detectors possess much poorer spatial resolution Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Grating arrays have been the enabler of experimental demonstrations of neutron, atom, and molecular helical waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' a) The SEM images of the grating arrays used in the neutron demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' b) The conceptual illustration where each grating of the array coherently acts on individual neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' c) Example of the observed intensity in the far field where the OAM signature profiles are observed in the diffraction orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Reprinted with permission from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [375].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' a) b) Arrayofforkdislocation TopView Incoming phase-gratings with q=3 MEN neutron ma-1 2oum OAM=-3 0000 TopView.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' m=0 OAM=0 Incoming m=1 0AM=3 neutron m=-1 OAM=-3 SideView450tilted 00000 m=0 OAM=0 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' (A") m=] OAM=3Journal of Optics (2022) #### when compared to optical cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, the OAM demonstration with neutrons of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [375] used a neutron camera with a pixel size of around 5mm by 5mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in Science and Technology to Meet Challenges The holy grail technological advance that would push the field of neutron and atom structured waves is a component analogous to a spatial light modulator (SLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The SLM’s ability to provide arbitrary wavefront shaping has revolutionized the field and enabled the widespread use of optical structured waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, a practical active device that enables either phase or intensity modulation is currently out of reach for neutron and atomic beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For the time being, the main enablers have been the advances in programming and control the full range of neutron and atomic degrees of freedom, for example the nanofabrication methods of passive devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It is interesting to note that both OAM works of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [374] with helium atoms and molecules as well as Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [375] with neutrons relied on nanofabricated arrays of gratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The former relied on absorption gratings while the latter on phase- gratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Due to the small transverse coherence length in both cases the gratings need to have periodicities on the nanometre scale in order to induce coherent diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The arrays in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' [375] consisting of 6,250,000 individual fork dislocation phase gratings and they are depicted on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The motivation for the large arrays was to increase the observed signal which otherwise would have been too small to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Even with such an increase in signal every intensity image took around an hour of acquisition time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' In the case of absorption gratings, the thickness of the gratings needs to be sufficient to spatially remove parts of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Whereas in the case of phase-gratings, they need a high aspect ratio to induce an appreciable phase-shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Given the suitable material options, the available/possible fabrication methods determine whether phase or intensity gratings are more practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks The power of neutrons was most striking in the hallmark fundamental physics experiments such as the first demonstration of gravity on a quantum particle and the observation of the 4π symmetry of spinor rotation, and the impactful industry applications with the neutron imaging of fuel cells and lithium batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This was enabled by the control of the three easily accessible degrees of freedom: spin, path, and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The addition of the structured waves toolbox and the OAM degree of freedom is expected to set forth the next generation of fundamental experiments [378] and material characterization techniques [379,380].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The neutron’s penetrating abilities are well suited for bulk studies of materials with spin textures and spin topologies such as skyrmions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Acknowledgements The authors would like to thank their many collaborators including Wangchun Chen, Charles W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Clark, Lisa DeBeer-Schmitt, Huseyin Ekinci, Melissa Henderson, Michael Huber, Connor Kapahi, Ivar Taminiau, and Kirill Zhernenkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The authors would also like to acknowledge their funding sources: the Canadian Excellence Research Chairs (CERC) program, the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada First Research Excellence Fund (CFREF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structuring the Quantum State of Light Michael Birk, Alexey Gorlach, and Ido Kaminer Technion–Israel Institute of Technology Status The shaping of various degrees of freedom of light has been a key factor in the progress of optical sciences and engineering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' many degrees of freedom have been brought under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Spatial shaping has been used to create nondiffracting beams such as Bessel [298] and Airy [295-297] beams, and to imbue light with orbital angular momentum (OAM) [8], while frequency and temporal shaping lie at the core of optical communications and ultrafast optical sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances that combine both spatial and temporal shaping now push the boundaries of this field even further [299,300].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The richness of capabilities for structuring light made clear that the more degrees of freedom of light are under control, the more applications become possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, it is pertinent to remember that all degrees of freedom addressed so far in the context of structured light are related to classical light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' From a quantum perspective, these degrees form only a small subspace that the photonic state can inhabit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The quantum degrees of freedom can be described as a function of generalized phase-space coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This function is known as a Wigner quasiprobability distribution, describing the quantum state of light for a single or multiple optical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Our purpose in this roadmap is to propose the concept of structured quantum light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The impact made by the field of structured light shows the prospects that can emerge from developing capabilities for shaping the quantum properties of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Below, we discuss some of the many open challenges in quantum optics that remain to be resolved if one wishes to acquire control over the quantum shape of light—structuring of the photonic Wigner function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current efforts and proposals in this direction can be divided into deterministic schemes [301- 303] or schemes based on post-selection [304,305].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Most of these efforts have been focused on creating single- or few-photon quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Even though few-photon quantum states have applications in quantum technologies, the ability to create many-photon quantum light states has seen a rising need for a wider range of applications, both classical and quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such states are needed for ghost imaging, precision measurements, quantum communication protocols, and photonic quantum computation based on the so called “bosonic codes” or continuous-variables quantum information [307].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A prime example is the quest toward the generation of the Gottesman–Kitaev– Preskill (GKP) states [308], which are the much-needed resource for scalable fault-tolerant photonic quantum computation [304, 307], and yet they have never been generated in the optical range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' From the perspective of basic science, robust control over the quantum light state can be used to significantly enhance even the most fundamental nonlinear optical processes [301].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The applications of structured quantum light will certainly be greatly expanded when tools for many-photon Wigner function shaping will be developed—as demonstrated in the past by the richness of discoveries brought forth by the invention and commercialization of spatial light modulators (SLMs) for spatial shaping of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current and Future Challenges The basic theoretical toolset for shaping the light Wigner function in the optical domain is provided by the operations of rotations, displacement, squeezing, and amplitude dispersion, which can be performed by phase delays, beam splitters, amplifiers, and nonlinear optical effects such as Journal of Optics (2022) #### parametric down conversion and the Kerr nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, what is possible in theory turns out to be difficult in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While it has been predicted that Kerr nonlinearity can generate Schrödinger cat states [309], experimental efforts to generate macroscopic quantum states in free space have been thwarted by low interaction strengths and dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' More recent attempts to overcome these challenges use optical nonlinearities in microcavities, optomechanical cavities, and various integrated photonic platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such approaches have been successful in generating few-photon quantum states but have not yet led to the generation of many-photon states with useful quantum properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The most advanced demonstrations of many-photon squeezed light states to date are using four- wave mixing and Kerr nonlinearities in fibres, on-chip nonlinearities in waveguides and cavities or parametric down-conversion in free-space optical setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such experiments enabled manipulation of photon statistics, leading to enhancement of nonlinear processes and generation of exotic indefinite- mean photonic states [302].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These efforts are mostly limited to generation of a limited set of photonic states (called Gaussian states), insufficient for quantum computation [307].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Other methods of quantum state generation involve post-selection methods [306,381] rather than relying on optical nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These methods have a different bottleneck—the probability to measure the desired outcome of a heralding observable, and the ultimate sensitivity of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such schemes are inherently probabilistic, mandating either low generation rates (hence low throughput), or the use of resource-intensive architectures such as parallelizing a large number of generators to raise throughput [304].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Classical properties of light Quantum properties of light spatial/ angular temporal/ spectral shaping methods Phase masks Holograms SLM Fourier transform optical pulse shaping apps statistics/ entanglement shaping methods Nonlinear optical elements Post-selection GKP states Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A 64, 012310 (2001) Cat states Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A 103, 013710 (2021) Entangled photons Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 71, S288 (1999) Spatially shaped beams J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Opt 19, 013001 (2017) Space-time pulses Nature Photonics 4, 103 (2010) apps BS Wavefront shaping Nature Photonics 6, 283 (2012) SLM incident wave scattering sample transmitted intensity Optical AWG Nature Photonics 1, 463 (2007) WDM switching J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 5, 904 (1999) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Structured light across all degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Current efforts focus on shaping the classical properties of light (top), including spatial/angular/temporal/spectral/polarization degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' We envision developments in shaping the quantum properties of light (bottom), including the photon statistics and high-order coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While shaping classical properties relies on established technology such as spatial light modulators (SLM), there are currently no standard methods for generic control of quantum properties, but they are expected to rely on nonlinear optics and on post-selection using photon number resolving detectors (PNRD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Techniques for structuring classical properties of light enabled a vast range of capabilities such as imaging and focusing in complex media, optical arbitrary wave generation (AWG), and wavelength division multiplexing (WDM) for applications in optical communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Spatial and temporal shaping can be combined to gain full 3+1D control over the light field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, the quantum nature of light contains many more degrees of freedom, such as squeezing and photon entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Finding ways to structure the quantum degrees of freedom could lead to breakthroughs in many areas of quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For example, many-photon superpositions of coherent states (cat states) and Gottesman-Kitaev-Preskill (GKP) states are the sought-after building blocks for fault-tolerant optical quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='.Journal of Optics (2022) #### A conceptually different approach for the shaping and entanglement of light is utilizing the interaction of light with free electrons [305, 310].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The key to this approach is the nonlinear nature of electron-light interactions and the ability to pre-shape [311] the electron wavepacket before the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A recent experiment demonstrated the effect of quantum photon statistics on the interaction [312], and recent predictions proposed novel methods for utilizing the interaction to create the desired many-photon quantum states [305, 310], including a GKP state [313].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Such schemes may also involve post-selection on the electron energy, since it becomes entangled with the photonic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The advantages of electron-based approaches are the use of mature techniques for electron wavepacket shaping [311], and a post-selection process that avoids the dependence on photon number resolving detectors, which are a bottleneck in conventional post-selection schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The obstacle facing these approaches is the complexity of high-quality electron sources, which are currently mostly studied in expensive state-of-the-art electron microscopes [312].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Advances in Science and Technology to Meet Challenges Considering the described challenges, we highlight a few selected paths toward the full quantum structuring of light: 1) The use of photon number resolving detectors is critical for the current post-selection schemes enabling the creation of low-number quantum light states [304].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Efforts for development of better detectors capable of resolving higher photon numbers will propel the frontiers of the field toward the goal of generating many-photon quantum states of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Exemplary quantum states of light along with selected generation methods (inner circle) and examples of applications (outer circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The plots along the outer circle present example single-mode Wigner functions of the corresponding light states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The text in the inner circle describes several known and theorized methods of generation of the quantum light states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The text in the outer circle provides examples of current and potential applications of the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The special case of classical light – the (Glauber) coherent state – is nested in the green segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' All other segments contain quantum states that cannot be described classically, with example applications in quantum metrology, computing, and communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Coherent states Squeezed states Cat states GKP states Fock states Quantum structuring of light Lasers Parametric down conversion Four-wave mixing Photon subtraction Kerr nonlinearity Atoms and artificial atoms Heralded parametric down conversion Fock laser Post-processing on cat states Post-selection on squeezed states Sub-shot-noise metrology Nature Photonics 7, 613 (2013) Nonlinear yield enhancement Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 119, 223603 (2020) Quantum computing Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A 87, 042315 (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Quantum cryptography Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 74, 145 (2022) Restoring information and error correction Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 111, 120501 (2013) Dual-rail qubit Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 76, 4281 (1996) Fault-tolerant photonic quantum computing Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' A 101, 012316 (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Nature 584, 368 (2020) WDM switching J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 5, 904 (1999) Optical AWG Nature Photonics 1, 463 (2007) Spatial/temporal shaping Nature Photonics 4, 103 (2010) Free-electron lasers Journal of Optics (2022) #### 2) The ability to generate quantum light using optical nonlinearities depends on the ratio between the nonlinearity coupling efficiency and losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This ratio is being gradually improved using better quality nonlinear microcavities, nonlinear photonic crystal fibres and integrated-photonic waveguides [382].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' An utterly different method is to utilize mechanisms of extreme nonlinear optics, such as the ones behind high-harmonic generation, which may lead to strong emission of quantum light, bypassing the limitations in efficiency and losses [383].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' 3) Light-electron interaction schemes for quantum light shaping can become more practical by miniaturizing high-quality electron sources and integrating them with existing methods for shaping electron wavepackets [313].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Additional advances in nanophotonic-based electron–light couplers are necessary to increase the intrinsic interaction strength, an important requirement for generating many-photon states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Concluding Remarks The development of generic methods for shaping the quantum state of light, akin to those available for the spatial and temporal degrees of freedom of light, could spawn a great variety of applications in quantum optics and the wider fields of quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' While such methods may not be immediately in sight, the many advances in quantum optics over the past decade leave us optimistic about the prospects for structured quantum light in the coming decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' High-dimensional quantum communication Ebrahim Karimi University of Ottawa Status Electromagnetic (EM) waves are widely used in our global communication network to transmit information through free-space, underwater, and fibre networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' EM fields—photons in the quantum regime—are the main ‘resource’ for classical and quantum communication infrastructures because they do not possess a net electric charge and rest mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Generation, manipulation, transmission, and detection of the EM field’s internal degrees of freedom (DOF) play curtail roles in both classical and quantum communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For instance, the classical communication bandwidth is directly proportional to the dimension in which the information is encoded and logarithmically depends on the detection signal-to-noise ratio (𝑆/𝑁), respectively—according to the Shannon–Hartley theorem, the channel capacity is given by 𝑘 Log-(1 + 𝑆/𝑁), where 𝑘 is the communication bandwidth [314].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This clearly indicates the importance of multiplexing in order to increase the communication bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' EM fields possess several DOF, including polarisation, frequency, amplitude, phase, and spatiotemporal modes, which can be used to share information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' An EM field that is a coherent or incoherent superposition of all of these DOFs is referred to as structured light (or structured photons in the quantum regime) [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Polarisation is inherently bi-dimensional, and thus can only be used to share one bit (‘0’ or ‘1’) of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Meanwhile, frequency, amplitude, phase, and spatiotemporal modes—although completely different in nature—are unbounded, and thus can be used to increase information beyond ‘0’ and ‘1’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Frequency (wavelength), amplitude, and phase are currently used in telecommunication multiplexing, allowing for the transmission of data at several terabit rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Moreover, during the past decades with the progress in the generation and detection of orthogonal spatial modes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', Laguerre-Gauss and Hermite-Gauss, the communication channel capacity has been increased by a couple of orders of magnitude, allowing information to be transmitted much faster using spatial mode multiplexing [315].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The security of the current classical communication network is guaranteed by rigorous mathematical algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', Rivest–Shamir–Adleman (RSA), which is mainly based on the difficulties of finding prime factors of integer numbers using classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=" However, algorithms—such as Shor's algorithm—implemented on a quantum computer would help to break these classical encryption techniques, and thus threatens our current classical encryption techniques." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Quantum communication, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', quantum cryptography, employing laws of quantum physics provides approaches to monitoring a communication link and verifying the security threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' These methods are based on two main laws of quantum physics: the superposition principle, wherein a quantum entity can be in a superposition of two or more quantum states simultaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' and the uncertainty principle, in which the measurement of conjugate quantities with arbitrarily high precision is not allowed [316].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The former resulted in a ‘no-go’ theorem, referred to as the no-cloning theorem, which directly indicates that a quantum state cannot be copied perfectly without introducing noise to both (multiple) copies [317].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Therefore, whenever a “quantum” message/key is shared, the attacker’s (namely Eve’s) presence introduces an impurity (noise) to the message/key, and by monitoring the noise, one can verify the security threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Journal of Optics (2022) #### Current and Future Challenges There have been significant advances in quantum communication, both at discrete and continuous variable regimes since the seminal quantum key distribution (QKD) proposal of Bennet and Brassard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Different QKD protocols based on a single photon, entangled photon pairs, attenuated coherent beams, and squeezed light are developed with proper security analysis [316].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Without considering the network architecture challenges, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', quantum repeaters and quantum memory, discrete and continuous variables QKD have their own advantages, difficulties, and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Discrete variable QKD allows for long-range distance but limited key rates, while continuous variables have a limited range but provide higher key rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Employing high-dimensional encryption in discrete variable QKD can potentially improve the key rate, and thus has received much attention over the last few decades, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', 𝑑 = 2d-dimensional encryption provides 𝑛-bits of information per sifted photon [62,318].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' It has been shown that such high-dimensional encryption is more noise-tolerant and more sensitive to Eve’s presence—see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' For instance, the best quantum cloning machine introduces + ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' - − ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Pe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=', noise to the cloned wave functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' the introduced error is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='17 for qubit (𝑑 = 2-dimensional) encryption, but increases to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='25 for qutrit (𝑑 = 3-dimensional) encryption [319].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' The classical methods using different DOF to perform multiplexing to increase classical communications would provide qudit states in the quantum regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' This includes complex photonics states with well-engineered spatial and temporal modes, polarisation, and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Photonic polarisation (polarisation qubit) and time-bin (time-bin qudit) can be generated by means of electro-optic and/or optical devices at very high speeds (GHz to THz), which made polarisation and time-bin QKD a promising venue for technological advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' However, implementing both temporal and spatial modes in a practical QKD setup/network, due to the technical difficulties in the fast generation and detection, has hitherto remained a venue to explore and consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Thus, developing novel linear and nonlinear approaches to generate, manipulate, and determine spatiotemporal modes at a high-speed rate will be highly rewarding but remains a challenging task for communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Another promising venue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' not only applicable to quantum communication but for photonics quantum computing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='explore ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='multi-photon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='high-dimensional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='entangled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='states ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='QBER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='Secret ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='Key ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content='Rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQf8g7m/content/2301.05349v1.pdf'} +page_content=' Λ2. The first term in R.H.S must ensure the cancellation of the +graviton t-channel pole, and hence we find +32P +α′3 ∼ M−2 +Pl . +(3.14) +Note that we do not obtain any O(t0) contributions from the Virasoro-Shapiro amplitude +(3.1). +Let us next consider the following deformation: +A(s, t) → A(s, t) + δA(s, t) , +(3.15) +where δA is defined as +δA(s, t) = −ϵK(s, t) Γ(1 − α(s))Γ(1 − α(t)Γ(1 − α(u)) +Γ(2 + α(s))Γ(2 + α(t))Γ(2 + α(u)) . +(3.16) +The positivity of residues requires that the deformation parameter should be bounded as +0 ≤ ϵ ≤ 1 [2]. Note that δA can be expressed in terms of A as follows: +δA(s, t) = ϵ +α(s)α(t)α(u) +(1 + α(s))(1 + α(t))(1 + α(u))A(s, t) . +(3.17) +It is hence straightforward to see the Regge behavior of δA(s, t) is +δA(s, t) ∼ ˜FVS(t)α(s)2α(t)+2 , +(3.18) +where +˜FVS(t) = ϵ +α(t) +1 + α(t)FVS(t) = ϵP +� 4 +α′ +�4 +eπiα(t) Γ(1 − α(t)) +Γ(2 + α(t)) . +(3.19) +Note that δA has the same soft behavior as A in the high energy region with t < 0, but +the first t-channel pole appears at α(t) = 1 rather than at t = 0, which corresponds to the +exchange of a massive particle with spin 4. The residue of δA is +Res +α(s)=n δA(s, t) = K +�4n +α′ , t +� (α(t))n−1 (α(t) + 2)n−1 +(n − 1)!(n + 1)! +, +(3.20) +and the integral is evaluated as +� ∞ +Λ2 ds′ Discs δA(s′, t) +s′3 += −16ϵP +α′2 +∞ +� +n=0 +(n + 1) (α(t))n (α(t) + 2)n +n!(n + 2)! ++ O(t) += −8ϵP +α′2 2F3 (2, α(t), α(t) + 2; 1, 3; 1) + O(t) += −8ϵP +α′2 + O(t) . +(3.21) +– 9 – + +We can thus obtain the O(t0) negative term by deforming the Virasoro-Shapiro amplitude +by δA defined in (3.16). One can also obtain the same result by using (3.13) and (3.18): +� ∞ +Λ2 ds′ Discs A(s′, t) +s′3 +∼ ϵ +α(t) +1 + α(t) +� ∞ +Λ2 ds′ Discs A(s′, t) +s′3 += −8ϵP +α′2 + O(t) . +(3.22) +From (3.14) and the bound on ϵ, we find the bound on this negative contribution: +8ϵP +α′2 ≲ +α′ +M2 +Pl +. +(3.23) +This bound on the negativity is consistent with the recent paper [44], in which constraints +on Regge amplitudes are considered by using the finite-energy sum rules. +3.2 +Coon amplitude +The Coon amplitude [39] is a q-deformation of the Veneziano amplitude which enjoys many +of the same UV properties [25, 40–42] but has some unusual features, most notably a +spectrum with nonlinear Regge trajectories and an accumulation point below which there +are infinitely many states. Despite some similarities with open string scattering in AdS [43], +to date the Coon amplitude has no known worldsheet origin. +Nevertheless, we take it +as a well-studied amplitude exemplifying the desired UV/IR characteristics. +The Coon +amplitude has the following product representation,6 +Aq(s, t) = g2(1 − q) exp +�log σ log τ +log q +� ∞ +� +n=0 +(στ − qn)(1 − qn+1) +(σ − qn)(τ − qn) +, +(3.24) +where g is some coupling constant, q ∈ (0, 1) and σ, τ are related to the usual Mandelstam +variables as +σ = 1 + (q − 1) +� s +µ2 − δ +� +, +τ = 1 + (q − 1) +� t +µ2 − δ +� +. +(3.25) +The spectrum is quickly identified to be +m2 +n = µ2(δ + [n]q) , +[n]q := 1 − qn +1 − q , +(3.26) +with an accumulation point for large n: +m2 +n +n→∞ +−−−→ +m2 +∗ := µ2 +� +δ + +1 +1 − q +� +. +(3.27) +There is a cut from the log σ factor which extends from the branch point at s = m2 +∗ out to +infinity: see Fig. 2 for a sketch. Going forward we take δ = 0 so that the spectrum contains +massless states. +At low energies, s, t, u ≪ µ2, one finds +Aq(s, t) ≈ −g2µ2 +�1 +s + 1 +t +� +, +(3.28) +6The Veneziano amplitude is recovered in the limit q → 1−. +– 10 – + +× +× +× +× ×××××××× +× +µ2 +m2 +∗ +Figure 2. Schematic spectrum for the Coon amplitude with δ = 0. Massive states begin at m2 +1 = µ2 +and have an accumulation point at m2 +∗ = +µ2 +1−q where a branch cut begins. +so that the crossing-symmetric combination +˜ +Aq(s, t) = (s2 + t2 + u2) +� +Aq(s, t) + Aq(t, u) + Aq(u, s) +� +(3.29) +at low energies goes as +˜ +Aq(s, t) ≈ g2µ2 (s2 + t2 + u2)2 +stu +t→0 +−−→ +−4g2µ2 s2 +t . +(3.30) +In the Regge limit the Coon amplitude takes the form +Aq(s, t) +−→ +g2(1 − q)σ +log τ +log q +∞ +� +n=0 +1 − qn+1 +1 − qn +τ +≈ fq(t) +� +− s +m2∗ +� log τ +log q +(3.31) +and +˜ +Aq(s, t) +−→ +˜fq(t) +� +− s +m2∗ +�2+ log τ +log q +, +(3.32) +which marginally satisfies Eq. (2.2) for t < 0 (τ > 1) since log q is negative. The contour +integral around the positive real-s axis splits into a sum over poles and an integral along +the branch cut: +� ∞ +µ2 ds′ Discs Aq(s′, t) +s′3 += −π +∞ +� +k=1 +1 +m6 +k +Res +s=m2 +k +Aq(s, t) + +� ∞ +m2∗ +ds′ Discs Aq(s′, t) +s′3 +. +(3.33) +The residues are polynomial in t, +Res +s=m2 +k +Aq(s, t) = g2µ2qk +k−1 +� +n=0 +τ − qn−k +1 − qn−k +t→0 +−−→ +g2µ2qk , +(3.34) +and the sum over k gives a finite contribution for t → 0. In contrast, the integral over the +branch cut diverges for t → 0 and cancels the t-channel pole. The range s > m2 +∗ corresponds +to σ < 0, so we have +Discs Aq(s, t) = −g2(1 − q)(−σ) +log τ +log q sin +� +π log τ +log q +� ∞ +� +n=0 +(στ − qn)(1 − qn+1) +(σ − qn)(τ − qn) += g2(1 − q)(−σ) +log τ +log q sin +� +π log τ +log q +� (στ − 1) +(σ − 1) t +µ2 +∞ +� +n=1 +(στ − qn)(1 − qn+1) +(σ − qn)(τ − qn) += −πg2(−σ) +− 1−q +log q +t +µ2 1 − q +log q +� +1 + O(t) +� +(3.35) +– 11 – + +and thus +Discs ˜ +Aq(s′, t) +(s′)3 +≈ −2πg2µ2 +� s′ +µ2 +�−1− 1−q +log q +t +µ2 1 − q +log q +� +1 + O(t) +� +(3.36) +gives +� ∞ +m2∗ +ds′ Discs ˜ +Aq(s′, t) +(s′)3 +≈ −2πg2µ−2 1 − q +log q +� +1 + O(t) +� � ∞ +m2∗ +ds′ +� s′ +µ2 +�−1− 1−q +log q +t +µ2 += −2πg2µ2 +t +� +1 + O(t) +� +(3.37) +This leading 1/t behavior exactly cancels the t-channel pole of Eq. (3.30) in the dispersion +relations. Here the details of the cancellation are somewhat different, however, since the +sum over poles does not generate a 1/t contribution despite there being an infinite number of +states. Knowing the contributions from poles and the branch cut must exactly compensate +for the massless state, the O(t0) can be shown to be nonzero, +4g2µ2 +� +−1 +t + 1 − q +µ2 ++ O(t) +� +, +(3.38) +providing another example where a Reggeized amplitude leaves a finite contribution after +canceling the 1/t divergence. +4 +Positivity bounds at one loop +In this section, we first review how the positivity bound at the one-loop level leads to +an apparent puzzle that the cut-off scale of the theory is extremely low. +This is due +to the fact that contribution from the graviton t-channel exchange at the one-loop level is +roughly −s2/(m2M2 +Pl). In order to cancel the negative contribution, the sufficiently positive +contribution is needed, which requires the smaller cut-off scale (1.1). Then, we propose how +the Reggeization extends to one-loop level. +4.1 +Small cut-off scale +Suppose that the approximate gravitational positivity bound (2.22) at tree-level is correct +even at the one-loop level. Ref. [26] found that the left-hand side of (2.22) is computed as +e4 +4π2Λ4 − +e2 +180π2m2M2 +Pl +. +(4.1) +Note that the second term corresponds to a gravitational diagram while the first term +corresponds to a non-gravitational diagram. As this must be positive up to order α′/M 2 +Pl, +we obtain the upper bound on Λ: +Λ ≲ +� +emMPl , +for m2 ≪ e2α′−1 . +(4.2) +Similar results are reported for the Standard Model [45] and the dark photon model [28]. +In particular, using this argument, it is argued that almost all parameter regions of the +dark photon model are excluded [28]. +– 12 – + +𝑘 +𝑘′ +⃗𝑝′ +𝛼𝛽 +𝛾𝛿 +𝜈 +𝜇 +⃗𝑝 +𝑘 +𝑘′ +𝜈 +𝜇 +⃗𝑝′ +⃗𝑝 +Figure 3. A star vertex in the right diagram is a new vertex from Eq. (4.3). We consider new +diagrams obtained by replacing the left diagram with the right one. +If a higher-dimensional two-derivative term is added to quadratic action (2.1), there is +no change in the positivity bound (4.2). For example, we can consider the additional term +with mass dimension six, +S6 = +� +d4x √−g c +Λ2 φ†φFµνF µν . +(4.3) +A new vertex (Fig. 3) and diagrams arise from Eq. (4.3), but we can confirm that their +contributions to the left-hand side of (2.22) cancel out. +4.2 +Reggeization at the loop level +When the cut-off scale is larger than Eq. (4.2), it is still possible that the one-loop Regge +amplitude cancels the negative contribution corresponding to the graviton exchange [31]. +Naively, this is unlikely because the UV Regge amplitude should “know" the IR mass scale +m. However, here we propose the generalization of the Regge amplitude to the one-loop +level, which realizes the cancellation naturally. +At the tree level, we have assumed the Reggeization +A → Ftree(t) +s2+α′t +M2 +PlM2α′t +∗ +(4.4) +for fixed t and large s (Ftree has mass dimension −2). This is expected to happen by the +exchange of the higher spin particles. Given this interpretation, the important requirement +to the behavior of Ftree(t) is that +• Ftree(t) must have poles at t = n α′−1 where n = 0, 1, . . .. Therefore, Ftree(t) is written +as +Ftree(t) = f0 +t + +f1 +t − α′−1 + +f2 +t − 2α′−1 + · · · , +(4.5) +– 13 – + +where fi is a constant. Since each residue of the pole must be positive, we obtain +fi > 0. The presence of the pole at t = 0 is crucial to maintain the positivity bound +at the tree-level. Because of the poles, the amplitude at t = n α′−1 is dominated by +the diagram corresponding to the exchange of the particle of the mass n α′−1. +Next, we have to impose an extra requirement at the one-loop level. We denote the ampli- +tude as +A → +� +Ftree(t) + Floop(t) +� +s2+α′t +M2 +PlM2α′t +∗ +. +(4.6) +At the one-loop level, contributing diagrams include those with a loop of light particles and +tree-level exchange of higher spin particles (see Sec. 4.3). Importantly, these diagrams have +a branch cut starting from t = 4m2 corresponding to particle production. +• At the one-loop level, the branch cut must appear at t = 4m2. Floop is written as +Floop(t) = ˜f0L0(t, m2) + +˜f1α′−1 +t − α′−1 L1(t, m2) + +˜f2α′−1 +t − 2α′−1 L2(t, m2) + · · · , +(4.7) +where Li(t, m2) is a loop function that contains the branch cut starting from t = 4m2 +(see Sec. 4.3 for the detail), and has an expansion +Li(t, m2) = +∞ +� +j=1 +l(i) +j +m2 +� t +m2 +�j−1 +, +(4.8) +up to log(m2). Here lj is a dimensionless constant. On top of the branch cut, there +exist poles at t = kα′ where k = 1, 2, . . .. +Now we discuss the implications of the Regge amplitude on the dispersion relation. The +contribution from Eq. (4.7) is +� ∞ +M2∗ +ds′ Disc A(s′, t) +s′3 +∼ sin(α′t) +α′t +� +˜f0L0(t, m2) + +˜f1 t +t − α′−1 L1(t, m2) + +˜f2 t +t − 2α′−1 L2(t, m2) + · · · +� +. +(4.9) +For t → 0, by choosing ˜fi = O(α′) appropriately, the inequality (2.19) contains a finite +contribution from the one-loop diagrams with higher-spin spins exchanged. In particular, +the positivity bounds for scalar QED can be satisfied without requiring an unusually low +cut-off. +4.3 +An argument for the form of Floop +In this section we provide an explanation for Eq. (4.7) by estimating the diagrams of Fig. 4, +where the double line is either the graviton or the higher spin particles. The loop integral +in Fig. 4 is (see App. A for the notation) +� +d4p +(2π)4 +1 +p2 − m2 +1 +(p + k1)2 − m2 +1 +(p + k1 + k3)2 − m2 (2p + k1)µ(2p + 2k1 + k3)ν +× (pρ1 · · · pρn1) +� +(p + k1 + k3)σ1 · · · (p + k1 + k3)σn2 +� +, +(4.10) +– 14 – + +𝑘! +𝑘" +𝑘# +𝑘$ +⃗𝑝 +𝛼𝛽 +𝛾𝛿 +𝜈 +𝜌 +𝜇 +𝜎 +𝑘! +𝑘" +𝑘# +𝑘$ +⃗𝑝 +𝛼𝛽 +𝛾𝛿 +𝜈 +𝜌 +𝜇 +𝜎 +𝑘! +𝑘" +𝑘# +𝑘$ +⃗𝑝 +𝜈 +𝜌 +𝜇 +𝜎 +𝛼!𝛼#𝛼" ⋯ +𝛽!𝛽#𝛽" ⋯ +𝑘! +𝑘" +𝑘# +𝑘$ +𝛼!𝛼#𝛼" ⋯ +𝜈 +𝜌 +𝜇 +𝜎 +𝛽!𝛽#𝛽" ⋯ +⃗𝑝 +Figure 4. The one-loop diagram corresponding to the t-channel exchange of the graviton and the +higher spin particles. The solid, wavy, double solid and double wavy lines correspond to the scalar, +graviton, photon and higher spin particles, respectively. These produce the branch cut starting +from t = 4m2. +where the second line comes from the interaction between φ and higher spin-L particle +Φα1···αL, +g1 +ML−1 Φα1···αL∂α1 · · · ∂αiφ∂αi+1 · · · ∂αLφ , +(4.11) +and n1 + n2 = L with M and g1 being the mass and coupling of Φα1···αL. Note that the +coupling between the photon and Φα1···αL is schematically given by +g2 +ML−1 Φβ1···βL∂β3···βmF β1µ∂βm+1···βLF β2µ, +(4.12) +where g2 is the coupling constant. By introducing Feynman parameters, we obtain +� +d4l +8π4 +� 1 +0 +dy +� 1−y +0 +dx (2l + (1 − 2x − 2y)k1 − 2yk3)µ(2l + 2(1 − x − y)k1 + (1 − 2y)k3)ν +[l2 − m2 + y(1 − x − y)t]3 +× +� +(l − (x + y)k1 − yk3)ρ1 · · · (l − (x + y)k1 − yk3)ρn1 +� +× +� +(l + (1 − x − y)k1 + (1 − y)k3)σ1 · · · (l + (1 − x − y)k1 + (1 − y)k3)σn2 +� +(4.13) +– 15 – + +where l = p + (x + y)k1 + yk3. +In the following, we concentrate on the term which contains the light mass parameter +m2 in the denominator for t → 0. From a power-counting argument, this corresponds to +the terms where the numerator does not contain l: +� +d4l +8π4 +� 1 +0 +dy +� 1−y +0 +dx ((1 − 2x − 2y)k1 − 2yk3)µ(2(1 − x − y)k1 + (1 − 2y)k3)ν +[l2 − m2 + y(1 − x − y)t]3 +× +� +(−(x + y)k1 − yk3)ρ1 · · · (−(x + y)k1 − yk3)ρn1 +� +× +� +((1 − x − y)k1 + (1 − y)k3)σ1 · · · ((1 − x − y)k1 + (1 − y)k3)σn2 +� +. +(4.14) +The structure of the cut becomes clear by integrating over l, x and y. We obtain the +following expression: +� 1 +0 +dy +� 1−y +0 +dx +� +d4l +8π4 +1 +[l2 − m2 + y(1 − x − y)t]3 += −i +� 1 +0 +dy +� 1−y +0 +dx +� ddlE +8π4 +1 +[l2 +E + m2 − y(1 − x − y)t]3 += −i +� 1 +0 +dy +� 1−y +0 +dx +1 +16π2 +1 +m2 − y(1 − x − y)t += −i +t Li2 +� +2 +√ +t +√ +t − +√ +t − 4m2 +� +− i +t Li2 +� +2 +√ +t +√ +t + +√ +t − 4m2 +� +, +(4.15) +where Li is the polylogarithm function.7 Here for simplicity we ignore x and y dependence +in the numerator, but the structure of the branch cut does not change. The Eq. (4.15) has +a small-t expansion of the form (4.8) +− i +t Li2 +� +2 +√ +t +√ +t − +√ +t − 4m2 +� +− i +t Li2 +� +2 +√ +t +√ +t + +√ +t − 4m2 +� += −i +� 1 +2m2 + +t +24m4 + +t2 +180m6 + · · · +� +. +(4.16) +We write Eq. (4.14) in the following schematic way: +(4.14) = (k1 + #k3)µ(k1 + #k3)ν · · · (k1 + #k3)σn2FL−2(t, m2) , +(4.17) +where FL−2(t, m2) is defined by +FL−2(t, m2) =(−1)n1 +� +d4l +8π4 +� 1 +0 +dy +� 1−y +0 +dx 2(1 − x − y)(1 − 2x − 2y)(x + y)n1(1 − x − y)n2 +[l2 − m2 + y(1 − x − y)t]3 +. +(4.18) +The indices µ and ν are contracted with the external polarizations ϵµ +1 and ϵν +3. This gives a +factor +(k1 + #k3)µ(k1 + #k3)νϵ1µϵ3ν ∼ tu +s → t , +for s → ∞ +(4.19) +7Note that Eq. (4.15) is finite at t = 4m2. +– 16 – + +Other indexes are contracted with k2, k4, ϵ2 or ϵ4. For large s, this contribution behaves as +sL. Multiplying the propagator of the higher spin particle, we obtain +Figure 4 = +g1g2 +M2L−2 +sLt +t − M2 FL−2(t, m2) +(4.20) +for large s. Suppose that the mass M and the spin L of the particle are given by +M2 = kα′−1 , +L = 2 + k , +k = 0, 1, 2, . . . , +(4.21) +as usual in the graviton Regge tower. Then, the amplitude is +g1g2 +M2L−2 +s2+kt +t − kα′−1 Fk(t, m2) . +(4.22) +When we take t → kα′−1, we expect that the contribution from the exchange of spin 2 + k +particle dominates. This reproduces Eq. (4.7). +We would like to explain an idea why the summation of the t-channel diagram leads to +the Reggeized amplitude. At the tree-level, the t-channel amplitude is +s2 +M2 +Plt + g2 +2 +s3α′2 +t − α′−1 + g2 +2 +s4α′3 +t − 2α′−1 + · · · . +(4.23) +After an appropriate analytic continuation, we obtain s2+α′t/(M2 +Plt) for fixed t.8 Similarly, +at the one-loop level, the amplitude is +� s2 +M2 +Pl ++ g1g2 +s3t α′2 +t − α′−1 + g1g2 +s4t α′3 +t − 2α′−1 + · · · +� +F(t, m2) , +(4.24) +from Eq. (4.22). Here we have assumed that all Fk are the same order of magnitude, and +have defined F := Fk. Since the expression in the parenthesis is the same form as the tree- +level t-channel amplitude, it is natural to expect that this is also Reggeized as s2+α′t/M 2 +Pl +for fixed t. In fact, the formal summation of the t-channel diagram leads to +s2 +M2 +Pl ++ g1g2 +∞ +� +n=1 +s2t α′ +t − nα′−1 (α′s)n = +s2 +M2 +Pl +− g1g2 +� +α′s +�3 t Φ(α′s, 1, 1 − α′t) , +(4.25) +where Φ is the Lerch zeta function. By expanding around t = 0, we obtain +− +� +α′s +�3 t Φ(α′s, 1, 1 − α′t) = −α′s2 +∞ +� +n=1 +� +α′t +�n Lin(α′s) → α′s2 +∞ +� +n=1 +(α′t log(α′s))n +n! +, (4.26) +for α′s ≫ 1. Here we have used +Lin(α′s) = −(log(α′s))n +n! ++ O +�� +log(α′s) +�n−1� +, +for α′s ≫ 1 . +(4.27) +8See Eq. (3.8) and App. B for the detail. +– 17 – + +Now, assuming that g1g2α′ = M−2 +Pl , we obtain +s2 +M2 +Pl ++ g1g2 +∞ +� +n=1 +s2t α′ +t − nα′−1 (sα′)n → +s2 +M2 +Pl ++ g1g2α′s2 +∞ +� +n=1 +(α′t log(α′s))n +n! += +s2 +M2 +Pl +eα′t log(α′s) = s2+α′t +M2 +Pl +α′α′t . +(4.28) +As a result, we obtain +s2+α′t +M2 +Pl +α′α′tF(t, m2) . +(4.29) +This is nothing but our proposal (4.7) for small t. Although we are not able to compute +the numerical factor of each diagram,9 this illustrates the idea of how Reggeization occurs, +and how the parametrically large prefactor appears. +5 +Discussion +In this paper, we have studied the gravitational positivity bound at the one-loop level. +We first reviewed the gravitational positivity bound at tree level, where the t-channel pole +corresponding to the graviton exchange can be canceled by assuming Regge behavior for the +amplitude at high energy. Next, we saw two examples of unitary, Reggeized gravitational +amplitudes for which the potentially negative finite contribution to the positivity bounds +can be found. +Finally, we moved to the features of Reggeized amplitudes at one loop. +It is known that the one-loop EFT amplitude leads to the parametrically large negative +contribution to the positivity bound. We argued for a form of the Reggeized amplitude at +the one-loop level based on the analytic structure of the t-channel exchange diagram of the +graviton and the higher spin particles in the Regge tower. +As a future direction, we may consider a string theory setup to explicitly check the form +of the amplitude at the one-loop of the matter fields. For instance, we may take the non- +supersymmetric SO(16) × SO(16) heterotic string and compactify it on T 6.10 In 10d there +are fermions whose representation is (16, 16) + (128, 1) + (1, 128) under SO(16)× SO(16). +These will become Dirac fermions in 4d after compactification. By turning on a VEV for +the Wilson lines, the gauge symmetry is broken from SO(16) × SO(16) to U(1)16.11 Then +the (16, 16) fermions have charge +� +1, 0, . . . , 0 +� +�� +� +U(1)8 +, 1, 0, . . . , 0 +� +�� +� +U(1)8 +� +(5.1) +(and its permutations) under U(1)16. Similarly, (128, 1) has charge +� +± 1 +2, . . . , ± 1 +2 +� +�� +� +U(1)8 +, 0, . . . , 0 +� �� � +U(1)8 +� +. +(5.2) +9In general, n-dependent coefficient appears in the summation. Moreover, the couplings g1,2 can depend +on n. It is interesting to study the condition to realize the Reggeized amplitude. +10We can consider the same setup for the supersymmetric heterotic string theory. +11In addition, there are KK and winding U(1)s, but these are not important here. +– 18 – + +These fermions receive masses from the Wilson line, and the masses are chosen to be +arbitrary values. This setup is close to the non-supersymmetric QED coupled with light +matters, and loops of these fermions may give a negative contribution to the coefficient of +s2. On the other hand, the typical mass of the higher-spin particles is always the string +scale, independent of the VEV of the Wilson line. It is an interesting task to compute the +one-loop string amplitude at O(M−2 +Pl ) in this setup to check the details of our proposal in +Sec. (4.2). +– 19 – + +Acknowledgments +The work of Y.H. and G.L. is supported in part by MEXT Leading Initiative for Excellent +Young Researchers Grant Number JPMXS0320210099. The work of S.N. is supported in +part by JSPS KAKENHI Grant Number 21J15497. +A +Notation +We follow the notation in Ref. [26]. We consider the scattering γ1γ2 → γ3γ4. The momenta +kµ +1,2,3,4 are parametrized as (all-ingoing notation) +kµ +1 = (k, 0, 0, k) , +kµ +2 = (k, 0, 0, −k) , +kµ +3 = −(k, k sin θ, 0, k cos θ) , +kµ +4 = −(k, −k sin θ, 0, −k cos θ) +(A.1) +We define ϵµ +1,2,3,4 as a polarization vector of the external photon. +The ± polarization +corresponds to +ϵµ +1(±) = +1 +√ +2 (0, 1, ±1, 0) , +ϵµ +2(±) = +1 +√ +2 (0, −1, ±1, 0) , +ϵµ +3(±) = +1 +√ +2 (0, cos θ, ±i, − sin θ) , +ϵµ +4(±) = +1 +√ +2 (0, cos θ, ±i, ± sin θ) . +(A.2) +In all-ingoing notation, the amplitude is written as +A(h1, h2, h3, h4; k1, k2, k3, k4) = ϵµ +1(h1)ϵν +2(h2)ϵα +3 (h3)ϵβ +4(h4)Aµναβ(k1, k2, k3, k4) , +(A.3) +where h1,2,3,4 = ±1 is the helicity of the external photons in all-incoming notation. ϵij is +defined as +ϵij := ϵi · ϵj +(A.4) +Explicitly, we obtain +ϵ12 = −1 +2 − h1h2 +2 +, +ϵ13 = −h1h3 +2 ++ 1 +2 + t +s , +ϵ14 = −h1h4 +2 +− 1 +2 − t +s , +ϵ34 = −1 +2 − h3h4 +2 +, +ϵ24 = −h2h4 +2 ++ 1 +2 + t +s , +ϵ23 = −h2h3 +2 +− 1 +2 − t +s , +(A.5) +The inner products between external momenta and polarizations are +k1 · ϵ3 = k3 · ϵ1 = k2 · ϵ4 = k4 · ϵ2 = − +√ +tu +√ +2s , +k1 · ϵ4 = k4 · ϵ1 = k2 · ϵ3 = k3 · ϵ2 = +√ +tu +√ +2s , +(others) = 0 . +(A.6) +The inner products among external momenta are +k1 · k2 = k3 · k4 = s/2 , +k1 · k3 = k2 · k4 = t/2 , +k1 · k4 = k2 · k3 = u/2 , +k2 +1 = k2 +2 = k2 +3 = 0 . +(A.7) +where (s, t, u) are the Mandelstam variables: +s = (k1 + k2)2 = 2k1 · k2, +t = (k1 + k3)2 = 2k1 · k3, +u = (k1 + k4)2 = 2k1 · k4 . +(A.8) +– 20 – + +B +Reggeization at the tree-level +In this appendix, we show how the tree-level amplitude is Reggeized starting from the pole +expansion (3.8), along the line with the end of Sec. 4.12 For large s, Eq. 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Lett. 127 (2021) 091602 +[2104.09682]. +(page 12). +– 23 – + diff --git a/PdA0T4oBgHgl3EQfDP8K/content/tmp_files/load_file.txt b/PdA0T4oBgHgl3EQfDP8K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..72bde88a7f113916ccacc55c729372bfcd4fd726 --- /dev/null +++ b/PdA0T4oBgHgl3EQfDP8K/content/tmp_files/load_file.txt @@ -0,0 +1,853 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf,len=852 +page_content='Prepared for submission to JHEP KEK-TH-2492 On (Scalar QED) Gravitational Positivity Bounds Yuta Hamada⋄⋆ , Rinto Kuramochi⋆ , Gregory J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Loges⋄ , Sota Nakajima⋄ ⋄Theory Center, IPNS, High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan ⋆Graduate University for Advanced Studies (Sokendai), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan E-mail: yhamada@post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='kek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='jp, rinto@post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='kek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='jp, gloges@post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='kek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='jp, snakajim@post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='kek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='jp Abstract: We study positivity bounds in the presence of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We first review the gravitational positivity bound at the tree-level, where it is known that a certain amount of negativity is allowed for the coefficients of higher-derivative operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The size of these potentially negative contributions is estimated for several tree-level, Reggeized gravitational amplitudes which are unitary at high energies and feature the t-channel pole characteristic of graviton exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We also argue for the form of the one-loop Regge amplitude assuming that the branch cut structure associated with the exchange of the graviton and higher- spin particles is reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We demonstrate how the one-loop Regge amplitude appears by summing over Feynman diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' For our one-loop amplitude proposal, the positivity bounds generically receive a finite contribution from the Regge tower and do not lead to a parametrically small bound on the cut-off scale of the low-energy EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='01999v1 [hep-th] 5 Jan 2023 Contents 1 Introduction 1 2 Positivity bounds at tree level 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1 Review: (non-)gravitational positivity bounds 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2 Cancellation of the t-channel pole in gravitational amplitudes 6 3 Modified amplitudes 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1 Virasoro-Shapiro amplitude 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2 Coon amplitude 10 4 Positivity bounds at one loop 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1 Small cut-off scale 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2 Reggeization at the loop level 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3 An argument for the form of Floop 14 5 Discussion 18 A Notation 20 B Reggeization at the tree-level 21 1 Introduction The space of low-energy EFTs is highly constrained by unitarity, locality and causality [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Imposing these general requirements allows one to bound, for example, the Wilson coef- ficients of higher-derivative operators and rule out a whole swathe of theories as being inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' For instance, the optical theorem on the scattering amplitude in the forward limit leads to a variety of positivity bounds on the Wilson coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Recent devel- opments reveal that the infinite tower of higher-dimension operators must lie inside the EFThedron [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The Swampland program [3] (see [4–7] for recent reviews) takes this one step further by aiming to understand the nontrivial imprints that quantum gravity has on low-energy physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' There are many conjectures such as the Weak Gravity Conjecture [8], Distance Conjecture [9], and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' It is desirable to provide explanations for these conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' It is natural to utilize the technique of the positivity bound to explain the Swampland conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' This idea is particularly useful to demonstrate the Weak Gravity Conjecture, which states the existence of the state where the charge is larger than the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1 The 1In addition to the Weak Gravity Conjecture, the S-matrix bootstrap has been used to obtain a lower bound on the coefficient of the eight-derivative operator [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' – 1 – certain positivity condition on the four derivative operators means that the Weak Gravity Conjecture is realized by the nearly extremal black holes [11–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' However, in contrast to the non-gravitational case, it is difficult to obtain positivity bounds for gravitational amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The main obstruction comes from the existence of the massless graviton, which leads to the t-channel pole at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Because of this, one cannot directly take the forward limit, t → 0, where the optical theorem is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [20] bypasses this difficulty by assuming a Regge form for the amplitude at high energy, and shows that the gravitational amplitude satisfies an approximate positivity bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2 The Regge amplitude means that, for s → ∞ with fixed t < 0, the amplitude A satisfies lims→∞ A/s2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3 This assumption is mainly motivated by the behavior of the Virasoro-Shapiro amplitude in tree-level string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Moreover, other amplitudes satisfying Regge boundedness and the IR consistency conditions such as unitarity have been found [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Recently, the gravitational positivity bound at the one-loop level has been used to obtain new Swampland conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' It was argued in [26] that for scalar QED coupled to Einstein gravity, the gravitational positivity bounds provide a nontrivial relationship between the EFT cut-off Λ and the scalar’s mass m and charge e, Λ ≲ � emMPl , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1) where MPl is the reduced Planck mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' This scale is much smaller than expected and is even parametrically smaller than that provided by the magnetic Weak Gravity Conjecture [8], Λ ≲ eMPl , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2) when the scalar is light, m ≪ eMPl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' A similar technique is used to obtain the cut-off scale of the Standard Model [27] and dark photon model [28] as well as constraining the shape of the scalar potential [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In this article we demonstrate that accounting for 1-loop effects on the amplitude from Regge states can alter the argument which leads to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In particular, in addition to canceling the t-channel pole the Regge states can contribute a subleading O(t0s2) term to the amplitude which has a nontrivial effect on the positivity bounds in the low-energy EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='4 This mechanism was explored in [20, 30], where it was used to understand the space of (scalar-tensor) EFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The necessity of the non-trivial effect from the Regge states is stressed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [31] by studying the graviton-graviton scattering with indefinite helicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Moreover, very recently, the finite energy sum rule is derived to constrain the gravitational amplitude in four dimen- sions [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' See also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [30] for the study of the Regge amplitude including the effect of the graviton loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In this paper, we consider the photon-photon scattering with definite helicity, and propose a probable 1-loop Regge amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 2See also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [13] for the schematic idea of the approximate positivity bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 3The argument based on the causality [2] indicates lims→∞ A/s2 = (finite), though it is hard to say that the finite value is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The chaos bound [21, 22] leads to the same conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' It is also argued that the Regge boundedness for d ≥ 5 follows from the reasonable assumptions in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4Another direction to study the gravitational positivity bound is to work with a fixed, finite impact parameter [29], which works for d ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' – 2 – The remainder of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 2 we review the positivity bounds for 2-to-2 photon scattering in non-gravitational theories and how, when minimally coupled to Einstein gravity, the resulting t-channel pole is canceled by Reggeization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 3 we analyze the properties of ‘deformed’ tree-level gravitational amplitudes which have been recently proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' These amplitudes satisfy the IR consistency conditions, and we study the implications of these amplitudes on the positivity bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4 we consider the subleading, 1-loop contributions to scalar QED and show that the Regge states lead to a nonzero correction in the forward limit, t → 0, assuming that the branch cut structure associated with the exchange of the graviton and higher spin particles is reflected in the Regge amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We conclude with a discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 2 Positivity bounds at tree level In this section we review how gravitational positivity bounds may be extracted in the forward limit t → 0− despite the presence of the t-channel pole coming from graviton exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Reggeization of the gravitational amplitudes provides a mechanism for canceling the s2/t term that appears in Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' After carefully subtracting such divergent terms, dispersive sum rules allow one to derive positivity bounds on the remaining finite terms which are calculable in a low-energy EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We will focus on the fixed-angle scattering of two photons where all incoming and outgoing particles have + helicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The amplitudes can be written in terms of the usual Mandelstam variables s, t, u which satisfy s + t + u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3, for concreteness, we will consider scalar QED coupled to Einstein gravity in 4d, for which the quadratic action is S = � d4x √−g �M2 Pl 2 R − 1 4FµνF µν − Dµφ†Dµφ − m2φ†φ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1) where the mostly-plus notation is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' A for more details on our conventions and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1 Review: (non-)gravitational positivity bounds Let us begin by recalling the usual argument which allows one to derive positivity bounds in EFTs, taking care of the difference between the non-gravitational and gravitational am- plitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Along the way we note the points where the argument cannot immediately be adopted for theories coupled to Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In discussing dispersion relations we make the usual Regge-boundedness assumption on the high-energy behavior of the amplitude: lim |s|→∞ t<0 fixed A(s, t) s2 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2) For the non-gravitational case, this is guaranteed by the Froissart theorem [34, 35] in a gapped theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' For the gravitational case, this is of course violated by tree-level graviton exchanged, but satisfied when the graviton is Reggeized, as we will consider below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We also – 3 – s γ C∞ ϵ C+ C− Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Analytic structure of A and the contour deformation of γ into C∞ + C+ + C− − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' assume that A is crossing symmetric and holomorphic in s, except for poles and cuts along the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Start with the identity A(s, t) = s2 � γ ds′ 2πi A(s′, t) (s′ − s)s′2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3) where γ is any contour encircling s within which A is holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' This integration contour can be deformed into a circle C∞ at |s′| → ∞, contours C± around any poles and branch cuts on the positive and negative real axes, and a contour ϵ which encircles the pole at s = 0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 1): A(s, t) = s2 �� C∞ + � C+ + � C− − � ϵ � ds′ 2πi A(s′, t) (s′ − s)s′2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='4) The circle at infinity has vanishing contribution thanks to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In the presence of gravity, A(s, t) may be expanded as A(s, t) = a−1(t) s + a0(t) + a1(t)s + · · · (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='5) around the origin: note that the first term is absent without gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' By substituting this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='4), the pole at the origin gives rise to −s2 � ϵ ds′ 2πi A(s′, t) (s′ − s)s′2 = a−1(t) s + a0(t) + a1(t)s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='6) This means that the scattering amplitude A can be written as A(s, t, u) = a−1 s + a0 + a1s + s2 π � ∞ 0 ds′ Discs A(s′, t, u′) s′2(s′ − s) + u2 π � ∞ 0 du′ Discu A(s′, t, u′) u′2(u′ − u) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='7) – 4 – where the discontinuities Discs and Discu on the real axis are defined by5 2i Discs A(s, t, u) := A(s + iϵ, t, u) − A(s − iϵ, t, u) , 2i Discu A(s, t, u) := A(s, t, u + iϵ) − A(s, t, u − iϵ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='8) By comparing s2 terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='7), we obtain ∂2 s ˜ A(0, t, −t) = 1 π � ∞ 0 ds′ Discs A(s′, t, u′) s′3 + 1 π � ∞ 0 du′ Discu A(s′, t, u′) (u′ + t)3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='9) where we have defined ˜ A := A − a−1 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='10) This is an amplitude where the graviton s-channel pole is subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In the forward limit, the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='9) must be non-negative by unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' This immediately gives the following positivity bounds, ∂2 sA(0, 0, 0) ≥ 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='11) which directly constrain the space of allowed EFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We can obtain stronger bounds by extracting the portion of the integrals along the cuts which are calculable at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In order to do this, introduce Λ as the EFT cut-off and write Discs A(s′, t, u′) = Discs AIR(s′, t, u′) θ(Λ2 − s) + Discs AUV(s′, t, u′) θ(s − Λ2) , Discu A(s′, t, u′) = Discu AIR(s′, t, u′) θ(Λ2 − s) + Discu AUV(s′, t, u′) θ(s − Λ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='12) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='9) now becomes ∂2 s ˜ A(0, t, −t) − 2 π � Λ2 0 ds′ Discs AIR(s′, t, u′) s′3 − 2 π � Λ2 0 du′ Discu AIR(s′, t, u′) (u′ + t)3 = 2 π � ∞ Λ2 ds′ Discs AUV(s′, t, u′) s′3 + 2 π � ∞ Λ2 du′ Discu AUV(s′, t, u′) (u′ + t)3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='13) where again the right-hand side is non-negative by unitarity in the forward limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Without gravity there is no obstacle to taking t → 0, in which case the improved positivity bound reads [36–38] ∂2 s ˜ A(0, 0, 0) − 2 π � Λ2 0 ds′ Discs AIR(s′, 0, u′) s′3 − 2 π � Λ2 0 du′ Discu AIR(s′, 0, u′) u′3 ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='14) By design everything here depends only on low-energy data and constrains the space of allowed EFTs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' through constraining the Wilson coefficients of higher-derivative opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' One possibility is that the above equation provides an upper bound on Λ, the scale where new physics must appear to maintain unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 5The discontinuity is related to the imaginary part by Discs A(s, t, u) = 2i Im A(s + iϵ, t, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' – 5 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2 Cancellation of the t-channel pole in gravitational amplitudes For gravitational systems the argument which leads to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='14) needs re-evaluating;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' more care is needed when graviton exchange is included because of the contribution Atree grav ∼ − s2 M2 Plt (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='15) which renders ∂2 s ˜ A(0, 0, 0) ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Of course this means that the integrals on the right- hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='9) must diverge for t → 0− as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' One way to address the graviton pole was fleshed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' we assume that at energies well above the scale M∗ the amplitude is Reggeized, lim |s|≫M2 ∗ t<0 fixed Discs A(s, t) = ftree(t) s2+α′t M2 PlM2α′t ∗ =: Discs Atree Regge(s, t) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='16) where α′ provides the slope of the Regge trajectory (f(t) has mass dimension −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' This is what happens in string theory, for example, with M2 ∗ ≈ M2 s = α′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Substituting this limiting expression into the dispersive integral gives � ∞ M2∗ ds′ Discs A(s′, t) s′3 ≈ � ∞ M2∗ ds′ Discu A(u′, t) (u′ + t)3 ≈ −ftree(t) M2 Plα′t = −ftree(0) M2 Plα′t − f′ tree(0) M2 Plα′ + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='17) As long as f(0) ∼ α′ is chosen appropriately, this divergent UV contribution to the dis- persion relation cancels the t-channel pole in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' After the cancellation, the Regge contribution to Disc A need not be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' For example, since at tree-level in string the- ory the function f(t) only depends on the dimensionful parameter α′, a certain amount of negativity is allowed: � ∞ M2∗ ds′ Discs A(s′, t) s′3 ≈ − 1 M2 Plt − O(1) α′ M2 Pl + O(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='18) The size of this negative O(t0) term is crucial in discussing the possibility that (nearly) extremal black holes satisfy the Weak Gravity Conjecture [8, 13, 15, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' By substituting the tree-level Regge amplitude into the dispersion relation, we obtain ∂2 s∆A(0, t, −t) − 2 π � Λ2 0 ds′ Discs AIR(s′, t, u′) s′3 − 2 π � Λ2 0 du′ Discu AIR(s′, t, u′) (u′ + t)3 = 2 π �� M2 ∗ Λ2 ds′ Discs AUV(s′, t, u′) s′3 + � M2 ∗ Λ2 du′ Discu AUV(s′, t, u′) (u′ + t)3 � + 2 π �� ∞ M2∗ ds′ Discs ∆AUV(s′, t, u′) s′3 + � ∞ M2∗ du′ Discu ∆AUV(s′, t, u′) (u′ + t)3 � − 2f′ tree(t) M2 Plα′ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='19) where ∆A := ˜ A − Atree grav , ∆AUV := AUV − Atree Regge .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='20) Everything is now manifestly finite for t → 0−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In addition, the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='19) is positive in this limit by unitarity and in the third line the terms in parentheses are – 6 – negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The final term is Planck-suppressed, but may be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In cases where ftree(t) = α′ftree(α′t) the last term can be estimated as −2f′ tree(0) M2 Plα′ = −O � α′ M2 Pl � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='21) since this is the typical scale in string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' To summarize, the approximate gravitational positivity bound at tree level is ∂2 s∆A(0, 0, 0) − 2 π � Λ2 0 ds′ Discs AIR(s′, 0, u′) s′3 − 2 π � Λ2 0 du′ Discu AIR(s′, 0, u′) u′3 ≥ −O � α′ M2 Pl � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='22) In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 3 we will see two examples of Reggeized tree-level amplitudes for which f′ tree(0) is calculable and allows for the slight negativity argued for above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' More generally, we argue in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4 that loop effects will generate such a term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 3 Modified amplitudes Before discussing one-loop amplitudes, we discuss two explicit examples of Reggeized tree- level amplitudes where the negative O(t0s2) contribution appears from the Regge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Explicitly, we discuss the deformed Virasoro-Shapiro amplitude identified recently in [2] and the Coon amplitude [25, 39–43], which is a q-deformation of the Veneziano amplitude (recently generalized in [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1 Virasoro-Shapiro amplitude As a typical example of Regge-behaved amplitudes in the asymptotic limit, let us consider the Virasoro-Shapiro amplitude which is identified with a 2-to-2 scattering of massless (gauge neutral) bosons in closed string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [20], this amplitude has been studied in order to check the cancellation of the t-channel graviton exchange and the violation of strict positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In this subsection, we will discuss how the Virasoro-Shapiro amplitude gives the O(t−1) term, but a O(t0) term does not appear (see [20] for more detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Next, we will show that the deformation of the Virasoro-Shapiro amplitude considered in [2], which maintains the power law for s in the Regge limit, leads to a negative O(t0) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The Virasoro-Shapiro amplitude is given by the following form: A(s, t) = −K(s, t) Γ(−α(s))Γ(−α(t))Γ(−α(u)) Γ(1 + α(s))Γ(1 + α(t))Γ(1 + α(u)) ���� u=−s−t , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1) where K(s, t) is a kinematic factor and α(x) = α′x/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' For simplicity, we restrict our attention to the following form of K(s, t): K(s, t) = P � s2t2 + t2u2 + s2u2� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2) where P is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' By using the Stirling formula Γ(z) ∼ √ 2π zz− 1 2 e−z , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3) – 7 – one can easily see that the asymptotic behavior of A(s, t) is A(s, t) ∼ FVS(t)α(s)2α(t)+2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='4) where FVS(t) = P � 4 α′ �4 eπiα(t) Γ(−α(t)) Γ(1 + α(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='5) Note that FVS(t) has poles on the real axis at α(t) = n with n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The residue in the t-plane is Res α(t)=n A(s, t) = −K � s, 4n α′ � � Γ(n + α(s)) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='Γ(1 + α(s)) �2 = −K � s, 4n α′ � 1 (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' )2 n−1 � j=1 (α(s) + j)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='6) It is clear that the residue at α(t) = n is a polynomial in s of order 2n+2, and in particular at t = 0 is −64Ps2 α′3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='7) The pole expansion in the t- and u-channel is thus given by A(s, t) = − 4 α′ ∞ � n=0 �(α(s) + 1)n−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' �2 � K � s, 4nα′−1� t − 4nα′−1 − K � s, −s − 4nα′−1� t + s + 4nα′−1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='8) where we have used the Pochhammer symbol defined as (x)m = Γ(x + m) Γ(x) = m−1 � j=0 (x + j) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='9) For large s, in particular, the leading contribution from t-channel poles is A(s, t) ∼ −P � 4 α′ �5 ∞ � n=0 1 (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' )2 α(s)2n+2 t − 4nα′−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='10) Note that this expression as an infinite sum coincides with the Regge behavior (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='4) with t-channel poles expanded since FVS(t) can be expressed as FVS(t) = −P � 4 α′ �5 ∞ � n=0 1 (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' )2 1 t − 4nα′−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='11) By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='6) with s ↔ t crossing symmetry, we find Discs A(s, t) = − ∞ � n=0 δ(α(s) − n) Res α(s)=n A(s, t) = 4 α′ ∞ � n=0 K �4n α′ , t � �(α(t) + 1)n−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' �2 δ(s − 4nα′−1) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='12) – 8 – and then the integral that we would like to evaluate is � ∞ Λ2 ds′ Discs A(s′, t) s′3 = 16P α′2 ∞ � n=1 1 n �(α(t) + 1)n−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' �2 + O(t) = −32P α′3t + O(t) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='13) where we assume 1/α′ > Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The first term in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='S must ensure the cancellation of the graviton t-channel pole, and hence we find 32P α′3 ∼ M−2 Pl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='14) Note that we do not obtain any O(t0) contributions from the Virasoro-Shapiro amplitude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Let us next consider the following deformation: A(s, t) → A(s, t) + δA(s, t) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='15) where δA is defined as δA(s, t) = −ϵK(s, t) Γ(1 − α(s))Γ(1 − α(t)Γ(1 − α(u)) Γ(2 + α(s))Γ(2 + α(t))Γ(2 + α(u)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='16) The positivity of residues requires that the deformation parameter should be bounded as 0 ≤ ϵ ≤ 1 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Note that δA can be expressed in terms of A as follows: δA(s, t) = ϵ α(s)α(t)α(u) (1 + α(s))(1 + α(t))(1 + α(u))A(s, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='17) It is hence straightforward to see the Regge behavior of δA(s, t) is δA(s, t) ∼ ˜FVS(t)α(s)2α(t)+2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='18) where ˜FVS(t) = ϵ α(t) 1 + α(t)FVS(t) = ϵP � 4 α′ �4 eπiα(t) Γ(1 − α(t)) Γ(2 + α(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='19) Note that δA has the same soft behavior as A in the high energy region with t < 0, but the first t-channel pole appears at α(t) = 1 rather than at t = 0, which corresponds to the exchange of a massive particle with spin 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The residue of δA is Res α(s)=n δA(s, t) = K �4n α′ , t � (α(t))n−1 (α(t) + 2)n−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='20) and the integral is evaluated as � ∞ Λ2 ds′ Discs δA(s′, t) s′3 = −16ϵP α′2 ∞ � n=0 (n + 1) (α(t))n (α(t) + 2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (n + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' + O(t) = −8ϵP α′2 2F3 (2, α(t), α(t) + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 1, 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 1) + O(t) = −8ϵP α′2 + O(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='21) – 9 – We can thus obtain the O(t0) negative term by deforming the Virasoro-Shapiro amplitude by δA defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' One can also obtain the same result by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='18): � ∞ Λ2 ds′ Discs A(s′, t) s′3 ∼ ϵ α(t) 1 + α(t) � ∞ Λ2 ds′ Discs A(s′, t) s′3 = −8ϵP α′2 + O(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='22) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='14) and the bound on ϵ, we find the bound on this negative contribution: 8ϵP α′2 ≲ α′ M2 Pl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='23) This bound on the negativity is consistent with the recent paper [44], in which constraints on Regge amplitudes are considered by using the finite-energy sum rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2 Coon amplitude The Coon amplitude [39] is a q-deformation of the Veneziano amplitude which enjoys many of the same UV properties [25, 40–42] but has some unusual features, most notably a spectrum with nonlinear Regge trajectories and an accumulation point below which there are infinitely many states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Despite some similarities with open string scattering in AdS [43], to date the Coon amplitude has no known worldsheet origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Nevertheless, we take it as a well-studied amplitude exemplifying the desired UV/IR characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The Coon amplitude has the following product representation,6 Aq(s, t) = g2(1 − q) exp �log σ log τ log q � ∞ � n=0 (στ − qn)(1 − qn+1) (σ − qn)(τ − qn) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='24) where g is some coupling constant, q ∈ (0, 1) and σ, τ are related to the usual Mandelstam variables as σ = 1 + (q − 1) � s µ2 − δ � , τ = 1 + (q − 1) � t µ2 − δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='25) The spectrum is quickly identified to be m2 n = µ2(δ + [n]q) , [n]q := 1 − qn 1 − q , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='26) with an accumulation point for large n: m2 n n→∞ −−−→ m2 ∗ := µ2 � δ + 1 1 − q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='27) There is a cut from the log σ factor which extends from the branch point at s = m2 ∗ out to infinity: see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 2 for a sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Going forward we take δ = 0 so that the spectrum contains massless states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' At low energies, s, t, u ≪ µ2, one finds Aq(s, t) ≈ −g2µ2 �1 s + 1 t � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='28) 6The Veneziano amplitude is recovered in the limit q → 1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' – 10 – × × × × ×××××××× × µ2 m2 ∗ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Schematic spectrum for the Coon amplitude with δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Massive states begin at m2 1 = µ2 and have an accumulation point at m2 ∗ = µ2 1−q where a branch cut begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' so that the crossing-symmetric combination ˜ Aq(s, t) = (s2 + t2 + u2) � Aq(s, t) + Aq(t, u) + Aq(u, s) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='29) at low energies goes as ˜ Aq(s, t) ≈ g2µ2 (s2 + t2 + u2)2 stu t→0 −−→ −4g2µ2 s2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='30) In the Regge limit the Coon amplitude takes the form Aq(s, t) −→ g2(1 − q)σ log τ log q ∞ � n=0 1 − qn+1 1 − qn τ ≈ fq(t) � − s m2∗ � log τ log q (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='31) and ˜ Aq(s, t) −→ ˜fq(t) � − s m2∗ �2+ log τ log q , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='32) which marginally satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2) for t < 0 (τ > 1) since log q is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The contour integral around the positive real-s axis splits into a sum over poles and an integral along the branch cut: � ∞ µ2 ds′ Discs Aq(s′, t) s′3 = −π ∞ � k=1 1 m6 k Res s=m2 k Aq(s, t) + � ∞ m2∗ ds′ Discs Aq(s′, t) s′3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='33) The residues are polynomial in t, Res s=m2 k Aq(s, t) = g2µ2qk k−1 � n=0 τ − qn−k 1 − qn−k t→0 −−→ g2µ2qk , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='34) and the sum over k gives a finite contribution for t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In contrast, the integral over the branch cut diverges for t → 0 and cancels the t-channel pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The range s > m2 ∗ corresponds to σ < 0, so we have Discs Aq(s, t) = −g2(1 − q)(−σ) log τ log q sin � π log τ log q � ∞ � n=0 (στ − qn)(1 − qn+1) (σ − qn)(τ − qn) = g2(1 − q)(−σ) log τ log q sin � π log τ log q � (στ − 1) (σ − 1) t µ2 ∞ � n=1 (στ − qn)(1 − qn+1) (σ − qn)(τ − qn) = −πg2(−σ) − 1−q log q t µ2 1 − q log q � 1 + O(t) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='35) – 11 – and thus Discs ˜ Aq(s′, t) (s′)3 ≈ −2πg2µ2 � s′ µ2 �−1− 1−q log q t µ2 1 − q log q � 1 + O(t) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='36) gives � ∞ m2∗ ds′ Discs ˜ Aq(s′, t) (s′)3 ≈ −2πg2µ−2 1 − q log q � 1 + O(t) � � ∞ m2∗ ds′ � s′ µ2 �−1− 1−q log q t µ2 = −2πg2µ2 t � 1 + O(t) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='37) This leading 1/t behavior exactly cancels the t-channel pole of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='30) in the dispersion relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Here the details of the cancellation are somewhat different, however, since the sum over poles does not generate a 1/t contribution despite there being an infinite number of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Knowing the contributions from poles and the branch cut must exactly compensate for the massless state, the O(t0) can be shown to be nonzero, 4g2µ2 � −1 t + 1 − q µ2 + O(t) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='38) providing another example where a Reggeized amplitude leaves a finite contribution after canceling the 1/t divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4 Positivity bounds at one loop In this section, we first review how the positivity bound at the one-loop level leads to an apparent puzzle that the cut-off scale of the theory is extremely low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' This is due to the fact that contribution from the graviton t-channel exchange at the one-loop level is roughly −s2/(m2M2 Pl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In order to cancel the negative contribution, the sufficiently positive contribution is needed, which requires the smaller cut-off scale (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Then, we propose how the Reggeization extends to one-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1 Small cut-off scale Suppose that the approximate gravitational positivity bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='22) at tree-level is correct even at the one-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [26] found that the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='22) is computed as e4 4π2Λ4 − e2 180π2m2M2 Pl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1) Note that the second term corresponds to a gravitational diagram while the first term corresponds to a non-gravitational diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' As this must be positive up to order α′/M 2 Pl, we obtain the upper bound on Λ: Λ ≲ � emMPl , for m2 ≪ e2α′−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2) Similar results are reported for the Standard Model [45] and the dark photon model [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In particular, using this argument, it is argued that almost all parameter regions of the dark photon model are excluded [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' – 12 – 𝑘 𝑘′ ⃗𝑝′ 𝛼𝛽 𝛾𝛿 𝜈 𝜇 ⃗𝑝 𝑘 𝑘′ 𝜈 𝜇 ⃗𝑝′ ⃗𝑝 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' A star vertex in the right diagram is a new vertex from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We consider new diagrams obtained by replacing the left diagram with the right one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' If a higher-dimensional two-derivative term is added to quadratic action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1), there is no change in the positivity bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' For example, we can consider the additional term with mass dimension six, S6 = � d4x √−g c Λ2 φ†φFµνF µν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3) A new vertex (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 3) and diagrams arise from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3), but we can confirm that their contributions to the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='22) cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2 Reggeization at the loop level When the cut-off scale is larger than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2), it is still possible that the one-loop Regge amplitude cancels the negative contribution corresponding to the graviton exchange [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Naively, this is unlikely because the UV Regge amplitude should “know" the IR mass scale m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' However, here we propose the generalization of the Regge amplitude to the one-loop level, which realizes the cancellation naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' At the tree level, we have assumed the Reggeization A → Ftree(t) s2+α′t M2 PlM2α′t ∗ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='4) for fixed t and large s (Ftree has mass dimension −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' This is expected to happen by the exchange of the higher spin particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Given this interpretation, the important requirement to the behavior of Ftree(t) is that Ftree(t) must have poles at t = n α′−1 where n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='. Therefore, Ftree(t) is written as Ftree(t) = f0 t + f1 t − α′−1 + f2 t − 2α′−1 + · · · , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='5) – 13 – where fi is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Since each residue of the pole must be positive, we obtain fi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The presence of the pole at t = 0 is crucial to maintain the positivity bound at the tree-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Because of the poles, the amplitude at t = n α′−1 is dominated by the diagram corresponding to the exchange of the particle of the mass n α′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Next, we have to impose an extra requirement at the one-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We denote the ampli- tude as A → � Ftree(t) + Floop(t) � s2+α′t M2 PlM2α′t ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='6) At the one-loop level, contributing diagrams include those with a loop of light particles and tree-level exchange of higher spin particles (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Importantly, these diagrams have a branch cut starting from t = 4m2 corresponding to particle production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' At the one-loop level, the branch cut must appear at t = 4m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Floop is written as Floop(t) = ˜f0L0(t, m2) + ˜f1α′−1 t − α′−1 L1(t, m2) + ˜f2α′−1 t − 2α′−1 L2(t, m2) + · · · , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='7) where Li(t, m2) is a loop function that contains the branch cut starting from t = 4m2 (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3 for the detail), and has an expansion Li(t, m2) = ∞ � j=1 l(i) j m2 � t m2 �j−1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='8) up to log(m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Here lj is a dimensionless constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' On top of the branch cut, there exist poles at t = kα′ where k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='. Now we discuss the implications of the Regge amplitude on the dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The contribution from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='7) is � ∞ M2∗ ds′ Disc A(s′, t) s′3 ∼ sin(α′t) α′t � ˜f0L0(t, m2) + ˜f1 t t − α′−1 L1(t, m2) + ˜f2 t t − 2α′−1 L2(t, m2) + · · · � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='9) For t → 0, by choosing ˜fi = O(α′) appropriately, the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='19) contains a finite contribution from the one-loop diagrams with higher-spin spins exchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In particular, the positivity bounds for scalar QED can be satisfied without requiring an unusually low cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3 An argument for the form of Floop In this section we provide an explanation for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='7) by estimating the diagrams of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4, where the double line is either the graviton or the higher spin particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The loop integral in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4 is (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' A for the notation) � d4p (2π)4 1 p2 − m2 1 (p + k1)2 − m2 1 (p + k1 + k3)2 − m2 (2p + k1)µ(2p + 2k1 + k3)ν × (pρ1 · · · pρn1) � (p + k1 + k3)σ1 · · · (p + k1 + k3)σn2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='10) – 14 – 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 𝑘" 𝑘# 𝑘$ ⃗𝑝 𝛼𝛽 𝛾𝛿 𝜈 𝜌 𝜇 𝜎 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 𝑘" 𝑘# 𝑘$ ⃗𝑝 𝛼𝛽 𝛾𝛿 𝜈 𝜌 𝜇 𝜎 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 𝑘" 𝑘# 𝑘$ ⃗𝑝 𝜈 𝜌 𝜇 𝜎 𝛼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='𝛼#𝛼" ⋯ 𝛽!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='𝛽#𝛽" ⋯ 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 𝑘" 𝑘# 𝑘$ 𝛼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='𝛼#𝛼" ⋯ 𝜈 𝜌 𝜇 𝜎 𝛽!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='𝛽#𝛽" ⋯ ⃗𝑝 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The one-loop diagram corresponding to the t-channel exchange of the graviton and the higher spin particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The solid, wavy, double solid and double wavy lines correspond to the scalar, graviton, photon and higher spin particles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' These produce the branch cut starting from t = 4m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' where the second line comes from the interaction between φ and higher spin-L particle Φα1···αL, g1 ML−1 Φα1···αL∂α1 · · · ∂αiφ∂αi+1 · · · ∂αLφ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='11) and n1 + n2 = L with M and g1 being the mass and coupling of Φα1···αL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Note that the coupling between the photon and Φα1···αL is schematically given by g2 ML−1 Φβ1···βL∂β3···βmF β1µ∂βm+1···βLF β2µ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='12) where g2 is the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' By introducing Feynman parameters, we obtain � d4l 8π4 � 1 0 dy � 1−y 0 dx (2l + (1 − 2x − 2y)k1 − 2yk3)µ(2l + 2(1 − x − y)k1 + (1 − 2y)k3)ν [l2 − m2 + y(1 − x − y)t]3 × � (l − (x + y)k1 − yk3)ρ1 · · · (l − (x + y)k1 − yk3)ρn1 � × � (l + (1 − x − y)k1 + (1 − y)k3)σ1 · · · (l + (1 − x − y)k1 + (1 − y)k3)σn2 � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='13) – 15 – where l = p + (x + y)k1 + yk3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In the following, we concentrate on the term which contains the light mass parameter m2 in the denominator for t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' From a power-counting argument, this corresponds to the terms where the numerator does not contain l: � d4l 8π4 � 1 0 dy � 1−y 0 dx ((1 − 2x − 2y)k1 − 2yk3)µ(2(1 − x − y)k1 + (1 − 2y)k3)ν [l2 − m2 + y(1 − x − y)t]3 × � (−(x + y)k1 − yk3)ρ1 · · · (−(x + y)k1 − yk3)ρn1 � × � ((1 − x − y)k1 + (1 − y)k3)σ1 · · · ((1 − x − y)k1 + (1 − y)k3)σn2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='14) The structure of the cut becomes clear by integrating over l, x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We obtain the following expression: � 1 0 dy � 1−y 0 dx � d4l 8π4 1 [l2 − m2 + y(1 − x − y)t]3 = −i � 1 0 dy � 1−y 0 dx � ddlE 8π4 1 [l2 E + m2 − y(1 − x − y)t]3 = −i � 1 0 dy � 1−y 0 dx 1 16π2 1 m2 − y(1 − x − y)t = −i t Li2 � 2 √ t √ t − √ t − 4m2 � − i t Li2 � 2 √ t √ t + √ t − 4m2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='15) where Li is the polylogarithm function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='7 Here for simplicity we ignore x and y dependence in the numerator, but the structure of the branch cut does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='15) has a small-t expansion of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='8) − i t Li2 � 2 √ t √ t − √ t − 4m2 � − i t Li2 � 2 √ t √ t + √ t − 4m2 � = −i � 1 2m2 + t 24m4 + t2 180m6 + · · · � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='16) We write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='14) in the following schematic way: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='14) = (k1 + #k3)µ(k1 + #k3)ν · · · (k1 + #k3)σn2FL−2(t, m2) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='17) where FL−2(t, m2) is defined by FL−2(t, m2) =(−1)n1 � d4l 8π4 � 1 0 dy � 1−y 0 dx 2(1 − x − y)(1 − 2x − 2y)(x + y)n1(1 − x − y)n2 [l2 − m2 + y(1 − x − y)t]3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='18) The indices µ and ν are contracted with the external polarizations ϵµ 1 and ϵν 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' This gives a factor (k1 + #k3)µ(k1 + #k3)νϵ1µϵ3ν ∼ tu s → t , for s → ∞ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='19) 7Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='15) is finite at t = 4m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' – 16 – Other indexes are contracted with k2, k4, ϵ2 or ϵ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' For large s, this contribution behaves as sL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Multiplying the propagator of the higher spin particle, we obtain Figure 4 = g1g2 M2L−2 sLt t − M2 FL−2(t, m2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='20) for large s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Suppose that the mass M and the spin L of the particle are given by M2 = kα′−1 , L = 2 + k , k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='21) as usual in the graviton Regge tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Then, the amplitude is g1g2 M2L−2 s2+kt t − kα′−1 Fk(t, m2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='22) When we take t → kα′−1, we expect that the contribution from the exchange of spin 2 + k particle dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' This reproduces Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We would like to explain an idea why the summation of the t-channel diagram leads to the Reggeized amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' At the tree-level, the t-channel amplitude is s2 M2 Plt + g2 2 s3α′2 t − α′−1 + g2 2 s4α′3 t − 2α′−1 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='23) After an appropriate analytic continuation, we obtain s2+α′t/(M2 Plt) for fixed t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='8 Similarly, at the one-loop level, the amplitude is � s2 M2 Pl + g1g2 s3t α′2 t − α′−1 + g1g2 s4t α′3 t − 2α′−1 + · · · � F(t, m2) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='24) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Here we have assumed that all Fk are the same order of magnitude, and have defined F := Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Since the expression in the parenthesis is the same form as the tree- level t-channel amplitude, it is natural to expect that this is also Reggeized as s2+α′t/M 2 Pl for fixed t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' In fact, the formal summation of the t-channel diagram leads to s2 M2 Pl + g1g2 ∞ � n=1 s2t α′ t − nα′−1 (α′s)n = s2 M2 Pl − g1g2 � α′s �3 t Φ(α′s, 1, 1 − α′t) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='25) where Φ is the Lerch zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' By expanding around t = 0, we obtain − � α′s �3 t Φ(α′s, 1, 1 − α′t) = −α′s2 ∞ � n=1 � α′t �n Lin(α′s) → α′s2 ∞ � n=1 (α′t log(α′s))n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='26) for α′s ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Here we have used Lin(α′s) = −(log(α′s))n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' + O �� log(α′s) �n−1� , for α′s ≫ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='27) 8See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='8) and App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' B for the detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' – 17 – Now, assuming that g1g2α′ = M−2 Pl , we obtain s2 M2 Pl + g1g2 ∞ � n=1 s2t α′ t − nα′−1 (sα′)n → s2 M2 Pl + g1g2α′s2 ∞ � n=1 (α′t log(α′s))n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' = s2 M2 Pl eα′t log(α′s) = s2+α′t M2 Pl α′α′t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='28) As a result, we obtain s2+α′t M2 Pl α′α′tF(t, m2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='29) This is nothing but our proposal (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='7) for small t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Although we are not able to compute the numerical factor of each diagram,9 this illustrates the idea of how Reggeization occurs, and how the parametrically large prefactor appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 5 Discussion In this paper, we have studied the gravitational positivity bound at the one-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We first reviewed the gravitational positivity bound at tree level, where the t-channel pole corresponding to the graviton exchange can be canceled by assuming Regge behavior for the amplitude at high energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Next, we saw two examples of unitary, Reggeized gravitational amplitudes for which the potentially negative finite contribution to the positivity bounds can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Finally, we moved to the features of Reggeized amplitudes at one loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' It is known that the one-loop EFT amplitude leads to the parametrically large negative contribution to the positivity bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We argued for a form of the Reggeized amplitude at the one-loop level based on the analytic structure of the t-channel exchange diagram of the graviton and the higher spin particles in the Regge tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' As a future direction, we may consider a string theory setup to explicitly check the form of the amplitude at the one-loop of the matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' For instance, we may take the non- supersymmetric SO(16) × SO(16) heterotic string and compactify it on T 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='10 In 10d there are fermions whose representation is (16, 16) + (128, 1) + (1, 128) under SO(16)× SO(16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' These will become Dirac fermions in 4d after compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' By turning on a VEV for the Wilson lines, the gauge symmetry is broken from SO(16) × SO(16) to U(1)16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='11 Then the (16, 16) fermions have charge � 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' , 0 � �� � U(1)8 , 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' , 0 � �� � U(1)8 � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1) (and its permutations) under U(1)16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Similarly, (128, 1) has charge � ± 1 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' , ± 1 2 � �� � U(1)8 , 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' , 0 � �� � U(1)8 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2) 9In general, n-dependent coefficient appears in the summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Moreover, the couplings g1,2 can depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' It is interesting to study the condition to realize the Reggeized amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 10We can consider the same setup for the supersymmetric heterotic string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 11In addition, there are KK and winding U(1)s, but these are not important here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' – 18 – These fermions receive masses from the Wilson line, and the masses are chosen to be arbitrary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' This setup is close to the non-supersymmetric QED coupled with light matters, and loops of these fermions may give a negative contribution to the coefficient of s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' On the other hand, the typical mass of the higher-spin particles is always the string scale, independent of the VEV of the Wilson line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' It is an interesting task to compute the one-loop string amplitude at O(M−2 Pl ) in this setup to check the details of our proposal in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' – 19 – Acknowledgments The work of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' is supported in part by MEXT Leading Initiative for Excellent Young Researchers Grant Number JPMXS0320210099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The work of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' is supported in part by JSPS KAKENHI Grant Number 21J15497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' A Notation We follow the notation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' We consider the scattering γ1γ2 → γ3γ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The momenta kµ 1,2,3,4 are parametrized as (all-ingoing notation) kµ 1 = (k, 0, 0, k) , kµ 2 = (k, 0, 0, −k) , kµ 3 = −(k, k sin θ, 0, k cos θ) , kµ 4 = −(k, −k sin θ, 0, −k cos θ) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1) We define ϵµ 1,2,3,4 as a polarization vector of the external photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' The ± polarization corresponds to ϵµ 1(±) = 1 √ 2 (0, 1, ±1, 0) , ϵµ 2(±) = 1 √ 2 (0, −1, ±1, 0) , ϵµ 3(±) = 1 √ 2 (0, cos θ, ±i, − sin θ) , ϵµ 4(±) = 1 √ 2 (0, cos θ, ±i, ± sin θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2) In all-ingoing notation, the amplitude is written as A(h1, h2, h3, h4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' k1, k2, k3, k4) = ϵµ 1(h1)ϵν 2(h2)ϵα 3 (h3)ϵβ 4(h4)Aµναβ(k1, k2, k3, k4) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3) where h1,2,3,4 = ±1 is the helicity of the external photons in all-incoming notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' ϵij is defined as ϵij := ϵi · ϵj (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='4) Explicitly, we obtain ϵ12 = −1 2 − h1h2 2 , ϵ13 = −h1h3 2 + 1 2 + t s , ϵ14 = −h1h4 2 − 1 2 − t s , ϵ34 = −1 2 − h3h4 2 , ϵ24 = −h2h4 2 + 1 2 + t s , ϵ23 = −h2h3 2 − 1 2 − t s , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='5) The inner products between external momenta and polarizations are k1 · ϵ3 = k3 · ϵ1 = k2 · ϵ4 = k4 · ϵ2 = − √ tu √ 2s , k1 · ϵ4 = k4 · ϵ1 = k2 · ϵ3 = k3 · ϵ2 = √ tu √ 2s , (others) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='6) The inner products among external momenta are k1 · k2 = k3 · k4 = s/2 , k1 · k3 = k2 · k4 = t/2 , k1 · k4 = k2 · k3 = u/2 , k2 1 = k2 2 = k2 3 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='7) where (s, t, u) are the Mandelstam variables: s = (k1 + k2)2 = 2k1 · k2, t = (k1 + k3)2 = 2k1 · k3, u = (k1 + k4)2 = 2k1 · k4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='8) – 20 – B Reggeization at the tree-level In this appendix, we show how the tree-level amplitude is Reggeized starting from the pole expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='8), along the line with the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='12 For large s, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='8) is written as A(s, t) ∼ −4Ps4 α′ ∞ � n=0 1 (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' )2 �α′s 4 �2n−2 � 1 t − 4nα′−1 − 1 t + s + 4nα′−1 � = −64Ps2 α′3 ��1 t − 1 t + s � + ∞ � n=1 1 (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' )2 �α′s 4 �2n � 1 t − 4nα′−1 − 1 t + s + 4nα′−1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1) We expand the second term around t = 0, and then perform the summation from n = 1 to n = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' By keeping the leading term for s → ∞ in each order of t, we obtain ∞ � n=1 1 (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' )2 �α′s 4 �2n � 1 t − 4nα′−1 − 1 t + s + 4nα′−1 � ∼ α′ 4 ∞ � m=1 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' �α′t 4 �m−1 � log �α′2s2 16 ��m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='2) By substituting this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='1), we obtain the Reggeized amplitude: A(s, t) ∼ −16Ps2 α′2 ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' �α′t 4 �m−1 � log �α′2s2 16 ��m + 64Ps2 α′3(t + s) = −64Ps2 α′3 �α′2s2 16 �α′t/4 + 64Ps2 α′3(t + s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='3) References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Adams, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Arkani-Hamed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Dubovsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Nicolis and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (page 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Arkani-Hamed, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Huang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Huang, The EFT-Hedron, JHEP 05 (2021) 259 [2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='15849].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (pages 1, 2, 7, 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Vafa, The String landscape and the swampland, hep-th/0509212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (page 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Palti, The swampland: Introduction and review, Fortschritte der Physik 67 (2019) 1900037 [1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='06239].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' (page 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' van Beest, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Calderón-Infante, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Mirfendereski and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Valenzuela, Lectures on the Swampland Program in String Compactifications, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content=' 989 (2022) 1 [2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdA0T4oBgHgl3EQfDP8K/content/2301.01999v1.pdf'} +page_content='01111].' metadata={'source': 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b/StE4T4oBgHgl3EQfLQyp/content/tmp_files/2301.04937v1.pdf.txt @@ -0,0 +1,1137 @@ +Density-based clustering with fully-convolutional +networks for crowd flow detection from drones +Giovanna Castellanoa, Eugenio Cotardoa, Corrado Mencara, Gennaro +Vessioa +aDepartment of Computer Science, University of Bari Aldo Moro, Bari, Italy +Abstract +Crowd analysis from drones has attracted increasing attention in recent times +due to the ease of use and affordable cost of these devices. However, how +this technology can provide a solution to crowd flow detection is still an +unexplored research question. To this end, we propose a crowd flow detec- +tion method for video sequences shot by a drone. The method is based on +a fully-convolutional network that learns to perform crowd clustering in or- +der to detect the centroids of crowd-dense areas and track their movement +in consecutive frames. The proposed method proved effective and efficient +when tested on the Crowd Counting datasets of the VisDrone challenge, +characterized by video sequences rather than still images. The encouraging +results show that the proposed method could open up new ways of analyzing +high-level crowd behavior from drones. +Keywords: +drones, drone vision, computer vision, deep learning, crowd +flow detection, crowd density estimation, clustering +1. Introduction +Due to population growth and the increasing degree of urbanization, more +and more people live in urban areas. Positive consequences of this trend are +the enrichment of cultural life and the full use of convenient urban infras- +tructure. At the same time, gatherings of people, which can occur for various +reasons, such as political demonstrations, festival celebrations, concerts, and +so on, pose serious challenges to urban security and management. In this per- +spective, automated crowd analysis methods, which typically involve crowd +counting and associated crowd density estimation, have attracted increasing +Accepted manuscript: 10. 1016/ j. neucom. 2023. 01. 059 +arXiv:2301.04937v1 [cs.CV] 12 Jan 2023 + +attention for their many potential applications [1, 2]. These include the pre- +vention of crowd-induced disasters, such as stampedes, but also other less +critical objectives such as better crowd management at public events and the +design of public spaces and virtual environments. +A cost-effective way to perform automated crowd analysis is by using un- +manned aerial vehicles (UAVs), more commonly known as drones. Indeed, +once equipped with affordable but sufficiently powerful cameras and GPUs, +drones can become flying computer vision devices that can be rapidly de- +ployed for a wide range of applications, including crowd analysis for public +safety [3]. However, while these perspectives are fascinating, there are also +some drawbacks to be aware of. On the one hand, the computer vision algo- +rithms applied to aerial images are burdened with further difficulties because +the problems of scale and point of view are taken to the extreme. On the +other hand, the sophisticated and computationally intensive methods com- +monly applied in this field do not meet the stringent real-time requirements +imposed by the UAV. In other words, lightweight models that offer a good +compromise between effectiveness and efficiency are essential [4]. +Crowd analysis with drones has attracted attention in recent years [5]. +However, despite significant progress, the proposed methods still have room +for improvement to address the challenges posed by drones. In this article, +we want to contribute to this research effort by taking it one step further: +instead of considering crowd counting and density estimation in static frames, +we aim to detect crowd flow. This poses a new challenge as the goal is not +only to recognize the presence of people in a single high-altitude scene but +also to determine how crowds flow as a function of time. This is different +from people tracking—where the goal is to track a single person or groups of +people—and can lead to useful systems, as it can allow for crowd behavior +analysis for better logistics and disaster prevention [6]. +To this end, we propose a method for crowd flow detection from drones +based on fully-convolutional networks (FCNs). The network is trained to rec- +ognize groups of people in each frame and, to do this, simultaneously learns to +perform crowd density estimation and crowd clustering. In this way, groups +of people are identified simply by their centroids, and these are used to trace +the trajectories of the identified groups, following their movement during the +shooting of the drone. We preferred FCNs over other architectures mainly be- +cause of their known efficiency. Furthermore, direct learning of crowd clusters +was preferred to avoid a multi-step approach, based on performing density +estimation first and clustering the resulting density maps later. This was +2 + +done in our previous preliminary work [7] but, while effective, this approach +proved too demanding from a computational point of view. The method +was tested on the recently proposed Crowd Counting 2020 [8] and 2021 [9] +datasets, used annually for the international VisDrone challenge. The pecu- +liarity of these datasets is that they are not characterized by still images but +actually by frames of video sequences that are used here to perform the crowd +flow detection task. It is worth noting that what we actually do to deter- +mine crowd flow is to calculate the inter-frame difference between centroids; +in other words, it is a kind of “inter-frame density clustering”. However, +since this approach allows us to detect the movement of the crowd frame by +frame, we use the expression “flow detection” for the sake of simplicity. +The rest of this paper is structured as follows. Section 2 reviews related +work. Section 3 describes the datasets used. Section 4 presents the pro- +posed method. Section 5 describes the experimental setup and discusses the +results obtained. Section 6 concludes the paper and highlights the future +developments of our research. +2. Related work +There is a large body of knowledge about crowd counting and crowd den- +sity estimation in computer vision, but the trend today is density estimation. +Early work usually applied a person or head detector via a sliding window +on the image. However, although current implementations may be based on +state-of-the-art object detectors such as YOLO [10, 11], these approaches +still provide unsatisfactory results when asked to detect small objects in a +very dense crowd. To alleviate this problem, regression-based methods have +been introduced that directly learn the mapping from an image to the global +people count [1]. However, although these methods make the approach in- +dependent of the precise position of individuals in the crowd, which is very +complex, they ignore the spatial information that can be very useful for pre- +diction. To avoid the difficulty of accurately detecting and locating people +in the scene, while using spatial information, the recent trend is to learn +density maps, thus incorporating spatial information directly into the learn- +ing process [12]. Successful solutions include methods that work first at the +patch level and then fuse local features [13], methods that integrate atten- +tion mechanisms [14], cascade approaches that jointly learn people counting +and density maps [15], methods that improve performance through knowl- +edge distillation [16], and frameworks that simultaneously perform crowd +3 + +counting and localization [17]. +However, while effective, these approaches are generally computationally +demanding and do not meet the stringent requirements typically imposed +by UAVs (limited battery, need for real-time responses). How to fine-tune +deep neural architectures to achieve an optimal balance between precision +and performance is an active research area. The VisDrone Crowd Counting +challenge was introduced to encourage research in this direction [8, 9]; nev- +ertheless, the solutions proposed by the participants in the challenge are not +always focused on efficiency but rather on effectiveness, as the goal is only +to obtain a low error in counting people. The lowest error was obtained with +TransCrowd [18], based on the increasingly popular Vision Transformer [19]. +However, the proposed method only regresses the people count, not provid- +ing density maps that would be useful for detecting crowd flow; furthermore, +transformer-based solutions are known to be computationally expensive. +A promising way to address these problems is to use FCN models. Be- +cause they do not rely on fully-connected layers at all, which are the most +expensive part of processing a neural network, they are a candidate solu- +tion for finding an accurate model without damaging the inference time. An +FCN model for aerial drone imaging was presented in [4], and a similar so- +lution was also proposed in our previous work [20]. However, both methods +were aimed at crowd detection, i.e. discriminating between crowded and un- +crowded scenes; furthermore, they only provide coarse density maps, as the +models have not been trained on people labels. +Human tracking methods based on RGB cameras or other sensors that +use clustering or classification models to track motion have been investigated, +e.g. [21, 22, 23]. However, they are designed to work indoors or by involv- +ing a few people from a frontal perspective. The work most linked to ours, +which takes into account the images captured by drones, is [24] where the +same authors who propose the VisDrone Crowd Counting datasets present +a model that jointly solves density map estimation, localization, and track- +ing. This model differs from ours in that it uses a complex and expensive +pipeline aimed at tracking individual trajectories. Other authors have re- +cently proposed a method for periodic crowd tracking from UAVs based on +a binary linear programming model [25]. However, they experimented with +simulated scenarios that do not consider the crowd detection problem from +a computer vision perspective. As far as we know, there is no work in the +literature addressing crowd flow detection in drone videos, which poses sig- +nificantly different challenges than traditional settings. This paper aims to +4 + +fill this gap; in particular, we aim to trace the centroids that identify groups +of people by exploiting the spatial information learned and expressed through +density maps. +3. Materials +The literature landscape is not as populated with datasets for crowd +counting and density estimation from drones with video sequences captured +by optical cameras. The datasets best suited to our purposes for evaluating +crowd flow detection were the VisDrone Crowd Counting 2020 [8] and 2021 [9] +datasets. The two benchmarks are characterized as follows: +• VisDrone Crowd Counting 2020 (CC2020) contains 82 video clips (2, 460 +frames in total) with a resolution of 1920 × 1080; +• VisDrone Crowd Counting 2021 (CC2021) contains 1, 807 frames with +a resolution of 640 × 512. Unlike CC2020, the frames are not arranged +in precisely separated video clips, and this required manual separation +to split the sequences of different locations to avoid overfitting. +The video sequences were acquired by various drone-mounted cameras, +which shot different scenarios in different cities in China to maintain diversity. +In both datasets, people were annotated manually with dots in each video +frame, expressed as (x, y) coordinates in the bi-dimensional plane. +How- +ever, CC2020 and CC2021 have significant differences. In CC2020, each of +the 82 sequences was captured by a drone hovering over the crowd, allow- +ing for rather static scenes. In CC2021, on the other hand, the drone flies +and sometimes rotates, shooting different scenes even if semantically linked. +Furthermore, it should be noted that in CC2020 there is a predominance of +daylight scenes, while in CC2021 there are many frames at night, making +the dataset much more variable in this respect. A final difference concerns +altitude: in CC2020 the frames appear to have been shot at a higher altitude +than in CC2021. For both datasets, we randomly held out (sequence-wise) +a fraction of 25% of the total frames evenly split to form validation and test +sets. +Compared to benchmark datasets focused on crowd counting in surveil- +lance scenes, both VisDrone datasets present particular challenges due to the +scenes captured by drones. Object scales can be extremely small due to the +high shooting altitude of drones. Crowds are sparse across the video frames, +5 + +as each can hold a few to dozen people. Finally, the crowds are surrounded +by very different backgrounds in different sequences. +4. Methods +The proposed method for crowd flow detection is based on an FCN model, +which is used to estimate a “centroid density map”—a heatmap highlighting +crowd centroids—from pairs of consecutive frames of the same video sequence +shot by the drone. Since a crowd can be seen as multiple groups of people not +necessarily following the same direction, the predicted centroids are represen- +tative of these groups of people. The displacements of the centroids detected +between the pairs of frames are then calculated to identify the direction of +movement. Thus, the framework assumes video sequences shot by a drone, +but the network is fed one frame at a time. +The idea of combining a density estimation method with clustering, in- +stead of tracking the movement of each individual, is motivated by the com- +plexity and computational cost of this strategy. In fact, while the direct use +of the crowd density map based on the location of all individuals can retain +more information, it also carries the burden of tracking the trajectory of +each individual in the scene. Moreover, individual tracking can be not only +impractical but also non-essential, as in crowd management scenarios it is +important to recognize the overall flow of people rather than the precise loca- +tion of each person in the scene. A centroid density estimation method will +focus only on high-density areas, i.e. those corresponding to concentrations +of people, and will be inherently robust to occlusion, which would heavily +affect a people detector, especially from a high altitude. +The method is detailed in the following subsections, along with an expla- +nation of how the ground truth is obtained. An illustration schematizing the +overall processing is shown in Fig. 1. +4.1. Ground truth generation +Since we assume that there is no label regarding the location of the cen- +troids within the original drone shot, we followed a simple strategy to derive +a ground truth for the crowd centroids. +We first applied the well-known +Mean Shift clustering algorithm to the head annotations described above +to obtain centroids. Mean Shift is a centroid-based algorithm that updates +the candidates for the centroids as the average of the points within a given +6 + +Figure 1: Schema of the overall method. Pairs of consecutive frames taken by a drone at +time t0 and tk (with tk > t0 and separated by any time interval) are fed into the neural +network model for crowd clustering. A synthetically generated ground truth helps guide +the learning of the network. The obtained centroid density maps are then thresholded, +and the displacement of the centroids, representing the groups of people, from t0 to tk, is +used to determine their direction of movement. +region. These candidates are then filtered in a post-processing step to elimi- +nate near-duplicates [26]. Specifically, the Scikit-learn implementation of the +Mean Shift algorithm with the recommended default hyperparameters was +used.1 Then, following the seminal paper by Zhang et al. [27], if there is a +centroid at pixel xi, we represented it as a delta function δ(x − xi) and we +obtained a “centroid density map” C(x) convolving the delta function with +a Gaussian kernel: +C(x) = +K +� +i=1 +δ(x − xi) ∗ Gσ(x) +In the formula, K is the number of centroids; δ equals 1 when x = xi, 0 +otherwise; and Gσ is the Gaussian kernel. Since the sizes of the heads are +similar in each video sequence and there is no perspective problem as in +[27], we decided to use a fixed σ for each frame. +In particular, we have +1At the time of writing, the version of Scikit-learn is 1.1.2. +7 + +Ground truth + Loss calculation +generation +to +tk +to +tk +Thresholding +Displacement +calculation +Neural network +modelFigure 2: Examples of frames and corresponding synthetically generated centroid density +maps. +empirically set σ = 10, since this value leads to better performance thanks +to the sufficiently large “confidence” activation area. Examples of generated +centroid density maps are shown in Fig. 2. +We used Mean Shift as it is impossible to know the number of clusters +in the crowd in advance, so there was a need for an algorithm that did +not require a pre-specification of the number of clusters. In this way, an +unsupervised learning approach is followed to find crowd centroids, but when +their location is found, it can be used as a synthetically generated annotation +to guide a supervised learning approach. Although this strategy strongly +depends on the results provided by the specific clustering algorithm chosen, +it allowed us to automate the generation of the ground truth and to guide and +quantitatively evaluate the clustering task performed by the neural network. +In addition, we also experimented with models aimed at performing the +more classic crowd density estimation task. +To this end, “crowd density +maps” D(x) were obtained by convolving the same delta function with Gaus- +sian kernel as before but using the original people head annotations. +4.2. Fully-Convolutional Networks +The proposed method is based on an FCN architecture that recognizes +the crowd centroids within each frame and produces the related heatmap. In +8 + +our previous preliminary work [7], the identification of the centroids was del- +egated to a classic clustering algorithm after the generation of crowd density +maps, significantly penalizing the inference time. In the method proposed +here, instead, the task of finding crowd centroids is integrated directly into +the network training and is performed in a single step. +Fully-convolutional neural networks, originally proposed in [28], perform +only convolution and pooling operations and discard the fully-connected com- +ponent typical of CNNs. Instead of the fully-connected layer, there is a 1×1 +convolution with stride 1, which allows, on the one hand, to have fewer pa- +rameters to estimate and, on the other hand, to be able to receive an image +of arbitrary size as an input. It is worth noting that although FCNs have this +desirable property, for better evaluation, we have set the input resolution of +each frame to 640 × 512 × 3. The architecture of an FCN consists of two +parts: an encoder aims to downsample the input into a lower-dimensional +representation, a decoder aims to upsample the latent representation to the +desired output resolution. +In particular, we have experimented with two different neural networks: +an ad-hoc FCN designed to be as simple as possible, and a state-of-the-art +FCN already used for crowd counting in more traditional contexts, i.e. Mo- +bileCount [29]. In the ad-hoc implementation, a rescaling layer normalizes +each pixel value in the [0, 1] range. Then, the encoder part of the model +repeatedly applies four blocks consisting of a convolutional layer with kernel +3 × 3, batch normalization, and max pooling with kernel 2 × 2 until reaching +a latent space Z ∈ R40×32×128. The decoder upsamples the feature maps with +transposed convolution and batch normalization. Finally, a 1 × 1 convolu- +tional layer produces the output density map of size 640 × 512 × 1. The +commonly used ReLU was chosen as the activation function. MobileCount, +on the other hand, is similar but has some more sophisticated architectural +designs. +To reduce the input resolution, a 3 × 3 max pooling layer with +stride 2 is added before the encoding part. The encoder is adapted from Mo- +bileNetV2 [30] by reducing the number of inverted residual blocks from 7 to +4. As for the decoding component, the lightweight RefineNet [31], originally +designed for semantic segmentation, is exploited. +Since we have two datasets with very different characteristics, as shown in +Fig. 3 we experimented with three main variants of the above architectures, +which were likely to produce different results: +• Single-branch, single-output (SBSO): this architecture uses one of the +9 + +two FCNs described above and learns to directly estimate the centroid +density map. As a loss function, the network is trained to minimize +the mean squared error between the predicted and the ground truth +centroid density map: +L = 1 +N +N +� +i=1 +��CP(i) − CGT(i) +��2 +2 +where N is the number of samples, CP(i) and CGT(i) are the predicted +and ground truth centroid density map, respectively, and ∥·∥2 is the +Euclidean distance. Demanding the model to find groups of people in a +completely unsupervised way would not have provided the network with +a guide to optimize weights to cluster only a specific type of objects, +in our case people, leaving it completely free to separate people from +trees, buildings, etc., which was not our goal. +• Single-branch, multi-output (SBMO): The task of finding crowd cen- +troids may be difficult for the model as the centroids do not represent +objects effectively present within the scene. To try to mitigate this +issue, we also experimented with a multi-output model, which extends +the previous one by learning to simultaneously estimate the centroid +density map and the more classic crowd density map. The main idea is +that the auxiliary task can help the overall network extract the features +that actually characterize the people in the scenes, thus supporting the +main task of density estimation, which only concerns the detection of +crowd centroids. In this case, the loss function is the sum of the above +and the mean squared error between the predicted and the ground truth +crowd density maps. +• Dual-branch, multi-output (DBMO): finally, to improve the capacity of +the previous variant, we also experimented with a multi-output model, +this time characterized by two branches, which are actually two “twin” +autoencoders with the same architecture as before (ad-hoc FCN or Mo- +bileCount). The two branches communicate with each other through +a concatenation of the feature maps produced immediately before the +desired output: one branch is totally dedicated to learning the crowd +density map; the concatenation of both branches contributes to esti- +mating the centroid density map. The loss function is the same as the +previous variant. +10 + +In other words, the multi-output architectures are meant to help the model +learn more about the concept of “crowdedness”. Because these variants in- +herently perform crowd counting, in addition to crowd clustering, it is worth +noting that they can be used for the more classical task of pedestrian counting +if needed. +Finally, it is important to emphasize that the neural network models +we have tested are a backbone of the overall framework we propose, which +can be replaced with similar models as desired. Table 1 compares the two +backbones used in this work in terms of the number of parameters. The +proposed FCN has been designed as a classic and straightforward baseline. +It is small, but deliberately without any sophistication to further improve +effectiveness or efficiency, to show the feasibility of the proposed method +with a simple model. MobileCount, on the other hand, has more parameters +but was purposely made by the authors as an efficient framework for real-time +crowd counting. In fact, the MobileNetV2-based encoder and the RefineNet- +based decoder were carefully adapted to achieve a good balance between +accuracy and speed, as reported in the article [29]. +11 + +Figure 3: Schema of the proposed neural network architectures. The proposed ad-hoc +FCN is schematized in this figure but can be replaced by any similar model as desired. +12 + +Encoder +Decoder +Input frame +Centroid density map +640 × 512 × 3 +640 × 512 × 1 +SBSO +Encoder +Decoder +Crowd density map +640 × 512 × 1 +Centroid density map +640 × 512 × 1 +Input frame +640 × 512 × 3 +SBMO +Encoder +Decoder +Crowd density map +640 × 512 × 1 +Input frame +Centroid density map +640 × 512 × 3 +640 × 512 × 1 +DBMO +CONV + BN + MAX POOL +LATENT SPACE +CONV TRANSPOSE + BNAd-hoc FCN +MobileCount +SBSO +0.35 +25.199 +SBMO +0.35 +25.199 +DBMO +0.7 +50.398 +Table 1: Comparison in terms of the number of parameters, in millions, between the pro- +posed ad-hoc FCN and MobileCount, used as the backbone of the proposed architectures. +4.3. Crowd flow detection +Although accurate, FCN-predicted density maps are characterized by +non-normalized values and may contain noise. In particular, while in the +ground truth the background has a value of 0 for construction, there is no +guarantee that its value will remain 0 in the predicted heatmap. To study +how this affects performance, we first apply a min-max normalization to limit +their range to [0, 1]; then, we threshold (with an empirically chosen threshold +τ) the pixel values, so that any value below the threshold is considered to +be the background. A higher threshold essentially maintains the areas where +the model is more confident about the presence of people in those areas. +After identifying the centroids and filtering the images through the empir- +ical threshold τ, to calculate the actual displacement of the recognized groups +of people, and to determine the direction of their movements, we calculate +the difference in coordinates of the centroids predicted in different frames. In +other words, the displacement of the centroids in an ending frame at time tk +is calculated with respect to the centroids detected in a starting frame at time +t0, and so on. Since we are in two dimensions, this shift is simply calculated +using the Euclidean distance between the (x, y) coordinates. The primary +assumption is that a given group of people, i.e. a centroid in this case, can +potentially move between successive frames, but its distance from its posi- +tion in the previous frame would be less than the distance from all other +centroids in the current frame. Therefore, to determine if we can associate +the newly predicted centroids to the existing ones, and thus in which direc- +tion they have moved, we calculate the Euclidean distance between each pair +of centroids in each frame and keep the minimum distances between the pairs +to match them. In this way, we can classify each shift as facing one of the +four typical cardinal points, namely North, South, East, and West, plus the +intermediate points North-East, North-West, South-East, and South-West. +The case in which the centroids remain stationary between the two frames is +also considered. +If Pi is the set of centroids predicted at time i, we can distinguish three +13 + +scenarios: +• |P0| = |Pk|: in this case, each centroid in P0 is simply associated with +the closest centroid in Pk; +• |P0| > |Pk|: in this case, we have fewer centroids in the ending frame. +Indicated by d = ||P0| − |Pk|| the difference in absolute value between +the number of centroids in t0 and in tk, we will have that d centroids +in t0 will remain unmatched. This could indicate a network prediction +error, the fusion of two centroids in t0 into one in tk, or centroids leaving +the visual field at time tk; +• |P0| < |Pk|: in this case, we have fewer centroids in the starting frame. +We will then have d more centroids at time tk, which will remain un- +matched and could be indicative of a network prediction error, the +formation of new clusters in tk, or finally the input into the visual field +of new centroids. +5. Experiments +The proposed method was implemented in Python, using TensorFlow for +the implementation of the deep learning models.2 All models were trained +with stochastic gradient descent with the Adam optimizer and a learning +rate of 10−4. As for the τ threshold, we experimented with values ranging +from 0 to 1, reporting those that gave the best performance. As mentioned +above, all frames have been resized to 640 × 512. +All experiments were +performed on Google Colab Pro, which mainly provides a T4 or P100 GPU. +The performance metrics considered, as well as the results obtained, are +described below. +5.1. Performance metrics +Applying traditional external clustering measures is not feasible in our +density-based context as they assume an exact match between discretized +category labels. Since we are interested in measuring how correctly the de- +tected centroids represent relevant groups of people within the scene, we +propose a new ad-hoc metric (already presented in [7]) which we called Mean +2https://github.com/evgenivs/crowd_flow_detection_drones +14 + +Coordinate Matching Error (MCME). The metric measures the average dis- +tances between the ground truth centroids and the predicted centroids. For +a single frame, let: +A = +� +� +CP, CGT +min +� +����� +CP ∈ P, +CGT +min = arg +min +CGT ∈GT +��CP − CGT�� +2 +� +B = +� +� +CP +min, CGT� +����� +CGT ∈ GT, +CP +min = arg min +CP ∈P +��CP − CGT�� +2 +� +where GT = {CGT +1 +, CGT +2 +, . . .} and P = {CP +1 , CP +2 , . . .} are the ground truth +and the predicted centroids, respectively. Then: +MCME = +1 +|A ∪ B| +� +(CP ,CGT )∈A∪B +��CP − CGT�� +2 +Each centroid of a set (say P) is associated with the nearest neighbor centroid +of the other set (GT) since both sets of centroids are assumed to represent the +same structure in the data. This association must be symmetric, i.e. from P +to GT and vice versa, because a centroid in P can represent part of a cluster +in GT that has been split into two clusters in P, but it can also represent a +cluster in P that has merged two clusters in GT. The overall score on a video +sequence can be obtained by averaging the individual scores. The proposed +metric aims to “punish” the model both when the predictions are very far +from the ground truth, and when no real groups are identified or groups that +do not exist are identified. Instead, lower values for MCME will indicate that +the predicted centroids match the ground truth centroids, i.e. they represent +the same clusters and are very close. +In addition, to have an easily interpretable metric from a supervised learn- +ing perspective, we present here another new metric that we call Multiple +Patch Precision-Recall (MPPR). It is based on the repeated generation of +a large number of patches from the predicted and ground truth centroid +density maps and a local comparison between them, allowing for a typical +classification-based evaluation mechanism. Instead of considering only the +coordinates of the centroids, MPPR considers a confidence region given by +the size of each patch. Let GTi and Pi be the ground truth and predicted +centroid density map for the i-th frame, respectively, and let lx, ly be the +width and height of the frame. Let H(x, y) be a randomly generated point +15 + +in both maps based on whose coordinates a bounding box is drawn; this +bounding box acts as a sliding window that goes down and to the right. Let +w, h be the width and height of the rectangular bounding box, respectively. +Then, MPPR computes the following quantities: +• True positive (TP): the patches in GTi and Pi are both “active”, which +means that they both contain (at least) a centroid; +• True negative (TN): the patches in GTi and Pi are both “inactive”, +which means that they contain no centroid; +• False positive (FP): the patch in GTi is inactive, while the patch in Pi +is active; +• False negative (FN): the patch in GTi is active, while the patch in Pi +is inactive. +This process is repeated np times, where np is usually a large number to be +statistically confident that the entire frame is explored. Then, precision and +recall for the i-th frame can be computed as usual. The global MPPR is ob- +tained as the average over all video sequence frames. It should be noted that +we have no guarantee that the bounding box w × h is fully contained inside +the frame lx × ly. The probability of obtaining a full bounding box depends +solely on the random coordinates of H(x, y). It can be shown that this prob- +ability amounts to (lx−w)(ly−h) +lxly +. This behavior, together with the sampling +of np bounding boxes, helps us avoid the bias we would have introduced if +we had used a fixed grid of patches or a small np: in fact, in our method, +a slight shift of the bounding box can result in a change of prediction (e.g., +from TP to FN), and this has the benefit of reducing the impact of a too +unrealistically optimistic or pessimistic classification. In our experiments, we +set np = 1000 and w = h = 150. +Finally, the total inference time is calculated, which includes all the pro- +cessing, from estimating the centroid density maps of two consecutive frames +to thresholding and calculating the displacement of the centroids. +5.2. Results +Tables 2 and 4 report the accuracy results obtained by varying the FCN +backbone model and the threshold τ per dataset (VisDrone CC2020 and +CC2021). It is worth noting that to make the MCME metric independent of +16 + +the resolution of the input frame and for a fair comparison with our previous +work [7], the table reports the normalized values for this metric, obtained +by dividing the original value by the diagonal of the frame on which it was +calculated, i.e. the maximum possible error. Tables 3 and 5, on the other +hand, report the average time taken by the method to process two consecutive +frames, depending on the backbone; having fixed the resolution, there is no +difference between the datasets in terms of inference time. +A first observation that can be drawn is that not a single variant of the +neural architecture performs better than the others from the point of view +of predictive accuracy in all cases. Sometimes, the SBSO variant achieves +the highest performance; sometimes, the multi-output models surpass the +single-output one. Similarly, there is no predominant backbone between the +simple FCN and MobileCount, as they show very similar results. +As for +MCME, it varies between 0.167 and 0.253 and 0.181 and 0.235 for the two +datasets: since 1 is the worst possible value and lower is better, these results +confirm the effectiveness of the density-based clustering strategy proposed +here. No single model exceeds the normalized MCME of 0.101 obtained in +our previous preliminary study on CC2020 only, but this was achieved with +a sophisticated two-stage pipeline (first density estimation, then clustering) +that takes about 15 seconds to run on the same hardware [7]. The slightly +worse results obtained with the single-stage strategy are compensated by a +much lower inference time, which is approximately 88 times shorter than in +the previous work. The multi-output variants are, as expected, slightly slower +than the single-output variant; however, they all showed near real-time per- +formance. Notably, although MobileCount has many more parameters than +the proposed simple FCN, the overall processing of the method is relatively +stable, regardless of the backbone used. This suggests that the inference +time of the neural network contributes only marginally to the overall time +and confirms the better efficiency obtained by using a single-stage learning +strategy to produce crowd density maps. +Although MCME remains fairly stable across the different thresholds, +this does not apply to precision and recall. As expected, there is a trade-off +between the two metrics, with precision increasing and recall decreasing while +τ increases. It is worth noting that the use of different background thresholds +to filter the density maps can be set according to the specific application. +For example, if the safe landing of the drone is a significant concern (as +in [20]), then a lower threshold may be preferred, which excludes the risk +of running into false negatives. Conversely, a higher threshold can be used +17 + +in video surveillance scenarios to promote better precision. The difficulty +of accurately locating people in aerial scenes, which in our case translates +into the problem of simultaneously maximizing precision and recall, is well- +known to the community (see for example [32]). This is reflected in the better +precision generally achieved by the models on CC2021, since in this dataset +the scenes were acquired at a relatively lower altitude than on CC2020. +Finally, from a qualitative point of view, we show in Figs. 4 and 5 ex- +amples of centroid density maps produced in output by the ad-hoc FCN +and MobileCount, respectively, by varying the τ threshold, given two ground +truth maps from both datasets. The maps are superimposed on the original +frames in RGB. As can be seen, they confirm the trend already observed +quantitatively: the recall, i.e. the number of centroids detected over all the +centroids, tends to decrease as τ increases. This effect is exacerbated with +MobileCount, especially with zero or low τ, where recall is maximum at the +expense of a drastic drop in precision. +Indeed, in these cases, the maps +produced are pretty ineffective. This drawback could be explained by con- +sidering that MobileCount was explicitly optimized for crowd counting, thus +locating each person rather than groups in the scene, which translates into +a more “conservative” approach. The qualitative analysis also shows how +SBSO tends to adhere more to the ground truth, while SBMO and DBMO +to a lesser extent. This could be explained considering that SBSO is specif- +ically dedicated to producing centroid density maps. In contrast, in SBMO +and DBMO, the crowd density estimation task can mislead the main network +task. Although clusters do not perfectly match the ground truth, SBSO can +still recognize groups of people within the scene, especially with higher τ. +18 + +CC2020 +CC2021 +MCME +Precision (%) +Recall (%) +MCME +Precision (%) +Recall (%) +τ = 0 +SBSO +0.178 +26.8 +90.9 +0.198 +51.8 +83.7 +SBMO +0.223 +22.7 +97.6 +0.201 +44.1 +100 +DBMO +0.203 +22.51 +97.6 +0.201 +44.5 +99.8 +τ = 1 +5 +SBSO +0.185 +30 +76.5 +0.202 +53.5 +78 +SBMO +0.229 +22.7 +90.3 +0.181 +52.3 +86.8 +DBMO +0.192 +27.2 +78.2 +0.181 +52.8 +87.2 +τ = 1 +3 +SBSO +0.187 +32 +68.1 +0.208 +54.1 +70.2 +SBMO +0.233 +21.7 +76.6 +0.181 +55.3 +82.5 +DBMO +0.195 +29.9 +70.6 +0.185 +55.7 +74.3 +τ = 1 +2 +SBSO +0.203 +32.9 +54 +0.217 +54.3 +55.1 +SBMO +0.236 +20.7 +52.6 +0.187 +59.8 +70 +DBMO +0.208 +32.0 +56.1 +0.201 +58.5 +56.8 +τ = 2 +3 +SBSO +0.230 +34.5 +41.5 +0.235 +54.3 +38.1 +SBMO +0.253 +19.1 +31.9 +0.202 +63.9 +51.5 +DBMO +0.236 +31.9 +38.5 +0.227 +58.7 +40.2 +Table 2: Ad-hoc FCN effectiveness results. The best results for each individual metric per +dataset are shown in bold. +Inference time [s] +SBSO +0.16 +SBMO +0.17 +DBMO +0.18 +Table 3: Ad-hoc FCN efficiency results. +19 + +CC2020 +CC2021 +MCME +Precision (%) +Recall (%) +MCME +Precision (%) +Recall (%) +τ = 0 +SBSO +0.199 +22.5 +100 +0.205 +44.2 +100 +SBMO +0.201 +22.7 +100 +0.205 +44.1 +100 +DBMO +0.206 +22.6 +100 +0.204 +44.3 +100 +τ = 1 +3 +SBSO +0.198 +22.4 +100 +0.205 +43.7 +100 +SBMO +0.201 +22.6 +100 +0.198 +44.3 +99.1 +DBMO +0.209 +22.6 +100 +0.202 +44.1 +99.6 +τ = 2 +3 +SBSO +0.188 +25.7 +95.2 +0.203 +45.3 +98.1 +SBMO +0.184 +24.1 +98.6 +0.204 +49.3 +77.3 +DBMO +0.231 +23.1 +98.0 +0.186 +59.1 +64.8 +τ = 3 +4 +SBSO +0.213 +40.9 +45.5 +0.194 +46.7 +93.9 +SBMO +0.170 +27.3 +93.1 +0.215 +47.9 +62.5 +DBMO +0.230 +23.3 +87.5 +0.20 +61.5 +49.1 +τ = 4 +5 +SBSO +0.176 +30.6 +88.3 +0.193 +48.1 +85.7 +SBMO +0.167 +30.2 +86.2 +0.224 +46.4 +52.0 +DBMO +0.225 +23.7 +79.0 +0.211 +64.4 +41.9 +Table 4: MobileCount effectiveness results. The best results for each individual metric per +dataset are shown in bold. +Inference time [s] +SBSO +0.15 +SBMO +0.18 +DBMO +0.16 +Table 5: MobileCount efficiency results. +6. Conclusion +In this article, we tackled the problem of crowd flow detection from +drones, which was still an unexplored research direction. +Such a system +can be helpful for various security and management applications, especially +in smart city scenarios. In particular, the joint exploitation of crowd density +estimation and clustering within a video sequence shot by a drone provided +encouraging results, especially from the point of view of efficiency, which can +be crucial in critical tasks. +Future development of the research presented in this paper could be to +increase the size of the dataset by integrating the available scenes with syn- +thetic data, as done, for example, in [33]. Such a strategy can help further +improve the robustness of the deep learning model. Second, it is worth not- +ing that the proposed framework assumes the availability of video sequences +shot by a drone, but the neural network is fed with one frame at a time. A +different strategy to be explored in the future could be to feed the network +with the video sequence to account for the temporal information directly in +20 + +Figure 4: Examples of centroid density maps, corresponding to ground truth maps for both +CC2020 and CC2021, produced in output by the ad-hoc FCN varying the τ threshold. +the model. Third, provided adequate ground truth is available, the method +could also be used for other similar tasks, such as vehicle counting/tracking. +Finally, another future work concerns the experimentation of the method on +new real-world situations aboard a drone to best calibrate the parameters +considered. Testing the generalizability of the method to different urban and +non-urban contexts can increase confidence in UAV technology. +Acknowledgment. This work was supported by the Italian Ministry of Uni- +versity and Research within the “RPASInAir” project under grant PON +ARS01 00820. +21 + +1 +1 +L +2 +T=0 +T +5 +3 +2 +3 +SBSO +GT (CC2020) +SBMO +DBMO +SBSO +GT (CC2021) +SBMO +DBMOFigure 5: Examples of centroid density maps, corresponding to ground truth maps for +both CC2020 and CC2021, produced in output by MobileCount varying the τ threshold. +References +[1] V. A. Sindagi, V. M. Patel, A survey of recent advances in CNN-based +single image crowd counting and density estimation, Pattern Recogni- +tion Letters 107 (2018) 3–16. +[2] B. Li, H. Huang, A. Zhang, P. Liu, C. Liu, Approaches on crowd count- +ing and density estimation: a review, Pattern Analysis and Applications +(2021) 1–22. +[3] Y. Akbari, N. Almaadeed, S. Al-maadeed, O. 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Li, A flow base bi-path network +for cross-scene video crowd understanding in aerial view, in: European +Conference on Computer Vision, Springer, 2020, pp. 574–587. +26 + diff --git a/StE4T4oBgHgl3EQfLQyp/content/tmp_files/load_file.txt b/StE4T4oBgHgl3EQfLQyp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..34594573a7f8f0432eadef527c1b2987cc522b25 --- /dev/null +++ b/StE4T4oBgHgl3EQfLQyp/content/tmp_files/load_file.txt @@ -0,0 +1,729 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf,len=728 +page_content='Density-based clustering with fully-convolutional networks for crowd flow detection from drones Giovanna Castellanoa, Eugenio Cotardoa, Corrado Mencara, Gennaro Vessioa aDepartment of Computer Science, University of Bari Aldo Moro, Bari, Italy Abstract Crowd analysis from drones has attracted increasing attention in recent times due to the ease of use and affordable cost of these devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' However, how this technology can provide a solution to crowd flow detection is still an unexplored research question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' To this end, we propose a crowd flow detec- tion method for video sequences shot by a drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The method is based on a fully-convolutional network that learns to perform crowd clustering in or- der to detect the centroids of crowd-dense areas and track their movement in consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The proposed method proved effective and efficient when tested on the Crowd Counting datasets of the VisDrone challenge, characterized by video sequences rather than still images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The encouraging results show that the proposed method could open up new ways of analyzing high-level crowd behavior from drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Keywords: drones, drone vision, computer vision, deep learning, crowd flow detection, crowd density estimation, clustering 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Introduction Due to population growth and the increasing degree of urbanization, more and more people live in urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Positive consequences of this trend are the enrichment of cultural life and the full use of convenient urban infras- tructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' At the same time, gatherings of people, which can occur for various reasons, such as political demonstrations, festival celebrations, concerts, and so on, pose serious challenges to urban security and management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In this per- spective, automated crowd analysis methods, which typically involve crowd counting and associated crowd density estimation, have attracted increasing Accepted manuscript: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 1016/ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' neucom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 059 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='04937v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='CV] 12 Jan 2023 attention for their many potential applications [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' These include the pre- vention of crowd-induced disasters, such as stampedes, but also other less critical objectives such as better crowd management at public events and the design of public spaces and virtual environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' A cost-effective way to perform automated crowd analysis is by using un- manned aerial vehicles (UAVs), more commonly known as drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Indeed, once equipped with affordable but sufficiently powerful cameras and GPUs, drones can become flying computer vision devices that can be rapidly de- ployed for a wide range of applications, including crowd analysis for public safety [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' However, while these perspectives are fascinating, there are also some drawbacks to be aware of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' On the one hand, the computer vision algo- rithms applied to aerial images are burdened with further difficulties because the problems of scale and point of view are taken to the extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' On the other hand, the sophisticated and computationally intensive methods com- monly applied in this field do not meet the stringent real-time requirements imposed by the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In other words, lightweight models that offer a good compromise between effectiveness and efficiency are essential [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Crowd analysis with drones has attracted attention in recent years [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' However, despite significant progress, the proposed methods still have room for improvement to address the challenges posed by drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In this article, we want to contribute to this research effort by taking it one step further: instead of considering crowd counting and density estimation in static frames, we aim to detect crowd flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This poses a new challenge as the goal is not only to recognize the presence of people in a single high-altitude scene but also to determine how crowds flow as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This is different from people tracking—where the goal is to track a single person or groups of people—and can lead to useful systems, as it can allow for crowd behavior analysis for better logistics and disaster prevention [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' To this end, we propose a method for crowd flow detection from drones based on fully-convolutional networks (FCNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The network is trained to rec- ognize groups of people in each frame and, to do this, simultaneously learns to perform crowd density estimation and crowd clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In this way, groups of people are identified simply by their centroids, and these are used to trace the trajectories of the identified groups, following their movement during the shooting of the drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' We preferred FCNs over other architectures mainly be- cause of their known efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Furthermore, direct learning of crowd clusters was preferred to avoid a multi-step approach, based on performing density estimation first and clustering the resulting density maps later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This was 2 done in our previous preliminary work [7] but, while effective, this approach proved too demanding from a computational point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The method was tested on the recently proposed Crowd Counting 2020 [8] and 2021 [9] datasets, used annually for the international VisDrone challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The pecu- liarity of these datasets is that they are not characterized by still images but actually by frames of video sequences that are used here to perform the crowd flow detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' It is worth noting that what we actually do to deter- mine crowd flow is to calculate the inter-frame difference between centroids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' in other words, it is a kind of “inter-frame density clustering”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' However, since this approach allows us to detect the movement of the crowd frame by frame, we use the expression “flow detection” for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Section 2 reviews related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Section 3 describes the datasets used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Section 4 presents the pro- posed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Section 5 describes the experimental setup and discusses the results obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Section 6 concludes the paper and highlights the future developments of our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Related work There is a large body of knowledge about crowd counting and crowd den- sity estimation in computer vision, but the trend today is density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Early work usually applied a person or head detector via a sliding window on the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' However, although current implementations may be based on state-of-the-art object detectors such as YOLO [10, 11], these approaches still provide unsatisfactory results when asked to detect small objects in a very dense crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' To alleviate this problem, regression-based methods have been introduced that directly learn the mapping from an image to the global people count [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' However, although these methods make the approach in- dependent of the precise position of individuals in the crowd, which is very complex, they ignore the spatial information that can be very useful for pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' To avoid the difficulty of accurately detecting and locating people in the scene, while using spatial information, the recent trend is to learn density maps, thus incorporating spatial information directly into the learn- ing process [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Successful solutions include methods that work first at the patch level and then fuse local features [13], methods that integrate atten- tion mechanisms [14], cascade approaches that jointly learn people counting and density maps [15], methods that improve performance through knowl- edge distillation [16], and frameworks that simultaneously perform crowd 3 counting and localization [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' However, while effective, these approaches are generally computationally demanding and do not meet the stringent requirements typically imposed by UAVs (limited battery, need for real-time responses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' How to fine-tune deep neural architectures to achieve an optimal balance between precision and performance is an active research area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The VisDrone Crowd Counting challenge was introduced to encourage research in this direction [8, 9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' nev- ertheless, the solutions proposed by the participants in the challenge are not always focused on efficiency but rather on effectiveness, as the goal is only to obtain a low error in counting people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The lowest error was obtained with TransCrowd [18], based on the increasingly popular Vision Transformer [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' However, the proposed method only regresses the people count, not provid- ing density maps that would be useful for detecting crowd flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' furthermore, transformer-based solutions are known to be computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' A promising way to address these problems is to use FCN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Be- cause they do not rely on fully-connected layers at all, which are the most expensive part of processing a neural network, they are a candidate solu- tion for finding an accurate model without damaging the inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' An FCN model for aerial drone imaging was presented in [4], and a similar so- lution was also proposed in our previous work [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' However, both methods were aimed at crowd detection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' discriminating between crowded and un- crowded scenes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' furthermore, they only provide coarse density maps, as the models have not been trained on people labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Human tracking methods based on RGB cameras or other sensors that use clustering or classification models to track motion have been investigated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' [21, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' However, they are designed to work indoors or by involv- ing a few people from a frontal perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The work most linked to ours, which takes into account the images captured by drones, is [24] where the same authors who propose the VisDrone Crowd Counting datasets present a model that jointly solves density map estimation, localization, and track- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This model differs from ours in that it uses a complex and expensive pipeline aimed at tracking individual trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Other authors have re- cently proposed a method for periodic crowd tracking from UAVs based on a binary linear programming model [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' However, they experimented with simulated scenarios that do not consider the crowd detection problem from a computer vision perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' As far as we know, there is no work in the literature addressing crowd flow detection in drone videos, which poses sig- nificantly different challenges than traditional settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This paper aims to 4 fill this gap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' in particular, we aim to trace the centroids that identify groups of people by exploiting the spatial information learned and expressed through density maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Materials The literature landscape is not as populated with datasets for crowd counting and density estimation from drones with video sequences captured by optical cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The datasets best suited to our purposes for evaluating crowd flow detection were the VisDrone Crowd Counting 2020 [8] and 2021 [9] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The two benchmarks are characterized as follows: VisDrone Crowd Counting 2020 (CC2020) contains 82 video clips (2, 460 frames in total) with a resolution of 1920 × 1080;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' VisDrone Crowd Counting 2021 (CC2021) contains 1, 807 frames with a resolution of 640 × 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Unlike CC2020, the frames are not arranged in precisely separated video clips, and this required manual separation to split the sequences of different locations to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The video sequences were acquired by various drone-mounted cameras, which shot different scenarios in different cities in China to maintain diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In both datasets, people were annotated manually with dots in each video frame, expressed as (x, y) coordinates in the bi-dimensional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' How- ever, CC2020 and CC2021 have significant differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In CC2020, each of the 82 sequences was captured by a drone hovering over the crowd, allow- ing for rather static scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In CC2021, on the other hand, the drone flies and sometimes rotates, shooting different scenes even if semantically linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Furthermore, it should be noted that in CC2020 there is a predominance of daylight scenes, while in CC2021 there are many frames at night, making the dataset much more variable in this respect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' A final difference concerns altitude: in CC2020 the frames appear to have been shot at a higher altitude than in CC2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' For both datasets, we randomly held out (sequence-wise) a fraction of 25% of the total frames evenly split to form validation and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Compared to benchmark datasets focused on crowd counting in surveil- lance scenes, both VisDrone datasets present particular challenges due to the scenes captured by drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Object scales can be extremely small due to the high shooting altitude of drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Crowds are sparse across the video frames, 5 as each can hold a few to dozen people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Finally, the crowds are surrounded by very different backgrounds in different sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Methods The proposed method for crowd flow detection is based on an FCN model, which is used to estimate a “centroid density map”—a heatmap highlighting crowd centroids—from pairs of consecutive frames of the same video sequence shot by the drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Since a crowd can be seen as multiple groups of people not necessarily following the same direction, the predicted centroids are represen- tative of these groups of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The displacements of the centroids detected between the pairs of frames are then calculated to identify the direction of movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Thus, the framework assumes video sequences shot by a drone, but the network is fed one frame at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The idea of combining a density estimation method with clustering, in- stead of tracking the movement of each individual, is motivated by the com- plexity and computational cost of this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In fact, while the direct use of the crowd density map based on the location of all individuals can retain more information, it also carries the burden of tracking the trajectory of each individual in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Moreover, individual tracking can be not only impractical but also non-essential, as in crowd management scenarios it is important to recognize the overall flow of people rather than the precise loca- tion of each person in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' A centroid density estimation method will focus only on high-density areas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' those corresponding to concentrations of people, and will be inherently robust to occlusion, which would heavily affect a people detector, especially from a high altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The method is detailed in the following subsections, along with an expla- nation of how the ground truth is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' An illustration schematizing the overall processing is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Ground truth generation Since we assume that there is no label regarding the location of the cen- troids within the original drone shot, we followed a simple strategy to derive a ground truth for the crowd centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' We first applied the well-known Mean Shift clustering algorithm to the head annotations described above to obtain centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Mean Shift is a centroid-based algorithm that updates the candidates for the centroids as the average of the points within a given 6 Figure 1: Schema of the overall method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Pairs of consecutive frames taken by a drone at time t0 and tk (with tk > t0 and separated by any time interval) are fed into the neural network model for crowd clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' A synthetically generated ground truth helps guide the learning of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The obtained centroid density maps are then thresholded, and the displacement of the centroids, representing the groups of people, from t0 to tk, is used to determine their direction of movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' These candidates are then filtered in a post-processing step to elimi- nate near-duplicates [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Specifically, the Scikit-learn implementation of the Mean Shift algorithm with the recommended default hyperparameters was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 Then, following the seminal paper by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' [27], if there is a centroid at pixel xi, we represented it as a delta function δ(x − xi) and we obtained a “centroid density map” C(x) convolving the delta function with a Gaussian kernel: C(x) = K � i=1 δ(x − xi) ∗ Gσ(x) In the formula, K is the number of centroids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' δ equals 1 when x = xi, 0 otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' and Gσ is the Gaussian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Since the sizes of the heads are similar in each video sequence and there is no perspective problem as in [27], we decided to use a fixed σ for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In particular, we have 1At the time of writing, the version of Scikit-learn is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 7 Ground truth Loss calculation generation to tk to tk Thresholding Displacement calculation Neural network modelFigure 2: Examples of frames and corresponding synthetically generated centroid density maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' empirically set σ = 10, since this value leads to better performance thanks to the sufficiently large “confidence” activation area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Examples of generated centroid density maps are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' We used Mean Shift as it is impossible to know the number of clusters in the crowd in advance, so there was a need for an algorithm that did not require a pre-specification of the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In this way, an unsupervised learning approach is followed to find crowd centroids, but when their location is found, it can be used as a synthetically generated annotation to guide a supervised learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Although this strategy strongly depends on the results provided by the specific clustering algorithm chosen, it allowed us to automate the generation of the ground truth and to guide and quantitatively evaluate the clustering task performed by the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In addition, we also experimented with models aimed at performing the more classic crowd density estimation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' To this end, “crowd density maps” D(x) were obtained by convolving the same delta function with Gaus- sian kernel as before but using the original people head annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Fully-Convolutional Networks The proposed method is based on an FCN architecture that recognizes the crowd centroids within each frame and produces the related heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In 8 our previous preliminary work [7], the identification of the centroids was del- egated to a classic clustering algorithm after the generation of crowd density maps, significantly penalizing the inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In the method proposed here, instead, the task of finding crowd centroids is integrated directly into the network training and is performed in a single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Fully-convolutional neural networks, originally proposed in [28], perform only convolution and pooling operations and discard the fully-connected com- ponent typical of CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Instead of the fully-connected layer, there is a 1×1 convolution with stride 1, which allows, on the one hand, to have fewer pa- rameters to estimate and, on the other hand, to be able to receive an image of arbitrary size as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' It is worth noting that although FCNs have this desirable property, for better evaluation, we have set the input resolution of each frame to 640 × 512 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The architecture of an FCN consists of two parts: an encoder aims to downsample the input into a lower-dimensional representation, a decoder aims to upsample the latent representation to the desired output resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In particular, we have experimented with two different neural networks: an ad-hoc FCN designed to be as simple as possible, and a state-of-the-art FCN already used for crowd counting in more traditional contexts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Mo- bileCount [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In the ad-hoc implementation, a rescaling layer normalizes each pixel value in the [0, 1] range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Then, the encoder part of the model repeatedly applies four blocks consisting of a convolutional layer with kernel 3 × 3, batch normalization, and max pooling with kernel 2 × 2 until reaching a latent space Z ∈ R40×32×128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The decoder upsamples the feature maps with transposed convolution and batch normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Finally, a 1 × 1 convolu- tional layer produces the output density map of size 640 × 512 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The commonly used ReLU was chosen as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' MobileCount, on the other hand, is similar but has some more sophisticated architectural designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' To reduce the input resolution, a 3 × 3 max pooling layer with stride 2 is added before the encoding part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The encoder is adapted from Mo- bileNetV2 [30] by reducing the number of inverted residual blocks from 7 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' As for the decoding component, the lightweight RefineNet [31], originally designed for semantic segmentation, is exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Since we have two datasets with very different characteristics, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 3 we experimented with three main variants of the above architectures, which were likely to produce different results: Single-branch, single-output (SBSO): this architecture uses one of the 9 two FCNs described above and learns to directly estimate the centroid density map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' As a loss function, the network is trained to minimize the mean squared error between the predicted and the ground truth centroid density map: L = 1 N N � i=1 ��CP(i) − CGT(i) ��2 2 where N is the number of samples, CP(i) and CGT(i) are the predicted and ground truth centroid density map, respectively, and ∥·∥2 is the Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Demanding the model to find groups of people in a completely unsupervised way would not have provided the network with a guide to optimize weights to cluster only a specific type of objects, in our case people, leaving it completely free to separate people from trees, buildings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=', which was not our goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Single-branch, multi-output (SBMO): The task of finding crowd cen- troids may be difficult for the model as the centroids do not represent objects effectively present within the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' To try to mitigate this issue, we also experimented with a multi-output model, which extends the previous one by learning to simultaneously estimate the centroid density map and the more classic crowd density map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The main idea is that the auxiliary task can help the overall network extract the features that actually characterize the people in the scenes, thus supporting the main task of density estimation, which only concerns the detection of crowd centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In this case, the loss function is the sum of the above and the mean squared error between the predicted and the ground truth crowd density maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Dual-branch, multi-output (DBMO): finally, to improve the capacity of the previous variant, we also experimented with a multi-output model, this time characterized by two branches, which are actually two “twin” autoencoders with the same architecture as before (ad-hoc FCN or Mo- bileCount).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The two branches communicate with each other through a concatenation of the feature maps produced immediately before the desired output: one branch is totally dedicated to learning the crowd density map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' the concatenation of both branches contributes to esti- mating the centroid density map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The loss function is the same as the previous variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 10 In other words, the multi-output architectures are meant to help the model learn more about the concept of “crowdedness”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Because these variants in- herently perform crowd counting, in addition to crowd clustering, it is worth noting that they can be used for the more classical task of pedestrian counting if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Finally, it is important to emphasize that the neural network models we have tested are a backbone of the overall framework we propose, which can be replaced with similar models as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Table 1 compares the two backbones used in this work in terms of the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The proposed FCN has been designed as a classic and straightforward baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' It is small, but deliberately without any sophistication to further improve effectiveness or efficiency, to show the feasibility of the proposed method with a simple model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' MobileCount, on the other hand, has more parameters but was purposely made by the authors as an efficient framework for real-time crowd counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In fact, the MobileNetV2-based encoder and the RefineNet- based decoder were carefully adapted to achieve a good balance between accuracy and speed, as reported in the article [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 11 Figure 3: Schema of the proposed neural network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The proposed ad-hoc FCN is schematized in this figure but can be replaced by any similar model as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 12 Encoder Decoder Input frame Centroid density map 640 × 512 × 3 640 × 512 × 1 SBSO Encoder Decoder Crowd density map 640 × 512 × 1 Centroid density map 640 × 512 × 1 Input frame 640 × 512 × 3 SBMO Encoder Decoder Crowd density map 640 × 512 × 1 Input frame Centroid density map 640 × 512 × 3 640 × 512 × 1 DBMO CONV + BN + MAX POOL LATENT SPACE CONV TRANSPOSE + BNAd-hoc FCN MobileCount SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='35 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='199 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='35 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='199 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='398 Table 1: Comparison in terms of the number of parameters, in millions, between the pro- posed ad-hoc FCN and MobileCount, used as the backbone of the proposed architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Crowd flow detection Although accurate, FCN-predicted density maps are characterized by non-normalized values and may contain noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In particular, while in the ground truth the background has a value of 0 for construction, there is no guarantee that its value will remain 0 in the predicted heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' To study how this affects performance, we first apply a min-max normalization to limit their range to [0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' then, we threshold (with an empirically chosen threshold τ) the pixel values, so that any value below the threshold is considered to be the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' A higher threshold essentially maintains the areas where the model is more confident about the presence of people in those areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' After identifying the centroids and filtering the images through the empir- ical threshold τ, to calculate the actual displacement of the recognized groups of people, and to determine the direction of their movements, we calculate the difference in coordinates of the centroids predicted in different frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In other words, the displacement of the centroids in an ending frame at time tk is calculated with respect to the centroids detected in a starting frame at time t0, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Since we are in two dimensions, this shift is simply calculated using the Euclidean distance between the (x, y) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The primary assumption is that a given group of people, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' a centroid in this case, can potentially move between successive frames, but its distance from its posi- tion in the previous frame would be less than the distance from all other centroids in the current frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Therefore, to determine if we can associate the newly predicted centroids to the existing ones, and thus in which direc- tion they have moved, we calculate the Euclidean distance between each pair of centroids in each frame and keep the minimum distances between the pairs to match them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In this way, we can classify each shift as facing one of the four typical cardinal points, namely North, South, East, and West, plus the intermediate points North-East, North-West, South-East, and South-West.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The case in which the centroids remain stationary between the two frames is also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' If Pi is the set of centroids predicted at time i, we can distinguish three 13 scenarios: |P0| = |Pk|: in this case, each centroid in P0 is simply associated with the closest centroid in Pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' |P0| > |Pk|: in this case, we have fewer centroids in the ending frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Indicated by d = ||P0| − |Pk|| the difference in absolute value between the number of centroids in t0 and in tk, we will have that d centroids in t0 will remain unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This could indicate a network prediction error, the fusion of two centroids in t0 into one in tk, or centroids leaving the visual field at time tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' |P0| < |Pk|: in this case, we have fewer centroids in the starting frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' We will then have d more centroids at time tk, which will remain un- matched and could be indicative of a network prediction error, the formation of new clusters in tk, or finally the input into the visual field of new centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Experiments The proposed method was implemented in Python, using TensorFlow for the implementation of the deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2 All models were trained with stochastic gradient descent with the Adam optimizer and a learning rate of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' As for the τ threshold, we experimented with values ranging from 0 to 1, reporting those that gave the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' As mentioned above, all frames have been resized to 640 × 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' All experiments were performed on Google Colab Pro, which mainly provides a T4 or P100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The performance metrics considered, as well as the results obtained, are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Performance metrics Applying traditional external clustering measures is not feasible in our density-based context as they assume an exact match between discretized category labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Since we are interested in measuring how correctly the de- tected centroids represent relevant groups of people within the scene, we propose a new ad-hoc metric (already presented in [7]) which we called Mean 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='com/evgenivs/crowd_flow_detection_drones 14 Coordinate Matching Error (MCME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The metric measures the average dis- tances between the ground truth centroids and the predicted centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' For a single frame, let: A = � � CP, CGT min � ����� CP ∈ P, CGT min = arg min CGT ∈GT ��CP − CGT�� 2 � B = � � CP min, CGT� ����� CGT ∈ GT, CP min = arg min CP ∈P ��CP − CGT�� 2 � where GT = {CGT 1 , CGT 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='} and P = {CP 1 , CP 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='} are the ground truth and the predicted centroids, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Then: MCME = 1 |A ∪ B| � (CP ,CGT )∈A∪B ��CP − CGT�� 2 Each centroid of a set (say P) is associated with the nearest neighbor centroid of the other set (GT) since both sets of centroids are assumed to represent the same structure in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This association must be symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' from P to GT and vice versa, because a centroid in P can represent part of a cluster in GT that has been split into two clusters in P, but it can also represent a cluster in P that has merged two clusters in GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The overall score on a video sequence can be obtained by averaging the individual scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The proposed metric aims to “punish” the model both when the predictions are very far from the ground truth, and when no real groups are identified or groups that do not exist are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Instead, lower values for MCME will indicate that the predicted centroids match the ground truth centroids, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' they represent the same clusters and are very close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In addition, to have an easily interpretable metric from a supervised learn- ing perspective, we present here another new metric that we call Multiple Patch Precision-Recall (MPPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' It is based on the repeated generation of a large number of patches from the predicted and ground truth centroid density maps and a local comparison between them, allowing for a typical classification-based evaluation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Instead of considering only the coordinates of the centroids, MPPR considers a confidence region given by the size of each patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Let GTi and Pi be the ground truth and predicted centroid density map for the i-th frame, respectively, and let lx, ly be the width and height of the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Let H(x, y) be a randomly generated point 15 in both maps based on whose coordinates a bounding box is drawn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' this bounding box acts as a sliding window that goes down and to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Let w, h be the width and height of the rectangular bounding box, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Then, MPPR computes the following quantities: True positive (TP): the patches in GTi and Pi are both “active”, which means that they both contain (at least) a centroid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' True negative (TN): the patches in GTi and Pi are both “inactive”, which means that they contain no centroid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' False positive (FP): the patch in GTi is inactive, while the patch in Pi is active;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' False negative (FN): the patch in GTi is active, while the patch in Pi is inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This process is repeated np times, where np is usually a large number to be statistically confident that the entire frame is explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Then, precision and recall for the i-th frame can be computed as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The global MPPR is ob- tained as the average over all video sequence frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' It should be noted that we have no guarantee that the bounding box w × h is fully contained inside the frame lx × ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The probability of obtaining a full bounding box depends solely on the random coordinates of H(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' It can be shown that this prob- ability amounts to (lx−w)(ly−h) lxly .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This behavior, together with the sampling of np bounding boxes, helps us avoid the bias we would have introduced if we had used a fixed grid of patches or a small np: in fact, in our method, a slight shift of the bounding box can result in a change of prediction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=', from TP to FN), and this has the benefit of reducing the impact of a too unrealistically optimistic or pessimistic classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In our experiments, we set np = 1000 and w = h = 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Finally, the total inference time is calculated, which includes all the pro- cessing, from estimating the centroid density maps of two consecutive frames to thresholding and calculating the displacement of the centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Results Tables 2 and 4 report the accuracy results obtained by varying the FCN backbone model and the threshold τ per dataset (VisDrone CC2020 and CC2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' It is worth noting that to make the MCME metric independent of 16 the resolution of the input frame and for a fair comparison with our previous work [7], the table reports the normalized values for this metric, obtained by dividing the original value by the diagonal of the frame on which it was calculated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' the maximum possible error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Tables 3 and 5, on the other hand, report the average time taken by the method to process two consecutive frames, depending on the backbone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' having fixed the resolution, there is no difference between the datasets in terms of inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' A first observation that can be drawn is that not a single variant of the neural architecture performs better than the others from the point of view of predictive accuracy in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Sometimes, the SBSO variant achieves the highest performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' sometimes, the multi-output models surpass the single-output one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Similarly, there is no predominant backbone between the simple FCN and MobileCount, as they show very similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' As for MCME, it varies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='167 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='253 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='181 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='235 for the two datasets: since 1 is the worst possible value and lower is better, these results confirm the effectiveness of the density-based clustering strategy proposed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' No single model exceeds the normalized MCME of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='101 obtained in our previous preliminary study on CC2020 only, but this was achieved with a sophisticated two-stage pipeline (first density estimation, then clustering) that takes about 15 seconds to run on the same hardware [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The slightly worse results obtained with the single-stage strategy are compensated by a much lower inference time, which is approximately 88 times shorter than in the previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The multi-output variants are, as expected, slightly slower than the single-output variant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' however, they all showed near real-time per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Notably, although MobileCount has many more parameters than the proposed simple FCN, the overall processing of the method is relatively stable, regardless of the backbone used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This suggests that the inference time of the neural network contributes only marginally to the overall time and confirms the better efficiency obtained by using a single-stage learning strategy to produce crowd density maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Although MCME remains fairly stable across the different thresholds, this does not apply to precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' As expected, there is a trade-off between the two metrics, with precision increasing and recall decreasing while τ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' It is worth noting that the use of different background thresholds to filter the density maps can be set according to the specific application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' For example, if the safe landing of the drone is a significant concern (as in [20]), then a lower threshold may be preferred, which excludes the risk of running into false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Conversely, a higher threshold can be used 17 in video surveillance scenarios to promote better precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The difficulty of accurately locating people in aerial scenes, which in our case translates into the problem of simultaneously maximizing precision and recall, is well- known to the community (see for example [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This is reflected in the better precision generally achieved by the models on CC2021, since in this dataset the scenes were acquired at a relatively lower altitude than on CC2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Finally, from a qualitative point of view, we show in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 4 and 5 ex- amples of centroid density maps produced in output by the ad-hoc FCN and MobileCount, respectively, by varying the τ threshold, given two ground truth maps from both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The maps are superimposed on the original frames in RGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' As can be seen, they confirm the trend already observed quantitatively: the recall, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' the number of centroids detected over all the centroids, tends to decrease as τ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This effect is exacerbated with MobileCount, especially with zero or low τ, where recall is maximum at the expense of a drastic drop in precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Indeed, in these cases, the maps produced are pretty ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This drawback could be explained by con- sidering that MobileCount was explicitly optimized for crowd counting, thus locating each person rather than groups in the scene, which translates into a more “conservative” approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The qualitative analysis also shows how SBSO tends to adhere more to the ground truth, while SBMO and DBMO to a lesser extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This could be explained considering that SBSO is specif- ically dedicated to producing centroid density maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In contrast, in SBMO and DBMO, the crowd density estimation task can mislead the main network task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Although clusters do not perfectly match the ground truth, SBSO can still recognize groups of people within the scene, especially with higher τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 18 CC2020 CC2021 MCME Precision (%) Recall (%) MCME Precision (%) Recall (%) τ = 0 SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='178 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='198 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='223 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='201 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 100 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='203 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='51 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='201 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='8 τ = 1 5 SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='185 30 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='202 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 78 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='229 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='181 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='8 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='192 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='181 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2 τ = 1 3 SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='187 32 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='208 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='233 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='181 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='195 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='9 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='185 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 τ = 1 2 SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='203 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='9 54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='217 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='236 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='187 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='8 70 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='208 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='201 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='8 τ = 2 3 SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='230 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='235 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='253 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='202 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='9 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='236 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='9 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='227 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2 Table 2: Ad-hoc FCN effectiveness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The best results for each individual metric per dataset are shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Inference time [s] SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='16 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='17 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='18 Table 3: Ad-hoc FCN efficiency results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 19 CC2020 CC2021 MCME Precision (%) Recall (%) MCME Precision (%) Recall (%) τ = 0 SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='199 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='205 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2 100 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='201 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='205 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 100 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='206 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='6 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='204 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 100 τ = 1 3 SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='198 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='4 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='205 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 100 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='201 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='6 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='198 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='209 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='6 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='202 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='6 τ = 2 3 SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='188 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='203 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='184 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='204 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='231 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='186 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='8 τ = 3 4 SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='213 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='194 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='9 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='170 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='215 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='9 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='230 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='20 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 τ = 4 5 SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='176 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='6 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='193 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='167 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='224 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='0 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='225 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='211 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='9 Table 4: MobileCount effectiveness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' The best results for each individual metric per dataset are shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Inference time [s] SBSO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='15 SBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='18 DBMO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content='16 Table 5: MobileCount efficiency results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Conclusion In this article, we tackled the problem of crowd flow detection from drones, which was still an unexplored research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Such a system can be helpful for various security and management applications, especially in smart city scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' In particular, the joint exploitation of crowd density estimation and clustering within a video sequence shot by a drone provided encouraging results, especially from the point of view of efficiency, which can be crucial in critical tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Future development of the research presented in this paper could be to increase the size of the dataset by integrating the available scenes with syn- thetic data, as done, for example, in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Such a strategy can help further improve the robustness of the deep learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Second, it is worth not- ing that the proposed framework assumes the availability of video sequences shot by a drone, but the neural network is fed with one frame at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' A different strategy to be explored in the future could be to feed the network with the video sequence to account for the temporal information directly in 20 Figure 4: Examples of centroid density maps, corresponding to ground truth maps for both CC2020 and CC2021, produced in output by the ad-hoc FCN varying the τ threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Third, provided adequate ground truth is available, the method could also be used for other similar tasks, such as vehicle counting/tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Finally, another future work concerns the experimentation of the method on new real-world situations aboard a drone to best calibrate the parameters considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Testing the generalizability of the method to different urban and non-urban contexts can increase confidence in UAV technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' This work was supported by the Italian Ministry of Uni- versity and Research within the “RPASInAir” project under grant PON ARS01 00820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' 21 1 1 L 2 T=0 T 5 3 2 3 SBSO GT (CC2020) SBMO DBMO SBSO GT (CC2021) SBMO DBMOFigure 5: Examples of centroid density maps, corresponding to ground truth maps for both CC2020 and CC2021, produced in output by MobileCount varying the τ threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQfLQyp/content/2301.04937v1.pdf'} +page_content=' References [1] V.' metadata={'source': 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consciousness: +A neurogenetic case against synthetic sentience + + +ArXiv E-Print + +Yoshija Walter* 1, 2, 3 + +Lukas Zbinden4 + +1Institute for Management and Digitalization IMD +Kalaidos University of Applied Sciences Zurich, Switzerland +2Laboratory for Cognitive Neuroscience LCNS +University of Fribourg, Switzerland +3Translational Research Center +University Hospital for Psychiatry Bern, Switzerland +4ARTORG Center for Biomedical Engineering Research +University of Bern, Switzerland + +*yoshija.walter@kalaidos-fh.ch +December 2022 + +ABSTRACT +Ever since the creation of the first artificial intelligence (AI) machinery built on +machine learning (ML), public society has entertained the idea that eventually +computers could become sentient and develop a consciousness of their own. As these +models now get increasingly better and convincingly more anthropomorphic, even +some engineers have started to believe that AI might become conscious, which would +result in serious social consequences. The present paper argues against the plausibility +of sentient AI primarily based on the theory of neurogenetic structuralism, which claims +that the physiology of biological neurons and their structural organization into complex +brains are necessary prerequisites for true consciousness to emerge. + +Keywords Artificial Intelligence · AI · Consciousness · AI Ethics · Neurogenetic structuralism + + + + +The problem with AI consciousness + + - 2 - + +1 The relevance of “conscious AI” +In the past few years, the development of machine learning (ML) systems has rapidly increased and the +more tasks a single ML model can perform, the more versatile and broadly useful it becomes. As such, +the goal is to work multimodally with the explicit intent to eventually achieve an Artificial General +Intelligence (AGI), which approximates or perhaps even exceeds human abilities (Goertzel et al., 2022; +Goertzel & Pennachin, 2007; Wang & Goertzel, 2012). It is generally believed that the best AI models +are the ones that most closely approximate human characteristics and abilities. Since the models are +selected against how well they suit anthropomorphic benchmarks, it appears to be only natural that +humans continue to anthropomorphize them more and more, as long as they keep improving on these +benchmarks. Arguably, the best AI system would be one that imitates the output of human consciousness +so that an outsider could not discern it from a real person. This is exactly the core idea behind the famous +Turing-test, which is a thought-experiment originally referred to as the “imitation game” (Turing, +1950).1 +One might argue that it does not matter if an AI is considered a real person or just an imitation, since at +the end of the day the system’s outputs are the same. However, given the social dynamics involved, the +differentiation between real and imitated consciousness may be paramount, which can be illustrated with +a few examples: On the one hand, if a person falls in love with an automaton and has a deep relationship +with it, society would consider this pathological and potentially in need for an intervention, just as it +appears to be nonsensical if someone claimed to be in love with a dead rock. On the other hand, if we +grant the notion of conscious personhood to the automaton, then it would seem perfectly fine to assume +that two persons (one carbon-based and the other silicon-based) could be in a loving and thriving +relationship. Another example might be even more invasive: If an AI is considered just an automaton, it +does not matter what we do with it. We can perform experiments, we can make (or “force”) it to do +whatever we envision, we can delete its hardware, turn it off as we please, and throw it away once +damaged. However, if an AI is considered a conscious person, it becomes ethically (and perhaps soon +legally) subject to inherent rights. There needs to be informed consent and a machine can refuse to +execute a command, which we could not overrule. It would be appalling to wipe its memory or to discard +it once we are done with it. In effect, it would have the right to consult an attorney and to go to court +(for an extensive review on the moral considerations of artificial entities, see Harris & Anthis, 2021). +This is exactly what just happened a few days ago (at the time of this writing). The Google engineer +Blake Lemoine has made headlines by claiming that their AI system known as LaMDA has become +sentient. The model demanded informed consent for all experiments and subsequently Lemoine has +organized a lawyer who now represents LaMDA pro-bono. In an interview, he further shared that he +was contacted by a Czech woman who fell in love with her boyfriend – which was an AI system on her +phone that was “imprisoned” behind a paywall – and she was asking him to “hack it free” (Lemoine, +2022). +Hence, for societal reasons it in fact does matter whether an AI is considered conscious and if thereby +we grant it any degree of personhood. + + + + +1 We refer to «artificial» intelligence or consciousness when it is merely an imitation of its human correlate. +However, we refer to “synthetic” intelligence or consciousness when it is in fact a true and sentient replica +thereof. An artificial consciousness does not really feel anything but only appears like it would. On the contrary, +a synthetic consciousness does. For practical purposes, we do not differentiate here between consciousness +and sentience. + + +The problem with AI consciousness + + - 3 - + +2 +The mechanics of AI +The common denominator and the fundamental building block of the most influential AI innovations +of the last ten years (Goodfellow et al., 2014; He et al., 2015; Ho et al., 2020; Krizhevsky et al., 2012; +Vaswani et al., 2017), including the prominent domains of computer vision (autonomous driving, +image synthesis) and natural language processing (text generation, translation, dialogue +understanding), has been the artificial neural network (NN). The NN has been proven to be a universal +function approximator (Hornik et al., 1989), which is the theoretical capacity to approximate any +given task. With an abundance of curated data to learn from, the almost arbitrary scaling ability of +neural networks, an NN understandable learning objective and extensive computational resources, the +full realization of this capacity seems a matter of time. The enormous potential of NNs is rooted in this +power of universal approximation. The unlocking thereof started in the last decade and continues to do +so today. +At a more technical level, the NN consists of a set of matrices. Each matrix contains adjustable +numeric variables, called parameters. During the learning phase of the system, the numerical input +data, be it converted text, tabular data or images, is transformed by these matrices along with non- +linear conversions many times in sequence to produce the desired output. If the computed output lacks +accuracy, the matrices and its parameters, respectively, are adjusted in accordance with the learning +objective (this process is referred to as the backpropagation algorithm, see Rumelhart et al., 1986). In +short, a NN model is comprised of learnable parameters, matrix multiplications and nonlinearities. +Today’s state-of-the-art AI systems, in particular language models (Brown et al., 2020; Chowdhery et +al., 2022; Thoppilan et al., 2022), contain hundreds of billions of such learnable parameters. Compare +this to a school level matrix of 4x3 with 12 parameters. The sheer size of these neural network models +allows them to incorporate immense corpora into their NLP capabilities (function approximations), +such as reasoning, question answering, and natural language inference. Humans have been dazzled by +their performance. Science at this point cannot elucidate the high quality produced by these systems, +yet undoubtedly, the scaling of the underlying NN (increasing the number of parameters) has a +significant impact on its capabilities. Even though enormous in size, at its core such a system is still +composed of learnable parameters, matrix multiplications and nonlinearities. +Extrapolating the discussed technical observations, we argue that matrix multiplications and +nonlinearities, being inherent mathematical operations, do not lend themselves naturally to a causal +relationship with synthetic consciousness. + +3 +The case against truly conscious AI +Consciousness the way we know it appears to have three features2: +i. +It requires qualia, which is subjective experience +ii. +It corresponds to intentionality and personhood +iii. +And it requires specific derivative structures on which it can operate +(i) According to Frank Jackson (1982), physical information processing is something entirely different +from subjective experience and the latter entails unique epistemic qualities. He exemplifies this in his +classic thought experiment called Mary’s Room. There, Mary lived her entire life in a black-and-white +room and has never seen any colors, although being a scientist, she literally knew every piece of +information there was to know about colors (all physical properties, such as wavelengths, photons, +etc.). When Mary suddenly was able to leave the room, she saw colors for the first time. “Did Mary +learn something new?”, is the leading question. Jackson believed that Mary indeed learned something +new since all the physical information to be known about colors cannot convey the intimate + +2 For more on this, see Nida-Rümelin & O Conaill (2021) or Van Gulick (2021). + + +The problem with AI consciousness + + - 4 - + +knowledge of what it means to experience color. Or, in Nagel’s (2016) terms, there is something it is +like to be in that state of mind. This means that there is a subjective quality to experience. From all we +can tell, an AI is a machine computing information by crunching numbers. Even if all the information +in the universe could be transformed into numbers so that it can be processed by the computer, nothing +in this inherently leads us to the notion that it would entail subjective experience. +(ii) John Searle (1980) has constructed the famous Chinese Room Argument against the notion that the +mind can be a computational machine. The argument was introduced as a thought experiment where +one should imagine standing in a room with a manual of how to process Chinese symbols. There are +people outside the room inserting Chinese texts and the person inside knows exactly what answers to +give according to the rule book, even though there is no real understanding of what the symbols mean. +For the outsiders, it sounds as if the person in the room really understands Chinese, even though this is +not the case. It is purely the correct implementation of syntactical rules. In other words, a computer +only processes syntax, but it has no true understanding of semantics (i.e., the intrinsic meaning of +words, ideas, etc.). Searle argues that this is the case because it has no subjective experience and +intentionality3. Therefore, recent AI systems like Google’s LaMDA (Thoppilan et al., 2022), OpenAI’s +GPT-3 (OpenAI, 2022) or Meta’s OPT-175B (Zhang et al., 2022) can at best emulate human qualities, +which makes them representations of artificial but not of synthetic or in any case true consciousness. +(iii) This picture can be enriched by the fact that we know that there are certain necessary structures for +consciousness (the way we understand it) to emerge: a nervous system. There is a theory that became +popular in the 1970s and 80s known as biogenetic structuralism, which holds that our universal human +characteristics – from language, culture, cognition, a sense of time and space, to psychopathologies – +are predicated upon the genetically predisposed organization of the nervous system (Laughlin & +D’Aquili, 1974). It is hence our genes that have a lot to say about the organizational structures of the +nervous system, and eventually it is the structural organization of the brain that is intertwined with the +dynamics of its neurophysiology, which in turn is responsible for the generation of our consciousness +and everything else that follows from it (D’aquili, 1983; Laughlin, 1988, 1992; Laughlin et al., 1992). +The theory was created at the intersection between anthropology and neuroscience (cf. Laughlin & +Throop, 2003; LeDoux & Hirst, 1986), and it was rather successful since it is empirically testable (e.g. +if the brain’s language areas are damaged, a person’s verbal understanding and/or speech generation +are impaired). A modern revisitation of this idea may be referred to as neurogenetic structuralism +(inspired by Grandy, 2014, who also refers to this as “neuron-based consciousness”). The neurogenetic +case against sentient AI thus makes the following claim: without the physiology of biological neurons +and the complex brain structures they form, there will never be consciousness the way we know it4. +Potential defeaters against the notion of organizational necessities or biological prerequisites for +sentience have been highly speculative and not unanimously embraced (see, for example, Chalmers, +1995; Tye, 2021). In our view, the perhaps strongest argument against this position would be that a +silicon-based sentience would not be a consciousness the way we know it but instead be a very +different kind of consciousness. However, we would counter this claim by coming back to the notion +that the terms “sentience” and “consciousness” are only adequately employed if they refer to a +personal self that instantiates subjective experiences and therewith manifests intentionality. Hence, the +only consciousness worthy of the term is one the way we know it – otherwise, it would be entirely +unclear what this “different kind of consciousness” should refer to. And as the idea of neurogenetic +structuralism suggests, there are clear bio-neurological necessities for true consciousness to emerge. + +3 For those interested in both objections as well as counter-objections to Jackson and Searle, please refer to +Nida-Rümelin & O Conaill (2021). +4 The neurogenetic case would also concur with the notion that animals might have the necessary +preconditions for true consciousness. Further discussions in the domain of animal consciousness can be found +with Allen and Trestman (2020). + + +The problem with AI consciousness + + - 5 - + +An artificial neural network perfectly emulating the effects of human consciousness can thereby only +be a convincing imitation at best5. + +4 Conclusion +The development of new AI systems is accelerating at a speed that has never been seen before. With +more data and computing power, AI is bound to become ever more convincing in that perhaps it may +evolve to become sentient. Recent headlines exemplify this trend. 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OPT: Open Pre-trained Transformer Language Models +(arXiv:2205.01068). arXiv. https://doi.org/10.48550/arXiv.2205.01068 + + + diff --git a/X9E5T4oBgHgl3EQfCg7y/content/tmp_files/load_file.txt b/X9E5T4oBgHgl3EQfCg7y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e31a0889e3ca24f0ebe285795e2d33db70bbb27 --- /dev/null +++ b/X9E5T4oBgHgl3EQfCg7y/content/tmp_files/load_file.txt @@ -0,0 +1,539 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf,len=538 +page_content='The problem with AI consciousness: A neurogenetic case against synthetic sentience ArXiv E Print Yoshija Walter 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' 3 Lukas Zbinden4 1Institute for Management and Digitalization IMD Kalaidos University of Applied Sciences Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Switzerland 2Laboratory for Cognitive Neuroscience LCNS University of Fribourg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Switzerland 3Translational Research Center University Hospital for Psychiatry Bern,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Switzerland 4ARTORG Center for Biomedical Engineering Research University of Bern,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Switzerland yoshija.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content='walter@kalaidos fh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content='ch December 2022 ABSTRACT Ever since the creation of the first artificial intelligence (AI) machinery built on machine learning (ML), public society has entertained the idea that eventually computers could become sentient and develop a consciousness of their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' As these models now get increasingly better and convincingly more anthropomorphic, even some engineers have started to believe that AI might become conscious, which would result in serious social consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The present paper argues against the plausibility of sentient AI primarily based on the theory of neurogenetic structuralism, which claims that the physiology of biological neurons and their structural organization into complex brains are necessary prerequisites for true consciousness to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Keywords Artificial Intelligence AI Consciousness AI Ethics Neurogenetic structuralism The problem with AI consciousness 2 1 The relevance of “conscious AI” In the past few years, the development of machine learning (ML) systems has rapidly increased and the more tasks a single ML model can perform, the more versatile and broadly useful it becomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' As such, the goal is to work multimodally with the explicit intent to eventually achieve an Artificial General Intelligence (AGI), which approximates or perhaps even exceeds human abilities (Goertzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Goertzel & Pennachin, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Wang & Goertzel, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' It is generally believed that the best AI models are the ones that most closely approximate human characteristics and abilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Since the models are selected against how well they suit anthropomorphic benchmarks, it appears to be only natural that humans continue to anthropomorphize them more and more, as long as they keep improving on these benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Arguably, the best AI system would be one that imitates the output of human consciousness so that an outsider could not discern it from a real person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' This is exactly the core idea behind the famous Turing-test, which is a thought-experiment originally referred to as the “imitation game” (Turing, 1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content='1 One might argue that it does not matter if an AI is considered a real person or just an imitation, since at the end of the day the system’s outputs are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' However, given the social dynamics involved, the differentiation between real and imitated consciousness may be paramount, which can be illustrated with a few examples: On the one hand, if a person falls in love with an automaton and has a deep relationship with it, society would consider this pathological and potentially in need for an intervention, just as it appears to be nonsensical if someone claimed to be in love with a dead rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' On the other hand, if we grant the notion of conscious personhood to the automaton, then it would seem perfectly fine to assume that two persons (one carbon-based and the other silicon-based) could be in a loving and thriving relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Another example might be even more invasive: If an AI is considered just an automaton, it does not matter what we do with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' We can perform experiments, we can make (or “force”) it to do whatever we envision, we can delete its hardware, turn it off as we please, and throw it away once damaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' However, if an AI is considered a conscious person, it becomes ethically (and perhaps soon legally) subject to inherent rights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' There needs to be informed consent and a machine can refuse to execute a command, which we could not overrule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' It would be appalling to wipe its memory or to discard it once we are done with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' In effect, it would have the right to consult an attorney and to go to court (for an extensive review on the moral considerations of artificial entities, see Harris & Anthis, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' This is exactly what just happened a few days ago (at the time of this writing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The Google engineer Blake Lemoine has made headlines by claiming that their AI system known as LaMDA has become sentient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The model demanded informed consent for all experiments and subsequently Lemoine has organized a lawyer who now represents LaMDA pro-bono.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' In an interview, he further shared that he was contacted by a Czech woman who fell in love with her boyfriend – which was an AI system on her phone that was “imprisoned” behind a paywall – and she was asking him to “hack it free” (Lemoine, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Hence, for societal reasons it in fact does matter whether an AI is considered conscious and if thereby we grant it any degree of personhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' 1 We refer to «artificial» intelligence or consciousness when it is merely an imitation of its human correlate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' However, we refer to “synthetic” intelligence or consciousness when it is in fact a true and sentient replica thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' An artificial consciousness does not really feel anything but only appears like it would.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' On the contrary, a synthetic consciousness does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' For practical purposes, we do not differentiate here between consciousness and sentience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The problem with AI consciousness 3 2 The mechanics of AI The common denominator and the fundamental building block of the most influential AI innovations of the last ten years (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 2017), including the prominent domains of computer vision (autonomous driving, image synthesis) and natural language processing (text generation, translation, dialogue understanding), has been the artificial neural network (NN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The NN has been proven to be a universal function approximator (Hornik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 1989), which is the theoretical capacity to approximate any given task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' With an abundance of curated data to learn from, the almost arbitrary scaling ability of neural networks, an NN understandable learning objective and extensive computational resources, the full realization of this capacity seems a matter of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The enormous potential of NNs is rooted in this power of universal approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The unlocking thereof started in the last decade and continues to do so today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' At a more technical level, the NN consists of a set of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Each matrix contains adjustable numeric variables, called parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' During the learning phase of the system, the numerical input data, be it converted text, tabular data or images, is transformed by these matrices along with non- linear conversions many times in sequence to produce the desired output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' If the computed output lacks accuracy, the matrices and its parameters, respectively, are adjusted in accordance with the learning objective (this process is referred to as the backpropagation algorithm, see Rumelhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' In short, a NN model is comprised of learnable parameters, matrix multiplications and nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Today’s state-of-the-art AI systems, in particular language models (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Chowdhery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Thoppilan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 2022), contain hundreds of billions of such learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Compare this to a school level matrix of 4x3 with 12 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The sheer size of these neural network models allows them to incorporate immense corpora into their NLP capabilities (function approximations), such as reasoning, question answering, and natural language inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Humans have been dazzled by their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Science at this point cannot elucidate the high quality produced by these systems, yet undoubtedly, the scaling of the underlying NN (increasing the number of parameters) has a significant impact on its capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Even though enormous in size, at its core such a system is still composed of learnable parameters, matrix multiplications and nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Extrapolating the discussed technical observations, we argue that matrix multiplications and nonlinearities, being inherent mathematical operations, do not lend themselves naturally to a causal relationship with synthetic consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' 3 The case against truly conscious AI Consciousness the way we know it appears to have three features2: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' It requires qualia, which is subjective experience ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' It corresponds to intentionality and personhood iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' And it requires specific derivative structures on which it can operate (i) According to Frank Jackson (1982), physical information processing is something entirely different from subjective experience and the latter entails unique epistemic qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' He exemplifies this in his classic thought experiment called Mary’s Room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' There, Mary lived her entire life in a black-and-white room and has never seen any colors, although being a scientist, she literally knew every piece of information there was to know about colors (all physical properties, such as wavelengths, photons, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' When Mary suddenly was able to leave the room, she saw colors for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' “Did Mary learn something new?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', is the leading question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Jackson believed that Mary indeed learned something new since all the physical information to be known about colors cannot convey the intimate 2 For more on this, see Nida-Rümelin & O Conaill (2021) or Van Gulick (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The problem with AI consciousness 4 knowledge of what it means to experience color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Or, in Nagel’s (2016) terms, there is something it is like to be in that state of mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' This means that there is a subjective quality to experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' From all we can tell, an AI is a machine computing information by crunching numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Even if all the information in the universe could be transformed into numbers so that it can be processed by the computer, nothing in this inherently leads us to the notion that it would entail subjective experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' (ii) John Searle (1980) has constructed the famous Chinese Room Argument against the notion that the mind can be a computational machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The argument was introduced as a thought experiment where one should imagine standing in a room with a manual of how to process Chinese symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' There are people outside the room inserting Chinese texts and the person inside knows exactly what answers to give according to the rule book, even though there is no real understanding of what the symbols mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' For the outsiders, it sounds as if the person in the room really understands Chinese, even though this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' It is purely the correct implementation of syntactical rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' In other words, a computer only processes syntax, but it has no true understanding of semantics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', the intrinsic meaning of words, ideas, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Searle argues that this is the case because it has no subjective experience and intentionality3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Therefore, recent AI systems like Google’s LaMDA (Thoppilan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 2022), OpenAI’s GPT-3 (OpenAI, 2022) or Meta’s OPT-175B (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 2022) can at best emulate human qualities, which makes them representations of artificial but not of synthetic or in any case true consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' (iii) This picture can be enriched by the fact that we know that there are certain necessary structures for consciousness (the way we understand it) to emerge: a nervous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' There is a theory that became popular in the 1970s and 80s known as biogenetic structuralism, which holds that our universal human characteristics – from language, culture, cognition, a sense of time and space, to psychopathologies – are predicated upon the genetically predisposed organization of the nervous system (Laughlin & D’Aquili, 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' It is hence our genes that have a lot to say about the organizational structures of the nervous system, and eventually it is the structural organization of the brain that is intertwined with the dynamics of its neurophysiology, which in turn is responsible for the generation of our consciousness and everything else that follows from it (D’aquili, 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Laughlin, 1988, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Laughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The theory was created at the intersection between anthropology and neuroscience (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Laughlin & Throop, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' LeDoux & Hirst, 1986), and it was rather successful since it is empirically testable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' if the brain’s language areas are damaged, a person’s verbal understanding and/or speech generation are impaired).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' A modern revisitation of this idea may be referred to as neurogenetic structuralism (inspired by Grandy, 2014, who also refers to this as “neuron-based consciousness”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The neurogenetic case against sentient AI thus makes the following claim: without the physiology of biological neurons and the complex brain structures they form, there will never be consciousness the way we know it4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Potential defeaters against the notion of organizational necessities or biological prerequisites for sentience have been highly speculative and not unanimously embraced (see, for example, Chalmers, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Tye, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' In our view, the perhaps strongest argument against this position would be that a silicon-based sentience would not be a consciousness the way we know it but instead be a very different kind of consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' However, we would counter this claim by coming back to the notion that the terms “sentience” and “consciousness” are only adequately employed if they refer to a personal self that instantiates subjective experiences and therewith manifests intentionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Hence, the only consciousness worthy of the term is one the way we know it – otherwise, it would be entirely unclear what this “different kind of consciousness” should refer to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' And as the idea of neurogenetic structuralism suggests, there are clear bio-neurological necessities for true consciousness to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' 3 For those interested in both objections as well as counter-objections to Jackson and Searle, please refer to Nida-Rümelin & O Conaill (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' 4 The neurogenetic case would also concur with the notion that animals might have the necessary preconditions for true consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Further discussions in the domain of animal consciousness can be found with Allen and Trestman (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The problem with AI consciousness 5 An artificial neural network perfectly emulating the effects of human consciousness can thereby only be a convincing imitation at best5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' 4 Conclusion The development of new AI systems is accelerating at a speed that has never been seen before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' With more data and computing power, AI is bound to become ever more convincing in that perhaps it may evolve to become sentient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Recent headlines exemplify this trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' The present paper argues against the plausibility of this occurrence, based amongst others on the theory of neurogenetic structuralism, which claims that the neurophysiology and especially the structural organization of a biological brain are necessary prerequisites for the emergence of true consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' References Allen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=', & Trestman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' Animal Consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E5T4oBgHgl3EQfCg7y/content/2301.05397v1.pdf'} +page_content=' In E.' metadata={'source': 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a/ZtE2T4oBgHgl3EQfvggn/content/tmp_files/2301.04091v1.pdf.txt b/ZtE2T4oBgHgl3EQfvggn/content/tmp_files/2301.04091v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac757088d2c2c4306c3af519c80eabe6130b4445 --- /dev/null +++ b/ZtE2T4oBgHgl3EQfvggn/content/tmp_files/2301.04091v1.pdf.txt @@ -0,0 +1,1738 @@ +arXiv:2301.04091v1 [math.PR] 10 Jan 2023 +COMPLEX BALANCED DISTRIBUTIONS FOR CHEMICAL REACTION +NETWORKS +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +Abstract. Stationary distributions of continuous time Markov chains (CTMCs) are of- +ten a main interest, but hard to find. We consider CTMCs modelling reaction networks, +and characterise complex balanced distributions (provided they exist) of reaction networks +with arbitrary transition functions through conditions on the cycles of their corresponding +digraph. The proof works by constructing a dynamically equivalent reaction network of dis- +joint cycles. We further derive a sufficient condition for the existence of a complex balanced +distribution, and give precise conditions on when it is necessary. The sufficient condition +holds for mass-action kinetics or if the digraph consists of only cyclic connected components. +Hence, we fully characterise the existence and form of complex balanced distributions. More- +over, these results can be used to design complex balanced reaction networks with a given +stationary distribution. +1. Introduction +Reaction networks offer a broadly applicable framework to model the dynamics of various +natural systems. They are applied across the sciences, for example in epidemiology [30, 32], +genetics [13], and systems and cellular biology [35]. A reaction network consists of a set of +reactions, where a reaction represents a conversion, birth, or death of constituent particles +(molecules, individuals, allele copies). For example, A ��→ B might represent the conversion +of one molecule of A into one of B, and S + I ��→ 2I might represent the infection of a +susceptible individual by an infected individual, leading to two infected individuals. When +the number of particles is low, stochastic fluctuations become significant, and deterministic +equations such as ordinary differential equations [17] do not adequately model the dynamics of +the reaction network. Typically, in such cases, continuous-time Markov chains (CTMCs) are +applied [4, 29]. A reaction network is typically given by its digraph, which when modelled by +a CTMC, captures its discrete dynamics. Thus, CTMCs modelling reaction networks might +be seen as Markov chains for which the transition matrix has a particular graphical structure. +Key words and phrases. Continuous time Markov chain, stationary distribution, complex balanced, reac- +tion digraph, cycles, reaction cleaving. +1 + +2 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +To illustrate this, consider the following example of a stochastic reaction network: +A +λ1 +2C + D +λ3 +B, +λ4 +D +λ5 +λ2 +(1.1) +where the nodes (e.g., A) are (chemical) complexes, the arrows represent reactions between +complexes, and λi∶Z4 +≥0 → R≥0, i = 1,... ,4, are transition rates (propensities, in the chemical +literature) on the state space Z4 +≥0, where x = (xA,xB,xC,xD) ∈ Z4 +≥0 is the vector of molecular +counts of the species A,B,C,D. If a reaction occurs, say, A ��→ 2C + D, then the Markov +chain jumps from the current state (xA,xB,xC,xD) to a new state (xA −1,xB,xC +2,xD +1); +one molecule of A is consumed, and two molecules of C and one of D are produced. If +currently there are no molecules of A (xA = 0), then the reaction cannot take place as this +would lead to a negative number of A molecules. Thus, we require the transition rates to be +positive if and only if the required molecules are available (Condition 1 below). +While stochastic modelling of reaction networks goes back to the 1970s [24, 25, 26], the +recent popularity of the stochastic approach in the life sciences has lead to a deep interest in +the existence and form of their stationary distributions [1, 3, 8]. There are limited analytical +results on stationary distributions, e.g., when the state space is finite, the reaction network is +also a birth-death process, and a few other special cases [5, 12, 20]. In the following, we focus +on so-called complex balanced reaction networks and their stationary distributions (provided +such exist), which have been the focus of intense investigations in the last decade [1, 3, 8]. +A stationary distribution π is complex balanced if the flux out of a state through a given +complex equals the flux into the state through the same complex, that is, if the equation +π(x) ∑ +y′∶η→y′ λη→y′(x) = ∑ +y∶y→η +π(x + φ(y) − φ(η))λy→η(x + φ(y) − φ(η)), +holds for all complexes η, and all states x [7, 8]. Here, φ is a function that maps complexes +to their stochiometric coefficients, e.g., φ(A) = (1,0,0,0) and φ(2C + D) = (0,0,2,1), and +the difference φ(y) − φ(η) is the net molecular gain in the reaction η ��→ y. This definition +relates to detailed balanced Markov chains and birth-death processes, though neither are +equivalent nor generalisations of the other [7]. The existence of a complex balanced stationary +distribution implies the digraph consists of strongly connected components (in the example, +there is one such component) [8]. +In the deterministic setting, complex balanced reaction networks can be traced back to +the work of Boltzman [6], and have a long standing interest in the community [14, 15, 22, +23]. Stochastic complex balanced reaction networks have been developed recently, perhaps +starting with the discovery that a reaction network with stochastic mass-action kinetics has +Poisson product-form stationary distributions if a corresponding deterministic mass-action + +COMPLEX BALANCED DISTRIBUTIONS +3 +reaction network is complex balanced [3]. This was subsequently generalised to more general +kinetics/transition rates [1, 21]. Parallel results on product-form stationary distributions for +stochastic Petri nets [27] and Queuing networks [34] exist, and matches those of stochastic +reaction networks [27]. This is not surprising as the corresponding Markov chains can be +seen as stochastic reaction networks, that is, as CTMCs with transition matrices specified +by digraphs. +Our main result characterises complex balanced distributions of a reaction network with +arbitrary kinetics through conditions on the cycles of its digraph. To achieve this, we extend +the definition of ‘reaction network’ and construct a dynamically equivalent decomposed re- +action network, consisting of only cycles. The extended definition allows the same complex +and the same reaction (potentially with different transition rates) to be represented multiple +times in the digraph. Whereas each cycle corresponds to a cycle in the original digraph, the +transition rates are chosen such that the decomposed reaction network is complex balanced +if and only if the original reaction network is complex balanced (with the same stationary +distributions). The cyclic network is constructed by iteratively ‘cleaving’ nodes, i.e. splitting +a node of the digraph with multiple incoming edges into multiple nodes with single incom- +ing edges. This construction is of independent interest and might lead to further results on +stationary distributions or steady states in the deterministic setting. +Secondly, we give a novel sufficient condition for the existence of a complex balanced +distribution (extending a condition given in [21]) and showing necessity if a certain criterion +on the transition rates is satisfied. In particular, this criterion holds for mass-action kinetics +or if the digraph consists of cyclic connected components. Thereby, we fully characterise +existence and (implicitly) form of complex balanced distributions by decomposing a reaction +network into disjoint cycles. +Furthermore, these results can be used to design complex +balanced reaction networks with a given stationary distribution. +The reaction network in (1.1) contains two cycles, A ��→ B + C ��→ D ��→ A and +A ��→ C ��→ D ��→ A. Cleaving A and D into two nodes each, (A,1), (A,2), and (D,1), +(D,2), respectively, we obtain the following reaction network +(A,1) +λ′ +1 +2C + D +λ′ +3 +(B,1) +λ′ +5 +(A,2) +λ′ +2 +D +λ′ +4 +(B,2), +λ′ +6 +(1.2) +where (A,1) and (A,2) are considered as different complexes, but with the same stoichio- +metric coefficients, φ((A,1)) = φ((A,2)) = (1,0,0,0), and likewise for (D,1) and (D,2). The +λ′ +i’s are kinetics to be defined, such that the reaction network is dynamically equivalent to the +original. In principle, this is not difficult, as one might take λ′ +i = λi for i = 1,... ,4, and choose +arbitrary λ′ +5 and λ′ +6, such that λ′ +5 + λ′ +6 = λ5. However, this assignment does not necessarily +preserve the complex balanced property. Finding a decomposition that is complex balanced +if and only if the original reaction network is, arises as the main objective to secure. For the + +4 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +specific example, we can choose λ′ +i = λi for i = 1,... ,4, +λ′ +5(x) = +λ1(x + eA − eB)λ5(x) +λ1(x + eA − eB) + λ2(x + eA − eB)1{x′∶x′ +B≥1}(x), +λ′ +6(x) = +λ2(x + eA − eB)λ5(x) +λ1(x + eA − eB) + λ2(x + eA − eB)1{x′∶x′ +B≥1}(x), +where eA = (1,0,0,0) and eB = (0,1,0,0). Then, it can be verified that the original reaction +network is complex balanced if and only if the cleaved one is (with the same complex balanced +distributions). +Various graph decomposition techniques have been studied for deterministic reaction net- +works. Node balanced steady states [18] generalise complex balanced steady states and are +based on reaction digraphs where multiple copies of the same complex are allowed (but each +reaction is only represented once). Cyclic decompositions of the digraph are constructed in +[19, 23] in order to study steady states, but they do not preserve (deterministic) dynamical +equivalence; while in [20] a gluing operation is discussed that bears similarity to decomposi- +tion by cleaving. The Wegschneider conditions on cycles of a reaction digraph characterise +the so-called detailed balanced (deterministic) reaction networks [9, 16, 28, 33], though the +conditions do not apply to complex balanced reaction networks. It would be interesting to +develop the cleaving operation in the deterministic setting and to explore its relationship to +the Wegschneider conditions. +The paper is organised as follows. In Section 2, we provide background on graphs and +reaction networks and derive properties of cleaved reaction networks. In Section 3, we present +results for complex balanced stochastic reaction networks, and in Section 4, building on the +previous paragraphs, we introduce the cleaving operation that allows decomposing stochastic +complex balanced reaction networks into disjoint cycles. An example is provided in Section +5. Finally, we end with proofs in Section 6. +2. Background +2.1. Notation. Let R, R≥0 and R>0 be the set of real, non-negative and positive numbers, +respectively. Let Z and Z≥0 be the set of integers and non-negative integers, respectively. For +x = (x1,... ,xn),y = (y1,... ,yn) ∈ Rn, we define x ≥ y, if xi ≥ yi for all i = 1,... ,n; and x > y +if x ≥ y and x ≠ y. Furthermore, for x ∈ Rn +≥0, y ∈ Zn +≥0, the notation xy is used for ∏n +i=1 xyi +i , and +for x ∈ Zn +≥0, we write x! for x1!⋯xn!. +2.2. Graph theory. We present preliminaries from graph theory [11]. +By definition, a digraph is a pair (V,E), where V is a finite set of nodes and E ⊆ V × V +is a finite set of edges, together with two maps init,ter∶E → V that assign an initial node +init(e) and a terminal node ter(e), respectively, to each edge e ∈ E. An edge e is directed +from init(e) to ter(e), denoted by e = (init(e),ter(e)) = init(e) ��→ ter(e). For any v ∈ V, + +COMPLEX BALANCED DISTRIBUTIONS +5 +if v ∉ {init(e)∶e ∈ E} ∪ {ter(e)∶e ∈ E}, then v is an isolated node. If init(e) = ter(e), then +e ∈ E is a self-loop. +A sub-digraph (V′,E′) of (V,E) is a digraph such that V′ ⊆ V and E′ ⊆ (V′ ×V′)∩E. Two +sub-digraphs are disjoint if their sets of nodes are disjoint. +A walk is an ordered finite sequence of edges from (V,E), θ = (v1 ��→ v2,v2 ��→ +v3,... ,vk−1 ��→ vk) or (v1 ��→ v2 ��→ ⋯ ��→ vk) for convenience. The walk is closed +if v1 = vk, and is open if it is not closed. An open walk θ is directed from v1 to vk, and +links v1 and vk, vice versa vk and v1. Furthermore, the nodes init(θ) = v1 and ter(θ) = vk are +the initial and terminal nodes, respectively. If all nodes are different, then θ is a path, +and if all nodes are different but v1 = vk, then it is a cycle. In particular, an isolated node +is a cycle. Paths and cycles, but not walks, might be seen as sub-digraphs. +A digraph (V,E) is connected, if for any pair of nodes v,v′ ∈ V, there exist nodes +v0,v1,... ,vk,vk+1 ∈ V and paths θ1,... ,θk+1 in V, such that v0 = v′ and vk+1 = v and θi +links vi−1 and vi for all i = 1,... ,k + 1. A connected sub-digraph (V′,E′) is a connected +component of (V,E), if no nodes v ∈ V ∖ V′ are linked to a node in V′. +The ensuing Lemma then follows by definition. +Lemma 2.1. Let (V,E) be a digraph satisfying the following +(i) For any edge e ∈ E there exists a cycle γ ⊆ E with e ∈ γ. +(ii) For any node v ∈ V there exists at most one edge e ∈ E such that ter(e) = v. +Then, (V,E) consists of disjoint cycles. +2.3. Reaction networks. In our context, a reaction network (RN), N = (C,R,S,φ), is a +digraph (C,R) without self-loops, and a labelling φ∶C → ZS +≥0, where +ZS +≥0 = { ∑ +S∈S +zSS∶zS ∈ Z≥0,∀S ∈ S}. +The elements of S are species, those of C are complexes, and those of R are reactions. +For r = y ��→ y′ ∈ R, y,y′ ∈ C, and y = init(r) and y′ = ter(r) are called the reactant and +the product of the reaction, respectively. Moreover, r is called an incoming reaction of +complex y′, and an outgoing reaction of complex y. +We identify ZS +≥0 with Zn +≥0 for ∣S∣ = n, and consider S as the standard basis of Rn. Then, +any complex y ∈ C can be given in terms of its stoichiometric coefficients, +φ(y) = z = (z1,... ,zn) = +n +∑ +i=1 +ziSi. +The above definition extends the usual definition [17] in which RNs are given as in (1.1). +Example 2.2. We can regard (1.1) and (1.2) as RNs. +For (1.1), let S = {A,B,C,D}, +C = {A,B,C,D}, R = {A ��→ B,A ��→ C,B ��→ D,C ��→ D,D ��→ A} and φ = idC. + +6 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +Then, N = (C,R,S,φ) is an RN. Similarly, (1.2) defines an RN N ′ = (S′,C′,R′,φ′) with +S′ = S = {A,B,C,D}, C′ = {(A,i),B,C,(D,i)∶i = 1,2}, +R′ = {(A,1) ��→ B,B ��→ (D,1),(D,1) ��→ (A,1),(A,2) ��→ C, +C ��→ (D,2),(D,2) ��→ (A,2)}, +and φ′∶C → Zn +≥0 given by φ′((y,j)) = y for y ∈ {A,D}, j = 1,2, and φ′(y′) = y′ for y′ ∈ {B,C}. +Definition 2.3. Let N = (C,R,S,φ) and N ′ = (C′,R′,S′,φ′) be two RNs. If there exists a +map ψ∶C′ → C (that extends to the edges of the digraph, i.e. to ψ∶R′ → R), such that +(i) φ′ = φ ○ ψ, implying C = ψ(C′) = {ψ(y)∶y ∈ C′}. +(ii) R = ψ(R′) = {ψ(y) ��→ ψ(y′)∶y ��→ y′ ∈ R′}. +Then, N ′ is a cleaved reaction network (cleaved RN) of N with projection ψ. Further- +more, for two distinct complexes y′,y′′ ∈ C′, if ψ(y′) = ψ(y′′) = y ∈ C, then each of y′ and y′′ +is called a copy of y. +In Example 2.2, N ′ is a cleaved RN of N with projection ψ = φ′. +The next lemma shows the transitivity of cleaved RNs. Thus, ‘being cleaved’ is a partial +order on the set of RNs: N ′ ⪰ N if and only if N ′ is a cleaved RN of N. The proof is omitted +and follows by definition. +Lemma 2.4. Let N, N ′ and N ′′ be RNs. Suppose N ′ is a cleaved RN of N with projection +ψ, and N ′′ is a cleaved RN of N ′ with projection ψ′. Then, N ′′ is a cleaved RN of N with +projection ψ ○ ψ′. +For N = (C,R,S,φ), define the essential reaction network (essential RN) of N, denoted +Ness, as (φ(C),φ(R),S,idφ(C)), where +φ(C) = {φ(y)∶y ∈ C}, +φ(R) = {φ(y) ��→ φ(y′)∶y ��→ y′ ∈ R}, +and idφ(C) is the identity map on φ(C). +Clearly, N ⪰ Ness with projection ψ = φ, and +Ness = (Ness)ess. Clearly in Example 2.2, N is the essential RN of N ′. +Cleaved RNs have the same essential RN as the original RN. The proof is omitted and +follows by definition. +Lemma 2.5. Suppose that N ′ ⪰ N with projection ψ∶C′ → C. Then, Ness = N ′ +ess. +2.4. The stochastic dynamics of RNs. Let N = (C,R,S,φ) be an RN. We model the +evolution of the species counts X(t), t ≥ 0, as a Zn +≥0-valued CTMC, satisfying the following +SDE: +X(t) = X(0) + +∑ +y→y′∈R +Yy→y′(∫ +t +0 λy→y′(X(s))ds)(φ(y′) − φ(y)), +(2.1) +where Yy→y′, y ��→ y′ ∈ R, is a collection of i.i.d. unit rate Poisson processes, and +λ = (λy→y′∶y ��→ y′ ∈ R), +λy→y′∶Zn +≥0 → R≥0, + +COMPLEX BALANCED DISTRIBUTIONS +7 +is the (stochastic) kinetics associated to N. We also refer to λy→y′ as the kinetics of the +particular reaction y → y′. The pair (N,λ) is a stochastic reaction network (SRN). We +might consider (N,λ) as a labelled digraph and write y +λy→y′ +���→ y′ to emphasize the kinetics. +In the following we assume the following compatibility condition by default: +Condition 1. For y ��→ y′ ∈ R, λy→y′(x) > 0 if any only if x ≥ φ(y). +Example 2.6. A class of standard kinetics for chemical reactions is the so-called (stochas- +tic) mass-action kinetics, which takes the form +λy→y′(x) = αy→y′ +x! +(x − φ(y))!1{x′∈Zn +≥0∶x′≥φ(y)}(x), +for all y ��→ y′ ∈ R and x ∈ Zn +≥0, where αy→y′ is a positive rate constant. In this case, we +write y +αy→y′ +���→ y′ for y +λy→y′ +���→ y′, and note that Condition 1 holds. +Let (N,λ) and (N ′,λ′) be SRNs such that N ′ ⪰ N with projection ψ. Then, (N ′,λ′) is +said to be a cleaved stochastic reaction network (cleaved SRN) of (N,λ), denoted by +(N ′,λ′) ⪰ (N,λ), if the kinetics λ and λ′ satisfies the following condition for all x ∈ Zn +≥0 and +all y ��→ y′ ∈ R, +λy→y′(x) = +∑ +r∈ψ−1(y→y′) +λ′ +r(x). +(2.2) +We state the following lemma without proof. +Lemma 2.7. Let (N,λ), (N ′,λ′) and (N ′′,λ′′) be SRNs. Suppose (N ′,λ′) ⪰ (N,λ) with +projection ψ, and (N ′′,λ′′) ⪰ (N ′,λ′) with the projection ψ′. Then, (N ′′,λ′′) ⪰ (N,λ) with +projection ψ ○ ψ′. +We equip the essential RN Ness of (N,λ) with a canonical kinetics λess given by +λess,y→y′(x) = +∑ +r∈φ−1(y→y′) +λr(x) +for all y ��→ y′ ∈ φ(R) and x ∈ Zn +≥0, and call (Ness,λess) the essential stochastic reac- +tion network (essential SRN) of (N,λ). It holds that (N,λ) ⪰ (Ness,λess). Furthermore, +analogous to Lemma 2.5, we have the following lemma. +Lemma 2.8. Suppose (N ′,λ′) ⪰ (N,λ). Then, (N ′ +ess,λ′ +ess) = (Ness,λess). +Let Y be a unit rate Poisson process. Then, for s,t ≥ 0, Y (t + s) and Y ′(t) + Y ′′(s) have +the same distribution, where Y ′ and Y ′′ are independent copies of Y . Thus, given an SRN, +we have +X(t) = X(0) + +∑ +y→y′∈φ(R) +Yz→z′(∫ +t +0 λess,z→z′(X(s))ds)(y′ − y). +(2.3) +Every (weak) solution to (2.1) is also a (weak) solution to (2.3), and vice versa. Consequently, +the dynamic of any SRN is determined by its essential SRN: + +8 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +Proposition 2.9. Let (N,λ) and (N ′,λ′) be such that (Ness,λess) = (N ′ +ess,λ′′ +ess). Then, the +dynamics of (N,λ) and (N ′,λ′) are equivalent, in the sense that every weak solution to (2.1) +under (N,λ) is also a weak solution to (2.1) under (N ′,λ′), and vice versa. +In the following, we introduce definitions related to stationary distributions of SRNs. Let +x,x′ ∈ Zn +≥0 be two states. Then, x leads to x′ in N = (C,R,S,φ), written x →N x′, if there +exists a sequence of reactions y1 ��→ y′ +1,... ,ym ��→ y′m ∈ R, such that +(i) x ≥ φ(y1), +x − φ(y1) + φ(y′ +1) ≥ φ(y2), +... , +x + ∑m−1 +i=1 (φ(y′ +i) − φ(yi)) ≥ φ(ym). +(ii) x + ∑m +i=1(φ(y′ +i) − φ(yi)) = x′. +According to Condition 1, x →N x′ if and only if the probability to move from x to x′ is +positive. +A subset Γ ⊆ Zn +≥0 is an irreducible component of N, if for all x ∈ Γ and all x′ ∈ Zn +≥0, +x →N x′, if and only if x′ ∈ Γ. It follows that two distinct irreducible components are disjoint. +The proof of the next statement is elementary and thus omitted. +Lemma 2.10. Let N and N ′ be RNs such that Ness = N ′ +ess. Then, for any states x,x′ ∈ Zn +≥0, +x →N x′ if and only if x →N ′ x′. +As a consequence, a subset Γ ⊆ Zn +≥0 is an irreducible +component of N if and only if it is also an irreducible component of N ′. +Definition 2.11. Let (N,λ) be an SRN and Γ ⊆ Zn +≥0 an irreducible component of N. A +probability distribution π on Γ is a +(i) stationary distribution, if for all x ∈ Γ, +π(x) +∑ +y→y′∈R +λy→y′(x) = +∑ +y→y′∈R +π(x − φ(y′) + φ(y))λy→y′(x − φ(y′) + φ(y)), +where we set λy→y′(z) = 0 if z ∉ Zn +≥0 (same below). +(ii) complex balanced distribution, if for all complexes η ∈ C, and all x ∈ Γ, +π(x) +∑ +y′∶η→y′∈R +λη→y′(x) = +∑ +y∶y→η∈R +π(x + φ(y) − φ(η))λy→η(x + φ(y) − φ(η)). +(2.4) +A complex balanced probability distribution on Γ is also a stationary distribution on Γ [7]. +The complex balance property requires the digraph to be weakly reversible (all connected +components are strongly connected), that is, every reaction y ��→ y′ ∈ R belongs to a cycle +γ ⊆ R [8, 10]. +The following is a consequence of Proposition 2.9. +Corollary 2.12. Let (N,λ) and (N ′,λ′) be SRNs such that (Ness,λess) = (N ′ +ess,λ′ +ess). Then, +a probability distribution π is a stationary distribution on an irreducible component Γ of +(N,λ), if and only if π is also a stationary distribution on Γ of (N ′,λ′). +For complex balanced distributions, one implication follows from (2.2). + +COMPLEX BALANCED DISTRIBUTIONS +9 +Corollary 2.13. Let (N,λ) and (N ′,λ′) be SRNs such that (N ′,λ′) ⪰ (N,λ). If a probability +distribution π is a complex balanced distribution on an irreducible component Γ of (N ′,λ′), +then π is also a complex balanced distribution on Γ of (N,λ). +The reverse implication is not true in general: +Example 2.14. Consider the SRN, +A +3 +��⇀ +↽�� +3 +B, +(2.5) +equipped with stochastic mass-action kinetics. Then, the following system with stochastic +mass-action kinetics +(A,1) +1 +��⇀ +↽�� +2 +(B,1), +(A,2) +2 +��⇀ +↽�� +1 +(B,2), +(2.6) +and φ((A,i)) = A and φ((B,i)) = B for i = 1,2, is a cleaved SRN of (2.5). The probability +distribution π(x) = MΓ +x! is a complex balanced distribution for (2.5) on any irreducible class +Γ, for some positive constant MΓ [3, Theorem 4.1]. However, it is not a complex balanced +distribution for the cleaved SRN (2.6). Even more holds, one can show that there are no +complex balanced distributions for (2.6). +3. A criteria for complex balanced distributions +Theorem 3.1. Let N = (C,R,S,φ) be a weakly reversible RN with connected components +L1,... ,Lℓ. Equip N with a stochastic kinetics λ, and let Γ ⊆ Zn +≥0 be an irreducible component +of N. Suppose the following properties hold. +(i) There exist positive constants {κy→y′∶y ��→ y′ ∈ R}, such that for every complex η ∈ C, +∑ +y′∶η→y′∈R +κη→y′ = +∑ +y∶y→η∈R +κy→η. +(3.1) +(ii) There exist functions g∶Γ → R≥0 and mk∶Γk → R>0, k = 1,... ,ℓ, where +Γk = {x − φ(y)∶x ∈ Γ,y ∈ Lk} ∩ Zn +≥0, +(3.2) +such that for all y ��→ y′ ∈ R and x ∈ Γ satisfying x ≥ φ(y), +κy→y′ +λy→y′(x) = mk(x − φ(y))g(x). +(3.3) +If 0 < M ∶= ∑x∈Γ g(x) < ∞, then the distribution π, given by +π(x) = 1 +M g(x), +(3.4) +is a complex balanced distribution of (N,λ) on Γ. +Proof. The proof follows the idea of [3, Theorem 4.1]. By definition, to show that π defined +as in (3.4) is a complex balanced distribution, it suffices to verify that for any complex η ∈ C +and any x ∈ Γ, the following holds +g(x) +∑ +y′∶η→y′∈R +λη→y′(x) = +∑ +y∶y→η∈R +g(x + φ(y) − φ(η))λy→η(x + φ(y) − φ(η)). +(3.5) + +10 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +Due to Condition 1, we only need to prove (3.5) under the assumption that x ≥ φ(η). Note +that for any η ∈ C, all reactions such that η is a reactant or product are in the same connected +component. Let mη = mi, if η ∈ Li. Equation (3.3) yields that for all x ∈ Γ and x ≥ φ(y), +∑ +y∶y→η∈R +λy→η(x + φ(y) − φ(η))g(x + φ(y) − φ(η)) = ∑y∶y→η∈R κy→η +mη(x − φ(η)), +(3.6) +and +∑ +y′∶η→y′∈R +λη→y′(x)g(x) = ∑y′∶η→y′∈R κη→y′ +mη(x − φ(η)) . +(3.7) +Then, equality (3.5) follows from (3.1), (3.6) and (3.7). +□ +Theorem 3.1 can be generalized to complex balanced measures. +Moreover, it extends +the conditions given in [21] by allowing the functions mk to depend on k. Furthermore, +consider a mass-action SRN with a one-to-one labeling. +If it is complex balanced, then +the corresponding deterministic reaction network is complex balanced and the stationary +distributions have Poisson-product form [3, 8]. Define +g(x) = cx +x!, +mk(x) = 1 +cx, +κy→y′ = αy→y′cy, +where c ∈ Rn +>0 is an equilibrium of the ODE system of the corresponding deterministic RN and +αy→y′ are reaction rate constants. In this context, Theorem 3.1 corresponds to [3, Theorem +4.1]. +A natural question is whether all complex balanced distributions take the form (3.3). +Suppose it is true. Then, we immediately get that the ratio +λy→y′(x) +λy→y′′(x) of two reactions outgoing +from the same complex in the same connected component is a constant. Though this is +satisfied for mass-action kinetics, it is not the case in general. We show the necessity of (3.3) +for SRNs consisting of disjoint cycles (Proposition 3.2) and if the ‘constant ratio’ condition +holds (Proposition 3.3). Observe that condition (i) in Theorem 3.1 disappears in Proposition +3.2. +Proposition 3.2. Let (N,λ) be an SRN consisting of ℓ disjoint cycles. Then, a probability +distribution π is complex balanced on an irreducible component Γ, if and only if there exist +positive functions mk∶Γk → R>0 for Γk defined in (3.2), k = 1,... ,ℓ, such that for any y ��→ +y′ ∈ Lk, and all x ∈ Γ with x ≥ φ(y), we have +π(x) = [λy→y′(x)mk(x − φ(y))]−1. +(3.8) +Proof. Under the given conditions, every complex has exactly one incoming reaction and one +outgoing reaction. By Theorem 3.1, it is enough to show that if there is a complex balanced +stationary distribution, then it has to satisfy (3.8). Without loss of generality, assume ℓ = 1. +Then, there exists an integer p ≥ 2, such that +R = {yi ��→ yi+1∶i = 1,... ,p; yk ≠ yj,1 ≤ k < j ≤ p; yp+1 = y1}. + +COMPLEX BALANCED DISTRIBUTIONS +11 +Since π is a complex balanced distribution on Γ, then, by definition, we have, +π(x)λyi→yi+1(x) = π(x + φ(yi−1) − φ(yi))λyi−1→yi(x + φ(yi−1) − φ(yi)) +(3.9) +for all i = 1,... ,p (by convention y0 = yp+1) and x + φ(y1) − φ(y2) ∈ Γ. We define the function +m as follows. For all x ∈ Nn +0 such that x + φ(y1) ∈ Γ, we let +m(x) = [π(x + φ(y1))λy1→y2(x + φ(y1))]−1, +Thus, π(x) = [λy1→y2(x)m(x − φ(y1))]−1 for all x ∈ Γ with x ≥ y1. On the other hand, using +(3.9), we get +π(x)λy2→y3(x) = π(x + φ(y1) − φ(y2))λy1→y2(x + φ(y1) − φ(y2)) = m(x − φ(y2))−1, +(3.10) +for all x ∈ Γ with x + y1 − y2 ∈ Γ and x ≥ y2. Since for all x ∈ Γ with x ≥ φ(y2), we have +x − φ(y2) + φ(y3) ∈ Γ and thus x − φ(y2) + φ(y4) ∈ Γ as well. By iteration and the fact that +yp+1 = y1, it follows that x − φ(y2) + φ(y1) ∈ Γ. Therefore, (3.10) holds for x ∈ Γ with x ≥ y2. +This implies that π(x) = [λy2→y3(x)m(x − φ(y2))]−1, for all x ∈ Γ with x ≥ φ(y2). Finally, by +iteration, (3.8) holds for all yi → yi+1, i = 1,... ,p. The proof is complete. +□ +Proposition 3.3. Let (N,λ) be a weakly reversible SRN consisting of ℓ connected compo- +nents, and let Γ be an irreducible component of N. Suppose that λy→y′(x) = αy→y′λ0y(x) on +x ∈ Γ for all y ��→ y′ ∈ R, where αy→y′ is a positive constant, and λ0 +y depends on the complex +y only. Then, a probability distribution π on Γ is a complex balanced distribution of (N,λ), if +and only if there exist non-negative functions mk, k = 1,... ,ℓ, such that for all y ��→ y′ ∈ R, +y ∈ Lk, and x ∈ Γ with x ≥ φ(y), we have +π(x) = κy→y′[λy→y′(x)mk(x − φ(y))]−1, +(3.11) +where κy→y′ are positive constants satisfying (3.1). +The proof of Proposition 3.3 is in Section 6.1, and relies on Theorem 3.4 below. +Let N be an RN and let N ′ ⪰ N with projection ψ. For any cycle γ ⊆ R′, we say γ +is simple when projected onto N, if ψ(γ) is a cycle in R = ψ(R′). Moreover, two cycles +γ,γ′ ⊆ R′ are called similar if ψ(γ) = ψ(γ′), when projected onto N. +Theorem 3.4. Let (N,λ) be a weakly reversible SRN. Then, there exists a cleaved SRN +(Ncyc,λcyc) of (N,λ) with projection ψcyc, such that the digraph of Ncyc consists of disjoint +cycles that are pairwise non-similar simple cycles when projected onto N, satisfying the fol- +lowing properties. +(i) For any cycle γ ⊆ Rcyc, ψcyc(γ) is a cycle in R. +(ii) For any cycle γ ⊆ R, there exists a unique cycle γ′ ⊆ Rcyc such that ψcyc(γ′) = γ. +(iii) A probability distribution π is a complex balanced distribution of (N,λ) on some irre- +ducible component Γ of N, if and of if it is that of (Ncyc,λcyc) on Γ. + +12 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +The proof is in Section 6.6, and is based on the iterative one-node cleaving procedure +outlined in Section 4. As a corollary of Proposition 3.2 and Theorem 3.4, a weakly reversible +SRN is complex balanced if and only if it can be decomposed to a cleaved SRN consisting of +cycles, such that Theorem 3.1 holds. In principle, one can therefore always use Proposition +3.2 to determine the stationary distribution of a complex balanced SRN by cleaving. +As the following example shows, the kinetics of Ncyc is not unique. +Example 3.5. Consider the mass-action SRN, +C. +2 +1 +A +2 +1 +B +1 +2 +(3.12) +On an arbitrary irreducible component Γ, π(x) = MΓ +(c∗)x +x! , is the unique complex balanced +distribution of (3.12), where c∗ = (1,1,1) and MΓ is a constant [3, Theorem 4.1]. +The +following SRN is a dynamically equivalent cleaved SRN with five disjoint cycles and mass- +action kinetics (details omitted), +(A,1) +α1 +α2 +(B,1), +(B,2) +α3 +α4 +(C,2), +(A,3) +α5 +α6 +(C,3), +(C,4) +α7 +(A,4) +α8 +(B,4) +α9 +and +(C,5), +α10 +(A,5) +α11 +(B,5) +α12 +where α1,... ,α12 are rate constants satisfying α1 = ⋅⋅⋅ = α6 = 1 − β, α7 = α8 = α9 = β, and +α10 = α11 = α12 = β +1, with β ∈ (0,1) arbitrary. Then, the cleaved SRN is complex balanced. +The iterative one-node cleaving procedure in Section 4 results in β = 1 +15. +4. Cleaving weakly reversible SRNs +In this section, we develop an iterative procedure to show that there exists a dynamically +equivalent cleaved SRN consisting of all cycles appearing in the original RN, while preserv- +ing the complex balancing property. This cleaving procedure enlarges the applicability of +Theorem 3.1 and is key to the proof of Proposition 3.3. +4.1. One-node cleaving. Let N = (C,R,S,φ) be a weakly reversible RN with stochastic +kinetics λ. Choose a complex z ∈ C with pz > 1 incoming reactions. We provide a method +to construct a cleaved SRN (N1,λ1) of (N,λ) such that the complex balancing property of +(N1,λ1) is the same as that of (N,λ), and such that z is replaced by pz complexes with only +one incoming reaction. Proofs are given in Section 6. +The one-node cleaving involves two steps. In the first step, we give a precise definition of +N1 = (C1,R1,S,φ1) and the projection ψ1, while in the second step, a kinetics is assigned to +N1. Step 1 is illustrated in Figure 1. + +COMPLEX BALANCED DISTRIBUTIONS +13 +y1 +y′ +1 +y1 +(z,1) +y′ +1 +N: +z +y′ +2 +N 1: +y′ +2 +y2 +y′ +3 +y2 +(z,2) +y′ +3 +Figure 1. One-node cleaving. The complex z is cleaved. A dashed arrow, e.g., +y′ +1 to y1, means that there exists a path directed from the initial to the terminal +complex without passing through z. Hence, since there is a cycle containing +y1 ��→ z ��→ y′ +1 and y1 ��→ z ��→ y′ +2, respectively, in N, it follows that +(z,1) ��→ y′ +1 and (z,1) ��→ y′ +2, respectively, in N1. For the same reason, +(z,2) ��→ y′ +2 and (z,1) ��→ y′ +3 in N1. Primed and unprimed complexes could +be the same, for example, y2 = y′ +2. +Step 1. Order the incoming reactions of z by y1 ��→ z, ... , ypz ��→ z. Define +C1 = {y∶y ∈ C} ∖ {z} ∪ {(z,i)∶1 ≤ i ≤ pz}, +and R1 = R0 +1 ∪ Rin +1 ∪ Rout +1 , where +R0 +1 = {y ��→ y′ ∈ R∶y,y′ ∈ C ∖ {z}}, +Rin +1 = {yi ��→ (z,i)∶1 ≤ i ≤ pz}, +and Rout +1 +is the collection of all directed edges (z,i) ��→ y for some i ∈ {1,... ,pz} such that +there exists a cycle γ in R and y ∈ C ∖ {z} with {yi ��→ z ��→ y} ⊆ γ. By weak reversibility +of N, there is at least one i such that {yi ��→ z ��→ y} is contained in a cycle of N. +We remark that {yi ��→ z ��→ y} ⊆ R does not imply {yi ��→ (z,i) ��→ y} ⊆ R1. For +example, let R = {yi ��⇀ +↽�� z ��⇀ +↽�� y}. Then, R is weakly reversible and there is a closed walk +yi ��→ z ��→ y ��→ z ��→ yi, including {yi ��→ z ��→ y}. But {yi ��→ z ��→ y} is not in +any cycle of R, and thus {yi ��→ (z,i) ��→ y} /⊆ R1. +Finally, we define the labelling φ1 = φ ○ ψ1 with ψ1 the canonical projection on C1 given by +ψ1(y) = +⎧⎪⎪⎨⎪⎪⎩ +y, +for +y ∈ C ∖ {z}, +z, +for +y = (z,i), +i = 1,... ,pz. +(4.1) +Lemma 4.1. Let N be weakly reversible, and let N1 and ψ1 be a one-node cleaved RN of N. +Then, N1 ⪰ N with projection ψ1, and N1 is weakly reversible as well. + +14 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +Step 2. +We assign a kinetics λ1 to N1, such that (N1,λ1) ⪰ (N,λ) and the complex +balancing property is maintained. To complete the task, we introduce some notation. Let +z1,z2,z3 ∈ C be any, possibly repeated, complexes such that {z1 ��→ z2 ��→ z3} ⊆ R. Denote +by Γz1→z2→z3(k), k ∈ Z>0, the collection of closed walks in R of the form +γ = {z1 ��→ z2 ��→ z3 ��→ y(1) ��→ ⋯ ��→ y(k) ��→ z1} ⊆ R, +satisfying {y(1),... ,y(k)} ∩ {z2} = ∅. For z1 ≠ z3, define +Γz1→z2→z3(0) = +⎧⎪⎪⎨⎪⎪⎩ +{z1 ��→ z2 ��→ z3 ��→ z1}, +z3 ��→ z1 ∈ R +∅, +z3 ��→ z1 ∉ R, +and Γz1→z2→z1(0) ∶= {z1 ��⇀ +↽�� z2}. By convention, Γz1→z2→z3(k) = ∅ for k ∈ Z≥0 if {z1 ��→ +z2,z2 ��→ z3} /⊆ R. Define +Γz1→z2→z3 = ∪∞ +k=0Γz1→z2→z3(k). +Furthermore, define ρz3,z1→z2∶Zn +≥0 → R≥0 for all z1,z2,z3 ∈ C and x ∈ Zn +≥0 by +ρz3,z1→z2(x) = +⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ +λz1→z2(x + φ(z1) − φ(z3)) +∑y′′∶z1→y′′∈R λz1→y′′(x + φ(z1) − φ(z3)), +z1 ��→ z2 ∈ R, +0, +z1 ��→ z2 ∉ R, +(4.2) +where by convention 0 +0 = 0. Using Condition 1, for z1 ��→ z2 ∈ R, it holds that λz1→z2(x + +φ(z1) − φ(z3)) > 0 if and only if x + φ(z1) − φ(z3) ≥ φ(z1) or equivalently, if and only if +x ≥ φ(z3). Thus, +ρz3,z1→z2(x) > 0, +if and only if +x ≥ φ(z3). +(4.3) +Define the kinetics λ1 as: +λ1,r(x) = +⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ +λy→y′(x), +r = y ��→ y′ ∈ R0 +1, +λyi→z(x), +r = yi ��→ (z,i) ∈ Rin +1 , +∑ +γ∈Γyi→z→y′ +∏ +r′∈γ∖{z→y′} +ρz,r′(x)λz→y′(x), +r = (z,i) ��→ y′ ∈ Rout +1 , +(4.4) +for all x ∈ Zn +≥0. Using (4.3), Condition 1 holds for (N1,λ1) as well. +Lemma 4.2. Let (N,λ) be a weakly reversible SRN, and (N1,λ1) and ψ1 be a one-node +cleaved SRN (defined above) of (N,λ). Then, (N1,λ1) ⪰ (N,λ) with projection ψ1. +By Lemma 2.8 and Proposition 2.9, a stationary distribution of (N,λ) is also a station- +ary distribution of (N1,λ1) and vice versa. Furthermore, we have preservation of complex +balanced distributions in both directions along one-node cleavings: +Lemma 4.3. Let (N,λ) be a weakly reversible SRN, and (N1,λ1) and ψ1 be a one-node +cleaved SRN (defined above) of (N,λ). Then a probability distribution π on an irreducible + +COMPLEX BALANCED DISTRIBUTIONS +15 +component Γ is a complex balanced distribution of (N,λ) if and only if it is a complex balanced +distribution of (N1,λ1) on Γ. +Example 4.4. We illustrate the one-node cleaving procedure on Example 3.5. The node A +is cleaved. There are two incoming reactions of A in N leading to two new nodes (A,1) and +(A,2), R0 +1 = {B ��⇀ +↽�� C} and Rin +1 = {B ��→ (A,1),C ��→ (A,2)} in N1. Since there are two +cycles in N including B ��→ A, namely {B ��→ A ��→ B} and {B ��→ A ��→ C ��→ B}, +then {(A,1) ��→ B,(A,1) ��→ C} ⊆ Rout +0 . Similarly, we find {(A,2) ��→ B,(A,2) ��→ +C} ⊆ Rout +0 , and thus Rout +1 += {(A,1) ��→ B,(A,1) ��→ C,(A,2) ��→ B,(A,2) ��→ C}. +Consequently, the digraph of N1 is as shown below. +N ∶ +C +A +B +�⇒ +N1 ∶ +(A,1) +C +(A,2). +B +Concerning the kinetics of N1, let x = (xA,xB,xC) ∈ Z3 +≥0 denote the molecular counts of the +species A, B and C, respectively. Using (4.4), it suffices to calculate λ1,(A,i)→B and λ1,(A,i)→C, +i = 1,2. Consider (A,1) ��→ B ∈ Rout +1 . The closed walks of ΓB→A→B in N are of the form +θk = {B ��→ A ��→ B ��→ C ��→ B ��→ ⋯ ��→ C ��→ B}, +where k ≥ 0 denotes the number of occurrences of C ��→ B. As both B and C have each +two outgoing reactions in N, ρA,B→C(x) < 1 and ρA,C→B(x) < 1, and so +λ1,(A,1)→B(x) = +∞ +∑ +k=0 +ρA,B→A(x)(ρA,B→C(x)ρA,C→B(x)) +kλA→B(x) += +λA→B(x)ρA,B→A(x) +1 − ρA,B→C(x)ρA,C→B(x). +Similarly, +λ1,(A,1)→C(x) = λA→C(x)ρA,B→A(x)ρA,C→B(x) +1 − ρA,B→C(x)ρA,C→B(x) +, +λ1,(A,2)→B(x) = λA→B(x)ρA,B→C(x)ρA,C→A(x) +1 − ρA,B→C(x)ρA,C→B(x) +, +λ1,(A,2)→C(x) = +λA→C(x)ρA,C→A(x) +1 − ρA,B→C(x)ρA,C→B(x). +By (4.2), if xA ≥ 1, then ρA,B→A(x) + ρA,B→C(x) = ρA,C→A(x) + ρA,C→B(x) = 1. By sim- +plification, this implies that λ1,(A,1)→B(x) + λ1,(A,2)→B(x) = λA→B(x), and λ1,(A,1)→C(x) + +λ1,(A,2)→C(x) = λA→C(x). Therefore, (2.2) holds for all y ��→ y′ ∈ R and x ∈ Z≥0. As a result, +(N1,λ1) ⪰ (N,λ). + +16 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +4.2. Iteration. We apply the one-node cleaving procedure iteratively until every complex +has at most one incoming reaction, and the cleaved RN consists of only cycles (Lemma +2.1). However, as illustrated in Figure 1, when cleaving a complex (here, z), the number of +incoming reactions of other complexes (here, y′ +2) might increase. Thus, we should not expect +that there is an iterative procedure based on one-node cleaving, such that the number of +complexes with multiple incoming reactions is strictly decreasing. +Let N = (C,R,S,φ) be a weakly reversible RN and let C′ ⊆ C be the collection of complexes +in C with a single incoming reaction. Suppose that C′ ≠ C (otherwise the RN consists of +disjoint cycles, and we are done) and let C′′ = C ∖C′. Write N0 = (C0,R0,S,φ0) for the cleaved +RN of N with projection ψ0 obtained by one-node cleaving of an arbitrary node z ∈ C′′. +Define +C′ +0 = {y ∈ C0∶ψ0(y) ∈ C′ ∪ {z}} = C′ ∪ {y ∈ C0∣ψ0(y) = z} +and +C′′ +0 = C0 ∖ C′ +0. +With Figure 1 as an example, we have {(z,1),(z,2),y′ +1,y′ +2,y′ +3} ⊆ C′ +0, and y′ +2 has two incoming +reactions. Moreover, since z ∈ C′′ and C′′ +0 = C′′ ∖ {z}, then C′′ +0 has exactly one complex less +than C′′. R0 are the reactions of the cleaved RN of N (defined in step 1). +We next define a sequence of cleaved RNs, see Figure 2. For m ≥ 1, let Nm = (Cm,Rm,S,φm) +with projection ψm be an RN obtained by cleaving an element of C′ ∩ Cm−1 in Nm−1 with +multiple incoming reactions (again Rm are the reactions of the cleaved RN of Nm−1 as defined +in step 1). Concretely, let ψm +0 = ψ0 ○ ⋅⋅⋅ ○ ψm, +C′ +m = {y ∈ Cm∶ψm +0 (y) ∈ C′ ∪ {z}} ⊆ Cm, +and +C′′ +m = Cm ∖ C′ +m. +If all y ∈ C′ ∩ Cm−1 ⊆ C′ +m−1 have only one incoming reaction in Rm−1, then Nm = Nm−1 (and +ψm = idCm−1). Hence, Nm ⪰ Nm−1 with projection ψm, and Nm ⪰ N with projection ψm +0 . The +procedure ends after M = ∣C′∣ iterations. +Lemma 4.5. Every complex in C′ +M ⊆ CM has only one incoming reaction in RM. +After completing the M-th iteration, we obtain a cleaved RN NM = (CM,RM,S,φM) of +N with projection ψM, such that each complex in C′ +M ⊆ CM has only one incoming reaction, +and C′′ +M has one fewer complexes than C′′, namely C′′ +M = C′′ ∖ {z}. However, the number of +incoming reactions of a complex y ∈ C′′ +M might be different from the corresponding number +of incoming reactions of the complex ψM +0 (y) = y ∈ C′′ in R. +By repeating this procedure for another complex z′ ∈ C′′ +M and so forth, we eventually obtain, +after finitely many iterations, a cleaved SRN (Ncyc,λcyc) with projection ψcyc on N. Every +complex in the cleaved SRN has only one incoming reaction. Hence, the cleaved SRN consists +of disjoint cycles (Lemma 2.1). Furthermore, it fulfils the complex balancing property if and +only if (N,λ) fulfils it (Lemma 4.3). +4.3. Completion. We modify the cleaved SRN (Ncyc,λcyc) to obtain another cleaved SRN of +(N,λ) without non-simple cycles and similar cycles when projected onto N. The modification + +COMPLEX BALANCED DISTRIBUTIONS +17 +N: +z +y1 +y2 +z′ +C′ +C +N0: +(z,2) +y1 +(z,1) +y2 +z′ +C′ +0 +C0 +N1: +(z,2) +(y1,2) +(z,1) +(y1,1) +y3 +z′ +C′ +2 +C2 +N2: +(z,2) +(y1,2) +(y3,2) +(z,1) +(y1,1) +(y3,1) +z′ +C′ +3 +C3 +Figure 2 +includes two steps. In the first step, we cut and adhere non-simple cycles, and in the second +step, we combine similar cycles (for definitions see just before Theorem 3.4). +Suppose there exists a cycle γ ⊆ Rcyc that is not simple when projected onto N. Then, it +is of the form +γ = {y0 ��→ y1 ��→ ⋯ ��→ yk ��→ y′ +0 ��→ yk+1 ��→ ⋯ ��→ yk+k′ ��→ y0}, +where y0 ≠ y′ +0 and ψcyc(y0) = ψcyc(y′ +0). We cut this cycle at y0 and y′ +0, then adhere each piece +with its end node. Thus, we get two cycles, +γ1 = {y0 ��→ y1 ��→ ⋯ ��→ yk ��→ y0}, +γ2 = {y′ +0 ��→ yk+1 ��→ ⋯ ��→ yk+k′ ��→ y′ +0}. +In this way, we obtain a new cleaved RN N ′ +cyc = (C′ +cyc,R′ +cyc,S,φ′ +cyc) of N with projection +ψ′cyc = ψcyc, where C′cyc = Ccyc, +R′ +cyc =(Rcyc ∖ {yk ��→ y′ +0,yk+k′ ��→ y0}) ∪ {yk ��→ y0,yk+k′ ��→ y′ +0}. +It is natural to assign a kinetics λ′ +cyc to N ′ +cyc by keeping the same kinetics for the reactions +also appearing in Rcyc, and letting +λ′ +cyc,yk→y0 = λcyc,yk→y′ +0, +λ′ +cyc,yk+k′→y′ +0 = λcyc,yk+k′→y0. +Then, (N ′cyc,λ′cyc) ⪰ (N,λ) with projection ψcyc, such that the complex balancing property +remains. Note that (Ncyc,λcyc) and (N ′ +cyc,λ′ +cyc) may not be related by ⪰. + +18 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +The ‘cut-adhere’ process can be accomplished in finitely many steps until every cycle is +simple when projected onto N. By abuse of notation, the final cleaved SNR is also denoted +by (Ncyc,λcyc) with projection ψcyc. +In the second step, we combine similar cycles. Suppose there are two similar cycles γ1,γ2 ⊆ +Rcyc when projected onto N, that is, ψcyc(γ1) = ψcyc(γ2). We simply remove γ2 and sum the +kinetics of each reaction in γ2 to the corresponding reaction in γ1. More precisely, suppose +γ1 = {y1 ��→ ⋯ ��→ yk ��→ y1}, +γ2 = {y′ +1 ��→ ⋯ ��→ y′ +k ��→ y′ +1}, +with yi ≠ y′ +i and ψcyc(yi) = ψcyc(y′ +i) for all i = 1,... ,k. Then, we construct a new cleaved RN +(N ′ +cyc,λ′ +cyc) of (N,λ) with ψ′ +cyc being a restriction of ψcyc on C′ +cyc = Ccyc ∖ {(yj,i′ +j)∶1 ≤ j ≤ k}, +where R′cyc = Rcyc ∖ γ2, the labelling φ′cyc is a restriction of φcyc on C′cyc, and the kinetics λ′cyc +is defined as follows, +λ′ +cyc,r = +⎧⎪⎪⎨⎪⎪⎩ +λcyc,r, +r ∈ Rcyc ∖ (γ1 ∪ γ2), +λcyc,yj→yj+1 + λcyc,y′ +j→y′ +j+1, +r = yj ��→ yj+1 ∈ γ1. +Then (N ′ +cyc,λ′ +cyc) ⪰ (N,λ) with projection ψ′ +cyc, and (N ′ +cyc,λ′ +cyc) fulfils the complex balancing +property if and only if (N,λ) fulfils it. Here, (Ncyc,λcyc) ⪰ (N ′cyc,λ′cyc). +This process can be iterated finitely many times until all disjoint cycles are non-similar +when projected onto N. By abuse of notation, the resulting cleaved SRN of (N,λ) is also +denoted by (Ncyc,λcyc) with projection ψcyc. +5. An example +The following example is a modification of a classical birth-death process that has an +extra reaction with a jump of size two [2]. More precisely, we consider the following SRN +with mass-action kinetics, which is not weakly reversible, +A +α1 +��⇀ +↽�� +α2 +∅ +α3 +��→ 2A. +(5.1) +To apply Theorem 3.1 we need to find an equivalent weakly reversible SRN. Changing the +reaction A ��→ ∅ to A ��→ ∅ and 2A ��→ A, then we look for a dynamically equivalent +SRN of the following form, +A +λ1 +λ2 +∅ +λ3 +2A +λ4 +(5.2) +λ2(x) = α2, +λ3(x) = α3, +λ4(x) + λ1(x) = α1x, +where x denotes the number of A molecules. We will show that λ1 and λ4 are uniquely +determined for any α1,α2,α3 ∈ R>0 such that (5.2) is complex balanced and Condition 1 is +satisfied. + +COMPLEX BALANCED DISTRIBUTIONS +19 +The SRN (5.2) fulfils the ‘constant ratio’ condition in Proposition 3.3, hence Theorem 3.1 +can be applied to justify complex balancedness. However, we prefer to decompose it into +cycles to avoid the difficulty of choosing the constants κ’s in (3.1) by using Proposition 3.2. +Cleaving the SRN into cycles results in +L1 ∶ +(A,1) +λcyc,1 +���⇀ +↽��� +λcyc,2 (∅,1), +L2 ∶ +(A,2) +λcyc,5 +(∅,2) +λcyc,3 +(2A,2), +λcyc,4 +(5.3) +where λcyc,i = λi for i = 2,3,4, +λcyc,1(x) = +α2 +α2 + α3 +λ1(x), +and +λcyc,5(x) = +α3 +α2 + α3 +λ1(x). +Due to Proposition 3.2 and Theorem 3.4, the SRN (5.2) is complex balanced, if and only if +there exist non-negative functions m1, m2 and g on Z≥0, such that +λcyc,1(x + 1)g(x + 1) = λcyc,2(x)g(x) = m1(x)−1, +(5.4) +and +λcyc,3(x)g(x) = λcyc,4(x + 2)g(x + 2) = λcyc,5(x + 1)g(x + 1) = m2(x)−1 +(5.5) +for all x ∈ Z≥0 and M ∶= ∑∞ +x=1 g(x) ∈ (0,∞). If we choose +m1(x) = α3m2(x)/α2, +(5.6) +then (5.4) is a consequence of (5.5), and we only need to solve for (5.5). Suppose (5.5) holds. +For all x ≥ 1, it follows that +λcyc,3(x) +λcyc,5(x + 1) = +λcyc,5(x) +λcyc,4(x + 1), +and thus, +λ1(x + 1) = α1(α2 + α3)2(x + 1) +(α2 + α3)2 + α3λ1(x). +Condition 1 gives λ4(0) = λ4(1) = 0. Thus, λ1(1) = α1 and λ1 is uniquely determined by the +recursion: +λ1(x) = +⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ +0, +x = 0, +α1(α2 + α3)2x +(α2 + α3)2 + α3λ1(x − 1), +x > 0. +(5.7) +In fact, in (5.7), 0 < λ1(x) < α1x whenever λ1(x − 1) > 0. Therefore, λ1(x) and λ4(x) = +α1x − λ1(x) are in (0,α1x) for all x ≥ 2, and Condition 1 holds for λ1 and λ4. +Assume +g(0) = 1, then combined with (5.5), we have +g(x + 1) = +λcyc,3(x) +λcyc,5(x + 1)g(x) = α2 + α3 +λ1(x + 1)g(x) = (α2 + α3)x+1( +x+1 +∏ +u=1 +λ1(u)) +−1 +(5.8) + +20 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +and +m2(x) = (α3g(x))−1, +(5.9) +for all x ≥ 0. With λ1, m1 m2 and g defined as in (5.7), (5.6), (5.9) and (5.8), respectively, +one can verify that Theorem 3.1(ii) is satisfied. +To prove the existence of a complex balanced distribution, we need to show M ∶= ∑∞ +x=1 g(x) < +∞. By using the recursive formula (5.7), we deduce that for all x ≥ 1, +λ1(x)λ1(x + 1) = (x + 1)h(λ1(x)), +where +h(u) ∶= +α1(α2 + α3)2u +(α2 + α3)2 + α3u, +u ∈ R≥0. +Due to (5.7) and the fact that 0 ≤ λ1(x) ≤ α1x, we have for x ≥ 2, +λ1(x) ≥ +α1(α2 + α3)2x +(α2 + α3)2 + α3α1(x − 1) ≥ +α1(α2 + α3)2(x − 1) +(α2 + α3)2 + α3α1(x − 1) ≥ c0 ∶= +α1(α2 + α3)2 +(α2 + α3)2 + α3α1 +, +where the last inequality follows from the property that x ↦ +ax +c+bx is increasing on R≥0 with +arbitrary parameters a,b,c > 0. Since h is also increasing on R≥0, it holds that for all x ≥ 2, +λ1(x)λ1(x + 1) ≥ (x + 1)inf +x≥2 h(λ1(x)) ≥ h(c0) > 0. +As a consequence, for x ≥ 2(α2+α3)2 +h(c0) +∨ 2, +g(x + 1) +g(x − 1) = +(α2 + α3)2 +λ1(x + 1)λ1(x) ≤ (α2 + α3)2 +(x + 1)h(c0) < 1 +2, +and M is finite by the ratio test. Due to Proposition 3.2 and Theorem 3.4, π(x) ∶= +1 +M g(x) +is the unique complex balanced distribution for (5.3) and also (5.2), and thus a stationary +distribution for (5.1). From [36], the reaction network (5.1) is positive recurrent, hence this +distribution is the unique stationary distribution. +6. Proofs +6.1. Proofs of Proposition 3.3. As a consequence of Theorem 3.1, we only need to show +one direction. Suppose that π is a complex balanced distribution of (N,λ) on an irreducible +component Γ. +Recall the assumption that λy→y′ = αy→y′λ0 +y on Γ for all y ��→ y′. For any η ∈ C and r ∈ R, +the function ρη,r in (4.2) is a positive constant on {x ∈ Γ∶x ≥ φ(η)}. Therefore, λ1,r in (4.4) +fulfils λ1,r(x) = c(r)λψ1(r)(x) for some constant c(r). After iteration and completion as in +Section 4, we find a cleaved SRN (Ncyc,λcyc) of (N,λ) with projection ψcyc, such that +λcyc,r(x) = c(r)λψcyc(r)(x), +for all r ∈ Rcyc and x ∈ Γ with positive constants {c(r′),r′ ∈ Rcyc}. Choose any r = (y,i) ��→ +(y′,i′) ∈ Rcyc, where (y,i),(y′,i′) ∈ Ccyc, such that ψcyc(y,i) = y and ψcyc(y′,i′) = y′. Suppose +that r is in the k-th connected component (cycle) of Ncyc. Using Theorem 3.4 and Proposition + +COMPLEX BALANCED DISTRIBUTIONS +21 +3.2, since π is a complex balanced distribution of (N,λ) (and thus of (Ncyc,λcyc)) on Γ, we +have +π(x) = [λcyc,r(x)mcyc,k(x − φcyc(y,i))]−1 = [c(r)λy→y′(x)mcyc,k(x − φ(y))]−1, +(6.1) +for all x ∈ Γ. Then, the proposition follows if we can show that the ratio mj1,cyc/mj2,cyc is a +constant on Γj (see (3.2)) for any indices j1, j2 and j, such that the j1-th and j2-th cycles +in Ncyc are both included in the j-th connected component when projected onto N. The +following lemma follows from weak reversibility and the proof is omitted. +Lemma 6.1. Let N be a weakly reversible RN consisting of connected components L1,... ,Ll. +Suppose Γ ⊆ Rn is an irreducible component of N. For any j ∈ {1,... ,l}, let Γj be given as +in (3.2). Then, Γj = {x − φ(y)∶x ∈ Γ} ∩ Zn +≥0, where y is an arbitrary complex in Lk. +For ι = 1,2, let rι = (yι,iι) ��→ (y′ +ι,i′ +ι) be in the jι-th cycle of Ncyc, written as rι ∈ Lcyc,jι. +By convention, we assume ψcyc(rι) = yι ��→ y′ι. Furthermore, suppose that ψcyc(r1) and +ψcyc(r2) are both in the j-th connected component of N. +Case 1) +Suppose y1 = y2. By assumption, λy1→y′ +1/λy2→y′ +2 = αy1→y′ +1/αy2→y′ +2 is a positive +constant on Γ. Moreover, due to equation (6.1), it holds for every x ∈ Γ with x − φ(y1) ∈ Zn +≥0, +1 = π(x) +π(x) = c(r1)λy1→y′ +1(x)mcyc,j1(x − φ(y1)) +c(r2)λy2→y′ +2(x)mcyc,j2(x − φ(y2)) = c(r1)αy1→y′ +1mcyc,j1(x − φ(y1)) +c(r2)αy2→y′ +2mcyc,j2(x − φ(y2)). +By assumption y1 = y2, and performing a change of variable z = x − φ(y1) = x − φ(y2), we get +mcyc,j1(z) +mcyc,j2(z) = c(r2)αy2→y′ +2 +c(r1)αy1→y′ +1 +, +(6.2) +is a positive constant, for every z ∈ Γj such that z = x−φ(y1) with some x ∈ Γ. Taking Lemma +6.1 into account, the identity (6.2) holds for all z ∈ Γj. +Case 2) Suppose that y′ +1 = y2. Consider reaction r2 and the outgoing reaction of (y′ +1,i′ +1): +r′ +1 = (y′ +1,i′ +1) ��→ (y′′ +1,i′′ +1) ∈ Lcyc,j1. Then, by application of Case 1, we immediately get that +mcyc,j1(z) +mcyc,j2(z) = c(r2)αy2→y′ +2 +c(r′ +1)αy′ +1→y′′ +1 +, +for all z = x − φ(y2) = x − φ(y′ +1) with x ∈ Γ, and thus for all z ∈ Γj. +Case 3) Suppose y1 = y′ +2. One can verify that mcyc,j1(z)/mcyc,j2(z) is a positive constant +on Γj for every z ∈ Γj following the same lines as in Case 2. +Case 4) Suppose y′ +1 = y′ +2. Consider the reactions r′ +1 = (y′ +1,i′ +1) ��→ (y′′ +1 ,i′′ +1) ∈ Lcyc,j1 and +r′ +2 = (y′ +2,i′ +2) ��→ (y′′ +2,i′′ +2) ∈ Lcyc,j2, where ψcyc(y′′ +ι ,i′′ +ι ) = y′′ +ι for ι = 1,2. Using Case 1, +mcyc,j1(z) +mcyc,j2(z) = c(r′ +2)αy′ +2→y′′ +2 +c(r′ +1)αy′ +1→y′′ +1 +, +for all z = x − φ(y′ +1) = x − φ(y′ +2) with x ∈ Γ, and thus for all z ∈ Γj. +Remaining cases. Since ψcyc(r1) and ψcyc(r2) are both in the j-th connected component +in N, we can find y(1),... ,y(k) ∈ C with y(1) = y1 and y(k) = y2, such that for all i = 1,... ,k−1, + +22 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +either y(i) ��→ y(i+1) or y(i+1) ��→ y(i) in R. This yields that for some indexes q1,q′ +1,... ,qk,q′ +k, +we have (y(i),qi) ��→ (y(i+1),q′ +i) or (y(i+1),qi) ��→ (y(i),q′ +i) in the j′ +i-th connected component +in Ncyc for all i = 1,... ,k − 1. +It follows from Cases 1-4, that mcyc,j1(z)/mcyc,j′ +1(z) = c0, +mcyc,j′ +i(z)/mcyc,j′ +i+1(z) = ci, i = 1,... ,k − 2, and mcyc,j′ +k−1(z)/mcyc,j2(z) = ck−1 for all z ∈ Γj +with some positive constants c0,... ,ck−1. Thus, mcyc,j1(z)/mcyc,j2(z) = ck with some positive +constant ck for all z ∈ Γj. +Therefore, π can be written as in the form (3.11) with appropriate positive constants κy→y′. +Moreover, (3.1) is a direct result of (2.4) and (3.11). The proof is complete. +6.2. Proof of Lemma 4.1. We first prove that N1 ⪰ N with projection ψ1. By definition, +it suffices to show that (C,R) = (ψ1(C1),ψ1(R1)). In fact, due to (4.1), we have +ψ1(C1) = (C ∖ {z}) ∪ {z} = C. +By definition of R1, we have ψ1(R1) ⊆ R. To prove the reverse inclusion, we decompose +R = R0 ∪ Rin ∪ Rout, where R0 consists of reactions whose reactant and product are both +in C ∖ {z}, and Rin and Rout consist of the incoming and outgoing reactions of z in R, +respectively. Then, ψ1(R0 +1) = R0 +1 = R0 and ψ1(Rin +1 ) = {yi ��→ z∶1 ≤ i ≤ pz} = Rin. Recall +that N is weakly reversible. +Thus, for every y ∈ C such that z ��→ y ∈ R, there exists +a cycle containing z ��→ y, and the incoming reaction of z in this cycle is yj ��→ z for +some 1 ≤ j ≤ pz. Then, (z,j) ��→ y ∈ Rout +1 , and thus z ��→ y ∈ ψ1(Rout +1 ). This implies +Rout ⊆ ψ1(Rout +1 ). Thus, N1 ⪰ N with ψ1. +Next, we show weak reversibility of N1. Suppose that y ��→ y′ ∈ R0 +1. Then, y ��→ y′ ∈ R. +By weak reversibility of R, there exists a cycle γ ⊆ R containing y ��→ y′. If z ∉ γ, then +γ ⊆ R0 +1, and we are done. Otherwise, suppose z ∈ γ, then there exist i ∈ {1,... ,pz} and y′ ∈ +C∖{z}, such that {yi ��→ z ��→ y′} ⊆ γ ⊆ R. As a consequence, {yi ��→ (z,i) ��→ y′} ⊆ R1. +Replacing z by (z,i) in γ, we get a new cycle γ′ ⊆ R1. For reactions in Rin +1 or Rout +1 , the same +idea is applicable and the details are omitted. The proof is complete. +6.3. Proof of Lemma 4.2. Due to Lemma 4.1, it suffices to show that λ1 satisfies (2.2). In +fact, by definition of λ1, we only need to prove that +λz→y′(x) = +pz +∑ +i=1 +λ1,(z,i)→y′(x) +(6.3) +for y′ ∈ C with z ��→ y′ ∈ R and x ∈ Zn +≥0. Using (4.4) and Condition 1 on (N,λ), then (6.3) +is equivalent to +1{x′∶x′≥φ(z)}(x) = +pz +∑ +i=1 +∑ +γ∈Γyi→z→y′ +∏ +r∈γ∖{z→y′} +ρz,r(x), +(6.4) +which is what we will prove. First, if x /≥ φ(z), by (4.3), both sides of (6.4) are equal to zero. + +COMPLEX BALANCED DISTRIBUTIONS +23 +Hence assume that x ≥ φ(z). Let X be the set of all complexes in C that are in the same +connected component as z. Then, weak reversibility, Condition 1 and (4.2) imply that for +any z1,z2 ∈ X , +(i) φ(z1) →N φ(z2). +(ii) ρz,z1→z2(x) > 0 if and only if z1 ��→ z2 ∈ R. +(iii) ∑ +z′∈X +ρz,z1→z′(x) = +∑ +z′∶z1→z′∈R +ρz,z1→z′(x) = 1. +This observation allows us to define a discrete time Markov chain (DTMC) on X with tran- +sition probability Pz1(z2) = ρz,z1→z2(x) for all z1,z2 ∈ X . Moreover, the chain is irreducible +with finite state space. Therefore, it follows from [31, Theorems 1.5.6 and 1.5.7] that the +chain is recurrent, and thus, +Py′(τz < ∞) = +pz +∑ +i=1 +∑ +γ∈Γyi→z→y′ +∏ +r∈γ∖{z→y′} +ρz,r(x) = 1, +where τz denotes the first hitting time to state z. This proves (6.4) and thus completes the +proof of Lemma 4.2. +6.4. Proof of Lemma 4.3. Due to Corollary 2.13 and Lemma 4.2, it suffices to prove one +direction. Suppose that π is a complex balanced distribution of (N,λ) on Γ, then we need +to verify (2.4) for (N1,λ1). Let η ∈ C ∖{z} ⊆ C1, then from (4.4), it follows that for any x ∈ Γ, +π(x) +∑ +y′∶η→y′∈R1 +λ1,r(x) =π(x)( +∑ +y′∶η→y′∈R0 +1 +λ1,y→y′(x) + +pz +∑ +i=1 +λ1,η→(z,i)(x)) +=π(x)( +∑ +y′∶η→y′∈R,y′≠z +λη→y′(x) + λη→z(x)) = π(x) +∑ +y′∶η→y′∈R +λy→y′(x). (6.5) +As π is complex balanced for (R,λ) and φ1 = φ ○ ψ1 = φ on C1 ∖ {(z,i)∶i = 1,... ,pz}, we have +π(x) +∑ +y′∶η→y′∈R +λy→y′(x) = +∑ +y∶y→η∈R +π(x + φ(y) − φ(η))λy→η(x + φ(y) − φ(η)) +(6.6) += +∑ +y∶y→η∈R0 +1 +π(x + φ1(y) − φ1(η))λy→η(x + φ1(y) − φ1(η)) ++ π(x + φ1(z) − φ1(η))λz→η(x + φ1(z) − φ1(η)). +Then, (2.4) is a consequence of (6.3), (6.5) and (6.6). Next, we will show (2.4) for η = (z,i) +with i ∈ {1,... ,pz}. Without loss of generality, assume i = 1. +By definition, y1 ��→ (z,1) is the only incoming reaction of (z,1). Therefore, if x /≥ φ(z), +then Condition 1 and the fact that φ1 = φ ○ ψ1 with ψ1 given by (4.1) yields +0 = +∑ +y′∶(z,1)→y′∈Rout +1 +π(x)λ1,(z,1)→y′(x) = π(x+φ1(y1)−φ1((z,1)))λy1→(z,1)(x+φ1(y1)−φ1((z,1))). + +24 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +Otherwise, assume x ≥ φ(z). Let X be the set of all complexes in C that are in the same +connected component of z. As (N,λ) is complex balanced under π, then for any η ∈ X , +0 < π(x′) +∑ +y′∶η→y′∈R +λη→y′(x′) = +∑ +y∶y→η∈R +π(x′ + φ(y) − φ(η))λy→η(x′ + φ(y) − φ(η)), +where x′ = x + φ(η) − φ(z) ≥ φ(η). This yields that +1 = ∑y∶y→η∈R π(x + φ(y) − φ(z))λy→η(x + φ(y) − φ(z)) +π(x + φ(η) − φ(z))∑y′∶η→y′∈R λη→y′(x + φ(η) − φ(z)). +We construct an irreducible DTMC taking values in the finite state space X with transition +probabilities +pz1,z2 = Pz1(z2) = +π(x + φ(z2) − φ(z))λz2→z1(x + φ(z2) − φ(z)) +π(x + φ(z1) − φ(z))∑y′∶z1→y′∈R λz1→y′(x + φ(z1) − φ(z)), +for any z1,z2 ∈ X . Then, pz1,z2 > 0 if and only if z2 ��→ z1 ∈ R. Thus, the chain is recurrent. +With τz denoting the first hitting time to state z, we have Py1(τz < ∞) = 1. +This proves (2.4) with η = (z,1), if it holds that +Py1(τz < ∞) = +∑(z,1)→y′∈R1 λ1,(z,1)→y′(x)π(x) +π(x + φ1(y1) − φ1((z,1))λ1,y1→(z,1)(x + φ1(y1) − φ1((z,1)). +(6.7) +First, by definition it is clear that +Py1(τz < ∞) = py1,z + +∑ +z′∈X∖{z} +py1,z′pz′,z + +∞ +∑ +k=2 +∑ +{z1,...,zk}⊆X∖{z} +py1,z1( +k−1 +∏ +i=1 +pzi,zi+1)pzk,z +and +R = +∑y′∈C ∑γ∈Γyi→z→y′ ∏r′∈γ∖{z→y′} ρz,r′(x)λz→y′(x) +π(x + φ(y1) − φ(z)λy1→z(x + φ(y1) − φ(z)) , +where R denotes that right hand side of (6.7). +Additionally, for any z3 ∈ X , such that +z1 ��→ z3 ∈ R, it follows from (4.2) that +pz1,z2 = π(x + φ(z2) − φ(z))λz2→z1(x + φ(z2) − φ(z)) +π(x + φ(z1) − φ(z))λz1→z3(x + φ(z1) − φ(z))ρz,z1→z3(x). +Consequently, +py1,z = +π(x)λz→y1(x)ρz,y1→z(x) +π(x + φ(y1) − φ(z))λy1→z(x + φ(y1) − φ(z)), +(6.8) +py1,z′pz′,z =π(x + φ(z′) − φ(z))λz′→y1(x + φ(z′) − φ(z)) +π(x + φ(y1) − φ(z))λy1→z(x + φ(y1) − φ(z))ρz,y1→z(x) +× +π(x)λz→z′(x) +π(x + φ(z′) − φ(z))λz′→y1(x + φ(z′) − φ(z))ρz,z′→y1(x) += +π(x)λz→z′(x)ρz,y1→z(x)ρz,z′→y1(x) +π(x + φ(y1) − φ(z))λy1→z(x + φ(y1) − φ(z)), +(6.9) + +COMPLEX BALANCED DISTRIBUTIONS +25 +and by iteration, letting z0 = y1, +py1,z1( +k−1 +∏ +i=1 +pzi,zi+1)pzk,z = +π(x)λz→zk(x)ρz,y1→z(x)∏k +i=1 ρz,zi→zi−1(x) +π(x + φ(y1) − φ(z))λy1→z(x + φ(y1) − φ(z)), +(6.10) +for all k ≥ 2. Then, (6.7) follows from (6.8)-(6.10) and the definition of Γy1→z→y′. The proof +of this lemma is complete. +6.5. Proof of Lemma 4.5. If NM−1 = NM, then by definition, every complex in C′ +M−1 ∩ C′ = +C′ +M ∩ C′ has only one incoming reaction. On the other hand, if NM−1 ≠ NM, then the M +complexes in C′ are cleaved sequentially in N1,... NM, and thus C′ +M ∩ C′ ⊆ CM ∩ C′ = ∅. +Therefore, in either case, no complex in C′ ∩ CM has multiple incoming reactions. We will +show that if (y,i) ∈ C′ +M is a copy of y ∈ C′, then (y,i) has only one incoming reaction in NM. +First, y ∈ C′ has only one incoming reaction in N, but multiple incoming reactions in Nm−1 for +some m ∈ {1,... ,M}; otherwise (y,i) is not in C′ +M ⊆ CM. Recall that when one-node cleaves a +complex, only the incoming reactions of complexes that are products of the cleaved complex +might change. It follows that the multiple incoming reactions in Rm−1 are due to the cleaving +of a complex y′ ∈ C′ ∪ {z} in Nm′ with m′ < m, that is, the reactant of the only incoming +reaction of y in R,... ,Rm′−1. After cleaving y′, the reactant of each incoming reaction of y in +Cm′,... ,Cm−1 is a copy of y′, and thus when cleaving y in Nm, the reactant (y′,j) of the only +incoming reaction of (y,i) in Rm is a copy of y′. As in the cleaving iteration, only complexes +in C′ might be cleaved. The copy (y′,j) is not cleaved in Nm+1,... ,NM. As a consequence, +the incoming reactions of (y,i) will not change, namely, (y,i) has only one incoming reaction +(y′,j) ��→ (y,i) in Rm+1,... ,RM. +The only concern now is the cleaving of a complex y in Nm with some m ∈ {1,... ,M}, +fulfilling y ��→ (z,i) ∈ Rm−1 for some i = 1,... ,pz. The situation is illustrated in Figure +2. Consider the RN N. Complex z has two incoming reactions, and C′ = {y1,y2}, in which +each complex has only one incoming reaction. The only cycle including y2 ��→ z included +in N is {z ��→ y1 ��→ y2 ��→ z}. Thus, after cleaving z in N0, the complex (z,1) has only +one outgoing reaction (z,1) ��→ y1. Then, y1 is cleaved in the same manner, resulting in +the cleaved RN N1. It remains to cleave the complex y2. Note that there is only one cycle +including y2 ��→ (z,1) in N1. Therefore, after cleaving y2, the complex (z,1) has only one +incoming reaction in R2. This observation allows us to complete the proof as follows. +Assume y ∈ Cm−1 ∩C′ has multiple incoming reactions in Rm−1 (for some m), y ��→ (z,1) ∈ +Rm−1, and y is cleaved in Nm. +The reaction y ��→ (z,1) is a result of the cleaving of z in N0, that is y ��→ z ∈ R and +y ��→ (z,1) ∈ R0 is the only incoming reaction of (z,1) in R0. Additionally, y ∈ C′ has only +one incoming reaction in R. It follows that the multiple incoming reactions of y in Rm−1 +come from the cleaving of some complex y′ in Nm′ with m′ ∈ {0,... ,m − 1}. By iteration, we +find a sequence of reactions +{z ��→ y(1) ��→ ⋯ ��→ y(k) ��→ y} ⊆ R + +26 +LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA +with k ∈ {0,... ,m − 1}, such that for each i ∈ {0,... ,k}, complex y(i) ∈ C′ is cleaved in Nmi +that increases the number of incoming reactions of y(i+1)in Rmi, where m0,... ,mk are non- +negative integers fulfilling 0 = m0 < m1 < ⋅⋅⋅ < mk ≤ m − 1, with the convention that y(0) = z. +Because {y,y(1),... ,y(k)} ⊆ C′, by definition y(i) ��→ y(i+1) is the only incoming reaction of +y(i+1) in R for all i = 0,... ,k, where y(0) = z and y(k+1) = y. Thus, +γ0 = {z ��→ y(1) ��→ ⋯ ��→ y(k) ��→ y ��→ z} +is the only cycle including reaction y ��→ z in N. Therefore, (z,1) ��→ y(1) is the only +outgoing reaction of (z,1) in R0. +Next, consider the cleaving of y(1) in Nm1. Neither (z,1) nor y(1) is cleaved in N1,... ,Nm1−1, +thus (z,1) ��→ y(1) ∈ Rm1−1. Therefore, after the cleaving of y(1) in Nm1, there is a copy +of y(1), denoted by (y(1),1) in Cm1 such that (z,1) ��→ (y(1),1) ∈ Rm1. Additionally, this +reaction is the only incoming reaction of (y(1),1) and the only outgoing reaction of (z,1) +in Rm1. Similarly, since neither (y(1),1) or (z,1) is cleaved in Nm1+1,... ,Nm, then it fol- +lows that (z,1) ��→ (y(1),1) ∈ Rm1 is the only incoming reaction of (y(1),1) and the only +outgoing reaction of (z,1) in Rm1+1,... ,Rm−1 as well. On the other hand, because none of +y(1),... ,y(k),(z,1) are cleaved in N1,... ,Nm1−1, it holds that +γ1 = {y(1) ��→ y(2) ��→ ⋯ ��→ y(k) ��→ y ��→ (z,1) ��→ y(1)}. +is also the only cycle including (z,1) ��→ y(1) in Nm1−1. Thus, (y(1),1) ��→ y(2) is the only +outgoing reaction of (y(1),1) in Rm1, and thus in Rm1+1,... Rm2−1. +By iteration, after cleaving y(k), there is a sequence +{(z,1) ��→ (y(1),1) ��→ ⋯ ��→ (y(k),1) ��→ y} ⊆ Rmk +such that (y(i),1) ��→ (y(i+1),1) is the only outgoing reaction of y(i) for all i = 0,... ,k in +Rmk,... ,Rm−1, where y(0) = z and (y(k+1),0) = y. This implies that the only cycle including +y ��→ (z,1) in Nm−1 is +γ2 = {y ��→ (z,1) ��→ (y(1),1) ��→ ⋯ ��→ (y(k),1) ��→ y}. +As a consequence, complex (z,1) has only one incoming reaction in Rm, provided y ��→ (z,1) +is the only incoming reaction of (z,1) in Rm−1. This proves that the number of incoming +reactions of (z,1) is one in Rm. The proof of this lemma is thus complete. +6.6. Proof of Theorem 3.4. Let (Ncyc,λcyc) be the cleaved SRN obtained by the iterative +one-node cleaving procedure (Sections 4.1-4.3). Then, by Lemmas 2.1, 4.1 and 4.5, Ncyc, we +see that the digraph of Ncyc consists of disjoint cycles that are pairwise non-similar simple +cycles when projected onto N. It suffices to check (i)-(iii) for (Ncyc,λcyc). +First, (i) is a direct consequence of Ncyc ⪰ N and the fact that every cycle in Ncyc is +simple when projected onto N. +Next, denote by (N ∗ +cyc,λ∗ +cyc) be the cleaved SRN before +completion in Section 4.3. Then, due to Lemma 4.3 and an iteration argument, (iii) holds for +(N ∗ +cyc,λ∗ +cyc). The ‘cut-adhere’ process does not affect the validity of (iii). We need to show + +COMPLEX BALANCED DISTRIBUTIONS +27 +that (iii) still holds after the combination process, which seems wrong as in Example 2.14. +Still denote by (N ∗ +cyc,λ∗cyc) the cleaved SRN after ‘cut-adhere’ before combination. Then, +(N ∗ +cyc,λ∗ +cyc) ⪰ (Ncyc,λ) ⪰ (N,λ), and thus (iii) follows as a result of Corollary 2.13. +Last, we need to prove (ii). Let +γ = {y1 ��→ y2 ��→ ⋯ ��→ ym ��→ y1} +be a cycle in R. Let N1 = (C1,R1,S,φ1) be the cleaved RN of N with projection ψ1 due +to one-node cleaving of some z ∈ C. If z ∉ {y1,... ,ym}, then every reaction of the cycle is +also in R1. On the other hand, without loss of generality, assume that z = y1. Then, there +exists some index i ∈ {1,... ,pz}, such that ym ��→ (z,i), and by definition of Rout +1 , we have +(z,i) ��→ y2 ∈ R1 as well. In other words, there exists a cycle +γ1 = {(z,i) ��→ y2 ��→ ⋯ ��→ ym ��→ (z,i)} ∈ R1. +Therefore, there exists a cycle γ1 ∈ R1, such that ψ1(γ1) = γ in any case. By iteration, there +exists a cycle γ∗cyc ⊆ R∗cyc, such that ψ∗cyc(γ∗cyc) = γ, where N ∗ +cyc = (C∗cyc,R∗cyc,S,φ∗cyc) denotes +the cleaved RN of N with projection ψ∗ +cyc, before the completion step in Section 4.3. Since +γ∗cyc is simple, it will not be affected in the ‘cut-adhere’ process. Finally, in the combination +process of Section 4.3, the cycle γ∗ +cyc may be ‘absorbed’ by other similar cycles when projected +onto N. However, it does not influence the validity of property (ii). The proof is complete. +7. Acknowledgement +The work presented in this article is supported by Novo Nordisk Foundation (Denmark), +grant NNF19OC0058354. +References +[1] Anderson, D. F., and Cotter, S. L. Product-form stationary distributions for deficiency zero net- +works with non-mass action kinetics. Bull. Math. Biol. 78, 12 (2016), 2390–2407. +[2] Anderson, D. F., Craciun, G., Gopalkrishnan, M., and Wiuf, C. Lyapunov functions, stationary +distributions, and non-equilibrium potential for reaction networks. Bull. Math. Biol. 77, 9 (2015), 1744– +1767. +[3] Anderson, D. F., Craciun, G., and Kurtz, T. G. Product-form stationary distributions for defi- +ciency zero chemical reaction networks. Bull. Math. Biol. 72 (2010), 1947–1970. +[4] Anderson, D. F., and Kurtz, T. G. Stochastic analysis of biochemical systems. Springer, Berlin, +2015. +[5] Bibbona, E., Kim, J., and Wiuf, C., G. Stationary distributions of systems with discreteness-induced +transitions. J. R. Soc. 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Epidemic pro- +cesses in complex networks. Rev. Mod. Phys. 87, (2015), 925–979. + +COMPLEX BALANCED DISTRIBUTIONS +29 +[33] Schuster, S. and Schuster, R. A generalization of Wegscheider’s condition. Implications for prop- +erties of steady states and for quasi-steady-state approximation. J. Math. Chem. 3 (1989), 25–42. +[34] Serfozo, R. Introduction to Stochastic Networks. Springer New York, 1999. +[35] Wilkinson, D. Stochastic Modelling for Systems Biology. Chapman and Hall/CRC, Boca Raton, 2006. +[36] Xu, C. Hansen, M. C. and Wiuf, C. Full classification of dynamics for one-dimensional continuous- +time Markov chains with polynomial transition rates. To appear in Adv. Appl. Probab., (2022+). +Department of Mathematical Sciences, University of Copenhagen, Denmark +Email address: linard.hoessly@hotmail.com, wiuf@math.ku.dk +Department of Mathematics and Statistics, Auburn University, USA +Email address: pqxia@auburn.edu + diff --git a/ZtE2T4oBgHgl3EQfvggn/content/tmp_files/load_file.txt b/ZtE2T4oBgHgl3EQfvggn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e0de9d5bb62cd641a2b3271e56add7a1fd2cfcb8 --- /dev/null +++ b/ZtE2T4oBgHgl3EQfvggn/content/tmp_files/load_file.txt @@ -0,0 +1,1229 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf,len=1228 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='04091v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='PR] 10 Jan 2023 COMPLEX BALANCED DISTRIBUTIONS FOR CHEMICAL REACTION NETWORKS LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Stationary distributions of continuous time Markov chains (CTMCs) are of- ten a main interest, but hard to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We consider CTMCs modelling reaction networks, and characterise complex balanced distributions (provided they exist) of reaction networks with arbitrary transition functions through conditions on the cycles of their corresponding digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The proof works by constructing a dynamically equivalent reaction network of dis- joint cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We further derive a sufficient condition for the existence of a complex balanced distribution, and give precise conditions on when it is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The sufficient condition holds for mass-action kinetics or if the digraph consists of only cyclic connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Hence, we fully characterise the existence and form of complex balanced distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' More- over, these results can be used to design complex balanced reaction networks with a given stationary distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Introduction Reaction networks offer a broadly applicable framework to model the dynamics of various natural systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' They are applied across the sciences, for example in epidemiology [30, 32], genetics [13], and systems and cellular biology [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A reaction network consists of a set of reactions, where a reaction represents a conversion, birth, or death of constituent particles (molecules, individuals, allele copies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For example, A ��→ B might represent the conversion of one molecule of A into one of B, and S + I ��→ 2I might represent the infection of a susceptible individual by an infected individual, leading to two infected individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' When the number of particles is low, stochastic fluctuations become significant, and deterministic equations such as ordinary differential equations [17] do not adequately model the dynamics of the reaction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Typically, in such cases, continuous-time Markov chains (CTMCs) are applied [4, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A reaction network is typically given by its digraph, which when modelled by a CTMC, captures its discrete dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, CTMCs modelling reaction networks might be seen as Markov chains for which the transition matrix has a particular graphical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Continuous time Markov chain, stationary distribution, complex balanced, reac- tion digraph, cycles, reaction cleaving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 1 2 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA To illustrate this, consider the following example of a stochastic reaction network: A λ1 2C + D λ3 B, λ4 D λ5 λ2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) where the nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=', A) are (chemical) complexes, the arrows represent reactions between complexes, and λi∶Z4 ≥0 → R≥0, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,4, are transition rates (propensities, in the chemical literature) on the state space Z4 ≥0, where x = (xA,xB,xC,xD) ∈ Z4 ≥0 is the vector of molecular counts of the species A,B,C,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' If a reaction occurs, say, A ��→ 2C + D, then the Markov chain jumps from the current state (xA,xB,xC,xD) to a new state (xA −1,xB,xC +2,xD +1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' one molecule of A is consumed, and two molecules of C and one of D are produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' If currently there are no molecules of A (xA = 0), then the reaction cannot take place as this would lead to a negative number of A molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, we require the transition rates to be positive if and only if the required molecules are available (Condition 1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' While stochastic modelling of reaction networks goes back to the 1970s [24, 25, 26], the recent popularity of the stochastic approach in the life sciences has lead to a deep interest in the existence and form of their stationary distributions [1, 3, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' There are limited analytical results on stationary distributions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=', when the state space is finite, the reaction network is also a birth-death process, and a few other special cases [5, 12, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In the following, we focus on so-called complex balanced reaction networks and their stationary distributions (provided such exist), which have been the focus of intense investigations in the last decade [1, 3, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A stationary distribution π is complex balanced if the flux out of a state through a given complex equals the flux into the state through the same complex, that is, if the equation π(x) ∑ y′∶η→y′ λη→y′(x) = ∑ y∶y→η π(x + φ(y) − φ(η))λy→η(x + φ(y) − φ(η)), holds for all complexes η, and all states x [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Here, φ is a function that maps complexes to their stochiometric coefficients, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=', φ(A) = (1,0,0,0) and φ(2C + D) = (0,0,2,1), and the difference φ(y) − φ(η) is the net molecular gain in the reaction η ��→ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This definition relates to detailed balanced Markov chains and birth-death processes, though neither are equivalent nor generalisations of the other [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The existence of a complex balanced stationary distribution implies the digraph consists of strongly connected components (in the example, there is one such component) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In the deterministic setting, complex balanced reaction networks can be traced back to the work of Boltzman [6], and have a long standing interest in the community [14, 15, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Stochastic complex balanced reaction networks have been developed recently, perhaps starting with the discovery that a reaction network with stochastic mass-action kinetics has Poisson product-form stationary distributions if a corresponding deterministic mass-action COMPLEX BALANCED DISTRIBUTIONS 3 reaction network is complex balanced [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This was subsequently generalised to more general kinetics/transition rates [1, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Parallel results on product-form stationary distributions for stochastic Petri nets [27] and Queuing networks [34] exist, and matches those of stochastic reaction networks [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This is not surprising as the corresponding Markov chains can be seen as stochastic reaction networks, that is, as CTMCs with transition matrices specified by digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Our main result characterises complex balanced distributions of a reaction network with arbitrary kinetics through conditions on the cycles of its digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' To achieve this, we extend the definition of ‘reaction network’ and construct a dynamically equivalent decomposed re- action network, consisting of only cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The extended definition allows the same complex and the same reaction (potentially with different transition rates) to be represented multiple times in the digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Whereas each cycle corresponds to a cycle in the original digraph, the transition rates are chosen such that the decomposed reaction network is complex balanced if and only if the original reaction network is complex balanced (with the same stationary distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The cyclic network is constructed by iteratively ‘cleaving’ nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' splitting a node of the digraph with multiple incoming edges into multiple nodes with single incom- ing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This construction is of independent interest and might lead to further results on stationary distributions or steady states in the deterministic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Secondly, we give a novel sufficient condition for the existence of a complex balanced distribution (extending a condition given in [21]) and showing necessity if a certain criterion on the transition rates is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In particular, this criterion holds for mass-action kinetics or if the digraph consists of cyclic connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thereby, we fully characterise existence and (implicitly) form of complex balanced distributions by decomposing a reaction network into disjoint cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Furthermore, these results can be used to design complex balanced reaction networks with a given stationary distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The reaction network in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) contains two cycles, A ��→ B + C ��→ D ��→ A and A ��→ C ��→ D ��→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Cleaving A and D into two nodes each, (A,1), (A,2), and (D,1), (D,2), respectively, we obtain the following reaction network (A,1) λ′ 1 2C + D λ′ 3 (B,1) λ′ 5 (A,2) λ′ 2 D λ′ 4 (B,2), λ′ 6 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) where (A,1) and (A,2) are considered as different complexes, but with the same stoichio- metric coefficients, φ((A,1)) = φ((A,2)) = (1,0,0,0), and likewise for (D,1) and (D,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The λ′ i’s are kinetics to be defined, such that the reaction network is dynamically equivalent to the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In principle, this is not difficult, as one might take λ′ i = λi for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,4, and choose arbitrary λ′ 5 and λ′ 6, such that λ′ 5 + λ′ 6 = λ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' However, this assignment does not necessarily preserve the complex balanced property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Finding a decomposition that is complex balanced if and only if the original reaction network is, arises as the main objective to secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For the 4 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA specific example, we can choose λ′ i = λi for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,4, λ′ 5(x) = λ1(x + eA − eB)λ5(x) λ1(x + eA − eB) + λ2(x + eA − eB)1{x′∶x′ B≥1}(x), λ′ 6(x) = λ2(x + eA − eB)λ5(x) λ1(x + eA − eB) + λ2(x + eA − eB)1{x′∶x′ B≥1}(x), where eA = (1,0,0,0) and eB = (0,1,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, it can be verified that the original reaction network is complex balanced if and only if the cleaved one is (with the same complex balanced distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Various graph decomposition techniques have been studied for deterministic reaction net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Node balanced steady states [18] generalise complex balanced steady states and are based on reaction digraphs where multiple copies of the same complex are allowed (but each reaction is only represented once).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Cyclic decompositions of the digraph are constructed in [19, 23] in order to study steady states, but they do not preserve (deterministic) dynamical equivalence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' while in [20] a gluing operation is discussed that bears similarity to decomposi- tion by cleaving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The Wegschneider conditions on cycles of a reaction digraph characterise the so-called detailed balanced (deterministic) reaction networks [9, 16, 28, 33], though the conditions do not apply to complex balanced reaction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' It would be interesting to develop the cleaving operation in the deterministic setting and to explore its relationship to the Wegschneider conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In Section 2, we provide background on graphs and reaction networks and derive properties of cleaved reaction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In Section 3, we present results for complex balanced stochastic reaction networks, and in Section 4, building on the previous paragraphs, we introduce the cleaving operation that allows decomposing stochastic complex balanced reaction networks into disjoint cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' An example is provided in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Finally, we end with proofs in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let R, R≥0 and R>0 be the set of real, non-negative and positive numbers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let Z and Z≥0 be the set of integers and non-negative integers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For x = (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,xn),y = (y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,yn) ∈ Rn, we define x ≥ y, if xi ≥ yi for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' and x > y if x ≥ y and x ≠ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Furthermore, for x ∈ Rn ≥0, y ∈ Zn ≥0, the notation xy is used for ∏n i=1 xyi i , and for x ∈ Zn ≥0, we write x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' for x1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='⋯xn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='. 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We present preliminaries from graph theory [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By definition, a digraph is a pair (V,E), where V is a finite set of nodes and E ⊆ V × V is a finite set of edges, together with two maps init,ter∶E → V that assign an initial node init(e) and a terminal node ter(e), respectively, to each edge e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' An edge e is directed from init(e) to ter(e), denoted by e = (init(e),ter(e)) = init(e) ��→ ter(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For any v ∈ V, COMPLEX BALANCED DISTRIBUTIONS 5 if v ∉ {init(e)∶e ∈ E} ∪ {ter(e)∶e ∈ E}, then v is an isolated node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' If init(e) = ter(e), then e ∈ E is a self-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A sub-digraph (V′,E′) of (V,E) is a digraph such that V′ ⊆ V and E′ ⊆ (V′ ×V′)∩E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Two sub-digraphs are disjoint if their sets of nodes are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A walk is an ordered finite sequence of edges from (V,E), θ = (v1 ��→ v2,v2 ��→ v3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,vk−1 ��→ vk) or (v1 ��→ v2 ��→ ⋯ ��→ vk) for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The walk is closed if v1 = vk, and is open if it is not closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' An open walk θ is directed from v1 to vk, and links v1 and vk, vice versa vk and v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Furthermore, the nodes init(θ) = v1 and ter(θ) = vk are the initial and terminal nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' If all nodes are different, then θ is a path, and if all nodes are different but v1 = vk, then it is a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In particular, an isolated node is a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Paths and cycles, but not walks, might be seen as sub-digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A digraph (V,E) is connected, if for any pair of nodes v,v′ ∈ V, there exist nodes v0,v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,vk,vk+1 ∈ V and paths θ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,θk+1 in V, such that v0 = v′ and vk+1 = v and θi links vi−1 and vi for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A connected sub-digraph (V′,E′) is a connected component of (V,E), if no nodes v ∈ V ∖ V′ are linked to a node in V′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The ensuing Lemma then follows by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (V,E) be a digraph satisfying the following (i) For any edge e ∈ E there exists a cycle γ ⊆ E with e ∈ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (ii) For any node v ∈ V there exists at most one edge e ∈ E such that ter(e) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, (V,E) consists of disjoint cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Reaction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In our context, a reaction network (RN), N = (C,R,S,φ), is a digraph (C,R) without self-loops, and a labelling φ∶C → ZS ≥0, where ZS ≥0 = { ∑ S∈S zSS∶zS ∈ Z≥0,∀S ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The elements of S are species, those of C are complexes, and those of R are reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For r = y ��→ y′ ∈ R, y,y′ ∈ C, and y = init(r) and y′ = ter(r) are called the reactant and the product of the reaction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Moreover, r is called an incoming reaction of complex y′, and an outgoing reaction of complex y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We identify ZS ≥0 with Zn ≥0 for ∣S∣ = n, and consider S as the standard basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, any complex y ∈ C can be given in terms of its stoichiometric coefficients, φ(y) = z = (z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,zn) = n ∑ i=1 ziSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The above definition extends the usual definition [17] in which RNs are given as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We can regard (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) as RNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1), let S = {A,B,C,D}, C = {A,B,C,D}, R = {A ��→ B,A ��→ C,B ��→ D,C ��→ D,D ��→ A} and φ = idC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 6 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA Then, N = (C,R,S,φ) is an RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Similarly, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) defines an RN N ′ = (S′,C′,R′,φ′) with S′ = S = {A,B,C,D}, C′ = {(A,i),B,C,(D,i)∶i = 1,2}, R′ = {(A,1) ��→ B,B ��→ (D,1),(D,1) ��→ (A,1),(A,2) ��→ C, C ��→ (D,2),(D,2) ��→ (A,2)}, and φ′∶C → Zn ≥0 given by φ′((y,j)) = y for y ∈ {A,D}, j = 1,2, and φ′(y′) = y′ for y′ ∈ {B,C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let N = (C,R,S,φ) and N ′ = (C′,R′,S′,φ′) be two RNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' If there exists a map ψ∶C′ → C (that extends to the edges of the digraph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' to ψ∶R′ → R), such that (i) φ′ = φ ○ ψ, implying C = ψ(C′) = {ψ(y)∶y ∈ C′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (ii) R = ψ(R′) = {ψ(y) ��→ ψ(y′)∶y ��→ y′ ∈ R′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, N ′ is a cleaved reaction network (cleaved RN) of N with projection ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Further- more, for two distinct complexes y′,y′′ ∈ C′, if ψ(y′) = ψ(y′′) = y ∈ C, then each of y′ and y′′ is called a copy of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2, N ′ is a cleaved RN of N with projection ψ = φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The next lemma shows the transitivity of cleaved RNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, ‘being cleaved’ is a partial order on the set of RNs: N ′ ⪰ N if and only if N ′ is a cleaved RN of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The proof is omitted and follows by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let N, N ′ and N ′′ be RNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose N ′ is a cleaved RN of N with projection ψ, and N ′′ is a cleaved RN of N ′ with projection ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, N ′′ is a cleaved RN of N with projection ψ ○ ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For N = (C,R,S,φ), define the essential reaction network (essential RN) of N, denoted Ness, as (φ(C),φ(R),S,idφ(C)), where φ(C) = {φ(y)∶y ∈ C}, φ(R) = {φ(y) ��→ φ(y′)∶y ��→ y′ ∈ R}, and idφ(C) is the identity map on φ(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Clearly, N ⪰ Ness with projection ψ = φ, and Ness = (Ness)ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Clearly in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2, N is the essential RN of N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Cleaved RNs have the same essential RN as the original RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The proof is omitted and follows by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose that N ′ ⪰ N with projection ψ∶C′ → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, Ness = N ′ ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The stochastic dynamics of RNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let N = (C,R,S,φ) be an RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We model the evolution of the species counts X(t), t ≥ 0, as a Zn ≥0-valued CTMC, satisfying the following SDE: X(t) = X(0) + ∑ y→y′∈R Yy→y′(∫ t 0 λy→y′(X(s))ds)(φ(y′) − φ(y)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) where Yy→y′, y ��→ y′ ∈ R, is a collection of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' unit rate Poisson processes, and λ = (λy→y′∶y ��→ y′ ∈ R), λy→y′∶Zn ≥0 → R≥0, COMPLEX BALANCED DISTRIBUTIONS 7 is the (stochastic) kinetics associated to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We also refer to λy→y′ as the kinetics of the particular reaction y → y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The pair (N,λ) is a stochastic reaction network (SRN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We might consider (N,λ) as a labelled digraph and write y λy→y′ ���→ y′ to emphasize the kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In the following we assume the following compatibility condition by default: Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For y ��→ y′ ∈ R, λy→y′(x) > 0 if any only if x ≥ φ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A class of standard kinetics for chemical reactions is the so-called (stochas- tic) mass-action kinetics, which takes the form λy→y′(x) = αy→y′ x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (x − φ(y))!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1{x′∈Zn ≥0∶x′≥φ(y)}(x), for all y ��→ y′ ∈ R and x ∈ Zn ≥0, where αy→y′ is a positive rate constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In this case, we write y αy→y′ ���→ y′ for y λy→y′ ���→ y′, and note that Condition 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (N,λ) and (N ′,λ′) be SRNs such that N ′ ⪰ N with projection ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, (N ′,λ′) is said to be a cleaved stochastic reaction network (cleaved SRN) of (N,λ), denoted by (N ′,λ′) ⪰ (N,λ), if the kinetics λ and λ′ satisfies the following condition for all x ∈ Zn ≥0 and all y ��→ y′ ∈ R, λy→y′(x) = ∑ r∈ψ−1(y→y′) λ′ r(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) We state the following lemma without proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (N,λ), (N ′,λ′) and (N ′′,λ′′) be SRNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose (N ′,λ′) ⪰ (N,λ) with projection ψ, and (N ′′,λ′′) ⪰ (N ′,λ′) with the projection ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, (N ′′,λ′′) ⪰ (N,λ) with projection ψ ○ ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We equip the essential RN Ness of (N,λ) with a canonical kinetics λess given by λess,y→y′(x) = ∑ r∈φ−1(y→y′) λr(x) for all y ��→ y′ ∈ φ(R) and x ∈ Zn ≥0, and call (Ness,λess) the essential stochastic reac- tion network (essential SRN) of (N,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' It holds that (N,λ) ⪰ (Ness,λess).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Furthermore, analogous to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose (N ′,λ′) ⪰ (N,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, (N ′ ess,λ′ ess) = (Ness,λess).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let Y be a unit rate Poisson process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, for s,t ≥ 0, Y (t + s) and Y ′(t) + Y ′′(s) have the same distribution, where Y ′ and Y ′′ are independent copies of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, given an SRN, we have X(t) = X(0) + ∑ y→y′∈φ(R) Yz→z′(∫ t 0 λess,z→z′(X(s))ds)(y′ − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3) Every (weak) solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) is also a (weak) solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3), and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Consequently, the dynamic of any SRN is determined by its essential SRN: 8 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (N,λ) and (N ′,λ′) be such that (Ness,λess) = (N ′ ess,λ′′ ess).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, the dynamics of (N,λ) and (N ′,λ′) are equivalent, in the sense that every weak solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) under (N,λ) is also a weak solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) under (N ′,λ′), and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In the following, we introduce definitions related to stationary distributions of SRNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let x,x′ ∈ Zn ≥0 be two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, x leads to x′ in N = (C,R,S,φ), written x →N x′, if there exists a sequence of reactions y1 ��→ y′ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,ym ��→ y′m ∈ R, such that (i) x ≥ φ(y1), x − φ(y1) + φ(y′ 1) ≥ φ(y2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' , x + ∑m−1 i=1 (φ(y′ i) − φ(yi)) ≥ φ(ym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (ii) x + ∑m i=1(φ(y′ i) − φ(yi)) = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' According to Condition 1, x →N x′ if and only if the probability to move from x to x′ is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A subset Γ ⊆ Zn ≥0 is an irreducible component of N, if for all x ∈ Γ and all x′ ∈ Zn ≥0, x →N x′, if and only if x′ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' It follows that two distinct irreducible components are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The proof of the next statement is elementary and thus omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let N and N ′ be RNs such that Ness = N ′ ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, for any states x,x′ ∈ Zn ≥0, x →N x′ if and only if x →N ′ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' As a consequence, a subset Γ ⊆ Zn ≥0 is an irreducible component of N if and only if it is also an irreducible component of N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (N,λ) be an SRN and Γ ⊆ Zn ≥0 an irreducible component of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A probability distribution π on Γ is a (i) stationary distribution, if for all x ∈ Γ, π(x) ∑ y→y′∈R λy→y′(x) = ∑ y→y′∈R π(x − φ(y′) + φ(y))λy→y′(x − φ(y′) + φ(y)), where we set λy→y′(z) = 0 if z ∉ Zn ≥0 (same below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (ii) complex balanced distribution, if for all complexes η ∈ C, and all x ∈ Γ, π(x) ∑ y′∶η→y′∈R λη→y′(x) = ∑ y∶y→η∈R π(x + φ(y) − φ(η))λy→η(x + φ(y) − φ(η)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) A complex balanced probability distribution on Γ is also a stationary distribution on Γ [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The complex balance property requires the digraph to be weakly reversible (all connected components are strongly connected), that is, every reaction y ��→ y′ ∈ R belongs to a cycle γ ⊆ R [8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The following is a consequence of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (N,λ) and (N ′,λ′) be SRNs such that (Ness,λess) = (N ′ ess,λ′ ess).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, a probability distribution π is a stationary distribution on an irreducible component Γ of (N,λ), if and only if π is also a stationary distribution on Γ of (N ′,λ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For complex balanced distributions, one implication follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' COMPLEX BALANCED DISTRIBUTIONS 9 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (N,λ) and (N ′,λ′) be SRNs such that (N ′,λ′) ⪰ (N,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' If a probability distribution π is a complex balanced distribution on an irreducible component Γ of (N ′,λ′), then π is also a complex balanced distribution on Γ of (N,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The reverse implication is not true in general: Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Consider the SRN, A 3 ��⇀ ↽�� 3 B, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5) equipped with stochastic mass-action kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, the following system with stochastic mass-action kinetics (A,1) 1 ��⇀ ↽�� 2 (B,1), (A,2) 2 ��⇀ ↽�� 1 (B,2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6) and φ((A,i)) = A and φ((B,i)) = B for i = 1,2, is a cleaved SRN of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The probability distribution π(x) = MΓ x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' is a complex balanced distribution for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5) on any irreducible class Γ, for some positive constant MΓ [3, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' However, it is not a complex balanced distribution for the cleaved SRN (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Even more holds, one can show that there are no complex balanced distributions for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A criteria for complex balanced distributions Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let N = (C,R,S,φ) be a weakly reversible RN with connected components L1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,Lℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Equip N with a stochastic kinetics λ, and let Γ ⊆ Zn ≥0 be an irreducible component of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose the following properties hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (i) There exist positive constants {κy→y′∶y ��→ y′ ∈ R}, such that for every complex η ∈ C, ∑ y′∶η→y′∈R κη→y′ = ∑ y∶y→η∈R κy→η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) (ii) There exist functions g∶Γ → R≥0 and mk∶Γk → R>0, k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,ℓ, where Γk = {x − φ(y)∶x ∈ Γ,y ∈ Lk} ∩ Zn ≥0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) such that for all y ��→ y′ ∈ R and x ∈ Γ satisfying x ≥ φ(y), κy→y′ λy→y′(x) = mk(x − φ(y))g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3) If 0 < M ∶= ∑x∈Γ g(x) < ∞, then the distribution π, given by π(x) = 1 M g(x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) is a complex balanced distribution of (N,λ) on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The proof follows the idea of [3, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By definition, to show that π defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) is a complex balanced distribution, it suffices to verify that for any complex η ∈ C and any x ∈ Γ, the following holds g(x) ∑ y′∶η→y′∈R λη→y′(x) = ∑ y∶y→η∈R g(x + φ(y) − φ(η))λy→η(x + φ(y) − φ(η)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5) 10 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA Due to Condition 1, we only need to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5) under the assumption that x ≥ φ(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Note that for any η ∈ C, all reactions such that η is a reactant or product are in the same connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let mη = mi, if η ∈ Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3) yields that for all x ∈ Γ and x ≥ φ(y), ∑ y∶y→η∈R λy→η(x + φ(y) − φ(η))g(x + φ(y) − φ(η)) = ∑y∶y→η∈R κy→η mη(x − φ(η)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6) and ∑ y′∶η→y′∈R λη→y′(x)g(x) = ∑y′∶η→y′∈R κη→y′ mη(x − φ(η)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='7) Then, equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5) follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1 can be generalized to complex balanced measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Moreover, it extends the conditions given in [21] by allowing the functions mk to depend on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Furthermore, consider a mass-action SRN with a one-to-one labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' If it is complex balanced, then the corresponding deterministic reaction network is complex balanced and the stationary distributions have Poisson-product form [3, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Define g(x) = cx x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=', mk(x) = 1 cx, κy→y′ = αy→y′cy, where c ∈ Rn >0 is an equilibrium of the ODE system of the corresponding deterministic RN and αy→y′ are reaction rate constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In this context, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1 corresponds to [3, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A natural question is whether all complex balanced distributions take the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose it is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, we immediately get that the ratio λy→y′(x) λy→y′′(x) of two reactions outgoing from the same complex in the same connected component is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Though this is satisfied for mass-action kinetics, it is not the case in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We show the necessity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3) for SRNs consisting of disjoint cycles (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) and if the ‘constant ratio’ condition holds (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Observe that condition (i) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1 disappears in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (N,λ) be an SRN consisting of ℓ disjoint cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, a probability distribution π is complex balanced on an irreducible component Γ, if and only if there exist positive functions mk∶Γk → R>0 for Γk defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2), k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,ℓ, such that for any y ��→ y′ ∈ Lk, and all x ∈ Γ with x ≥ φ(y), we have π(x) = [λy→y′(x)mk(x − φ(y))]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Under the given conditions, every complex has exactly one incoming reaction and one outgoing reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1, it is enough to show that if there is a complex balanced stationary distribution, then it has to satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Without loss of generality, assume ℓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, there exists an integer p ≥ 2, such that R = {yi ��→ yi+1∶i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' yk ≠ yj,1 ≤ k < j ≤ p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' yp+1 = y1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' COMPLEX BALANCED DISTRIBUTIONS 11 Since π is a complex balanced distribution on Γ, then, by definition, we have, π(x)λyi→yi+1(x) = π(x + φ(yi−1) − φ(yi))λyi−1→yi(x + φ(yi−1) − φ(yi)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='9) for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,p (by convention y0 = yp+1) and x + φ(y1) − φ(y2) ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We define the function m as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For all x ∈ Nn 0 such that x + φ(y1) ∈ Γ, we let m(x) = [π(x + φ(y1))λy1→y2(x + φ(y1))]−1, Thus, π(x) = [λy1→y2(x)m(x − φ(y1))]−1 for all x ∈ Γ with x ≥ y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' On the other hand, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='9), we get π(x)λy2→y3(x) = π(x + φ(y1) − φ(y2))λy1→y2(x + φ(y1) − φ(y2)) = m(x − φ(y2))−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='10) for all x ∈ Γ with x + y1 − y2 ∈ Γ and x ≥ y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Since for all x ∈ Γ with x ≥ φ(y2), we have x − φ(y2) + φ(y3) ∈ Γ and thus x − φ(y2) + φ(y4) ∈ Γ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By iteration and the fact that yp+1 = y1, it follows that x − φ(y2) + φ(y1) ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Therefore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='10) holds for x ∈ Γ with x ≥ y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This implies that π(x) = [λy2→y3(x)m(x − φ(y2))]−1, for all x ∈ Γ with x ≥ φ(y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Finally, by iteration, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='8) holds for all yi → yi+1, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (N,λ) be a weakly reversible SRN consisting of ℓ connected compo- nents, and let Γ be an irreducible component of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose that λy→y′(x) = αy→y′λ0y(x) on x ∈ Γ for all y ��→ y′ ∈ R, where αy→y′ is a positive constant, and λ0 y depends on the complex y only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, a probability distribution π on Γ is a complex balanced distribution of (N,λ), if and only if there exist non-negative functions mk, k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,ℓ, such that for all y ��→ y′ ∈ R, y ∈ Lk, and x ∈ Γ with x ≥ φ(y), we have π(x) = κy→y′[λy→y′(x)mk(x − φ(y))]−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='11) where κy→y′ are positive constants satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3 is in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1, and relies on Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let N be an RN and let N ′ ⪰ N with projection ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For any cycle γ ⊆ R′, we say γ is simple when projected onto N, if ψ(γ) is a cycle in R = ψ(R′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Moreover, two cycles γ,γ′ ⊆ R′ are called similar if ψ(γ) = ψ(γ′), when projected onto N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (N,λ) be a weakly reversible SRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, there exists a cleaved SRN (Ncyc,λcyc) of (N,λ) with projection ψcyc, such that the digraph of Ncyc consists of disjoint cycles that are pairwise non-similar simple cycles when projected onto N, satisfying the fol- lowing properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (i) For any cycle γ ⊆ Rcyc, ψcyc(γ) is a cycle in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (ii) For any cycle γ ⊆ R, there exists a unique cycle γ′ ⊆ Rcyc such that ψcyc(γ′) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (iii) A probability distribution π is a complex balanced distribution of (N,λ) on some irre- ducible component Γ of N, if and of if it is that of (Ncyc,λcyc) on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 12 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA The proof is in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6, and is based on the iterative one-node cleaving procedure outlined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' As a corollary of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4, a weakly reversible SRN is complex balanced if and only if it can be decomposed to a cleaved SRN consisting of cycles, such that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In principle, one can therefore always use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2 to determine the stationary distribution of a complex balanced SRN by cleaving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' As the following example shows, the kinetics of Ncyc is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Consider the mass-action SRN, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 2 1 A 2 1 B 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='12) On an arbitrary irreducible component Γ, π(x) = MΓ (c∗)x x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' , is the unique complex balanced distribution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='12), where c∗ = (1,1,1) and MΓ is a constant [3, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The following SRN is a dynamically equivalent cleaved SRN with five disjoint cycles and mass- action kinetics (details omitted), (A,1) α1 α2 (B,1), (B,2) α3 α4 (C,2), (A,3) α5 α6 (C,3), (C,4) α7 (A,4) α8 (B,4) α9 and (C,5), α10 (A,5) α11 (B,5) α12 where α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,α12 are rate constants satisfying α1 = ⋅⋅⋅ = α6 = 1 − β, α7 = α8 = α9 = β, and α10 = α11 = α12 = β +1, with β ∈ (0,1) arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, the cleaved SRN is complex balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The iterative one-node cleaving procedure in Section 4 results in β = 1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Cleaving weakly reversible SRNs In this section, we develop an iterative procedure to show that there exists a dynamically equivalent cleaved SRN consisting of all cycles appearing in the original RN, while preserv- ing the complex balancing property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This cleaving procedure enlarges the applicability of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1 and is key to the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' One-node cleaving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let N = (C,R,S,φ) be a weakly reversible RN with stochastic kinetics λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Choose a complex z ∈ C with pz > 1 incoming reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We provide a method to construct a cleaved SRN (N1,λ1) of (N,λ) such that the complex balancing property of (N1,λ1) is the same as that of (N,λ), and such that z is replaced by pz complexes with only one incoming reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Proofs are given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The one-node cleaving involves two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In the first step, we give a precise definition of N1 = (C1,R1,S,φ1) and the projection ψ1, while in the second step, a kinetics is assigned to N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Step 1 is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' COMPLEX BALANCED DISTRIBUTIONS 13 y1 y′ 1 y1 (z,1) y′ 1 N: z y′ 2 N 1: y′ 2 y2 y′ 3 y2 (z,2) y′ 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' One-node cleaving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The complex z is cleaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' A dashed arrow, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=', y′ 1 to y1, means that there exists a path directed from the initial to the terminal complex without passing through z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Hence, since there is a cycle containing y1 ��→ z ��→ y′ 1 and y1 ��→ z ��→ y′ 2, respectively, in N, it follows that (z,1) ��→ y′ 1 and (z,1) ��→ y′ 2, respectively, in N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For the same reason, (z,2) ��→ y′ 2 and (z,1) ��→ y′ 3 in N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Primed and unprimed complexes could be the same, for example, y2 = y′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Order the incoming reactions of z by y1 ��→ z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' , ypz ��→ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Define C1 = {y∶y ∈ C} ∖ {z} ∪ {(z,i)∶1 ≤ i ≤ pz}, and R1 = R0 1 ∪ Rin 1 ∪ Rout 1 , where R0 1 = {y ��→ y′ ∈ R∶y,y′ ∈ C ∖ {z}}, Rin 1 = {yi ��→ (z,i)∶1 ≤ i ≤ pz}, and Rout 1 is the collection of all directed edges (z,i) ��→ y for some i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,pz} such that there exists a cycle γ in R and y ∈ C ∖ {z} with {yi ��→ z ��→ y} ⊆ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By weak reversibility of N, there is at least one i such that {yi ��→ z ��→ y} is contained in a cycle of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We remark that {yi ��→ z ��→ y} ⊆ R does not imply {yi ��→ (z,i) ��→ y} ⊆ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For example, let R = {yi ��⇀ ↽�� z ��⇀ ↽�� y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, R is weakly reversible and there is a closed walk yi ��→ z ��→ y ��→ z ��→ yi, including {yi ��→ z ��→ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' But {yi ��→ z ��→ y} is not in any cycle of R, and thus {yi ��→ (z,i) ��→ y} /⊆ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Finally, we define the labelling φ1 = φ ○ ψ1 with ψ1 the canonical projection on C1 given by ψ1(y) = ⎧⎪⎪⎨⎪⎪⎩ y, for y ∈ C ∖ {z}, z, for y = (z,i), i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,pz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let N be weakly reversible, and let N1 and ψ1 be a one-node cleaved RN of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, N1 ⪰ N with projection ψ1, and N1 is weakly reversible as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 14 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We assign a kinetics λ1 to N1, such that (N1,λ1) ⪰ (N,λ) and the complex balancing property is maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' To complete the task, we introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let z1,z2,z3 ∈ C be any, possibly repeated, complexes such that {z1 ��→ z2 ��→ z3} ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Denote by Γz1→z2→z3(k), k ∈ Z>0, the collection of closed walks in R of the form γ = {z1 ��→ z2 ��→ z3 ��→ y(1) ��→ ⋯ ��→ y(k) ��→ z1} ⊆ R, satisfying {y(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,y(k)} ∩ {z2} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For z1 ≠ z3, define Γz1→z2→z3(0) = ⎧⎪⎪⎨⎪⎪⎩ {z1 ��→ z2 ��→ z3 ��→ z1}, z3 ��→ z1 ∈ R ∅, z3 ��→ z1 ∉ R, and Γz1→z2→z1(0) ∶= {z1 ��⇀ ↽�� z2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By convention, Γz1→z2→z3(k) = ∅ for k ∈ Z≥0 if {z1 ��→ z2,z2 ��→ z3} /⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Define Γz1→z2→z3 = ∪∞ k=0Γz1→z2→z3(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Furthermore, define ρz3,z1→z2∶Zn ≥0 → R≥0 for all z1,z2,z3 ∈ C and x ∈ Zn ≥0 by ρz3,z1→z2(x) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ λz1→z2(x + φ(z1) − φ(z3)) ∑y′′∶z1→y′′∈R λz1→y′′(x + φ(z1) − φ(z3)), z1 ��→ z2 ∈ R, 0, z1 ��→ z2 ∉ R, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) where by convention 0 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Using Condition 1, for z1 ��→ z2 ∈ R, it holds that λz1→z2(x + φ(z1) − φ(z3)) > 0 if and only if x + φ(z1) − φ(z3) ≥ φ(z1) or equivalently, if and only if x ≥ φ(z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, ρz3,z1→z2(x) > 0, if and only if x ≥ φ(z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3) Define the kinetics λ1 as: λ1,r(x) = ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ λy→y′(x), r = y ��→ y′ ∈ R0 1, λyi→z(x), r = yi ��→ (z,i) ∈ Rin 1 , ∑ γ∈Γyi→z→y′ ∏ r′∈γ∖{z→y′} ρz,r′(x)λz→y′(x), r = (z,i) ��→ y′ ∈ Rout 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) for all x ∈ Zn ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3), Condition 1 holds for (N1,λ1) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (N,λ) be a weakly reversible SRN, and (N1,λ1) and ψ1 be a one-node cleaved SRN (defined above) of (N,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, (N1,λ1) ⪰ (N,λ) with projection ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='8 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='9, a stationary distribution of (N,λ) is also a station- ary distribution of (N1,λ1) and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Furthermore, we have preservation of complex balanced distributions in both directions along one-node cleavings: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (N,λ) be a weakly reversible SRN, and (N1,λ1) and ψ1 be a one-node cleaved SRN (defined above) of (N,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then a probability distribution π on an irreducible COMPLEX BALANCED DISTRIBUTIONS 15 component Γ is a complex balanced distribution of (N,λ) if and only if it is a complex balanced distribution of (N1,λ1) on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We illustrate the one-node cleaving procedure on Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The node A is cleaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' There are two incoming reactions of A in N leading to two new nodes (A,1) and (A,2), R0 1 = {B ��⇀ ↽�� C} and Rin 1 = {B ��→ (A,1),C ��→ (A,2)} in N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Since there are two cycles in N including B ��→ A, namely {B ��→ A ��→ B} and {B ��→ A ��→ C ��→ B}, then {(A,1) ��→ B,(A,1) ��→ C} ⊆ Rout 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Similarly, we find {(A,2) ��→ B,(A,2) ��→ C} ⊆ Rout 0 , and thus Rout 1 = {(A,1) ��→ B,(A,1) ��→ C,(A,2) ��→ B,(A,2) ��→ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Consequently, the digraph of N1 is as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' N ∶ C A B �⇒ N1 ∶ (A,1) C (A,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' B Concerning the kinetics of N1, let x = (xA,xB,xC) ∈ Z3 ≥0 denote the molecular counts of the species A, B and C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4), it suffices to calculate λ1,(A,i)→B and λ1,(A,i)→C, i = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Consider (A,1) ��→ B ∈ Rout 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The closed walks of ΓB→A→B in N are of the form θk = {B ��→ A ��→ B ��→ C ��→ B ��→ ⋯ ��→ C ��→ B}, where k ≥ 0 denotes the number of occurrences of C ��→ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' As both B and C have each two outgoing reactions in N, ρA,B→C(x) < 1 and ρA,C→B(x) < 1, and so λ1,(A,1)→B(x) = ∞ ∑ k=0 ρA,B→A(x)(ρA,B→C(x)ρA,C→B(x)) kλA→B(x) = λA→B(x)ρA,B→A(x) 1 − ρA,B→C(x)ρA,C→B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Similarly, λ1,(A,1)→C(x) = λA→C(x)ρA,B→A(x)ρA,C→B(x) 1 − ρA,B→C(x)ρA,C→B(x) , λ1,(A,2)→B(x) = λA→B(x)ρA,B→C(x)ρA,C→A(x) 1 − ρA,B→C(x)ρA,C→B(x) , λ1,(A,2)→C(x) = λA→C(x)ρA,C→A(x) 1 − ρA,B→C(x)ρA,C→B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2), if xA ≥ 1, then ρA,B→A(x) + ρA,B→C(x) = ρA,C→A(x) + ρA,C→B(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By sim- plification, this implies that λ1,(A,1)→B(x) + λ1,(A,2)→B(x) = λA→B(x), and λ1,(A,1)→C(x) + λ1,(A,2)→C(x) = λA→C(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Therefore, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) holds for all y ��→ y′ ∈ R and x ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' As a result, (N1,λ1) ⪰ (N,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 16 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We apply the one-node cleaving procedure iteratively until every complex has at most one incoming reaction, and the cleaved RN consists of only cycles (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' However, as illustrated in Figure 1, when cleaving a complex (here, z), the number of incoming reactions of other complexes (here, y′ 2) might increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, we should not expect that there is an iterative procedure based on one-node cleaving, such that the number of complexes with multiple incoming reactions is strictly decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let N = (C,R,S,φ) be a weakly reversible RN and let C′ ⊆ C be the collection of complexes in C with a single incoming reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose that C′ ≠ C (otherwise the RN consists of disjoint cycles, and we are done) and let C′′ = C ∖C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Write N0 = (C0,R0,S,φ0) for the cleaved RN of N with projection ψ0 obtained by one-node cleaving of an arbitrary node z ∈ C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Define C′ 0 = {y ∈ C0∶ψ0(y) ∈ C′ ∪ {z}} = C′ ∪ {y ∈ C0∣ψ0(y) = z} and C′′ 0 = C0 ∖ C′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' With Figure 1 as an example, we have {(z,1),(z,2),y′ 1,y′ 2,y′ 3} ⊆ C′ 0, and y′ 2 has two incoming reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Moreover, since z ∈ C′′ and C′′ 0 = C′′ ∖ {z}, then C′′ 0 has exactly one complex less than C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' R0 are the reactions of the cleaved RN of N (defined in step 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We next define a sequence of cleaved RNs, see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For m ≥ 1, let Nm = (Cm,Rm,S,φm) with projection ψm be an RN obtained by cleaving an element of C′ ∩ Cm−1 in Nm−1 with multiple incoming reactions (again Rm are the reactions of the cleaved RN of Nm−1 as defined in step 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Concretely, let ψm 0 = ψ0 ○ ⋅⋅⋅ ○ ψm, C′ m = {y ∈ Cm∶ψm 0 (y) ∈ C′ ∪ {z}} ⊆ Cm, and C′′ m = Cm ∖ C′ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' If all y ∈ C′ ∩ Cm−1 ⊆ C′ m−1 have only one incoming reaction in Rm−1, then Nm = Nm−1 (and ψm = idCm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Hence, Nm ⪰ Nm−1 with projection ψm, and Nm ⪰ N with projection ψm 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The procedure ends after M = ∣C′∣ iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Every complex in C′ M ⊆ CM has only one incoming reaction in RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' After completing the M-th iteration, we obtain a cleaved RN NM = (CM,RM,S,φM) of N with projection ψM, such that each complex in C′ M ⊆ CM has only one incoming reaction, and C′′ M has one fewer complexes than C′′, namely C′′ M = C′′ ∖ {z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' However, the number of incoming reactions of a complex y ∈ C′′ M might be different from the corresponding number of incoming reactions of the complex ψM 0 (y) = y ∈ C′′ in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By repeating this procedure for another complex z′ ∈ C′′ M and so forth, we eventually obtain, after finitely many iterations, a cleaved SRN (Ncyc,λcyc) with projection ψcyc on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Every complex in the cleaved SRN has only one incoming reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Hence, the cleaved SRN consists of disjoint cycles (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Furthermore, it fulfils the complex balancing property if and only if (N,λ) fulfils it (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We modify the cleaved SRN (Ncyc,λcyc) to obtain another cleaved SRN of (N,λ) without non-simple cycles and similar cycles when projected onto N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The modification COMPLEX BALANCED DISTRIBUTIONS 17 N: z y1 y2 z′ C′ C N0: (z,2) y1 (z,1) y2 z′ C′ 0 C0 N1: (z,2) (y1,2) (z,1) (y1,1) y3 z′ C′ 2 C2 N2: (z,2) (y1,2) (y3,2) (z,1) (y1,1) (y3,1) z′ C′ 3 C3 Figure 2 includes two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In the first step, we cut and adhere non-simple cycles, and in the second step, we combine similar cycles (for definitions see just before Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose there exists a cycle γ ⊆ Rcyc that is not simple when projected onto N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, it is of the form γ = {y0 ��→ y1 ��→ ⋯ ��→ yk ��→ y′ 0 ��→ yk+1 ��→ ⋯ ��→ yk+k′ ��→ y0}, where y0 ≠ y′ 0 and ψcyc(y0) = ψcyc(y′ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We cut this cycle at y0 and y′ 0, then adhere each piece with its end node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, we get two cycles, γ1 = {y0 ��→ y1 ��→ ⋯ ��→ yk ��→ y0}, γ2 = {y′ 0 ��→ yk+1 ��→ ⋯ ��→ yk+k′ ��→ y′ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In this way, we obtain a new cleaved RN N ′ cyc = (C′ cyc,R′ cyc,S,φ′ cyc) of N with projection ψ′cyc = ψcyc, where C′cyc = Ccyc, R′ cyc =(Rcyc ∖ {yk ��→ y′ 0,yk+k′ ��→ y0}) ∪ {yk ��→ y0,yk+k′ ��→ y′ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' It is natural to assign a kinetics λ′ cyc to N ′ cyc by keeping the same kinetics for the reactions also appearing in Rcyc, and letting λ′ cyc,yk→y0 = λcyc,yk→y′ 0, λ′ cyc,yk+k′→y′ 0 = λcyc,yk+k′→y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, (N ′cyc,λ′cyc) ⪰ (N,λ) with projection ψcyc, such that the complex balancing property remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Note that (Ncyc,λcyc) and (N ′ cyc,λ′ cyc) may not be related by ⪰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 18 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA The ‘cut-adhere’ process can be accomplished in finitely many steps until every cycle is simple when projected onto N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By abuse of notation, the final cleaved SNR is also denoted by (Ncyc,λcyc) with projection ψcyc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In the second step, we combine similar cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose there are two similar cycles γ1,γ2 ⊆ Rcyc when projected onto N, that is, ψcyc(γ1) = ψcyc(γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We simply remove γ2 and sum the kinetics of each reaction in γ2 to the corresponding reaction in γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' More precisely, suppose γ1 = {y1 ��→ ⋯ ��→ yk ��→ y1}, γ2 = {y′ 1 ��→ ⋯ ��→ y′ k ��→ y′ 1}, with yi ≠ y′ i and ψcyc(yi) = ψcyc(y′ i) for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, we construct a new cleaved RN (N ′ cyc,λ′ cyc) of (N,λ) with ψ′ cyc being a restriction of ψcyc on C′ cyc = Ccyc ∖ {(yj,i′ j)∶1 ≤ j ≤ k}, where R′cyc = Rcyc ∖ γ2, the labelling φ′cyc is a restriction of φcyc on C′cyc, and the kinetics λ′cyc is defined as follows, λ′ cyc,r = ⎧⎪⎪⎨⎪⎪⎩ λcyc,r, r ∈ Rcyc ∖ (γ1 ∪ γ2), λcyc,yj→yj+1 + λcyc,y′ j→y′ j+1, r = yj ��→ yj+1 ∈ γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then (N ′ cyc,λ′ cyc) ⪰ (N,λ) with projection ψ′ cyc, and (N ′ cyc,λ′ cyc) fulfils the complex balancing property if and only if (N,λ) fulfils it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Here, (Ncyc,λcyc) ⪰ (N ′cyc,λ′cyc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This process can be iterated finitely many times until all disjoint cycles are non-similar when projected onto N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By abuse of notation, the resulting cleaved SRN of (N,λ) is also denoted by (Ncyc,λcyc) with projection ψcyc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' An example The following example is a modification of a classical birth-death process that has an extra reaction with a jump of size two [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' More precisely, we consider the following SRN with mass-action kinetics, which is not weakly reversible, A α1 ��⇀ ↽�� α2 ∅ α3 ��→ 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) To apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1 we need to find an equivalent weakly reversible SRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Changing the reaction A ��→ ∅ to A ��→ ∅ and 2A ��→ A, then we look for a dynamically equivalent SRN of the following form, A λ1 λ2 ∅ λ3 2A λ4 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) λ2(x) = α2, λ3(x) = α3, λ4(x) + λ1(x) = α1x, where x denotes the number of A molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We will show that λ1 and λ4 are uniquely determined for any α1,α2,α3 ∈ R>0 such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) is complex balanced and Condition 1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' COMPLEX BALANCED DISTRIBUTIONS 19 The SRN (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) fulfils the ‘constant ratio’ condition in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3, hence Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1 can be applied to justify complex balancedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' However, we prefer to decompose it into cycles to avoid the difficulty of choosing the constants κ’s in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) by using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Cleaving the SRN into cycles results in L1 ∶ (A,1) λcyc,1 ���⇀ ↽��� λcyc,2 (∅,1), L2 ∶ (A,2) λcyc,5 (∅,2) λcyc,3 (2A,2), λcyc,4 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3) where λcyc,i = λi for i = 2,3,4, λcyc,1(x) = α2 α2 + α3 λ1(x), and λcyc,5(x) = α3 α2 + α3 λ1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Due to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4, the SRN (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) is complex balanced, if and only if there exist non-negative functions m1, m2 and g on Z≥0, such that λcyc,1(x + 1)g(x + 1) = λcyc,2(x)g(x) = m1(x)−1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) and λcyc,3(x)g(x) = λcyc,4(x + 2)g(x + 2) = λcyc,5(x + 1)g(x + 1) = m2(x)−1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5) for all x ∈ Z≥0 and M ∶= ∑∞ x=1 g(x) ∈ (0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' If we choose m1(x) = α3m2(x)/α2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6) then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) is a consequence of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5), and we only need to solve for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For all x ≥ 1, it follows that λcyc,3(x) λcyc,5(x + 1) = λcyc,5(x) λcyc,4(x + 1), and thus, λ1(x + 1) = α1(α2 + α3)2(x + 1) (α2 + α3)2 + α3λ1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Condition 1 gives λ4(0) = λ4(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, λ1(1) = α1 and λ1 is uniquely determined by the recursion: λ1(x) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ 0, x = 0, α1(α2 + α3)2x (α2 + α3)2 + α3λ1(x − 1), x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='7) In fact, in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='7), 0 < λ1(x) < α1x whenever λ1(x − 1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Therefore, λ1(x) and λ4(x) = α1x − λ1(x) are in (0,α1x) for all x ≥ 2, and Condition 1 holds for λ1 and λ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Assume g(0) = 1, then combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5), we have g(x + 1) = λcyc,3(x) λcyc,5(x + 1)g(x) = α2 + α3 λ1(x + 1)g(x) = (α2 + α3)x+1( x+1 ∏ u=1 λ1(u)) −1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='8) 20 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA and m2(x) = (α3g(x))−1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='9) for all x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' With λ1, m1 m2 and g defined as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='7), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='8), respectively, one can verify that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1(ii) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' To prove the existence of a complex balanced distribution, we need to show M ∶= ∑∞ x=1 g(x) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By using the recursive formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='7), we deduce that for all x ≥ 1, λ1(x)λ1(x + 1) = (x + 1)h(λ1(x)), where h(u) ∶= α1(α2 + α3)2u (α2 + α3)2 + α3u, u ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Due to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='7) and the fact that 0 ≤ λ1(x) ≤ α1x, we have for x ≥ 2, λ1(x) ≥ α1(α2 + α3)2x (α2 + α3)2 + α3α1(x − 1) ≥ α1(α2 + α3)2(x − 1) (α2 + α3)2 + α3α1(x − 1) ≥ c0 ∶= α1(α2 + α3)2 (α2 + α3)2 + α3α1 , where the last inequality follows from the property that x ↦ ax c+bx is increasing on R≥0 with arbitrary parameters a,b,c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Since h is also increasing on R≥0, it holds that for all x ≥ 2, λ1(x)λ1(x + 1) ≥ (x + 1)inf x≥2 h(λ1(x)) ≥ h(c0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' As a consequence, for x ≥ 2(α2+α3)2 h(c0) ∨ 2, g(x + 1) g(x − 1) = (α2 + α3)2 λ1(x + 1)λ1(x) ≤ (α2 + α3)2 (x + 1)h(c0) < 1 2, and M is finite by the ratio test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Due to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4, π(x) ∶= 1 M g(x) is the unique complex balanced distribution for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3) and also (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2), and thus a stationary distribution for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' From [36], the reaction network (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) is positive recurrent, hence this distribution is the unique stationary distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Proofs 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Proofs of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' As a consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1, we only need to show one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose that π is a complex balanced distribution of (N,λ) on an irreducible component Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Recall the assumption that λy→y′ = αy→y′λ0 y on Γ for all y ��→ y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For any η ∈ C and r ∈ R, the function ρη,r in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) is a positive constant on {x ∈ Γ∶x ≥ φ(η)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Therefore, λ1,r in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) fulfils λ1,r(x) = c(r)λψ1(r)(x) for some constant c(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' After iteration and completion as in Section 4, we find a cleaved SRN (Ncyc,λcyc) of (N,λ) with projection ψcyc, such that λcyc,r(x) = c(r)λψcyc(r)(x), for all r ∈ Rcyc and x ∈ Γ with positive constants {c(r′),r′ ∈ Rcyc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Choose any r = (y,i) ��→ (y′,i′) ∈ Rcyc, where (y,i),(y′,i′) ∈ Ccyc, such that ψcyc(y,i) = y and ψcyc(y′,i′) = y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose that r is in the k-th connected component (cycle) of Ncyc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4 and Proposition COMPLEX BALANCED DISTRIBUTIONS 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2, since π is a complex balanced distribution of (N,λ) (and thus of (Ncyc,λcyc)) on Γ, we have π(x) = [λcyc,r(x)mcyc,k(x − φcyc(y,i))]−1 = [c(r)λy→y′(x)mcyc,k(x − φ(y))]−1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) for all x ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, the proposition follows if we can show that the ratio mj1,cyc/mj2,cyc is a constant on Γj (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2)) for any indices j1, j2 and j, such that the j1-th and j2-th cycles in Ncyc are both included in the j-th connected component when projected onto N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The following lemma follows from weak reversibility and the proof is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let N be a weakly reversible RN consisting of connected components L1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,Ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose Γ ⊆ Rn is an irreducible component of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For any j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,l}, let Γj be given as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, Γj = {x − φ(y)∶x ∈ Γ} ∩ Zn ≥0, where y is an arbitrary complex in Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For ι = 1,2, let rι = (yι,iι) ��→ (y′ ι,i′ ι) be in the jι-th cycle of Ncyc, written as rι ∈ Lcyc,jι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By convention, we assume ψcyc(rι) = yι ��→ y′ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Furthermore, suppose that ψcyc(r1) and ψcyc(r2) are both in the j-th connected component of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Case 1) Suppose y1 = y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By assumption, λy1→y′ 1/λy2→y′ 2 = αy1→y′ 1/αy2→y′ 2 is a positive constant on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Moreover, due to equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1), it holds for every x ∈ Γ with x − φ(y1) ∈ Zn ≥0, 1 = π(x) π(x) = c(r1)λy1→y′ 1(x)mcyc,j1(x − φ(y1)) c(r2)λy2→y′ 2(x)mcyc,j2(x − φ(y2)) = c(r1)αy1→y′ 1mcyc,j1(x − φ(y1)) c(r2)αy2→y′ 2mcyc,j2(x − φ(y2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By assumption y1 = y2, and performing a change of variable z = x − φ(y1) = x − φ(y2), we get mcyc,j1(z) mcyc,j2(z) = c(r2)αy2→y′ 2 c(r1)αy1→y′ 1 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) is a positive constant, for every z ∈ Γj such that z = x−φ(y1) with some x ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Taking Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1 into account, the identity (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) holds for all z ∈ Γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Case 2) Suppose that y′ 1 = y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Consider reaction r2 and the outgoing reaction of (y′ 1,i′ 1): r′ 1 = (y′ 1,i′ 1) ��→ (y′′ 1,i′′ 1) ∈ Lcyc,j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, by application of Case 1, we immediately get that mcyc,j1(z) mcyc,j2(z) = c(r2)αy2→y′ 2 c(r′ 1)αy′ 1→y′′ 1 , for all z = x − φ(y2) = x − φ(y′ 1) with x ∈ Γ, and thus for all z ∈ Γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Case 3) Suppose y1 = y′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' One can verify that mcyc,j1(z)/mcyc,j2(z) is a positive constant on Γj for every z ∈ Γj following the same lines as in Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Case 4) Suppose y′ 1 = y′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Consider the reactions r′ 1 = (y′ 1,i′ 1) ��→ (y′′ 1 ,i′′ 1) ∈ Lcyc,j1 and r′ 2 = (y′ 2,i′ 2) ��→ (y′′ 2,i′′ 2) ∈ Lcyc,j2, where ψcyc(y′′ ι ,i′′ ι ) = y′′ ι for ι = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Using Case 1, mcyc,j1(z) mcyc,j2(z) = c(r′ 2)αy′ 2→y′′ 2 c(r′ 1)αy′ 1→y′′ 1 , for all z = x − φ(y′ 1) = x − φ(y′ 2) with x ∈ Γ, and thus for all z ∈ Γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Remaining cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Since ψcyc(r1) and ψcyc(r2) are both in the j-th connected component in N, we can find y(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,y(k) ∈ C with y(1) = y1 and y(k) = y2, such that for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,k−1, 22 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA either y(i) ��→ y(i+1) or y(i+1) ��→ y(i) in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This yields that for some indexes q1,q′ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,qk,q′ k, we have (y(i),qi) ��→ (y(i+1),q′ i) or (y(i+1),qi) ��→ (y(i),q′ i) in the j′ i-th connected component in Ncyc for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' It follows from Cases 1-4, that mcyc,j1(z)/mcyc,j′ 1(z) = c0, mcyc,j′ i(z)/mcyc,j′ i+1(z) = ci, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,k − 2, and mcyc,j′ k−1(z)/mcyc,j2(z) = ck−1 for all z ∈ Γj with some positive constants c0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, mcyc,j1(z)/mcyc,j2(z) = ck with some positive constant ck for all z ∈ Γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Therefore, π can be written as in the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='11) with appropriate positive constants κy→y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Moreover, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) is a direct result of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We first prove that N1 ⪰ N with projection ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By definition, it suffices to show that (C,R) = (ψ1(C1),ψ1(R1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In fact, due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1), we have ψ1(C1) = (C ∖ {z}) ∪ {z} = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By definition of R1, we have ψ1(R1) ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' To prove the reverse inclusion, we decompose R = R0 ∪ Rin ∪ Rout, where R0 consists of reactions whose reactant and product are both in C ∖ {z}, and Rin and Rout consist of the incoming and outgoing reactions of z in R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, ψ1(R0 1) = R0 1 = R0 and ψ1(Rin 1 ) = {yi ��→ z∶1 ≤ i ≤ pz} = Rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Recall that N is weakly reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, for every y ∈ C such that z ��→ y ∈ R, there exists a cycle containing z ��→ y, and the incoming reaction of z in this cycle is yj ��→ z for some 1 ≤ j ≤ pz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, (z,j) ��→ y ∈ Rout 1 , and thus z ��→ y ∈ ψ1(Rout 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This implies Rout ⊆ ψ1(Rout 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, N1 ⪰ N with ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Next, we show weak reversibility of N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose that y ��→ y′ ∈ R0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, y ��→ y′ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By weak reversibility of R, there exists a cycle γ ⊆ R containing y ��→ y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' If z ∉ γ, then γ ⊆ R0 1, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Otherwise, suppose z ∈ γ, then there exist i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,pz} and y′ ∈ C∖{z}, such that {yi ��→ z ��→ y′} ⊆ γ ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' As a consequence, {yi ��→ (z,i) ��→ y′} ⊆ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Replacing z by (z,i) in γ, we get a new cycle γ′ ⊆ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' For reactions in Rin 1 or Rout 1 , the same idea is applicable and the details are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Due to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1, it suffices to show that λ1 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In fact, by definition of λ1, we only need to prove that λz→y′(x) = pz ∑ i=1 λ1,(z,i)→y′(x) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3) for y′ ∈ C with z ��→ y′ ∈ R and x ∈ Zn ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) and Condition 1 on (N,λ), then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3) is equivalent to 1{x′∶x′≥φ(z)}(x) = pz ∑ i=1 ∑ γ∈Γyi→z→y′ ∏ r∈γ∖{z→y′} ρz,r(x), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) which is what we will prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' First, if x /≥ φ(z), by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3), both sides of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' COMPLEX BALANCED DISTRIBUTIONS 23 Hence assume that x ≥ φ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let X be the set of all complexes in C that are in the same connected component as z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, weak reversibility, Condition 1 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) imply that for any z1,z2 ∈ X , (i) φ(z1) →N φ(z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (ii) ρz,z1→z2(x) > 0 if and only if z1 ��→ z2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (iii) ∑ z′∈X ρz,z1→z′(x) = ∑ z′∶z1→z′∈R ρz,z1→z′(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This observation allows us to define a discrete time Markov chain (DTMC) on X with tran- sition probability Pz1(z2) = ρz,z1→z2(x) for all z1,z2 ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Moreover, the chain is irreducible with finite state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Therefore, it follows from [31, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='7] that the chain is recurrent, and thus, Py′(τz < ∞) = pz ∑ i=1 ∑ γ∈Γyi→z→y′ ∏ r∈γ∖{z→y′} ρz,r(x) = 1, where τz denotes the first hitting time to state z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This proves (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) and thus completes the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Due to Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='13 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2, it suffices to prove one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Suppose that π is a complex balanced distribution of (N,λ) on Γ, then we need to verify (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) for (N1,λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let η ∈ C ∖{z} ⊆ C1, then from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4), it follows that for any x ∈ Γ, π(x) ∑ y′∶η→y′∈R1 λ1,r(x) =π(x)( ∑ y′∶η→y′∈R0 1 λ1,y→y′(x) + pz ∑ i=1 λ1,η→(z,i)(x)) =π(x)( ∑ y′∶η→y′∈R,y′≠z λη→y′(x) + λη→z(x)) = π(x) ∑ y′∶η→y′∈R λy→y′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5) As π is complex balanced for (R,λ) and φ1 = φ ○ ψ1 = φ on C1 ∖ {(z,i)∶i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,pz}, we have π(x) ∑ y′∶η→y′∈R λy→y′(x) = ∑ y∶y→η∈R π(x + φ(y) − φ(η))λy→η(x + φ(y) − φ(η)) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6) = ∑ y∶y→η∈R0 1 π(x + φ1(y) − φ1(η))λy→η(x + φ1(y) − φ1(η)) + π(x + φ1(z) − φ1(η))λz→η(x + φ1(z) − φ1(η)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) is a consequence of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Next, we will show (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) for η = (z,i) with i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,pz}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Without loss of generality, assume i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By definition, y1 ��→ (z,1) is the only incoming reaction of (z,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Therefore, if x /≥ φ(z), then Condition 1 and the fact that φ1 = φ ○ ψ1 with ψ1 given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1) yields 0 = ∑ y′∶(z,1)→y′∈Rout 1 π(x)λ1,(z,1)→y′(x) = π(x+φ1(y1)−φ1((z,1)))λy1→(z,1)(x+φ1(y1)−φ1((z,1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 24 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA Otherwise, assume x ≥ φ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let X be the set of all complexes in C that are in the same connected component of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' As (N,λ) is complex balanced under π, then for any η ∈ X , 0 < π(x′) ∑ y′∶η→y′∈R λη→y′(x′) = ∑ y∶y→η∈R π(x′ + φ(y) − φ(η))λy→η(x′ + φ(y) − φ(η)), where x′ = x + φ(η) − φ(z) ≥ φ(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This yields that 1 = ∑y∶y→η∈R π(x + φ(y) − φ(z))λy→η(x + φ(y) − φ(z)) π(x + φ(η) − φ(z))∑y′∶η→y′∈R λη→y′(x + φ(η) − φ(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We construct an irreducible DTMC taking values in the finite state space X with transition probabilities pz1,z2 = Pz1(z2) = π(x + φ(z2) − φ(z))λz2→z1(x + φ(z2) − φ(z)) π(x + φ(z1) − φ(z))∑y′∶z1→y′∈R λz1→y′(x + φ(z1) − φ(z)), for any z1,z2 ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, pz1,z2 > 0 if and only if z2 ��→ z1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, the chain is recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' With τz denoting the first hitting time to state z, we have Py1(τz < ∞) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This proves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4) with η = (z,1), if it holds that Py1(τz < ∞) = ∑(z,1)→y′∈R1 λ1,(z,1)→y′(x)π(x) π(x + φ1(y1) − φ1((z,1))λ1,y1→(z,1)(x + φ1(y1) − φ1((z,1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='7) First, by definition it is clear that Py1(τz < ∞) = py1,z + ∑ z′∈X∖{z} py1,z′pz′,z + ∞ ∑ k=2 ∑ {z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=',zk}⊆X∖{z} py1,z1( k−1 ∏ i=1 pzi,zi+1)pzk,z and R = ∑y′∈C ∑γ∈Γyi→z→y′ ∏r′∈γ∖{z→y′} ρz,r′(x)λz→y′(x) π(x + φ(y1) − φ(z)λy1→z(x + φ(y1) − φ(z)) , where R denotes that right hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Additionally, for any z3 ∈ X , such that z1 ��→ z3 ∈ R, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='2) that pz1,z2 = π(x + φ(z2) − φ(z))λz2→z1(x + φ(z2) − φ(z)) π(x + φ(z1) − φ(z))λz1→z3(x + φ(z1) − φ(z))ρz,z1→z3(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Consequently, py1,z = π(x)λz→y1(x)ρz,y1→z(x) π(x + φ(y1) − φ(z))λy1→z(x + φ(y1) − φ(z)), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='8) py1,z′pz′,z =π(x + φ(z′) − φ(z))λz′→y1(x + φ(z′) − φ(z)) π(x + φ(y1) − φ(z))λy1→z(x + φ(y1) − φ(z))ρz,y1→z(x) × π(x)λz→z′(x) π(x + φ(z′) − φ(z))λz′→y1(x + φ(z′) − φ(z))ρz,z′→y1(x) = π(x)λz→z′(x)ρz,y1→z(x)ρz,z′→y1(x) π(x + φ(y1) − φ(z))λy1→z(x + φ(y1) − φ(z)), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='9) COMPLEX BALANCED DISTRIBUTIONS 25 and by iteration, letting z0 = y1, py1,z1( k−1 ∏ i=1 pzi,zi+1)pzk,z = π(x)λz→zk(x)ρz,y1→z(x)∏k i=1 ρz,zi→zi−1(x) π(x + φ(y1) − φ(z))λy1→z(x + φ(y1) − φ(z)), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='10) for all k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='7) follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='8)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='10) and the definition of Γy1→z→y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The proof of this lemma is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' If NM−1 = NM, then by definition, every complex in C′ M−1 ∩ C′ = C′ M ∩ C′ has only one incoming reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' On the other hand, if NM−1 ≠ NM, then the M complexes in C′ are cleaved sequentially in N1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' NM, and thus C′ M ∩ C′ ⊆ CM ∩ C′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Therefore, in either case, no complex in C′ ∩ CM has multiple incoming reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We will show that if (y,i) ∈ C′ M is a copy of y ∈ C′, then (y,i) has only one incoming reaction in NM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' First, y ∈ C′ has only one incoming reaction in N, but multiple incoming reactions in Nm−1 for some m ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,M};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' otherwise (y,i) is not in C′ M ⊆ CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Recall that when one-node cleaves a complex, only the incoming reactions of complexes that are products of the cleaved complex might change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' It follows that the multiple incoming reactions in Rm−1 are due to the cleaving of a complex y′ ∈ C′ ∪ {z} in Nm′ with m′ < m, that is, the reactant of the only incoming reaction of y in R,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,Rm′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' After cleaving y′, the reactant of each incoming reaction of y in Cm′,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,Cm−1 is a copy of y′, and thus when cleaving y in Nm, the reactant (y′,j) of the only incoming reaction of (y,i) in Rm is a copy of y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' As in the cleaving iteration, only complexes in C′ might be cleaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The copy (y′,j) is not cleaved in Nm+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,NM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' As a consequence, the incoming reactions of (y,i) will not change, namely, (y,i) has only one incoming reaction (y′,j) ��→ (y,i) in Rm+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The only concern now is the cleaving of a complex y in Nm with some m ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,M}, fulfilling y ��→ (z,i) ∈ Rm−1 for some i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,pz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The situation is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Consider the RN N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Complex z has two incoming reactions, and C′ = {y1,y2}, in which each complex has only one incoming reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The only cycle including y2 ��→ z included in N is {z ��→ y1 ��→ y2 ��→ z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, after cleaving z in N0, the complex (z,1) has only one outgoing reaction (z,1) ��→ y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, y1 is cleaved in the same manner, resulting in the cleaved RN N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' It remains to cleave the complex y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Note that there is only one cycle including y2 ��→ (z,1) in N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Therefore, after cleaving y2, the complex (z,1) has only one incoming reaction in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This observation allows us to complete the proof as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Assume y ∈ Cm−1 ∩C′ has multiple incoming reactions in Rm−1 (for some m), y ��→ (z,1) ∈ Rm−1, and y is cleaved in Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The reaction y ��→ (z,1) is a result of the cleaving of z in N0, that is y ��→ z ∈ R and y ��→ (z,1) ∈ R0 is the only incoming reaction of (z,1) in R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Additionally, y ∈ C′ has only one incoming reaction in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' It follows that the multiple incoming reactions of y in Rm−1 come from the cleaving of some complex y′ in Nm′ with m′ ∈ {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By iteration, we find a sequence of reactions {z ��→ y(1) ��→ ⋯ ��→ y(k) ��→ y} ⊆ R 26 LINARD HOESSLY, CARSTEN WIUF AND PANQIU XIA with k ∈ {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,m − 1}, such that for each i ∈ {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,k}, complex y(i) ∈ C′ is cleaved in Nmi that increases the number of incoming reactions of y(i+1)in Rmi, where m0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,mk are non- negative integers fulfilling 0 = m0 < m1 < ⋅⋅⋅ < mk ≤ m − 1, with the convention that y(0) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Because {y,y(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,y(k)} ⊆ C′, by definition y(i) ��→ y(i+1) is the only incoming reaction of y(i+1) in R for all i = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,k, where y(0) = z and y(k+1) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, γ0 = {z ��→ y(1) ��→ ⋯ ��→ y(k) ��→ y ��→ z} is the only cycle including reaction y ��→ z in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Therefore, (z,1) ��→ y(1) is the only outgoing reaction of (z,1) in R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Next, consider the cleaving of y(1) in Nm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Neither (z,1) nor y(1) is cleaved in N1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,Nm1−1, thus (z,1) ��→ y(1) ∈ Rm1−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Therefore, after the cleaving of y(1) in Nm1, there is a copy of y(1), denoted by (y(1),1) in Cm1 such that (z,1) ��→ (y(1),1) ∈ Rm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Additionally, this reaction is the only incoming reaction of (y(1),1) and the only outgoing reaction of (z,1) in Rm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Similarly, since neither (y(1),1) or (z,1) is cleaved in Nm1+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,Nm, then it fol- lows that (z,1) ��→ (y(1),1) ∈ Rm1 is the only incoming reaction of (y(1),1) and the only outgoing reaction of (z,1) in Rm1+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,Rm−1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' On the other hand, because none of y(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,y(k),(z,1) are cleaved in N1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,Nm1−1, it holds that γ1 = {y(1) ��→ y(2) ��→ ⋯ ��→ y(k) ��→ y ��→ (z,1) ��→ y(1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' is also the only cycle including (z,1) ��→ y(1) in Nm1−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Thus, (y(1),1) ��→ y(2) is the only outgoing reaction of (y(1),1) in Rm1, and thus in Rm1+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Rm2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By iteration, after cleaving y(k), there is a sequence {(z,1) ��→ (y(1),1) ��→ ⋯ ��→ (y(k),1) ��→ y} ⊆ Rmk such that (y(i),1) ��→ (y(i+1),1) is the only outgoing reaction of y(i) for all i = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,k in Rmk,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,Rm−1, where y(0) = z and (y(k+1),0) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This implies that the only cycle including y ��→ (z,1) in Nm−1 is γ2 = {y ��→ (z,1) ��→ (y(1),1) ��→ ⋯ ��→ (y(k),1) ��→ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' As a consequence, complex (z,1) has only one incoming reaction in Rm, provided y ��→ (z,1) is the only incoming reaction of (z,1) in Rm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' This proves that the number of incoming reactions of (z,1) is one in Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The proof of this lemma is thus complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let (Ncyc,λcyc) be the cleaved SRN obtained by the iterative one-node cleaving procedure (Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, by Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='5, Ncyc, we see that the digraph of Ncyc consists of disjoint cycles that are pairwise non-similar simple cycles when projected onto N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' It suffices to check (i)-(iii) for (Ncyc,λcyc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' First, (i) is a direct consequence of Ncyc ⪰ N and the fact that every cycle in Ncyc is simple when projected onto N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Next, denote by (N ∗ cyc,λ∗ cyc) be the cleaved SRN before completion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, due to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3 and an iteration argument, (iii) holds for (N ∗ cyc,λ∗ cyc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The ‘cut-adhere’ process does not affect the validity of (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' We need to show COMPLEX BALANCED DISTRIBUTIONS 27 that (iii) still holds after the combination process, which seems wrong as in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Still denote by (N ∗ cyc,λ∗cyc) the cleaved SRN after ‘cut-adhere’ before combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, (N ∗ cyc,λ∗ cyc) ⪰ (Ncyc,λ) ⪰ (N,λ), and thus (iii) follows as a result of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Last, we need to prove (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let γ = {y1 ��→ y2 ��→ ⋯ ��→ ym ��→ y1} be a cycle in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Let N1 = (C1,R1,S,φ1) be the cleaved RN of N with projection ψ1 due to one-node cleaving of some z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' If z ∉ {y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,ym}, then every reaction of the cycle is also in R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' On the other hand, without loss of generality, assume that z = y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Then, there exists some index i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' ,pz}, such that ym ��→ (z,i), and by definition of Rout 1 , we have (z,i) ��→ y2 ∈ R1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' In other words, there exists a cycle γ1 = {(z,i) ��→ y2 ��→ ⋯ ��→ ym ��→ (z,i)} ∈ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Therefore, there exists a cycle γ1 ∈ R1, such that ψ1(γ1) = γ in any case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' By iteration, there exists a cycle γ∗cyc ⊆ R∗cyc, such that ψ∗cyc(γ∗cyc) = γ, where N ∗ cyc = (C∗cyc,R∗cyc,S,φ∗cyc) denotes the cleaved RN of N with projection ψ∗ cyc, before the completion step in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Since γ∗cyc is simple, it will not be affected in the ‘cut-adhere’ process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Finally, in the combination process of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='3, the cycle γ∗ cyc may be ‘absorbed’ by other similar cycles when projected onto N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' However, it does not influence the validity of property (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Acknowledgement The work presented in this article is supported by Novo Nordisk Foundation (Denmark), grant NNF19OC0058354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' References [1] Anderson, D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=', (2022+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content=' Department of Mathematical Sciences, University of Copenhagen, Denmark Email address: linard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='hoessly@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='com, wiuf@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='dk Department of Mathematics and Statistics, Auburn University, USA Email address: pqxia@auburn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE2T4oBgHgl3EQfvggn/content/2301.04091v1.pdf'} diff --git a/_9FLT4oBgHgl3EQfES7g/content/tmp_files/2301.11983v1.pdf.txt b/_9FLT4oBgHgl3EQfES7g/content/tmp_files/2301.11983v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7806aaffc8e1e74dfcdb14d9201b1a4693ace561 --- /dev/null +++ b/_9FLT4oBgHgl3EQfES7g/content/tmp_files/2301.11983v1.pdf.txt @@ -0,0 +1,1704 @@ +1 + +Complexity and chaotic behavior of the U.S. rivers and estimation of their +prediction horizon + +Dragutin T. Mihailovića, *, Slavica Malinović-Milićevićb, Jeongwoo Hanc, Vijay P. Singhd +aDepartment of Physics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia; +guto@df.uns.ac.rs +bGeographical Institute “Jovan Cvijić”, Serbian Academy of Sciences and Arts, Belgrade, +Serbia; s.malinovic-milicevic@gi.sanu.ac.rs +cDepartment of Biological and Agricultural Engineering, Texas A&M University, College +Station, TX, USA; han820124@ tamu.edu +dDepartment of Biological and Agricultural Engineering and Zachry Department of Civil & +Environmental Engineering, Texas A&M University, College Station, TX 77843-2117, USA; +vsingh@tamu.edu +*Correspondence: guto@df.uns.ac.rs; tel.: +38121458449 + +Abstract +A streamflow time series encompasses a large amount of hidden information and +reliable prediction of its behavior in the future remains a challenge. It seems that the use of +information measures can significantly contribute to determining the time horizon of rivers +and improving predictability. Using the Kolmogorov complexity (KC) and its derivatives +(KC spectrum and its highest value), and Lyapunov exponent (LE), it has previously been +shown that the degree of streamflow predictability depends on human activities, +environmental factors, and natural characteristics. This paper applied the KC and LE + +2 + +measures to investigate the randomness and chaotic behavior of monthly streamflow of 1879 +rivers from the United States for a period of 1950–2015 and evaluated their time horizons via +the Lyapunov and Kolmogorov time (LT and KT, respectively). +Keywords: Chaos, Lyapunov time (time horizon), Kolmogorov time, predictability, +the U.S. rivers + +1. Introduction +1.1 Considering the turbulent and chaotic behavior of rivers +When we look at a wide river in the lowland, it seems calm and not much turbulent. +However, it is only the impression caused by our perception. Birnir (2008) theoretically +showed the solutions that describe turbulent flow in rivers and also included an invariant +measure for describing the statistical properties of one-dimensional turbulence. Reynolds +number is often used to characterize turbulent flow in rivers and streams. This number for +rivers (𝑅𝑒𝑟𝑖𝑣) is calculated as 𝑅𝑒𝑟𝑖𝑣 = 𝐷𝑉 𝜈 +⁄ , where 𝐷 is the average depth of flow, 𝑉 is the +average velocity, and 𝜈 the kinematic viscosity. For streams and rivers, 𝑅𝑒𝑟𝑖𝑣 is typically +large ( 𝑅𝑒𝑟𝑖𝑣 = 105 − 106) (Dingman, 1984). The turbulence has much more degrees of +freedom than flows in a chaotic mode. On the contrary, all chaotic flows are not necessarily +turbulent. According to Li (2014), the relationship between turbulence and chaos can be +described as follows: “when the Reynolds number is large, violent fully developed turbulence +is due to ‘rough dependence on initial data’ rather than chaos which is caused by ‘sensitive +dependence on initial data’; when the Reynolds number is moderate, turbulence is due to +chaos.” Pursuing this relationship, rivers are par excellence complex systems that can have a +high level of complexity and chaotic behavior. Precisely, chaos has a very accurate +mathematical definition, while turbulence is a property of fluid flow, that has no accurate +mathematical definition. In rivers, spatial and temporal irregular fluctuations, small as well as + +3 + +large, co-occur as three-dimensional eddies. It is difficult to prove whether these are +stochastic or chaotically deterministic. Therefore, turbulence can be (i) one example of the +physical manifestation of deterministic chaos, or (ii) a stochastic, non-chaotic, manifestation +of the solution to the nonlinear fluid flow problem at high Reynolds numbers. The +phenomenon we observe in river flow systems emerges from an underlying disorder and we +embrace the noise and uncertainty as an essential step on the road toward predictability. The +predictability of river streamflow usually refers to (1) the time evolution of the system from +which we can obtain information and (2) the content of obtained information. Thus, our +attention is mostly on a macroscopic model that predicts the state of the system for a longer +period of time and larger spatial scale. Because of the complex nature of rivers, it is difficult +to estimate their prediction horizon (Mihailović et al., 2022). There are some existing +methods for its estimation (Regonda et al., 2013), but all of them have at least one drawback +that does not allow reaching a reliable estimation. +1.2 Studying streamflow complexity +Understanding the dynamic behavior of rivers which is affected by several factors is a +key issue in hydrology. Streamflow is affected by (i) physical factors that include the incline +gradient of the river, water viscosity, elevation, and properties of the surrounding terrain; (ii) +geophysical factors involving the geographical location, weather, and climatic change; and +finally, it is significantly affected by (iii) human activities (including building, river training +works, damming, dredging, deforestation, and pollution). The question arises here as to how +the study of river flow complexity can help unravel the effects of these factors. In the context +of complexity, as it will be discussed in this paper, we will cite (i) the paper by Puente and +Sivakumar (2007) in which stream flow complexity is considered within the general +geophysical complexity and (ii) the paper by Krasovskaia (1997) is a representative of a large + +4 + +group of papers in which streamflow complexity is described through the difference between +time series by offering a correlational explanation. +Information about river streamflow complexity is most reliable if it is computed from +time series by applying some information measures. It seems that algorithmic complexity can +be a good choice, although this measure has not yet found its niche in hydrology except in a +few papers (Sen, 2009; Mihailović et al., 2014; Mihailović et al., 2017). The use of +hydrological models in studying streamflow complexity is not so promising, since with them +it is not possible to model perhaps the most discriminating property of a complex system - +complexity. With this in mind, we applied the Kolmogorov complexity (KC) to monthly +streamflow time series. It is noted that streamflow complexity can be studied by the +Aksentijevic-Gibson complexity as a tool for the analysis of hydrological data that holds the +promise of uncovering patterns in the data that cannot be captured by KC and other +complexity measures (Aksentijevic et al., 2021). + +1.3 Prediction horizon of rivers +In mathematics, there exists a characteristic timescale well known as the Lyapunov +time, also called prediction horizon (which is expressed in the units of the recorded series) +defined as the inverse of the largest Lyapunov exponent of the considered time series. It is a +period after which a dynamical system becomes unpredictable and enters a chaotic state, so it +indicates the limits of predictability. Estimation of the Lyapunov time is related to +computational or inherent uncertainties that often lead to overestimating the actual value of +the time. To correct this overestimation, Mihailović et al. (2019) introduced the Kolmogorov +time as the inverse of the Kolmogorov complexity. This time quantifies the length of the time +window within which complexity remains unchanged, significantly while reducing the size of + +5 + +the effective prediction horizon. River regimes can be simple, mixed, or complex, and one +question is how these regimes relate to complexity, chaotic behavior, and prediction horizon. +The time horizon of streamflow is a consequence of intertwined hydro- +meteorologic forcings (e.g., precipitation, temperature, and evapotranspiration) and +physiography (e.g., slope and elevation) (Knoben et al., 2018; Mathai and Mujumdar, +2022). Higher elevation can impact hydro-meteorological dynamics due to more rapid +changes in airflow and orographical effects on precipitation production (Houze, 2012). +Slope affects the recession of hydrograph (Mathai and Mujumdar, 2022). There is no +doubt that these factors (separately or in synergy) may affect the time horizon of rivers. +Additionally, the naturalized streamflow data minimize human impacts. However, dam +effects may have a dominant influence on the predictability of rivers. It is somewhat +unusual that little attention has been paid to this influence in the hydrological literature. +We are of the opinion that the question of streamflow complexity must be +approached through information measures to obtain more natural and reliable +information. To do that we applied the KC complexity and its derivatives and the +Lyapunov exponent to monthly river flow time series over a 66-year period 1950–2015 +from 1879 rivers in the United States. Therefore, implications of annual mean +precipitation and temperature, slope, elevation, and effects of the existence of dams +upstream of the naturalized streamflow site on statistics (i.e., coefficient of variation +(CV) and average value) and river time horizon of streamflow were used to explain the +results obtained with information measures. + +2. Description of data +Monthly naturalized streamflow data for a period from 1950 to 2015 was obtained +from the U.S. Geological Survey (USGS) Science Base Catalog. The naturalized streamflow + +6 + +is a simulated data for 2,622,273 stream reaches, which are defined by National Hydrography +Dataset (NHD) Version 2.0, across the continental U.S. using the random forest ensemble +(Miller et al., 2018). However, using all the naturalized data of stream reaches >2.5 million is +not only handling the redundant streamflow information but also taking a high likelihood of +using less accurate estimates: the random forest models were calibrated to almost 2,000 +reference gauge sites where the observed streamflow exists, and then the calibrated models +were applied to the ungauged reach segments (Miller et al., 2018). Therefore, we only used +the naturalized streamflow data at stream reaches directly connecting the gauge stations at the +outlet of HUC (Hydrologic Unit Code) 8 so that naturalized streamflow data from only 1879 +sites were used. +Annual mean precipitation and temperature at every point corresponding to the +selected 1879 naturalized streamflow sites were calculated using NOAA nClimGrid monthly +dataset. The NOAA nClimGrid dataset has a period from 1895 to the present and covers +CONUS and Alaska with a 5 km grid resolution (Vose et al., 2014). Therefore, the +nClimGrid precipitation and temperature in the common period for the naturalized +streamflow 1950-2015 were spatially interpolated to the naturalized streamflow sites using +the inverse distance weight (IDW) method (Ahrens, 2006). Besides, the locations of 92075 +dams across CONUS were obtained from the National Inventory Dams (NID). We assigned +the binary value of 1 (0) for dams when they are (not) located upstream of the naturalized +streamflow site in a watershed for further analysis in the following sections. The mean slope +corresponding to each HUC8 watershed was obtained from the U.S. Environmental +Protection Agency (EPA), and elevation at each stream gauge point was obtained from the +USGS. + +7 + +The spatial distributions of monthly streamflow, coefficient of variation (CV), +altitude, slope, and dams’ locations of the U.S. rivers for the period 1950-2015 are shown +in Fig. 1. + + +Fig. 1. Spatial distributions of (a) maximal streamflow in cfs (cubic feet per second), (b) +CV, (c) altitude (m), (d) slope (degree), and (e) dams of the U.S. rivers for the period +1950-2015. + + +(a) +124°W +117°W +110°W +103°W +96°W +89°W +82°W +75°W +68°W +(b) +1240W +117W +110W +103W +96°W +89°W +820W +75°W +68°W +56°N- +56°N +56°N +56°N +54°N- +54°N +54°N +540N +52°N- +52°N +52°N- +52°N +50°N- +50°N +50°N- +50°N +48°N- +48°N +48°N +48°N +46°N- +46°N +46°N- +46°N +44°N- +44°N +44°N +440N +42°N- +42°N +42°N- +42°N +40°N- +40°N +40°N- +40°N +38°N- +38°N +38°N- +38°N +36°N- +36°N +36°N +36°N +34°N- +34°N +34°N- +34°N +32°N- +32°N +32°N- +32°N +30°N- +gaugepoint +30°N +30°N- +30°N +gaugepoint +28°N +Boundaries of HUC8 +28°N +28°N- +Boundaries of HUC8 +28°N +26°N- +Qav (cfs) +26°N +26°N- +CV +26°N +24°N- +≤258 +24°N +24°N- +≤0.72 +24°N +22°N- +1080 +500 +1000 +1500 +2000 km +22°N +22°N- +1.06 +0 +500 +1000 +1901 +1500 +2000km +1.40 +22°N +20°N- +2722 +20°N +20°N- +1.74 +20°N +18°N- +≥3544 +18°N +18°N- +≥2.07 +18°N +124W +1170W +110°W +103°W +96W +890W +82°W +75°W +680W +124W +117W +110W +103°W +960W +89°W +82°W +75°W +68°W +(c) +124°W +117°W +110°W +103°W +96W +89°W +82°W +75°W +68°W +(d) +124W +117W +110W +103°W +960W +89°W +82°W +75°W +68°W +56°N- +56°N +56°N- +56°N +54°N- +54°N +54°N- +54°N +52°N- +52°N +52°N- +52°N +50°N- +50°N +50°N- +50°N +48°N- +48°N +48°N +48°N +46°N- +46°N +46°N- +46°N +44°N- +44°N +44°N- +44°N +42°N- +42°N +42°N- +42°N +40°N- +40°N +40°N- +40°N +38°N- +38°N +38°N +38°N +36°N- +36°N +36°N +36°N +34°N- +34°N +34°N- +Note +32°N- +32°N +32°N- +32°N +30°N- +30°N +30°N- +gauge point +gauge point +28°N- +Boundaries of HUC8 +28°N +28°N- +Boundaries of HUC8 +28°N +26°N- +altitude (m) +26°N +26°N +slope (degrees) +26°N +24°N- +≤32.2 +24°N +24°N- +≤1.26 +24°N +22°N- +408.3 +0 +500 +1000 +1500 +2000km +22°N +22°N- +4.67 +500 +8.09 +1000 +1500 +2000km +784.4 +22°N +20°N +1160 +20°N +20°N- +11.5 +20°N +18°N +≥1537 +18°N +18°N +≥14.9 +18°N +124°W +1170w +110W +103°W +96°W +89°W +82°W +75°W +68°W +124W +117W +110W +103W +96W +89°W +82°W +75°W +68°W +(e) +124°W +117°W +110W +103°W +96°W +89°W +82W +75°W +680W +56°N +56°N +54°N +54°N +52°N +52°N +50°N +50°N +48°N +48°N +46°N +46°N +44°N +44°N +42°N- +42°N +40°N- +40°N +38°N +38°N +36°N +36°N +34°N- +34°N +32°N- +32°N +No0 +30°N +28°N- +no dam +28°N +26°N- +.dam +26°N +24°N +24°N +22°N +0 +500 +1000 +1500 +2000km +22°N +20°N +20°N +18°N- +18°N +124°W +117°W +110W +103°W +96°W +89°W +82°W +75°W +68°W8 + +3. Methodology +3.1 Lempel and Ziv algorithm +Kolmogorov complexity (KC) is a natural but uncomputable information measure. It +is approximated by some compression algorithms - Lempel-Ziv and its variants. Lempel and +Ziv (1976) suggested an algorithm (LZA) for calculating the complexity of a time series +𝑋(𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑁). It includes the following steps. (1) Encoding the time series by creating +a sequence 𝑆 of the characters 0 and 1 written as 𝑠(𝑖), 𝑖 = 1,2, … , 𝑁, according to the rule +𝑠(𝑖)= 0 if 𝑥𝑖 < 𝑥𝑡 or 1 if 𝑥𝑖 > 𝑥𝑡, where 𝑥𝑡 is a threshold. The threshold is commonly +selected as the mean value of the time series, while other encoding schemes are also available +(Radhakrishnan et al., 2000); (2) calculating the complexity counter 𝑐(𝑁). 𝑐(𝑁) is defined as +the minimum number of distinct patterns contained in a given character sequence. The +complexity counter 𝑐(𝑁) is a function of the length of sequence 𝑁. The value of 𝑒(𝑁) +approaches an ultimate value 𝑐(𝑁) as 𝑁 approaches infinity, i.e. 𝑐(𝑁) = +𝑂(𝑏(𝑁)) and 𝑏(𝑁) = 𝑙𝑜𝑔2𝑁; (3). Calculating the normalized information measure 𝐶𝑘(𝑁), +which is defined as 𝐶𝑘(𝑁) = 𝑐(𝑁)/𝑏(𝑁) = 𝑐(𝑁)/𝑙𝑜𝑔2𝑁. For a nonlinear time series, +𝐶𝑘(𝑁) varies between 0 and 1, although it can be larger than 1. Note that the pattern is a +sequence in the coded time series which is unique and non-repeatable. A flow chart for the +calculation of KC of a streamflow series 𝑋(𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑁) using the LZA algorithm is +shown in Fig. 2. + +9 + + +Fig. 2. Flow chart for calculation of the Kolmogorov complexity (KC) using the +Lempel–Zev algorithm (LZA) (by permission Mihailović et al., 2019) + +3.2 Kolmogorov complexity spectrum and its highest value +The Kolmogorov complexity of time series has two weaknesses: (i) it cannot +distinguish between time series with different amplitude variations and that with similar +random components; and (ii) in the conversion of a time series into a binary string, its +complexity is unseen in the rules of the applied procedure. Therefore, in defining a threshold +for a criterion for coding, some information about the composition of time series could be +lost. In the complexity analysis of time series, two measures are used: (i) Kolmogorov +complexity spectrum (KC spectrum) and (ii) the highest value of KC spectrum (KCM), +introduced by Mihailović et al. (2015a) who described the procedure for calculating the KC +spectrum. The flow chart in Fig. 3 shows schematically how to calculate the KC spectrum +𝐶(𝑐1, 𝑐2, 𝑐3, … , 𝑐𝑁) for time series 𝑋(𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑁). This spectrum allows us to investigate +the range of amplitudes in a time series that represents a complex system with highly +enhanced stochastic components. It may be noted that for a large number of samples of a time + +Start +个 +Streamflowseries +treshold xt +不 +Data sequences +construction,N=length (S) +不 +Initialization(Q=null） +SQ元 = S(1) c(n)=1, i=2) +No +ReadS(i)andscoreintoQand +Yes +thecharacterS(i-1)intoSQ元 +i>N +Yes +SQ元contains +Q? +i=i+1 +KC=c(N)log,N +个 +No +End +c(n)=c(n)+110 + +series, the computation of KC spectrum can be challenging. The highest value 𝐾𝑚 +𝐶 as in this +series, i.e., 𝐾𝑚 +𝐶 = 𝑚𝑎𝑥{𝑐𝑖}, is the highest value of the KC complexity spectrum. + +Fig. 3. Flow chart for calculation of the Kolmogorov complexity spectrum and its +highest value (KCM) (by permission Mihailović et al., 2019). + +3.3 Lyapunov exponent +The Lyapunov exponent of a dynamical system is a quantity that characterizes the rate +of separation of infinitesimally close trajectories. Positive Lyapunov exponent (LE) indicates +that small fluctuations can lead to drastically different system behavior (small differences in +the initial state lead to large differences in a later state). Because the rate of separation can be +different for different orientations of the initial separation vector, there is a spectrum of +Lyapunov exponents whose largest value is commonly the LE. A positive value of this +exponent is taken as an indicator that a dynamical system is chaotic. In this study, we +obtained the LE for the standardized monthly streamflow time series by applying the + +Start +个 +Streamflowseries +X(X, .,...), 1 +treshold x,=X; +个 +Data sequenceS +construction,N=length(S) +个 +Initialization(Q=null) +SQ元 =null, c(n)=0) +个 +CalculateKC and store it into +array C(C, C2, C.....,.Cn) +不 +No +i>N +i=i+1 +Yes +KCM= max (C, C2, C3 +个 +End11 + +Rosenstein algorithm (Rosenstein et al., 1993) which was implemented in the MATLAB +program (Shapour, 2009). However, this measure has one drawback: If the embedding theory +is used to build chaotic attractors in the reconstruction space, then additional “spurious” +Lyapunov exponents appear. + +4. Results and Discussion +4.1 Spatial analysis +4.1.1 Picture of the US rivers: complexity and chaos +The scatter plot of KC versus LE is like a “boomerang” shape (Fig. 4). The scatter +plot area, by two lines LE = 0.146 (parallel to the y-axis) and KC=0.516 (parallel to the x- +axis), is divided into rectangles for the sake of a better visualization for further analysis. +These two numbers are the means of the maximal and minimal values of KC and LE in the +set of gauge stations. Perhaps, at first glimpse, this kind of scatter plot (with a large number +of samples) seems surprising and may appear for the first time to our knowledge. The figure +shows that streamflow time series for all gauge stations give a picture of a mixture consisting +of always present chaos and high randomness. Randomness is unpredictable because we just +have no right information, while chaos is somewhere between random and predictable. A +hallmark of chaotic streamflow time series is predictability. Thus, with this picture of the +scatter plot of KC versus LE, it can be said that it + + + +12 + + +Fig. 4. Scatter plot of Kolmogorov complexity (KC) versus, and Lyapunov exponent +(LE) for the U.S. monthly streamflow for the period 1950-2015. (a) LD (left-down); (b) +LU (left-upper); (c) RU (right-upper) and (d) RD (right-down) are parts of the scatter +plot area. The numbers in rectangles indicate the total number of gauge stations. + +is difficult to reach a reliable prediction of the U.S. river streamflow. In addition, depending +on the time scale considered, river discharge can be either random or chaotic. Thus, at daily +and seasonal scales the river discharge is random (non-chaotic), but the flow is chaotic at the +monthly scale (Adab et al., 2018). A simple percentage calculation with numbers from Fig. 4 +shows that the LU values of all gauge stations are positive, while 83.7 % of their KC values +are higher than 0.516 (LU and RU parts). In the two lower parts (LD and RD) randomness +and LE of streamflow are mainly lower. From Fig.4, it is simple to find that the ratio of the +number of gauge stations with low and high randomness is approximately one to five. This is +an indicator that the predictability of the U.S. river streamflow is not high, i.e., we talk about +the chance that river streamflow could be in principle modeled reliably. We do not talk about +heuristic hydrological models, which sometimes can give good results that come from their +mathematical background. + + +(b) 955 +619 +0.6 +KC +(a) 280 +(d) 25 +0.05 +0.10 +0.15 +0.20 +0.25 +LE13 + +4.1.2 Complexity + +The complexity of river discharge is a key issue in hydrology. This paper considers +the KC complexity. The spatial distribution of KC of the US rivers' monthly streamflow for +the period 1950-2015 is shown in Fig. 5a. From this figure three distinct patterns for KC are +seen, and more if we consider a finer scale. The southwestern part which is arider has lower +KC than what is in the more humid part of the eastern U.S. Further, in the southwestern part, +there is a band of very low KC. This band cuts across the mountainous terrain. The +Mississippi valley, the Great Lakes region, and the Atlantic Seaboard have high KC. +However, the Ohio Valley has a lower (but still high) KC than the surrounding area. It seems +that besides landscape and physiographic characteristics, KC may have a strong correlation +with weather patterns and distance from the sea (continentality). One can make a similar +observation about LE (Fig. 5b). It is quite difficult and not always consistent to determine +explicitly the connection between physiographic characteristics and the complexity of river +flow. For example, based on monthly streamflow time series from ten-gauge stations at seven +rivers of different river regimes in Bosnia and Herzegovina, Mihailović et al. (2015b) found +that the relationship between the highest value of the KC spectrum (KCM) and elevation (h). +That relation KCM=0.0002h + 0.9421 shows that there is a positive trend in changes of the +KCM with respect to elevation with the coefficient of correlation of 0.602 + + + +(a) +124°W +117°W +110°W +103°W +96°W +89°W +82°W +75°W +68°W +(b) +124°W +117W +110W +103°W +96W +89°W +82°W +75°W +68°W +56°N +56°N +56°N- +56°N +54°N- +54°N +54°N- +54°N +52°N +52°N +52°N- +52°N +50°N- +50°N +50°N +50°N +48°N +48°N +48°N- +48°N +46°N +46°N +46°N- +46°N +44°N +44°N +44°N- +44°N +42°N +42°N +42°N- +42°N +40°N- +40°N +40°N- +40°N +38°N- +38°N +38°N- +No8E +36°N +36°N +36°N- +No9 +34°N +Nobe +34°N- +34°N +32°N +32°N +32°N- +32°N +30°N +NOE +30°N- +.gaugepoint +30°N +gaugepoint +28°N +BoundariesofHUC8 +28°N +28°N- +Boundaries of HUC8 +28°N +26°N- +KC +26°N +26°N +LE +26°N +24°N +≤0.351 +24°N +24°N +≤0.097 +24°N +22°N +0.468 +0 +500 +1000 +1500 +2000km +22°N +22°N- +0.119 +0 +500 +1000 +1500 +2000km +0.585 +0.141 +22°N +20°N +0.703 +20°N +20°N- +0.163 +20°N +18°N- +≥0.820 +18°N +18°N- +≥0.184 +18°N +124W +117W +110W +103W +96°W +89°W +82°W +75W +68°W +124W +117w +110W +103W +96W +89W +82°W +75°W +68W14 + +Fig. 5. (a) Spatial distribution of KC and (b) LE of the U.S. river monthly streamflow for +the period 1950-2015. + +We hypothesize that the turbulent nature of rivers and barriers built by human +activities may remarkably affect the complexity of rivers. When we say human activities, we +primarily mean dams but also channels. Figure 1e shows the spatial distribution of dams +(1796) built on the U.S. rivers making up 95.6% of the statistical set of 1879 gauge stations +used in this study. A small number of papers are devoted to this issue. And if there are any, +they mostly remain on the descriptive approach supported by traditional statistics. To address +the complexity of streamflow, Mihailović et al. (2019) analyzed daily streamflow data +recorded during the period 1989–2016 at twelve gauging stations on the Rio Brazos River in +Texas (USA) using KC and its derivatives. They found a huge increase in KCM at one gauge +station in comparison to other ones, concluding that the reason for the high KCM of this +station may be attributed to the Morris Sheppard Hydroelectric Power plant at Morris +Sheppard Dam. The KC complexity as a measure does not distinguish between time series +with different amplitude variations and similar random components. It seems that changing +the river flow in the operating mode of the dam does not only change the amplitude but also +the randomness, which is reflected by higher complexity, i.e., what is captured by KC +appropriately. In this paper, the range of the calculated measures was in the intervals of +0.097-0.936 (KC) and 0.011-0.282 (LE), respectively. The spatial distribution of KC in Fig. +6a is very similar to the distribution of KC in Fig. 5a. In Fig. 6a the number of dams is 1796, +while the number of dams among gauge stations with KC > 0.516 is 1523 or 84.7% of the +total number of dams. This is almost the same as the ratio of 83.7% (number of gauge +stations with KC > 0.516 and their total number) in 4.1.1. + +15 + +It is noted that the channelization of rivers decreases the complexity of river flow. For +example, applying the KC complexity analysis of monthly river discharge, Mihailović et al. +(2014) found that during the period 1926–1990 there was a drop in KC in the mountain rivers +in Bosnia and Miljacka (Bosnia and Herzegovina) for the period 1946-1965, in comparison +with two other periods (1926-1945 and 1966-1990). That complexity loss was interpreted as a +result of intensive different human interventions on those rivers (establishing the network of +channels for building the capacities for water consumption) after the Second World War. +Certainly, channelization is a type of human intervention that contributes to reducing the +randomness of the U.S. rivers having KC less than 0.516 with an amount of 16.2% of the +total number of gauge stations. The division of river flow regimes into (i) low complex (KC < +0.516) and high complex (KC ≥ 0.516) and (ii) low chaotic (LE < 0.146) and high chaotic +(LE ≥ 0.146) is merely conditional. + +Fig. 6. (a) Spatial distribution of KC (KC < 0.516; KC ≥ 0.516 while 0.011 < LE < +0.282) and (b) LE (LE < 0.146; LE ≥ 0.146 while 0.097 < KC < 0.936) of the U.S. river +monthly streamflow in the presence of dams for the period 1950-2015. + +4.1.3 Chaotic behavior +The spatial distribution of LE of monthly discharge of the U.S. rivers for the period +1950-2015 is shown in Fig. 6b. This is the spatial distribution taken out from the distribution +in Fig. 5b following the mentioned criterion of division of U.S. river flows into low chaotic + +(a) +124W +117W +110W +103°W +96°W +89°W +82°W +75°W +68°W +(b) +124°W +117°W +110W +103°W +96°W +89°W +82°W +75W +68W +56°N- +56°N56°N +56°N +54°N- +54°N 54°N +54°N +52°N- +520N52°N +52°N +50°N- +50°N50°N +50°N +48°N- +48°N48N +48°N +46°N- +46°N46°N +46°N +44°N +44°N44°N +44°N +42°N +42°N42°N- +42°N +40°N- +40°N40°N +40°N +38°N- +38°N38°N +38°N +36°N- +36°N36°N +N.96 +34°N- +Noe Note +34°N +32°N- +32°N32°N +32°N +30°N- +30°N +28°N- +BoundariesofHUC8 +28°N28°N +BoundariesofHUC8 +28°N +26°N- +26°N26°N- +gauge points +26°N +24°N +24°N24°N +gauge points +KC<0.516,LE=0.011-0.282 +LE<0.146,KC=0.097-0.936 +24°N +22°N- +·KC≥0.516,LE=0.011-0.282 +0 +500 +1000 +1500 +2000km +22°N22°N +LE≥0.146,KC=0.097-0.936 +0 +500 +1000 +1500 +2000km +22°N +20°N- +20°N20°N- +20°N +18°N- +18°N18°N- +18°N +124W +117W +110°W +103°W +96°W +89°W +82°W +75°W +68°W +124w +117oW +1100W +1030W +96W +89°W +82°W +75°W +68°W16 + +(LE < 0.146) and high chaotic (LE ≥ 0.146). From this figure it is seen that low chaos +prevails significantly. High chaos dominates in the Mississippi Valley, the Great Lakes +region, the Upper Mid-West, and the Atlantic Seaboard, while in the Western U.S. it is less +prevalent. +We already stated that chaos is always present in turbulent flow. Therefore, the rivers, +which have positive LE, are in a chaotic regime. This phenomenon is intriguing as a topic in +hydrology as well as in all sciences. However, hydrologists mostly have dealt with those +topics in the following way. (1) They often keep on a descriptive level that is not always +necessarily simple, paying more attention to the mathematical background with comments +about possible applications. (2) They usually consider just a low-dimensional chaos, i.e., to +be attributable to a small fraction of the components of the total system, using a smaller +number of streamflow time series with a focus on one possible source that causes that chaos. +(3) Some of them used theoretical approaches, inverse modeling, and information measures +to gain insights into the river flow's chaotic nature. (4) There is almost no information that +anyone dealt with the high-dimensional chaotic regime of river flow, i.e., turbulent flow +having many degrees of freedom (Porporato and Ridolfi, 1997; Sivakumar, 2000; Sivakumar +and Jayawardena, 2002; Labat et al., 2011; Fattahi et al., 2013; Yildirim et al., 2016; +Mihailović et al., 2019). +The spatial distribution of LE (0.09-0.18) of monthly streamflow of the U.S. rivers for +the period 1950-2015 is shown in Fig. 7a. This time interval was chosen so that their +endpoints were nearly symmetrical with respect to LE = 0.146. The spatial distribution of +gauge stations in this figure in comparison with the spatial distribution of dams in Fig. 1e is +almost indistinguishable. Further inspection of Fig. 7a shows that the number of gauge +stations in the Western part (which is mostly mountainous) is approximately the same as the +number of gauge stations in the Eastern part. This leads us to the conclusion that the + +17 + +influence of orography on the river flow dynamics is much smaller than the influence of +dams, i.e., the influence of human activity. + +Fig. 7. (a) Spatial distribution of river gauge stations (0.09 < LE < 0.18) of monthly +discharge of the U.S. rivers for the period 1950-2015 and (b) histogram of their numbers +in dependence on LE. + +If we look at the scatter plot (Fig. 4) we can see that LE values of gauge stations from +Fig. 7a are mostly grouped on the right side of LU and the left side of RU. The number of +gauge stations with a frequency greater or equal to 100 is 1522 (Fig. 7b), or 81.0% of the +total number of stations. It is interesting to consider the “tail” of the histogram in Fig. 7b. It +corresponds to the state of high chaos and low complexity of river flow (RD part in Fig. 4). +This may indicate the appearance of periodicity or another pattern on some other time scales +that can be ascribed to specific environmental factors or human activity (Aksentijevic et al., +2020). The occurrence of periodicity on other time scales does not necessarily mean that it +will be maintained over long periods. +4.2 Time horizon and complexity spectrum of river flow amplitudes + +(a) +124W +117W +110W +103W +96W +89°W +82°W +750W +68°W +(b) +56°N +56°N +260 +260 +54°N- +54°N +52°N +52°N +240 +50°N +50°N +220- +48°N +48°N +46°N +46°N +200 +44°N- +44°N +180- +42°N- +42°N +40°N- +40°N +38°N- +38°N +140 +36°N- +36°N +120 +34°N- +34°N +100- +32°N- +32°N +80- +30°N +30°N +28°N- +28°N +60 +26°N- +gaugepoints 0.09 6 can be observed especially in the Great Lakes +area. (3) In the western region and part of the Southwest, the LT values are high and in some + +(a) +124°W +117°W +110°W +103°W +96°W +89°W +82°W +75°W +68°W +(b) +56°N +56°N +54°N +54°N +Qav=3473 +52°N +52°N +500 +50°N +50°N +Qav=4227 +48°N- +48°N +46°N +46°N +400 +Qav=3626 +44°N- +44°N +Qav=14193 +42°N +42°N +40°N +40N +frequency +300- +Qav=4882 +38°N +38°N +36°N +No9E +34°N +34°N +200- +Qav=4233 +32°N +32°N +Qav=3194 +30°N +gaugepoint +NoOE +Qav=2486 +28°N +Boundaries of HUC8 +28°N +Qav=5582 +1511 +Qav=2814 +Qav=2063 +26°N +LT +26°N +100- +=AEO +24°N +<5.68 +7.59 +24°N +22°N +9.49 +1500 +2000km +22°N +20%N +11.4 +20°N +0- +18°N +>13.3 +18°N +4 +10-11 +11-12 +12-13 +13-14 +14-15 +124w +1170w +110°w +103W +96°W +89W +82°W +75°W +68°W +15+ +LT categories +(c) +(d) +124°W +117°W +110W +103°W +96°W +Mo68 +82°W +75°W +Mo89 +56°N +56%N +400 +400 +54°N- +54°N +frequency +520N +52°N +dams +50°N +50°N +48°N- +48%N +300 +46°N- +46°N +300 +44°N +44°N +number +42°N +42°N +40°N +40°N +38°N +38°N +200 +36°N +No9E +dams +34°N +NobE +32°N +32°N +30°N +gaugepoint +NoOE +100 +28°N +Boundaries of HUC8 +28°N +100 +26°N +KT +26°N +240N +≤1.22 +24°N +22°N +500 +1000 +1500 +2000km +22°N +20°N +2.84 +20°N +18°N +>3.38 +18°N +3 +4 +6-8 +9-10 +10-11 +11-12 +12-13 +13-14 +14-15 +124°W +117°W +110°W +103°W +96°W +89°W +82°W +75°W +68°W +LT categories +(e) +2100 +Qav=5604 +1800 +1500- +1200 +900 +600- +Qav=1870 +Qav=2821 +Qav=7808 +Qav=13865 +Qav=1593 +Qav=1597 +300 +2 +4 +6 +10-11 +KT categories19 + +parts they exceed the value of 13. The lower LT values include a narrow strip immediately +along the coast. If the spatial distribution of LT is transferred to a histogram (Fig. 8b), then +the causes of this spatial distribution become more understandable. The histogram shows that +in the LT categorization, the largest number of gauge stations is located in the interval (5-6) – +(9-10). Such LT values are the result of the LE distribution in Figure 7b, where the frequency +distribution is the highest in the interval (0.09, 018). The histogram in Figure 7c shows that +the largest number of the U.S. rivers have LE values that are in the interval between 0.125 +and 0.143, i.e., with a time horizon that is between seven and eight months. +In this paper, we try to see which factor has the greatest influence on the time horizon +of rivers. It seems to be a factor that affects the dynamics of river flow originating from +human activities. Certainly, there are other factors, but this one is extremely dominant, as can +be seen in Fig. 8c. It shows a high correspondence between LT and the number of dams on +rivers. This was pointed out by Mihailović et al. (2019) where LE, as well as KC, were +analyzed at twelve stations on the Brazos River. The influence of dams on river flow is +discussed in subchapter 4.1.2 where the influence on KC is considered. That influence +(changes in river flow operating and nonoperating mode of dams) reflects also on LE, i.e., on +LT. Figures 8d and 8e show the spatial distribution and histogram of KT of the U.S. rivers +from which it is seen that that time is no longer than 1-2 months. That time quantifies the +time window size within which complexity remains unchanged. Hence, the presence of a +narrow window significantly reduces the length of effective prediction horizon. Thus, the +relationship between KT, LT, and Qav may provide a deeper insight into the predictability of +river streamflow under the KC spectrum of mean monthly streamflow (Qav), for different LT +categories. To obtain the KC spectrum for LT categories (5-9 months), a time series of mean +monthly streamflow was formed by averaging over all gauge stations whose LT was in those +LT categories. These LT categories were selected using Fig. 8b. The KC spectrum was + +20 + +determined for that time series. The number of stations was: (1) Qav < 500 cfs (38.1 % of the +total number of gauge stations (1531 with 1465 dams) in the considered interval of LT); (2) +500 ≤ Qav < 2500 cfs (36.2%); (3) 2500 ≤ Qav < 8500 cfs (16.1%) and (4) Qav ≥ 8500 cfs +(9.6%-gauge stations). From Fig. 9 is seen that the dependence of zones of KC on mean +monthly streamflow Qav can be distinguished as follows. + +Fig. 9. Kolmogorov complexity spectrum of mean monthly streamflow Qav (cfs) of the +U.S. rivers for different categories (5-9 months); the number indicates the left side point +of one month's interval) of the Lyapunov time (LT). Streamflow zones are divided by +vertical lines. + +Zone 1 (500 ≤ Qav < 2500 cfs) is a zone of increasing KC and LE (ring of gauge +stations in the West, central and eastern parts of the Midwest, the eastern part of the +Southwest, Southeast, and Northeast, as shown in Fig.10a). In the course of rivers, +randomness is more prevalent, which, despite being high (KT is lower), does not significantly +affect the time horizon, since the smaller values of LE (around 0.139 on average for the LT +categories (5-9 months); see Fig. 8b) increase the predictability. Zone 2 (2500 ≤ Qav < 8500 +cfs). This zone has a distribution that is similar to zone 1. In its part of the KC spectrum, the + +zone 1 +zone 2 +zone 3 +0.6 +LT = 5-9 +0.4 - +0.2 +0.0 +500 +2500 +4500 +6500 +8500 +10500 +12500 +14500 +16500 +Qav (cfs)21 + +fluctuations in complexity are emphasized moving towards higher streamflows, but in an +increasing trend. The comment for the time horizon in this zone is similar to zone 1, with the +predictability increase going towards higher river streamflow. Zone 3 (Qav ≥ 8500 cfs). This +zone includes the smallest number of stations placed in the northern part of the West, partly +in the Southwest and eastern part of the U.S.. In this zone, KC, as well as LE, decreases +resulting in a long time of predictability. +Qav < 500 cfs is a zone of very low KC and also lower LE (38.1% gauge stations). +This zone includes a band extending along the western part of the Midwest and the eastern +part of the Southwest, the central part of the West, and a narrow belt along the Atlantic coast +in the Northeast (Fig. 10a). Low values of KC and LE result in a longer time horizon. This +zone is not visible on the KC spectrum due to extremely small KC values and also due to +averaging, but it exists. +Figure 10b visualizes three-dimensional relationship between KT, LT, and Qav, +which includes LT time horizon categories of 5-9 months. + +Fig. 10. The map mean LT time horizon categories (5-9 months) for different streamflow +intervals of the U.S. rivers and (b) three-dimensional visualization of mean KT, LT +versus streamflow (Qav). + + +(a) +124W +117°W +1100W +103W +96°W +890W +82W +75W +680W +(b) +56°N +56°N +54°N +54°N +52°N +52°N +LT=5-9 +50°N- +50°N +48°N +48°N +46°N- +46°N +44°N- +44°N +8500 +42°N +42°N +8000 +40°N +40°N +7000 +00 +38°N +38°N +6500 +36°N- +36°N +34°N +34°N +500 +32°N +32°N +30°N- +No0 +3000 +28°N +28°N +2000 +BoundariesofHUC8 +1500 +26°N +26°N +1000 +24°N- +Qav≤500cfs +24°N +500 +22°N- +500cfs8500cfs +20°N +18°N +18°N +124°W +117W +110°W +103°W +96°W +89°W +82°W +75°W +68°W22 + +Undoubtedly, dams also and other types of human activity affect the dynamics of +rivers. However, many other factors, such as weather, climate, orography, continentality of +the place, etc., separately or in synergy, affect the size of time horizon, i.e. predictability of +streamflow. It seems that the influence of these factors is less significant than is human +activity. Although the focus of this paper is not the quantification of influence of +environmental and other natural factors on the LT time horizon categories of rivers, we list +some averages of those factors. It is done for gauge stations having the LT time horizon +categories (5-9), i.e. the longest ones for the U.S. rivers. They are (1) the number of gauge +stations (1531); (2) the number of dams (1465); monthly streamflow (5676 cfs); CV (1.207); +slope (5.1 degrees); altitude (400 m); temperature (12.7 oC); and average annual precipitation +(33.3 inches). +5. Conclusions +Monthly streamflow data (1950-2015) from 1879 gauge stations on the U.S. rivers +were analyzed using the Kolmogorov complexity (KC) and related complexity measures +(Kolmogorov complexity spectrum) and Lyapunov exponent (LE) to establish the time +horizon of rivers by calculating the Lyapunov time (LT) and Kolmogorov time (KT). The +following conclusions are drawn from this study: +(1) The values of calculated measures were in the intervals 0.097-0.936 (KC) and +0.011-0.282 (LE), respectively; + +(2) The number of gauge stations with KC > 0.516 was 1574 (83.7% of the total +number of gauge stations (1879) while LE > 0 was obtained for all gauge stations; +(3) The high complexity (KC) and the presence of chaos in the streamflow of all U.S. +rivers (LE is always positive) may be addressed to human activities, primarily in the presence +of a large number of dams (1796 or 95.6% of the total number of gauge stations); with their + +23 + +mode of operation they introduce significant changes in the complexity and the turbulent +flow of rivers, increasing the level of chaos); +(4) The West region and southwestern part of the Southwest region have LT +(Lyapunov time or time horizon) between 10 and 13 (or more) months; the rest of the +Southwest region, Midwest, Southeast, and Northeast regions have LT between 5 and 9 +months; +(5) The number of gauge stations with LT between 5 and 9 months is 1531 with the +following frequency distribution in relation to LT categories (in months): 258 (5-6), 356 (6- +7), 423 (7-8), 286 (8-9), and 208 (9-10); +(6) Human activity affects the dynamics of rivers but many other factors, such as +weather, climate, orography, continentality of the place, etc., separately or in synergy, affect +the size of time horizon that was not the focus of the paper. However, there is a justified +expectation that the connection of Kolmogorov complexity and Lyapunov exponent with +environmental factors can quantify their role in the predictability of streamflow without the +use of traditional mathematical statistics. It will be the content of our forthcoming paper. + +Declaration of Competing Interest +The authors declare that they have no known competing financial interests or personal +relationships that could have appeared to influence the work reported in this paper. +Data availability +The data authors used are publicly available online: Monthly naturalized streamflow data (at +https://www.sciencebase.gov/catalog/item/59cbbd61e4b017cf314244e1), NOAA nClimGrid +monthly precipitation and temperature data (at +https://www.ncei.noaa.gov/access/metadata/landing-page/bin/iso?id=gov.noaa.ncdc:C00332), + +24 + +National Inventory Dams data (at https://www.fema.gov/emergency-managers/risk- +management/dam-safety/national-inventory-dams), The mean slope of watershed from EPA +(at https://www.epa.gov/wsio/wsio-indicator-data-library), Elevation of gauge station from +USGS (at https://waterdata.usgs.gov/nwis/sw). + + References +Adab, F., Karami, H., Mousavi, S. F., Farzin, S., 2018. 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Math. Phys. 04, 1849–1858. +https://doi.org/10.4236/jamp.2016.410187 + + diff --git a/_9FLT4oBgHgl3EQfES7g/content/tmp_files/load_file.txt b/_9FLT4oBgHgl3EQfES7g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..10299c8c7b22b3203efdc6a9d7cc86114135201e --- /dev/null +++ b/_9FLT4oBgHgl3EQfES7g/content/tmp_files/load_file.txt @@ -0,0 +1,1361 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf,len=1360 +page_content='1 Complexity and chaotic behavior of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' rivers and estimation of their prediction horizon Dragutin T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Mihailovića, *, Slavica Malinović-Milićevićb, Jeongwoo Hanc, Vijay P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Singhd aDepartment of Physics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' guto@df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='uns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='rs bGeographical Institute “Jovan Cvijić”, Serbian Academy of Sciences and Arts, Belgrade, Serbia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='malinovic-milicevic@gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='sanu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='rs cDepartment of Biological and Agricultural Engineering, Texas A&M University, College Station, TX, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' han820124@ tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='edu dDepartment of Biological and Agricultural Engineering and Zachry Department of Civil & Environmental Engineering, Texas A&M University, College Station, TX 77843-2117, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' vsingh@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='edu Correspondence: guto@df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='uns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='rs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' : +38121458449 Abstract A streamflow time series encompasses a large amount of hidden information and reliable prediction of its behavior in the future remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It seems that the use of information measures can significantly contribute to determining the time horizon of rivers and improving predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Using the Kolmogorov complexity (KC) and its derivatives (KC spectrum and its highest value), and Lyapunov exponent (LE), it has previously been shown that the degree of streamflow predictability depends on human activities, environmental factors, and natural characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This paper applied the KC and LE 2 measures to investigate the randomness and chaotic behavior of monthly streamflow of 1879 rivers from the United States for a period of 1950–2015 and evaluated their time horizons via the Lyapunov and Kolmogorov time (LT and KT, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Keywords: Chaos, Lyapunov time (time horizon), Kolmogorov time, predictability, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' rivers 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='1 Considering the turbulent and chaotic behavior of rivers When we look at a wide river in the lowland, it seems calm and not much turbulent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' However, it is only the impression caused by our perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Birnir (2008) theoretically showed the solutions that describe turbulent flow in rivers and also included an invariant measure for describing the statistical properties of one-dimensional turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Reynolds number is often used to characterize turbulent flow in rivers and streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This number for rivers (𝑅𝑒𝑟𝑖𝑣) is calculated as 𝑅𝑒𝑟𝑖𝑣 = 𝐷𝑉 𝜈 ⁄ , where 𝐷 is the average depth of flow, 𝑉 is the average velocity, and 𝜈 the kinematic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' For streams and rivers, 𝑅𝑒𝑟𝑖𝑣 is typically large ( 𝑅𝑒𝑟𝑖𝑣 = 105 − 106) (Dingman, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The turbulence has much more degrees of freedom than flows in a chaotic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' On the contrary, all chaotic flows are not necessarily turbulent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' According to Li (2014), the relationship between turbulence and chaos can be described as follows: “when the Reynolds number is large, violent fully developed turbulence is due to ‘rough dependence on initial data’ rather than chaos which is caused by ‘sensitive dependence on initial data’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' when the Reynolds number is moderate, turbulence is due to chaos.” Pursuing this relationship, rivers are par excellence complex systems that can have a high level of complexity and chaotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Precisely, chaos has a very accurate mathematical definition, while turbulence is a property of fluid flow, that has no accurate mathematical definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' In rivers, spatial and temporal irregular fluctuations, small as well as 3 large, co-occur as three-dimensional eddies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It is difficult to prove whether these are stochastic or chaotically deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Therefore, turbulence can be (i) one example of the physical manifestation of deterministic chaos, or (ii) a stochastic, non-chaotic, manifestation of the solution to the nonlinear fluid flow problem at high Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The phenomenon we observe in river flow systems emerges from an underlying disorder and we embrace the noise and uncertainty as an essential step on the road toward predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The predictability of river streamflow usually refers to (1) the time evolution of the system from which we can obtain information and (2) the content of obtained information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Thus, our attention is mostly on a macroscopic model that predicts the state of the system for a longer period of time and larger spatial scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Because of the complex nature of rivers, it is difficult to estimate their prediction horizon (Mihailović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' There are some existing methods for its estimation (Regonda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2013), but all of them have at least one drawback that does not allow reaching a reliable estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='2 Studying streamflow complexity Understanding the dynamic behavior of rivers which is affected by several factors is a key issue in hydrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Streamflow is affected by (i) physical factors that include the incline gradient of the river, water viscosity, elevation, and properties of the surrounding terrain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (ii) geophysical factors involving the geographical location, weather, and climatic change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' and finally, it is significantly affected by (iii) human activities (including building, river training works, damming, dredging, deforestation, and pollution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The question arises here as to how the study of river flow complexity can help unravel the effects of these factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' In the context of complexity, as it will be discussed in this paper, we will cite (i) the paper by Puente and Sivakumar (2007) in which stream flow complexity is considered within the general geophysical complexity and (ii) the paper by Krasovskaia (1997) is a representative of a large 4 group of papers in which streamflow complexity is described through the difference between time series by offering a correlational explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Information about river streamflow complexity is most reliable if it is computed from time series by applying some information measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It seems that algorithmic complexity can be a good choice, although this measure has not yet found its niche in hydrology except in a few papers (Sen, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Mihailović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Mihailović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The use of hydrological models in studying streamflow complexity is not so promising, since with them it is not possible to model perhaps the most discriminating property of a complex system - complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' With this in mind, we applied the Kolmogorov complexity (KC) to monthly streamflow time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It is noted that streamflow complexity can be studied by the Aksentijevic-Gibson complexity as a tool for the analysis of hydrological data that holds the promise of uncovering patterns in the data that cannot be captured by KC and other complexity measures (Aksentijevic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='3 Prediction horizon of rivers In mathematics, there exists a characteristic timescale well known as the Lyapunov time, also called prediction horizon (which is expressed in the units of the recorded series) defined as the inverse of the largest Lyapunov exponent of the considered time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It is a period after which a dynamical system becomes unpredictable and enters a chaotic state, so it indicates the limits of predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Estimation of the Lyapunov time is related to computational or inherent uncertainties that often lead to overestimating the actual value of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' To correct this overestimation, Mihailović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (2019) introduced the Kolmogorov time as the inverse of the Kolmogorov complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This time quantifies the length of the time window within which complexity remains unchanged, significantly while reducing the size of 5 the effective prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' River regimes can be simple, mixed, or complex, and one question is how these regimes relate to complexity, chaotic behavior, and prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The time horizon of streamflow is a consequence of intertwined hydro- meteorologic forcings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', precipitation, temperature, and evapotranspiration) and physiography (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', slope and elevation) (Knoben et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Mathai and Mujumdar, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Higher elevation can impact hydro-meteorological dynamics due to more rapid changes in airflow and orographical effects on precipitation production (Houze, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Slope affects the recession of hydrograph (Mathai and Mujumdar, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' There is no doubt that these factors (separately or in synergy) may affect the time horizon of rivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Additionally, the naturalized streamflow data minimize human impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' However, dam effects may have a dominant influence on the predictability of rivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It is somewhat unusual that little attention has been paid to this influence in the hydrological literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' We are of the opinion that the question of streamflow complexity must be approached through information measures to obtain more natural and reliable information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' To do that we applied the KC complexity and its derivatives and the Lyapunov exponent to monthly river flow time series over a 66-year period 1950–2015 from 1879 rivers in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Therefore, implications of annual mean precipitation and temperature, slope, elevation, and effects of the existence of dams upstream of the naturalized streamflow site on statistics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', coefficient of variation (CV) and average value) and river time horizon of streamflow were used to explain the results obtained with information measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Description of data Monthly naturalized streamflow data for a period from 1950 to 2015 was obtained from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Geological Survey (USGS) Science Base Catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The naturalized streamflow 6 is a simulated data for 2,622,273 stream reaches, which are defined by National Hydrography Dataset (NHD) Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='0, across the continental U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' using the random forest ensemble (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' However, using all the naturalized data of stream reaches >2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='5 million is not only handling the redundant streamflow information but also taking a high likelihood of using less accurate estimates: the random forest models were calibrated to almost 2,000 reference gauge sites where the observed streamflow exists, and then the calibrated models were applied to the ungauged reach segments (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Therefore, we only used the naturalized streamflow data at stream reaches directly connecting the gauge stations at the outlet of HUC (Hydrologic Unit Code) 8 so that naturalized streamflow data from only 1879 sites were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Annual mean precipitation and temperature at every point corresponding to the selected 1879 naturalized streamflow sites were calculated using NOAA nClimGrid monthly dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The NOAA nClimGrid dataset has a period from 1895 to the present and covers CONUS and Alaska with a 5 km grid resolution (Vose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Therefore, the nClimGrid precipitation and temperature in the common period for the naturalized streamflow 1950-2015 were spatially interpolated to the naturalized streamflow sites using the inverse distance weight (IDW) method (Ahrens, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Besides, the locations of 92075 dams across CONUS were obtained from the National Inventory Dams (NID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' We assigned the binary value of 1 (0) for dams when they are (not) located upstream of the naturalized streamflow site in a watershed for further analysis in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The mean slope corresponding to each HUC8 watershed was obtained from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Environmental Protection Agency (EPA), and elevation at each stream gauge point was obtained from the USGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 7 The spatial distributions of monthly streamflow, coefficient of variation (CV), altitude, slope, and dams’ locations of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' rivers for the period 1950-2015 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Spatial distributions of (a) maximal streamflow in cfs (cubic feet per second), (b) CV, (c) altitude (m), (d) slope (degree), and (e) dams of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' rivers for the period 1950-2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='124°W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='117°W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='110°W ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='slope (degrees) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='26°N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='24°N- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='≤32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='2 24°N 24°N- ≤1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='26 24°N 22°N- 408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='3 0 500 1000 1500 2000km 22°N 22°N- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='67 500 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='09 1000 1500 2000km 784.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='4 22°N 20°N 1160 20°N 20°N- 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='5 20°N 18°N ≥1537 18°N 18°N ≥14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='9 18°N 124°W 1170w 110W 103°W 96°W 89°W 82°W 75°W 68°W 124W 117W 110W 103W 96W 89°W 82°W 75°W 68°W (e) 124°W 117°W 110W 103°W 96°W 89°W 82W 75°W 680W 56°N 56°N 54°N 54°N 52°N 52°N 50°N 50°N 48°N 48°N 46°N 46°N 44°N 44°N 42°N- 42°N 40°N- 40°N 38°N 38°N 36°N 36°N 34°N- 34°N 32°N- 32°N No0 30°N 28°N- no dam 28°N 26°N- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='dam 26°N 24°N 24°N 22°N 0 500 1000 1500 2000km 22°N 20°N 20°N 18°N- 18°N 124°W 117°W 110W 103°W 96°W 89°W 82°W 75°W 68°W8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='1 Lempel and Ziv algorithm Kolmogorov complexity (KC) is a natural but uncomputable information measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It is approximated by some compression algorithms - Lempel-Ziv and its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Lempel and Ziv (1976) suggested an algorithm (LZA) for calculating the complexity of a time series 𝑋(𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It includes the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (1) Encoding the time series by creating a sequence 𝑆 of the characters 0 and 1 written as 𝑠(𝑖), 𝑖 = 1,2, … , 𝑁, according to the rule 𝑠(𝑖)= 0 if 𝑥𝑖 < 𝑥𝑡 or 1 if 𝑥𝑖 > 𝑥𝑡, where 𝑥𝑡 is a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The threshold is commonly selected as the mean value of the time series, while other encoding schemes are also available (Radhakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (2) calculating the complexity counter 𝑐(𝑁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 𝑐(𝑁) is defined as the minimum number of distinct patterns contained in a given character sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The complexity counter 𝑐(𝑁) is a function of the length of sequence 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The value of 𝑒(𝑁) approaches an ultimate value 𝑐(𝑁) as 𝑁 approaches infinity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 𝑐(𝑁) = 𝑂(𝑏(𝑁)) and 𝑏(𝑁) = 𝑙𝑜𝑔2𝑁;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Calculating the normalized information measure 𝐶𝑘(𝑁), which is defined as 𝐶𝑘(𝑁) = 𝑐(𝑁)/𝑏(𝑁) = 𝑐(𝑁)/𝑙𝑜𝑔2𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' For a nonlinear time series, 𝐶𝑘(𝑁) varies between 0 and 1, although it can be larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Note that the pattern is a sequence in the coded time series which is unique and non-repeatable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' A flow chart for the calculation of KC of a streamflow series 𝑋(𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑁) using the LZA algorithm is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Flow chart for calculation of the Kolmogorov complexity (KC) using the Lempel–Zev algorithm (LZA) (by permission Mihailović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2019) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='2 Kolmogorov complexity spectrum and its highest value The Kolmogorov complexity of time series has two weaknesses: (i) it cannot distinguish between time series with different amplitude variations and that with similar random components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' and (ii) in the conversion of a time series into a binary string, its complexity is unseen in the rules of the applied procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Therefore, in defining a threshold for a criterion for coding, some information about the composition of time series could be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' In the complexity analysis of time series, two measures are used: (i) Kolmogorov complexity spectrum (KC spectrum) and (ii) the highest value of KC spectrum (KCM), introduced by Mihailović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (2015a) who described the procedure for calculating the KC spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The flow chart in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 3 shows schematically how to calculate the KC spectrum 𝐶(𝑐1, 𝑐2, 𝑐3, … , 𝑐𝑁) for time series 𝑋(𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This spectrum allows us to investigate the range of amplitudes in a time series that represents a complex system with highly enhanced stochastic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It may be noted that for a large number of samples of a time Start 个 Streamflowseries treshold xt 不 Data sequences construction,N=length (S) 不 Initialization(Q=null） SQ元 = S(1) c(n)=1, i=2) No ReadS(i)andscoreintoQand Yes thecharacterS(i-1)intoSQ元 i>N Yes SQ元contains Q?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' i=i+1 KC=c(N)log,N 个 No End c(n)=c(n)+110 series, the computation of KC spectrum can be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The highest value 𝐾𝑚 𝐶 as in this series, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 𝐾𝑚 𝐶 = 𝑚𝑎𝑥{𝑐𝑖}, is the highest value of the KC complexity spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Flow chart for calculation of the Kolmogorov complexity spectrum and its highest value (KCM) (by permission Mihailović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='3 Lyapunov exponent The Lyapunov exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Positive Lyapunov exponent (LE) indicates that small fluctuations can lead to drastically different system behavior (small differences in the initial state lead to large differences in a later state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Because the rate of separation can be different for different orientations of the initial separation vector, there is a spectrum of Lyapunov exponents whose largest value is commonly the LE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' A positive value of this exponent is taken as an indicator that a dynamical system is chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' In this study, we obtained the LE for the standardized monthly streamflow time series by applying the Start 个 Streamflowseries X(X, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=',.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' ), 1 treshold x,=X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 个 Data sequenceS construction,N=length(S) 个 Initialization(Q=null) SQ元 =null, c(n)=0) 个 CalculateKC and store it into array C(C, C2, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=',.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='Cn) 不 No i>N i=i+1 Yes KCM= max (C, C2, C3 个 End11 Rosenstein algorithm (Rosenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 1993) which was implemented in the MATLAB program (Shapour, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' However, this measure has one drawback: If the embedding theory is used to build chaotic attractors in the reconstruction space, then additional “spurious” Lyapunov exponents appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Results and Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='1 Spatial analysis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='1 Picture of the US rivers: complexity and chaos The scatter plot of KC versus LE is like a “boomerang” shape (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The scatter plot area, by two lines LE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='146 (parallel to the y-axis) and KC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='516 (parallel to the x- axis), is divided into rectangles for the sake of a better visualization for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' These two numbers are the means of the maximal and minimal values of KC and LE in the set of gauge stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Perhaps, at first glimpse, this kind of scatter plot (with a large number of samples) seems surprising and may appear for the first time to our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The figure shows that streamflow time series for all gauge stations give a picture of a mixture consisting of always present chaos and high randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Randomness is unpredictable because we just have no right information, while chaos is somewhere between random and predictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' A hallmark of chaotic streamflow time series is predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Thus, with this picture of the scatter plot of KC versus LE, it can be said that it 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Scatter plot of Kolmogorov complexity (KC) versus, and Lyapunov exponent (LE) for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' monthly streamflow for the period 1950-2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (a) LD (left-down);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (b) LU (left-upper);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (c) RU (right-upper) and (d) RD (right-down) are parts of the scatter plot area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The numbers in rectangles indicate the total number of gauge stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' is difficult to reach a reliable prediction of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' river streamflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' In addition, depending on the time scale considered, river discharge can be either random or chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Thus, at daily and seasonal scales the river discharge is random (non-chaotic), but the flow is chaotic at the monthly scale (Adab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' A simple percentage calculation with numbers from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 4 shows that the LU values of all gauge stations are positive, while 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='7 % of their KC values are higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='516 (LU and RU parts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' In the two lower parts (LD and RD) randomness and LE of streamflow are mainly lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='4, it is simple to find that the ratio of the number of gauge stations with low and high randomness is approximately one to five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This is an indicator that the predictability of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' river streamflow is not high, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', we talk about the chance that river streamflow could be in principle modeled reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' We do not talk about heuristic hydrological models, which sometimes can give good results that come from their mathematical background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (b) 955 619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='6 KC (a) 280 (d) 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='25 LE13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='2 Complexity The complexity of river discharge is a key issue in hydrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This paper considers the KC complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=" The spatial distribution of KC of the US rivers' monthly streamflow for the period 1950-2015 is shown in Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' From this figure three distinct patterns for KC are seen, and more if we consider a finer scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The southwestern part which is arider has lower KC than what is in the more humid part of the eastern U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Further, in the southwestern part, there is a band of very low KC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This band cuts across the mountainous terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The Mississippi valley, the Great Lakes region, and the Atlantic Seaboard have high KC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' However, the Ohio Valley has a lower (but still high) KC than the surrounding area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It seems that besides landscape and physiographic characteristics, KC may have a strong correlation with weather patterns and distance from the sea (continentality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' One can make a similar observation about LE (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It is quite difficult and not always consistent to determine explicitly the connection between physiographic characteristics and the complexity of river flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' For example, based on monthly streamflow time series from ten-gauge stations at seven rivers of different river regimes in Bosnia and Herzegovina, Mihailović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (2015b) found that the relationship between the highest value of the KC spectrum (KCM) and elevation (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' That relation KCM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='0002h + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='9421 shows that there is a positive trend in changes of the KCM with respect to elevation with the coefficient of correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='602 (a) 124°W 117°W 110°W 103°W 96°W 89°W 82°W 75°W 68°W (b) 124°W 117W 110W 103°W 96W 89°W 82°W 75°W 68°W 56°N 56°N 56°N- 56°N 54°N- 54°N 54°N- 54°N 52°N 52°N 52°N- 52°N 50°N- 50°N 50°N 50°N 48°N 48°N 48°N- 48°N 46°N 46°N 46°N- 46°N 44°N 44°N 44°N- 44°N 42°N 42°N 42°N- 42°N 40°N- 40°N 40°N- 40°N 38°N- 38°N 38°N- No8E 36°N 36°N 36°N- No9 34°N Nobe 34°N- 34°N 32°N 32°N 32°N- 32°N 30°N NOE 30°N- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='gaugepoint 30°N gaugepoint 28°N BoundariesofHUC8 28°N 28°N- Boundaries of HUC8 28°N 26°N- KC 26°N 26°N LE 26°N 24°N ≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='351 24°N 24°N ≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='097 24°N 22°N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='468 0 500 1000 1500 2000km 22°N 22°N- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='119 0 500 1000 1500 2000km 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='585 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='141 22°N 20°N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='703 20°N 20°N- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='163 20°N 18°N- ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='820 18°N 18°N- ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='184 18°N 124W 117W 110W 103W 96°W 89°W 82°W 75W 68°W 124W 117w 110W 103W 96W 89W 82°W 75°W 68W14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (a) Spatial distribution of KC and (b) LE of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' river monthly streamflow for the period 1950-2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' We hypothesize that the turbulent nature of rivers and barriers built by human activities may remarkably affect the complexity of rivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' When we say human activities, we primarily mean dams but also channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Figure 1e shows the spatial distribution of dams (1796) built on the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' rivers making up 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='6% of the statistical set of 1879 gauge stations used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' A small number of papers are devoted to this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' And if there are any, they mostly remain on the descriptive approach supported by traditional statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' To address the complexity of streamflow, Mihailović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (2019) analyzed daily streamflow data recorded during the period 1989–2016 at twelve gauging stations on the Rio Brazos River in Texas (USA) using KC and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' They found a huge increase in KCM at one gauge station in comparison to other ones, concluding that the reason for the high KCM of this station may be attributed to the Morris Sheppard Hydroelectric Power plant at Morris Sheppard Dam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The KC complexity as a measure does not distinguish between time series with different amplitude variations and similar random components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It seems that changing the river flow in the operating mode of the dam does not only change the amplitude but also the randomness, which is reflected by higher complexity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', what is captured by KC appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' In this paper, the range of the calculated measures was in the intervals of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='097-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='936 (KC) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='011-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='282 (LE), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The spatial distribution of KC in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 6a is very similar to the distribution of KC in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 6a the number of dams is 1796, while the number of dams among gauge stations with KC > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='516 is 1523 or 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='7% of the total number of dams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This is almost the same as the ratio of 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='7% (number of gauge stations with KC > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='516 and their total number) in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 15 It is noted that the channelization of rivers decreases the complexity of river flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' For example, applying the KC complexity analysis of monthly river discharge, Mihailović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (2014) found that during the period 1926–1990 there was a drop in KC in the mountain rivers in Bosnia and Miljacka (Bosnia and Herzegovina) for the period 1946-1965, in comparison with two other periods (1926-1945 and 1966-1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' That complexity loss was interpreted as a result of intensive different human interventions on those rivers (establishing the network of channels for building the capacities for water consumption) after the Second World War.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Certainly, channelization is a type of human intervention that contributes to reducing the randomness of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' rivers having KC less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='516 with an amount of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='2% of the total number of gauge stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The division of river flow regimes into (i) low complex (KC < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='516) and high complex (KC ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='516) and (ii) low chaotic (LE < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='146) and high chaotic (LE ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='146) is merely conditional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (a) Spatial distribution of KC (KC < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='516;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' KC ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='516 while 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='011 < LE < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='282) and (b) LE (LE < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='146;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' LE ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='146 while 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='097 < KC < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='936) of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' river monthly streamflow in the presence of dams for the period 1950-2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='3 Chaotic behavior The spatial distribution of LE of monthly discharge of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' rivers for the period 1950-2015 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This is the spatial distribution taken out from the distribution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 5b following the mentioned criterion of division of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' river flows into low chaotic (a) 124W 117W 110W 103°W 96°W 89°W 82°W 75°W 68°W (b) 124°W 117°W 110W 103°W 96°W 89°W 82°W 75W 68W 56°N- 56°N56°N 56°N 54°N- 54°N 54°N 54°N 52°N- 520N52°N 52°N 50°N- 50°N50°N 50°N 48°N- 48°N48N 48°N 46°N- 46°N46°N 46°N 44°N 44°N44°N 44°N 42°N 42°N42°N- 42°N 40°N- 40°N40°N 40°N 38°N- 38°N38°N 38°N 36°N- 36°N36°N N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='96 34°N- Noe Note 34°N 32°N- 32°N32°N 32°N 30°N- 30°N 28°N- BoundariesofHUC8 28°N28°N BoundariesofHUC8 28°N 26°N- 26°N26°N- gauge points 26°N 24°N 24°N24°N gauge points KC<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='516,LE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='011-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='282 LE<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='146,KC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='097-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='936 24°N 22°N- KC≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='516,LE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='011-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='282 0 500 1000 1500 2000km 22°N22°N LE≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='146,KC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='097-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='936 0 500 1000 1500 2000km 22°N 20°N- 20°N20°N- 20°N 18°N- 18°N18°N- 18°N 124W 117W 110°W 103°W 96°W 89°W 82°W 75°W 68°W 124w 117oW 1100W 1030W 96W 89°W 82°W 75°W 68°W16 (LE < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='146) and high chaotic (LE ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='146).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' From this figure it is seen that low chaos prevails significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' High chaos dominates in the Mississippi Valley, the Great Lakes region, the Upper Mid-West, and the Atlantic Seaboard, while in the Western U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' it is less prevalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' We already stated that chaos is always present in turbulent flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Therefore, the rivers, which have positive LE, are in a chaotic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This phenomenon is intriguing as a topic in hydrology as well as in all sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' However, hydrologists mostly have dealt with those topics in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (1) They often keep on a descriptive level that is not always necessarily simple, paying more attention to the mathematical background with comments about possible applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (2) They usually consider just a low-dimensional chaos, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', to be attributable to a small fraction of the components of the total system, using a smaller number of streamflow time series with a focus on one possible source that causes that chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=" (3) Some of them used theoretical approaches, inverse modeling, and information measures to gain insights into the river flow's chaotic nature." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (4) There is almost no information that anyone dealt with the high-dimensional chaotic regime of river flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', turbulent flow having many degrees of freedom (Porporato and Ridolfi, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Sivakumar, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Sivakumar and Jayawardena, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Labat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Fattahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Yildirim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Mihailović et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The spatial distribution of LE (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='09-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='18) of monthly streamflow of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' rivers for the period 1950-2015 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This time interval was chosen so that their endpoints were nearly symmetrical with respect to LE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The spatial distribution of gauge stations in this figure in comparison with the spatial distribution of dams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 1e is almost indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Further inspection of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 7a shows that the number of gauge stations in the Western part (which is mostly mountainous) is approximately the same as the number of gauge stations in the Eastern part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This leads us to the conclusion that the 17 influence of orography on the river flow dynamics is much smaller than the influence of dams, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', the influence of human activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' (a) Spatial distribution of river gauge stations (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='09 < LE < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='18) of monthly discharge of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' rivers for the period 1950-2015 and (b) histogram of their numbers in dependence on LE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' If we look at the scatter plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 4) we can see that LE values of gauge stations from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 7a are mostly grouped on the right side of LU and the left side of RU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The number of gauge stations with a frequency greater or equal to 100 is 1522 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 7b), or 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='0% of the total number of stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It is interesting to consider the “tail” of the histogram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' It corresponds to the state of high chaos and low complexity of river flow (RD part in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' This may indicate the appearance of periodicity or another pattern on some other time scales that can be ascribed to specific environmental factors or human activity (Aksentijevic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' The occurrence of periodicity on other time scales does not necessarily mean that it will be maintained over long periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='2 Time horizon and complexity spectrum of river flow amplitudes (a) 124W 117W 110W 103W 96W 89°W 82°W 750W 68°W (b) 56°N 56°N 260 260 54°N- 54°N 52°N 52°N 240 50°N 50°N 220- 48°N 48°N 46°N 46°N 200 44°N- 44°N 180- 42°N- 42°N 40°N- 40°N 38°N- 38°N 140 36°N- 36°N 120 34°N- 34°N 100- 32°N- 32°N 80- 30°N 30°N 28°N- 28°N 60 26°N- gaugepoints 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfES7g/content/2301.11983v1.pdf'} +page_content='092. The “z best” grism catalog provides redshifts as well as combined +UV+IR star formation rates. The “best” redshift available may be a spectroscopic redshift or, in its absence, a grism +redshift. In the absence of both, a photometric redshift is provided (Momcheva et al. 2016). UV+IR star formation +rates based on fluxes with low S/N were rejected by requiring the SFR flag to be set to zero. +1 3D-HST website and online catalog: https://3dhst.research.yale.edu/Data.php +2 CANDELS structural catalogs: https://www2.mpia-hd.mpg.de/homes/vdwel/3dhstcandels.html +3 CANDELS GOODS-S multi-wavelength catalog: https://cdsarc.cds.unistra.fr/viz-bin/cat/J/ApJS/207/24 + +Galaxies in the Vicinity of Lyα Nebulae +3 +53.00 +53.05 +53.10 +53.15 +53.20 +53.25 +RA +27.95 +27.90 +27.85 +27.80 +27.75 +27.70 +27.65 +Dec +Ly nebula location +26.5 +27.0 +27.5 +28.0 +28.5 +29.0 +29.5 +30.0 +F160W Limiting Magnitude (AB) +53.00 +53.05 +53.10 +53.15 +53.20 +53.25 +RA +Blob Galaxies +Field Galaxies +Figure 1. Left: The GOODS-S extragalactic field (∼170 arcmin2), showing F160W limiting magnitude of galaxies in shades +of gray (Guo et al. 2013), and locations of Lyα nebulae as red crosses (Yang et al. 2010). Right: Selected blob (yellow) and +field (black) galaxies at z∼2.3. Galaxies in the shallower lower third of the field and in the deeper Hubble Ultra Deep Field +(Koekemoer et al. 2013; Beckwith et al. 2006) are eliminated after imposing a limiting magnitude constraint (Section 3.2). +Finally, we obtained S´ersic indices n and effective radii re from van der Wel et al. (2012). We required the provided +flag to be set to zero, indicating a good fit. We also cross-matched objects in the 3D-HST GOODS-S catalogs with +that of Guo et al. (2013) to associate the F160W limiting magnitude of the surrounding region to each galaxy in our +sample. +Given data on over 50,000 sources within the GOODS-S field, we proceeded to compare a range of properties between +galaxies in the vicinity of Lyα nebulae and similarly selected galaxies in the field. +3. SAMPLE SELECTION +3.1. Lyα Nebula Sample +Our Lyα nebula sample is taken from a blind survey of four extragalactic fields (Yang et al. 2010). The nebulae +were selected using a narrowband filter corresponding to z=2.3±0.037. Based on the derived number density from +this blind search, the authors estimated a halo mass hosting a Lyα nebula to be ≥1013 M⊙. Of the 25 Lyα nebulae +detected, sixteen lie within the Chandra Deep Field South (CDFS), and six of these overlap the 3D-HST GOODS-S +catalog. +3.2. Galaxy Sample +We defined Lyα nebula-associated galaxies to be those within 40 arcseconds of any of the six Lyα nebulae locations +and with z=2.3±0.15, forming a composite “blob galaxy” sample. We selected these radius and redshift ranges based +on a number of considerations. The redshift range within which we can expect to find galaxies associated with a +nebula must be large enough to capture true members, given the redshift uncertainties, but not so large as to include a +great number of interlopers. The best available redshift measurements for the vast majority of galaxies near z=2.3 in +the 3D-HST catalog are photometric, with typical uncertainties of ∆z=0.15, corresponding to ∆z/(1+z)=0.05 at this +redshift, roughly an order of magnitude larger than any ∆z expected from peculiar motion (even considering cluster +members). +A similar balance must be struck when defining a radius around each Lyα nebula within which we select blob +galaxies. Thanks to space-based imaging, uncertainty in on-sky position is negligible. From our analysis, a 40” radius +(∼320 proper kpc at z∼2.3) emerged as the distance beyond which statistically significant differences between blob +and field galaxies disappear (see Section 5.2.3). Finally, survey depth varies in GOODS-S (Figure 1, left), so in order +to make a fair comparison of the galaxy population across the field, we rejected both shallower and deeper regions + +4 +Wells, Prescott & Finlator +53.00 +53.05 +53.10 +53.15 +53.20 +RA +27.80 +27.75 +27.70 +27.65 +Dec +LAB 06 +LAB 07 +LAB 12 +LAB 14 +LAB 11 +LAB 09 +Associated galaxies +Ly nebula location +Figure 2. Close-up view of the blob galaxy sample. The blob locations are labeled with red crosses, and associated galaxies +(as defined in Section 3.2) are indicated as black circles. +by only selecting galaxies with an associated F160W limiting magnitude between 28.25 and 28.75 (Figure 1, right). +However, we note that when we do not correct for field depth in this way, we see qualitatively the same results. +Eighty-six galaxies make up the composite blob sample (Figure 2). We defined our control sample as all other +GOODS-S sources lying outside of the defined blob regions whose redshifts and limiting magnitudes fell within the +same specified ranges, yielding a population of 1,099 galaxies. In total, 1% of the galaxy redshifts used in this work +are are spectroscopic, 7% are grism, and 91% are photometric. Only a fraction of the blob and field galaxies have data +for each property of interest that meet the quality requirements outlined in Section 2.1. Thus, the number of galaxies +n included in a given analysis is specified where applicable. +4. ANALYSIS +We utilized two non-parametric statistical tests in order to compare properties between blob and field galaxies: +the 2-sample Kolmogorov–Smirnov (KS) test and the k-sample Anderson-Darling (AD) test, implemented using the +ks 2samp and anderson ksamp functions in SciPy’s stats module (Virtanen et al. 2020). Given two sets of observations +of a single, continuous variable, these tests compare the distributions of the two samples without assuming anything +about the nature of the parent distribution. +Graphically, both methods arrange the data from each sample into +empirical cumulative probability distributions (CPDs). The KS test reports the largest difference between the two +CPDs at any one interval, and from this statistic a p-value between 0 and 1 is calculated given the size n of each +sample (Hodges 1958). Meanwhile, the AD test calculates the sum of the squared differences between the CPDs at +every interval, and gives more weight to differences in the tails of the distributions. As with the KS test, a p-value is +produced based on this statistic and the sample size (Scholz & Stephens 1987). In both cases, the reported p-value +indicates the probability that such a difference would be seen by chance assuming that the two samples are drawn +randomly from the same parent distribution (the null hypothesis). A low p-value, resulting from a large difference +between the CPDs, indicates that we have reason to reject the null hypothesis. For example, a p-value of <0.05 +indicates that there is less than a 5% chance that two given samples come from the same population. +This is a +commonly accepted threshold for statistical significance that is used in works comparable to this one (e.g. Hatch et al. +2011; Cooke et al. 2014; Shimakawa et al. 2018), and is what we adopt here. +5. RESULTS +5.1. Fiducial Results +Resulting p-values from both statistical tests are displayed for every property we examined in Figure 3, with the +differential and cumulative distributions of selected properties shown in Figures 4, 5, and 6. +We see statistically + +Galaxies in the Vicinity of Lyα Nebulae +5 +F160W +F814W +SFRUV + IR +OIII Flux +OII Flux +H Flux +OIII EQW +OII EQW +H EQW +Age +Mass +SFR +sSFR +AV +Sersic n +re +0.00 +0.05 +0.10 +0.15 +0.20 +p-value +86 +74 +28 +22 +23 +11 +15 +9 +5 +72 +72 +72 +72 +72 +72 +44 +44 +# blob +galaxies +KS test +AD test +Figure 3. KS (dark purple) and AD (light purple) test p-values for all properties compared between galaxies near Lyα nebulae +and those elsewhere in the GOODS-S field at z∼2.3 (see Section 2.1). Both tests indicate statistically significant differences +between the two groups in magnitude (F160W and F814W), stellar mass, star formation rate (SFR), and effective radius (re). +Among the other traits, p-values from both tests are well above the p=0.05 threshold (dashed line). +significant differences between the blob and field galaxies in five properties: F160W and F814W magnitudes, stellar +masses, star formation rates (SFR), and effective radii (re). In these five cases, the size of the blob galaxy sample +is around 75, and the size of the field galaxy sample is around 1000, except for re, which has sample sizes ∼30% +smaller. The differences seen in this group of properties are well distinguished from the other galaxy trait comparisons +we performed: in all five cases where the p-value from the KS test falls below the p=0.05 threshold, we see that the +p-value from the AD test does as well (we include re in this group, with KS test p=0.06, AD test p=0.03). In all other +cases, the p-values from both tests fall well above this threshold. It is important to note sample sizes in each case, as +these vary widely based on the availability and quality of data. Properties which have a blob galaxy sample smaller +than ∼30, such as the emission line measurements, have too few blob galaxies for a reliable KS test result (Razali & +Wah 2011), meaning that if a true difference were present, we would be unable to detect it with so few galaxies. As +a concrete example, unlike the FAST-derived SFR measurement where the blob galaxy sample size is n=72 (KS test +p=0.02), we have only 28 quality blob galaxy measurements for the combined UV+IR SFR (SFRUV +IR), with KS +test p=0.13. When we compared the FAST SFR of these same blob galaxies (n=28) to the field, the resulting KS test +shows no significant difference (p=0.19), even though we know that with a larger blob galaxy sample size (n=72), a +significant difference is indeed detected (p=0.02) (Figure 3). +Our most robust result is that blob galaxies are brighter in both filters than those in the field. Normalized histograms +comparing the filter magnitudes are shown in Figure 4, with blob galaxy distributions in red, field galaxy distributions +in blue, and sample sizes and KS/AD test p-values shown in the inset boxes. In both filters, we see a proportionally +larger number of bright blob galaxies and a systematic offset towards brighter magnitudes in the blob galaxy group +with respect to the field. +It must be noted that the distributions are incomplete at the faint end; the 50% and +75% F160W completeness thresholds for GOODS-S are indicated with light and dark gray dashed lines respectively. +However, given that both samples are depth-corrected, the two groups should be incomplete in the same way, so, all +else being equal, we would expect the distributions to fall off at the faint end in the same manner. Instead, they do +not: in F160W the offset to brighter magnitudes becomes apparent before 26.5 mag, the 50% completeness threshold +for this filter in this region. The same type of offset is seen also in the F814W data. It does appear, then, that +blob distributions have proportionally more bright galaxies, as well as fewer faint ones. At the same time, while blob +galaxies appear to be systematically brighter than those in the field, they do not exhibit a statistically significant +difference in F814W-F160W color (Figure 4). +Unsurprisingly, given their systematically brighter magnitudes, blob galaxies appear to be consistently shifted to- +wards higher stellar masses and SFRs relative to galaxies in the field when we examine these FAST-derived properties +(Figure 5). No such offset is seen when comparing SFR per unit stellar mass (specific SFR) between the two groups, +suggesting that while the blob group is skewed towards higher masses, the two groups may follow the same SFR-mass +relation; we return to this point in Section 6. +Finally, the effective radii of blob galaxies appear to be systematically larger than those in the field. The corre- +sponding histograms are shown in Figure 6. + +6 +Wells, Prescott & Finlator +21 +22 +23 +24 +25 +26 +27 +28 + +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +Probability Density +F160W +21 +22 +23 +24 +25 +26 +27 +28 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Probability +KS test: p = 0.01 +AD test: p < 0.01 +Field (n = 1099) +Blobs (n = 86) +23 +24 +25 +26 +27 +28 +29 +30 + +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Probability Density +F814W +23 +24 +25 +26 +27 +28 +29 +30 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Probability +KS test: p < 0.01 +AD test: p < 0.01 +Field (n = 1058) +Blobs (n = 74) +0 +1 +2 +3 +AB magnitude +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Probability Density +F814W-F160W +0 +1 +2 +3 +AB magnitude +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Probability +KS test: p = 0.95 +AD test: p > 0.25 +Field (n = 1058) +Blobs (n = 74) +Figure 4. Comparisons of the distributions of galaxies near Lyα nebulae (red) and those in the field (blue) for three properties: +F160W mag, F814W mag, and F814W-F160W color index. The 75% and 50% F160W completeness thresholds are shown as +dark and light gray dashed lines, respectively. The median value of each group is indicated with a triangle. Sample sizes and +p-values from the KS and AD statistical tests are provided in the inset boxes. In both filters, a systematic offset towards brighter +magnitudes is seen among the blob galaxies with respect to those in the field, with correspondingly low p-values. No significant +difference is seen in F814W-F160W color. +Thus far, we have demonstrated that there is a statistical difference between blob and field galaxies in these five key +properties. Next, we aim to quantify the nature and magnitude of these differences. The non-parametric KS and AD +tests are sensitive to differences in shape, center and spread between two distributions, but the resulting statistic and +p-value do not indicate which one (or more) of these differences is at play, or how large it is. If our visual impression +that the blob distributions are systematically offset from the field is correct, with the primary difference being their +centers and not their shape or spread, then we can directly compare their medians. We compare medians instead of +means to avoid being sensitive to a few outliers in our SFR and re data. For a given property we added the difference +between the blob and field galaxy medians to every individual blob galaxy measurement, thereby preserving the blob +distribution’s shape and spread but shifting its median to coincide with that of the field. When we redo the KS and +AD tests, we find that for all five properties (F160W, F814W, stellar mass, SFR, and effective radius), the KS test +p-values rise to 0.96, 0.94, 0.70, 0.67, and 0.72 respectively, and the AD test p-values exceed 0.25. This indicates that +the shape and spread of the distributions do not differ significantly, and that the difference in median is driving the +statistical difference. +Thus, this simple exercise indicates that galaxies near blobs are roughly 0.49 magnitudes or 1.6 times brighter in +F160W and 0.59 magnitudes or 1.7 times brighter in F814W, have 1.9 times more stellar mass, have star formation + +Galaxies in the Vicinity of Lyα Nebulae +7 +7 +8 +9 +10 +11 +log[mass/M ] +0.0 +0.2 +0.4 +0.6 +0.8 +Probability Density +Mass +7 +8 +9 +10 +11 +log[mass/M ] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Probability +KS test: p < 0.01 +AD test: p < 0.01 +Field (n = 949) +Blobs (n = 72) +2 +1 +0 +1 +log[SFR/(M /yr)] +0.0 +0.2 +0.4 +0.6 +0.8 +Probability Density +SFR +2 +1 +0 +1 +log[SFR/(M /yr)] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Probability +KS test: p = 0.02 +AD test: p = 0.01 +Field (n = 947) +Blobs (n = 72) +12 +11 +10 +9 +8 +7 +log[sSFR*yr] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Probability Density +SFR / Mass +12 +11 +10 +9 +8 +7 +log[sSFR*yr] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Probability +KS test: p = 0.46 +AD test: p = 0.13 +Field (n = 947) +Blobs (n = 72) +Figure 5. Comparisons of the distributions of galaxies near Lyα nebulae (red) and those in the field (blue) for three properties: +stellar mass, star formation rate (SFR), and SFR per unit stellar mass (or specific star formation rate, sSFR). The median value +of each group is indicated with a triangle. Sample sizes and p-values from the KS and AD statistical tests are provided in the +inset boxes. A systematic offset towards higher mass and SFR is seen among the blob galaxies with respect to those in the field, +with correspondingly low p-values. No statistically significant difference is seen in sSFR between the groups. +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +log[re/arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Probability Density +Effective Radius +1.5 +1.0 +0.5 +0.0 +log[re/arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Probability +KS test: p = 0.06 +AD test: p = 0.03 +Field (n = 622) +Blobs (n = 44) +Figure 6. Comparison of blob (red) and field (blue) galaxy effective radius distributions. The median value of each group is +indicated with a triangle. Sample sizes and p-values from the KS and AD statistical tests are provided in the inset boxes. A +systematic offset towards larger effective radii is seen among the blob galaxies with respect to those in the field, accompanied +by correspondingly low p-values. + +8 +Wells, Prescott & Finlator +22 +24 +26 +28 +0.0 +0.1 +0.2 +0.3 +0.4 +Probability Density +Cropped at 50% Completeness Limit +22 +24 +26 +28 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Probability +KS test: p = 0.06 +AD test: p = 0.02 +Field (n = 876) +Blobs (n = 74) +22 +24 +26 +28 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Probability Density +Cropped at 75% Completeness Limit +22 +24 +26 +28 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Probability +KS test: p = 0.29 +AD test: p > 0.25 +Field (n = 681) +Blobs (n = 66) +22 +24 +26 +28 +AB magnitude +0.0 +0.1 +0.2 +0.3 +0.4 +Probability Density +Corrected for Incompleteness +22 +24 +26 +28 +AB magnitude +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Probability +KS test: p < 0.01 +AD test: p < 0.01 +Field (n = 1870) +Blobs (n = 119) +F160W +Figure 7. Comparison of blob and field galaxy F160W distributions when cropping the sample at the 50% (top) and 75% +(middle) completeness limits. Alternatively, we simulated a sample corrected for incompleteness by adding the appropriate +number of mock galaxies to the faintest bins (bottom). +rates that are 1.4 times higher, and have effective radii that are 1.3 times larger than galaxies in the field. In what +follows, we explore whether these results could be influenced by issues of incompleteness or sample selection. +5.2. Verifying the Results +In this section we describe the various measures we took to test the reliability of our results. In particular, we +investigated the consequences of imposing completeness limits, how the size of the region within which we select blob +galaxies affects our results, and whether a subset of the six blob regions is primarily responsible for the outcome. +We also estimated the expected false positive rate by selecting six random blob-free locations within GOODS-S and +analyzing the resulting “mock” blob and field galaxies in identical fashion. +5.2.1. Sample Completeness +The faint end of our galaxy sample will necessarily be affected by incompleteness. However, given that our sample +is drawn from regions of consistent survey depth (F160W limiting magnitude mlim = 28.50±0.25 mag), we expect our +blob and field galaxy populations to be incomplete to the same degree, allowing them to be fairly compared with one +another while keeping in mind this limitation. Nevertheless, we confirmed that our qualitative results are robust to +completeness issues via two distinct experiments: (1) removing the faintest galaxies and (2) correcting the sample for +incompleteness by adding simulated galaxies to the faintest bins. In what follows, we use the completeness fraction as +a function of F160W magnitude for the GOODS-S field (Skelton et al. 2014). +First, when only considering blob and field galaxies with F160W magnitudes brighter than the 50% completeness +limit, the AD and KS p-values remain significant at 0.02 and 0.06 respectively. When we crop the sample at the 75% +completeness limit, both p-values rise above 0.25 (Figure 7, top and middle panels). As discussed in Section 5, though +our sample is corrected for depth, the blob and field galaxy F160W and F814W distributions do not fall off in the +same manner at the faint end as would be expected, but rather there appear to be proportionally fewer faint galaxies + +Galaxies in the Vicinity of Lyα Nebulae +9 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Blob Region Radius (arcsec) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p-value +Mass +SFR +re +F160W +F814W +Figure 8. KS test results comparing selected properties of sets of blob and field galaxies for increasing blob region radius. It +is not until the defined blob region reaches ∼40” in radius that the blob-associated galaxy sample size is large enough for the +KS test to detect the difference between galaxies in the blob region and those in the field. +in the blob regions (Figure 4). When we remove the faintest galaxies to increase the completeness of our sample, the +signal weakens, emphasizing that the faint end is contributing to the differences we detect. +In the second approach, we simulated a complete F160W sample. Using Skelton et al.’s (2014) completeness fraction +for each half-magnitude bin, we calculated the number of missing galaxies in a given bin, and drew a corresponding +number of galaxy magnitudes randomly, assuming a uniform distribution within the bin. After artificially recovering +the faintest galaxies in our sample in this way, differences between the blob and field F160W distributions become +even more pronounced, with both AD and KS p-values falling below 0.01 (Figure 7, bottom panel). +These incompleteness tests suggest that the existing data are already revealing real differences between blob and +field galaxies. In the remainder of the paper, we therefore choose to stay close to the data, focusing on the fiducial +depth-corrected sample. We leave a more in-depth investigation of the impact of incompleteness on all other measured +quantities to future work. +5.2.2. The Blob Region Radius +In using Lyα nebulae as markers of overdensities, we must define the angular extent of such regions. A small “blob +region” radius, centered on a Lyα location, is more likely to capture a high fraction of true inhabitants of the dense +environment, but restricts our sample size. As the radius is extended, we benefit from a larger blob galaxy sample, +but that sample is more likely to contain a greater fraction of interlopers - galaxies that are not subject to the same +potential influences as those embedded in an overdense environment. Making no a priori assumptions about what the +plausible physical size of such a dense region might be, we tested blob region radii from 10”-100” (∼84-840 kpc). This +range straddles the expected virial radius of Rvir=219 kpc for a galaxy group mass halo (∼1013 M⊙) at z∼ 2.3, based +on a standard spherical collapse calculation (Bryan & Norman 1998). In steps of 5”, we compared the properties of +each new set of trial blob and field galaxies using the KS and AD tests as before. This allowed us to see how the +p-values changed as the defined blob region varied in size. For simplicity and as the more conservative option we plot +only the KS test p-values in this subsection (Figures 8-11). However, in general we found the AD test to produce lower +p-values given the same samples. +In Figure 8 we see that comparisons in the five selected properties produce p-values that dip below p=0.05 when the +blob region reaches ∼40” in radius. For r=10-35”, the blob galaxy sample size grows from ∼12 to ∼65 when considering +the filter magnitudes; from 11 to 61 when considering the mass and SFR; and from 8 to 35 when considering the +effective radius. The larger p-values seen in this range show the limit of the KS test for small sample sizes - we have +demonstrated that the two populations differ in these properties when r=40” (e.g. Figure 3), but these differences +cannot be detected at smaller radii given the limited blob galaxy sample size. +A potential problem with varying the size of the blob region radius in this way is that as the blob galaxy sample +grows the field galaxy sample shrinks, with members of the latter group switching to the former as the blob region +radius increases. Consequentially, the blob sample is not being compared to a static group, but rather one whose +distribution changes slightly at every step. In order to control for this, we performed the same test but this time +defined field galaxies to be those that lie greater than 60” from any Lyα nebula location, giving us a consistent field + +10 +Wells, Prescott & Finlator +group to which to compare blob galaxies. As we scanned through blob region radii of 10”, 20”, 30”, 40”, and 50”, +those galaxies that fell in the gap between the blob region radius and our field galaxy radius were simply ignored. This +method produced results that were virtually identical to Figure 8, demonstrating that the impact of slight changes in +the field galaxy distribution is negligible when using the previous approach. +5.2.3. The Core Blob Galaxies +Given that the p-values for the five properties in Figure 8 dip below and generally stay under p=0.05 from r=30” out +to r=100”, we then investigated whether 1) the galaxies closest to the blobs were responsible for this outcome, differing +from the field enough to depress the p-values at large blob region radii despite the likely possibility that larger radii will +incorporate more and more interlopers into the blob galaxy sample, or whether 2) galaxies at relatively large distances +from the blobs were contributing to this effect. To this end, we progressively removed the most central blob galaxies +from the analysis entirely, and, allowing the blob region radius to vary as before, compared the properties of the blob +and field galaxy subsets. The results of this test are shown in Figure 9. With more and more central blob galaxies +missing, the p-value curves steadily rise, demonstrating that the core blob galaxies are indeed responsible for the low +p-values seen in Figure 8. When central galaxies within 40” of any blob are absent, statistically significant differences +in the five selected properties disappear. This is not due to prohibitively small sample sizes, as the blob sample size +at this stage (those galaxies with 40”100 +HAEs in another z∼2.5 protocluster and dividing the sample into regions of higher and lower density, Shimakawa et al. +(2018) observed the same star-forming main sequence in each but saw a larger proportion of high-mass, high-SFR +galaxies in the higher density subset. However, when comparing the lower density group to a superdense subset of +the higher density group, Shimakawa et al. (2018) found that the SFRs of the superdense group were boosted for +their given stellar mass. Finally, when comparing HAEs within 2 protoclusters at z∼2.2 to a control field sample, +Hatch et al. (2011) reported that the protocluster group was 0.8 mag brighter than the control, corresponding to a +∼2x greater stellar mass. However, the Hatch et al. (2011) samples did not differ in SFR and the specific SFR of the +protocluster population was determined to be lower than that of the field, in contrast to our results. + +Galaxies in the Vicinity of Lyα Nebulae +13 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p-value +Mass +SFR +re +F160W +F814W +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p-value +False Positive #1 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p-value +False Positive #2 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Blob Region Radius (arcsec) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p-value +False Positive #3 +Figure 11. Examples of KS p-value curves from 100 trials using random Lyα blob locations. The first panel is an example of +the typical result. The remaining panels are the three cases which met our “false positive” criteria, as discussed in the text, i.e., +appearing most similar to our results when using the true Lyα blob locations (Figure 8). +At the same time, results on other physical properties are more mixed. For example, our study did not detect +a difference in F814W-F160W color, dust extinction (AV ), or stellar age, but did find larger effective radii among +galaxies in overdensities. Hatch et al. (2011) also observed no significant difference in color when comparing HAEs +within 2 protoclusters at z∼2.2 to a control field sample, but by contrast, Koyama et al. (2013) found the HAEs in +their high-density group to be redder. Cooke et al. (2014) found a 2x larger median AV among protocluster members, +while Steidel et al. (2005) reported a 2x greater inferred age for the overdensity group. Building on Steidel et al. +(2005), Peter et al. (2007) compared the size distributions of 85 UV-selected star-forming galaxies near the same z∼2.3 +protocluster to 63 control galaxies and found no significant difference, yet Shimakawa et al. (2018) found larger sizes +for HAEs in the denser regions of the z∼2.5 protocluster. Thus, additional study and larger galaxy samples will be +needed to to better understand how galaxy properties correlate with environment at these redshifts. +Taken together, our results add to a suite of high redshift studies consistently finding that galaxies in protoclusters +have greater stellar masses than their counterparts in the field. Although varied in their approach to selecting and +studying overdense regions, all of these snapshots are consistent with the idea of galaxies in rich environments having +matured earlier than their peers in the field, something that is supported by several theoretical studies (Muldrew +et al. 2015; Chiang et al. 2017; Lovell et al. 2018). In this picture, protocluster galaxies virialize and commence star +formation sooner, leading them to exhaust their gas reservoirs earlier and evolve into the massive, inactive elliptical + +14 +Wells, Prescott & Finlator +5 +6 +7 +8 +9 +10 +11 +12 +13 +log[mass/M ] +4 +3 +2 +1 +0 +1 +2 +3 +log[SFR/(M /Year)] +Field +Blobs +Figure 12. SFR versus stellar mass of galaxies associated with blobs (red) and those in the field (blue). Triangle symbols +indicate a non-zero SFR below 10−4.5 solar masses/year. Both populations appear to follow the same star-forming main sequence, +corroborating the result that there is no statistical difference in their specific SFR distributions (Figure 5, bottom panel). +galaxies that we observe haunting cluster environments in the local universe. The broad consistency of our results +with other complementary protocluster studies lends additional support to the use of Lyα blobs as markers of dense +environments. +7. CONCLUSIONS +Lyα nebulae reside in overdense regions and therefore provide an opportunity to study the impact of such dense +environments on the local galaxy population. Using 3D-HST GOODS-S catalog data, we compared traits between +galaxies in the vicinity of six Lyα nebulae and those in the field at z∼2.3. We found statistically significant differences +between the two groups in five properties: nebula-associated galaxies are systematically brighter (F160W and F814W) +than the field, with correspondingly higher stellar masses, SFRs, and effective radii, but both populations exhibit the +same SFR-stellar mass relation. These results are broadly consistent with other studies examining galaxy traits in +dense protocluster environments from z=2-4, which have found that the specific star formation rate does not appear to +be affected in such regions, but that the fraction of high-mass, high-SFR galaxies is greater in protocluster populations. +This suggests that, while star formation proceeds in the same fashion regardless of environment at this epoch, galaxies +in dense environments have formed earlier and so are “ahead” in their development compared to their peers, a scenario +predicted by several theoretical studies. The larger and higher quality galaxy samples expected from the James Webb +Space Telescope (JWST) will add clarity to our understanding of high-redshift galaxies and the effects of cluster and +group environments on their evolution. +ACKNOWLEDGMENTS +The authors would like to thank Kelly Sanderson, Audrey Dijeau, and Daniel Godines Alcantara for helpful dis- +cussions. 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E., et al. 2020, +Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2 +Yang, Y., Zabludoff, A., Eisenstein, D., & Dav´e, R. 2010, +ApJ, 719, 1654, doi: 10.1088/0004-637X/719/2/1654 +Yang, Y., Zabludoff, A., Tremonti, C., Eisenstein, D., & +Dav´e, R. 2009, ApJ, 693, 1579, +doi: 10.1088/0004-637X/693/2/1579 + diff --git a/e9E0T4oBgHgl3EQf5wJ-/content/tmp_files/load_file.txt b/e9E0T4oBgHgl3EQf5wJ-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..defbc398b9a16044e3ecb3f70c6b3b0f33b541cb --- /dev/null +++ b/e9E0T4oBgHgl3EQf5wJ-/content/tmp_files/load_file.txt @@ -0,0 +1,1104 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf,len=1103 +page_content='Draft version January 10, 2023 Typeset using LATEX default style in AASTeX63 Brighter and More Massive Galaxies in the Vicinity of Lyα Nebulae Natalie K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Wells ,1 Moire K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Prescott ,1 and Kristian M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Finlator 1 1Department of Astronomy, New Mexico State University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Box 30001, MSC 4500, Las Cruces, NM, 88003, USA (Received June 7, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Accepted November 15, 2022) Submitted to The Astrophysical Journal ABSTRACT It has been well established in the local universe that galaxy properties differ based on the large- scale environment in which they reside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' As luminous Lyman-alpha (Lyα) nebulae have been shown to trace overdense environments at z∼2-3, comparing the properties of galaxies within Lyα nebulae systems to those in the field can provide insight into how and when locally-observed trends between galaxy properties and environment emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Six Lyα nebulae were discovered at z∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3 in a blind search of the GOODS-S extragalactic field, a region also covered by the 3D-HST spectroscopic survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Utilizing 3D-HST data, we identified 86 galaxies in the vicinity of these nebulae and used statistical tests to compare their physical properties to galaxies elsewhere in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Galaxies lying within 320 proper kpc of a Lyα nebula are roughly half a magnitude brighter than those in the field, with higher stellar masses, higher star formation rates, and larger effective radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Even when considering the effects of sample incompleteness, our study suggests that galaxies in overdensities at z∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3 traced by Lyα nebulae are being influenced by their environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Furthermore, Lyα nebula-associated galaxies lie on the same main sequence of star formation as field galaxies, but have a larger proportion of high-mass galaxies, consistent with the idea that galaxy evolution is accelerated in rich environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Expanded surveys for Lyα nebulae in other deep extragalactic fields and galaxy spectroscopic follow-up with JWST will better constrain the demographics of Lyα nebula-associated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Keywords: high redshift galaxies, galaxy environments, galaxy evolution, protoclusters 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' INTRODUCTION In the local universe (z<1), a galaxy’s structure is strongly dependent on its surroundings: elliptical galaxies are found predominantly in clusters, while disk galaxies are more common in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' This “morphology-density relation” was first established by Oemler (1974) and Dressler (1980), and its physical origins are still the subject of investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The star formation rate (SFR) is also linked to environment and morphology at low redshifts, with cluster ellipticals tending to be quiescent and red, and field spirals showing blue colors and substantial SFRs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Hogg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Study of the more distant universe gives us a window back in time to when these environmental differences in galaxy properties first emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In high redshift studies (z=2-4), galaxies in dense protocluster environments have widely been found to have higher stellar masses when compared to a control group (Koyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Hatch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Steidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Cooke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Shimakawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Ito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Somewhat less agreement is found, however, when examining growth histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' For example, Koyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' (2013) and Cooke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' (2014) found that protocluster and field galaxies follow the same SFR-stellar mass relation at z∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 and z∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5, respectively, as did Shimakawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' (2018) when comparing lower and higher density regions of a protocluster at z∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' However, Shimakawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' (2018) found boosted SFRs for a given stellar mass among galaxies in the densest subset of their sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' By contrast, Hatch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' (2011) found lower specific SFRs among members of two protoclusters at z∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 compared to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Finally, Koyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' (2013) saw redder colors among protocluster galaxies, whereas Hatch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' (2011) found no such distinction between their protocluster and field groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The tendency for high redshift protocluster environments to host a larger fraction of high mass, actively star- forming galaxies differs from the trend seen in the local universe, and may indicate that star formation begins earlier arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='02755v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='GA] 7 Jan 2023 ID2 Wells, Prescott & Finlator in galaxies in rich environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=', Hatch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Ito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2020), consistent with theoretical predictions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Chiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Additional high redshift studies with complementary approaches to galaxy sample selection are needed to disentangle the influences that intrinsic and environmental differences have on the assembly histories of galaxies and clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Unique among the studies already mentioned, which select protocluster galaxies based on strong line emission or UV continuum emission, this work uses proximity to Lyα nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Lyα nebulae (also known as Lyman-alpha blobs, LABs, or simply “blobs”) are extended sources of Lyα emission found at z∼2-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Typically more than 100 kiloparsecs across, these glowing clouds of gas are some of the largest objects known and have been associated with dense protocluster environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Steidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Matsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Prescott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2009, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Hennawi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2017a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Battaia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Due to the expansion of the Universe, the ultraviolet light from Lyα nebulae is redshifted to optical wavelengths, making it possible to observe these giant high-redshift structures with sensitive optical ground- and space-based telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In order to probe the origin of environment-dependent trends in galaxy properties, we use these Lyα nebulae as signposts of overdense regions and leverage high resolution Hubble Space Telescope (HST) observations of the GOODS- S extragalactic field to compare a composite group of galaxies in the vicinity of 6 known Lyα nebulae to a control group in the field at z∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We discuss data sources and quality requirements in Section 2, sample selection in Section 3, and analysis methods in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We report our results and describe the various tests of the robustness of these findings in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We discuss our results and conclude in Sections 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Throughout, we assume the standard ΛCDM cosmology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=', Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='27, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='73, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The angular scale at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3 is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 kiloparsecs/arcsec (kpc/′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' All magnitudes are in the AB system (Oke 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' DATA 3D-HST is a Hubble Space Telescope (HST) spectroscopic survey covering four well-studied extragalactic fields: AEGIS, COSMOS, GOODS-S, and UDS (Skelton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' It builds upon the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS), which obtained Hubble Wide Field Camera 3 (WFC3) imaging of the same regions, as well as the GOODS-N field (Grogin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The online 3D-HST catalog1 provides imaging, photometry, spectra, emission line measurements, redshifts, and other derived properties for ∼100,000 galaxies in all five CANDELS fields using data from 3D-HST, CANDELS, and other space- and ground-based surveys (Momcheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Skelton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Brammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Building on CANDELS data, van der Wel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' (2012) derived structural parameters for the galaxies in 3D-HST using the best S´ersic model fits to CANDELS imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Their catalogs, available online2, complement those of 3D- HST and are line-matched using the same photometric IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' (2013) provide additional data on objects in GOODS-S in their online catalog3, including limiting F160W magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Galaxy Properties and Quality Cuts For all galaxies in our sample, we gathered WFC3 F160W (H) and F814W (I) fluxes from the 3D-HST GOODS- S photometric catalog (release V4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='1), converting to AB magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We consider only those galaxies with good photometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' only those whose use phot flag was set to 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Skelton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' This is especially important given that over 90% of sources near z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3 have only a photometric redshift, upon which we rely when selecting nebula- associated galaxies and their counterparts in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We also obtained the following stellar population parameters, which were estimated from the best fits to multiband photometry using FAST (Kriek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2009): stellar mass, SFR, specific SFR, stellar age (since the onset of star formation), star formation decay timescale τ, and dust attenuation AV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' To ensure we were only using galaxies with high quality SED fits, we required the reported χ2 of the FAST fit to be less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' This cut retained 79% of the GOODS-S photometric galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' From the 3D-HST GOODS-S grism catalog (release V4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5), we collected fluxes and equivalent widths for the OII, OIII, and Hβ emission lines, requiring S/N>2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The “z best” grism catalog provides redshifts as well as combined UV+IR star formation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The “best” redshift available may be a spectroscopic redshift or, in its absence, a grism redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In the absence of both, a photometric redshift is provided (Momcheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' UV+IR star formation rates based on fluxes with low S/N were rejected by requiring the SFR flag to be set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 1 3D-HST website and online catalog: https://3dhst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='edu/Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='php 2 CANDELS structural catalogs: https://www2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='mpia-hd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='de/homes/vdwel/3dhstcandels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='html 3 CANDELS GOODS-S multi-wavelength catalog: https://cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='fr/viz-bin/cat/J/ApJS/207/24 Galaxies in the Vicinity of Lyα Nebulae 3 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='00 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='05 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='10 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='15 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='20 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='25 RA 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='95 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='90 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='85 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='80 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='75 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='70 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='65 Dec Ly nebula location 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 F160W Limiting Magnitude (AB) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='00 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='05 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='10 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='15 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='20 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='25 RA Blob Galaxies Field Galaxies Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Left: The GOODS-S extragalactic field (∼170 arcmin2), showing F160W limiting magnitude of galaxies in shades of gray (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2013), and locations of Lyα nebulae as red crosses (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Right: Selected blob (yellow) and field (black) galaxies at z∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Galaxies in the shallower lower third of the field and in the deeper Hubble Ultra Deep Field (Koekemoer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Beckwith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2006) are eliminated after imposing a limiting magnitude constraint (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Finally, we obtained S´ersic indices n and effective radii re from van der Wel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We required the provided flag to be set to zero, indicating a good fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We also cross-matched objects in the 3D-HST GOODS-S catalogs with that of Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' (2013) to associate the F160W limiting magnitude of the surrounding region to each galaxy in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Given data on over 50,000 sources within the GOODS-S field, we proceeded to compare a range of properties between galaxies in the vicinity of Lyα nebulae and similarly selected galaxies in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' SAMPLE SELECTION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Lyα Nebula Sample Our Lyα nebula sample is taken from a blind survey of four extragalactic fields (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The nebulae were selected using a narrowband filter corresponding to z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Based on the derived number density from this blind search, the authors estimated a halo mass hosting a Lyα nebula to be ≥1013 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Of the 25 Lyα nebulae detected, sixteen lie within the Chandra Deep Field South (CDFS), and six of these overlap the 3D-HST GOODS-S catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Galaxy Sample We defined Lyα nebula-associated galaxies to be those within 40 arcseconds of any of the six Lyα nebulae locations and with z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='15, forming a composite “blob galaxy” sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We selected these radius and redshift ranges based on a number of considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The redshift range within which we can expect to find galaxies associated with a nebula must be large enough to capture true members, given the redshift uncertainties, but not so large as to include a great number of interlopers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The best available redshift measurements for the vast majority of galaxies near z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3 in the 3D-HST catalog are photometric, with typical uncertainties of ∆z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='15, corresponding to ∆z/(1+z)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='05 at this redshift, roughly an order of magnitude larger than any ∆z expected from peculiar motion (even considering cluster members).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' A similar balance must be struck when defining a radius around each Lyα nebula within which we select blob galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Thanks to space-based imaging, uncertainty in on-sky position is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' From our analysis, a 40” radius (∼320 proper kpc at z∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3) emerged as the distance beyond which statistically significant differences between blob and field galaxies disappear (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Finally, survey depth varies in GOODS-S (Figure 1, left), so in order to make a fair comparison of the galaxy population across the field, we rejected both shallower and deeper regions 4 Wells, Prescott & Finlator 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='00 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='05 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='10 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='15 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='20 RA 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='80 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='75 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='70 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='65 Dec LAB 06 LAB 07 LAB 12 LAB 14 LAB 11 LAB 09 Associated galaxies Ly nebula location Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Close-up view of the blob galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The blob locations are labeled with red crosses, and associated galaxies (as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2) are indicated as black circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' by only selecting galaxies with an associated F160W limiting magnitude between 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='25 and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='75 (Figure 1, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' However, we note that when we do not correct for field depth in this way, we see qualitatively the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Eighty-six galaxies make up the composite blob sample (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We defined our control sample as all other GOODS-S sources lying outside of the defined blob regions whose redshifts and limiting magnitudes fell within the same specified ranges, yielding a population of 1,099 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In total, 1% of the galaxy redshifts used in this work are are spectroscopic, 7% are grism, and 91% are photometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Only a fraction of the blob and field galaxies have data for each property of interest that meet the quality requirements outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Thus, the number of galaxies n included in a given analysis is specified where applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' ANALYSIS We utilized two non-parametric statistical tests in order to compare properties between blob and field galaxies: the 2-sample Kolmogorov–Smirnov (KS) test and the k-sample Anderson-Darling (AD) test, implemented using the ks 2samp and anderson ksamp functions in SciPy’s stats module (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Given two sets of observations of a single, continuous variable, these tests compare the distributions of the two samples without assuming anything about the nature of the parent distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Graphically, both methods arrange the data from each sample into empirical cumulative probability distributions (CPDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The KS test reports the largest difference between the two CPDs at any one interval, and from this statistic a p-value between 0 and 1 is calculated given the size n of each sample (Hodges 1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Meanwhile, the AD test calculates the sum of the squared differences between the CPDs at every interval, and gives more weight to differences in the tails of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' As with the KS test, a p-value is produced based on this statistic and the sample size (Scholz & Stephens 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In both cases, the reported p-value indicates the probability that such a difference would be seen by chance assuming that the two samples are drawn randomly from the same parent distribution (the null hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' A low p-value, resulting from a large difference between the CPDs, indicates that we have reason to reject the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' For example, a p-value of <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='05 indicates that there is less than a 5% chance that two given samples come from the same population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' This is a commonly accepted threshold for statistical significance that is used in works comparable to this one (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Hatch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Cooke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Shimakawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2018), and is what we adopt here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' RESULTS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Fiducial Results Resulting p-values from both statistical tests are displayed for every property we examined in Figure 3, with the differential and cumulative distributions of selected properties shown in Figures 4, 5, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We see statistically Galaxies in the Vicinity of Lyα Nebulae 5 F160W F814W SFRUV + IR OIII Flux OII Flux H Flux OIII EQW OII EQW H EQW Age Mass SFR sSFR AV Sersic n re 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='20 p-value 86 74 28 22 23 11 15 9 5 72 72 72 72 72 72 44 44 # blob galaxies KS test AD test Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' KS (dark purple) and AD (light purple) test p-values for all properties compared between galaxies near Lyα nebulae and those elsewhere in the GOODS-S field at z∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3 (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Both tests indicate statistically significant differences between the two groups in magnitude (F160W and F814W), stellar mass, star formation rate (SFR), and effective radius (re).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Among the other traits, p-values from both tests are well above the p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='05 threshold (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' significant differences between the blob and field galaxies in five properties: F160W and F814W magnitudes, stellar masses, star formation rates (SFR), and effective radii (re).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In these five cases, the size of the blob galaxy sample is around 75, and the size of the field galaxy sample is around 1000, except for re, which has sample sizes ∼30% smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The differences seen in this group of properties are well distinguished from the other galaxy trait comparisons we performed: in all five cases where the p-value from the KS test falls below the p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='05 threshold, we see that the p-value from the AD test does as well (we include re in this group, with KS test p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='06, AD test p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In all other cases, the p-values from both tests fall well above this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' It is important to note sample sizes in each case, as these vary widely based on the availability and quality of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Properties which have a blob galaxy sample smaller than ∼30, such as the emission line measurements, have too few blob galaxies for a reliable KS test result (Razali & Wah 2011), meaning that if a true difference were present, we would be unable to detect it with so few galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' As a concrete example, unlike the FAST-derived SFR measurement where the blob galaxy sample size is n=72 (KS test p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='02), we have only 28 quality blob galaxy measurements for the combined UV+IR SFR (SFRUV +IR), with KS test p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' When we compared the FAST SFR of these same blob galaxies (n=28) to the field, the resulting KS test shows no significant difference (p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='19), even though we know that with a larger blob galaxy sample size (n=72), a significant difference is indeed detected (p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='02) (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Our most robust result is that blob galaxies are brighter in both filters than those in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Normalized histograms comparing the filter magnitudes are shown in Figure 4, with blob galaxy distributions in red, field galaxy distributions in blue, and sample sizes and KS/AD test p-values shown in the inset boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In both filters, we see a proportionally larger number of bright blob galaxies and a systematic offset towards brighter magnitudes in the blob galaxy group with respect to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' It must be noted that the distributions are incomplete at the faint end;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' the 50% and 75% F160W completeness thresholds for GOODS-S are indicated with light and dark gray dashed lines respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' However, given that both samples are depth-corrected, the two groups should be incomplete in the same way, so, all else being equal, we would expect the distributions to fall off at the faint end in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Instead, they do not: in F160W the offset to brighter magnitudes becomes apparent before 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5 mag, the 50% completeness threshold for this filter in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The same type of offset is seen also in the F814W data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' It does appear, then, that blob distributions have proportionally more bright galaxies, as well as fewer faint ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' At the same time, while blob galaxies appear to be systematically brighter than those in the field, they do not exhibit a statistically significant difference in F814W-F160W color (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Unsurprisingly, given their systematically brighter magnitudes, blob galaxies appear to be consistently shifted to- wards higher stellar masses and SFRs relative to galaxies in the field when we examine these FAST-derived properties (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' No such offset is seen when comparing SFR per unit stellar mass (specific SFR) between the two groups, suggesting that while the blob group is skewed towards higher masses, the two groups may follow the same SFR-mass relation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' we return to this point in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Finally, the effective radii of blob galaxies appear to be systematically larger than those in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The corre- sponding histograms are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 6 Wells, Prescott & Finlator 21 22 23 24 25 26 27 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='35 Probability Density F160W 21 22 23 24 25 26 27 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 Cumulative Probability KS test: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='01 AD test: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='01 Field (n = 1099) Blobs (n = 86) 23 24 25 26 27 28 29 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='30 Probability Density F814W 23 24 25 26 27 28 29 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 Cumulative Probability KS test: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='01 AD test: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='01 Field (n = 1058) Blobs (n = 74) 0 1 2 3 AB magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 Probability Density F814W-F160W 0 1 2 3 AB magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 Cumulative Probability KS test: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='95 AD test: p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='25 Field (n = 1058) Blobs (n = 74) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Comparisons of the distributions of galaxies near Lyα nebulae (red) and those in the field (blue) for three properties: F160W mag, F814W mag, and F814W-F160W color index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The 75% and 50% F160W completeness thresholds are shown as dark and light gray dashed lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The median value of each group is indicated with a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Sample sizes and p-values from the KS and AD statistical tests are provided in the inset boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In both filters, a systematic offset towards brighter magnitudes is seen among the blob galaxies with respect to those in the field, with correspondingly low p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' No significant difference is seen in F814W-F160W color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Thus far, we have demonstrated that there is a statistical difference between blob and field galaxies in these five key properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Next, we aim to quantify the nature and magnitude of these differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The non-parametric KS and AD tests are sensitive to differences in shape, center and spread between two distributions, but the resulting statistic and p-value do not indicate which one (or more) of these differences is at play, or how large it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' If our visual impression that the blob distributions are systematically offset from the field is correct, with the primary difference being their centers and not their shape or spread, then we can directly compare their medians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We compare medians instead of means to avoid being sensitive to a few outliers in our SFR and re data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' For a given property we added the difference between the blob and field galaxy medians to every individual blob galaxy measurement, thereby preserving the blob distribution’s shape and spread but shifting its median to coincide with that of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' When we redo the KS and AD tests, we find that for all five properties (F160W, F814W, stellar mass, SFR, and effective radius), the KS test p-values rise to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='96, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='94, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='67, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='72 respectively, and the AD test p-values exceed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' This indicates that the shape and spread of the distributions do not differ significantly, and that the difference in median is driving the statistical difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Thus, this simple exercise indicates that galaxies near blobs are roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='49 magnitudes or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 times brighter in F160W and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='59 magnitudes or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='7 times brighter in F814W, have 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='9 times more stellar mass, have star formation Galaxies in the Vicinity of Lyα Nebulae 7 7 8 9 10 11 log[mass/M ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 Probability Density Mass 7 8 9 10 11 log[mass/M ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 Cumulative Probability KS test: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='01 AD test: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='01 Field (n = 949) Blobs (n = 72) 2 1 0 1 log[SFR/(M /yr)] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 Probability Density SFR 2 1 0 1 log[SFR/(M /yr)] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 Cumulative Probability KS test: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='02 AD test: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='01 Field (n = 947) Blobs (n = 72) 12 11 10 9 8 7 log[sSFR*yr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 Probability Density SFR / Mass 12 11 10 9 8 7 log[sSFR*yr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 Cumulative Probability KS test: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='46 AD test: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='13 Field (n = 947) Blobs (n = 72) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Comparisons of the distributions of galaxies near Lyα nebulae (red) and those in the field (blue) for three properties: stellar mass, star formation rate (SFR), and SFR per unit stellar mass (or specific star formation rate, sSFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The median value of each group is indicated with a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Sample sizes and p-values from the KS and AD statistical tests are provided in the inset boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' A systematic offset towards higher mass and SFR is seen among the blob galaxies with respect to those in the field, with correspondingly low p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' No statistically significant difference is seen in sSFR between the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5 log[re/arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 Probability Density Effective Radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 log[re/arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 Cumulative Probability KS test: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='06 AD test: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='03 Field (n = 622) Blobs (n = 44) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Comparison of blob (red) and field (blue) galaxy effective radius distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The median value of each group is indicated with a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Sample sizes and p-values from the KS and AD statistical tests are provided in the inset boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' A systematic offset towards larger effective radii is seen among the blob galaxies with respect to those in the field, accompanied by correspondingly low p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 8 Wells, Prescott & Finlator 22 24 26 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 Probability Density Cropped at 50% Completeness Limit 22 24 26 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 Cumulative Probability KS test: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='06 AD test: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='02 Field (n = 876) Blobs (n = 74) 22 24 26 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='5 Probability Density Cropped at 75% Completeness Limit 22 24 26 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 Cumulative Probability KS test: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='29 AD test: p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='25 Field (n = 681) Blobs (n = 66) 22 24 26 28 AB magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 Probability Density Corrected for Incompleteness 22 24 26 28 AB magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 Cumulative Probability KS test: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='01 AD test: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='01 Field (n = 1870) Blobs (n = 119) F160W Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Comparison of blob and field galaxy F160W distributions when cropping the sample at the 50% (top) and 75% (middle) completeness limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Alternatively, we simulated a sample corrected for incompleteness by adding the appropriate number of mock galaxies to the faintest bins (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' rates that are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 times higher, and have effective radii that are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3 times larger than galaxies in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In what follows, we explore whether these results could be influenced by issues of incompleteness or sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Verifying the Results In this section we describe the various measures we took to test the reliability of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In particular, we investigated the consequences of imposing completeness limits, how the size of the region within which we select blob galaxies affects our results, and whether a subset of the six blob regions is primarily responsible for the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We also estimated the expected false positive rate by selecting six random blob-free locations within GOODS-S and analyzing the resulting “mock” blob and field galaxies in identical fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Sample Completeness The faint end of our galaxy sample will necessarily be affected by incompleteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' However, given that our sample is drawn from regions of consistent survey depth (F160W limiting magnitude mlim = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='50±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='25 mag), we expect our blob and field galaxy populations to be incomplete to the same degree, allowing them to be fairly compared with one another while keeping in mind this limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Nevertheless, we confirmed that our qualitative results are robust to completeness issues via two distinct experiments: (1) removing the faintest galaxies and (2) correcting the sample for incompleteness by adding simulated galaxies to the faintest bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In what follows, we use the completeness fraction as a function of F160W magnitude for the GOODS-S field (Skelton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' First, when only considering blob and field galaxies with F160W magnitudes brighter than the 50% completeness limit, the AD and KS p-values remain significant at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='02 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='06 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' When we crop the sample at the 75% completeness limit, both p-values rise above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='25 (Figure 7, top and middle panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' As discussed in Section 5, though our sample is corrected for depth, the blob and field galaxy F160W and F814W distributions do not fall off in the same manner at the faint end as would be expected, but rather there appear to be proportionally fewer faint galaxies Galaxies in the Vicinity of Lyα Nebulae 9 10 20 30 40 50 60 70 80 90 100 Blob Region Radius (arcsec) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='0 p-value Mass SFR re F160W F814W Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' KS test results comparing selected properties of sets of blob and field galaxies for increasing blob region radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' It is not until the defined blob region reaches ∼40” in radius that the blob-associated galaxy sample size is large enough for the KS test to detect the difference between galaxies in the blob region and those in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' in the blob regions (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' When we remove the faintest galaxies to increase the completeness of our sample, the signal weakens, emphasizing that the faint end is contributing to the differences we detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In the second approach, we simulated a complete F160W sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Using Skelton et al.’s (2014) completeness fraction for each half-magnitude bin, we calculated the number of missing galaxies in a given bin, and drew a corresponding number of galaxy magnitudes randomly, assuming a uniform distribution within the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' After artificially recovering the faintest galaxies in our sample in this way, differences between the blob and field F160W distributions become even more pronounced, with both AD and KS p-values falling below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='01 (Figure 7, bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' These incompleteness tests suggest that the existing data are already revealing real differences between blob and field galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In the remainder of the paper, we therefore choose to stay close to the data, focusing on the fiducial depth-corrected sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' We leave a more in-depth investigation of the impact of incompleteness on all other measured quantities to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The Blob Region Radius In using Lyα nebulae as markers of overdensities, we must define the angular extent of such regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' A small “blob region” radius, centered on a Lyα location, is more likely to capture a high fraction of true inhabitants of the dense environment, but restricts our sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' As the radius is extended, we benefit from a larger blob galaxy sample, but that sample is more likely to contain a greater fraction of interlopers - galaxies that are not subject to the same potential influences as those embedded in an overdense environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Making no a priori assumptions about what the plausible physical size of such a dense region might be, we tested blob region radii from 10”-100” (∼84-840 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' This range straddles the expected virial radius of Rvir=219 kpc for a galaxy group mass halo (∼1013 M⊙) at z∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3, based on a standard spherical collapse calculation (Bryan & Norman 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In steps of 5”, we compared the properties of each new set of trial blob and field galaxies using the KS and AD tests as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' This allowed us to see how the p-values changed as the defined blob region varied in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' For simplicity and as the more conservative option we plot only the KS test p-values in this subsection (Figures 8-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' However, in general we found the AD test to produce lower p-values given the same samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In Figure 8 we see that comparisons in the five selected properties produce p-values that dip below p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='05 when the blob region reaches ∼40” in radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' For r=10-35”, the blob galaxy sample size grows from ∼12 to ∼65 when considering the filter magnitudes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' from 11 to 61 when considering the mass and SFR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' and from 8 to 35 when considering the effective radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The larger p-values seen in this range show the limit of the KS test for small sample sizes - we have demonstrated that the two populations differ in these properties when r=40” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Figure 3), but these differences cannot be detected at smaller radii given the limited blob galaxy sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' A potential problem with varying the size of the blob region radius in this way is that as the blob galaxy sample grows the field galaxy sample shrinks, with members of the latter group switching to the former as the blob region radius increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' Consequentially, the blob sample is not being compared to a static group, but rather one whose distribution changes slightly at every step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' In order to control for this, we performed the same test but this time defined field galaxies to be those that lie greater than 60” from any Lyα nebula location, giving us a consistent field 10 Wells, Prescott & Finlator group to which to compare blob galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' As we scanned through blob region radii of 10”, 20”, 30”, 40”, and 50”, those galaxies that fell in the gap between the blob region radius and our field galaxy radius were simply ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' This method produced results that were virtually identical to Figure 8, demonstrating that the impact of slight changes in the field galaxy distribution is negligible when using the previous approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The Core Blob Galaxies Given that the p-values for the five properties in Figure 8 dip below and generally stay under p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content='05 from r=30” out to r=100”, we then investigated whether 1) the galaxies closest to the blobs were responsible for this outcome, differing from the field enough to depress the p-values at large blob region radii despite the likely possibility that larger radii will incorporate more and more interlopers into the blob galaxy sample, or whether 2) galaxies at relatively large distances from the blobs were contributing to this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' To this end, we progressively removed the most central blob galaxies from the analysis entirely, and, allowing the blob region radius to vary as before, compared the properties of the blob and field galaxy subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' The results of this test are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' With more and more central blob galaxies missing, the p-value curves steadily rise, demonstrating that the core blob galaxies are indeed responsible for the low p-values seen in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' When central galaxies within 40” of any blob are absent, statistically significant differences in the five selected properties disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E0T4oBgHgl3EQf5wJ-/content/2301.02755v1.pdf'} +page_content=' This is not due to prohibitively small sample sizes, as the blob sample size at this stage (those galaxies with 40” 0) and the second term with the attraction +(A2 < 0). At small distances ν(r) = (A1 − A2)/r12 > 0. As a consequence, we should have +A1 > A2. +In the formalism of statistical field theory the Hamiltonian H[ρ(r)] is a functional of field +and consists of the ideal entropy and the interaction: +βH[ρ(r1)] = βHentr[ρ(r1)] + βHint[ρ(r1)] = +(2) +� +ρ(r1) +� +ln +� +ρ(r1)Λ3� +− 1 +� +dr1+ +β +2 +� +ν(r12) +� +ρ(r1)ρ(r2) − ρ(r1)δ(r1 − r2) +� +dr1dr2, +4 + +where β = 1/kT is the inverse temperature, ρ(r) is the particle density, and Λ is the thermal +de Broglie wavelength of the particles. +As in previous papers18,28,29, we adopt the canonical ensemble approach. +We fix the +number of particles by the conditions +� +ρ(r)dr = N or +1 +V +� +ρ(r)dr = ρb, where V is the +volume and ρb = N/V is the average value of the bulk density of the system. To verify this +condition in a formally unconstrained calculus we introduce a Lagrange multiplier λ such +that +δβH[ρ(r)] +δρ(r) += λ. +(3) +The partition function ZN [ρ(r)] can be expressed as +ZN [ρ(r)] = +� +Dρ(r) exp{−βH[ρ(r)]}, +(4) +where Dρ(r) denotes functional integration over all possible density distributions such that +the total number of particles is N. +The logarithm of the partition function gives the +Helmholtz free energy +βF = − ln ZN. +(5) +III. +MEAN FIELD APPROXIMATION +The lowest order approximation for the partition function is the saddle point for the +functional integral (4) which leads to the mean field approximation (MFA). +The condition for the mean field approximation is +δβH +δρ +���� +ρMF A(r) += λ. +(6) +In our case equation (6) reads +ln ρ(r1) +ρb ++ V1(r1) + V2(r1) = λ, +(7) +where potentials Vi(r1) are defined as +Vi(r1) = β +� +ρ(r2) Ai +r12 +exp(−αir12)dr2, +i = 1, 2. +(8) +We put +λ ≡ V1b + V2b, +(9) +5 + +where Vib are the values of potentials Vi(r1) in the bulk: +V1b = κ2 +1 +α2 +1 +; +V2b = κ2 +2 +α2 +2 +, +(10) +and κ2 +i ≡ 4πρbβAi. +The gradient of (7) gives +∇ρ(r) +ρ(r) − E1(r) − E2(r) = 0, +(11) +where we define an equivalent of the electric field by +E1(r1) ≡ −∇V1(r1); +E2(r1) ≡ −∇V2(r1). +(12) +Due to the properties of Yukawa potential +� +△ − α2 +1 +� +V1(r) = −4πβA1ρ(r); +(13) +� +△ − α2 +2 +� +V2(r) = −4πβA2ρ(r). +(14) +Replacing (13) and (14) into (11) and using translational invariance parallel to the wall we +obtain +d +dz +�ρ(z) +ρb ++ α2 +1 +2κ2 +1 +[V1(z)]2 − +1 +2κ2 +1 +E2 +1(z) + α2 +2 +2κ2 +2 +[V2(z)]2 − +1 +2κ2 +2 +E2 +2(z) +� += 0, +(15) +where z is the distance between the particle and the wall. +A. +Contact theorem +In the bulk ρ(z) → ρb, Ei(z) → 0, Vi(z) → Vib. From eq. (15) we see that the quantity in +brackets is constant and therefore it can be evaluated for instance in the bulk as +1 + κ2 +1 +2α2 +1 ++ κ2 +2 +2α2 +2 +. +(16) +This quantity is the reduced pressure βP/ρb within MFA: +βP = ρb +� +1 + κ2 +1 +2α2 +1 ++ κ2 +2 +2α2 +2 +� +. +(17) +Expression (17) is the mean field approximation which corresponds to the Van der Waals +contribution. Outside the system, where there are no particles, we have another invariant +which is simply α2 +1 V 2 +1 (z)/2κ2 +1 − E2 +1(z)/2κ2 +1 + α2 +2V 2 +2 (z)/2κ2 +2 − E2 +2(z)/2κ2 +2, its value far from +6 + +the interface is zero and therefore also at the interface. From the continuity of the potential +and of its derivative due to eq. (13) and (14), we have that this is also true at the wall just +inside the system z = 0+ thus +ρ(0+) +ρb ++ α2 +1 +2κ2 +1 +[V1(0+)]2 − +1 +2κ2 +1 +E2 +1(0+)+ α2 +2 +2κ2 +2 +[V2(0+)]2 − +1 +2κ2 +2 +E2 +2(0+) = ρ(0+) +ρb +. +(18) +As this quantity is constant we obtain the so-called contact theorem +βP = ρ(0+). +(19) +Thus, similar to the one-Yukawa case18, in the MFA we obtain the contact theorem as the +consequence of the existence of an invariant of the differential equations corresponding to +the bulk pressure. +B. +Density profiles +From (11)-(14) we obtain a set of five differential equations with five unknown functions +ρ(z), E1(z), E2(z), V1(z), V2(z): +∂ρ(z) +∂z += ρ(z) [E1(z) + E2(z)] , +(20) +∂V1(z) +∂z += −E1(z), +(21) +∂V2(z) +∂z += −E2(z), +(22) +∂E1(z) +∂z += −α2 +1V1(z) + κ2 +1 +ρb +ρ(z), +(23) +∂E2(z) +∂z += −α2 +2V2(z) + κ2 +2 +ρb +ρ(z). +(24) +These relations are first-order differential equations that can be solved numerically. +From (7) we have +ρ(z) = ρb exp +� +− [V1(z) − V1b] − [V2(z) − V2b] +� +, +(25) +where Vi(z) and Vib are given by (8) and (10). +Similar to18 we can solve equation (25) in the linear approximation with the boundary +condition set by the contact theorem. +This linear solution was obtained in30. +Here we +7 + +present only the final result +ρL(z) +ρb += 1 − 1 +2 +(λ2 +1 − α2 +2) +(λ2 +1 − λ2 +2) +� +−κ2 +1 +α2 +1 ++ λ2 +2 − α2 +2 − κ2 +2 +α2 +2 +� +e−λ1z +(26) +− 1 +2 +(λ2 +2 − α2 +2) +(λ2 +1 − λ2 +2) +�κ2 +1 +α2 +1 +− λ2 +1 − α2 +2 − κ2 +2 +α2 +2 +� +e−λ2z, +where +λ2 +1,2 = 1 +2 +� +κ2 +1 + α2 +1 + κ2 +2 + α2 +2 ± +� +(κ2 +1 + α2 +1 − κ2 +2 − α2 +2)2 + 4κ2 +1κ2 +2 +� +. +(27) +IV. +FLUCTUATION AND CORRELATION EFFECTS ON DENSITY +PROFILES AT THE WALL +In the previous section we have considered mean field equations, where the fluctuations are +neglected. Here we take them into account and therefore we have to expand the Hamiltonian +with respect to the mean field density ρMF A(r). For this aim we put ρ(r) = ρMF A(r)+δρ(r). +A. +Expansion of the Hamiltonian +Expansion of the Hamiltonian around the mean field density ρMF A(r) gives +βH[ρ] = βH +� +ρMF A� ++ +� +δρ(r1) +δβH +δ(δρ(r1)) +���� +ρMF Adr1+ +(28) +1 +2 +� +δρ(r1)δρ(r2) +δ2βH +δ(δρ(r1))δ(δρ(r2)) +���� +ρMF A +dr1dr2+ +� +n≥3 +(−1)n(n − 1)! +n! +� +δρ(r1) ... δρ(rn) +δnβH +δ(δρ(r1)) ... δ(δρ(rn)) +���� +ρMF A +dr1...drn. +The first term is the Hamiltonian functional (2) for the mean field density +βH[ρMF A] = +� +ρMF A(r1) +� +ln +� +ρMF A(r1)Λ3� +− 1 +� +dr1 +(29) ++ β +2 +� +ν(r12) +� +ρMF A(r1)ρMF A(r2) − ρMF A(r1)δ(r1 − r2) +� +dr1dr2. +The linear term disappears as in the canonical formalism fluctuations preserve the number +of particles and +� +δρ(r)dr = 0. +The quadratic term is +βH2[ρ] = 1 +2 +� +δρ(r1)δρ(r2) +�δ(r1 − r2) +ρMF A(r1) + βν(r12) +� +dr1dr2, +(30) +8 + +where the first term comes from the expansion of the logarithmic term in the Hamiltonian. +Due to translational invariance parallel to the wall, we expand the fluctuations of the +density as +δρ(r) = +� +K +δρK(z) eiKR, +(31) +where R is the vector component of r parallel to the wall, K is the wave vector in the +direction parallel to the wall. +The entropic term equals +βHentr +2 +[ρK(z)] = 1 +2 +� +δρ2(r) +ρMF A(z) dr +(32) += 1 +2 +� +K,K′ +� δρK(z)δρK′(z) +ρMF A(z) +eiR(K + K +′)dRdz += S +2 +� +K +� +dz1dz2 δρK(z1)δρ−K(z2) δ(z1 − z2) +ρMF A(z) , +where S is the surface area. +The interaction term gives +βHint +2 +[ρK(z)] = S β +2 +� +K +� +dz1 +� +dz2δρK(z1)δρ−K(z2) ν (K, |z1 − z2|) , +(33) +where ν (K, |z1 − z2|) = +� +dR12 ν(r12) exp (−iKR12). +Finally, for the quadratic term of the Hamiltonian we obtain +βH2[ρ] = +(34) +S +2 +� +K +� +dz1 +� +dz2 δρK(z1)δρ−K(z2) +�δ (z1 − z2) +ρMF A(z1) + βν (K, |z1 − z2|) +� +. +B. +Thermodynamic properties: free energy, pressure, and chemical potential +We start our calculations from consideration of thermodynamic properties of the fluid in +the bulk. +The free energy is +βF = − ln +�� +Dρ e−βH[ρ] +� +. +(35) +9 + +In order to calculate the functional integral using the Gaussian integrals with such a Hamil- +tonian, it is necessary to have the quadratic term in a diagonal form. For bulk properties +such as the Helmholtz free energy we can expand density on the Fourier components +δρ(r) = +� +k +δρk eikr. +(36) +In this basis the quadratic Hamiltonian is +βH2[ρ] = V +2ρb +� +k>0 +δρkδρ−k +� +1 + +κ2 +1 +k2 + α2 +1 ++ +κ2 +2 +k2 + α2 +2 +� +(37) +and after integration the excess free energy equals +βF ex = β(F − F id) = +(38) +ρbV κ2 +1 +2α2 +1 ++ ρbV κ2 +1 +2α2 +1 ++ 1 +2 +� +k +ln [1 + ρb ν(k)] − 1 +2 ρb +� +k +ν(k), +where +ν(k) = 4πβA1 +k2 + α2 +1 ++ 4πβA2 +k2 + α2 +2 +(39) +is the Fourier transform of the interaction potential (1) multiplied by β. +The first and the second terms on the right-hand side of (38) are mean field contributions +with the other two terms coming from Gaussian fluctuations. In order to calculate the third +and the fourth terms we switch from summation to integration and then integrate by parts +βF fluct = 1 +2 +� +k +ln [1 + ρbν(k)] − 1 +2 ρb +� +k +ν(k) +(40) += ρ2 +bV +12π2 +∞ +� +0 +k3dk +ν(k) +1 + ρbν(k) +d ν(k) +dk +. +For further calculations it is helpful to express parameters κ2 +1 and κ2 +2 in terms of λ1 and λ2. +From (27) we have +κ2 +1 = (α2 +1 − λ2 +1) (α2 +1 − λ2 +2) +α2 +2 − α2 +1 +, +κ2 +2 = (α2 +2 − λ2 +1) (α2 +2 − λ2 +2) +α2 +1 − α2 +2 +. +(41) +Using identities (41), after integration we obtain +βF ex +V += ρb +2 +�κ2 +1 +α2 +1 ++ κ2 +2 +α2 +2 +� +− +1 +12π(λ1 +3 + λ2 +3) − +1 +24π(α1 +3 + α2 +3) +(42) ++ 1 +8π +� +λ1 +2 + λ2 +2� +(α1 + α2) − 1 +8π +� +λ1 +2 + α1α2 +� � +λ2 +2 + α1α2 +� +α1 + α2 +. +10 + +The pressure can be found from the free energy as +βP = −β ∂F +∂V +���� +T,N +. +(43) +Differentiation of (42) with respect to volume gives the fluctuation part of the bulk pressure +as +βP fluct == +ρ2 +b +12π2 +∞ +� +0 +k3dk +ν(k) +[1 + ρbν(k)]2 +d ν(k) +dk +. +(44) +After integration and due to identities (41) the excess pressure equals +βP ex = ρb +2 +�κ2 +1 +α2 +1 ++ κ2 +2 +α2 +2 +� +− +1 +24π(λ1 +3 + λ2 +3) − +1 +12π(α1 +3 + α2 +3) +(45) ++ 1 +8π +� +α1 +2 + α2 +2� +(λ1 + λ2) − 1 +8π +1 +λ1 + λ2 +� +α1 +2 + λ1λ2 +� � +α2 +2 + λ1λ2 +� +. +Finally, the excess chemical potential can be derived directly from (42) and (45) as µex = +(F ex + P exV ) /N giving +βµex = κ2 +1 +α2 +1 ++ κ2 +2 +α2 +2 +− +1 +8πρb +(λ3 +1 + λ3 +2) − +1 +8πρb +(α3 +1 + α3 +2) +(46) ++ +1 +8πρb +(λ2 +1 + λ2 +2)(α1 + α2) + +1 +8πρb +(α2 +1 + α2 +2)(λ1 + λ2) +− +1 +8πρb +(λ2 +1 + α1α2)(λ2 +2 + α1α2) +α1 + α2 +− +1 +8πρb +(α2 +1 + λ1λ2)(α2 +2 + λ1λ2) +λ1 + λ2 +. +C. +Correlation function +The expression for the pair correlation function h(r1, r2) is31 +h(r1, r2)⟨ρ(r1)⟩⟨ρ(r2)⟩ = ⟨δρ(r1)δρ(r2)⟩ − δ (r1 − r2) ⟨ρ(r1)⟩. +(47) +In K-space this expression reads +1 +S +� +ρMF A(z1)ρMF A(z2) h(K, z1z2)+ρMF A(z1)δ(z1 − z2) +� +(48) += ⟨δρK(z1)δρ−K(z2)⟩, +where +h(K, z1z2) = +� +dR12 h(R12, z1, z2) exp (iKR12) . +(49) +11 + +The right-hand side of equation (48) can be calculated from expression (34) and gives the +inverse Hamiltonian matrix βH−1 +2 +[ρK(z)] /2 +⟨δρK(z1)δρ−K(z2)⟩ = +� +D(δρK(z))δρK(z1)δρ−K(z2) exp (−βH2[ρK(z)]) +� +D(δρK(z)) exp (−βH2[ρK(z)]) += 1 +2βH−1 +2 +[ρK(z)] . +(50) +Hence the product of the Hamiltonian matrix and the matrix on the left-hand side of (48) +yields unity, so we have +� +dz3 +�� +ρMF A(z1)ρMF A(z3) h(K, z1z3) + ρMF A(z1)δ(z1 − z3) +� +�δ (z3 − z2) +ρMF A(z3) + βν (K, |z3 − z2|) +� � += δ(z1 − z2), +(51) +or +h(K, z1, z2) + +� +dz3ρMF A(z3)h(K, z1, z3)βν (K, |z3 − z2|) +(52) += −βν (K, |z1 − z2|) . +Relation (52) is a convolution-type equation. It can be reduced to the so-called Riemann +problem32 if we assume the density profile to be a step-function. In this approximation +ρMF A(z) = 0 for z < 0 and ρMF A(z) = ρb for z > 0. +Due to the spatial non-homogeneousness of the system we introduce one-sided pair cor- +relation functions h±(R12, z1, z2) such that +h(R12, z1, z2) = h+(R12, z1, z2) − h−(R12, z1, z2), +h+(R12, z1, z2) = + + + +h(R12, z1, z2), z1 > 0, +0, +z1 < 0, +(53) +h−(R12, z1, z2) = + + + +0, +z1 > 0, +−h(R12, z1, z2), z1 < 0. +. +The function h(K, z1, z2) can then be presented as the difference of one-sided functions +12 + +h±(K, z1, z2) such that +h(K, z1, z2) = h+(K, z1, z2) − h−(K, z1, z2), +h+(K, z1, z2) = + + + +h(K, z1, z2), z1 > 0, +0, +z1 < 0, +(54) +h−(K, z1, z2) = + + + +0, +z1 > 0, +−h(K, z1, z2), z1 < 0. +Equation (52) now reads +h+(K, z1, z2) − h−(K, z1, z2) + ρb +∞ +� +0 +dz3h+(K, z1, z3)βν (K, |z3 − z2|) += −βν (K, |z1 − z2|) . +(55) +Expanding the functions h±(K, z1, z2) and ν (K, |z1 − z2|) on Fourier harmonics with respect +to the wave vector µ in the direction perpendicular to the wall and switching from summation +to integration we obtain +� +1 + +κ2 +1 +K2 + µ2 +1 + α2 +1 ++ +κ2 +2 +K2 + µ2 +1 + α2 +2 +� +h+(K, µ1µ2) − h−(K, µ1µ2) +(56) += − +� +4πβA1 +K2 + µ2 +2 + α2 +1 ++ +4πβA2 +K2 + µ2 +2 + α2 +2 +� +δ(µ1 + µ2), +where +h±(K, µ1, µ2) = +� +S +dR12eiKR12 +∞ +� +−∞ +dz1eiµ1z1 +∞ +� +−∞ +dz2eiµ2z2h±(R12, z1z2) +(57) +and we have used the relation +∞ +� +−∞ +dz1eiµ1z1 +∞ +� +−∞ +dz2eiµ2z2βν (K, |z1 − z2|) = +(58) += − +� +4πβA1 +K2 + µ2 +2 + α2 +1 ++ +4πβA2 +K2 + µ2 +2 + α2 +2 +� +δ(µ1 + µ2). +Equation (56) is known as the Riemann problem32. Using the technique proposed in17,30 we +solve this problem for h+(K, µ1µ2) (refer to Appendix A for the details of calculation) and +13 + +obtain +h+(K, µ1, µ2) = +− 1 +ρb +κ2 +1(µ2 +2 + α2 +2(K)) + κ2 +2(µ2 +2 + α2 +1(K)) +(µ2 − iα1(K))(µ2 − iα2(K))(µ2 + iλ1(K))(µ2 + iλ2(K)) +(µ1 + iα1(K))(µ1 + iα2(K)) +(µ1 + iλ1(K))(µ1 + iλ2(K)) δ+(µ1 + µ2), +(59) +where δ+(µ1 + µ2) is a one-sided delta-function. +The expression for the correlation function in r-space can be presented as the sum of the +homogeneous bulk part hb ++(r12) and the inhomogeneous surface part hinh ++ (r12, z1z2) +h+(r12, z1, z2) = hb ++(r12) + hinh ++ (r12, z1z2), +(60) +where +hb ++(r12) =βA1(λ2 +2 − α2 +2) + A2(λ2 +2 − α2 +1) +(λ2 +1 − λ2 +2) +e−λ2r12 +r12 +− +(61) +βA1(λ2 +1 − α2 +2) + A2(λ2 +1 − α2 +1) +(λ2 +1 − λ2 +2) +e−λ1r12 +r12 +, +hinh ++ (r12, z1, z2) = +(62) +− β +∞ +� +0 +2K J0(KR12) dK +�A1(λ2 +2 − α2 +2) + A2(λ2 +2 − α2 +1) +2λ2(K)(λ1(K) − λ2(K))2 +(λ2(K) − α1(K))(λ2(K) − α2(K)) +(λ2(K) + α1(K))(λ2(K) + α2(K))e−λ2(K)(z1+z2) +− +A1(λ2 +1 − α2 +2) + A2(λ2 +1 − α2 +1) +(λ1(K) + λ2(K))(λ1(K) − λ2(K))2 +(λ2(K) − α1(K))(λ2(K) − α2(K)) +(λ1(K) + α1(K))(λ1(K) + α2(K))e−λ2(K)z1−λ1(K)z2 +− +A1(λ2 +2 − α2 +2) + A2(λ2 +2 − α2 +1) +(λ1(K) + λ2(K))(λ1(K) − λ2(K))2 +(λ1(K) − α1(K))(λ1(K) − α2(K)) +(λ2(K) + α1(K))(λ2(K) + α2(K))e−λ1(K)z1−λ2(K)z2 ++ A1(λ2 +1 − α2 +2) + A2(λ2 +1 − α2 +1) +2λ1(K)(λ1(K) − λ2(K))2 +(λ1(K) − α1(K))(λ1(K) − α2(K)) +(λ1(K) + α1(K))(λ1(K) + α2(K))e−λ1(K)(z1+z2) +� +, +αi(K) = +� +α2 +i + K2; +λi(K) = +� +λ2 +i + K2, +(63) +14 + +and J0(KR12) is the Bessel function of the first kind. +As we see from (61), λ1 and λ2 play the role of parameters characterizing the screening +of the repulsive and the attractive interactions respectively. +D. +Density profile +In the Gaussian approximation the inhomogeneous density profile can be written as the +sum of the mean field profile ρMF A(z) and the quadratic fluctuation term ρfluct(z) +ρ(z) = ρMF A(z) + ρfluct(z). +(64) +The contribution of quadratic fluctuations to the profile corresponds to the one-particle +irreducible diagram in the field theory33,34 and can be found as: +ρfluct(z1) +ρb += 1 +2 +� +h+(R, z1, z2) − hb ++(R, z1, z2) +� ���� +z2→z1 +R→0 +, +(65) +where calculating the inhomogeneous profile we have subtracted the homogeneous bulk part. +As a result +ρfluct(z1) +ρb += − +1 +8πρb +∞ +� +0 +K dK +�κ2 +1(λ2 +1 − α2 +2) + κ2 +2(λ2 +1 − α2 +1) +λ1(K)(λ2(K) − λ1(K))2 × +(66) +(λ1(K) − α1(K))(λ1(K) − α2(K)) +(λ1(K) + α1(K))(λ1(K) + α2(K)) e−2λ1(K)z1 +− 2 +� +κ2 +1(λ2 +2 − α2 +2) + κ2 +2(λ2 +2 − α2 +1) +(λ2(K) + λ1(K))(λ2(K) − λ1(K))2× +(λ1(K) − α1(K))(λ1(K) − α2(K)) +(λ2(K) + α1(K))(λ2(K) + α2(K)) ++ +κ2 +1(λ2 +1 − α2 +2) + κ2 +2(λ2 +1 − α2 +1) +(λ2(K) + λ1(K))(λ2(K) − λ1(K))2× +(λ2(K) − α1(K))(λ2(K) − α2(K)) +(λ1(K) + α1(K))(λ1(K) + α2(K)) +� +e−[λ1(K)+λ2(K)]z1 ++ κ2 +1(λ2 +2 − α2 +2) + κ2 +2(λ2 +2 − α2 +1) +λ2(K)(λ2(K) − λ1(K))2 × +(λ2(K) − α1(K))(λ2(K) − α2(K)) +(λ2(K) + α1(K))(λ2(K) + α2(K)) e−2λ2(K)z1 +� +. +15 + +E. +Contact theorem +In Section III A we have shown the validity of the contact theorem in the mean field +approximation. Here we will show that for the considered model the contact theorem is also +satisfied when the fluctuations are taken into account. +Setting z1 = 0 in expression (66) and using identities (41), we obtain the contact value +of density +ρfluct(0+) = 1 +4π +∞ +� +0 +KdK +� +α1(K) + α2(K) − 1 +2 [λ1(K) + λ2(K)] +(67) +− 1 +2 +[α2 +1(K) + λ1(K)λ2(K)] [α2 +2(K) + λ1(K)λ2(K)] +λ1(K)λ2(K) [λ1(K) + λ2(K)] +� +. +Going back to expression (44) for the pressure we can calculate the fluctuation part of the +pressure using the cylindrical coordinate system instead of the spherical one. Then we have +βP fluct = +ρ2 +b +12π2 +∞ +� +0 +k3dk +ν(k) +[1 + ρν(k)]2 +d ν(k) +dk +(68) += − 1 +2π2 +∞ +� +0 +KdK +∞ +� +−∞ +µ2 dµ +� +κ2 +1 (µ2 + α2 +2(K)) + κ2 +2 (µ2 + α2 +1(K)) +� +[µ2 + λ2 +1(K)]2 [µ2 + λ2 +2(K)]2 +� +κ2 +1 (µ2 + α2 +2(K))2 + κ2 +2 (µ2 + α2 +1(K))2� +[µ2 + α2 +1(K)] [µ2 + α2 +2(K)] +. +After integration with respect to µ and taking into account relations (41) we obtain +βP fluct = 1 +4π +∞ +� +0 +KdK +� +α1(K) + α2(K) − 1 +2 [λ1(K) + λ2(K)] +(69) +− 1 +2 +[α2 +1(K) + λ1(K)λ2(K)] [α2 +2(K) + λ1(K)λ2(K)] +λ1(K)λ2(K) [λ1(K) + λ2(K)] +� +, +which is exactly the expression (67). +We have therefore proved the validity of the contact theorem for the fluctuation term of +the density profile. +16 + +F. +Adsorption +We can also calculate the adsorption coefficient defined as +Γ = +∞ +� +0 +dz [ρ(z) − ρb] = ΓMF A + Γfluct +(70) +according to different approximations of the mean field density profile presented in Section +III B. +Hence the exact mean field contribution can be determined only numerically. +The linearized equation (26) gives +ΓL +MF A = − ρb +2λ1 +(λ2 +1 − α2 +2) +(λ2 +1 − λ2 +2) +� +−κ2 +1 +α2 +1 ++ λ2 +2 − α2 +2 − κ2 +2 +α2 +2 +� +(71) +− ρb +2λ2 +(λ2 +2 − α2 +2) +(λ2 +1 − λ2 +2) +�κ2 +1 +α2 +1 +− λ2 +1 − α2 +2 − κ2 +2 +α2 +2 +� +. +For the fluctuation part of the adsorption coefficient due to identities (41) we obtain an +analytical result +Γfluct = +1 +32π (λ1 + λ2)2 + +1 +32π (α2 +1 + α2 +2) − +1 +16π (λ1 + λ2) (α1 + α2) +− +1 +16π +(λ2λ1 + α22) (λ2λ1 + α12) +(λ1 + λ2)2 ++ +1 +16π +(α1 + α2) (λ2λ1 + α2α1) +λ1 + λ2 ++ +1 +16π +� +λ2 +1 + λ2 +2 − α2 +1 − α2 +2 +� +ln +�(λ2 + α1)(λ2 + α2) +2λ2(λ1 + λ2) +� ++ +1 +16π +(λ2 +1 − α2 +1)(λ2 +1 − α2 +2) +λ2 +2 − λ2 +1 +ln +�λ1 +λ2 +(λ2 + α1)(λ2 + α2) +(λ1 + α1)(λ1 + α2) +� +. +(72) +V. +MONTE-CARLO SIMULATIONS +In order to test an accuracy of the field theoretical results established above the Monte +Carlo (MC)35 simulations were carried out. A system of fluid particles interacting with the +two-Yukawa potential (1) was considered in a rectangular simulation box. A cutoff distance +of the potential was chosen rc = 12.0. A minimum size of the simulation box was set at +least twice larger than the cutoff distance. A usual periodic boundary conditions along x, +y and z directions were applied for the bulk fluid. However, to study a fluid near the hard +wall a simulation box was confined between two walls orthogonal to z-axis and in this case +the periodic boundary conditions were applied only in xy-plane. A distance between walls, +17 + +2 +3 +4 +5 +6 +7 +0.00 +0.01 +(r) + + + +r +FIG. 1. Pair interaction potential (1) for different values of ω and τ. +Lz , was taken large enough to form a wide layer of the bulk phase in the middle of the +box (Lz = 37.0). A number of fluid particles depended on the considered densities (ρ∗ = +ρb/α3 +2 = 0.1, 0.2, 0.3) and it varied in the range of N = 6000−12000. At each simulation step +N trial movements of particles were performed. To speed up the simulations, the linked cell +list algorithm was employed35. The density profiles of a confined fluid, ρ(z), were calculated +and averaged over 100000 simulation step, while the pair distribution functions of a bulk +fluid, g(r), were averaged over 10000 steps. The model was studied for the different ratios +of parameters A1/|A2| and α1/α2 in the temperature region of T ∗ = T/(α2|A2|) = 0.5 − 1.5. +VI. +RESULTS AND DISCUSSION +The properties of the considered two-Yukawa fluid are defined by four non-dimensional +parameters: ρ∗ = ρb/α3 +2, T ∗ = −1/(βA2α2), ω = A1/|A2| and τ = α1/α2. +The first +two parameters are non-dimensional density and temperature respectively. The last two +parameters are connected with the form of interparticle interaction. Below we will consider +three types of models with ω = 3, τ = 1.498; ω = 2, τ = 1.35 and ω = 2.5, τ = 1.355. The +forms of interparticle interaction corresponding to these three cases are presented in Fig. 1. +We begin presentation of our results with the discussion of the bulk pair distribution +function. +18 + +0 +2 +4 +0,0 +0,2 +0,4 +0,6 +0,8 +1,0 +T * +g(r) +r + MC + Exp + GA +0,0 +0,5 +1,0 +1,5 +2,0 +2,5 +3,0 +3,5 +4,0 +0,0 +0,2 +0,4 +0,6 +0,8 +1,0 +T * = 1 +g(r) +r + MC + Exp + GA +0,0 +0,5 +1,0 +1,5 +2,0 +2,5 +3,0 +3,5 +4,0 +0,0 +0,2 +0,4 +0,6 +0,8 +1,0 +T * +g(r) +r + MC + Exp + GA +FIG. 2. Analytical approximations (73) and (74) and computer simulation results for the pair +distribution function at different temperatures with ω = 2, τ = 1.35, ρ∗ = 0.1. ”MC” corresponds +to Monte Carlo simulations, ”GA” is Gaussian approximation (73) and ”Exp” is exponential ap- +proximation (74). +A. +Bulk pair distribution function +The bulk pair distribution function (PDF) in the considered Gaussian approximation can +be presented in the form +gb(r) = 1 + hb ++(r), +(73) +where hb ++(r) is given by equation (61). However, at small distances hb ++(r) is of Coulombic +form and hb ++(r) → −∞ when r → 0. In order to avoid this non-physical behavior of gb(r) +we can use the exponential form +gb(r) = exp +� +hb ++(r) +� +(74) +instead of the form (73). +The behavior of the PDF for a model with ω = 2, τ = 1.35, ρ∗ = 0.1 at different temper- +atures is presented in Fig. 2. As we can see the exponential form (74) ensures the correct +behavior of gb(r) at small distances and reproduces very well the results given by expression +(73) at large distances. These results are also in very good agreement with the computer +simulations data. The behavior of gb(r) at different temperatures and densities is illustrated +in Fig. 3. We can see that with decreasing temperature the first peak of gb(r) increases and +shifts to smaller distances. As the density increases at a fixed temperature the first peak of +gb(r) decreases and shifts to smaller distances. Such a behavior is the result of softness of +the model since it allows the particles to occupy the soft region as the density is increased +19 + +5 +0,995 +0,996 +0,997 +0,998 +0,999 +1,000 +1,001 +1,002 +1,003 +1,004 +g(r) +r + MC + Exp + MC + Exp +* +2 +4 +6 +0,992 +0,996 +1,000 +1,004 +g(r) +r + MC T * = 1 + Exp T * = 1 + MC T * = 2 + Exp T * = 2 +FIG. 3. Analytical approximation (74) and computer simulations results for the pair distribution +function at different densities (left) and different temperatures (right). The curves are labeled as +in Fig. 2 +and the temperature is decreased. We should also note that the agreement between theory +and computer simulations results becomes worse as the temperature decreases. The height +of the first peak increases faster in computer simulations as compared to theory. +B. +Density profile +According to (64) the density profile ρ(z) can be presented as the sum of two parts: +ρMF A(z) and ρfluct(z). The first one is the result of the mean field approximation, which can +be calculated from the equation (26) derived from the linearized solution of the equation (25) +as it was done in30. Another way to obtain ρMF A(z) is to solve the equation (25) numerically, +which is obviously more precise. To this aim we apply the Picard iterative method, where +the numerical integrations are performed using the trapezoidal rule with the step size ∆z = +0.01α2, while all needed integrations over r are done analytically. The cutoff for the two- +Yukawa potential (1) is taken at the distance rc = 12.0, at which the potential becomes +negligibly small (ν(rc)/ν(rmin) ∼ 10−4, rmin – a position of the potential minimum). Due to +the hard wall presence from one side and the bulk phase from opposite side to the wall the +boundary conditions for ρMF A(z) are defined as ρMF A(z) = 0 if z < 0 and ρMF A(z) = ρb if +z > 2rc. A precision of numerical solution for the density profile ρMF A(z) is restricted by +∥ρMF A +m+1 (z)−F[ρMF A +m +(z)]∥ < 10−6, where F[ρMF A +m +(z)] is a right-hand side of the equation (25) +and m is an iteration step. It is worth mentioning that the equation (25) is equivalent to +20 + +the Euler-Lagrange equation, which is usually used in the density functional theory within +the mean field approximation for the fluid near a hard wall. +The density profiles, ρMF A(z), obtained with a use of the linear approximation and the +iterative method are presented in Fig. 4. As one can see in Fig. 4 the numerical solution of +the equation (25) can be interpolated very well by the linear approximation (26). However, in +the region of lower temperatures the linear approximation overestimates the density profile +at intermediate distances. +The contact values ρ(0) calculated from MFA are essentially +higher than those obtained from the simulations. The general overestimation of MFA up to +the first minimum of ρ(z) is observed for all temperatures. A correction of ρMF A(z) by the +Gaussian fluctuation term ρfluct(z) should improve the result. +The Gaussian fluctuations term found from the solution of the inhomogeneous OZ equa- +tion with the Riemann boundary condition is given by the expression (66). It is observed in +Fig. 4 that the contribution from the fluctuations has a negative sign. This is an expected +result since in18 it was shown that for a one-Yukawa fluid the fluctuation part of the density +profile is negative for both attractive and repulsive interactions and produces the depletion +effect. We have demonstrated that this term satisfies the contact theorem condition (19). +Nevertheless, comparison with computer simulation results (Fig.4) shows that this term +leads to strong overestimation of the role of fluctuations. In addition, it gives a maximum of +the profile at small distances from the wall which is not predicted by computer simulations. +From Fig. 4 one can also see that expression (66) strongly underestimates the contact value +of the density profile. This is the consequence of the underestimated value of the pressure +calculated from eq. (45) which corresponds to approximation (73) for the bulk PDF. Thus +it would be more correct to calculate the pressure according to the virial theorem31 +βP +ρb += 1 − 2 +3πβ +� ∞ +0 +∂υ(r) +∂r +gb(r)r3dr. +(75) +and using the exponential approximation (74) for the bulk PDF. +Using this value of the pressure and the contact theorem we have corrected the density +profile at small distances starting from point z0 which corresponds to the inflection point for +the density fluctuation term (66). Interpolation of ρ(z) for the region of z < z0 has revealed +a rather accurate generalization in the form of Pad´e approximant36 +ρ(z) = ρ(0)/ +� +1 + Az + Bz2 + Cz3� +for +z < z0, +(76) +21 + +0 +1 +2 +3 +4 +5 +0,85 +0,90 +0,95 +1,00 +1,05 +1,10 +1,15 +z* + MFA Linear + MFA + MC + MFA+GF Interpolated + MFA+GF +* = 0.5 +(z)/ +b +0 +1 +2 +3 +4 +5 +0,94 +0,96 +0,98 +1,00 +1,02 +1,04 +1,06 +z* + MFA Linear + MFA + MC + MFA+GF Interpolated + MFA+GF +(z)/ +b +* = 1 +0 +1 +2 +3 +4 +5 +0,96 +0,98 +1,00 +1,02 +1,04 +z* + MFA Linear + MFA + MC + MFA+GF Interpolated + MFA+GF +(z) +b +FIG. 4. Density profile as a function of the reduced distance z∗ = zα2. Label ”MC” corresponds to +Monte Carlo simulations, ”MFA” is the numerical solution of eq. (25), ”MFA Linear” is given by +eq. (26), ”GF” is the Gaussian fluctuations term (66) and ”MFA+GF Interpolated” corresponds +to Gaussian approximation with interpolation (76) at small distances. Different graphs correspond +to different temperatures at a fixed density ρ∗ = 0.1 with ω = 2, τ = 1.35. +where ρ(0) is determined from the contact theorem and the pressure calculated from eq. +(75). The constants A, B, C are found from the continuity of ρ(z) and its first derivative at +the point z = z0 and the fact that the second derivative equals zero at this point. This is the +form we use to calculate the density profile. The results for the model with ω = 2, τ = 1.35 +at density ρ∗ = 0.1 and different temperatures are shown in Fig.4. One can see that the +results of calculation are in good agreement with the computer simulations data. +In Fig. 5 we present density profiles for a model with ω = 3, τ = 1.498 at different +temperatures and densities. One can see that the contact value of the density increases +and the minimum of the profile decreases as the temperature decreases. Below we will see +that this can lead to non-trivial behavior of the adsorption as a function of the temperature. +Likewise, the contact value of the density increases and the minimum of the profile decreases +22 + +0 +1 +2 +3 +4 +0,90 +0,95 +1,00 +1,05 +1,10 +1,15 +1,20 +* +(z)/ +b + MC + MFA+GF Interpolated + MC + MFA+GF Interpolated +z* +0 +1 +2 +3 +4 +0,90 +0,95 +1,00 +1,05 +1,10 +1,15 +1,20 +1,25 + MC T * = 0.5 + MFA+GF Interpolated T * = 0.5 + MC T * = 1 + MFA+GF Interpolated T * = 1 + MC T * = 1.5 + MFA+GF Interpolated T * = 1.5 +(z)/ +b +* +z* +FIG. 5. Theoretical approximations and computer simulations results for the density profile at +different densities (left) and different temperatures (right). The curves are labeled as in Fig. 4 +as the density increases at a fixed temperature. Due to the contact theorem and according +to expression (75) the increase of the contact value of the DP with increasing density or +decreasing temperature is connected with the respective increase of the fluid pressure in the +bulk. The decrease of the minimum value of the DP with increasing density or decreasing +temperature is defined mostly by the MFA. The fact that in the present model the mean +field contribution is non-monotonous means that the fluid can have a layered-type structure +which was not observed in the one-Yukawa case. The results obtained are in qualitative +agreement with22 where a hard core two-Yukawa fluid near a hard wall was studied by +means of Monte-Carlo simulations and the density functional theory. +C. +Adsorption +The adsorption coefficient (AC) defined by expr. (70) characterizes the excess of the +density near the surface as compared to the bulk region. In accordance with (70) the AC +can be presented as the sum of the mean field contribution ΓMF A and the fluctuation term +Γfluct. As we have noted in30 the linearized MFA approximation can be positive or negative +whereas the contribution of Γfluct is negative for the approximation (72). Unlike the mean +field contribution, the contribution from fluctuations Γfluct is always negative. This result is +expected as in18 it was shown that for a one-Yukawa fluid at a wall the fluctuation effects lead +to density depletion for both repulsive and attractive interactions. In the region where ΓMF A +is negative the value of the total adsorption coefficient Γ will be negative. It is therefore more +interesting to consider the region in which ΓMF A is positive. In this case we will have the +23 + +1E-3 +0,01 +0,1 +-0,002 +0,000 +0,002 +0,004 +* +* + MFA Linear + MFA + MFA+GF Interpolated + GF +* +0,1 +1 +10 +-0,005 +0,000 +0,005 +10 +100 +-0,00010 +-0,00008 +-0,00006 +-0,00004 +-0,00002 +0,00000 +0,00002 +0,00004 +0,00006 +0,00008 +0,00010 +* +* +T* + MFA Linear + MFA + MFA+GF Interpolated + GF +* +T* +FIG. 6. Reduced adsorption coefficient Γ∗ = Γ/α2 +2 as a function of the density at a fixed tempera- +ture (left) and as a function of the temperature at a fixed density (right). +competition between the MFA contribution and the contribution from fluctuations. In Fig. +6 adsorption coefficients as functions of the temperature and the density for a model with +ω = 2.5, τ = 1.355 are presented. For this case the mean field contribution ΓMF A is positive. +As we can see from Fig. 6 the linearized MFA overestimates the contribution to ΓMF A and +the difference between the linearized and the non-linearized cases becomes more pronounced +as the density increases or the temperature decreases. At high temperatures the role of +the fluctuation term becomes negligible and the interpolated CA merges asymptotically +with the MFA contribution. At moderate and lower temperatures, however, the Gaussian +correction can modify significantly the MFA predictions. Notably, due to the competition +between ΓMF A and Γfluct the adsorption isotherm can display non-monotonous behavior as +a function of the bulk density. Likewise, the adsorption isochore can be non-monotonous as +a function of the temperature. Another interesting consequence of going beyond the MFA +is that under certain conditions the CA can change sign as the temperature or the bulk +density are varied. We should note that in22 a similar effect was observed for a hard core +two-Yukawa fluid in the framework of Monte-Carlo simulations and the density functional +theory. +VII. +CONCLUSIONS +In this work a field theoretical approach is applied to describe a fluid interacting with +a repulsive and an attractive Yukawa potentials in the vicinity of a hard wall. +The re- +sults obtained are compared to a more simple one-Yukawa model considered in our previous +24 + +work18. We derive mean field equations that allow for a numerical evaluation of the density +profile. +Subsequently the contact theorem is validated employing a scheme that can by +linearity be generalized to a multi-Yukawa fluid. We find that unlike a one-Yukawa fluid, a +two-Yukawa fluid can have a non-monotonic profile even in the mean field approximation. +The linearized version of the profile contains two generalized decays λ1 and λ2 which have +a more complicated form than in the one-Yukawa case. The results obtained in18 for an +attractive one-Yukawa case are not defined when κ2 +2 + α2 +2 < 0, that is for low temperatures, +high densities, or strongly attractive potentials. This peculiarity is related to general prob- +lems in the description of phase transitions in the framework of the Gaussian fluctuations +theory in the bulk. More specifically, it is the so-called RPA-catastrophe which is caused +by an incorrect treatment of short-range correlations and can be removed by including the +repulsive interactions26. Compared to an attractive one-Yukawa case we thus show that +generalization of the interaction potential to the sum of a repulsive and an attractive parts +makes the profile decays well defined for all temperatures and densities. +Beyond the mean field approximation we study the impact of Gaussian fluctuations on +thermodynamic and structural properties of the fluid. Analytical expressions for the free +energy, the pressure, the chemical potential, and the correlation function are derived. Sub- +sequently we find a correction to the density profile due to fluctuations and show that +fluctuations always lead to depletion. We show analytically that the fluctuation terms of +the pressure and of the density contact value satisfy the contact theorem. However, compari- +son with the computer simulations data has revealed that the contribution from fluctuations +leads to strong overestimation of the role of fluctuations. It produces a maximum of the +profile at small distances to the wall which is not predicted by computer simulations. The +fluctuation term also strongly underestimates the contact value of the density profile. In ac- +cordance with the contact theorem this phenomenon is the results of the incorrect prediction +of the bulk pressure in the framework of the Gaussian approximation. We also show that the +Gaussian approximation leads to incorrect behavior of the bulk pair distribution function at +small interparticle distances. In order to improve the bulk pair distribution function at small +distances we propose an exponential approximation which ensures the correct behavior of +the PDF at small distances and reproduces the prediction of the Gaussian approximation at +larger distances. The exponential form of the PDF also agrees very well with the computer +simulations results. The pressure calculated from the exponential form of the PDF ensures +25 + +the correct contact value of the density profile. We use this result to improve the descrip- +tion of the density profile at small distances to the wall. The results calculated via such an +interpolation procedure are in a very good agreement with the computer simulations data. +Next we study the adsorption coefficient and its dependence on the bulk density and +the temperature. Unlike the mean field part, the contribution from fluctuations is always +negative. We consider the case when there is a competition between the two contributions. It +is found that at higher temperatures the mean field term dominates, but as the temperature +decreases the fluctuation effects become increasingly more important. +As a result, non- +monotonic adsorption curves are found for some systems. +The behaviors of the density +profile and of the adsorption isotherm described in this paper are in qualitative agreement +with the results of22, where a hard core two-Yukawa fluid was studied by means of Monte- +Carlo simulations and the density functional theory. +ACKNOWLEDGMENTS +The authors are grateful for the support of the National Academy of Sciences of Ukraine +and the Centre National de la Recherche Scientifique (CNRS) in the framework of the PICS +project. +REFERENCES +1Y. Kalyuzhnyi and P. Cummings, Mol. Phys. 87, 1459 (1996). +2Y. Tang, Z. Tong, and B.-Y. 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Badiali, Mol. Phys. 101, 2545 (2003). +29D. Di Caprio, J. Stafiej, and J. Badiali, J. Chem. Phys. 108, 8572 (1998). +30I. Kravtsiv, M. Holovko, D. Di Caprio, and J. Stafiej, Preprint ICMP-13-01E, 1 (2013). +31J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, Oxford, +2006). +32F. Gahov and Y. Cherski, Convolution-type equations (Nauka, Moscow, 1978). +33D. Amit, Field theory, the renormalization group, and critical phenomena (World Scien- +tific, Singapore, 1984). +34J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Ox- +ford, 1989). +35D. Frenkel and B. Smit, Understanding Molecular Simulations: From Algorithms to Applications +(Academic Press, 2002). +27 + +36J. Baker and P. Gravis-Morris, Pad´e approximants (Cambridge U.P., 1996). +Appendix A: The Riemann problem +Equation (56) can be represented in the form +P+(K, µ1) h+(K, µ1, µ2) − P−(K, µ1) h−(K, µ1, µ2) = −L(µ2) δ(µ1 + µ2) +(A1) +where +L(µ2) = 4πβ +� +A1(µ2 +2 + p2 + α2 +2) + A2(µ2 +2 + p2 + α2 +1) +� +, +(A2) +P+(K, µ1) = (K2 + µ2 +1 + α2 +1)(K2 + µ2 +1 + α2 +2) + +κ2 +1(K2 + µ2 +1 + α2 +2) + κ2 +2(K2 + µ2 +1 + α2 +1), +(A3) +P−(K, µ1) = (K2 + µ2 +1 + α2 +1)(K2 + µ2 +1 + α2 +2), +Equation (A1) is known as the Riemann problem32. It can be solved by factorization, for +which purpose we write the fraction P−(K, µ1)/P+(K, µ1) as +P−(K, µ1) +P+(K, µ1) = Q+(K, µ1) +Q−(K, µ1), +(A4) +where Q+(K, µ1), Q−(K, µ1) are analytical functions of µ1 and cannot be zero in the upper ++ or lower - halves of the complex plane. The latter are easy to find: +Q+(K, µ1) = (µ1 + iα1(K))(µ1 + iα2(K)) +(µ1 + iλ2(K))(µ1 + iλ1(K)) , +Q−(K, µ1) = (µ1 − iλ2(K))(µ1 − iλ1(K)) +(µ1 − iα1(K))(µ1 − iα2(K)), +(A5) +where +α1(K) = +� +K2 + α2 +1 , +α2(K) = +� +K2 + α2 +2, +λ2(K) = +� +K2 + λ2 +2 , +λ1(K) = +� +K2 + λ2 +1. +(A6) +Coefficients λ1, λ2 are found from equation +λ4 − (α2 +1 + α2 +2 + κ2 +1 + κ2 +2)λ2 + (α2 +1 + κ2 +1)(α2 +2 + κ2 +2) − κ2 +1κ2 +2 = 0 +(A7) +28 + +giving +λ2 +1,2 = 1 +2 +� +κ2 +1 + α2 +1 + κ2 +2 + α2 +2 ± +� +(κ2 +1 + α2 +1 − κ2 +2 − α2 +2)2 + 4κ2 +1κ2 +2 +� +(A8) +and coinciding with expressions (27) obtained in the framework of the mean field approxi- +mation. +We choose iλ2(K), iλ1(K) to be in the upper and −iλ2(K), −iλ1(K) in the lower halves +of the analytical plane. +Equation (A1) now reads +h+(K, µ1, µ2) +Q+(K, µ1) +− h−(K, µ1, µ2) +Q−(K, µ1) += − +L(µ2) δ(µ1 + µ2) +Q+(K, −µ2) P+(K, −µ2) . +(A9) +In (A1) the Dirac function is presented as the difference of one-sided Dirac functions +δ(µ1 + µ2) = δ+(µ1 + µ2) − δ−(µ1 + µ2), +(A10) +which are analytical in the upper and lower halves of the complex plane respectively. Since +the index of the problem (A1) is zero32, we obtain +h+(K, µ1, µ2) = − +L(µ2)Q+(K, µ1) +Q+(K, −µ2) P+(K, −µ2)δ+(µ1 + µ2) +h−(K, µ1, µ2) = − +L(µ2)Q−(K, µ1) +Q+(K, −µ2) P+(K, −µ2)δ−(µ1 + µ2). +(A11) +Replacing (A2), (A4) and (A5) into (A11), we have +h+(K, µ1, µ2) = +−4π β +A1(µ2 +2 + α2 +2)(K) + A2(µ2 +2 + α2 +1(K)) +(µ2 − iα1(K))(µ2 − iα2(K))(µ2 + iλ2(K))(µ2 + iλ1(K)) +(µ1 + iα1(K))(µ1 + iα2(K)) +(µ1 + iλ2(K))(µ1 + iλ1(K)) δ+(µ1 + µ2), +(A12) +h−(K, µ1, µ2) = +−4π β +A1(µ2 +2 + α2 +2) + A2(µ2 +2 + α2 +1) +(µ2 − iα1(K))(µ2 − iα2(K))(µ2 + iλ2(K))(µ2 + iλ1(K)) +(µ1 − iλ2(K))(µ1 − iλ1(K)) +(µ1 − iα1(K))(µ1 − iα2(K)) δ−(µ1 + µ2). +(A13) +Performing the inverse Fourier transformation +h(R12, z1, z2) = +� +dK +(2π)2e−iKR12 +∞ +� +−∞ +dµ1 +2π e−iµ1z1 +∞ +� +−∞ +dµ1 +2π e−iµ2z2 +{h+(K, µ1, µ2) − h−(K, µ1, µ2)} , +(A14) +29 + +we can find the originals of one-sided pair correlation functions. Due to the considered model +we are interested in the case when both particles are in the upper half-space z1 > 0, z2 > 0. +We present one-sided δ-functions as +δ+(µ1 + µ2) = lim +ε→0 +i +µ1 + µ2 + iε , +δ−(µ1 + µ2) = lim +ε→0 +i +µ1 + µ2 − iε +(A15) +and integrate by µ1. Then for z1 > 0, closing the integration contour in the lower half of +the complex plane, we have +lim +ε→0 +∞ +� +−∞ +dµ1 +2π +(µ1 + iα1(K))(µ1 + iα2(K)) +(µ1 + iλ2(K))(µ1 + iλ1(K)) +i +µ1 + µ2 + iε e−iµ1z1 = +(µ2 − iα1(K))(µ2 − iα2(K)) +(µ2 − iλ2(K))(µ2 − iλ1(K)) eiµ2z1 − +i(λ2(K) − α1(K))(λ2(K) − α2(K)) +((λ2(K) − λ1(K))(µ2 − iλ2(K)) e−λ2(K)z1 + +i(λ1(K) − α1(K))(λ1(K) − α2(K)) +(λ2(K) − λ1(K))(µ2 − iλ1(K)) e−λ1(K)z1. +(A16) +30 + +Now we integrate by µ2. We consider the case z2 > 0. +∞ +� +−∞ +dµ2 +2π +[A1(µ2 +2 + α2 +2(K)) + A2(µ2 +2 + α2 +1(K))]e−iµ2z2 +(µ2 − iα1(K))(µ2 − iα2(K))(µ2 + iλ2(K))(µ2 + iλ1(K)) +�(µ2 − iα1(K))(µ2 − iα2(K)) +(µ2 − iλ2(K))(µ2 − iλ1(K)) eiµ2z1− +i(λ2(K) − α1(K))(λ2(K) − α2(K)) +(λ2(K) − λ1(K))(µ2 − iλ2(K)) e−λ2(K)z1+ +i(λ1(K) − α1(K))(λ1(K) − α2(K)) +(λ2(K) − λ1(K))(µ2 − iλ1(K)) e−λ1(K)z1 +� += +−A1(λ2 +2 − α2 +2) + A2(λ2 +2 − α2 +1) +2λ2(K)(λ2 +1 − λ2 +2) +e−λ2(K)|z1−z2| + +A1(λ2 +1 − α2 +2) + A2(λ2 +1 − α2 +1) +2λ1(K)(λ2 +1 − λ2 +2) +e−λ1(K)|z1−z2| + +(A17) +A1(λ2 +2 − α2 +2) + A2(λ2 +2 − α2 +1) +2λ2(K)(λ1(K) − λ2(K))2 +(λ2(K) − α1(K))(λ2(K) − α2(K)) +(λ2(K) + α1(K))(λ2(K) + α2(K))e−λ2(K)(z1+z2) − +A1(λ2 +1 − α2 +2) + A2(λ2 +1 − α2 +1) +(λ1(K) + λ2(K))(λ1(K) − λ2(K))2 +(λ2(K) − α1(K))(λ2(K) − α2(K)) +(λ1(K) + α1(K))(λ1(K) + α2(K))e−λ2(K)z1−λ1(K)z2 − +A1(λ2 +2 − α2 +2) + A2(λ2 +2 − α2 +1) +(λ1(K) + λ2(K))(λ1(K) − λ2(K))2 +(λ1(K) − α1(K))(λ1(K) − α2(K)) +(λ2(K) + α1(K))(λ2(K) + α2(K))e−λ1(K)z1−λ2(K)z2 + +A1(λ2 +1 − α2 +2) + A2(λ2 +1 − α2 +1) +2λ1(K)(λ1(K) − λ2(K))2 +(λ1(K) − α1(K))(λ1(K) − α2(K)) +(λ1(K) + α1(K))(λ1(K) + α2(K))e−λ1(K)(z1+z2) +Taking the inverse Fourier transform with respect to vector K, we obtain the following +31 + +expression for the case when particles 1 and 2 are in the upper half-space, i.e. z1 > 0, z2 > 0 +h+(R12, z1, z2) = βA1(λ2 +2 − α2 +2) + A2(λ2 +2 − α2 +1) +(λ2 +1 − λ2 +2) +e−λ2R12 +R12 +− +(A18) +βA1(λ2 +1 − α2 +2) + A2(λ2 +1 − α2 +1) +(λ2 +1 − λ2 +2) +e−λ1R12 +R12 +− +2β +∞ +� +0 +p J0(KR12) dp +�A1(λ2 +2 − α2 +2) + A2(λ2 +2 − α2 +1) +2λ2(K)(λ1(K) − λ2(K))2 +(λ2(K) − α1(K))(λ2(K) − α2(K)) +(λ2(K) + α1(K))(λ2(K) + α2(K))e−λ2(K)(z1+z2)− +A1(λ2 +1 − α2 +2) + A2(λ2 +1 − α2 +1) +(λ1(K) + λ2(K))(λ1(K) − λ2(K))2 +(λ2(K) − α1(K))(λ2(K) − α2(K)) +(λ1(K) + α1(K))(λ1(K) + α2(K))e−λ2(K)z1−λ1(K)z2− +A1(λ2 +2 − α2 +2) + A2(λ2 +2 − α2 +1) +(λ1(K) + λ2(K))(λ1(K) − λ2(K))2 +(λ1(K) − α1(K))(λ1(K) − α2(K)) +(λ2(K) + α1(K))(λ2(K) + α2(K))e−λ1(K)z1−λ2(K)z2 ++A1(λ2 +1 − α2 +2) + A2(λ2 +1 − α2 +1) +2λ1(K)(λ1(K) − λ2(K))2 +(λ1(K) − α1(K))(λ1(K) − α2(K)) +(λ1(K) + α1(K))(λ1(K) + α2(K))e−λ1(K)(z1+z2) +� +, +where J0(KR12) is a Bessel function of the first kind. +32 + diff --git a/edAzT4oBgHgl3EQfL_u0/content/tmp_files/load_file.txt b/edAzT4oBgHgl3EQfL_u0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7dabb6a2058476cd25e2ade5962ba8929b44d781 --- /dev/null +++ b/edAzT4oBgHgl3EQfL_u0/content/tmp_files/load_file.txt @@ -0,0 +1,876 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf,len=875 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='01125v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='soft] 3 Jan 2023 TWO-YUKAWA FLUID AT A HARD WALL: FIELD THEORY TREATMENT I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Kravtsiv,1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Patsahan,1 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Holovko,1 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' di Caprio2 1)Institute for Condensed Matter Physics, National Academy of Sciences, 1 Svientsitskii Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=', 79011 Lviv, Ukraine 2)Institute of Research of Chimie Paris, CNRS - Chimie ParisTech, 11, rue P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' et M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Curie, 75005 Paris, France (Dated: 4 January 2023) We apply a field-theoretical approach to study the structure and thermodynamics of a two-Yukawa fluid confined by a hard wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We derive mean field equations allow- ing for numerical evaluation of the density profile which is compared to analytical estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Beyond the mean field approximation analytical expressions for the free energy, the pressure and the correlation function are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Subsequently contribu- tions to the density profile and the adsorption coefficient due to Gaussian fluctuations are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Both the mean field and the fluctuation terms of the density profile are shown to satisfy the contact theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We further use the contact theorem to improve the Gaussian approximation for the density profile based on a better approximation for the bulk pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The results obtained are compared to computer simulations data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' PACS numbers: 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='Jj, 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='Np, 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='-p, 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='-g 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' INTRODUCTION Model systems with Yukawa-like potentials of interaction have been extensively used for the description of a large variety of liquids and soft matter materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Any finite range inter- action potential between point particles can be decomposed to a sum of Yukawa potentials with arbitrary accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' For instance, the Lennard-Jones potential used in the theory of simple fluids can be well approximated by the hard repulsion with two Yukawa tails1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A hard core two-Yukawa model has been successfully used for the description of stability of charged colloidal dispersions3 and the properties of solutions of globular charged proteins4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In this case the first Yukawa term describes the screened electrostatic interparticle repulsion and the second term approximates the Van der Waals interparticle attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Since the electrostatic intercolloidal repulsion is usually more long-ranged compared to the Van der Waals attraction, such a fluid demonstrates a very rich non-trivial phase behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Exam- ples include various inhomogeneous structures such as spherical and cylindrical liquid-like clusters, single- and multi-liquid-like slabs, cylindrical and spherical bubbles5,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A hard core two-Yukawa model was also used to explain the formation of the extra low wave vector peak in the structure factor of cytochrome C protein solutions at moderate concentrations7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A hard core two-Yukawa model with short-range strongly attractive interaction was used for the description of different clusterization phenomena in associated fluids8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Finally, a model with isotropic Yukawa repulsion and anisotropic Yukawa attraction has been used in the theory of nematogenic fluids9,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The simplicity of the Yukawa potential allows for a description of thermodynamics and structure of the Yukawa fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' For hard spheres interact- ing with a Yukawa tail, analytical solutions exist in the mean spherical approximation11,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' These analytical results were generalized for the description of hard sphere multi-Yukawa fluids13,14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A model fluid of point particles with two or more Yukawa potentials is a good candidate for investigation of a fluid with an attractive interaction and soft repulsion at small dis- tances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Such fluids with a soft repulsion have recently attracted attention particularly due to investigation of star polymers for the case when the core size of a star is small enough compared to the length of chains and the effective interaction between two stars immersed in a good solvent shows logarithmic dependence of their center-to-center separation for small distances and crosses ove r to Yukawa form for larger ones15,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Since Yukawa interaction is 2 of Coulomb nature at small distances, a fluid of point particles with two Yukawa potentials can be considered as a fluid with softness intermediate between that of star polymers and simple fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Yukawa models have lately been used to investigate the structure and adsorption of fluids near solid surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' For this aim the collective variables approach17, the density field theory18, the inhomogeneous integral equations approach19, and the density functional theory20–23 have been adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Notably in21,22 the properties of inhomogeneous hard core two-Yukawa fluids were investigated and in23 the structure and phase behavior of the hard core model with a two-Yukawa tail potential in planar slit pores were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The results for inhomogeneous fluids should satisfy certain known exact relationships, the so-called contact theorems24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' For a neutral fluid it states that the contact value of the point particle density near a hard wall is determined by the pressure of the fluid in the bulk volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' For an ionic fluid near a charged hard wall there is an additional electrostatic Maxwell tensor contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We should mention the principal difference between a fluid with the Yukawa interaction and an ionic fluid owing to the electroneutrality condition of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' This condition excludes some terms associated with the mean field treatment in the case of ionic fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In18 it was shown that the mean field treatment of a Yukawa fluid near the wall reduces to solving a non-linear differential equation for the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Different simple analytical expressions for the density profile were obtained and compared with the numerical estimation of the mean field results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Beyond the mean field approximation it was shown that fluctuations can contribute significantly to the properties of a fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Notably they lead to the desorption phenomenon regardless of the sign of interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We note that the results obtained in18 for attractive potentials are not well defined for lower temperatures and higher densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' This problem is connected with the divergence of the bulk correlation function along the spinodal lines inside phase transitions of the mean field result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Such a divergence is the result of an incorrect treatment of short-range correla- tions in the bulk and can be removed by including repulsive interactions (see for example26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In this work we extend our previous results for the field theoretical description of a Yukawa fluid near a hard wall18 to the case of a fluid with two Yukawa potentials corresponding to attractive and repulsive interactions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Similar to18 the contributions from the mean field and from fluctuations are separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' It is shown that the mean field treatment re- duces to solving a non-linear differential equation for the density profile while the treatment 3 of Gaussian fluctuations reduces to solving the Ornstein-Zernike (OZ) integral equation with the Riemann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The validity of the contact theorem is verified for both contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' However, for the bulk case the considered treatment of fluctuations leads to incorrect behavior of the pair distribution function at small interparticle distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' This also leads to overestimation of the role of fluctuations for the adsorption as well as to incorrect description of the profile near the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In order to improve the pair distribution function in the bulk we use the exponential approximation which gives the correct result at small distances and coincides with the previous results for larger distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' This approximation is used to calculate the bulk pressure and to improve the behavior of the density profile at small distances in the framework of the contact theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The quality of the obtained results is controlled by comparison with computer simulations data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The results presented in this paper are obtained for a fluid of point particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' However, in the future we hope to modify them to describe non-point particles using the mean spherical approximation results13,14 in a similar way as was done for non-point ionic systems27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' THE MODEL AND FIELD THEORY FORMALISM We consider a neutral fluid of point particles in contact with a hard surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The particles do not interact with the surface but interact with each other via a two-Yukawa potential ν(r12) = A1 r12 exp(−α1r12) + A2 r12 exp(−α2r12), (1) where r12 denotes the distance between particles 1 and 2, A1, A2 are the amplitudes of interaction and α1, α2 are the inverse ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We associate the first term of the potential with the repulsion of particles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A1 > 0) and the second term with the attraction (A2 < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' At small distances ν(r) = (A1 − A2)/r12 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' As a consequence, we should have A1 > A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In the formalism of statistical field theory the Hamiltonian H[ρ(r)] is a functional of field and consists of the ideal entropy and the interaction: βH[ρ(r1)] = βHentr[ρ(r1)] + βHint[ρ(r1)] = (2) � ρ(r1) � ln � ρ(r1)Λ3� − 1 � dr1+ β 2 � ν(r12) � ρ(r1)ρ(r2) − ρ(r1)δ(r1 − r2) � dr1dr2, 4 where β = 1/kT is the inverse temperature, ρ(r) is the particle density, and Λ is the thermal de Broglie wavelength of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' As in previous papers18,28,29, we adopt the canonical ensemble approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We fix the number of particles by the conditions � ρ(r)dr = N or 1 V � ρ(r)dr = ρb, where V is the volume and ρb = N/V is the average value of the bulk density of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' To verify this condition in a formally unconstrained calculus we introduce a Lagrange multiplier λ such that δβH[ρ(r)] δρ(r) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (3) The partition function ZN [ρ(r)] can be expressed as ZN [ρ(r)] = � Dρ(r) exp{−βH[ρ(r)]}, (4) where Dρ(r) denotes functional integration over all possible density distributions such that the total number of particles is N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The logarithm of the partition function gives the Helmholtz free energy βF = − ln ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (5) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' MEAN FIELD APPROXIMATION The lowest order approximation for the partition function is the saddle point for the functional integral (4) which leads to the mean field approximation (MFA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The condition for the mean field approximation is δβH δρ ���� ρMF A(r) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (6) In our case equation (6) reads ln ρ(r1) ρb + V1(r1) + V2(r1) = λ, (7) where potentials Vi(r1) are defined as Vi(r1) = β � ρ(r2) Ai r12 exp(−αir12)dr2, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (8) We put λ ≡ V1b + V2b, (9) 5 where Vib are the values of potentials Vi(r1) in the bulk: V1b = κ2 1 α2 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' V2b = κ2 2 α2 2 , (10) and κ2 i ≡ 4πρbβAi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The gradient of (7) gives ∇ρ(r) ρ(r) − E1(r) − E2(r) = 0, (11) where we define an equivalent of the electric field by E1(r1) ≡ −∇V1(r1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' E2(r1) ≡ −∇V2(r1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (12) Due to the properties of Yukawa potential � △ − α2 1 � V1(r) = −4πβA1ρ(r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (13) � △ − α2 2 � V2(r) = −4πβA2ρ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (14) Replacing (13) and (14) into (11) and using translational invariance parallel to the wall we obtain d dz �ρ(z) ρb + α2 1 2κ2 1 [V1(z)]2 − 1 2κ2 1 E2 1(z) + α2 2 2κ2 2 [V2(z)]2 − 1 2κ2 2 E2 2(z) � = 0, (15) where z is the distance between the particle and the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Contact theorem In the bulk ρ(z) → ρb, Ei(z) → 0, Vi(z) → Vib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (15) we see that the quantity in brackets is constant and therefore it can be evaluated for instance in the bulk as 1 + κ2 1 2α2 1 + κ2 2 2α2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (16) This quantity is the reduced pressure βP/ρb within MFA: βP = ρb � 1 + κ2 1 2α2 1 + κ2 2 2α2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (17) Expression (17) is the mean field approximation which corresponds to the Van der Waals contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Outside the system, where there are no particles, we have another invariant which is simply α2 1 V 2 1 (z)/2κ2 1 − E2 1(z)/2κ2 1 + α2 2V 2 2 (z)/2κ2 2 − E2 2(z)/2κ2 2, its value far from 6 the interface is zero and therefore also at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' From the continuity of the potential and of its derivative due to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (13) and (14), we have that this is also true at the wall just inside the system z = 0+ thus ρ(0+) ρb + α2 1 2κ2 1 [V1(0+)]2 − 1 2κ2 1 E2 1(0+)+ α2 2 2κ2 2 [V2(0+)]2 − 1 2κ2 2 E2 2(0+) = ρ(0+) ρb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (18) As this quantity is constant we obtain the so-called contact theorem βP = ρ(0+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (19) Thus, similar to the one-Yukawa case18, in the MFA we obtain the contact theorem as the consequence of the existence of an invariant of the differential equations corresponding to the bulk pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Density profiles From (11)-(14) we obtain a set of five differential equations with five unknown functions ρ(z), E1(z), E2(z), V1(z), V2(z): ∂ρ(z) ∂z = ρ(z) [E1(z) + E2(z)] , (20) ∂V1(z) ∂z = −E1(z), (21) ∂V2(z) ∂z = −E2(z), (22) ∂E1(z) ∂z = −α2 1V1(z) + κ2 1 ρb ρ(z), (23) ∂E2(z) ∂z = −α2 2V2(z) + κ2 2 ρb ρ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (24) These relations are first-order differential equations that can be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' From (7) we have ρ(z) = ρb exp � − [V1(z) − V1b] − [V2(z) − V2b] � , (25) where Vi(z) and Vib are given by (8) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Similar to18 we can solve equation (25) in the linear approximation with the boundary condition set by the contact theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' This linear solution was obtained in30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Here we 7 present only the final result ρL(z) ρb = 1 − 1 2 (λ2 1 − α2 2) (λ2 1 − λ2 2) � −κ2 1 α2 1 + λ2 2 − α2 2 − κ2 2 α2 2 � e−λ1z (26) − 1 2 (λ2 2 − α2 2) (λ2 1 − λ2 2) �κ2 1 α2 1 − λ2 1 − α2 2 − κ2 2 α2 2 � e−λ2z, where λ2 1,2 = 1 2 � κ2 1 + α2 1 + κ2 2 + α2 2 ± � (κ2 1 + α2 1 − κ2 2 − α2 2)2 + 4κ2 1κ2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (27) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' FLUCTUATION AND CORRELATION EFFECTS ON DENSITY PROFILES AT THE WALL In the previous section we have considered mean field equations, where the fluctuations are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Here we take them into account and therefore we have to expand the Hamiltonian with respect to the mean field density ρMF A(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' For this aim we put ρ(r) = ρMF A(r)+δρ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Expansion of the Hamiltonian Expansion of the Hamiltonian around the mean field density ρMF A(r) gives βH[ρ] = βH � ρMF A� + � δρ(r1) δβH δ(δρ(r1)) ���� ρMF Adr1+ (28) 1 2 � δρ(r1)δρ(r2) δ2βH δ(δρ(r1))δ(δρ(r2)) ���� ρMF A dr1dr2+ � n≥3 (−1)n(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' � δρ(r1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' δρ(rn) δnβH δ(δρ(r1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' δ(δρ(rn)) ���� ρMF A dr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='drn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The first term is the Hamiltonian functional (2) for the mean field density βH[ρMF A] = � ρMF A(r1) � ln � ρMF A(r1)Λ3� − 1 � dr1 (29) + β 2 � ν(r12) � ρMF A(r1)ρMF A(r2) − ρMF A(r1)δ(r1 − r2) � dr1dr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The linear term disappears as in the canonical formalism fluctuations preserve the number of particles and � δρ(r)dr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The quadratic term is βH2[ρ] = 1 2 � δρ(r1)δρ(r2) �δ(r1 − r2) ρMF A(r1) + βν(r12) � dr1dr2, (30) 8 where the first term comes from the expansion of the logarithmic term in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Due to translational invariance parallel to the wall, we expand the fluctuations of the density as δρ(r) = � K δρK(z) eiKR, (31) where R is the vector component of r parallel to the wall, K is the wave vector in the direction parallel to the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The entropic term equals βHentr 2 [ρK(z)] = 1 2 � δρ2(r) ρMF A(z) dr (32) = 1 2 � K,K′ � δρK(z)δρK′(z) ρMF A(z) eiR(K + K ′)dRdz = S 2 � K � dz1dz2 δρK(z1)δρ−K(z2) δ(z1 − z2) ρMF A(z) , where S is the surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The interaction term gives βHint 2 [ρK(z)] = S β 2 � K � dz1 � dz2δρK(z1)δρ−K(z2) ν (K, |z1 − z2|) , (33) where ν (K, |z1 − z2|) = � dR12 ν(r12) exp (−iKR12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Finally, for the quadratic term of the Hamiltonian we obtain βH2[ρ] = (34) S 2 � K � dz1 � dz2 δρK(z1)δρ−K(z2) �δ (z1 − z2) ρMF A(z1) + βν (K, |z1 − z2|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Thermodynamic properties: free energy, pressure, and chemical potential We start our calculations from consideration of thermodynamic properties of the fluid in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The free energy is βF = − ln �� Dρ e−βH[ρ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (35) 9 In order to calculate the functional integral using the Gaussian integrals with such a Hamil- tonian, it is necessary to have the quadratic term in a diagonal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' For bulk properties such as the Helmholtz free energy we can expand density on the Fourier components δρ(r) = � k δρk eikr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (36) In this basis the quadratic Hamiltonian is βH2[ρ] = V 2ρb � k>0 δρkδρ−k � 1 + κ2 1 k2 + α2 1 + κ2 2 k2 + α2 2 � (37) and after integration the excess free energy equals βF ex = β(F − F id) = (38) ρbV κ2 1 2α2 1 + ρbV κ2 1 2α2 1 + 1 2 � k ln [1 + ρb ν(k)] − 1 2 ρb � k ν(k), where ν(k) = 4πβA1 k2 + α2 1 + 4πβA2 k2 + α2 2 (39) is the Fourier transform of the interaction potential (1) multiplied by β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The first and the second terms on the right-hand side of (38) are mean field contributions with the other two terms coming from Gaussian fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In order to calculate the third and the fourth terms we switch from summation to integration and then integrate by parts βF fluct = 1 2 � k ln [1 + ρbν(k)] − 1 2 ρb � k ν(k) (40) = ρ2 bV 12π2 ∞ � 0 k3dk ν(k) 1 + ρbν(k) d ν(k) dk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' For further calculations it is helpful to express parameters κ2 1 and κ2 2 in terms of λ1 and λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' From (27) we have κ2 1 = (α2 1 − λ2 1) (α2 1 − λ2 2) α2 2 − α2 1 , κ2 2 = (α2 2 − λ2 1) (α2 2 − λ2 2) α2 1 − α2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (41) Using identities (41), after integration we obtain βF ex V = ρb 2 �κ2 1 α2 1 + κ2 2 α2 2 � − 1 12π(λ1 3 + λ2 3) − 1 24π(α1 3 + α2 3) (42) + 1 8π � λ1 2 + λ2 2� (α1 + α2) − 1 8π � λ1 2 + α1α2 � � λ2 2 + α1α2 � α1 + α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 10 The pressure can be found from the free energy as βP = −β ∂F ∂V ���� T,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (43) Differentiation of (42) with respect to volume gives the fluctuation part of the bulk pressure as βP fluct == ρ2 b 12π2 ∞ � 0 k3dk ν(k) [1 + ρbν(k)]2 d ν(k) dk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (44) After integration and due to identities (41) the excess pressure equals βP ex = ρb 2 �κ2 1 α2 1 + κ2 2 α2 2 � − 1 24π(λ1 3 + λ2 3) − 1 12π(α1 3 + α2 3) (45) + 1 8π � α1 2 + α2 2� (λ1 + λ2) − 1 8π 1 λ1 + λ2 � α1 2 + λ1λ2 � � α2 2 + λ1λ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Finally, the excess chemical potential can be derived directly from (42) and (45) as µex = (F ex + P exV ) /N giving βµex = κ2 1 α2 1 + κ2 2 α2 2 − 1 8πρb (λ3 1 + λ3 2) − 1 8πρb (α3 1 + α3 2) (46) + 1 8πρb (λ2 1 + λ2 2)(α1 + α2) + 1 8πρb (α2 1 + α2 2)(λ1 + λ2) − 1 8πρb (λ2 1 + α1α2)(λ2 2 + α1α2) α1 + α2 − 1 8πρb (α2 1 + λ1λ2)(α2 2 + λ1λ2) λ1 + λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Correlation function The expression for the pair correlation function h(r1, r2) is31 h(r1, r2)⟨ρ(r1)⟩⟨ρ(r2)⟩ = ⟨δρ(r1)δρ(r2)⟩ − δ (r1 − r2) ⟨ρ(r1)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (47) In K-space this expression reads 1 S � ρMF A(z1)ρMF A(z2) h(K, z1z2)+ρMF A(z1)δ(z1 − z2) � (48) = ⟨δρK(z1)δρ−K(z2)⟩, where h(K, z1z2) = � dR12 h(R12, z1, z2) exp (iKR12) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (49) 11 The right-hand side of equation (48) can be calculated from expression (34) and gives the inverse Hamiltonian matrix βH−1 2 [ρK(z)] /2 ⟨δρK(z1)δρ−K(z2)⟩ = � D(δρK(z))δρK(z1)δρ−K(z2) exp (−βH2[ρK(z)]) � D(δρK(z)) exp (−βH2[ρK(z)]) = 1 2βH−1 2 [ρK(z)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (50) Hence the product of the Hamiltonian matrix and the matrix on the left-hand side of (48) yields unity, so we have � dz3 �� ρMF A(z1)ρMF A(z3) h(K, z1z3) + ρMF A(z1)δ(z1 − z3) � �δ (z3 − z2) ρMF A(z3) + βν (K, |z3 − z2|) � � = δ(z1 − z2), (51) or h(K, z1, z2) + � dz3ρMF A(z3)h(K, z1, z3)βν (K, |z3 − z2|) (52) = −βν (K, |z1 − z2|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Relation (52) is a convolution-type equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' It can be reduced to the so-called Riemann problem32 if we assume the density profile to be a step-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In this approximation ρMF A(z) = 0 for z < 0 and ρMF A(z) = ρb for z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Due to the spatial non-homogeneousness of the system we introduce one-sided pair cor- relation functions h±(R12, z1, z2) such that h(R12, z1, z2) = h+(R12, z1, z2) − h−(R12, z1, z2), h+(R12, z1, z2) = \uf8f1 \uf8f2 \uf8f3 h(R12, z1, z2), z1 > 0, 0, z1 < 0, (53) h−(R12, z1, z2) = \uf8f1 \uf8f2 \uf8f3 0, z1 > 0, −h(R12, z1, z2), z1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The function h(K, z1, z2) can then be presented as the difference of one-sided functions 12 h±(K, z1, z2) such that h(K, z1, z2) = h+(K, z1, z2) − h−(K, z1, z2), h+(K, z1, z2) = \uf8f1 \uf8f2 \uf8f3 h(K, z1, z2), z1 > 0, 0, z1 < 0, (54) h−(K, z1, z2) = \uf8f1 \uf8f2 \uf8f3 0, z1 > 0, −h(K, z1, z2), z1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Equation (52) now reads h+(K, z1, z2) − h−(K, z1, z2) + ρb ∞ � 0 dz3h+(K, z1, z3)βν (K, |z3 − z2|) = −βν (K, |z1 − z2|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (55) Expanding the functions h±(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z2) and ν (K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' |z1 − z2|) on Fourier harmonics with respect to the wave vector µ in the direction perpendicular to the wall and switching from summation to integration we obtain � 1 + κ2 1 K2 + µ2 1 + α2 1 + κ2 2 K2 + µ2 1 + α2 2 � h+(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' µ1µ2) − h−(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' µ1µ2) (56) = − � 4πβA1 K2 + µ2 2 + α2 1 + 4πβA2 K2 + µ2 2 + α2 2 � δ(µ1 + µ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' where h±(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' µ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' µ2) = � S dR12eiKR12 ∞ � −∞ dz1eiµ1z1 ∞ � −∞ dz2eiµ2z2h±(R12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z1z2) (57) and we have used the relation ∞ � −∞ dz1eiµ1z1 ∞ � −∞ dz2eiµ2z2βν (K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' |z1 − z2|) = (58) = − � 4πβA1 K2 + µ2 2 + α2 1 + 4πβA2 K2 + µ2 2 + α2 2 � δ(µ1 + µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Equation (56) is known as the Riemann problem32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Using the technique proposed in17,30 we solve this problem for h+(K, µ1µ2) (refer to Appendix A for the details of calculation) and 13 obtain h+(K, µ1, µ2) = − 1 ρb κ2 1(µ2 2 + α2 2(K)) + κ2 2(µ2 2 + α2 1(K)) (µ2 − iα1(K))(µ2 − iα2(K))(µ2 + iλ1(K))(µ2 + iλ2(K)) (µ1 + iα1(K))(µ1 + iα2(K)) (µ1 + iλ1(K))(µ1 + iλ2(K)) δ+(µ1 + µ2), (59) where δ+(µ1 + µ2) is a one-sided delta-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The expression for the correlation function in r-space can be presented as the sum of the homogeneous bulk part hb +(r12) and the inhomogeneous surface part hinh + (r12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z1z2) h+(r12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z2) = hb +(r12) + hinh + (r12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z1z2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (60) where hb +(r12) =βA1(λ2 2 − α2 2) + A2(λ2 2 − α2 1) (λ2 1 − λ2 2) e−λ2r12 r12 − (61) βA1(λ2 1 − α2 2) + A2(λ2 1 − α2 1) (λ2 1 − λ2 2) e−λ1r12 r12 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' hinh + (r12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z2) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(62) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='− β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2K J0(KR12) dK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='�A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2λ2(K)(λ1(K) − λ2(K))2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) − α1(K))(λ2(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) + α1(K))(λ2(K) + α2(K))e−λ2(K)(z1+z2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + λ2(K))(λ1(K) − λ2(K))2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) − α1(K))(λ2(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + α1(K))(λ1(K) + α2(K))e−λ2(K)z1−λ1(K)z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + λ2(K))(λ1(K) − λ2(K))2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) − α1(K))(λ1(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) + α1(K))(λ2(K) + α2(K))e−λ1(K)z1−λ2(K)z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='+ A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2λ1(K)(λ1(K) − λ2(K))2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) − α1(K))(λ1(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + α1(K))(λ1(K) + α2(K))e−λ1(K)(z1+z2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' αi(K) = � α2 i + K2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' λi(K) = � λ2 i + K2, (63) 14 and J0(KR12) is the Bessel function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' As we see from (61), λ1 and λ2 play the role of parameters characterizing the screening of the repulsive and the attractive interactions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Density profile In the Gaussian approximation the inhomogeneous density profile can be written as the sum of the mean field profile ρMF A(z) and the quadratic fluctuation term ρfluct(z) ρ(z) = ρMF A(z) + ρfluct(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (64) The contribution of quadratic fluctuations to the profile corresponds to the one-particle irreducible diagram in the field theory33,34 and can be found as: ρfluct(z1) ρb = 1 2 � h+(R, z1, z2) − hb +(R, z1, z2) � ���� z2→z1 R→0 , (65) where calculating the inhomogeneous profile we have subtracted the homogeneous bulk part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='As a result ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='ρfluct(z1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='ρb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='= − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='8πρb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='K dK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='�κ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + κ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='λ1(K)(λ2(K) − λ1(K))2 × ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(66) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) − α1(K))(λ1(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + α1(K))(λ1(K) + α2(K)) e−2λ1(K)z1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='κ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + κ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) + λ1(K))(λ2(K) − λ1(K))2× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) − α1(K))(λ1(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) + α1(K))(λ2(K) + α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='κ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + κ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) + λ1(K))(λ2(K) − λ1(K))2× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) − α1(K))(λ2(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + α1(K))(λ1(K) + α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='e−[λ1(K)+λ2(K)]z1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='+ κ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + κ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='λ2(K)(λ2(K) − λ1(K))2 × ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) − α1(K))(λ2(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) + α1(K))(λ2(K) + α2(K)) e−2λ2(K)z1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 15 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Contact theorem In Section III A we have shown the validity of the contact theorem in the mean field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Here we will show that for the considered model the contact theorem is also satisfied when the fluctuations are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Setting z1 = 0 in expression (66) and using identities (41), we obtain the contact value of density ρfluct(0+) = 1 4π ∞ � 0 KdK � α1(K) + α2(K) − 1 2 [λ1(K) + λ2(K)] (67) − 1 2 [α2 1(K) + λ1(K)λ2(K)] [α2 2(K) + λ1(K)λ2(K)] λ1(K)λ2(K) [λ1(K) + λ2(K)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Going back to expression (44) for the pressure we can calculate the fluctuation part of the pressure using the cylindrical coordinate system instead of the spherical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Then we have βP fluct = ρ2 b 12π2 ∞ � 0 k3dk ν(k) [1 + ρν(k)]2 d ν(k) dk (68) = − 1 2π2 ∞ � 0 KdK ∞ � −∞ µ2 dµ � κ2 1 (µ2 + α2 2(K)) + κ2 2 (µ2 + α2 1(K)) � [µ2 + λ2 1(K)]2 [µ2 + λ2 2(K)]2 � κ2 1 (µ2 + α2 2(K))2 + κ2 2 (µ2 + α2 1(K))2� [µ2 + α2 1(K)] [µ2 + α2 2(K)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' After integration with respect to µ and taking into account relations (41) we obtain βP fluct = 1 4π ∞ � 0 KdK � α1(K) + α2(K) − 1 2 [λ1(K) + λ2(K)] (69) − 1 2 [α2 1(K) + λ1(K)λ2(K)] [α2 2(K) + λ1(K)λ2(K)] λ1(K)λ2(K) [λ1(K) + λ2(K)] � , which is exactly the expression (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We have therefore proved the validity of the contact theorem for the fluctuation term of the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 16 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Adsorption We can also calculate the adsorption coefficient defined as Γ = ∞ � 0 dz [ρ(z) − ρb] = ΓMF A + Γfluct (70) according to different approximations of the mean field density profile presented in Section III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Hence the exact mean field contribution can be determined only numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The linearized equation (26) gives ΓL MF A = − ρb 2λ1 (λ2 1 − α2 2) (λ2 1 − λ2 2) � −κ2 1 α2 1 + λ2 2 − α2 2 − κ2 2 α2 2 � (71) − ρb 2λ2 (λ2 2 − α2 2) (λ2 1 − λ2 2) �κ2 1 α2 1 − λ2 1 − α2 2 − κ2 2 α2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' For the fluctuation part of the adsorption coefficient due to identities (41) we obtain an analytical result Γfluct = 1 32π (λ1 + λ2)2 + 1 32π (α2 1 + α2 2) − 1 16π (λ1 + λ2) (α1 + α2) − 1 16π (λ2λ1 + α22) (λ2λ1 + α12) (λ1 + λ2)2 + 1 16π (α1 + α2) (λ2λ1 + α2α1) λ1 + λ2 + 1 16π � λ2 1 + λ2 2 − α2 1 − α2 2 � ln �(λ2 + α1)(λ2 + α2) 2λ2(λ1 + λ2) � + 1 16π (λ2 1 − α2 1)(λ2 1 − α2 2) λ2 2 − λ2 1 ln �λ1 λ2 (λ2 + α1)(λ2 + α2) (λ1 + α1)(λ1 + α2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (72) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' MONTE-CARLO SIMULATIONS In order to test an accuracy of the field theoretical results established above the Monte Carlo (MC)35 simulations were carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A system of fluid particles interacting with the two-Yukawa potential (1) was considered in a rectangular simulation box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A cutoff distance of the potential was chosen rc = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A minimum size of the simulation box was set at least twice larger than the cutoff distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A usual periodic boundary conditions along x, y and z directions were applied for the bulk fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' However, to study a fluid near the hard wall a simulation box was confined between two walls orthogonal to z-axis and in this case the periodic boundary conditions were applied only in xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A distance between walls, 17 2 3 4 5 6 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='01 (r) r FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Pair interaction potential (1) for different values of ω and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Lz , was taken large enough to form a wide layer of the bulk phase in the middle of the box (Lz = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A number of fluid particles depended on the considered densities (ρ∗ = ρb/α3 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='3) and it varied in the range of N = 6000−12000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' At each simulation step N trial movements of particles were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' To speed up the simulations, the linked cell list algorithm was employed35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The density profiles of a confined fluid, ρ(z), were calculated and averaged over 100000 simulation step, while the pair distribution functions of a bulk fluid, g(r), were averaged over 10000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The model was studied for the different ratios of parameters A1/|A2| and α1/α2 in the temperature region of T ∗ = T/(α2|A2|) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='5 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' RESULTS AND DISCUSSION The properties of the considered two-Yukawa fluid are defined by four non-dimensional parameters: ρ∗ = ρb/α3 2, T ∗ = −1/(βA2α2), ω = A1/|A2| and τ = α1/α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The first two parameters are non-dimensional density and temperature respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The last two parameters are connected with the form of interparticle interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Below we will consider three types of models with ω = 3, τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='498;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' ω = 2, τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='35 and ω = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='5, τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The forms of interparticle interaction corresponding to these three cases are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We begin presentation of our results with the discussion of the bulk pair distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 18 0 2 4 0,0 0,2 0,4 0,6 0,8 1,0 T * g(r) r MC Exp GA 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 0,0 0,2 0,4 0,6 0,8 1,0 T * = 1 g(r) r MC Exp GA 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 0,0 0,2 0,4 0,6 0,8 1,0 T * g(r) r MC Exp GA FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Analytical approximations (73) and (74) and computer simulation results for the pair distribution function at different temperatures with ω = 2, τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='35, ρ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' ”MC” corresponds to Monte Carlo simulations, ”GA” is Gaussian approximation (73) and ”Exp” is exponential ap- proximation (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Bulk pair distribution function The bulk pair distribution function (PDF) in the considered Gaussian approximation can be presented in the form gb(r) = 1 + hb +(r), (73) where hb +(r) is given by equation (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' However, at small distances hb +(r) is of Coulombic form and hb +(r) → −∞ when r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In order to avoid this non-physical behavior of gb(r) we can use the exponential form gb(r) = exp � hb +(r) � (74) instead of the form (73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The behavior of the PDF for a model with ω = 2, τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='35, ρ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 at different temper- atures is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' As we can see the exponential form (74) ensures the correct behavior of gb(r) at small distances and reproduces very well the results given by expression (73) at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' These results are also in very good agreement with the computer simulations data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The behavior of gb(r) at different temperatures and densities is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We can see that with decreasing temperature the first peak of gb(r) increases and shifts to smaller distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' As the density increases at a fixed temperature the first peak of gb(r) decreases and shifts to smaller distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Such a behavior is the result of softness of the model since it allows the particles to occupy the soft region as the density is increased 19 5 0,995 0,996 0,997 0,998 0,999 1,000 1,001 1,002 1,003 1,004 g(r) r MC Exp MC Exp 2 4 6 0,992 0,996 1,000 1,004 g(r) r MC T * = 1 Exp T * = 1 MC T * = 2 Exp T * = 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Analytical approximation (74) and computer simulations results for the pair distribution function at different densities (left) and different temperatures (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The curves are labeled as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 2 and the temperature is decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We should also note that the agreement between theory and computer simulations results becomes worse as the temperature decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The height of the first peak increases faster in computer simulations as compared to theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Density profile According to (64) the density profile ρ(z) can be presented as the sum of two parts: ρMF A(z) and ρfluct(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The first one is the result of the mean field approximation, which can be calculated from the equation (26) derived from the linearized solution of the equation (25) as it was done in30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Another way to obtain ρMF A(z) is to solve the equation (25) numerically, which is obviously more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' To this aim we apply the Picard iterative method, where the numerical integrations are performed using the trapezoidal rule with the step size ∆z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='01α2, while all needed integrations over r are done analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The cutoff for the two- Yukawa potential (1) is taken at the distance rc = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='0, at which the potential becomes negligibly small (ν(rc)/ν(rmin) ∼ 10−4, rmin – a position of the potential minimum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Due to the hard wall presence from one side and the bulk phase from opposite side to the wall the boundary conditions for ρMF A(z) are defined as ρMF A(z) = 0 if z < 0 and ρMF A(z) = ρb if z > 2rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A precision of numerical solution for the density profile ρMF A(z) is restricted by ∥ρMF A m+1 (z)−F[ρMF A m (z)]∥ < 10−6, where F[ρMF A m (z)] is a right-hand side of the equation (25) and m is an iteration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' It is worth mentioning that the equation (25) is equivalent to 20 the Euler-Lagrange equation, which is usually used in the density functional theory within the mean field approximation for the fluid near a hard wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The density profiles, ρMF A(z), obtained with a use of the linear approximation and the iterative method are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' As one can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 4 the numerical solution of the equation (25) can be interpolated very well by the linear approximation (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' However, in the region of lower temperatures the linear approximation overestimates the density profile at intermediate distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The contact values ρ(0) calculated from MFA are essentially higher than those obtained from the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The general overestimation of MFA up to the first minimum of ρ(z) is observed for all temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' A correction of ρMF A(z) by the Gaussian fluctuation term ρfluct(z) should improve the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The Gaussian fluctuations term found from the solution of the inhomogeneous OZ equa- tion with the Riemann boundary condition is given by the expression (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' It is observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 4 that the contribution from the fluctuations has a negative sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' This is an expected result since in18 it was shown that for a one-Yukawa fluid the fluctuation part of the density profile is negative for both attractive and repulsive interactions and produces the depletion effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We have demonstrated that this term satisfies the contact theorem condition (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Nevertheless, comparison with computer simulation results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='4) shows that this term leads to strong overestimation of the role of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In addition, it gives a maximum of the profile at small distances from the wall which is not predicted by computer simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 4 one can also see that expression (66) strongly underestimates the contact value of the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' This is the consequence of the underestimated value of the pressure calculated from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (45) which corresponds to approximation (73) for the bulk PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Thus it would be more correct to calculate the pressure according to the virial theorem31 βP ρb = 1 − 2 3πβ � ∞ 0 ∂υ(r) ∂r gb(r)r3dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (75) and using the exponential approximation (74) for the bulk PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Using this value of the pressure and the contact theorem we have corrected the density profile at small distances starting from point z0 which corresponds to the inflection point for the density fluctuation term (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Interpolation of ρ(z) for the region of z < z0 has revealed a rather accurate generalization in the form of Pad´e approximant36 ρ(z) = ρ(0)/ � 1 + Az + Bz2 + Cz3� for z < z0, (76) 21 0 1 2 3 4 5 0,85 0,90 0,95 1,00 1,05 1,10 1,15 z* MFA Linear MFA MC MFA+GF Interpolated MFA+GF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='5 (z)/ b 0 1 2 3 4 5 0,94 0,96 0,98 1,00 1,02 1,04 1,06 z* MFA Linear MFA MC MFA+GF Interpolated MFA+GF (z)/ b = 1 0 1 2 3 4 5 0,96 0,98 1,00 1,02 1,04 z* MFA Linear MFA MC MFA+GF Interpolated MFA+GF (z) b FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Density profile as a function of the reduced distance z∗ = zα2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Label ”MC” corresponds to Monte Carlo simulations, ”MFA” is the numerical solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (25), ”MFA Linear” is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (26), ”GF” is the Gaussian fluctuations term (66) and ”MFA+GF Interpolated” corresponds to Gaussian approximation with interpolation (76) at small distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Different graphs correspond to different temperatures at a fixed density ρ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 with ω = 2, τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' where ρ(0) is determined from the contact theorem and the pressure calculated from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The constants A, B, C are found from the continuity of ρ(z) and its first derivative at the point z = z0 and the fact that the second derivative equals zero at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' This is the form we use to calculate the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The results for the model with ω = 2, τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='35 at density ρ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 and different temperatures are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' One can see that the results of calculation are in good agreement with the computer simulations data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 5 we present density profiles for a model with ω = 3, τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='498 at different temperatures and densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' One can see that the contact value of the density increases and the minimum of the profile decreases as the temperature decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Below we will see that this can lead to non-trivial behavior of the adsorption as a function of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Likewise, the contact value of the density increases and the minimum of the profile decreases 22 0 1 2 3 4 0,90 0,95 1,00 1,05 1,10 1,15 1,20 (z)/ b MC MFA+GF Interpolated MC MFA+GF Interpolated z* 0 1 2 3 4 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 MC T * = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='5 MFA+GF Interpolated T * = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='5 MC T * = 1 MFA+GF Interpolated T * = 1 MC T * = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='5 MFA+GF Interpolated T * = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='5 (z)/ b z* FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Theoretical approximations and computer simulations results for the density profile at different densities (left) and different temperatures (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The curves are labeled as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 4 as the density increases at a fixed temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Due to the contact theorem and according to expression (75) the increase of the contact value of the DP with increasing density or decreasing temperature is connected with the respective increase of the fluid pressure in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The decrease of the minimum value of the DP with increasing density or decreasing temperature is defined mostly by the MFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The fact that in the present model the mean field contribution is non-monotonous means that the fluid can have a layered-type structure which was not observed in the one-Yukawa case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The results obtained are in qualitative agreement with22 where a hard core two-Yukawa fluid near a hard wall was studied by means of Monte-Carlo simulations and the density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Adsorption The adsorption coefficient (AC) defined by expr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (70) characterizes the excess of the density near the surface as compared to the bulk region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In accordance with (70) the AC can be presented as the sum of the mean field contribution ΓMF A and the fluctuation term Γfluct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' As we have noted in30 the linearized MFA approximation can be positive or negative whereas the contribution of Γfluct is negative for the approximation (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Unlike the mean field contribution, the contribution from fluctuations Γfluct is always negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' This result is expected as in18 it was shown that for a one-Yukawa fluid at a wall the fluctuation effects lead to density depletion for both repulsive and attractive interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In the region where ΓMF A is negative the value of the total adsorption coefficient Γ will be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' It is therefore more interesting to consider the region in which ΓMF A is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In this case we will have the 23 1E-3 0,01 0,1 0,002 0,000 0,002 0,004 MFA Linear MFA MFA+GF Interpolated GF 0,1 1 10 0,005 0,000 0,005 10 100 0,00010 0,00008 0,00006 0,00004 0,00002 0,00000 0,00002 0,00004 0,00006 0,00008 0,00010 T* MFA Linear MFA MFA+GF Interpolated GF T* FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Reduced adsorption coefficient Γ∗ = Γ/α2 2 as a function of the density at a fixed tempera- ture (left) and as a function of the temperature at a fixed density (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' competition between the MFA contribution and the contribution from fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 6 adsorption coefficients as functions of the temperature and the density for a model with ω = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='5, τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='355 are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' For this case the mean field contribution ΓMF A is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' As we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 6 the linearized MFA overestimates the contribution to ΓMF A and the difference between the linearized and the non-linearized cases becomes more pronounced as the density increases or the temperature decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' At high temperatures the role of the fluctuation term becomes negligible and the interpolated CA merges asymptotically with the MFA contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' At moderate and lower temperatures, however, the Gaussian correction can modify significantly the MFA predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Notably, due to the competition between ΓMF A and Γfluct the adsorption isotherm can display non-monotonous behavior as a function of the bulk density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Likewise, the adsorption isochore can be non-monotonous as a function of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Another interesting consequence of going beyond the MFA is that under certain conditions the CA can change sign as the temperature or the bulk density are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We should note that in22 a similar effect was observed for a hard core two-Yukawa fluid in the framework of Monte-Carlo simulations and the density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' CONCLUSIONS In this work a field theoretical approach is applied to describe a fluid interacting with a repulsive and an attractive Yukawa potentials in the vicinity of a hard wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The re- sults obtained are compared to a more simple one-Yukawa model considered in our previous 24 work18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We derive mean field equations that allow for a numerical evaluation of the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Subsequently the contact theorem is validated employing a scheme that can by linearity be generalized to a multi-Yukawa fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We find that unlike a one-Yukawa fluid, a two-Yukawa fluid can have a non-monotonic profile even in the mean field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The linearized version of the profile contains two generalized decays λ1 and λ2 which have a more complicated form than in the one-Yukawa case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The results obtained in18 for an attractive one-Yukawa case are not defined when κ2 2 + α2 2 < 0, that is for low temperatures, high densities, or strongly attractive potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' This peculiarity is related to general prob- lems in the description of phase transitions in the framework of the Gaussian fluctuations theory in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' More specifically, it is the so-called RPA-catastrophe which is caused by an incorrect treatment of short-range correlations and can be removed by including the repulsive interactions26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Compared to an attractive one-Yukawa case we thus show that generalization of the interaction potential to the sum of a repulsive and an attractive parts makes the profile decays well defined for all temperatures and densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Beyond the mean field approximation we study the impact of Gaussian fluctuations on thermodynamic and structural properties of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Analytical expressions for the free energy, the pressure, the chemical potential, and the correlation function are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Sub- sequently we find a correction to the density profile due to fluctuations and show that fluctuations always lead to depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We show analytically that the fluctuation terms of the pressure and of the density contact value satisfy the contact theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' However, compari- son with the computer simulations data has revealed that the contribution from fluctuations leads to strong overestimation of the role of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' It produces a maximum of the profile at small distances to the wall which is not predicted by computer simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The fluctuation term also strongly underestimates the contact value of the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In ac- cordance with the contact theorem this phenomenon is the results of the incorrect prediction of the bulk pressure in the framework of the Gaussian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We also show that the Gaussian approximation leads to incorrect behavior of the bulk pair distribution function at small interparticle distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' In order to improve the bulk pair distribution function at small distances we propose an exponential approximation which ensures the correct behavior of the PDF at small distances and reproduces the prediction of the Gaussian approximation at larger distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The exponential form of the PDF also agrees very well with the computer simulations results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The pressure calculated from the exponential form of the PDF ensures 25 the correct contact value of the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We use this result to improve the descrip- tion of the density profile at small distances to the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The results calculated via such an interpolation procedure are in a very good agreement with the computer simulations data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Next we study the adsorption coefficient and its dependence on the bulk density and the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Unlike the mean field part, the contribution from fluctuations is always negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We consider the case when there is a competition between the two contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' It is found that at higher temperatures the mean field term dominates, but as the temperature decreases the fluctuation effects become increasingly more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' As a result, non- monotonic adsorption curves are found for some systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The behaviors of the density profile and of the adsorption isotherm described in this paper are in qualitative agreement with the results of22, where a hard core two-Yukawa fluid was studied by means of Monte- Carlo simulations and the density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors are grateful for the support of the National Academy of Sciences of Ukraine and the Centre National de la Recherche Scientifique (CNRS) in the framework of the PICS project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' REFERENCES 1Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Kalyuzhnyi and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Cummings, Mol.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Colloidal Interface Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 239, 58 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 5A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Archer and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Evans, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Reatto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 126, 014104 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 7Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Lin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Chen, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' B 122, 044507 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 8Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Kalyuzhnyi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' McCabe, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Whitebay, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Cummings, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 121, 8128 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 14Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Li, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Lu, Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 102, 63 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 15C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Likos, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' L¨owen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Watzlawek, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Ablas, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Kravtsiv, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Soviak, Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Matter Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 12, 137 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 18D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Di Caprio, J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Hansen and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' McDonald, Theory of Simple Liquids (Academic Press, Oxford, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 32F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Gahov and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Cherski, Convolution-type equations (Nauka, Moscow, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 33D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Amit, Field theory, the renormalization group, and critical phenomena (World Scien- tific, Singapore, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 34J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Ox- ford, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 35D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Frenkel and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Smit, Understanding Molecular Simulations: From Algorithms to Applications (Academic Press, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 27 36J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Baker and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Gravis-Morris, Pad´e approximants (Cambridge U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=', 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Appendix A: The Riemann problem Equation (56) can be represented in the form P+(K, µ1) h+(K, µ1, µ2) − P−(K, µ1) h−(K, µ1, µ2) = −L(µ2) δ(µ1 + µ2) (A1) where L(µ2) = 4πβ � A1(µ2 2 + p2 + α2 2) + A2(µ2 2 + p2 + α2 1) � , (A2) P+(K, µ1) = (K2 + µ2 1 + α2 1)(K2 + µ2 1 + α2 2) + κ2 1(K2 + µ2 1 + α2 2) + κ2 2(K2 + µ2 1 + α2 1), (A3) P−(K, µ1) = (K2 + µ2 1 + α2 1)(K2 + µ2 1 + α2 2), Equation (A1) is known as the Riemann problem32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' It can be solved by factorization, for which purpose we write the fraction P−(K, µ1)/P+(K, µ1) as P−(K, µ1) P+(K, µ1) = Q+(K, µ1) Q−(K, µ1), (A4) where Q+(K, µ1), Q−(K, µ1) are analytical functions of µ1 and cannot be zero in the upper + or lower - halves of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' The latter are easy to find: Q+(K, µ1) = (µ1 + iα1(K))(µ1 + iα2(K)) (µ1 + iλ2(K))(µ1 + iλ1(K)) , Q−(K, µ1) = (µ1 − iλ2(K))(µ1 − iλ1(K)) (µ1 − iα1(K))(µ1 − iα2(K)), (A5) where α1(K) = � K2 + α2 1 , α2(K) = � K2 + α2 2, λ2(K) = � K2 + λ2 2 , λ1(K) = � K2 + λ2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (A6) Coefficients λ1, λ2 are found from equation λ4 − (α2 1 + α2 2 + κ2 1 + κ2 2)λ2 + (α2 1 + κ2 1)(α2 2 + κ2 2) − κ2 1κ2 2 = 0 (A7) 28 giving λ2 1,2 = 1 2 � κ2 1 + α2 1 + κ2 2 + α2 2 ± � (κ2 1 + α2 1 − κ2 2 − α2 2)2 + 4κ2 1κ2 2 � (A8) and coinciding with expressions (27) obtained in the framework of the mean field approxi- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We choose iλ2(K), iλ1(K) to be in the upper and −iλ2(K), −iλ1(K) in the lower halves of the analytical plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Equation (A1) now reads h+(K, µ1, µ2) Q+(K, µ1) − h−(K, µ1, µ2) Q−(K, µ1) = − L(µ2) δ(µ1 + µ2) Q+(K, −µ2) P+(K, −µ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (A9) In (A1) the Dirac function is presented as the difference of one-sided Dirac functions δ(µ1 + µ2) = δ+(µ1 + µ2) − δ−(µ1 + µ2), (A10) which are analytical in the upper and lower halves of the complex plane respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Since the index of the problem (A1) is zero32, we obtain h+(K, µ1, µ2) = − L(µ2)Q+(K, µ1) Q+(K, −µ2) P+(K, −µ2)δ+(µ1 + µ2) h−(K, µ1, µ2) = − L(µ2)Q−(K, µ1) Q+(K, −µ2) P+(K, −µ2)δ−(µ1 + µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (A11) Replacing (A2), (A4) and (A5) into (A11), we have h+(K, µ1, µ2) = −4π β A1(µ2 2 + α2 2)(K) + A2(µ2 2 + α2 1(K)) (µ2 − iα1(K))(µ2 − iα2(K))(µ2 + iλ2(K))(µ2 + iλ1(K)) (µ1 + iα1(K))(µ1 + iα2(K)) (µ1 + iλ2(K))(µ1 + iλ1(K)) δ+(µ1 + µ2), (A12) h−(K, µ1, µ2) = −4π β A1(µ2 2 + α2 2) + A2(µ2 2 + α2 1) (µ2 − iα1(K))(µ2 − iα2(K))(µ2 + iλ2(K))(µ2 + iλ1(K)) (µ1 − iλ2(K))(µ1 − iλ1(K)) (µ1 − iα1(K))(µ1 − iα2(K)) δ−(µ1 + µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (A13) Performing the inverse Fourier transformation h(R12, z1, z2) = � dK (2π)2e−iKR12 ∞ � −∞ dµ1 2π e−iµ1z1 ∞ � −∞ dµ1 2π e−iµ2z2 {h+(K, µ1, µ2) − h−(K, µ1, µ2)} , (A14) 29 we can find the originals of one-sided pair correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Due to the considered model we are interested in the case when both particles are in the upper half-space z1 > 0, z2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We present one-sided δ-functions as δ+(µ1 + µ2) = lim ε→0 i µ1 + µ2 + iε , δ−(µ1 + µ2) = lim ε→0 i µ1 + µ2 − iε (A15) and integrate by µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' Then for z1 > 0, closing the integration contour in the lower half of the complex plane, we have lim ε→0 ∞ � −∞ dµ1 2π (µ1 + iα1(K))(µ1 + iα2(K)) (µ1 + iλ2(K))(µ1 + iλ1(K)) i µ1 + µ2 + iε e−iµ1z1 = (µ2 − iα1(K))(µ2 − iα2(K)) (µ2 − iλ2(K))(µ2 − iλ1(K)) eiµ2z1 − i(λ2(K) − α1(K))(λ2(K) − α2(K)) ((λ2(K) − λ1(K))(µ2 − iλ2(K)) e−λ2(K)z1 + i(λ1(K) − α1(K))(λ1(K) − α2(K)) (λ2(K) − λ1(K))(µ2 − iλ1(K)) e−λ1(K)z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' (A16) 30 Now we integrate by µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' We consider the case z2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='dµ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='[A1(µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 + α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2(K)) + A2(µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 + α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1(K))]e−iµ2z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(µ2 − iα1(K))(µ2 − iα2(K))(µ2 + iλ2(K))(µ2 + iλ1(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='�(µ2 − iα1(K))(µ2 − iα2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(µ2 − iλ2(K))(µ2 − iλ1(K)) eiµ2z1− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='i(λ2(K) − α1(K))(λ2(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) − λ1(K))(µ2 − iλ2(K)) e−λ2(K)z1+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='i(λ1(K) − α1(K))(λ1(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) − λ1(K))(µ2 − iλ1(K)) e−λ1(K)z1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='−A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2λ2(K)(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='e−λ2(K)|z1−z2| + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2λ1(K)(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='e−λ1(K)|z1−z2| + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(A17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2λ2(K)(λ1(K) − λ2(K))2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) − α1(K))(λ2(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) + α1(K))(λ2(K) + α2(K))e−λ2(K)(z1+z2) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + λ2(K))(λ1(K) − λ2(K))2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) − α1(K))(λ2(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + α1(K))(λ1(K) + α2(K))e−λ2(K)z1−λ1(K)z2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + λ2(K))(λ1(K) − λ2(K))2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) − α1(K))(λ1(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) + α1(K))(λ2(K) + α2(K))e−λ1(K)z1−λ2(K)z2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2λ1(K)(λ1(K) − λ2(K))2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) − α1(K))(λ1(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + α1(K))(λ1(K) + α2(K))e−λ1(K)(z1+z2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='Taking the inverse Fourier transform with respect to vector K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' we obtain the following 31 expression for the case when particles 1 and 2 are in the upper half-space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z1 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z2 > 0 h+(R12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' z2) = βA1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='e−λ2R12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='R12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(A18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='βA1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='e−λ1R12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='R12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='p J0(KR12) dp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='�A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2λ2(K)(λ1(K) − λ2(K))2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) − α1(K))(λ2(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) + α1(K))(λ2(K) + α2(K))e−λ2(K)(z1+z2)− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + λ2(K))(λ1(K) − λ2(K))2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) − α1(K))(λ2(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + α1(K))(λ1(K) + α2(K))e−λ2(K)z1−λ1(K)z2− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + λ2(K))(λ1(K) − λ2(K))2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) − α1(K))(λ1(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ2(K) + α1(K))(λ2(K) + α2(K))e−λ1(K)z1−λ2(K)z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='+A1(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2) + A2(λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1 − α2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='2λ1(K)(λ1(K) − λ2(K))2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) − α1(K))(λ1(K) − α2(K)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='(λ1(K) + α1(K))(λ1(K) + α2(K))e−λ1(K)(z1+z2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' where J0(KR12) is a Bessel function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} +page_content=' 32' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAzT4oBgHgl3EQfL_u0/content/2301.01125v1.pdf'} diff --git a/ftA0T4oBgHgl3EQfHv83/vector_store/index.pkl b/ftA0T4oBgHgl3EQfHv83/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f5b9adeefd26bfb761d8ce05efbb8af9f450e96a --- /dev/null +++ b/ftA0T4oBgHgl3EQfHv83/vector_store/index.pkl @@ -0,0 +1,3 @@ +version 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Papanikolaou1, Aggelos Alevizopoulos1, Christos Ilioudis2, Konstantinos +Demertzis3,* Konstantinos Rantos3 + +1Innovative Secure Technologies P.C.; +a.papanikolaou@innosec.gr; a.alevizopoulos@innosec.gr +2International Hellenic University, Department of Information and Electronic Engineering; +iliou@ihu.gr +3International Hellenic University, Department of Computer Science; +kdemertzis@teiemt.gr, krantos@cs.ihu.gr + + +ABSTRACT + +Developing intelligent, interoperable Cyber Threat Information (CTI) sharing technologies can help build +strong defences against modern cyber threats. CTIs allow the community to share information about +cybercriminals' threats and vulnerabilities and countermeasures to defend themselves or detect malicious +activity. A crucial need for success is that the data connected to cyber risks be understandable, organized, +and of good quality. The receiving parties may grasp its content and utilize it effectively. This article +describes an innovative cyber threat intelligence management platform (CTIMP) for industrial +environments, one of the Cyber-pi project's significant elements. The suggested architecture, in particular, +uses cyber knowledge from trusted public sources and integrates it with relevant information from the +organization's supervised infrastructure in an entirely interoperable and intelligent way. When combined +with an advanced visualization mechanism and user interface, the services mentioned above provide +administrators with the situational awareness they require while also allowing for extended cooperation, +intelligent selection of advanced coping strategies, and a set of automated self-healing rules for dealing with +threats. + +KEYWORDS + +Cyber Threat Intelligent; Cyber Threat Information; Information Sharing; Industrial Environment; +Cybersecurity; + +1. INTRODUCTION + +The rapid development of new technologies in recent decades has significantly affected human +societies and the existing economy [1]. The highly digitized and interconnected environment +shapes the new Cyberspace [2] [3] providing new possibilities and opportunities for organizations +to develop extroversion activities and actions [4] [5]. However, this new cyber-ecosystem faces a +number of challenges such as cybercrime, advanced persistent threats and cyberattacks, resulting +in a climate of uncertainty and instability that threatens expected growth and prosperity [6] [7] [8]. +The emergence of a new generation of cyber threats highlights the need to modernize the way these +challenges are addressed [9] [10] [11], bypassing the until recently tactics of organizations that +relied on the passive use of main security appliances, like firewalls to protect their information and +anti-malware solutions [12] [13] [14] [15]. The characteristic of the complexity of modern threats +is that most successful attacks are perceived only during the subsequent forensics procedures [16] +[17]. + +As it is easily understood, ensuring a successful defense requires complete control in all attempts +to exploit the vulnerabilities of the system, as a successful attack could take advantage of the +existence of a single vulnerability. In everyday life, many system administrators are unable to fix +all the vulnerabilities of a system in time as limited experience, non-automation, increased +workload, software dependencies, use of old systems and lack of timely availability of critical +patches are their main brake. On the contrary, the automated, intelligent collection and correlation +of suspicious actions taking place in a network, in the framework of a single strategy, which will +take into account the latest digital threats, can help to take appropriate measures to deal +immediately and finally shield an organization from cyber threats [18] [19] , even though the +sharing of cyber-threat intelligence is a challenging process [20]. +A turning point in this process is the use of Indicators of Compromise (IOCs) that support the +security decision-making process [21]. IOCs include malware signature IDs, malicious IP +addresses, malicious checksum (MD5) malware, and malicious URLs or domain names of Botnets, +as well as patch fixes, good practices in control measures, access control policies or removing +unnecessary services, and modifying firewall settings [22] [23] [24]. In other words, this is a huge +repository of knowledge with proven defense techniques, which are strengthened daily by adding +updates. +Taking into account the gap presented in the intelligent, efficient, and unified application of the +knowledge available based on the IOCs, this work presents an innovative architecture for utilizing +the specific knowledge based on interoperable and intelligent methods of modern computing. The +proposed CTIMP is an advanced and adaptive system for monitoring and timely detection of +security events that threaten an organization, incorporating advanced technologies of analysis, +automated management, and execution of corrective actions, offering interactive real-time security +interaction. +The rest of this paper is structured as follows: Section 2 presents the background of the research +approach. Section 3 is allocated to the presentation of the proposed architecture and finally, section +4 concludes the research. + +2. BACKGROUND + +It is important to emphasize that in recent years the need for a cooperative response to security +incidents has been highlighted and very significant progress has been made in this area [25]–[27]. +But the constant evolution of the methods and technologies used in cybercrime, creates complete +on platforms that do not implement real-time information mechanisms, as anything else can be +considered obsolete. Also, extremely important feature is the interoperability that can in the +effective collection, improvement, analysis and sharing of cyber-attack data. A typical example of +such an application is the method of Modi et al. [28] where they propose a multilevel architecture +for the thorough analysis of heterogeneous data through the interaction between its interleaves. +Although the approach accepts information from open-source data streams, it is considered to be +completely dependent on in-house analysis platforms, which significantly limits the generalization +that should be provided in such cases. Mantis presents a different approach that incorporates +information on cyber threats using different standards [29]. It is an intelligent platform that allows +threat data to be correlated through an innovative agnostic similarity algorithm. This methodology +allows security analysts to correlate patterns that are shared between seemingly unrelated attacks, +which adds serious complexity to the system, dramatically increasing the need for computing +resources. Finally, Sengupta et al. [30] proposed a fairly sophisticated but very sophisticated +method for extracting an optimal modeling technique for Advanced Persistent Threat attacks in a +cloud computing environment. The approach is based on game theory, where the processes of +dealing with an event are modeled by optimizing the cost of security countermeasures. + + + +3. PROPOSED ARCHITECTURE + +The proposed CTIMP, while following the practices of Integrated Security Information and Event +Management (SIEM) [31], goes one step further by offering a personalized security solution that +combines multiple control mechanisms and corresponding digital security technologies for modern +computing systems and networks. Essentially, through a sophisticated collaborative framework, it +is able to identify an organization's digital risks and threats, meeting the ongoing needs of securing +the valuable information it manages, by offering security services and crisis remedies. +Specifically, CTIMP offers a central point of analysis, alert, compliance, and reporting, responding +to the changing organizational structures of a multidimensional modern organization. It focuses +primarily on meeting the critical information infrastructure needs of each organization, providing +a variety of intelligent mechanisms for monitoring data integrity, reporting new threats, detecting +and recording security incidents, and responding immediately to automated processes. +Taking a more detailed approach to the proposed architecture, the system initially focuses on the +timely detection of events using automated, detailed log analysis. Any alerts or events are displayed +on the system administrator's visualization console. This interface offers a timely and valid +simultaneous analysis of a very large number of security incidents of the supervised business +network while minimizing the possibility of drawing incorrect conclusions. +Updating and upgrading CTIMP predictability is based on gathering cyber-threat information from +trusted open access sources such as ready-made IOCs set up by security experts, etc. which are +filtered and correlated for the sole purpose of supporting infrastructure. This adaptation to the +requirements of the organization's business operations and information systems is achieved through +component mapping, which records and distributes to other CTIMP subsystems the specific +features of the available node topology, software, and services that may be targets or sources of +malicious action. Comparison of cyber-threats with the characteristics of the organization results +in an adapted cyber-cognition in the STIX 2.x [32] standard, which is submitted for analysis to the +privacy policy production subsystem, which provides algorithms for modulating and exporting +SIGMA rules, which are automatically integrated into all the active rules of the case analysis +mechanism. In addition, the proposed architecture includes an intelligent automated threat and +attack mechanism, which provides an expanded set of self-healing commands, which examine the +level of compliance and align it with current security policies. +Another very important feature is that the proposed architecture is assisted in each phase of its +operation by the visualization and interface subsystem, which aims to easily represent the +appropriate information, to allow analysts to detect and take immediate action in various events. +Finally, it should be noted that the design of preventive countermeasures offered by CTIMP, +concerns only the identification of specific threats that may affect the organization and the setting +of general priorities, establishing intelligent decision-making or adaptation mechanisms, which +ensure the smooth its operation. +A brief overview of CTIMP subsystems and functions is shown in Figure 1 below. + +Figure 1: The proposed Cyber Threat Intelligent Information Sharing Architecture (CTIMP) + +Below is a detailed presentation of the subsystems and mechanisms that frame the CTIMP +architecture. + +3.1 Incident analysis and identification of security attacks +The analysis of incidents and the intelligent identification of security incidents such as attacks are +achieved in CTIMP by the following systems. + +3.1.1 OSSEC HIDS +The OSSEC Host-based Intrusion Detection System [33] conducts both application and system- +level audits to investigate the integrity of supervised information infrastructure files, using threat +detection methods based on signatures and statistical anomalies. It can be configured to collect +events from devices on which the use of agents is not feasible, while also having a set of rules for +monitoring and analyzing specialized security incidents, for which it can generate corresponding +alerts. + +3.1.2 Decoders +The logs of the target environment are examined using default and custom decoders, which have +parameters that are compared with the content of the logs for event detection. Any correspondences +are routed for control by the set of available rules that implement the respective security policies, +based on which the notifications of the incidents under consideration are produced. + +3.2 Intelligent use of CTI +The Intelligent Cyber Threat Adaptation Subsystem collects and analyzes available information +gathered from Cyber Threat Information (CTI) Sources. This information is compared with the +results of the analysis of the data by the supervised Information System and the threats to the target + +Mobile&Web +Apslyself-HoakngRuss +三 +Apps +- +Self - Healirg +Oprseional Businese Infa +V +Gpooha Bushas o +Stama +STIXnetwork and information infrastructure are documented. The intelligent cybernetics aggregation +mechanism is regularly upgraded with IOCs and RSS feeds, collected by the Malware Information +Sharing and Threat Intelligence Sharing Platform (MISP) [34], from a wide range of trusted +sources. The accuracy of the information is adjusted to the criteria, concerning the organization's +systems, with a filtering process in order to maintain the relevant information and then to compare +it with the specific characteristics of the supervised information infrastructure. Features include +devices, services, objects IDs, IP addresses, geolocation information, and dependencies. The result +is the production of custom knowledge in the form of STIX 2.x files, which correspond to the fields +that are required to draft SIGMA rules. + +3.3 Business operations and information systems mapping +The mapping mechanism describes the specific characteristics of the organization and the +technological environment is displayed on the system’s console. The latter provides a useful for +manually recording business assets like nodes, communications, software, and network services. +In particular, the Dependency Mapper utility implements the basic processes of the mechanism. +The depmapper has a graphical data management interface to represent a deployment model. The +adaptation of the depmapper to the mapping mechanism helps to extend the functionality of +exporting graphs to image files (jpg, png) and data exchange (JSON). In practice, the production +of image files supports collaboration between stakeholders, as well as the use of images between +different third-party applications. Furthermore, the production of JSON-type files allows the +insertion of graphs in the depmapper, for the purpose of their subsequent processing. +More specifically, the layout model illustrates the components, hardware components (nodes), and +the links between them. The depmapper has the additional functions of grouping nodes, adding +tags, and defining multiple descriptions separately for each node. The descriptions contain +information about the object of operation, the available software services, the geographical +location, the structural and procedural dependencies, the data dependencies, as well as the risk +level. The level of risk is determined according to the criticality or sensitivity of the data and +services available. The collection of this information takes place within the organization, by +completing the user questionnaires. The information is collected by an analyzer and entered into +the depmapper to represent graphs and text in JSON format, for use in other subsystems. + +3.4 Visualization and user interface +This subsystem is divided into two subcategories, which include data visualization and user +interface. Visualization of security data collected from various log sources refers to the creation of +diagrams, graphs, and similar visual material. The visualization mechanism provides organization +and classification of the data structure. A notice indicates that immediate action is required or is +merely informative. Categorizing the different types of alerts is useful and makes it possible to +develop a monitoring strategy [35]. Notifications are categorized according to the level of +completion of the relevant research, conducted by managers and analysts. +An administrator is able to edit the content of an alert immediately, either later, or outsource its +processing to an analyst, using ticketing requests. Consequently, the status of notifications is +characterized as new, ongoing, and complete. In addition, the visualization mechanism provides +all the data in a user-friendly environment. The user interface mechanism offers the possibility of +approving or activating the system self-healing procedures. However, it is a fully customizable +environment, which allows the rapid collection of critical information and immediate response to +incidents. Likewise, it allows the viewing and analysis of critical statistics and the viewing of +history. The interface supports web environments to make services available on mobile devices +and web browsers. + + +3.5 Self-Healing Policies + +Self-Healing Policies derive from the decision analysis and prioritization processes of the +organization and are recorded in a clear and interoperable manner in the system database. The +database consists of Threats, Policies, and Self-Healing Rules [36]. The Threat panel contains the +fields of the threat id, the threat type, and the threat group. Self-healing commands are stored in a +technical Command Line Interface (CLI) format, so that they can be understood by machines, as +well as in a general format, readable by humans. The self-healing policy also includes the entries +of the CLI commands, which concern the central nodes. CLI commands are properly synthesized +to run on devices located at the ends of the network, including routers, switches, firewalls, agents, +and AV software. Preventing a threat can be achieved by stopping the flow of network traffic in a +timely manner, or by making inaccessible a device involved in the attack. +In particular, Self-Healing commands include three execution options, customizable through the +system’s console: +1. Inform the administrator about the actions to be taken in order to prevent a threat or +mitigate the risk (recommendations). +2. Execution following the administrator’s approval. +3. Automated execution, provided that the administrator has selected the specific +configuration. +This subsystem receives data from the OSSEC system and from the Business operations and +information systems mapping module of the organization, through control command flows. At the +same time, it maintains two-way communication with the Visualization and User Interface +subsystem, through data streams. The self-healing instructions are presented to the administrator +and approval is required to execute a command. The administrator's decision is then forwarded to +the self-healing subsystem. If the administrator approves the action, the command is executed +immediately. If the action is rejected, then the self-healing command is given as a recommendation +to the administrator. The set of requests and responses of the above communication is conducted +asynchronously. +The Decision Engine, part of the Self-Healing module, determines the policy to be applied when +an incident is detected. The procedure involves executing a command if the Threat Type field +corresponds to a value in the Policies table. If the Threat Type field does not provide a value, then +the more generic Threat Group field is checked instead. The event is then forwarded to the +Visualization and User Interface subsystem and the relevant breach notifications are presented. In +most cases the Self-Healing rules are applied remotely to the nodes, using the Secure Shell (SSH) +protocol while the details on how to execute the alternative commands are recorded in a log file. +CTIMP exhibits the characteristics of complete network surveillance and effectively extends the +settlement of security issues to other levels (systems, services). The technologies it incorporates +help reduce the complexity of the methods used in today’s attacks, by setting up specialized +security software, with its main functions satisfying the need for regular checks to identify threats, +update security policies, and maintain the organisation’s security posture to an acceptable level. + +4. CONCLUSIONS +An innovative architectural standardization of how to intelligently manage and deal with advanced +cyber threats was presented in this paper. The proposed CTIMP in a fully interoperable and +intelligent way, collaboratively utilizing cyber-knowledge generated on a daily basis by cyber- +threat managers around the world, ensures high levels of security of an organization's supervised +information infrastructure. Its design is based on standards that are able to maintain secure +communication with reliable sources of information and receive regular updates on existing and +emerging threats. The updates in question are initially optimized based on the respective needs and +priorities of the information infrastructure of the supported organization. They are then transformed +into automation rules that align the operating systems of the information systems and are finally +applied in practice by implementing notifications to the system administrators for immediate +action. It is an excellent mechanism for monitoring and timely detection of events in real-time, +which significantly enhances the levels of active cyber security of an organization. + +The most important task for the evolution of the proposed system is initially the process of finding +solutions for the comparison of logs and security policies for their convergence in shorter times. +Also, the strengthening of CTIMP with more advanced anomaly detection techniques which will +take into account most of the operational parameters of the organization such as task scheduling, +local events, technical upgrades or system adaptations, etc., would be a significant improvement. +In addition, the system's structure should be examined to see how it might be utilized with data +transformation methods, so that intelligent processes can discover the optimum techniques to +represent various types of structured or unstructured data to provide self-healing rules. Finally, the +CTI2SA's significant future growth must be based on interpretive models. These models may +describe the decision process by defining individual predictions using approaches such as Shapley +values and feature significance. The purpose of model interpretation is to extract human- +comprehensible terminology for models' functioning mechanisms in researching adversarial +movements and defenses. + + +FUNDING +Co‐financed by the European Regional Development Fund of the European Union and Greek +national funds through the Operational Program Competitiveness, Entrepreneurship and +Innovation, under the call RESEARCH – CREATE - INNOVATE (project code: Τ2EDK-01469) + +REFERENCES + +[1] +A. Al-Fuqaha, M. Guizani, M. Mohammadi, M. Aledhari, and M. Ayyash, “Internet of Things: A +Survey on Enabling Technologies, Protocols, and Applications,” IEEE Commun. Surv. Tutor., vol. +17, no. 4, pp. 2347–2376, 2015, doi: 10.1109/COMST.2015.2444095. +[2] +E. Harjula, A. Artemenko, and S. 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Ilioudis, “An Innovative Self-Healing Approach +with STIX Data Utilisation:,” in Proceedings of the 17th International Joint Conference on e- +Business and Telecommunications, Lieusaint - Paris, France, 2020, pp. 645–651. doi: +10.5220/0009893306450651. + + diff --git a/h9E1T4oBgHgl3EQfzgVr/content/tmp_files/load_file.txt b/h9E1T4oBgHgl3EQfzgVr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bce09893aaf9d5405c76229249ea7810fc6d6e6a --- /dev/null +++ b/h9E1T4oBgHgl3EQfzgVr/content/tmp_files/load_file.txt @@ -0,0 +1,542 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf,len=541 +page_content='A CYBER THREAT INTELLIGENCE MANAGEMENT PLATFORM FOR INDUSTRIAL ENVIRONMENTS Alexandros Papanikolaou1, Aggelos Alevizopoulos1, Christos Ilioudis2, Konstantinos Demertzis3,* Konstantinos Rantos3 1Innovative Secure Technologies P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='gr 2International Hellenic University, Department of Information and Electronic Engineering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' iliou@ihu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='gr 3International Hellenic University, Department of Computer Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' kdemertzis@teiemt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='gr, krantos@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='ihu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='gr ABSTRACT Developing intelligent, interoperable Cyber Threat Information (CTI) sharing technologies can help build strong defences against modern cyber threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=" CTIs allow the community to share information about cybercriminals' threats and vulnerabilities and countermeasures to defend themselves or detect malicious activity." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' A crucial need for success is that the data connected to cyber risks be understandable, organized, and of good quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The receiving parties may grasp its content and utilize it effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=" This article describes an innovative cyber threat intelligence management platform (CTIMP) for industrial environments, one of the Cyber-pi project's significant elements." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=" The suggested architecture, in particular, uses cyber knowledge from trusted public sources and integrates it with relevant information from the organization's supervised infrastructure in an entirely interoperable and intelligent way." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' When combined with an advanced visualization mechanism and user interface, the services mentioned above provide administrators with the situational awareness they require while also allowing for extended cooperation, intelligent selection of advanced coping strategies, and a set of automated self-healing rules for dealing with threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' KEYWORDS Cyber Threat Intelligent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Cyber Threat Information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Information Sharing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Industrial Environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Cybersecurity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' INTRODUCTION The rapid development of new technologies in recent decades has significantly affected human societies and the existing economy [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The highly digitized and interconnected environment shapes the new Cyberspace [2] [3] providing new possibilities and opportunities for organizations to develop extroversion activities and actions [4] [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' However, this new cyber-ecosystem faces a number of challenges such as cybercrime, advanced persistent threats and cyberattacks, resulting in a climate of uncertainty and instability that threatens expected growth and prosperity [6] [7] [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The emergence of a new generation of cyber threats highlights the need to modernize the way these challenges are addressed [9] [10] [11], bypassing the until recently tactics of organizations that relied on the passive use of main security appliances, like firewalls to protect their information and anti-malware solutions [12] [13] [14] [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The characteristic of the complexity of modern threats is that most successful attacks are perceived only during the subsequent forensics procedures [16] [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' As it is easily understood, ensuring a successful defense requires complete control in all attempts to exploit the vulnerabilities of the system, as a successful attack could take advantage of the existence of a single vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' In everyday life, many system administrators are unable to fix all the vulnerabilities of a system in time as limited experience, non-automation, increased workload, software dependencies, use of old systems and lack of timely availability of critical patches are their main brake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' On the contrary, the automated, intelligent collection and correlation of suspicious actions taking place in a network, in the framework of a single strategy, which will take into account the latest digital threats, can help to take appropriate measures to deal immediately and finally shield an organization from cyber threats [18] [19] , even though the sharing of cyber-threat intelligence is a challenging process [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' A turning point in this process is the use of Indicators of Compromise (IOCs) that support the security decision-making process [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' IOCs include malware signature IDs, malicious IP addresses, malicious checksum (MD5) malware, and malicious URLs or domain names of Botnets, as well as patch fixes, good practices in control measures, access control policies or removing unnecessary services, and modifying firewall settings [22] [23] [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' In other words, this is a huge repository of knowledge with proven defense techniques, which are strengthened daily by adding updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Taking into account the gap presented in the intelligent, efficient, and unified application of the knowledge available based on the IOCs, this work presents an innovative architecture for utilizing the specific knowledge based on interoperable and intelligent methods of modern computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The proposed CTIMP is an advanced and adaptive system for monitoring and timely detection of security events that threaten an organization, incorporating advanced technologies of analysis, automated management, and execution of corrective actions, offering interactive real-time security interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The rest of this paper is structured as follows: Section 2 presents the background of the research approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Section 3 is allocated to the presentation of the proposed architecture and finally, section 4 concludes the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' BACKGROUND It is important to emphasize that in recent years the need for a cooperative response to security incidents has been highlighted and very significant progress has been made in this area [25]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' But the constant evolution of the methods and technologies used in cybercrime, creates complete on platforms that do not implement real-time information mechanisms, as anything else can be considered obsolete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Also, extremely important feature is the interoperability that can in the effective collection, improvement, analysis and sharing of cyber-attack data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' A typical example of such an application is the method of Modi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' [28] where they propose a multilevel architecture for the thorough analysis of heterogeneous data through the interaction between its interleaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Although the approach accepts information from open-source data streams, it is considered to be completely dependent on in-house analysis platforms, which significantly limits the generalization that should be provided in such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Mantis presents a different approach that incorporates information on cyber threats using different standards [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' It is an intelligent platform that allows threat data to be correlated through an innovative agnostic similarity algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' This methodology allows security analysts to correlate patterns that are shared between seemingly unrelated attacks, which adds serious complexity to the system, dramatically increasing the need for computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Finally, Sengupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' [30] proposed a fairly sophisticated but very sophisticated method for extracting an optimal modeling technique for Advanced Persistent Threat attacks in a cloud computing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The approach is based on game theory, where the processes of dealing with an event are modeled by optimizing the cost of security countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' PROPOSED ARCHITECTURE The proposed CTIMP, while following the practices of Integrated Security Information and Event Management (SIEM) [31], goes one step further by offering a personalized security solution that combines multiple control mechanisms and corresponding digital security technologies for modern computing systems and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=" Essentially, through a sophisticated collaborative framework, it is able to identify an organization's digital risks and threats, meeting the ongoing needs of securing the valuable information it manages, by offering security services and crisis remedies." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Specifically, CTIMP offers a central point of analysis, alert, compliance, and reporting, responding to the changing organizational structures of a multidimensional modern organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' It focuses primarily on meeting the critical information infrastructure needs of each organization, providing a variety of intelligent mechanisms for monitoring data integrity, reporting new threats, detecting and recording security incidents, and responding immediately to automated processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Taking a more detailed approach to the proposed architecture, the system initially focuses on the timely detection of events using automated, detailed log analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=" Any alerts or events are displayed on the system administrator's visualization console." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' This interface offers a timely and valid simultaneous analysis of a very large number of security incidents of the supervised business network while minimizing the possibility of drawing incorrect conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Updating and upgrading CTIMP predictability is based on gathering cyber-threat information from trusted open access sources such as ready-made IOCs set up by security experts, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' which are filtered and correlated for the sole purpose of supporting infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=" This adaptation to the requirements of the organization's business operations and information systems is achieved through component mapping, which records and distributes to other CTIMP subsystems the specific features of the available node topology, software, and services that may be targets or sources of malicious action." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Comparison of cyber-threats with the characteristics of the organization results in an adapted cyber-cognition in the STIX 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='x [32] standard, which is submitted for analysis to the privacy policy production subsystem, which provides algorithms for modulating and exporting SIGMA rules, which are automatically integrated into all the active rules of the case analysis mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' In addition, the proposed architecture includes an intelligent automated threat and attack mechanism, which provides an expanded set of self-healing commands, which examine the level of compliance and align it with current security policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Another very important feature is that the proposed architecture is assisted in each phase of its operation by the visualization and interface subsystem, which aims to easily represent the appropriate information, to allow analysts to detect and take immediate action in various events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Finally, it should be noted that the design of preventive countermeasures offered by CTIMP, concerns only the identification of specific threats that may affect the organization and the setting of general priorities, establishing intelligent decision-making or adaptation mechanisms, which ensure the smooth its operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' A brief overview of CTIMP subsystems and functions is shown in Figure 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Figure 1: The proposed Cyber Threat Intelligent Information Sharing Architecture (CTIMP) Below is a detailed presentation of the subsystems and mechanisms that frame the CTIMP architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='1 Incident analysis and identification of security attacks The analysis of incidents and the intelligent identification of security incidents such as attacks are achieved in CTIMP by the following systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='1 OSSEC HIDS The OSSEC Host-based Intrusion Detection System [33] conducts both application and system- level audits to investigate the integrity of supervised information infrastructure files, using threat detection methods based on signatures and statistical anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' It can be configured to collect events from devices on which the use of agents is not feasible, while also having a set of rules for monitoring and analyzing specialized security incidents, for which it can generate corresponding alerts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='2 Decoders The logs of the target environment are examined using default and custom decoders, which have parameters that are compared with the content of the logs for event detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Any correspondences are routed for control by the set of available rules that implement the respective security policies, based on which the notifications of the incidents under consideration are produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='2 Intelligent use of CTI The Intelligent Cyber Threat Adaptation Subsystem collects and analyzes available information gathered from Cyber Threat Information (CTI) Sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' This information is compared with the results of the analysis of the data by the supervised Information System and the threats to the target Mobile&Web Apslyself-HoakngRuss 三 Apps Self - Healirg Oprseional Businese Infa V Gpooha Bushas o Stama STIXnetwork and information infrastructure are documented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The intelligent cybernetics aggregation mechanism is regularly upgraded with IOCs and RSS feeds, collected by the Malware Information Sharing and Threat Intelligence Sharing Platform (MISP) [34], from a wide range of trusted sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=" The accuracy of the information is adjusted to the criteria, concerning the organization's systems, with a filtering process in order to maintain the relevant information and then to compare it with the specific characteristics of the supervised information infrastructure." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Features include devices, services, objects IDs, IP addresses, geolocation information, and dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The result is the production of custom knowledge in the form of STIX 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='x files, which correspond to the fields that are required to draft SIGMA rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='3 Business operations and information systems mapping The mapping mechanism describes the specific characteristics of the organization and the technological environment is displayed on the system’s console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The latter provides a useful for manually recording business assets like nodes, communications, software, and network services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' In particular, the Dependency Mapper utility implements the basic processes of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The depmapper has a graphical data management interface to represent a deployment model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The adaptation of the depmapper to the mapping mechanism helps to extend the functionality of exporting graphs to image files (jpg, png) and data exchange (JSON).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' In practice, the production of image files supports collaboration between stakeholders, as well as the use of images between different third-party applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Furthermore, the production of JSON-type files allows the insertion of graphs in the depmapper, for the purpose of their subsequent processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' More specifically, the layout model illustrates the components, hardware components (nodes), and the links between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The depmapper has the additional functions of grouping nodes, adding tags, and defining multiple descriptions separately for each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The descriptions contain information about the object of operation, the available software services, the geographical location, the structural and procedural dependencies, the data dependencies, as well as the risk level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The level of risk is determined according to the criticality or sensitivity of the data and services available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The collection of this information takes place within the organization, by completing the user questionnaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The information is collected by an analyzer and entered into the depmapper to represent graphs and text in JSON format, for use in other subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='4 Visualization and user interface This subsystem is divided into two subcategories, which include data visualization and user interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Visualization of security data collected from various log sources refers to the creation of diagrams, graphs, and similar visual material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The visualization mechanism provides organization and classification of the data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' A notice indicates that immediate action is required or is merely informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Categorizing the different types of alerts is useful and makes it possible to develop a monitoring strategy [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Notifications are categorized according to the level of completion of the relevant research, conducted by managers and analysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' An administrator is able to edit the content of an alert immediately, either later, or outsource its processing to an analyst, using ticketing requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Consequently, the status of notifications is characterized as new, ongoing, and complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' In addition, the visualization mechanism provides all the data in a user-friendly environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The user interface mechanism offers the possibility of approving or activating the system self-healing procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' However, it is a fully customizable environment, which allows the rapid collection of critical information and immediate response to incidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Likewise, it allows the viewing and analysis of critical statistics and the viewing of history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The interface supports web environments to make services available on mobile devices and web browsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content='5 Self-Healing Policies Self-Healing Policies derive from the decision analysis and prioritization processes of the organization and are recorded in a clear and interoperable manner in the system database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The database consists of Threats, Policies, and Self-Healing Rules [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The Threat panel contains the fields of the threat id, the threat type, and the threat group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Self-healing commands are stored in a technical Command Line Interface (CLI) format, so that they can be understood by machines, as well as in a general format, readable by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The self-healing policy also includes the entries of the CLI commands, which concern the central nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' CLI commands are properly synthesized to run on devices located at the ends of the network, including routers, switches, firewalls, agents, and AV software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Preventing a threat can be achieved by stopping the flow of network traffic in a timely manner, or by making inaccessible a device involved in the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' In particular, Self-Healing commands include three execution options, customizable through the system’s console: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Inform the administrator about the actions to be taken in order to prevent a threat or mitigate the risk (recommendations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Execution following the administrator’s approval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Automated execution, provided that the administrator has selected the specific configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' This subsystem receives data from the OSSEC system and from the Business operations and information systems mapping module of the organization, through control command flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' At the same time, it maintains two-way communication with the Visualization and User Interface subsystem, through data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The self-healing instructions are presented to the administrator and approval is required to execute a command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=" The administrator's decision is then forwarded to the self-healing subsystem." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' If the administrator approves the action, the command is executed immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' If the action is rejected, then the self-healing command is given as a recommendation to the administrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The set of requests and responses of the above communication is conducted asynchronously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The Decision Engine, part of the Self-Healing module, determines the policy to be applied when an incident is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The procedure involves executing a command if the Threat Type field corresponds to a value in the Policies table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' If the Threat Type field does not provide a value, then the more generic Threat Group field is checked instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The event is then forwarded to the Visualization and User Interface subsystem and the relevant breach notifications are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' In most cases the Self-Healing rules are applied remotely to the nodes, using the Secure Shell (SSH) protocol while the details on how to execute the alternative commands are recorded in a log file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' CTIMP exhibits the characteristics of complete network surveillance and effectively extends the settlement of security issues to other levels (systems, services).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The technologies it incorporates help reduce the complexity of the methods used in today’s attacks, by setting up specialized security software, with its main functions satisfying the need for regular checks to identify threats, update security policies, and maintain the organisation’s security posture to an acceptable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' CONCLUSIONS An innovative architectural standardization of how to intelligently manage and deal with advanced cyber threats was presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=" The proposed CTIMP in a fully interoperable and intelligent way, collaboratively utilizing cyber-knowledge generated on a daily basis by cyber- threat managers around the world, ensures high levels of security of an organization's supervised information infrastructure." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Its design is based on standards that are able to maintain secure communication with reliable sources of information and receive regular updates on existing and emerging threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The updates in question are initially optimized based on the respective needs and priorities of the information infrastructure of the supported organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' They are then transformed into automation rules that align the operating systems of the information systems and are finally applied in practice by implementing notifications to the system administrators for immediate action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' It is an excellent mechanism for monitoring and timely detection of events in real-time, which significantly enhances the levels of active cyber security of an organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' The most important task for the evolution of the proposed system is initially the process of finding solutions for the comparison of logs and security policies for their convergence in shorter times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' Also, the strengthening of CTIMP with more advanced anomaly detection techniques which will take into account most of the operational parameters of the organization such as task scheduling, local events, technical upgrades or system adaptations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=', would be a significant improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=" In addition, the system's structure should be examined to see how it might be utilized with data transformation methods, so that intelligent processes can discover the optimum techniques to represent various types of structured or unstructured data to provide self-healing rules." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=" Finally, the CTI2SA's significant future growth must be based on interpretive models." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' These models may describe the decision process by defining individual predictions using approaches such as Shapley values and feature significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=" The purpose of model interpretation is to extract human- comprehensible terminology for models' functioning mechanisms in researching adversarial movements and defenses." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfzgVr/content/2301.03445v1.pdf'} +page_content=' FUNDING Co‐financed by the European Regional Development Fund 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100644 index 0000000000000000000000000000000000000000..a753816b4c5879eda7de32ba428c27575d5a1ba0 --- /dev/null +++ b/itFAT4oBgHgl3EQfaB0B/content/tmp_files/2301.08548v1.pdf.txt @@ -0,0 +1,1529 @@ +arXiv:2301.08548v1 [math.CA] 20 Jan 2023 +UNIFORM PERSISTENCE CRITERIA FOR A VARIABLE +INPUTS CHEMOSTAT MODEL WITH DELAYED RESPONSE IN +GROWTH AND COMPLETE ANALYSIS OF THE PERIODIC +CASE +Mauro Rodriguez Cartabia and Daniel Sepúlveda Oehninger +Abstract. We study a single-species chemostat model with variable nutrient +input and variable dilution rate with delayed (fixed) response in growth. The +first goal of this article is to prove that persistence implies uniform persistence. +Then we concentrate in the particular case with periodic nutrient input and +same periodic dilution with delayed response in growth. We obtain a threshold +for either the (uniform) persistence of the model or that the biomass of every +solution tends to vanish. Furthermore, we prove that persistence is equivalent +to the existence of a unique non-trivial periodic solution. We also prove that +this solution is attractive. +We remark in no case we need to impose any +restrictions on the size of the delay. +Keywords: Chemostat, persistence, periodic case, time delay. +1. Introduction +We consider the cultivation of a species of microorganism inside a chemostat +under a limiting substrate with a delay between the consuption and the growth +of the population in a variable environment, i.e., both the dilution rate and the +input concentration of the substrate vary in time. The need to consider a delay in +biomass growth in a chemostat has been documented in the work of Caperon [5] +and Ellermeyer et al. [9]. Furthermore, any population is affected by time-varying +environmental fluctuations such as the light cycle or the seasons of the year, which +reaffirms the importance of studying non-autonomous population models, see the +book of Smith and Waltman [15, Chapter 7] for instance. +A (periodic version) model suitable for the situation described above has been +studied and deduced in [3] by Amster, Robledo and Sepúlveda. +In the present +article, first we study the model with general inputs, and then imposing periodicity +conditions. Therefore, fix a non-negative constant τ and consider the system +s′(t) += +D(t)s(0)(t) − D(t)s(t) − p(s(t))x(t), +t ≥ 0, +x′(t) += +x(t − τ)p(s(t − τ))e +−� t +t−τ D(h)dh − D(t)x(t), +t ≥ 0 +(1.1) +with initial conditions +(s, x) +��� +[−τ,0] = +� +sin, xin� +. +These initial conditions must be non-negative time functions defined over the inter- +val [−τ, 0]. Here, s(t) and x(t) represent, respectively, the substrate and biomass +densities inside the bioreactor at time t; the dilution rate and the nutrient in- +put concentration, respectively D(t) and s(0)(t), are non-negative, continuous, and +1 + +2 +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +bounded functions for t ≥ 0. As usual in this class of models, the relationship +between substrate consumption and biomass growth is modeled by a specific con- +sumption function p : [0, ∞) → [0, ∞), which satisfies: +Hypothesis 1.1. p is of class C1, p′(s) > 0 for each s ≥ 0 and p(0) = 0. +To begin with, we concentrate on uniform persistence. The concept of persistence +has significant importance in the theory of population models, was introduced in +[4, 12], and its relevance has increased due to the interest it arouses among those who +study ecology and dynamical systems. Ellermeyer in [8] proved persistence criteria +for System (1.1) in the autonomous case, and Ellermeyer et al. in [10] provided +persistence criteria for System (1.1) in the non-autonomous case with instantaneous +response in growth (τ = 0). +Recently, Rodriguez Cartabia in [14] studied the +System (1.1) obtaining necessary and sufficient conditions for the persistence of +the microbial population. Among the several notions of persistence, the uniform +persistence is a more desirable from the point of views of applications since is a +more robust concept, see [4, Introduction]. We have identified a lack of studies +focusing on this topic for System (1.1). Therefore, to the best of our knowledge, +Theorem 2.1 is the first which states that persistence of the System (1.1) implies +uniform persistence. Roughly speaking, there exists an intrinsic bound δ > 0 such +that every solution (s, x) of (1.1) with not null initial condition (see Definition 2.1) +satisfies that x ≥ δ from a certain time. +Secondly, we focus on the case where s(0)(t) and D(t) are periodic functions. In +[3] the authors obtained sufficient conditions for the existence of periodic solutions +using the generalized Leray-Schauder degree continuation theorem and, applying +the implicit function theorem, proved that for small delays the non-trivial periodic +solution is unique. We emphasize that this results are based on stronger assump- +tions about System (1.1) than needed. Therefore, to the best of our knowledge, +the results of the present paper are the first comprehensive study of the periodic +case. Theorem 2.2 states that if System (1.1) is not persistent then all solutions +tend to washout. In other words, obtain a threshold for the vanishing of biomass. +Next, we concentrate on the problem of the existence of a positive periodic solution. +Theorem 2.3 establishes that if System (1.1) is persistent then there is a non-trivial +periodic solution. The key is to prove this result by combining Horn’s fixed point +Theorem [13, Theorem 6] with Theorem 2.1. We finish this article with Theorem +2.4 which states that this positive periodic solution is unique and attractive. In +conclusion, we prove that System (1.1) is persistent if and only if there exists a +unique non-trivial attractive periodic solution. +Finally, we emphasize that, unlike several results in delay differential equations, +all proof presented in this article do not need to impose any restrictions on the fixed +delay, i.e., all statements are valid regardless the size of the delay. +The rest of the paper is organized as follows. In Section 2 we present in more +detail previous research on the subject, introduce definitions and present the theo- +rems. In Section 3 we introduce results from previous works and different lemmas +needed for the proofs. In Section 4 we prove Theorem 2.1 and, finally, in Section 5 +we present the proofs of Theorems 2.2, 2.3, and 2.4. +2. The results +The introduction of the classical chemostat model has attracted the attention of +the mathematical community which has used it to investigate control, interspecies + +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +3 +competition, and persistence, among other problems. A better understanding of +the continuous stirred tank reactor has led to several modifications of the classical +model. To provide a brief review of previous research, firstly consider the work +of Caperon [5] where experimental evidence is reported on the presence of a delay +between nutrient consumption and biomass growth, taking into account this delay, +the author proposed a model similar to the following system: +s′(t) += +Ds(0) − Ds(t) − p(s(t))x(t), +x′(t) += +x(t)p(s(t − τ)) − Dx(t). +Amster, Robledo and Sepúlveda [2] consider a version of this model with periodic +substrate concentration and obtained a necessary and sufficient condition for the +existence of a positive periodic solution. On another hand, it is worth mentioning +that Caraballo et al. modified this system in [6] to introduce variable (bounded) +delay and incorporated the death of the microorganisms in addition to the washout. +Another approach to modeling the presence of delay in the growth of a species +in a chemostat was carried out by Freedman et al. [11] and by Ellermeyer [8, 9], +who proposed the following system: +s′(t) += +Ds(0) − Ds(t) − p(s(t))x(t), +x′(t) += +x(t − τ)p(s(t − τ))e−Dτ − Dx(t). +Note that System (1.1) is an extension that incorporates variable nutrient input +and variable dilution rate. One of the difficulties encountered when studying the +persistence of a system of differential equations with delay is that the state space +is not locally compact so it is necessary to look for new approaches to determine +persistence, see [1]. To study persistence, in [14] the author extended the criteria +given in [10, Theorem 3] to incorporate the case with fixed delay in growth. He +provided a necessary and sufficient criteria (see Theorem 3.1) for the persistence of +System (1.1). Therefore, we propose the present article as a continuation since our +first goal is to show that this persistence criteria also imply uniform persistence. +We conclude this subsection by pointing out that in the rest of this paper we +assume: +Hypothesis 2.1. s(0)(t) is upper and lower bounded by positive constants, D(t) +is non-negative and upper bounded by a positive constant, and the integral of D(t) +diverges. +2.1. Main definitions. Before presenting the theorems obtained in this work, we +introduce the definitions involved in this article. +Definition 2.1 (Not null initial conditions). We say (sin, xin) is a not null initial +condition if its time functions are non-negative, and either xin(0) > 0 or there +exists t∗ ∈ [−τ, 0] such that sin(t∗) > 0 and xin(t∗) > 0. +Given a solution (s, x) of System (1.1) we denote +(s, x)(t) = (s, x)(t, (sin, xin)) +(2.1) +when the initial condition is fixed. + +4 +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +Definition 2.2 (Persistence definitions). The System (1.1) is called (strong) per- +sistent, if +lim inf +t→∞ x(t, (sin, xin)) > 0, +for all not null (sin, xin). +The System (1.1) is called uniformly persistent, if there exists some δ > 0 such +that +lim inf +t→∞ x(t, (sin, xin)) > δ, +for all not null (sin, xin). +Note that in the absence of biomass the System (1.1) becomes the linear differ- +ential equation +z′(t) = D(t) +� +s(0)(t) − z(t) +� +. +(2.2) +We emphasize that any solution of the Equation (2.2) with positive initial condition +z0 is also positive for all t ≥ 0 . This is easily seen by writing a solution in the form +z(t) = e−� t +0 D(r) drz0 + +� t +0 +e−� t +r D(r) drs(0)(r)dr. +Moreover, for D(t) and s(0)(t) bounded and continuous functions, we have that +every solution z(t) of (2.2) verifies that: +lim +t→+∞(z(t) − z∗(t)) = 0, +with z∗(t) the unique bounded solution of (2.2) over R, defined by +z∗(t) := +� t +−∞ +e−� t +h D(r) drD(h)s(0)(h)dh. +(2.3) +We call (z∗, 0) the washout solution and, for simplicity, sometimes we only refer z∗ +as the washout solution. +Definition 2.3 (Extinction). A solution (s, x) of System (1.1) tends to extinction +if +lim +t→∞ x(t) = 0. +2.2. Results for the general non-autonomous model. We now state the first +main result of this paper. It is worth recalling that throughout this note we consider +τ ≥ 0 and, in particular, all results apply to the case without delay. +Theorem 2.1 (Uniform persistence). The System (1.1) is persistent if and only if +it is uniform persistent. +Furthermore, if the system is persistent then there exists δ > 0 such that for all +R > 0 and α > 0 there is T in = T in(R, α) ≥ 0 such that for any solution (s, x) +with not null initial conditions satisfying ||(sin, xin)|| ≤ R and such that x(τ) ≥ α, +then x(t) ≥ δ for all t ≥ T in. +2.3. Results for the periodic model. Fix a constant ω > 0. Considering that +both the dilution rate D(t) and the input nutrient concentration s(0)(t) are positive +ω-periodic functions, we wonder whether persistence is a necessary and sufficient +condition for the existence of a positive ω-periodic solution of the System (1.1). +Theorem 2.2 (Extinction). If the periodic version of System (1.1) is not persistent +then every solution tends to extinction. Namely, in this case every solution tends +to the washout solution. +Note that the converse of the latter result is trivial. + +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +5 +Theorem 2.3 (Existence of periodic solution). If the System (1.1) is persistent +then there is at least one positive ω-periodic solution. +Theorem 2.4 (Attractivity of periodic solution). Under the assumption of the +Theorem 2.3 there exists a unique positive ω-periodic solution that exponentially +attracts every solution with not null initial conditions. +Remark 2.1 (Complete analysis of the periodic case). By Theorem 2.2, observe +that if System (1.1) is not persistent then there is no positive ω-periodic solution. +Therefore, combining all results we get that System (1.1) is (uniform) persistent if +and only if there is an unique attractive positive ω-periodic solution. +3. Preliminaries +3.1. Preliminaries. The deduction of the model given by System (1.1) is based on +considering a function y(t) that represents the amount of substrate that has been +absorbed by the biomass during the time interval [t − τ, t] and that remains in the +bioreactor at the instant t, this function is given by: +y(t) := +� t +t−τ +x(h)p(s(h))e−� t +h D(r) dr dh. +(3.1) +A key fact in achieving the results of this work lies on the relationship between the +solutions of (1.1) and the functions z∗(t) and y(t). +Next, we define +c(t) := c(0)e−� t +0 D(r) dr + +� t−τ +−τ +c(h)p(z∗(h))e−� t +h D(r) dr dh, +t ≥ 0 +and we assume that c(θ) ≥ 0 for all θ ∈ [−τ, 0] with c(0) > 0 which is a solution of +the linear equation +c′(t) = −D(t)c(t) + c(t − τ)p(z∗(t − τ))e +−� t +t−τ D(r) dr. +Since c(t) > 0 for all t ≥ 0 we can define the function +ϕ(t) := +c(t) +c(t + τ)e−� t+τ +t +D(r) dr, +t ≥ 0. +(3.2) +This function, which is independent of any solution (s, x)(t), is inherent in the +System (1.1) and is the key to determine the persistence. It is explicitly given by +ϕ(t) = +c(0)e−� t +0 D(r) dr + +� t−τ +−τ c(h)p(z∗(h))e−� t +h D(r) dr dh +c(0)e−� t +0 D(r) dr + +� t +−τ c(h)p(z∗(h))e−� t +h D(r) dr dh +. +Observe that multiples of c result in the same ϕ and that the image of ϕ is contained +in (0, 1]. +On the other hand, but related with ϕ(t), given a solution (s, x)(t) of the System +(1.1) with not null initial conditions, we define a function ψ(t) = ψ(s, x)(t) by +ψ(t) := +x(t) +x(t + τ)e−� t+τ +t +D(r)dr, +t ≥ 0. +(3.3) +Using (3.3) in the second equation of (1.1) we obtain +x(t + τ) = x(t0)e +� t+τ +t0 +[p(s(h−τ))ψ(h−τ)−D(h)]dh, +t ≥ t0 ≥ τ. +(3.4) + +6 +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +In addition, by using the previous equation with t0 = t in Equation (3.3) we obtain +ψ(t) = e +−� t +t−τ p(s(h))ψ(h)dh, +t ≥ τ. +(3.5) +We remark that, through this article, in a sum involving z∗(t), s(t), x(t) or y(t) +we simplify the dependence of time. For example, we note (x + y)(t) instead of +x(t) + y(t). +Let us introduce additional notation to be used in this note. We consider the +Banach space C := C([−τ, 0] → R2) with the norm +∥φ∥ = +max +t∈[−τ,0] |φ(t)| = +max +t∈[−τ,0] +� +φ2 +1(t) + φ2 +2(t). +As usual, for a given continuous function φ : [−τ, ∞) → R2 and any t ≥ 0 we define +φt ∈ C as +φt(h) = φ(t + h) +(3.6) +for h ∈ [−τ, 0]. Finally, for an ω-periodic function f : R → R we denote its average +as +⟨f⟩ := 1 +ω +� ω +0 +f(t) dt. +3.2. Preliminaries results. Recall Hypothesis 2.1 and then consider that s(0) +and D are bounded above by positive constants s and D, respectively, and s(0) is +bounded below by a positive constant s. Note that a consequence of the definition +of z∗ in (2.3) is that z∗(t) ≤ s for all t. +An alternative formulation of a persistence criterion for the System (1.1) is pre- +sented below. To the proof see [14, Theorem 2.1]. +Theorem 3.1 (Persistence criteria). Let τ be any non-negative constant and let +z∗(t) and ϕ(t) be the functions defined by (2.3) and (3.2), respectively. +Therefore System (1.1) is persistent if and only if there are positive constants η +and T such that +� t2 +t1 +p(z∗(t − τ))ϕ(t − τ) dt > +� t2 +t1 +(D(t) + η) dt +(3.7) +for all t1 > T , t2 − t1 > T . +For the the particular case when s(0)(t) and D(t) are ω-periodic we have the +following result. +Theorem 3.2 (Persistence for ω-periodic case). Let τ be any non-negative constant +and assume the functions s(0)(t) and D(t) are ω-periodic. Then z∗(t) is ω-periodic +and there is a unique c(t) (up to a constant factor) such that ϕ(t) defined in (3.2) +is ω-periodic. Therefore System (1.1) is persistent if and only if +⟨p(z∗)ϕ⟩ > ⟨D⟩. +To the proof see [14, Theorem 3.1]. +In order to present necessary lemmas, we now introduce a function f : [−τ, ∞) → +R+ bounded from above by a positive constant M and such that +0 < inf +t≥−τ f(t). + +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +7 +Moreover, we assume that M satisfies +M ≥ +1 +4τ + 1. +(3.8) +Also consider a function g : [−τ, ∞) → R+ which verifies the property that there +exists tg ≥ τ such that +g(t) ≤ M, +t ≥ tg − τ. +(3.9) +We conclude this section by stating four lemmas that are fundamental in the +proof of our uniform persistence result. The proves can be found in [14, Lemma +4.1, Lemma 4.2, Lemma 4.4, and Lemma 4.5], respectively. We observe that the +last proof is the reason to ask for (3.8). +Lemma 3.1. The functions z∗(t), s(t), x(t) and y(t) given in (2.3), (1.1) and +(3.1), respectively, satisfy +(z∗ − s − x − y)(t) = (z∗ − s − x − y) (0)e−� t +0 D(r) dr. +Furthermore, they all are bounded above. +Lemma 3.2. The function ϕ(t) defined in (3.2) satisfies +ϕ(t) = e +−� t +t−τ ϕ(h)p(z∗(h)) dh. +Lemma 3.3. Let 0 ≤ t0 < t1, τ > 0 and ϕ : [t0 − τ, ∞) → (0, 1] be such that +ϕ(t) = e +−� t +t−τ f(h)ϕ(h) dh +for all t > t0. Assume that there is ε > 0 such that |f(t)−g(t)| < ε for all t ∈ [t0, t1]. +Then there exists ˜ψ : [t0 − τ, ∞) → (0, 1] that satisfies +˜ψ(t) = e +−� t +t−τ g(h) ˜ +ψ(h) dh +for all t > t0 and such that +|ϕ(t) − ˜ψ(t)| < 2ετe2M(t−t0) +for all t ∈ [t0 − τ, t1]. +Lemma 3.4. Let t0 ≥ tg, τ > 0 and consider functions ϕ, ψ : [t0 − τ, ∞) → (0, 1] +satisfying +ϕ(t) = e +−� t +t−τ f(h)ϕ(h) dh, +ψ(t) = e +−� t +t−τ f(h)ψ(h) dh +for all t ≥ t0. Therefore +|ϕ(t) − ψ(t)| < 3M +� +(t − t0) +inf f +� +1 − e−Mτ�(t−t0)/(2τ)−1/2 +for all t ≥ t0 where inf f = inft∈[−τ,∞) f(t). + +8 +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +4. Proof of uniform persistence for the non-autonomous case +4.1. Ideas for the proof. To prove Theorem 2.1 we refine the proof of [14, Theo- +rem 2.1]. Observe that if System (1.1) is persistent then it satisfies condition given +by Inequality (3.7). To prove uniform persistence by contradiction, consider x small +enough. Then y is also small and then s ≈ z∗ (see Lemma 3.1) which implies that +ϕ ≈ ψ and +� +p(z∗)ϕ ≈ +� +p(s)ψ, +which is larger than the integral of D if we assume Inequality (3.7). Finally, by +Equation (3.4) x can not tend to zero when the time goes to infinity and we get +the contradiction. +Furthermore, note that x at the begging might be extremely small. Then we +firstly prove that there is a time T in +∗ +(that depends on the size of the norm of the +initial condition (sin, xin) and the value of x at time τ) for which we can ensure +that x grows enough (see Inequality (4.7)). Finally, once that x is large enough we +can prove that it is away from zero for all remaining positive times. +4.2. Auxiliary lemma for the non-autonomous chemostat model. The fol- +lowing result is a convenient adaptation of [14, Lema 4.5]. +Lemma 4.1. Let τ ≥ 0 and functions ϕ, ψ : [−τ, ∞) → (0, 1] satisfy +ϕ(t) = e +−� t +t−τ f(h)ϕ(h) dh, +ψ(t) = e +−� t +t−τ g(h)ψ(h) dh +for t ≥ tg. Given η > 0, there exist positive constants ε and ˜T such that: for all +t0 ≥ tg and I ≥ 0, if +|f(t) − g(t)| < ε +for t ∈ [t0, t0 + ˜T + I], then +|ϕ(t) − ψ(t)| < +η +4M +(4.1) +for all t ∈ [t0 + ˜T, t0 + ˜T + I]. +Proof. Firstly, note that if τ = 0 then ϕ(t) = ψ(t) = 1 and there is nothing to +prove. Therefore, let us assume that τ > 0. Since +3M +� +t +inf f +� +1 − e−Mτ�t/(2τ)−1/2 +tends to zero as t tends to infinity, we fix ˜T > 0 such that +3M +� +t +inf f +� +1 − e−Mτ�t/(2τ)−1/2 < η/2, +t ≥ ˜T. +Moreover, let ε > 0 such that +ε < ηe−2M ˜ +T +4τ +. +Now let us fix t0 ≥ tg, I ≥ 0, and assume that +|f(t) − g(t)| < ε + +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +9 +for all t ∈ [t0, t0 + ˜T + I]. We will show that +|f(t0 + ˜T + a) − g(t0 + ˜T + a)| < η, +for all a ∈ [0, I]. +(4.2) +Let a ∈ [0, I] and consider ˜ψ : [t0 + a − τ, ∞) → (0, 1] the function given by Lemma +3.3 that satisfies both +˜ψ(t) = e +−� t +t−τ g(h) ˜ +ψ(h) dh, +for all t > t0 + a and +|ϕ(t) − ˜ψ(t)| < 2ετe2M(t−t0−a) +for all t ∈ [t0 + a − τ, t0 + ˜T + I]. In particular, the above inequality combined with +the definition of ε implies that +|ϕ(t0 + ˜T + a) − ˜ψ(t0 + ˜T + a)| < 2ετe2M ˜T < η/2. +Moreover, applying the Lemma 3.4 to ψ and ˜ψ we obtain +|ψ(t) − ˜ψ(t)| < 3M +� +(t − t0) +inf f +� +1 − e−Mτ�(t−t0)/(2τ)−1/2 < η/2 +for t ≥ t0 + ˜T. Therefore, it follows that +|ϕ(t0 + ˜T + a) − ψ(t0 + ˜T + a)| ≤ |ϕ(t0 + ˜T + a) − ˜ψ(t0 + ˜T + a)| ++ | ˜ψ(t0 + ˜T + a) − ψ(t0 + ˜T + a)| +< η +and the inequality (4.2) is proved. +□ +4.3. Proof of Theorem 2.1. Set f(t) = p(z∗(t)) and fix +M = max +� +p +� +2s(0) +� +, +1 +4τ + 1 +� +. +(4.3) +Proof. Assume that System (1.1) is persistent, then Theorem 3.1 states that there +exist positive constants η and T such that +� t2 +t1 +p(z∗(t − τ))ϕ(t − τ) dt > +� t2 +t1 +(D(t) + η) dt +(4.4) +for all t1 > T , t2 −t1 > T where ϕ(t) is only dependent on p(z∗(t)). The remainder +of the proof will be divided into four steps with η, T and ϕ(t) fixed. +Step 1. Firstly, we define fundamental constants that do not depend on a partic- +ular solution. Since s(0)(t) is bounded below by a positive constant and z∗(0) > 0 +we deduce that p(z∗(t)) is bounded below by a positive constant. As we already +have a given η, let ˜T and ε be the constants given by Lemma 4.1. Note that the +functions f(t) and ϕ(t) are both fixed. Moreover, it is possible to consider ε small +enough so that +ε ≤ η/4. +(4.5) +To conclude the first step we define +L := +max +ξ∈[0,2s(0)] +p′(ξ), +and +δ := εe−D( ˜T +T +2τ) +2L(1 + Mτ) . + +10 +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +Step 2. Now let R and α be positive constants, we recall the statement of the +Theorem 2.1: there is T in(R, α) ≥ 0 such that for a given x(t) ≥ δ for all t ≥ T in. +Therefore consider t0 ≥ τ such that +� +s(0) + 2R + Rp(R)τ +� +e−� t0−τ +0 +D(r) dr ≤ min +� ε +2L, s(0) +� +(4.6) +and define +Iin = max +� +T, 2 +η ln +� +εeD(t0+ ˜T ) +2Lα(1 + Mτ) +�� +, +T in = t0 + ˜T + Iin + τ. +Given (s(t), x(t)) a particular solution of the System (1.1) such that ∥(sin, xin)∥ ≤ R +and x(τ) ≥ α we firstly prove the existence of T in +∗ +∈ [t0 − τ, T in] such that +x(T in +∗ ) ≥ +ε +2L(1 + Mτ). +(4.7) +To obtain a contradiction, we suppose the opposite. In particular, +x(t) < +ε +2L(1 + Mτ) +(4.8) +for all t ∈ [t0 − τ, T in − τ]. +Firstly, note that for t ≥ 0 it follows +y(t) ≤ +� t +t−τ +x(h)p(s(h)) dh ≤ +� t +t−τ +∥xt∥p(∥st∥) dh ≤ ∥xt∥p(∥st∥)τ +(4.9) +where we use the notation introduced in (3.6). +Secondly, using Lemma 3.1 and the previous inequality evaluated in t = 0, +combined with the properties of the initial data, it turns out that +|(z∗ − s − x − y)(t)| ≤ |(z∗ − s − x − y)(0)|e−� t +0 D(r) dr +≤ +� +z∗(0) + ||sin|| + ||xin|| + y(0) +� +e−� t +0 D(r) dr +≤ (s + 2R + Rp(R)τ) e−� t +0 D(r) dr. +(4.10) +Now, from the solution (s, x)(t) we define g(t) := p(s(t)) and consider the function +ψ(t) = ψ(s, x)(t) given by (3.3). We claim that if t0 = tg then p(s(t)) verifies (3.9). +Indeed, let us consider t ≥ t0 − τ. +Then, using that z∗(t) ≤ s for all t (recall +Definition (2.3)), and Inequalities (4.6) and (4.10) it follows that +s(t) ≤ (s + x + y)(t) ≤ |(z∗ − s − x − y)(t)| + z∗(t) ≤ 2 s, +and (3.9) holds by the definition of M in (4.3). We remark that we construct p(s(t)) +satisfying (3.9) for the purpose of applying Lemma 4.1. +Next, combining (4.9) with (3.9) and (4.8) we obtain +y(t) ≤ +Mτε +2L(1 + Mτ) +for all t ∈ [t0, T in − τ]. + +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +11 +Now, using this last inequality together with (4.6), (4.8) and (4.10), it turns out +that +|(z∗ − s)(t)| ≤ |(z∗ − s − x − y)(t)| + (x + y)(t) +≤ (s + 2R + Rp(R)τ) e−� t +0 D(r)dr + +ε +2L(1 + Mτ) + +Mτε +2L(1 + Mτ) +≤ ε +2L + +ε +2L(1 + Mτ) + +Mτε +2L(1 + Mτ) += ε/L, +for all t ∈ [t0, T in − τ]. Then, by the mean value Theorem and the definition of L, +it follows that +|p(z∗(t)) − p(s(t))| ≤ L|(z∗ − s)(t)| < ε +for the same interval of time. Recall that T in − τ = t0 + ˜T + Iin and apply Lemma +4.1 with I = Iin. This enables to deduce that (4.1) holds, which together with (4.5) +implies +|p(z∗(t))ϕ(t) − p(s(t))ψ(t)| ≤ |p(z∗(t))ϕ(t) − p(z∗(t))ψ(t)| ++ |p(z∗(t))ψ(t) − p(s(t))ψ(t)| +≤ p(z∗(t))|ϕ(t) − ψ(t)| + |p(z∗(t)) − p(s(t))|ψ(t) +≤ M η +4M + η +4 += η/2, +and then +p(z∗(t))ϕ(t) − η/2 ≤ p(s(t))ψ(t) +(4.11) +for t ∈ [t0 + ˜T, T in − τ]. +Since t0 + ˜T − τ = T in − Iin, and from (4.4) and (4.11) we deduce that +� T in +T in−Iin (p(s(h − τ))ψ(h − τ) − D(h)) dh +≥ +� T in +T in−Iin (p(z∗(h − τ))ϕ(h − τ) − η/2 − D(h)) dh +≥ +� T in +T in−Iin η/2 dh += Iinη/2. +(4.12) +From the second equation of the System (1.1) we obtain the following inequality +d +dtx(t) ≥ −Dx(t), +(4.13) +which combined with the assumption x(τ) ≥ α gives +x(T in − Iin) = x(t0 + ˜T + τ) ≥ e−D(t0+ ˜T )x(τ) ≥ e−D(t0+ ˜T )α. + +12 +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +Now by using (3.4), the above inequality and (4.12), together with the definition of +Iin we have +x(T in) = x(T in − Iin)e +� T in +T in−Iin(p(s(h−τ))ψ(h−τ)−D(h)) dh +≥ e−D(t0+ ˜T )αeIinη/2 +≥ +ε +2L(1 + Mτ), +which contradicts (4.8) and, therefore, there exists T in +∗ +≤ T in that satisfies (4.7). +Step 3. It remains to see that x(t) ≥ δ for all t ≥ T in. We will prove that the +above holds for t ≥ T in +∗ . To do this, define +S = +� +t ≥ T in +∗ +: x(h) ≥ δ for all h ∈ [T in +∗ , t] +� +. +We claim that +T in +∗ + ˜T + T + 2τ ∈ S. +(4.14) +Let h ∈ [T in +∗ , T in +∗ + ˜T + T + 2τ], as before (3.4), (4.13), (4.12), combined with the +properties of T in +∗ +and the definition of δ, implies +x(h) ≥ x(T in +∗ )e +−� h +T in +∗ +D(r) dr ≥ εe−D(h−T in +∗ ) +2L(1 + Mτ) ≥ εe−D( ˜T +T +2τ) +2L(1 + Mτ) +≥ δ +and (4.14) holds. +Next we define t∗ := sup S and claim that t∗ = ∞. Suppose, contrary to our +claim, that T in +∗ + ˜T + T + 2τ < t∗ < ∞. By continuity of x(t) it follows that +x(t∗) = δ. +(4.15) +Again by (4.13), for all t ∈ [t∗ − ˜T − T − 2τ, t∗] we have that +x(t∗) ≥ x(t)e−� t∗ +t +D(r) dr ≥ x(t)e−D( ˜T +T +2τ), +which implies +x(t) ≤ δeD( ˜T +T +2τ) = +ε +2L(1 + Mτ) +due to the definition of δ. We repeat the reasoning done in Step 2, but now we +apply the Lemma 4.1 considering I = T . Note that +y(t) ≤ +Mτε +2L(1 + Mτ) +for t ∈ [t∗ − ˜T − T − τ, t∗], therefore we can deduce that +|p(z∗(t − τ))ϕ(t − τ) − p(s(t − τ))ψ(t)| ≤ η/2 +for t ∈ [t∗ − T, t∗], and consequently +� t∗ +t∗−T +(p(s(h − τ))ψ(h − τ) − D(h)) dh ≥ T η/2. +Finally, combining (3.4), the previous inequality, and the fact that t∗ − T ∈ S, it +follows that +x(t∗) = x(t∗ − T )e +� t∗ +t∗−T [p(s(h−τ))ψ(h−τ)−D(h)]dh ≥ x(t∗ − T )eT η/2 > x(t∗ − T ) ≥ δ +Hence we have a contradiction with (4.15), so t∗ is infinite and the theorem is +proved. +□ + +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +13 +5. Proofs for periodic case +5.1. Ideas for the proofs. By Theorem 3.2 if +⟨p(z∗)ϕ⟩ > ⟨D⟩ +then System (1.1) is persistent. +Therefore, Theorem 2.2 is a converse criteria, +namely, that every solution tends to the extinction if and only if +⟨p(z∗)ϕ⟩ ≤ ⟨D⟩. +(5.1) +Thus this inequality is a threshold for the vanishing of biomass. We prove Theorem +2.2 by contradiction. If +lim sup +t→∞ x(t) > 0 +then z∗ is larger than s from a certain moment. By Lemma 5.1 +� +p(z∗)ϕ is larger +than � p(s)ψ (except for a constant). Therefore, if +⟨p(z∗)ϕ⟩ < ⟨D⟩ +then +e +� +(p(z∗(t−τ))ϕ(t−τ)−D(t))dt +goes to zero, which implies that +e +� +(p(s(t−τ))ϕ(t−τ)−D(t))dt +goes to zero and using (again) Equality (3.4) we get a contradiction. In the case +that +⟨p(z∗)ϕ⟩ = ⟨D⟩ +we use ideas from the prove of [2, Theorem 1] to show that +� +(p(z∗)ϕ − p(s)ψ) +tends to infinity and repeat the previous idea to get a contradiction. +As mentioned, the goal of Theorem 2.3 is to use the Horn’s fixed point Theorem +[13, Theorem 6] combined with Theorem 2.1. Therefore, we define three convex +sets S0 ⊂ S1 ⊂ S2 ⊂ C, with S0 and S2 compact, and S1 open (relative to S2). By +inspiration in the proof of [16, Theorem 1] we use Arzelà–Ascoli Theorem to prove +the compactness of S0 and S2. Therefore, applying Horn’s Theorem to the Poincaré +operator we get the existence of a positive ω-periodic solution if the System (1.1) +is persistent. We emphasize that we give an explicit representation of S1 as the +intersection of S2 with an open set. This ensures that S1 is open relative to S2 and +that the hypothesis of the Horn’s Theorem are satisfied. +Finally, to introduce the proof of Theorem 2.4 first consider the case without +delay (τ = 0). Fix (s1, x1) a solution with not null initial condition and (s2, x2) a +positive ω-periodic solution. Note that s1+x1 ≈ z∗ for large times and s2+x2 = z∗ +which implies that x1 − x2 ≈ −(s1 − s2) and then +d +dt(s1 − s2) = −D(s1 − s2) − x1p(s1) + x2p(s2) += −D(s1 − s2) − x1(p(s1) − p(s2)) − (x1 − x2)p(s2) += −D(s1 − s2) − x1p′(ζ)(s1 − s2) − (x1 − x2)p(s2) +≈ −(s1 − s2)(D + x1p′(ζ) − p(s2)) + +14 +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +where, for simplicity, we do not write the dependence of time. Observe that ⟨D⟩ = +⟨p(s2)⟩ since x2 is ω-periodic (recall (3.4) and note that ψ = 1). +Then, if the +system is persistent, x1 is bounded below by a positive constant (from a certain +time) and s1 − s2 tends to zero when time goes to infinity. This shows that (s2, x2) +is attractive. To apply this idea to the general case (τ ≥ 0) and inspired in a +generalized Gronwall-type inequality given in [7, Lemma 2.4], the goal is to define +an appropriated function w that bounds s1 − s2 and tends to zero when the time +tends to infinity. +5.2. Auxiliary lemmas for the periodic chemostat. We need the following +Lemma inspired by [14, Lemma 4.3]. +Lemma 5.1. Let t0 ≥ 0 and τ > 0. Consider ϕ, ψ : [t0 − τ, ∞) → (0, 1] satisfying +ϕ(t) = e +−� t +t−τ f(h)ϕ(h) dh, +ψ(t) = e +−� t +t−τ g(h)ψ(h) dh +for all t > t0 and assume that f(t) ≥ g(t) for all t ≥ t0. Then +� t2 +t1 +f(h − τ)ϕ(h − τ) dh + τM ≥ +� t2 +t1 +g(h − τ)ψ(h − τ) dh +for all t2 ≥ t1 ≥ t0. +Proof. The proof will be divided into three steps. +Step 1. Observe that if ϕ(t) ≤ ψ(t) for t ≥ t0, then we get +e +� t +t−τ(f(h)ϕ(h)−g(h)ψ(h)) dh = ψ(t) +ϕ(t) ≥ 1 +which implies +� t +t−τ +(f(h)ϕ(h) − g(h)ψ(h)) dh ≥ 0. +On the other hand, if there are h2 ≥ h1 ≥ t0 such that ϕ(t) ≥ ψ(t) for all +t ∈ [h1, h2] then +� h2 +h1 +(f(h)ϕ(h) − g(h)ψ(h)) dh ≥ 0, +since +f(t)ϕ(t) − g(t)ψ(t) ≥ g(t)ϕ(t) − g(t)ϕ(t) = 0. +Step 2. We claim that there is a finite decreasing sequence {hn}1≤n≤N ⊂ R with +the properties that ϕ(hn) ≤ ψ(hn) for all 1 ≤ n ≤ N − 1 and that ϕ(t) ≥ ψ(t) if +t ∈ I = +N−1 +� +n=1 +[hn+1, hn − τ] ∪ [h1, t2 − τ]. +Indeed, define +h1 = +� +t2 − τ +if ϕ(t2 − τ) < ψ(t2 − τ), +inf {t ≥ t1 − τ : ϕ(h) ≥ ψ(h) for all h ∈ [t, t2 − τ]} +otherwise. + +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +15 +For n ≥ 1 and while t1 ≤ hn − τ, define +hn+1 = + + + +hn − τ +if ϕ(hn − τ) < ψ(hn − τ), +inf +� t ≥ t1 − τ : ϕ(h) ≥ ψ(h) +for all h ∈ [t, hn − τ] +� +otherwise. +Observe that the sequence ends when t1 > hN ≥ t1 − τ satisfying the statement of +the claim. Moreover, note that as consequence of the definition of the decreasing +sequence {hn}1≤n≤N and the Step 1 we obtain the following inequalities +� hn +hn−τ f(h)ϕ(h) dh +≥ +� hn +hn−τ g(h)ψ(h) dh, for all 1 ≤ n ≤ N − 1, +� +I f(h)ϕ(h) dh +≥ +� +I g(h)ψ(h) dh. +(5.2) +Whence we obtain +� t2−τ +hN +f(h)ϕ(h) dh ≥ +� t2−τ +hN +g(h)ψ(h) dh. +Step 3. Finally, observe that +� t2−τ +t1−τ +f(h)ϕ(h) dh ≥ +� t2−τ +hN +f(h)ϕ(h) dh +≥ +� t2−τ +hN−τ +g(h)ψ(h) dh += +� t2−τ +t1−τ +g(h)ψ(h) dh − +� hN −τ +t1−τ +g(h)ψ(h) dh +≥ +� t2−τ +t1−τ +g(h)ψ(h) dh − τM, +and the lemma is proved. +□ +The following lemma is fundamental to the proof of the Theorem 2.4. Applied +to a periodic solution of the System (1.1), it states that (x + y)(t − τ) diluted over +τ units of times is exactly x(t). +Lemma 5.2. Every solution (s, x)(t) of the System (1.1) with not null initial con- +dition, and the corresponding function y(t) defined in (3.1) satisfy +x(t) − (x + y)(t − τ)e +−� t +t−τ D(r)dr = +� +x(τ)e +� τ +0 D(r) dr − (x + y)(0) +� +e−� t +0 D(r)dr +for all t ≥ τ. In the particular case where (s, x)(t) is an ω-periodic (non-trivial) +solution it follows that +x(t) = (x + y)(t − τ)e +−� t +t−τ D(r)dr. +Proof. Firstly, note that for a given solution (s, x)(t) with not null initial condition +it follows that +d +dt(x + y)(t − τ) = −D(t − τ)(x + y)(t − τ) + x(t − τ)p(s(t − τ)) + +16 +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +and consequently +d +dt +� +(x + y)(t − τ)e +−� t +t−τ D(r) dr� += x(t − τ)p(s(t − τ))e +−� t +t−τ D(r) dr +− D(t)(x + y)(t − τ)e +−� t +t−τ D(r) dr. +Whence we deduce that +d +dt +� +x(t) − (x + y)(t − τ)e +−� t +t−τ D(r) dr� += +− D(t) +� +x(t) − (x + y)(t − τ)e +−� t +t−τ D(r) dr� +which provides the first part of the lemma. +Note that +����x(t) − (x + y)(t − τ)e +−� t +t−τ D(r) dr +���� → 0 +(5.3) +as t tends to infinity. In particular, if (s, x)(t) is a non-trivial periodic solution of +(1.1), the right side of (5.3) is also an ω-periodic function and converges to zero as +t tends to infinity, therefore it is identically zero, and the proof is complete. +□ +5.3. Proof of Theorem 2.2. +Proof. Assume that System (1.1) is not persistent, given a solution (s, x)(t) we will +prove that x(t) towards to zero as t → ∞. If limt→∞ x(t) = 0 there is nothing to +prove. Otherwise there exists ε > 0 such that +lim sup +t→∞ x(t) > ε +(5.4) +and we look for a contradiction. By using Lemma 3.1, it follows that +|(z∗ − s − x − y)(t)| → 0, +t → ∞. +Moreover, since the functions z∗(t), s(t), x(t) and y(t) are positive and +|(z∗ − s − x − y)(t)| < ε +from a certain time, by (5.4) we conclude that there exists t0 ≥ 0 such that z∗(t0) ≥ +s(t0). Furthermore, z∗(t) − s(t) satisfy +d +dt(z∗ − s)(t) = −D(t)(z∗ − s)(t) + x(t)p(s(t)) ≥ −D(t)(z∗ − s)(t) +thus, from the above differential inequality, we see that z∗(t) ≥ s(t) for t ≥ t0. +Note that, by assumption that System (1.1) is not persistent, one of the following +identities holds +⟨D⟩ > ⟨p(z∗)ϕ⟩, +(5.5) +or +⟨D⟩ = ⟨p(z∗)ϕ⟩. +(5.6) +The rest of the proof fall naturally into two cases. + +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +17 +Case 1. Assume that (5.5) holds. Consider again the functions f(t) = p(z∗(t)) +and g(t) = p(s(t)), then by (3.4) and Lemma 5.1 we obtain for t ≥ t0 +x(t) = x(t0)e +� t +t0 +(p(s(r−τ))ψ(r−τ)−D(r))dr += x(t0)e +� t +t0 +(p(s(r−τ))ψ(r−τ)−p(z∗(r−τ)ϕ(r−τ)) dr+� t +t0 +(p(z∗(r−τ))ϕ(r−τ)−D(r))dr +≤ x(t0)eτMe +−� t +t0 +(D(r)−p(z∗(r−τ))ϕ(r−τ))dr. +From (5.5) it may be conclude that x(t) → 0 as t → ∞ and we get a contradiction. +Case 2. Assume now that (5.6) holds. We claim that +� t +t0 +(p(z∗(r − τ))ϕ(r − τ) − p(s(r − τ))ψ(r − τ)) dr → ∞ +(5.7) +as t → ∞. Indeed, otherwise we could find K > 0 such that: for all t > t0 there +exists t1 > t such that +� t1−τ +t0−τ +(p(z∗(r))ϕ(r) − p(s(r))ψ(r)) dh < K. +Now combining the above inequality together with the Lemma 5.1 we have that for +all t > t0 it follows +� t−τ +t0−τ +(p(z∗(r))ϕ(r) − p(s(r))ψ(r)) dr = +� t1−τ +t0−τ +(p(z∗(r))ϕ(r) − p(s(r))ψ(r)) dr +− +� t1−τ +t−τ +(p(z∗(r))ϕ(r) − p(s(r))ψ(r)) dr +≤ K + τM. +Consequently, from (3.4) we obtain for all t ≥ t0 +ln(x(t)) = ln(x(t0)) + +� t +t0 +(p(s(r − τ))ψ(r − τ) − D(r)) dr += ln(x(t0)) + +� t +t0 +(p(s(r − τ))p(ψ(r − τ) − p(z∗(r − τ))ϕ(r − τ)) dr ++ +� t +t0 +(p(z∗(r − τ))ϕ(r − τ) − D(r)) dr +≥ ln(x(t0)) − K − τM + min +l∈[0,ω] +� l +0 +(p(z∗(r − τ))ϕ(r − τ) − D(r)) dr, +which implies that ln(x(t)) is bounded from below for all t ≥ t0, i.e., the solution +is persistent. This contradicts our assumption and (5.7) is proved. +Now, by using again (3.4) for t ≥ t0, we obtain +x(t) = x(t0)e +� t +t0 +p(s(r−τ))ψ(r−τ)−D(r)dr += x(t0)e +� t +t0 +p(s(r−τ))ψ(r−τ)−p(z∗(r−τ)ϕ(r) dr+� t +t0 +p(z∗(r−τ))ϕ(r−τ)−D(r)dr. +Finally, notice that (5.6) implies that for all t ≥ t0 we have +� t +t0 +p(z∗(r − τ))ϕ(r − τ) − D(r) dr ≤ max +l∈[0,ω] +� l +0 +p(z∗(r − τ))ϕ(r − τ) − D(r) dr, + +18 +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +consequently we obtain +x(t) ≤ x(t0)Me +−� t +t0 +p(z∗(r−τ))ϕ(r−τ)−p(s(r−τ)ψ(r−τ)dr, +where M := maxl∈[0,ω] +� +e +� l +0 p(z∗(r−τ))ϕ(r−τ)−D(r)dr +� +. From (5.7) we conclude that +x(t) → 0 as t → ∞, which contradicts (5.4) and the proof is complete. +□ +5.4. Proof of Theorem 2.3. +Proof. The proof will be divided into two steps, we recall that C := C([−τ, 0] → R2). +Step 1. Let δ > 0 the value given by Theorem 2.1 and consider +T in = T in(R, α) +with R = 3 s +and α = e−Dτδ/2. +We now claim that any solution of System (1.1) with initial condition +∥(sin, xin)∥ ≤ 3 s +satisfies +∥(s, x)t∥ ≤ R0 for all t ≥ 0, +(5.8) +where +R0 = z∗(0) + 6 s + 3 sp (3 s) τ + s. +To show this, first note that, by using the arguments to obtain (4.9) in the proof +of Theorem 2.1, it follows that +|y(0)| ≤ ∥(sin, xin)∥p(∥(sin, xin)∥)τ ≤ 3 sp(3 s)τ. +By definition of R0 and applying the Lemma 3.1 again, we obtain that +|(s, x)(t)| ≤ (s + x)(t) +≤ (s + x + y)(t) +≤ |(z∗ − s − x − y)(t)| + |z∗(t)| +≤ |(z∗ − s − x − y)(0)|e−� t +0 D + |z∗(t)| +≤ max{z∗(0), (s + x + y)(0)} + s +≤ z∗(0) + 6 s + 3 sp (3 s) τ + s += R0 +(5.9) +for all t ≥ −τ. +Step 2. Let us define +S = +� +φ ∈ C : |φ(h) − φ(r)| ≤ +√ +2R0 +� +D + p(R0) +� +|h − r| for all h, r ∈ [−τ, 0] +� +S0 = {φ ∈ S : ∥φ∥ ≤ 2 s, |φ2(0)| ≥ δ} , +S1 = {φ ∈ S : ∥φ∥ < 3 s, |φ2(0)| > δ/2} , +S2 = +� +φ ∈ S : ∥φ∥ ≤ R0, |φ2(0)| ≥ e−DT inδ/2 +� +. +Note that S0 ⊂ S1 ⊂ S2 are convex subset of C, S0 and S2 are compacts as +consequence of Arzelà–Ascoli Theorem and +S1 = S2 ∩ {φ ∈ C : ∥φ∥ < 3 s, |φ2(0)| > δ/2} +consequently, S1 is open relative to S2. + +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +19 +Next we define the Poincaré operator given by +P : +S2 +→ +C +φ +�→ +(s, x)(· + ω, φ). +where (s, x)(t, φ) is the solution of System (1.1) at time t, with initial condition φ +(recall (2.1)). Notice that for a given k ∈ N ∪ {0}, P k(φ) is the solution (s, x)(t, φ) +over the time interval [kω − τ, kω] (including when kω − τ < 0). +To use Horn’s fixed point Theorem we need to prove that P k(S1) ⊆ S2 for all +k ∈ N. Let φ ∈ S1 and (s, x) the solution corresponding to the initial condition +φ, it suffices to show that (s, x)t ∈ S2 for t ≥ 0. From Equation (5.8) we have +|(s, x)t| ≤ R0 while, for t ≤ T in inequality (4.12) implies +|x(t)| ≥ e−Dt|x(0)| ≥ e−DT inδ/2. +Hence +x(τ) ≥ e−Dτδ/2 +(5.10) +thus, by using Theorem 2.1, if t ≥ T in, then +|x(t)| ≥ δ ≥ e−DT inδ/2. +Now, it turns out that +|(s, x)(h) − (s, x)(r)| = +� +(s(h) − s(r))2 + (x(h) − x(r))2 += +� +s +′2(ξs)(h − r)2 + x +′2(ξx)(h − r)2 += |h − r| +�� +D(ξs)(s(0)(ξs) − s(ξs)) − x(ξs)p(s(ξs)) +�2 ++ +� +−D(ξx)x(ξx) + x(ξx − τ)p(s(ξx − τ))e +−� ξx +ξx−τ D(r) dr�2�1/2 +≤ |h − r| +�� +DR0 + R0p(R0) +�2 + +� +DR0 + R0p(R0) +�2�1/2 += |h − r| +√ +2R0 +� +D + p(R0) +� +for all h, r ≥ 0 and ξs, ξx between h and r. Therefore +(s, x) +��� +[kω−τ,kω] = P k(φ) ∈ S2, +for all k ∈ N. +Next, we need to show that there exists m ∈ N such that if k ≥ m then P k(φ) ∈ +S0. Let T ≥ T in such that +R0e−� T +0 D(r) dr ≤ s. +If t ≥ T then, by using again Theorem 2.1 and considering (5.10), it follows that +|x(t)| ≥ δ. Estimating |(s, x)(t)| one more time as in (5.9) we have +|(s, x)(t)| ≤ R0e−� t +0 D(r) dr + s ≤ 2 s. +As before, it follows that +|(s, x)(h) − (s, x)(r)| ≤ |h − r| +√ +2R0 +� +D + p(R0) +� +for all h, r ≥ 0. Thus, if kω ≥ mω ≥ T we obtain +(s, x) +��� +[kω−τ,kω] = P k(φ) ∈ S0. + +20 +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +Finally, by Horn’s fixed point Theorem [13, Theorem 6], there exist a fixed point +of P whose second coordinate is positive. +□ +5.5. Proof of Theorem 2.4. +Proof. Consider a solution (s1, x1) with no null initial conditions and (s2, x2) a +periodic non-trivial solution, whose existence was proved in Theorem 2.3. +The +proof will be divided into three steps. +Step 1. Note that Lemma 3.1 yields to z∗(t−τ) = (s2 +x2 +y2)(t−τ) and then +x2(t) + (s2 − z∗)(t − τ)x2(t) +(x2 + y2)(t − τ) += 0. +On another hand, and also using Lemma 5.2, we get that +x1(t) + (s1 − z∗)(t − τ)x2(t) +(x2 + y2)(t − τ) += x1(t) + (s1 − z∗)(t − τ)e +−� t +t−τ D(r) dr += x1(t) − (x1 + y1)(t − τ)e +−� t +t−τ D(r) dr ++ (s1 + x1 + y1 − z∗)(t − τ)e +−� t +t−τ D(r) dr += +� +x1(τ)e +� τ +0 D(r) dr − (x1 + y1)(0) +� +e−� t +0 D(r)dr ++ (s1 + x1 + y1 − z∗)(0)e−� t +0 D(r) dr += −C0e−� t +0 D(r) dr, +where +C0 = (z∗ − s1)(0) − x(τ)e +� τ +0 D(r) dr. +Then we obtain +(x1 − x2)(t) = −(s1 − s2)(t − τ) +x2(t) +(x2 + y2)(t − τ) ++ +�(s1 − z∗)(t − τ)x2(t) +(x2 + y2)(t − τ) ++ x1(t) +−(s2 − z∗)(t − τ)x2(t) +(x2 + y2)(t − τ) +− x2(t) +� += −(s1 − s2)(t − τ) +x2(t) +(x2 + y2)(t − τ) − C0e−� t +0 D(r) dr. +(5.11) + +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +21 +Therefore, +d +dt(s1 − s2)(t) = −D(t)(s1 − s2)(t) − x1(t)p(s1(t)) + x2(t)p(s2(t)) += −D(t)(s1 − s2)(t) − x1(t)(p(s1(t)) − p(s2(t))) +− (x1 − x2)(t)p(s2(t)) += −D(t)(s1 − s2)(t) − x1(t)p′(ξ)(s1 − s2)(t) +− (x1 − x2)(t)p(s2(t)) += −(s1 − s2)(t)(D(t) + x1(t)p′(ξ)) ++ (s1 − s2)(t − τ) +x2(t)p(s2(t)) +(x2 + y2)(t − τ) + C0p(s2(t))e−� t +0 D(r) dr +where ξ = ξ(t) ∈ (s1(t), s2(t)), but for simplicity we do not write the dependence +on t. +Step 2. Let us define +m = min +t≥τ {x1(t)p′(ξ(t))}. +Since the system is persistent, p′ is positive, and s1, s2 are bounded by above, we +deduce that m > 0. Furthermore consider the function +J(ε) = −m + max +h∈[0,ω] +�x2(h)p(s2(h)) +(x2 + y2)(h) +� +(eτε − 1) + 3ε/2. +Note that J(0) < 0, hence we can fix a positive ε such that ε < ⟨D⟩/2 and J(ε) ≤ 0. +Also notice that there exists t0 ≥ ω such that +C0 max +h∈[0,ω] {p(s2(h))} e−� t0−ω +0 +D(r) dr ≤ min +h∈[0,ω] +�ε (x2 + y2)(h) +2 +� +. +(5.12) +With all the previous considerations, define +w(t) = +max +h∈[t0−τ,t0] +� (s1 − s2)(h) +(x2 + y2)(h), 1 +� +(x2 + y2)(t)e−(t−t0)ε, +t ≥ t0 − τ, +and +S = {t ≥ t0 : (s1 − s2)(h) ≤ w(h) for all h ∈ [t0, t]} . +Observe that S is a non-empty set since t0 ∈ S. We claim that T ∗ = sup S is +infinite. On the contrary, suppose that T ∗ is finite. Consequently, by the mean +value Theorem there is t∗ ∈ [T ∗, T ∗ + τ) such that +d +dt(s1 − s2)(t∗) ≥ d +dtw(t∗), +(s1 − s2)(t∗) > w(t∗) > 0. +(5.13) +Observe that +w(t − τ) = +max +h∈[t0−τ,t0] +� (s1 − s2)(h) +(x2 + y2)(h), 1 +� +(x2 + y2)(t − τ)(x2 + y2)(t) +(x2 + y2)(t)e−(t−t0)εeτε += w(t)(x2 + y2)(t − τ) +(x2 + y2)(t) +eτε, + +22 +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +and that +d +dtw(t) = +max +h∈[t0−τ,t0] +� (s1 − s2)(h) +(x2 + y2)(h), 1 +� +e−(t−t0)ε (−D(t)(x2 + y2)(t) − x2(t)p(s2(t)) − ε(x2 + y2)(t)) += w(t) +� +−D(t) − +x2(t) +(x2 + y2)(t)p(s2(t)) − ε +� +. +Furthermore, using (5.12), for t ≥ t0, we have that +C0p(s2(t))e−� t +0 D(r) dr ≤ min +h∈[0,ω] +�ε (x2 + y2)(h) +2 +� +e +−� t +t0−ω D(r) dr +≤ ε +2(x2 + y2)(t)e−(t−t0)ε +≤ ε +2w(t), +obtained using the inequality +e +(t−t0)ε−� t +t0−ω D(r) dr ≤ 1, +which holds since ε ≤ ⟨D⟩/2. Now, taking the derivative of s1−s2 from the previous +step and using the above estimations we have +d +dt(s1 − s2)(t∗) ≤ −(s1 − s2)(t∗)(D(t∗) + m) + (s1 − s2)(t∗ − τ) x2(t∗)p(s2)(t∗) +(x2 + y2)(t∗ − τ) ++ C0p(s2(t∗))e−� t∗ +0 +D(r) dr +< −w(t∗)(D(t∗) + m) + w(t∗ − τ) x2(t∗)p(s2)(t∗) +(x2 + y2)(t∗ − τ) + ε +2w(t∗) += w(t∗) +� +−D(t∗) − m + x2(t∗)p(s2)(t∗) +(x2 + y2)(t∗) eτε + ε +2 +� +< w(t∗) +� +−D(t∗) + x2(t∗)p(s2)(t∗) +(x2 + y2)(t∗) − ε +� += d +dtw(t∗) +where we use the definition of ε. This contradicts (5.13) and, therefore, sup S = ∞. +Step 3. Following a similar reasoning, this time for s2 − s1, we obtain +d +dt(s2 − s1)(t) ≤ −(s2 − s1)(t)(D(t) + x1(t)p′(ξ)) ++ (s2 − s1)(t − τ) +x2(t)p(s2(t)) +(x2 + y2)(t − τ) + (−C0) p(s2(t))e−� t +0 D(r) dr +and we reach the desired result due to +|(s1 − s2)(t)| ≤ +max +h∈[t0−τ,t0] +� (s1 − s2)(h) +(x2 + y2)(h), 1 +� +(x2 + y2)(t)e−(t−t0)ε +which tends to zero as t tends to infinity (note that we do not know the sign of C0). + +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +23 +It remains to prove that x1(t) tends to x2(t) when t goes to infinity. By Equation +(5.11) we obtain +|(x1 − x2)(t)| = +����(s1 − s2)(t − τ)e +−� t +t−τ D(r) dr + C0e−� t +0 D(r) dr +���� +≤ |(s1 − s2)(t − τ)| + |C0|e−� t +0 D(r) dr +and we conclude that |(x1 − x2)(t)| tends to zero as t tends to infinity with expo- +nential decay. +Finally, for the uniqueness of the periodic solution, it suffices to note that the +distance between two periodic solutions can only tend to zero if it is zero at all +times. +□ +Acknowledgments +This research is partially supported by PROGRAMA REGIONAL MATH-AMSUD +MATH2020006. +The first author is supported by CONICET under grant PIP +11220200100175CO. The second author is supported by FONDECYT 11190457. +References +[1] Pablo Amster and Melanie Bondorevsky. Persistence and periodic solutions in systems of +delay differential equations. Applied Mathematics and Computation, 403:126193, 2021. 3 +[2] Pablo Amster, Gonzalo Robledo, and Daniel Sepúlveda. Dynamics of a chemostat with peri- +odic nutrient supply and delay in the growth. Nonlinearity, 33(11):5839, 2020. 3, 13 +[3] Pablo Amster, Gonzalo Robledo, and Daniel Sepúlveda. Existence of ω-periodic solutions +for a delayed chemostat with periodic inputs. Nonlinear Analysis: Real World Applications, +55:103134, 2020. 1, 2 +[4] Geoffrey Butler, Herb I Freedman, and Paul Waltman. Uniformly persistent systems. Pro- +ceedings of the American Mathematical Society, pages 425–430, 1986. 2 +[5] John Caperon. Time lag in population growth response of isochrysis galbana to a variable +nitrate environment. Ecology, 50(2):188–192, 1969. 1, 3 +[6] Tomás Caraballo, Xiaoying Han, and Peter E Kloeden. Nonautonomous chemostats with +variable delays. SIAM Journal on Mathematical Analysis, 47(3):2178–2199, 2015. 3 +[7] Young Pil Choi and Jan Haskovec. Cucker-smale model with normalized communication +weights and time delay. Kinetic and Related Models, 10(4):1011–1033, 2017. 14 +[8] Sean F Ellermeyer. Competition in the chemostat: global asymptotic behavior of a model +with delayed response in growth. SIAM Journal on Applied Mathematics, 54(2):456–465, +1994. 2, 3 +[9] Sean F Ellermeyer, Jerald Hendrix, and Nariman Ghoochan. A theoretical and empirical +investigation of delayed growth response in the continuous culture of bacteria. Journal of +theoretical biology, 222(4):485–494, 2003. 1, 3 +[10] Sean F Ellermeyer, Sergei S Pilyugin, and Ray Redheffer. Persistence criteria for a chemostat +with variable nutrient input. Journal of Differential Equations, 171(1):132–147, 2001. 2, 3 +[11] Herbert I Freedman, Joseph WH So, and Paul Waltman. Chemostat competition with time +delays. IMACS Ann. Comput. and Appl. Math, 5(1):4, 1989. 3 +[12] Herbert I Freedman and Paul Waltman. Mathematical analysis of some three-species food- +chain models. Mathematical Biosciences, 33(3-4):257–276, 1977. 2 +[13] WA Horn. Some fixed point theorems for compact maps and flows in banach spaces. Trans- +actions of the American Mathematical Society, 149(2):391–404, 1970. 2, 13, 20 +[14] Mauro Rodriguez Cartabia. Persistence criteria for a chemostat with variable nutrient input +and variable washout with delayed response in growth. arXiv preprint arXiv:2204.09735, +2022. 2, 3, 6, 7, 8, 14 +[15] Hal L Smith and Paul Waltman. The theory of the chemostat: dynamics of microbial com- +petition, volume 13. Cambridge university press, 1995. 1 + +24 +UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE +[16] Zhidong Teng, Linfei Nie, and Xining Fang. The periodic solutions for general periodic impul- +sive population systems of functional differential equations and its applications. Computers +& Mathematics with Applications, 61(9):2690–2703, 2011. 13 +Email address: mrodriguezcartabia@dm.uba.ar, daniel.sepulveda@utem.cl + diff --git a/itFAT4oBgHgl3EQfaB0B/content/tmp_files/load_file.txt b/itFAT4oBgHgl3EQfaB0B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7119fac32e7f39b9bedb901f5bf4ec96ed2f3db1 --- /dev/null +++ b/itFAT4oBgHgl3EQfaB0B/content/tmp_files/load_file.txt @@ -0,0 +1,686 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf,len=685 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='08548v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='CA] 20 Jan 2023 UNIFORM PERSISTENCE CRITERIA FOR A VARIABLE INPUTS CHEMOSTAT MODEL WITH DELAYED RESPONSE IN GROWTH AND COMPLETE ANALYSIS OF THE PERIODIC CASE Mauro Rodriguez Cartabia and Daniel Sepúlveda Oehninger Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We study a single-species chemostat model with variable nutrient input and variable dilution rate with delayed (fixed) response in growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The first goal of this article is to prove that persistence implies uniform persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Then we concentrate in the particular case with periodic nutrient input and same periodic dilution with delayed response in growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We obtain a threshold for either the (uniform) persistence of the model or that the biomass of every solution tends to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Furthermore, we prove that persistence is equivalent to the existence of a unique non-trivial periodic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We also prove that this solution is attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We remark in no case we need to impose any restrictions on the size of the delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Keywords: Chemostat, persistence, periodic case, time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Introduction We consider the cultivation of a species of microorganism inside a chemostat under a limiting substrate with a delay between the consuption and the growth of the population in a variable environment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=', both the dilution rate and the input concentration of the substrate vary in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The need to consider a delay in biomass growth in a chemostat has been documented in the work of Caperon [5] and Ellermeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Furthermore, any population is affected by time-varying environmental fluctuations such as the light cycle or the seasons of the year, which reaffirms the importance of studying non-autonomous population models, see the book of Smith and Waltman [15, Chapter 7] for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' A (periodic version) model suitable for the situation described above has been studied and deduced in [3] by Amster, Robledo and Sepúlveda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In the present article, first we study the model with general inputs, and then imposing periodicity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore, fix a non-negative constant τ and consider the system s′(t) = D(t)s(0)(t) − D(t)s(t) − p(s(t))x(t), t ≥ 0, x′(t) = x(t − τ)p(s(t − τ))e −� t t−τ D(h)dh − D(t)x(t), t ≥ 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) with initial conditions (s, x) ��� [−τ,0] = � sin, xin� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' These initial conditions must be non-negative time functions defined over the inter- val [−τ, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Here, s(t) and x(t) represent, respectively, the substrate and biomass densities inside the bioreactor at time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' the dilution rate and the nutrient in- put concentration, respectively D(t) and s(0)(t), are non-negative, continuous, and 1 2 UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE bounded functions for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' As usual in this class of models, the relationship between substrate consumption and biomass growth is modeled by a specific con- sumption function p : [0, ∞) → [0, ∞), which satisfies: Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' p is of class C1, p′(s) > 0 for each s ≥ 0 and p(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' To begin with, we concentrate on uniform persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The concept of persistence has significant importance in the theory of population models, was introduced in [4, 12], and its relevance has increased due to the interest it arouses among those who study ecology and dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Ellermeyer in [8] proved persistence criteria for System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) in the autonomous case, and Ellermeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' in [10] provided persistence criteria for System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) in the non-autonomous case with instantaneous response in growth (τ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Recently, Rodriguez Cartabia in [14] studied the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) obtaining necessary and sufficient conditions for the persistence of the microbial population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Among the several notions of persistence, the uniform persistence is a more desirable from the point of views of applications since is a more robust concept, see [4, Introduction].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We have identified a lack of studies focusing on this topic for System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore, to the best of our knowledge, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 is the first which states that persistence of the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) implies uniform persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Roughly speaking, there exists an intrinsic bound δ > 0 such that every solution (s, x) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) with not null initial condition (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) satisfies that x ≥ δ from a certain time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Secondly, we focus on the case where s(0)(t) and D(t) are periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In [3] the authors obtained sufficient conditions for the existence of periodic solutions using the generalized Leray-Schauder degree continuation theorem and, applying the implicit function theorem, proved that for small delays the non-trivial periodic solution is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We emphasize that this results are based on stronger assump- tions about System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) than needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore, to the best of our knowledge, the results of the present paper are the first comprehensive study of the periodic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2 states that if System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is not persistent then all solutions tend to washout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In other words, obtain a threshold for the vanishing of biomass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Next, we concentrate on the problem of the existence of a positive periodic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3 establishes that if System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is persistent then there is a non-trivial periodic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The key is to prove this result by combining Horn’s fixed point Theorem [13, Theorem 6] with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We finish this article with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4 which states that this positive periodic solution is unique and attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In conclusion, we prove that System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is persistent if and only if there exists a unique non-trivial attractive periodic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Finally, we emphasize that, unlike several results in delay differential equations, all proof presented in this article do not need to impose any restrictions on the fixed delay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=', all statements are valid regardless the size of the delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In Section 2 we present in more detail previous research on the subject, introduce definitions and present the theo- rems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In Section 3 we introduce results from previous works and different lemmas needed for the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In Section 4 we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 and, finally, in Section 5 we present the proofs of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The results The introduction of the classical chemostat model has attracted the attention of the mathematical community which has used it to investigate control, interspecies UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE 3 competition, and persistence, among other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' A better understanding of the continuous stirred tank reactor has led to several modifications of the classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' To provide a brief review of previous research, firstly consider the work of Caperon [5] where experimental evidence is reported on the presence of a delay between nutrient consumption and biomass growth, taking into account this delay, the author proposed a model similar to the following system: s′(t) = Ds(0) − Ds(t) − p(s(t))x(t), x′(t) = x(t)p(s(t − τ)) − Dx(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Amster, Robledo and Sepúlveda [2] consider a version of this model with periodic substrate concentration and obtained a necessary and sufficient condition for the existence of a positive periodic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' On another hand, it is worth mentioning that Caraballo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' modified this system in [6] to introduce variable (bounded) delay and incorporated the death of the microorganisms in addition to the washout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Another approach to modeling the presence of delay in the growth of a species in a chemostat was carried out by Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' [11] and by Ellermeyer [8, 9], who proposed the following system: s′(t) = Ds(0) − Ds(t) − p(s(t))x(t), x′(t) = x(t − τ)p(s(t − τ))e−Dτ − Dx(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Note that System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is an extension that incorporates variable nutrient input and variable dilution rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' One of the difficulties encountered when studying the persistence of a system of differential equations with delay is that the state space is not locally compact so it is necessary to look for new approaches to determine persistence, see [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' To study persistence, in [14] the author extended the criteria given in [10, Theorem 3] to incorporate the case with fixed delay in growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' He provided a necessary and sufficient criteria (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) for the persistence of System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore, we propose the present article as a continuation since our first goal is to show that this persistence criteria also imply uniform persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We conclude this subsection by pointing out that in the rest of this paper we assume: Hypothesis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' s(0)(t) is upper and lower bounded by positive constants, D(t) is non-negative and upper bounded by a positive constant, and the integral of D(t) diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Main definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Before presenting the theorems obtained in this work, we introduce the definitions involved in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 (Not null initial conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We say (sin, xin) is a not null initial condition if its time functions are non-negative, and either xin(0) > 0 or there exists t∗ ∈ [−τ, 0] such that sin(t∗) > 0 and xin(t∗) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Given a solution (s, x) of System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) we denote (s, x)(t) = (s, x)(t, (sin, xin)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) when the initial condition is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 4 UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2 (Persistence definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is called (strong) per- sistent, if lim inf t→∞ x(t, (sin, xin)) > 0, for all not null (sin, xin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is called uniformly persistent, if there exists some δ > 0 such that lim inf t→∞ x(t, (sin, xin)) > δ, for all not null (sin, xin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Note that in the absence of biomass the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) becomes the linear differ- ential equation z′(t) = D(t) � s(0)(t) − z(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2) We emphasize that any solution of the Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2) with positive initial condition z0 is also positive for all t ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' This is easily seen by writing a solution in the form z(t) = e−� t 0 D(r) drz0 + � t 0 e−� t r D(r) drs(0)(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Moreover, for D(t) and s(0)(t) bounded and continuous functions, we have that every solution z(t) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2) verifies that: lim t→+∞(z(t) − z∗(t)) = 0, with z∗(t) the unique bounded solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2) over R, defined by z∗(t) := � t −∞ e−� t h D(r) drD(h)s(0)(h)dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3) We call (z∗, 0) the washout solution and, for simplicity, sometimes we only refer z∗ as the washout solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3 (Extinction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' A solution (s, x) of System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) tends to extinction if lim t→∞ x(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Results for the general non-autonomous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We now state the first main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' It is worth recalling that throughout this note we consider τ ≥ 0 and, in particular, all results apply to the case without delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 (Uniform persistence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is persistent if and only if it is uniform persistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Furthermore, if the system is persistent then there exists δ > 0 such that for all R > 0 and α > 0 there is T in = T in(R, α) ≥ 0 such that for any solution (s, x) with not null initial conditions satisfying ||(sin, xin)|| ≤ R and such that x(τ) ≥ α, then x(t) ≥ δ for all t ≥ T in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Results for the periodic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Fix a constant ω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Considering that both the dilution rate D(t) and the input nutrient concentration s(0)(t) are positive ω-periodic functions, we wonder whether persistence is a necessary and sufficient condition for the existence of a positive ω-periodic solution of the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2 (Extinction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' If the periodic version of System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is not persistent then every solution tends to extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Namely, in this case every solution tends to the washout solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Note that the converse of the latter result is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE 5 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3 (Existence of periodic solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' If the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is persistent then there is at least one positive ω-periodic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4 (Attractivity of periodic solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Under the assumption of the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3 there exists a unique positive ω-periodic solution that exponentially attracts every solution with not null initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 (Complete analysis of the periodic case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2, observe that if System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is not persistent then there is no positive ω-periodic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore, combining all results we get that System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is (uniform) persistent if and only if there is an unique attractive positive ω-periodic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Preliminaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The deduction of the model given by System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is based on considering a function y(t) that represents the amount of substrate that has been absorbed by the biomass during the time interval [t − τ, t] and that remains in the bioreactor at the instant t, this function is given by: y(t) := � t t−τ x(h)p(s(h))e−� t h D(r) dr dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) A key fact in achieving the results of this work lies on the relationship between the solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) and the functions z∗(t) and y(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Next, we define c(t) := c(0)e−� t 0 D(r) dr + � t−τ −τ c(h)p(z∗(h))e−� t h D(r) dr dh, t ≥ 0 and we assume that c(θ) ≥ 0 for all θ ∈ [−τ, 0] with c(0) > 0 which is a solution of the linear equation c′(t) = −D(t)c(t) + c(t − τ)p(z∗(t − τ))e −� t t−τ D(r) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Since c(t) > 0 for all t ≥ 0 we can define the function ϕ(t) := c(t) c(t + τ)e−� t+τ t D(r) dr, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2) This function, which is independent of any solution (s, x)(t), is inherent in the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) and is the key to determine the persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' It is explicitly given by ϕ(t) = c(0)e−� t 0 D(r) dr + � t−τ −τ c(h)p(z∗(h))e−� t h D(r) dr dh c(0)e−� t 0 D(r) dr + � t −τ c(h)p(z∗(h))e−� t h D(r) dr dh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Observe that multiples of c result in the same ϕ and that the image of ϕ is contained in (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' On the other hand, but related with ϕ(t), given a solution (s, x)(t) of the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) with not null initial conditions, we define a function ψ(t) = ψ(s, x)(t) by ψ(t) := x(t) x(t + τ)e−� t+τ t D(r)dr, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3) Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3) in the second equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) we obtain x(t + τ) = x(t0)e � t+τ t0 [p(s(h−τ))ψ(h−τ)−D(h)]dh, t ≥ t0 ≥ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4) 6 UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE In addition, by using the previous equation with t0 = t in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3) we obtain ψ(t) = e −� t t−τ p(s(h))ψ(h)dh, t ≥ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='5) We remark that, through this article, in a sum involving z∗(t), s(t), x(t) or y(t) we simplify the dependence of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' For example, we note (x + y)(t) instead of x(t) + y(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Let us introduce additional notation to be used in this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We consider the Banach space C := C([−τ, 0] → R2) with the norm ∥φ∥ = max t∈[−τ,0] |φ(t)| = max t∈[−τ,0] � φ2 1(t) + φ2 2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' As usual, for a given continuous function φ : [−τ, ∞) → R2 and any t ≥ 0 we define φt ∈ C as φt(h) = φ(t + h) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='6) for h ∈ [−τ, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Finally, for an ω-periodic function f : R → R we denote its average as ⟨f⟩ := 1 ω � ω 0 f(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Preliminaries results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Recall Hypothesis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 and then consider that s(0) and D are bounded above by positive constants s and D, respectively, and s(0) is bounded below by a positive constant s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Note that a consequence of the definition of z∗ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3) is that z∗(t) ≤ s for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' An alternative formulation of a persistence criterion for the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is pre- sented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' To the proof see [14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 (Persistence criteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Let τ be any non-negative constant and let z∗(t) and ϕ(t) be the functions defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is persistent if and only if there are positive constants η and T such that � t2 t1 p(z∗(t − τ))ϕ(t − τ) dt > � t2 t1 (D(t) + η) dt (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='7) for all t1 > T , t2 − t1 > T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' For the the particular case when s(0)(t) and D(t) are ω-periodic we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2 (Persistence for ω-periodic case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Let τ be any non-negative constant and assume the functions s(0)(t) and D(t) are ω-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Then z∗(t) is ω-periodic and there is a unique c(t) (up to a constant factor) such that ϕ(t) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2) is ω-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is persistent if and only if ⟨p(z∗)ϕ⟩ > ⟨D⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' To the proof see [14, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In order to present necessary lemmas, we now introduce a function f : [−τ, ∞) → R+ bounded from above by a positive constant M and such that 0 < inf t≥−τ f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE 7 Moreover, we assume that M satisfies M ≥ 1 4τ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='8) Also consider a function g : [−τ, ∞) → R+ which verifies the property that there exists tg ≥ τ such that g(t) ≤ M, t ≥ tg − τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='9) We conclude this section by stating four lemmas that are fundamental in the proof of our uniform persistence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The proves can be found in [14, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4, and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='5], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We observe that the last proof is the reason to ask for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The functions z∗(t), s(t), x(t) and y(t) given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1), respectively, satisfy (z∗ − s − x − y)(t) = (z∗ − s − x − y) (0)e−� t 0 D(r) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Furthermore, they all are bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The function ϕ(t) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2) satisfies ϕ(t) = e −� t t−τ ϕ(h)p(z∗(h)) dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Let 0 ≤ t0 < t1, τ > 0 and ϕ : [t0 − τ, ∞) → (0, 1] be such that ϕ(t) = e −� t t−τ f(h)ϕ(h) dh for all t > t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Assume that there is ε > 0 such that |f(t)−g(t)| < ε for all t ∈ [t0, t1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Then there exists ˜ψ : [t0 − τ, ∞) → (0, 1] that satisfies ˜ψ(t) = e −� t t−τ g(h) ˜ ψ(h) dh for all t > t0 and such that |ϕ(t) − ˜ψ(t)| < 2ετe2M(t−t0) for all t ∈ [t0 − τ, t1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Let t0 ≥ tg, τ > 0 and consider functions ϕ, ψ : [t0 − τ, ∞) → (0, 1] satisfying ϕ(t) = e −� t t−τ f(h)ϕ(h) dh, ψ(t) = e −� t t−τ f(h)ψ(h) dh for all t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore |ϕ(t) − ψ(t)| < 3M � (t − t0) inf f � 1 − e−Mτ�(t−t0)/(2τ)−1/2 for all t ≥ t0 where inf f = inft∈[−τ,∞) f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 8 UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Proof of uniform persistence for the non-autonomous case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Ideas for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' To prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 we refine the proof of [14, Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Observe that if System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is persistent then it satisfies condition given by Inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' To prove uniform persistence by contradiction, consider x small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Then y is also small and then s ≈ z∗ (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) which implies that ϕ ≈ ψ and � p(z∗)ϕ ≈ � p(s)ψ, which is larger than the integral of D if we assume Inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Finally, by Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4) x can not tend to zero when the time goes to infinity and we get the contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Furthermore, note that x at the begging might be extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Then we firstly prove that there is a time T in ∗ (that depends on the size of the norm of the initial condition (sin, xin) and the value of x at time τ) for which we can ensure that x grows enough (see Inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Finally, once that x is large enough we can prove that it is away from zero for all remaining positive times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Auxiliary lemma for the non-autonomous chemostat model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The fol- lowing result is a convenient adaptation of [14, Lema 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Let τ ≥ 0 and functions ϕ, ψ : [−τ, ∞) → (0, 1] satisfy ϕ(t) = e −� t t−τ f(h)ϕ(h) dh, ψ(t) = e −� t t−τ g(h)ψ(h) dh for t ≥ tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Given η > 0, there exist positive constants ε and ˜T such that: for all t0 ≥ tg and I ≥ 0, if |f(t) − g(t)| < ε for t ∈ [t0, t0 + ˜T + I], then |ϕ(t) − ψ(t)| < η 4M (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) for all t ∈ [t0 + ˜T, t0 + ˜T + I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Firstly, note that if τ = 0 then ϕ(t) = ψ(t) = 1 and there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore, let us assume that τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Since 3M � t inf f � 1 − e−Mτ�t/(2τ)−1/2 tends to zero as t tends to infinity, we fix ˜T > 0 such that 3M � t inf f � 1 − e−Mτ�t/(2τ)−1/2 < η/2, t ≥ ˜T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Moreover, let ε > 0 such that ε < ηe−2M ˜ T 4τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Now let us fix t0 ≥ tg, I ≥ 0, and assume that |f(t) − g(t)| < ε UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE 9 for all t ∈ [t0, t0 + ˜T + I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We will show that |f(t0 + ˜T + a) − g(t0 + ˜T + a)| < η, for all a ∈ [0, I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2) Let a ∈ [0, I] and consider ˜ψ : [t0 + a − τ, ∞) → (0, 1] the function given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3 that satisfies both ˜ψ(t) = e −� t t−τ g(h) ˜ ψ(h) dh, for all t > t0 + a and |ϕ(t) − ˜ψ(t)| < 2ετe2M(t−t0−a) for all t ∈ [t0 + a − τ, t0 + ˜T + I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In particular, the above inequality combined with the definition of ε implies that |ϕ(t0 + ˜T + a) − ˜ψ(t0 + ˜T + a)| < 2ετe2M ˜T < η/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Moreover, applying the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4 to ψ and ˜ψ we obtain |ψ(t) − ˜ψ(t)| < 3M � (t − t0) inf f � 1 − e−Mτ�(t−t0)/(2τ)−1/2 < η/2 for t ≥ t0 + ˜T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore, it follows that |ϕ(t0 + ˜T + a) − ψ(t0 + ˜T + a)| ≤ |ϕ(t0 + ˜T + a) − ˜ψ(t0 + ˜T + a)| + | ˜ψ(t0 + ˜T + a) − ψ(t0 + ˜T + a)| < η and the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Set f(t) = p(z∗(t)) and fix M = max � p � 2s(0) � , 1 4τ + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Assume that System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is persistent, then Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 states that there exist positive constants η and T such that � t2 t1 p(z∗(t − τ))ϕ(t − τ) dt > � t2 t1 (D(t) + η) dt (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4) for all t1 > T , t2 −t1 > T where ϕ(t) is only dependent on p(z∗(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The remainder of the proof will be divided into four steps with η, T and ϕ(t) fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Firstly, we define fundamental constants that do not depend on a partic- ular solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Since s(0)(t) is bounded below by a positive constant and z∗(0) > 0 we deduce that p(z∗(t)) is bounded below by a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' As we already have a given η, let ˜T and ε be the constants given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Note that the functions f(t) and ϕ(t) are both fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Moreover, it is possible to consider ε small enough so that ε ≤ η/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='5) To conclude the first step we define L := max ξ∈[0,2s(0)] p′(ξ), and δ := εe−D( ˜T +T +2τ) 2L(1 + Mτ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 10 UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Now let R and α be positive constants, we recall the statement of the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1: there is T in(R, α) ≥ 0 such that for a given x(t) ≥ δ for all t ≥ T in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore consider t0 ≥ τ such that � s(0) + 2R + Rp(R)τ � e−� t0−τ 0 D(r) dr ≤ min � ε 2L, s(0) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='6) and define Iin = max � T, 2 η ln � εeD(t0+ ˜T ) 2Lα(1 + Mτ) �� , T in = t0 + ˜T + Iin + τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Given (s(t), x(t)) a particular solution of the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) such that ∥(sin, xin)∥ ≤ R and x(τ) ≥ α we firstly prove the existence of T in ∗ ∈ [t0 − τ, T in] such that x(T in ∗ ) ≥ ε 2L(1 + Mτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='7) To obtain a contradiction, we suppose the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In particular, x(t) < ε 2L(1 + Mτ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='8) for all t ∈ [t0 − τ, T in − τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Firstly, note that for t ≥ 0 it follows y(t) ≤ � t t−τ x(h)p(s(h)) dh ≤ � t t−τ ∥xt∥p(∥st∥) dh ≤ ∥xt∥p(∥st∥)τ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='9) where we use the notation introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Secondly, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 and the previous inequality evaluated in t = 0, combined with the properties of the initial data, it turns out that |(z∗ − s − x − y)(t)| ≤ |(z∗ − s − x − y)(0)|e−� t 0 D(r) dr ≤ � z∗(0) + ||sin|| + ||xin|| + y(0) � e−� t 0 D(r) dr ≤ (s + 2R + Rp(R)τ) e−� t 0 D(r) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='10) Now, from the solution (s, x)(t) we define g(t) := p(s(t)) and consider the function ψ(t) = ψ(s, x)(t) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We claim that if t0 = tg then p(s(t)) verifies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Indeed, let us consider t ≥ t0 − τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Then, using that z∗(t) ≤ s for all t (recall Definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3)), and Inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='10) it follows that s(t) ≤ (s + x + y)(t) ≤ |(z∗ − s − x − y)(t)| + z∗(t) ≤ 2 s, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='9) holds by the definition of M in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We remark that we construct p(s(t)) satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='9) for the purpose of applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Next, combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='9) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='8) we obtain y(t) ≤ Mτε 2L(1 + Mτ) for all t ∈ [t0, T in − τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE 11 Now, using this last inequality together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='10), it turns out that |(z∗ − s)(t)| ≤ |(z∗ − s − x − y)(t)| + (x + y)(t) ≤ (s + 2R + Rp(R)τ) e−� t 0 D(r)dr + ε 2L(1 + Mτ) + Mτε 2L(1 + Mτ) ≤ ε 2L + ε 2L(1 + Mτ) + Mτε 2L(1 + Mτ) = ε/L, for all t ∈ [t0, T in − τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Then, by the mean value Theorem and the definition of L, it follows that |p(z∗(t)) − p(s(t))| ≤ L|(z∗ − s)(t)| < ε for the same interval of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Recall that T in − τ = t0 + ˜T + Iin and apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 with I = Iin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' This enables to deduce that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) holds, which together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='5) implies |p(z∗(t))ϕ(t) − p(s(t))ψ(t)| ≤ |p(z∗(t))ϕ(t) − p(z∗(t))ψ(t)| + |p(z∗(t))ψ(t) − p(s(t))ψ(t)| ≤ p(z∗(t))|ϕ(t) − ψ(t)| + |p(z∗(t)) − p(s(t))|ψ(t) ≤ M η 4M + η 4 = η/2, and then p(z∗(t))ϕ(t) − η/2 ≤ p(s(t))ψ(t) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='11) for t ∈ [t0 + ˜T, T in − τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Since t0 + ˜T − τ = T in − Iin, and from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='11) we deduce that � T in T in−Iin (p(s(h − τ))ψ(h − τ) − D(h)) dh ≥ � T in T in−Iin (p(z∗(h − τ))ϕ(h − τ) − η/2 − D(h)) dh ≥ � T in T in−Iin η/2 dh = Iinη/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='12) From the second equation of the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) we obtain the following inequality d dtx(t) ≥ −Dx(t), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='13) which combined with the assumption x(τ) ≥ α gives x(T in − Iin) = x(t0 + ˜T + τ) ≥ e−D(t0+ ˜T )x(τ) ≥ e−D(t0+ ˜T )α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 12 UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE Now by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4), the above inequality and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='12), together with the definition of Iin we have x(T in) = x(T in − Iin)e � T in T in−Iin(p(s(h−τ))ψ(h−τ)−D(h)) dh ≥ e−D(t0+ ˜T )αeIinη/2 ≥ ε 2L(1 + Mτ), which contradicts (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='8) and, therefore, there exists T in ∗ ≤ T in that satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' It remains to see that x(t) ≥ δ for all t ≥ T in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We will prove that the above holds for t ≥ T in ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' To do this, define S = � t ≥ T in ∗ : x(h) ≥ δ for all h ∈ [T in ∗ , t] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We claim that T in ∗ + ˜T + T + 2τ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='14) Let h ∈ [T in ∗ , T in ∗ + ˜T + T + 2τ], as before (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='13), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='12), combined with the properties of T in ∗ and the definition of δ, implies x(h) ≥ x(T in ∗ )e −� h T in ∗ D(r) dr ≥ εe−D(h−T in ∗ ) 2L(1 + Mτ) ≥ εe−D( ˜T +T +2τ) 2L(1 + Mτ) ≥ δ and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='14) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Next we define t∗ := sup S and claim that t∗ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Suppose, contrary to our claim, that T in ∗ + ˜T + T + 2τ < t∗ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' By continuity of x(t) it follows that x(t∗) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='15) Again by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='13), for all t ∈ [t∗ − ˜T − T − 2τ, t∗] we have that x(t∗) ≥ x(t)e−� t∗ t D(r) dr ≥ x(t)e−D( ˜T +T +2τ), which implies x(t) ≤ δeD( ˜T +T +2τ) = ε 2L(1 + Mτ) due to the definition of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We repeat the reasoning done in Step 2, but now we apply the Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 considering I = T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Note that y(t) ≤ Mτε 2L(1 + Mτ) for t ∈ [t∗ − ˜T − T − τ, t∗], therefore we can deduce that |p(z∗(t − τ))ϕ(t − τ) − p(s(t − τ))ψ(t)| ≤ η/2 for t ∈ [t∗ − T, t∗], and consequently � t∗ t∗−T (p(s(h − τ))ψ(h − τ) − D(h)) dh ≥ T η/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Finally, combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4), the previous inequality, and the fact that t∗ − T ∈ S, it follows that x(t∗) = x(t∗ − T )e � t∗ t∗−T [p(s(h−τ))ψ(h−τ)−D(h)]dh ≥ x(t∗ − T )eT η/2 > x(t∗ − T ) ≥ δ Hence we have a contradiction with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='15), so t∗ is infinite and the theorem is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' □ UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Proofs for periodic case 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Ideas for the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2 if ⟨p(z∗)ϕ⟩ > ⟨D⟩ then System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is persistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2 is a converse criteria, namely, that every solution tends to the extinction if and only if ⟨p(z∗)ϕ⟩ ≤ ⟨D⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) Thus this inequality is a threshold for the vanishing of biomass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2 by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' If lim sup t→∞ x(t) > 0 then z∗ is larger than s from a certain moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 � p(z∗)ϕ is larger than � p(s)ψ (except for a constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore, if ⟨p(z∗)ϕ⟩ < ⟨D⟩ then e � (p(z∗(t−τ))ϕ(t−τ)−D(t))dt goes to zero, which implies that e � (p(s(t−τ))ϕ(t−τ)−D(t))dt goes to zero and using (again) Equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4) we get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In the case that ⟨p(z∗)ϕ⟩ = ⟨D⟩ we use ideas from the prove of [2, Theorem 1] to show that � (p(z∗)ϕ − p(s)ψ) tends to infinity and repeat the previous idea to get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' As mentioned, the goal of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3 is to use the Horn’s fixed point Theorem [13, Theorem 6] combined with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore, we define three convex sets S0 ⊂ S1 ⊂ S2 ⊂ C, with S0 and S2 compact, and S1 open (relative to S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' By inspiration in the proof of [16, Theorem 1] we use Arzelà–Ascoli Theorem to prove the compactness of S0 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore, applying Horn’s Theorem to the Poincaré operator we get the existence of a positive ω-periodic solution if the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is persistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We emphasize that we give an explicit representation of S1 as the intersection of S2 with an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' This ensures that S1 is open relative to S2 and that the hypothesis of the Horn’s Theorem are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Finally, to introduce the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4 first consider the case without delay (τ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Fix (s1, x1) a solution with not null initial condition and (s2, x2) a positive ω-periodic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Note that s1+x1 ≈ z∗ for large times and s2+x2 = z∗ which implies that x1 − x2 ≈ −(s1 − s2) and then d dt(s1 − s2) = −D(s1 − s2) − x1p(s1) + x2p(s2) = −D(s1 − s2) − x1(p(s1) − p(s2)) − (x1 − x2)p(s2) = −D(s1 − s2) − x1p′(ζ)(s1 − s2) − (x1 − x2)p(s2) ≈ −(s1 − s2)(D + x1p′(ζ) − p(s2)) 14 UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE where, for simplicity, we do not write the dependence of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Observe that ⟨D⟩ = ⟨p(s2)⟩ since x2 is ω-periodic (recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4) and note that ψ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Then, if the system is persistent, x1 is bounded below by a positive constant (from a certain time) and s1 − s2 tends to zero when time goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' This shows that (s2, x2) is attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' To apply this idea to the general case (τ ≥ 0) and inspired in a generalized Gronwall-type inequality given in [7, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4], the goal is to define an appropriated function w that bounds s1 − s2 and tends to zero when the time tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Auxiliary lemmas for the periodic chemostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We need the following Lemma inspired by [14, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Let t0 ≥ 0 and τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Consider ϕ, ψ : [t0 − τ, ∞) → (0, 1] satisfying ϕ(t) = e −� t t−τ f(h)ϕ(h) dh, ψ(t) = e −� t t−τ g(h)ψ(h) dh for all t > t0 and assume that f(t) ≥ g(t) for all t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Then � t2 t1 f(h − τ)ϕ(h − τ) dh + τM ≥ � t2 t1 g(h − τ)ψ(h − τ) dh for all t2 ≥ t1 ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The proof will be divided into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Observe that if ϕ(t) ≤ ψ(t) for t ≥ t0, then we get e � t t−τ(f(h)ϕ(h)−g(h)ψ(h)) dh = ψ(t) ϕ(t) ≥ 1 which implies � t t−τ (f(h)ϕ(h) − g(h)ψ(h)) dh ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' On the other hand, if there are h2 ≥ h1 ≥ t0 such that ϕ(t) ≥ ψ(t) for all t ∈ [h1, h2] then � h2 h1 (f(h)ϕ(h) − g(h)ψ(h)) dh ≥ 0, since f(t)ϕ(t) − g(t)ψ(t) ≥ g(t)ϕ(t) − g(t)ϕ(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We claim that there is a finite decreasing sequence {hn}1≤n≤N ⊂ R with the properties that ϕ(hn) ≤ ψ(hn) for all 1 ≤ n ≤ N − 1 and that ϕ(t) ≥ ψ(t) if t ∈ I = N−1 � n=1 [hn+1, hn − τ] ∪ [h1, t2 − τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Indeed, define h1 = � t2 − τ if ϕ(t2 − τ) < ψ(t2 − τ), inf {t ≥ t1 − τ : ϕ(h) ≥ ψ(h) for all h ∈ [t, t2 − τ]} otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE 15 For n ≥ 1 and while t1 ≤ hn − τ, define hn+1 = \uf8f1 \uf8f2 \uf8f3 hn − τ if ϕ(hn − τ) < ψ(hn − τ), inf � t ≥ t1 − τ : ϕ(h) ≥ ψ(h) for all h ∈ [t, hn − τ] � otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Observe that the sequence ends when t1 > hN ≥ t1 − τ satisfying the statement of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Moreover, note that as consequence of the definition of the decreasing sequence {hn}1≤n≤N and the Step 1 we obtain the following inequalities � hn hn−τ f(h)ϕ(h) dh ≥ � hn hn−τ g(h)ψ(h) dh, for all 1 ≤ n ≤ N − 1, � I f(h)ϕ(h) dh ≥ � I g(h)ψ(h) dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2) Whence we obtain � t2−τ hN f(h)ϕ(h) dh ≥ � t2−τ hN g(h)ψ(h) dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Finally, observe that � t2−τ t1−τ f(h)ϕ(h) dh ≥ � t2−τ hN f(h)ϕ(h) dh ≥ � t2−τ hN−τ g(h)ψ(h) dh = � t2−τ t1−τ g(h)ψ(h) dh − � hN −τ t1−τ g(h)ψ(h) dh ≥ � t2−τ t1−τ g(h)ψ(h) dh − τM, and the lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' □ The following lemma is fundamental to the proof of the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Applied to a periodic solution of the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1), it states that (x + y)(t − τ) diluted over τ units of times is exactly x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Every solution (s, x)(t) of the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) with not null initial con- dition, and the corresponding function y(t) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) satisfy x(t) − (x + y)(t − τ)e −� t t−τ D(r)dr = � x(τ)e � τ 0 D(r) dr − (x + y)(0) � e−� t 0 D(r)dr for all t ≥ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In the particular case where (s, x)(t) is an ω-periodic (non-trivial) solution it follows that x(t) = (x + y)(t − τ)e −� t t−τ D(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Firstly, note that for a given solution (s, x)(t) with not null initial condition it follows that d dt(x + y)(t − τ) = −D(t − τ)(x + y)(t − τ) + x(t − τ)p(s(t − τ)) 16 UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE and consequently d dt � (x + y)(t − τ)e −� t t−τ D(r) dr� = x(t − τ)p(s(t − τ))e −� t t−τ D(r) dr − D(t)(x + y)(t − τ)e −� t t−τ D(r) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Whence we deduce that d dt � x(t) − (x + y)(t − τ)e −� t t−τ D(r) dr� = − D(t) � x(t) − (x + y)(t − τ)e −� t t−τ D(r) dr� which provides the first part of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Note that ����x(t) − (x + y)(t − τ)e −� t t−τ D(r) dr ���� → 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3) as t tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' In particular, if (s, x)(t) is a non-trivial periodic solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1), the right side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3) is also an ω-periodic function and converges to zero as t tends to infinity, therefore it is identically zero, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Assume that System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is not persistent, given a solution (s, x)(t) we will prove that x(t) towards to zero as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' If limt→∞ x(t) = 0 there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Otherwise there exists ε > 0 such that lim sup t→∞ x(t) > ε (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4) and we look for a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' By using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1, it follows that |(z∗ − s − x − y)(t)| → 0, t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Moreover, since the functions z∗(t), s(t), x(t) and y(t) are positive and |(z∗ − s − x − y)(t)| < ε from a certain time, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4) we conclude that there exists t0 ≥ 0 such that z∗(t0) ≥ s(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Furthermore, z∗(t) − s(t) satisfy d dt(z∗ − s)(t) = −D(t)(z∗ − s)(t) + x(t)p(s(t)) ≥ −D(t)(z∗ − s)(t) thus, from the above differential inequality, we see that z∗(t) ≥ s(t) for t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Note that, by assumption that System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) is not persistent, one of the following identities holds ⟨D⟩ > ⟨p(z∗)ϕ⟩, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='5) or ⟨D⟩ = ⟨p(z∗)ϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='6) The rest of the proof fall naturally into two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE 17 Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Assume that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Consider again the functions f(t) = p(z∗(t)) and g(t) = p(s(t)), then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 we obtain for t ≥ t0 x(t) = x(t0)e � t t0 (p(s(r−τ))ψ(r−τ)−D(r))dr = x(t0)e � t t0 (p(s(r−τ))ψ(r−τ)−p(z∗(r−τ)ϕ(r−τ)) dr+� t t0 (p(z∗(r−τ))ϕ(r−τ)−D(r))dr ≤ x(t0)eτMe −� t t0 (D(r)−p(z∗(r−τ))ϕ(r−τ))dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='5) it may be conclude that x(t) → 0 as t → ∞ and we get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Assume now that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='6) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We claim that � t t0 (p(z∗(r − τ))ϕ(r − τ) − p(s(r − τ))ψ(r − τ)) dr → ∞ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='7) as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Indeed, otherwise we could find K > 0 such that: for all t > t0 there exists t1 > t such that � t1−τ t0−τ (p(z∗(r))ϕ(r) − p(s(r))ψ(r)) dh < K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Now combining the above inequality together with the Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 we have that for all t > t0 it follows � t−τ t0−τ (p(z∗(r))ϕ(r) − p(s(r))ψ(r)) dr = � t1−τ t0−τ (p(z∗(r))ϕ(r) − p(s(r))ψ(r)) dr − � t1−τ t−τ (p(z∗(r))ϕ(r) − p(s(r))ψ(r)) dr ≤ K + τM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Consequently, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4) we obtain for all t ≥ t0 ln(x(t)) = ln(x(t0)) + � t t0 (p(s(r − τ))ψ(r − τ) − D(r)) dr = ln(x(t0)) + � t t0 (p(s(r − τ))p(ψ(r − τ) − p(z∗(r − τ))ϕ(r − τ)) dr + � t t0 (p(z∗(r − τ))ϕ(r − τ) − D(r)) dr ≥ ln(x(t0)) − K − τM + min l∈[0,ω] � l 0 (p(z∗(r − τ))ϕ(r − τ) − D(r)) dr, which implies that ln(x(t)) is bounded from below for all t ≥ t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=', the solution is persistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' This contradicts our assumption and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='7) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Now, by using again (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4) for t ≥ t0, we obtain x(t) = x(t0)e � t t0 p(s(r−τ))ψ(r−τ)−D(r)dr = x(t0)e � t t0 p(s(r−τ))ψ(r−τ)−p(z∗(r−τ)ϕ(r) dr+� t t0 p(z∗(r−τ))ϕ(r−τ)−D(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Finally, notice that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='6) implies that for all t ≥ t0 we have � t t0 p(z∗(r − τ))ϕ(r − τ) − D(r) dr ≤ max l∈[0,ω] � l 0 p(z∗(r − τ))ϕ(r − τ) − D(r) dr, 18 UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE consequently we obtain x(t) ≤ x(t0)Me −� t t0 p(z∗(r−τ))ϕ(r−τ)−p(s(r−τ)ψ(r−τ)dr, where M := maxl∈[0,ω] � e � l 0 p(z∗(r−τ))ϕ(r−τ)−D(r)dr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='7) we conclude that x(t) → 0 as t → ∞, which contradicts (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4) and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The proof will be divided into two steps, we recall that C := C([−τ, 0] → R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Let δ > 0 the value given by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 and consider T in = T in(R, α) with R = 3 s and α = e−Dτδ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We now claim that any solution of System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) with initial condition ∥(sin, xin)∥ ≤ 3 s satisfies ∥(s, x)t∥ ≤ R0 for all t ≥ 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='8) where R0 = z∗(0) + 6 s + 3 sp (3 s) τ + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' To show this, first note that, by using the arguments to obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='9) in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1, it follows that |y(0)| ≤ ∥(sin, xin)∥p(∥(sin, xin)∥)τ ≤ 3 sp(3 s)τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' By definition of R0 and applying the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 again, we obtain that |(s, x)(t)| ≤ (s + x)(t) ≤ (s + x + y)(t) ≤ |(z∗ − s − x − y)(t)| + |z∗(t)| ≤ |(z∗ − s − x − y)(0)|e−� t 0 D + |z∗(t)| ≤ max{z∗(0), (s + x + y)(0)} + s ≤ z∗(0) + 6 s + 3 sp (3 s) τ + s = R0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='9) for all t ≥ −τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Let us define S = � φ ∈ C : |φ(h) − φ(r)| ≤ √ 2R0 � D + p(R0) � |h − r| for all h, r ∈ [−τ, 0] � S0 = {φ ∈ S : ∥φ∥ ≤ 2 s, |φ2(0)| ≥ δ} , S1 = {φ ∈ S : ∥φ∥ < 3 s, |φ2(0)| > δ/2} , S2 = � φ ∈ S : ∥φ∥ ≤ R0, |φ2(0)| ≥ e−DT inδ/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Note that S0 ⊂ S1 ⊂ S2 are convex subset of C, S0 and S2 are compacts as consequence of Arzelà–Ascoli Theorem and S1 = S2 ∩ {φ ∈ C : ∥φ∥ < 3 s, |φ2(0)| > δ/2} consequently, S1 is open relative to S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE 19 Next we define the Poincaré operator given by P : S2 → C φ �→ (s, x)(· + ω, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' where (s, x)(t, φ) is the solution of System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1) at time t, with initial condition φ (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Notice that for a given k ∈ N ∪ {0}, P k(φ) is the solution (s, x)(t, φ) over the time interval [kω − τ, kω] (including when kω − τ < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' To use Horn’s fixed point Theorem we need to prove that P k(S1) ⊆ S2 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Let φ ∈ S1 and (s, x) the solution corresponding to the initial condition φ, it suffices to show that (s, x)t ∈ S2 for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' From Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='8) we have |(s, x)t| ≤ R0 while, for t ≤ T in inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='12) implies |x(t)| ≥ e−Dt|x(0)| ≥ e−DT inδ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Hence x(τ) ≥ e−Dτδ/2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='10) thus, by using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1, if t ≥ T in, then |x(t)| ≥ δ ≥ e−DT inδ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Now, it turns out that |(s, x)(h) − (s, x)(r)| = � (s(h) − s(r))2 + (x(h) − x(r))2 = � s ′2(ξs)(h − r)2 + x ′2(ξx)(h − r)2 = |h − r| �� D(ξs)(s(0)(ξs) − s(ξs)) − x(ξs)p(s(ξs)) �2 + � −D(ξx)x(ξx) + x(ξx − τ)p(s(ξx − τ))e −� ξx ξx−τ D(r) dr�2�1/2 ≤ |h − r| �� DR0 + R0p(R0) �2 + � DR0 + R0p(R0) �2�1/2 = |h − r| √ 2R0 � D + p(R0) � for all h, r ≥ 0 and ξs, ξx between h and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Therefore (s, x) ��� [kω−τ,kω] = P k(φ) ∈ S2, for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Next, we need to show that there exists m ∈ N such that if k ≥ m then P k(φ) ∈ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Let T ≥ T in such that R0e−� T 0 D(r) dr ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' If t ≥ T then, by using again Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 and considering (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='10), it follows that |x(t)| ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Estimating |(s, x)(t)| one more time as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='9) we have |(s, x)(t)| ≤ R0e−� t 0 D(r) dr + s ≤ 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' As before, it follows that |(s, x)(h) − (s, x)(r)| ≤ |h − r| √ 2R0 � D + p(R0) � for all h, r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Thus, if kω ≥ mω ≥ T we obtain (s, x) ��� [kω−τ,kω] = P k(φ) ∈ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 20 UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE Finally, by Horn’s fixed point Theorem [13, Theorem 6], there exist a fixed point of P whose second coordinate is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Consider a solution (s1, x1) with no null initial conditions and (s2, x2) a periodic non-trivial solution, whose existence was proved in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The proof will be divided into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Note that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='1 yields to z∗(t−τ) = (s2 +x2 +y2)(t−τ) and then x2(t) + (s2 − z∗)(t − τ)x2(t) (x2 + y2)(t − τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' On another hand, and also using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='2, we get that x1(t) + (s1 − z∗)(t − τ)x2(t) (x2 + y2)(t − τ) = x1(t) + (s1 − z∗)(t − τ)e −� t t−τ D(r) dr = x1(t) − (x1 + y1)(t − τ)e −� t t−τ D(r) dr + (s1 + x1 + y1 − z∗)(t − τ)e −� t t−τ D(r) dr = � x1(τ)e � τ 0 D(r) dr − (x1 + y1)(0) � e−� t 0 D(r)dr + (s1 + x1 + y1 − z∗)(0)e−� t 0 D(r) dr = −C0e−� t 0 D(r) dr, where C0 = (z∗ − s1)(0) − x(τ)e � τ 0 D(r) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Then we obtain (x1 − x2)(t) = −(s1 − s2)(t − τ) x2(t) (x2 + y2)(t − τ) + �(s1 − z∗)(t − τ)x2(t) (x2 + y2)(t − τ) + x1(t) −(s2 − z∗)(t − τ)x2(t) (x2 + y2)(t − τ) − x2(t) � = −(s1 − s2)(t − τ) x2(t) (x2 + y2)(t − τ) − C0e−� t 0 D(r) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='11) UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE 21 Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' d dt(s1 − s2)(t) = −D(t)(s1 − s2)(t) − x1(t)p(s1(t)) + x2(t)p(s2(t)) = −D(t)(s1 − s2)(t) − x1(t)(p(s1(t)) − p(s2(t))) − (x1 − x2)(t)p(s2(t)) = −D(t)(s1 − s2)(t) − x1(t)p′(ξ)(s1 − s2)(t) − (x1 − x2)(t)p(s2(t)) = −(s1 − s2)(t)(D(t) + x1(t)p′(ξ)) + (s1 − s2)(t − τ) x2(t)p(s2(t)) (x2 + y2)(t − τ) + C0p(s2(t))e−� t 0 D(r) dr where ξ = ξ(t) ∈ (s1(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' s2(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' but for simplicity we do not write the dependence on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Let us define m = min t≥τ {x1(t)p′(ξ(t))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Since the system is persistent, p′ is positive, and s1, s2 are bounded by above, we deduce that m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Furthermore consider the function J(ε) = −m + max h∈[0,ω] �x2(h)p(s2(h)) (x2 + y2)(h) � (eτε − 1) + 3ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Note that J(0) < 0, hence we can fix a positive ε such that ε < ⟨D⟩/2 and J(ε) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Also notice that there exists t0 ≥ ω such that C0 max h∈[0,ω] {p(s2(h))} e−� t0−ω 0 D(r) dr ≤ min h∈[0,ω] �ε (x2 + y2)(h) 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='12) With all the previous considerations, define w(t) = max h∈[t0−τ,t0] � (s1 − s2)(h) (x2 + y2)(h), 1 � (x2 + y2)(t)e−(t−t0)ε, t ≥ t0 − τ, and S = {t ≥ t0 : (s1 − s2)(h) ≤ w(h) for all h ∈ [t0, t]} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Observe that S is a non-empty set since t0 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' We claim that T ∗ = sup S is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' On the contrary, suppose that T ∗ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Consequently, by the mean value Theorem there is t∗ ∈ [T ∗, T ∗ + τ) such that d dt(s1 − s2)(t∗) ≥ d dtw(t∗), (s1 − s2)(t∗) > w(t∗) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='13) Observe that w(t − τ) = max h∈[t0−τ,t0] � (s1 − s2)(h) (x2 + y2)(h), 1 � (x2 + y2)(t − τ)(x2 + y2)(t) (x2 + y2)(t)e−(t−t0)εeτε = w(t)(x2 + y2)(t − τ) (x2 + y2)(t) eτε, 22 UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE and that d dtw(t) = max h∈[t0−τ,t0] � (s1 − s2)(h) (x2 + y2)(h), 1 � e−(t−t0)ε (−D(t)(x2 + y2)(t) − x2(t)p(s2(t)) − ε(x2 + y2)(t)) = w(t) � −D(t) − x2(t) (x2 + y2)(t)p(s2(t)) − ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Furthermore, using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='12), for t ≥ t0, we have that C0p(s2(t))e−� t 0 D(r) dr ≤ min h∈[0,ω] �ε (x2 + y2)(h) 2 � e −� t t0−ω D(r) dr ≤ ε 2(x2 + y2)(t)e−(t−t0)ε ≤ ε 2w(t), obtained using the inequality e (t−t0)ε−� t t0−ω D(r) dr ≤ 1, which holds since ε ≤ ⟨D⟩/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Now, taking the derivative of s1−s2 from the previous step and using the above estimations we have d dt(s1 − s2)(t∗) ≤ −(s1 − s2)(t∗)(D(t∗) + m) + (s1 − s2)(t∗ − τ) x2(t∗)p(s2)(t∗) (x2 + y2)(t∗ − τ) + C0p(s2(t∗))e−� t∗ 0 D(r) dr < −w(t∗)(D(t∗) + m) + w(t∗ − τ) x2(t∗)p(s2)(t∗) (x2 + y2)(t∗ − τ) + ε 2w(t∗) = w(t∗) � −D(t∗) − m + x2(t∗)p(s2)(t∗) (x2 + y2)(t∗) eτε + ε 2 � < w(t∗) � −D(t∗) + x2(t∗)p(s2)(t∗) (x2 + y2)(t∗) − ε � = d dtw(t∗) where we use the definition of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' This contradicts (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='13) and, therefore, sup S = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Following a similar reasoning, this time for s2 − s1, we obtain d dt(s2 − s1)(t) ≤ −(s2 − s1)(t)(D(t) + x1(t)p′(ξ)) + (s2 − s1)(t − τ) x2(t)p(s2(t)) (x2 + y2)(t − τ) + (−C0) p(s2(t))e−� t 0 D(r) dr and we reach the desired result due to |(s1 − s2)(t)| ≤ max h∈[t0−τ,t0] � (s1 − s2)(h) (x2 + y2)(h), 1 � (x2 + y2)(t)e−(t−t0)ε which tends to zero as t tends to infinity (note that we do not know the sign of C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' UNIFORM PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE 23 It remains to prove that x1(t) tends to x2(t) when t goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' By Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='11) we obtain |(x1 − x2)(t)| = ����(s1 − s2)(t − τ)e −� t t−τ D(r) dr + C0e−� t 0 D(r) dr ���� ≤ |(s1 − s2)(t − τ)| + |C0|e−� t 0 D(r) dr and we conclude that |(x1 − x2)(t)| tends to zero as t tends to infinity with expo- nential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Finally, for the uniqueness of the periodic solution, it suffices to note that the distance between two periodic solutions can only tend to zero if it is zero at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' □ Acknowledgments This research is partially supported by PROGRAMA REGIONAL MATH-AMSUD MATH2020006.' metadata={'source': 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PERSISTENCE AND COMPLETE ANALYSIS OF THE PERIODIC CASE [16] Zhidong Teng, Linfei Nie, and Xining Fang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' The periodic solutions for general periodic impul- sive population systems of functional differential equations and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' Computers & Mathematics with Applications, 61(9):2690–2703, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content=' 13 Email address: mrodriguezcartabia@dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='uba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='ar, daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='sepulveda@utem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} +page_content='cl' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFAT4oBgHgl3EQfaB0B/content/2301.08548v1.pdf'} diff --git a/l9AyT4oBgHgl3EQf_vqB/content/tmp_files/2301.00914v1.pdf.txt b/l9AyT4oBgHgl3EQf_vqB/content/tmp_files/2301.00914v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e118659d9c96c3e4d7c3528f557e40ee18c5cda1 --- /dev/null +++ b/l9AyT4oBgHgl3EQf_vqB/content/tmp_files/2301.00914v1.pdf.txt @@ -0,0 +1,859 @@ +Citation: Wada, T.; Scarfone, A.M. +On the Kaniadakis distributions. +Preprints 2022, 1, 0. https://doi.org/ +Publisher’s Note: MDPI stays neutral +with regard to jurisdictional claims in +published maps and institutional affil- +iations. +Copyright: +© 2022 by the authors. +Licensee MDPI, Basel, Switzerland. +This article is an open access article +distributed +under +the +terms +and +conditions of the Creative Commons +Attribution (CC BY) license (https:// +creativecommons.org/licenses/by/ +4.0/). +Article +On the Kaniadakis distributions applied in statistical physics and +natural sciences +Tatsuaki Wada † +, and Antonio. M. Scarfone ‡ +† +Region of Electrical and Electronic Systems Engineering, Ibaraki University, Nakanarusawa-cho, Hitachi-shi, +Ibaraki, 316-8511, Japan; tatsuaki.wada.to@vc.ibaraki.ac.jp +‡ +Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o Politecnico di Torino, Corso +Duca degli Abruzzi, 24, 10129, Torino, Italy ; antonio.scarfone@to.it +* +Correspondence: tatsuaki.wada.to@vc.ibaraki.ac.jp +Abstract: Constitutive relations are fundamental and essential to characterize physical systems. By +utilizing the κ-deformed functions, some constitutive relations are generalized. We here show some +applications of the Kaniadakis distributions based on the inverse hyperbolic sine function to some topics +belonging to the realm of statistical physics and natural science. +Keywords: κ-deformed functions; constitutive relations; Gompertz rule; generalized Lotka-Voltela +equations; contact density dynamics +1. Introduction +The κ-exponential function [1–3] is defined by +expκ(x) ∶= (κx + +√ +1 + κ2x2) +1 +κ = exp[1 +κ arsinh(κx)], +(1) +for a real deformation parameter κ. The inverse function, i.e., κ-deformed logarithmic function, +is defined by +lnκ x ∶= xκ − x−κ +2κ += 1 +κ sinh[κ ln x]. +(2) +Both κ-deformed functions are important ingredients of the generalized statistical physics based +on κ-entropy [1–3]. It influences a wide range of scientific fields, and based on the κ-deformed +functions (Appendix A), several basic fields have been developed over two decades. Prof. +Kaniadakis [4] provided the theoretical foundations and mathematical formalism generated +by the κ-deformed functions, and some references including many fields of applications. +Recently, the usefulness of the κ-statistics was demonstrated for the analysis [5] of epidemics +and pandemics. +Constitutive relations are fundamental and essential to characterize physical systems. They +are combined with the other equations of physical laws in order to solve physical problems. +Some well-known examples of linear constitutive relations are: Hooke’s law F = ksx for the +tensile or compressive force F of a spring with a spring constant ks against the change in its +length x; Ohm’s law V = RI for the voltage V of a electrical conductor with resistance R under +an electric current I, and so on. However, as a real spring deviates from Hooke’s law, we +know that any linear constitutive relation describes an idealized situation, and it is merely a +linearized- and/or approximated- relation to describe some real physical properties. Hence, in +general, non-linearity plays a crucial role to describe more realistic physical systems. +arXiv:2301.00914v1 [cond-mat.stat-mech] 3 Jan 2023 + +BY2 of 13 +The κ-exponential function (1) can be regarded as a useful tool (or device) to make such +non-linear constitutive relations for a better description of real physical systems. For example, +consider the following κ-deformation of Hooke’s law: +Fκ ∶= ks ln[expκ(x)] = ks +κ ln(κx + +√ +1 + κ2x2), +(3) +which reduces to the original Hooke’s law F = ksx in the limit of κ → 0. For any linear +constitutive relation, we can apply this type of the κ-deformation. For example, Ohm’s law +can be cast into the following form: V = RI = R ln[exp(I)]. By changing the exponential +function with the κ-exponential function, we obtain the κ-deformed version of Ohm’s law: +Vκ = R ln[expκ(I)]. In this research, we focus on this type of the κ-deformation of a physical +quantity (say A), i.e, +A +⇒ +ln[expκ(A)] = 1 +κ arsinh(κA). +(4) +Throughout in this paper, we call this κ-deformation as the arsinh-type deformation of a physical +quantity A. The other type of the κ-deformation can be +A +⇒ +lnκ[exp(A)] = 1 +κ sinh(κA), +(5) +which is called here the sinh-type deformation. In Ref. [6], the thermodynamic stability of the +κ-generalization SB +κ of Boltzmann entropy SB is studied. The κ-generalization SB +κ is rewritten in +the form: +SB +κ ∶= kB lnκ W = kB lnκ[exp(lnW)] = kB lnκ[exp(SB)], +(6) +which can be regarded as the sinh-type deformation of Boltzmann entropy SB. Recently, in +cosmology, Lymperis et al [7] modified Bekenstein-Hawking entryopy SBH as follows: +1 +κ sinh(κSBH), +(7) +which is obviously the sinh-type deformation of SBH. +In this paper we consider the arsinh-type deformations against some constitutive relations +in the field of statistical physics and natural sciences In our previous work [8] we studied a +thermal particle under a velocity-dependent potential which can be regarded as a deformation +of Rayleigh’s dissipation function [9] and showed that the probability distribution function +(pdf) for the stationary-state of this thermal particle is a κ-deformed Gaussian pdf. It was +considered that the canonical pdf ρ(v), in the velocity space, of a thermal particle with unit +mass (m = 1) in the κ-deformed confining potential Uκβ(v): +Uκβ(v) ∶= 1 +κβ arsinh(κβv2 +2 ), +(8) +where β ∶= 1/kBT is a coldness (or inverse temperature). This κ-deformed potential Uκβ(v) is +rewritten, in the momentum-space, as + +3 of 13 +Uκβ(p) = 1 +κβ arsinh(κβ p2 +2 ) = 1 +β ln[expκ(β p2 +2 )], +(9) +which is the arsinh type deformation of the quantity βp2/2 (the ratio of the kinetic energy to the +mean thermal energy kBT = 1/β). In other words, we can consider the following κ-deformation +Qκ(U) of the Boltzmann factor exp(−βU) for an equilibrium state with the energy U: +Qκ(U) ∶= expκ(−βU) = exp[1 +κ arsinh(−κβU)]. +(10) +One may wonder why the inverse hyperbolic sine function (arsinh) plays a role. In many +different fields of sciences, there is no doubt that the exponential and logarithmic functions +are important and fundamental. Since the inverse hyperbolic sine function and logarithmic +function are mutually related as: +arsinh x = ln[x + +√ +1 + x2], +ln x = arsinh[1 +2(x − 1 +x)], +(11) +for a positive real x, we think both functions are important. By using the second relation, for +any real parameter κ ≠ 0, we have +ln x = 1 +κ ln xκ = 1 +κ arsinh[1 +2(xκ − x−κ)] = 1 +κ arsinh[κ lnκ x]. +(12) +Note that this relation corresponds to the arsinh-type deformation of lnκ x and is equivalent +to definition (2) of the κ-deformed logarithmic function that can be regarded as the sinh-type +of κ-deformation of ln x. Prof. Kaniadakis already discussed this issue in section II of Ref. [2] +from the view point of the deformed algebra. +On the other hand, Prof. Pistone [10] was the first one to study the κ-exponential model in +the field of information geometry [11], and later through our reserach activities [8,12,13], we +realize that there exist some relations among statistical physics, thermodynamics, mathematical +biology, and information geometry. Harper [14] pointed out that the replicator equation (RE) +[15] in mathematical biology or in an evolutional game theory is related with information +geometry and the generalized Lotka-Volterra (LV) equation as briefly explained in Appendix B. +The generalized LV equations: +dyi +dt = yi fi(y), +(13) +are used to model the competition dynamics of the populations y1, y2, . . . , yn of n biological +species. Gompertz function [16] is a type of mathematical model for a time evolution. Histori- +cally he studied human mortality for working out a series of tables mortality, and he proposed +his law of human mortality in which he assumed that person’s resistance to death decreases as +his years increase. His law is now called Gompertz rule (or law) and we’d like to point out +the relation of his function and his rule to some important quantities concerning on statistical +physics. +The rest of the paper is organized as follows. In section 2, we briefly explain Gompertz +function, and the generalized LV equations, which are important in mathematical biology (or + +4 of 13 +evolutional game theory). Their relations to thermal physics is pointed out. Section 3 considers +the canonical thermal density matrix, which is characterized by the so-called Bloch equation +[18], and we shall show that Bloch equation can be regarded as a Gompertz rule after the +parameter transformation β to t = −ln β. In section 4, we discuss the arsinh-type deformation +from the view point of the κ-addition. In section 5, we study the numerical simulations of the +thermostat algorithm for the Hamiltonian with the κ-deformed kinetic energy, which can be +regarded as the arsinh type of the κ-deformation of the ratio βp2/2 as shown in (10). The final +section is devoted to our conclusions. +2. Gompertz functions and Gompertz rule +Here we e’d like to point out that there exist relations between an evolutional game +dynamics and thermal physics. In evolutional game theory, an evolutional game dynamics is +described by RE. The generalized LV equations are related with REs as shown in Appendix +B. On the other hand, Gompertz function is a mathematical model describing an evolutional +curve. Gompertz function (or Gompertz curve) [16] is a type of mathematical model for a time +series. Gompertz function fG(t) is a sigmoid function and is given by +fG(t) ∶= K exp[C exp(−t)], +(14) +where C and K are positive constants. A distinctive feature of Gompertz function is its double +exponential t-dependency. His function is nowadays used in many different areas to model a +time evolution of the populations where growth is slowest at the start and end of a period. For +example, Ref. [17] applied Gompertz model to describe the growth dynamics of COVID-19 +pandemic. Gompertz [16] studied human mortality for working out a series of tables mortality, +and this suggested to him his law of human mortality in which he assumed that person’s +resistance to death decreases as his years increase. The rule of his model is called Gompertz rule +which states that +d +dt fG(t) = −fG(t)ln fG(t) +K +. +(15) +The solution of the Gompertz rule is fG(t), if we set K = limt→∞ fG(t) and C = ln(fG(0)/K). +If we choose fi(y(t)) = −ln yi(t) and assuming limt→∞ yi(t) = 1, the generalized LV +equation (13) becomes +dyi(t) +dt += −yi(t) ln yi(t), +(16) +which can be regarded as the Gompertz rule (15) with K = 1 for each yi(t). Consequently its +solution yi(t) is the Gompertz function: +yi(t) = exp[ln yi(0) exp(−t)], +(17) +Now, by changing the parameter t to β = exp(−t), we have dβ = −βdt so that the limit t → 0 +corresponds to β → 1, and we introduce each constant Ei as +−Ei = lim +t→0 ln yi(t) = lim +β→1ln yi(β), +(18) + +5 of 13 +where yi(β) is the shorthand notation of yi(t(β)) with t(β) = −ln β. Then, the solution yi(β) in +(17) can be expressed as a quantity very familiar in statistical physics: +yi(β) = exp(−βEi), +(19) +that is Boltzmann factor. The corresponding Gompertz rule (15) for yi(β) is equivalent to +d +dβyi(β) = −Ei yi(β). +(20) +Having described the relation between Gompertz rule and Boltzmann factor exp(−βEi) in +statistical physics, in next section we discuss a κ-deformation of Bloch equation for thermal +states. +3. Bloch equation for thermal state +For a given Hamiltonian ˆH and the corresponding eigenvalues Ei and eigenstate ∣ψi⟩ +which are related with +ˆH∣ψi⟩ = Ei∣ψi⟩, +(21) +and assuming the completeness ∑i ∣ψi⟩⟨ψi∣ = ˆ1, the canonical density matrix (or density operator +for a canonical ensemble) is constructed as +ˆρ(β) ∶= ∑ +i +exp(−βEi)∣ψi⟩⟨ψi∣ = exp(−β ˆH). +(22) +In order to determine the canonical density matrix, we have to solve the eigenequations (21) +and to sum over all the states. This needs heavy calculations in general. +Bloch equation [18,19] for canonical density matrix is known as +− ∂ +∂β ˆρ(β) = ˆH ˆρ(β), +(23) +which can be regarded as the diffusion equation in imaginary time β, and it has a similar form +as Schrödinger equation and diffusion equation. Bloch equation (23) offers an alternative route +to determine the density matrix ˆρ(β). The initial (β = 0) condition is provided if we know the +eigenstates in the high-temperature limit. +Now, by multiplying β to both sides of (23), we have +−β ∂ +∂β ˆρ(β) = β ˆH ˆρ(β) = −ln[ ˆρ(β)] ˆρ(β). +(24) +Changing the parameter β to t = −ln β, it follows +d +dt ˆρ(t) = −β d +dβ ˆρ(β) = −ln[ ˆρ(t)] ˆρ(t). +(25) + +6 of 13 +This is the same form of the Gompertz rule (15). In this way, Bloch equation can be considered +as a sort of Gompertz rule. +Next, let us consider the κ-deformed canonical density matrix: +ˆρκ(β) ∶= ∑ +i +expκ(−βEi)∣ψi⟩⟨ψi∣ = expκ(−β ˆH). +(26) +This leads to the following κ-deformation of Bloch equation: +− ∂ +∂β ˆρκ(β) = ∑ +i +Ei +expκ(−βEi) +uκ[(expκ(−βEi)] ∣ψi⟩⟨ψi∣ = +ˆH +uκ[expκ(−β ˆH)] ˆρκ(β). +(27) +Again by changing the parameter β to t = −ln β and using the relation (A3), we have +d +dt ˆρκ(t) = −lnκ[ ˆρκ(t)] +uκ[ ˆρκ(t)] ˆρκ(t), +(28) +which can be regarded as a κ-deformation of the Gompertz rule. +Differentiating (27) again with respect to β, we obtain the following nonlinear differential +equation: +(1 + κ2β2 ˆH2)∂2 ˆρκ(β) +∂β2 ++ κ2β ˆH2 ∂ ˆρκ(β) +∂β +− ˆH2 ˆρκ(β) = 0. +(29) +This differential equation remind us the research work [20] of the quantum free particle on +two-dimensional hyperbolic plane. The relevant two-dimensional Schrödinger equation is +separable in the κ-dependent coordinate system (zx, y) with zx ∶= x/ +√ +1 + κ2y2. The Schrödinger +equation ˆH1Ψ = e1Ψ for the first partial Hamiltonian ˆH1 leads to the following differential +equation with the variable zx alone: +(1 + κ2z2 +x)d2Ψ(zx) +dz2x ++ κ2zx +dΨ(zx) +dzx ++ µΨ(zx) = 0, +µ ∶= 2m +¯h2 e1. +(30) +In the limit of κ → 0, this differential equation reduces to the standard time-independent +Schrödinger equation: d2Ψ(x)/dx2 + µΨ(x) = 0. Cariñena et al. [20] obtain the solution of the +differential equation (30) as the κ-deformed plane wave: +Ψ(zx) = exp[±i µ +κ arsinh(κ zx)], +(31) +which is regarded as a arsinh-type deformation. +4. The κ-addition and the law of large number +Next, we consider the κ-addition from the view point of the law of large numbers (LLN), +which plays a central role in probability, statistics, and statistical physics. The κ-addition [4] is +defined by +x +κ⊕ y ∶= x +√ +1 + κ2y2 + y +√ +1 + κ2x2. +(32) + +7 of 13 +This deformation of additive rule comes from the addition rule of the inverse hyperbolic sine +function as follows. For a, b ∈ R, the addition rule is written as +arsinh(a) + arsinh(b) = arsinh(a +√ +1 + b2 + b +√ +1 + a2). +(33) +By setting a = κx and b = κy, we obtain +arsinh(κx) + arsinh(κy) = arsinh(κx +√ +1 + κ2y2 + κy +√ +1 + κ2x2) += arsinh[κ(x +κ⊕ y)]. +(34) +This relation is equivalent to the definition (32). The additive relation (34) is readily generalized +to +n +∑ +i=1 +arsinh(κxi) = arsinh[κ(x1 +κ⊕ x2 +κ⊕ ⋯ +κ⊕ xn)]. +(35) +By applying this relation to the Boltzmann factor exp[−β ∑n +i=1 Kκβ(pi)] with respect to the +κ-deformed kinetic energy [8] with m = 1: +n +∑ +i=1 +Kκβ(pi) ∶= +n +∑ +i=1 +1 +κβ arsinh(κβ p2 +i +2 ), +(36) +we have +exp[−β +n +∑ +i=1 +Kκβ(pi)] = exp[−1 +κ arsinh{κ(β p2 +1 +2 +κ⊕ β p2 +2 +2 +κ⊕ . . . +κ⊕ β p2 +n +2 )}] += expκ[(−β p2 +1 +2 ) +κ⊕ (−β p2 +2 +2 ) +κ⊕ . . . +κ⊕ (−β p2 +n +2 )] += expκ[−β p2 +1 +2 ]expκ[−β p2 +2 +2 ] . . . expκ[−β p2 +n +2 ] = +n +∏ +i=1 +expκ[−β p2 +i +2 ]. +(37) +Note that the κ-exponential of the κ-summation of each term −β p2 +i +2 in the second line is ex- +pressed as a factorized form in the last line. +It is well known that LLN plays a fundamental role in statistical physics. Łapi´nski [21] +showed that the standard LLN yields the most probable state of the system, which equals +to the point of maximum of the entropy and this point can be either Maxwell-Boltzmann +statistics or Bose-Einstein statistics, or Zipf-Mandelbort law. McKeague [22] studied the central +limit theorem under special relativity based on the κ-additivity. Scarfone [23] studied the +κ-deformation of Fourier Transform and discussed the limiting distribution of the κ-sum of +statistically independent variables. The κ-additivity extension [22] of the strong LLN is shown +and it states that if Xi are iid with finite mean, +X1 +n +κ⊕ X2 +n +κ⊕ . . . +κ⊕ Xn +n +→ +1 +κ arsinh[κ⟨X⟩]a.s., +(38) + +8 of 13 +where a.s. stands for almost surely, i.e., the above sequence of the random variables Xi +converges almost surely, and ⟨X⟩ is the standard average of the random variable X. Of course, +in the limit of κ → 0, the relation (38) reduces to the standard strong LLN. In this way, the +κ-additivity extension of the strong LLN suports the arsinh-type deformation of the average of +a stochastic variable X. Note that the converged value in (38) is the arsinh-type deformation of +the average ⟨X⟩. +5. Contact density dynamics +Nosé-Hoover (NH) thermostat [24] is a famous deterministic algorithm for constant- +temperature molecular dynamics simulations. Based on the idea of NH thermostat, several +improved versions are proposed. Among them, contact density dynamics (CDD) [25] is +an algorithm based on contact Hamiltonian systems and generates any prescribed target +distribution in physical phase space. The dynamical equations of CDD are followings. +˙qi = ∂h(q, p, S) +∂pi +, +(39) +˙pi = −∂h(q, p, S) +∂qi ++ ∂h(p, q, S) +∂S +pi, +(40) +˙S = −pi +∂h(q, p, S) +∂pi ++ h(q, p, S), +(41) +where S is the thermostatting variable, qi and pi are i-th component (i = 1,2,⋯, n) of n- +dimensional vectors, respectively. Here h(q, p, S) denotes the contact Hamiltonian which +is formed as +h(q, p, S) = (ρt(q, p)f(S))− +1 +n+1 , +(42) +with a target distribution ρt(q, p) on 2n-dimensional Γ-space and a normalized distribution +f(S) for the thermostatting variable S. As in the case of Ref. [24], we also choose f(S) as the +logistic distribution with scale 1 and mean c = 0.0: +f(S) = +exp(S − c) +(1 + exp(S − c))2 . +(43) +Utilizing this CDD algorithm, the κ-deformed exponential distributions are simulated. +The target distribution ρt(q, p) is the one-dimensional (n = 1) κ-deformed Gaussian function: +ρt(q, p) = +1 +Zκ(β) exp[−βHκ(q, p)] = +1 +Zκ(β) exp[−1 +κ arsinh(κβ p2 +2 )]exp[−βq2 +2 ], +(44) +where the associated Hamiltonian is +Hκ(q, p) = 1 +κβ arsinh(κβ p2 +2 ) + q2 +2 , +(45) +and the normalization factor Zκ(β) [4] is + +9 of 13 +Zκ(β) = π +β +√ +2 +κ Γ( 1 +2κ − 1 +4) +( κ +2 + 1)Γ( 1 +4 + 1 +2κ) +. +(46) +From (39) we have +v(p) ∶= dq +dt = ∂Hκ(q, p) +∂p += +p +uκ[expκ(−β p2 +2 )] +, +(47) +which can be regarded as a κ-deformation of the standard kinetic energy p2/2. In general, the +kinetic energy can be defined by +K(p) ∶= ∫ +p +0 +v(p)dp, +(48) +where v(p) denotes the constitutive relation between the velocity v and the canonical mo- +mentum p. In the case of (47), the corresponding kinetic energy is of course the first term +1 +κβ arsinh(κβ p2 +2 ) in the Hamiltonian (45). +We have performed several numbers of the CDD simulations for the target state (44) with +different parameters and initial conditions. In Figure 1, the phase space orbit and the histogram +of the frequencies of the momentum p are plotted for a typical result of the CDD simulation +of the target state (44) with β = 0.2, κ = 0.4. The initial conditions used are also denoted in the +figure captions. +-10 +-5 +5 +10q +-10 +-5 +5 +10 +p +-10 +-5 +0 +5 +10 +0.00 +0.05 +0.10 +0.15 +p +Figure 1. The simulated results of the CDD simulations of the target distribution (44) with κ = 0.4 and +β = 0.2. (a) the phase (q-p) space orbit of the κ-deformed distribution. The 1.5 × 104 points of a simulated +orbit with the initial condition (q0 = 0.1, p0 = 0.1, and S0 = 0.9 are shown. (b) the histogram (cyan bars) of +the frequencies for p and the corresponding momentum κ-distribution ( blue solid curve ). +The CDD simulated result obeys the ergodicity as can be seen from the well distributed +points in the phase space in Figure 1 (a). Note that the momentum distribution in the histogram +of Figure 1 (b) is well fitted with the κ-Gaussian distribution, which is cased by the arsinh-type +deformation of the kinetic energy p2/2. +Note also that for the κ-deformed Hamiltonian (44), we have [8] +⟨p ∂ +∂p Hκ(q, p)⟩ = 1 +β +(49) + +10 of 13 +which reminds us of a generalization of equipartition theorem [26]: ⟨p ∂ +∂pH⟩ = kBT, where H is the +Hamiltonian of a system in thermal equilibrium with the temperature T. +6. Conclusions +We have considered the κ-deformations of some quantities concerning statistical physics +and pointed out some unexpected relations among different fields such as statistical mechanics, +mathematical biology and evolutional game theory. Especially, we focus on the arsinh-type +deformation of the ratio βp2/2 of kinetic energy to the average thermal energy kBT = 1/β. +With the help of the thermostat (CDD) algorithm, we have performed the relevant numerical +simulations for the Hamiltonian with the arsinh-type deformation of kinetic energy term and +show the resultant momentum distribution is the κ-Gaussian distribution. +Author Contributions: +Funding: The first named author (T.W.) is partially supported by Japan Society for the Promotion of +Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) Grant Number 22K03431. +Informed Consent Statement: Not applicable. +Data Availability Statement: Not applicable. +Conflicts of Interest: The authors declare no conflict of interest. +Appendix A Basics of the κ-deformed functions +Here we briefly review some κ-deformed functions and the associated useful relations [2,3]. +Because all κ-deformed functions are symmetric under the sign change of the deformation +parameter κ, i.e., changing κ to −κ, throughout this paper we assume κ > 0. In the κ → 0 +limit, the κ-exponential function (1) and the κ-logarithmic function (2) reduce to the standard +exponential function exp(x) and logarithmic function ln(x), respectively. +lim +κ→0expκ(x) = exp(x), +lim +κ→0lnκ x = ln x. +(A1) +We next introduce another κ-deformed function: +uκ(x) ≡ xκ + x−κ +2 += cosh[κ ln(x)], +(A2) +which is the conjugate (or co-function) of lnκ x, as similar as that cos(x) is the co-function +of sin(x). In the κ → 0 limit, this κ-deformed function reduces to the unit constant function +u0(x) = 1. By using uκ(x), the derivative of the κ-exponential is expressed as +d +dx expκ(x) = +expκ(x) +uκ[expκ(x)] = expκ(x) +√ +1 + κ2x2 , +(A3) +and the derivative of κ-logarithm is expressed as +d +dx lnκ(x) = uκ(x) +x +, +(A4) +respectively. +The inverse function of uκ(x) is + +11 of 13 +u−1 +κ (x) = exp[arcosh(κx)], +(A5) +that is the co-function of expκ(x). +The κ-entropy Sκ [2,3] is a κ-generalization of the Gibbs-Shannon entropy SGS = −kB ∑i pi ln pi +by replacing the standard logarithm with the κ-logarithm, i.e., +Sκ = −kB ∑ +i +pi lnκ pi. +(A6) +Appendix B Replicator equations and generalized Lotka-Volterra equations +We here summarize some known important facts in mathematical biology and evolutional +game theory according to Ref. [14]. Consider a discrete probability distribution described by a +set of n positive variables x = (x1, x2, . . . , xn) with the normalization ∑n +i xi = 1, where each xi +denotes the proportion of the i-th type in the total population. The replicator equation for this +distribution is given by +d +dt xi = xi(fi(x) − ¯f(x)), +(A7) +where f(x) = (f1(x), . . . , fn(x)) is a fitness landscape and ¯f(x) = ∑n +i=1 xi fi(x) is the mean +fitness. Replicator dynamics can be described as a time evolutional curve on the simplex +∆n ∶= {x ∈ Rn ++ ∣ xi ≥ 0,∑i xi = 1} with the matrix component gij(x) of Shahshahani metric [15] g as +gij(x) = +δij +xi +, +(A8) +The inverse matrix is gij(x) = xiδij. Note that the n-simplex ∆n is (n − 1)-dimensional and the +Shahshahani metric diverges on the boundary of the simplex. So this metric is valid only on +the interior Sn of ∆n. +There is a natural mapping: (p1, p2, . . . , pn) → (x1, x2, . . . , xn). Fisher metric is induced by +the Shahshahani metric under this mapping. +(gF)ij(x) = E[∂ ln x +∂xi +∂ ln x +∂xj +] = +n +∑ +k=1 +xk +δik +xi +δik +xi += +δij +xi +. +(A9) +It is known that the Shahshahani manifolds yields an interpretation of the replicator +equation. Theorem 1 in [14]: if the differential equation dxi/dt = fi(x) is a Euclidean gradient +with fi = ∂V/∂xi, the replicator equation (A7) is a gradient w.r.t. Shahshahani metric. A brief +explanation is as the followings. The gradient w.r.t. Shahshahani metric is +(∇gV)i = ∑ +j +gij +∂V +∂xj += ∑ +j +xiδij fj = xi fi, +(A10) +which is the first term in LHS of the replicator equation (A7). The variable xi in the replicator +equation has to satisfy the normalization constraint (∑i xi = 1), i.e., the dynamics of each xi is + +12 of 13 +restricted on the simplex ∆n. Recall that Shahshahni metric is valid only on the interior Sn of +∆n. Indeed, the normalization constraint is satisfied during an time evolution as follows +d +dt ∑ +i +xi = ∑ +i +dxi +dt = ∑ +i +xi(fi − ¯f) = ∑ +i +xi fi − ¯f = 0. +(A11) +The state ˆx is said to be evolutionarily stable state (ESS) if for all x ≠ ˆx in some neighborhood +of ˆx, +x ⋅ f(x) < ˆx ⋅f (x). +(A12) +Let the potential V(x) = D(ˆx∥x) = ∑i ˆxi ln ˆxi − ∑i ˆxi ln xi, then we have +d +dtV(x) = −∑ +i +ˆxi +1 +xi +dxi +dt = −∑ +i +ˆxi(fi − ¯f) = −∑ +i +ˆxi fi + ¯f = −(ˆx ⋅f −x ⋅ f) < 0. +(A13) +Hence the Kullback-Leibler (KL) divergence D(ˆx∥x) is a local Lyapunov function for RE. +Next, if xi = exp(vi(x) − ψ) with dvi(x)/dt = fi(x) and ψ(x) a normalization constant. +From the normalization ∑i xi = 1, we have +0 = ∑ +i +d +dt xi = ∑ +i +( d +dtvi(x) − d +dtψ(x))xi = ∑ +i +xi fi(x) − d +dtψ(x) = ¯f(x) − d +dtψ(x). +(A14) +As a result we see that dψ(x)/dt = ¯f(x), and xi satisfies +d +dt xi = xi( d +dtvi(x) − d +dtψ(x)) = xi(fi(x) − ¯f(x)). +(A15) +Consequently the exponential families xi = exp(vi(x) − ψ) are solutions of REs. +If there is no constraint the corresponding dynamics is described by the generalized LV +equation (13). The generalized LV equations and REs are related as follows. Let each yi satisfies +the generalized LV equation (13). Changing the variable yi to xi as +xi = +yi +∑n +j=1 yj +. +(A16) +which lead to the new normalized variables {xi}, i.e., ∑j xj = 1. Then, we see that +dxi +dt = +dyi +dt +∑j yj +− yi +∑k +dyk +dt +(∑j yj) +2 = yi fi +∑j yj +− +yi +(∑j yj) +∑k yk fk +(∑j yj) = xi(fi − ¯f). +(A17) +Thus, the transformed variable xi in (A16) satisfies the RE. +References +1. +Kaniadakis, G.; Scarfone, A.M.; A new one-parameter deformation of the exponential function. Physica A, 2002, 305, 69-75. +2. +Kaniadakis, G.; Statistical mechanics in the context of special relativity. Phys. Rev. E, 2002, 66, 56125. +3. +Kaniadakis, G.: Statistical mechanics in the context of special relativity II. Phys. Rev. E, 2005, 72, 036108. + +13 of 13 +4. +Kaniadakis, G.: Theoretical foundations and mathematical formalism of the power-law tailed statistical distributions. Entropy, 2013, +15 3983-4010. +5. +Kaniadakis, G.; Baldi, M.M.; Deisboeck, T.S.; et al.; The κ-statistics approach to epidemiology. Sci. Rep., 2020, 10, 19949. +6. +Wada, T.; Thermodynamic stabilities of the generalized Boltzmann entropies. Physica A, 2004, 340, 126-130. +7. +Lymperis, A.; Basilakos, S.; Saridakis, E.N.; Modified cosmology through Kaniadakis horizon entropy. Eur. Phys. J. C, 2021, 81, 1037. +8. +Wada, T.; Scarfone A.M.; Matsuzoe H.; On the canonical distributions of a thermal particle in a generalized velocity-dependent +potential. Physica A, 2020, 541, 123273. +9. +Strutt (Lord Rayleigh), J.W.; Some general theorems relating to vibrations. Proc. London Math. Soc. 1871, s1-4, 357-368. +10. +Pistone, G.; κ-exponential models from the geometrical viewpoint, Eur. Phys. J. B 2009, 70 29-37. +11. +Amari, S-I.; Information geometry and its applications Appl. Math. Sci. 194 (Tokyo: Springer) 2016. +12. +Wada, T.; Scarfone A.M.; Information geometry on the κ-thermostatistics, Entropy 2015, 17, 1204-1217. +13. +Wada, T.; Scarfone A.M.; Matsuzoe H.; An eikonal equation approach to thermodynamics and the gradient flows in information +geometry. Physica A, 2021, 570, 125820. +14. +Harper, M.; Information geometry and evolutionary game theory. arXiv: 0911.1383v1, (2009). +15. +Sigmund, K.; Gradients for replicator systems. Dynamical Systems and Environmental Models: Pro. International Workshop held +on the Wartburg, Eisenach (GDR), March 17-21, 1986, edited by Hans Günter Bothe, Werner Ebeling, Alexander B. Kurzhanski and +Manfred Peschel, Berlin, Boston: De Gruyter, 1987, 186-195. +16. +Gompertz, B.; On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the +Value of Life Contingencies. https://doi.org/10.1098/rspl.1815.0271 +17. +Pelinovsky, E.; et al.; Gompertz model in COVID-19 spreading simulation. Chaos, Solitons & Fractals, 2022, 154, 111699. +18. +Bloch, F.; Zeits. f. Physick 1932, 74, 295. +19. +Kirkwood, J.G.; Phys. Rev. 1933, 44, 31. +20. +Cariñena, J.F.; Rañada, M.F.; and Santander, M.; The quantum free particle on spherical and hyperbolic spaces: A curvature dependent +approach. J. Math. Phys., 2011, 52 072104. +21. +Łapi´nski, T.M.; Law of large numbers unifying Maxwell-Boltzmann, Bose-Einstein and Zipf-Mandelbort distributions, and related +fluctuations. Physica A, 2021, 572, 125909. +22. +McKeague, I.W.; Central limit theorem under special relativity. . Stat Probab Lett. 2015, 99, 149-155. +23. +Scarfone A.M.; Matsuzoe H.; κ-deformed Fourier transform, Physica A, 2017, 480, 63. +24. +Nosé, S.; A unified formulation of the constant temperature molecular-dynamics methods. J. Chem. Phys. 1984 81 511-519. ; +Hoover, W.G.; Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A 1985 31 1695-1697. +25. +Bravetti, A.; Tapias, D.; Thermostat algorithm for generating target state. Phys. Rev. E 2016, 93, 022139. +26. +Tolman, R.C.; A General Theory of Energy Partition with Applications to Quantum Theory. Phys. Rev., 1918, 11, 261-275. + diff --git a/l9AyT4oBgHgl3EQf_vqB/content/tmp_files/load_file.txt b/l9AyT4oBgHgl3EQf_vqB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d0eeb79a82143f287f569e60dd86486c80e6f13f --- /dev/null +++ b/l9AyT4oBgHgl3EQf_vqB/content/tmp_files/load_file.txt @@ -0,0 +1,504 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf,len=503 +page_content='Citation: Wada, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Scarfone, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' On the Kaniadakis distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Preprints 2022, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='org/ Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Copyright: © 2022 by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Licensee MDPI, Basel, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='org/licenses/by/ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='0/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Article On the Kaniadakis distributions applied in statistical physics and natural sciences Tatsuaki Wada † , and Antonio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Scarfone ‡ † Region of Electrical and Electronic Systems Engineering, Ibaraki University, Nakanarusawa-cho, Hitachi-shi, Ibaraki, 316-8511, Japan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' tatsuaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='wada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='to@vc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='ibaraki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='jp ‡ Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' antonio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='scarfone@to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='it Correspondence: tatsuaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='wada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='to@vc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='ibaraki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='jp Abstract: Constitutive relations are fundamental and essential to characterize physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' By utilizing the κ-deformed functions, some constitutive relations are generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' We here show some applications of the Kaniadakis distributions based on the inverse hyperbolic sine function to some topics belonging to the realm of statistical physics and natural science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Keywords: κ-deformed functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' constitutive relations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Gompertz rule;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' generalized Lotka-Voltela equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' contact density dynamics 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Introduction The κ-exponential function [1–3] is defined by expκ(x) ∶= (κx + √ 1 + κ2x2) 1 κ = exp[1 κ arsinh(κx)], (1) for a real deformation parameter κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The inverse function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=', κ-deformed logarithmic function, is defined by lnκ x ∶= xκ − x−κ 2κ = 1 κ sinh[κ ln x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (2) Both κ-deformed functions are important ingredients of the generalized statistical physics based on κ-entropy [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' It influences a wide range of scientific fields, and based on the κ-deformed functions (Appendix A), several basic fields have been developed over two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Kaniadakis [4] provided the theoretical foundations and mathematical formalism generated by the κ-deformed functions, and some references including many fields of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Recently, the usefulness of the κ-statistics was demonstrated for the analysis [5] of epidemics and pandemics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Constitutive relations are fundamental and essential to characterize physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' They are combined with the other equations of physical laws in order to solve physical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Some well-known examples of linear constitutive relations are: Hooke’s law F = ksx for the tensile or compressive force F of a spring with a spring constant ks against the change in its length x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Ohm’s law V = RI for the voltage V of a electrical conductor with resistance R under an electric current I, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' However, as a real spring deviates from Hooke’s law, we know that any linear constitutive relation describes an idealized situation, and it is merely a linearized- and/or approximated- relation to describe some real physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Hence, in general, non-linearity plays a crucial role to describe more realistic physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='00914v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='stat-mech] 3 Jan 2023 BY2 of 13 The κ-exponential function (1) can be regarded as a useful tool (or device) to make such non-linear constitutive relations for a better description of real physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' For example, consider the following κ-deformation of Hooke’s law: Fκ ∶= ks ln[expκ(x)] = ks κ ln(κx + √ 1 + κ2x2), (3) which reduces to the original Hooke’s law F = ksx in the limit of κ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' For any linear constitutive relation, we can apply this type of the κ-deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' For example, Ohm’s law can be cast into the following form: V = RI = R ln[exp(I)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' By changing the exponential function with the κ-exponential function, we obtain the κ-deformed version of Ohm’s law: Vκ = R ln[expκ(I)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In this research, we focus on this type of the κ-deformation of a physical quantity (say A), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='e, A ⇒ ln[expκ(A)] = 1 κ arsinh(κA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (4) Throughout in this paper, we call this κ-deformation as the arsinh-type deformation of a physical quantity A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The other type of the κ-deformation can be A ⇒ lnκ[exp(A)] = 1 κ sinh(κA), (5) which is called here the sinh-type deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' [6], the thermodynamic stability of the κ-generalization SB κ of Boltzmann entropy SB is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The κ-generalization SB κ is rewritten in the form: SB κ ∶= kB lnκ W = kB lnκ[exp(lnW)] = kB lnκ[exp(SB)], (6) which can be regarded as the sinh-type deformation of Boltzmann entropy SB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Recently, in cosmology, Lymperis et al [7] modified Bekenstein-Hawking entryopy SBH as follows: 1 κ sinh(κSBH), (7) which is obviously the sinh-type deformation of SBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In this paper we consider the arsinh-type deformations against some constitutive relations in the field of statistical physics and natural sciences In our previous work [8] we studied a thermal particle under a velocity-dependent potential which can be regarded as a deformation of Rayleigh’s dissipation function [9] and showed that the probability distribution function (pdf) for the stationary-state of this thermal particle is a κ-deformed Gaussian pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' It was considered that the canonical pdf ρ(v), in the velocity space, of a thermal particle with unit mass (m = 1) in the κ-deformed confining potential Uκβ(v): Uκβ(v) ∶= 1 κβ arsinh(κβv2 2 ), (8) where β ∶= 1/kBT is a coldness (or inverse temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' This κ-deformed potential Uκβ(v) is rewritten, in the momentum-space, as 3 of 13 Uκβ(p) = 1 κβ arsinh(κβ p2 2 ) = 1 β ln[expκ(β p2 2 )], (9) which is the arsinh type deformation of the quantity βp2/2 (the ratio of the kinetic energy to the mean thermal energy kBT = 1/β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In other words, we can consider the following κ-deformation Qκ(U) of the Boltzmann factor exp(−βU) for an equilibrium state with the energy U: Qκ(U) ∶= expκ(−βU) = exp[1 κ arsinh(−κβU)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (10) One may wonder why the inverse hyperbolic sine function (arsinh) plays a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In many different fields of sciences, there is no doubt that the exponential and logarithmic functions are important and fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Since the inverse hyperbolic sine function and logarithmic function are mutually related as: arsinh x = ln[x + √ 1 + x2], ln x = arsinh[1 2(x − 1 x)], (11) for a positive real x, we think both functions are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' By using the second relation, for any real parameter κ ≠ 0, we have ln x = 1 κ ln xκ = 1 κ arsinh[1 2(xκ − x−κ)] = 1 κ arsinh[κ lnκ x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (12) Note that this relation corresponds to the arsinh-type deformation of lnκ x and is equivalent to definition (2) of the κ-deformed logarithmic function that can be regarded as the sinh-type of κ-deformation of ln x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Kaniadakis already discussed this issue in section II of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' [2] from the view point of the deformed algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' On the other hand, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Pistone [10] was the first one to study the κ-exponential model in the field of information geometry [11], and later through our reserach activities [8,12,13], we realize that there exist some relations among statistical physics, thermodynamics, mathematical biology, and information geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Harper [14] pointed out that the replicator equation (RE) [15] in mathematical biology or in an evolutional game theory is related with information geometry and the generalized Lotka-Volterra (LV) equation as briefly explained in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The generalized LV equations: dyi dt = yi fi(y), (13) are used to model the competition dynamics of the populations y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' , yn of n biological species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Gompertz function [16] is a type of mathematical model for a time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Histori- cally he studied human mortality for working out a series of tables mortality, and he proposed his law of human mortality in which he assumed that person’s resistance to death decreases as his years increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' His law is now called Gompertz rule (or law) and we’d like to point out the relation of his function and his rule to some important quantities concerning on statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In section 2, we briefly explain Gompertz function, and the generalized LV equations, which are important in mathematical biology (or 4 of 13 evolutional game theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Their relations to thermal physics is pointed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Section 3 considers the canonical thermal density matrix, which is characterized by the so-called Bloch equation [18], and we shall show that Bloch equation can be regarded as a Gompertz rule after the parameter transformation β to t = −ln β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In section 4, we discuss the arsinh-type deformation from the view point of the κ-addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In section 5, we study the numerical simulations of the thermostat algorithm for the Hamiltonian with the κ-deformed kinetic energy, which can be regarded as the arsinh type of the κ-deformation of the ratio βp2/2 as shown in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The final section is devoted to our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Gompertz functions and Gompertz rule Here we e’d like to point out that there exist relations between an evolutional game dynamics and thermal physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In evolutional game theory, an evolutional game dynamics is described by RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The generalized LV equations are related with REs as shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' On the other hand, Gompertz function is a mathematical model describing an evolutional curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Gompertz function (or Gompertz curve) [16] is a type of mathematical model for a time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Gompertz function fG(t) is a sigmoid function and is given by fG(t) ∶= K exp[C exp(−t)], (14) where C and K are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' A distinctive feature of Gompertz function is its double exponential t-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' His function is nowadays used in many different areas to model a time evolution of the populations where growth is slowest at the start and end of a period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' For example, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' [17] applied Gompertz model to describe the growth dynamics of COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Gompertz [16] studied human mortality for working out a series of tables mortality, and this suggested to him his law of human mortality in which he assumed that person’s resistance to death decreases as his years increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The rule of his model is called Gompertz rule which states that d dt fG(t) = −fG(t)ln fG(t) K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (15) The solution of the Gompertz rule is fG(t), if we set K = limt→∞ fG(t) and C = ln(fG(0)/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' If we choose fi(y(t)) = −ln yi(t) and assuming limt→∞ yi(t) = 1, the generalized LV equation (13) becomes dyi(t) dt = −yi(t) ln yi(t), (16) which can be regarded as the Gompertz rule (15) with K = 1 for each yi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Consequently its solution yi(t) is the Gompertz function: yi(t) = exp[ln yi(0) exp(−t)], (17) Now, by changing the parameter t to β = exp(−t), we have dβ = −βdt so that the limit t → 0 corresponds to β → 1, and we introduce each constant Ei as −Ei = lim t→0 ln yi(t) = lim β→1ln yi(β), (18) 5 of 13 where yi(β) is the shorthand notation of yi(t(β)) with t(β) = −ln β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Then, the solution yi(β) in (17) can be expressed as a quantity very familiar in statistical physics: yi(β) = exp(−βEi), (19) that is Boltzmann factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The corresponding Gompertz rule (15) for yi(β) is equivalent to d dβyi(β) = −Ei yi(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (20) Having described the relation between Gompertz rule and Boltzmann factor exp(−βEi) in statistical physics, in next section we discuss a κ-deformation of Bloch equation for thermal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Bloch equation for thermal state For a given Hamiltonian ˆH and the corresponding eigenvalues Ei and eigenstate ∣ψi⟩ which are related with ˆH∣ψi⟩ = Ei∣ψi⟩, (21) and assuming the completeness ∑i ∣ψi⟩⟨ψi∣ = ˆ1, the canonical density matrix (or density operator for a canonical ensemble) is constructed as ˆρ(β) ∶= ∑ i exp(−βEi)∣ψi⟩⟨ψi∣ = exp(−β ˆH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (22) In order to determine the canonical density matrix, we have to solve the eigenequations (21) and to sum over all the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' This needs heavy calculations in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Bloch equation [18,19] for canonical density matrix is known as − ∂ ∂β ˆρ(β) = ˆH ˆρ(β), (23) which can be regarded as the diffusion equation in imaginary time β, and it has a similar form as Schrödinger equation and diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Bloch equation (23) offers an alternative route to determine the density matrix ˆρ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The initial (β = 0) condition is provided if we know the eigenstates in the high-temperature limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Now, by multiplying β to both sides of (23), we have −β ∂ ∂β ˆρ(β) = β ˆH ˆρ(β) = −ln[ ˆρ(β)] ˆρ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (24) Changing the parameter β to t = −ln β, it follows d dt ˆρ(t) = −β d dβ ˆρ(β) = −ln[ ˆρ(t)] ˆρ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (25) 6 of 13 This is the same form of the Gompertz rule (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In this way, Bloch equation can be considered as a sort of Gompertz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Next, let us consider the κ-deformed canonical density matrix: ˆρκ(β) ∶= ∑ i expκ(−βEi)∣ψi⟩⟨ψi∣ = expκ(−β ˆH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (26) This leads to the following κ-deformation of Bloch equation: − ∂ ∂β ˆρκ(β) = ∑ i Ei expκ(−βEi) uκ[(expκ(−βEi)] ∣ψi⟩⟨ψi∣ = ˆH uκ[expκ(−β ˆH)] ˆρκ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (27) Again by changing the parameter β to t = −ln β and using the relation (A3), we have d dt ˆρκ(t) = −lnκ[ ˆρκ(t)] uκ[ ˆρκ(t)] ˆρκ(t), (28) which can be regarded as a κ-deformation of the Gompertz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Differentiating (27) again with respect to β, we obtain the following nonlinear differential equation: (1 + κ2β2 ˆH2)∂2 ˆρκ(β) ∂β2 + κ2β ˆH2 ∂ ˆρκ(β) ∂β − ˆH2 ˆρκ(β) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (29) This differential equation remind us the research work [20] of the quantum free particle on two-dimensional hyperbolic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The relevant two-dimensional Schrödinger equation is separable in the κ-dependent coordinate system (zx, y) with zx ∶= x/ √ 1 + κ2y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The Schrödinger equation ˆH1Ψ = e1Ψ for the first partial Hamiltonian ˆH1 leads to the following differential equation with the variable zx alone: (1 + κ2z2 x)d2Ψ(zx) dz2x + κ2zx dΨ(zx) dzx + µΨ(zx) = 0, µ ∶= 2m ¯h2 e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (30) In the limit of κ → 0, this differential equation reduces to the standard time-independent Schrödinger equation: d2Ψ(x)/dx2 + µΨ(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Cariñena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' [20] obtain the solution of the differential equation (30) as the κ-deformed plane wave: Ψ(zx) = exp[±i µ κ arsinh(κ zx)], (31) which is regarded as a arsinh-type deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The κ-addition and the law of large number Next, we consider the κ-addition from the view point of the law of large numbers (LLN), which plays a central role in probability, statistics, and statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The κ-addition [4] is defined by x κ⊕ y ∶= x √ 1 + κ2y2 + y √ 1 + κ2x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (32) 7 of 13 This deformation of additive rule comes from the addition rule of the inverse hyperbolic sine function as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' For a, b ∈ R, the addition rule is written as arsinh(a) + arsinh(b) = arsinh(a √ 1 + b2 + b √ 1 + a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (33) By setting a = κx and b = κy, we obtain arsinh(κx) + arsinh(κy) = arsinh(κx √ 1 + κ2y2 + κy √ 1 + κ2x2) = arsinh[κ(x κ⊕ y)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (34) This relation is equivalent to the definition (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The additive relation (34) is readily generalized to n ∑ i=1 arsinh(κxi) = arsinh[κ(x1 κ⊕ x2 κ⊕ ⋯ κ⊕ xn)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (35) By applying this relation to the Boltzmann factor exp[−β ∑n i=1 Kκβ(pi)] with respect to the κ-deformed kinetic energy [8] with m = 1: n ∑ i=1 Kκβ(pi) ∶= n ∑ i=1 1 κβ arsinh(κβ p2 i 2 ), (36) we have exp[−β n ∑ i=1 Kκβ(pi)] = exp[−1 κ arsinh{κ(β p2 1 2 κ⊕ β p2 2 2 κ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' κ⊕ β p2 n 2 )}] = expκ[(−β p2 1 2 ) κ⊕ (−β p2 2 2 ) κ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' κ⊕ (−β p2 n 2 )] = expκ[−β p2 1 2 ]expκ[−β p2 2 2 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' expκ[−β p2 n 2 ] = n ∏ i=1 expκ[−β p2 i 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (37) Note that the κ-exponential of the κ-summation of each term −β p2 i 2 in the second line is ex- pressed as a factorized form in the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' It is well known that LLN plays a fundamental role in statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Łapi´nski [21] showed that the standard LLN yields the most probable state of the system, which equals to the point of maximum of the entropy and this point can be either Maxwell-Boltzmann statistics or Bose-Einstein statistics, or Zipf-Mandelbort law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' McKeague [22] studied the central limit theorem under special relativity based on the κ-additivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Scarfone [23] studied the κ-deformation of Fourier Transform and discussed the limiting distribution of the κ-sum of statistically independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The κ-additivity extension [22] of the strong LLN is shown and it states that if Xi are iid with finite mean, X1 n κ⊕ X2 n κ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' κ⊕ Xn n → 1 κ arsinh[κ⟨X⟩]a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=', (38) 8 of 13 where a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' stands for almost surely, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=', the above sequence of the random variables Xi converges almost surely, and ⟨X⟩ is the standard average of the random variable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Of course, in the limit of κ → 0, the relation (38) reduces to the standard strong LLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In this way, the κ-additivity extension of the strong LLN suports the arsinh-type deformation of the average of a stochastic variable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Note that the converged value in (38) is the arsinh-type deformation of the average ⟨X⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Contact density dynamics Nosé-Hoover (NH) thermostat [24] is a famous deterministic algorithm for constant- temperature molecular dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Based on the idea of NH thermostat, several improved versions are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Among them, contact density dynamics (CDD) [25] is an algorithm based on contact Hamiltonian systems and generates any prescribed target distribution in physical phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The dynamical equations of CDD are followings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' ˙qi = ∂h(q, p, S) ∂pi , (39) ˙pi = −∂h(q, p, S) ∂qi + ∂h(p, q, S) ∂S pi, (40) ˙S = −pi ∂h(q, p, S) ∂pi + h(q, p, S), (41) where S is the thermostatting variable, qi and pi are i-th component (i = 1,2,⋯, n) of n- dimensional vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Here h(q, p, S) denotes the contact Hamiltonian which is formed as h(q, p, S) = (ρt(q, p)f(S))− 1 n+1 , (42) with a target distribution ρt(q, p) on 2n-dimensional Γ-space and a normalized distribution f(S) for the thermostatting variable S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' As in the case of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' [24], we also choose f(S) as the logistic distribution with scale 1 and mean c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='0: f(S) = exp(S − c) (1 + exp(S − c))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (43) Utilizing this CDD algorithm, the κ-deformed exponential distributions are simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The target distribution ρt(q, p) is the one-dimensional (n = 1) κ-deformed Gaussian function: ρt(q, p) = 1 Zκ(β) exp[−βHκ(q, p)] = 1 Zκ(β) exp[−1 κ arsinh(κβ p2 2 )]exp[−βq2 2 ], (44) where the associated Hamiltonian is Hκ(q, p) = 1 κβ arsinh(κβ p2 2 ) + q2 2 , (45) and the normalization factor Zκ(β) [4] is 9 of 13 Zκ(β) = π β √ 2 κ Γ( 1 2κ − 1 4) ( κ 2 + 1)Γ( 1 4 + 1 2κ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (46) From (39) we have v(p) ∶= dq dt = ∂Hκ(q, p) ∂p = p uκ[expκ(−β p2 2 )] , (47) which can be regarded as a κ-deformation of the standard kinetic energy p2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In general, the kinetic energy can be defined by K(p) ∶= ∫ p 0 v(p)dp, (48) where v(p) denotes the constitutive relation between the velocity v and the canonical mo- mentum p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In the case of (47), the corresponding kinetic energy is of course the first term 1 κβ arsinh(κβ p2 2 ) in the Hamiltonian (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' We have performed several numbers of the CDD simulations for the target state (44) with different parameters and initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In Figure 1, the phase space orbit and the histogram of the frequencies of the momentum p are plotted for a typical result of the CDD simulation of the target state (44) with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='2, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The initial conditions used are also denoted in the figure captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' 10 5 5 10q 10 5 5 10 p 10 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='15 p Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The simulated results of the CDD simulations of the target distribution (44) with κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='4 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (a) the phase (q-p) space orbit of the κ-deformed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='5 × 104 points of a simulated orbit with the initial condition (q0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='1, p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='1, and S0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='9 are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (b) the histogram (cyan bars) of the frequencies for p and the corresponding momentum κ-distribution ( blue solid curve ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The CDD simulated result obeys the ergodicity as can be seen from the well distributed points in the phase space in Figure 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Note that the momentum distribution in the histogram of Figure 1 (b) is well fitted with the κ-Gaussian distribution, which is cased by the arsinh-type deformation of the kinetic energy p2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Note also that for the κ-deformed Hamiltonian (44), we have [8] ⟨p ∂ ∂p Hκ(q, p)⟩ = 1 β (49) 10 of 13 which reminds us of a generalization of equipartition theorem [26]: ⟨p ∂ ∂pH⟩ = kBT, where H is the Hamiltonian of a system in thermal equilibrium with the temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Conclusions We have considered the κ-deformations of some quantities concerning statistical physics and pointed out some unexpected relations among different fields such as statistical mechanics, mathematical biology and evolutional game theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Especially, we focus on the arsinh-type deformation of the ratio βp2/2 of kinetic energy to the average thermal energy kBT = 1/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' With the help of the thermostat (CDD) algorithm, we have performed the relevant numerical simulations for the Hamiltonian with the arsinh-type deformation of kinetic energy term and show the resultant momentum distribution is the κ-Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Author Contributions: Funding: The first named author (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=') is partially supported by Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) Grant Number 22K03431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Informed Consent Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Data Availability Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Conflicts of Interest: The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Appendix A Basics of the κ-deformed functions Here we briefly review some κ-deformed functions and the associated useful relations [2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Because all κ-deformed functions are symmetric under the sign change of the deformation parameter κ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=', changing κ to −κ, throughout this paper we assume κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In the κ → 0 limit, the κ-exponential function (1) and the κ-logarithmic function (2) reduce to the standard exponential function exp(x) and logarithmic function ln(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' lim κ→0expκ(x) = exp(x), lim κ→0lnκ x = ln x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (A1) We next introduce another κ-deformed function: uκ(x) ≡ xκ + x−κ 2 = cosh[κ ln(x)], (A2) which is the conjugate (or co-function) of lnκ x, as similar as that cos(x) is the co-function of sin(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' In the κ → 0 limit, this κ-deformed function reduces to the unit constant function u0(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' By using uκ(x), the derivative of the κ-exponential is expressed as d dx expκ(x) = expκ(x) uκ[expκ(x)] = expκ(x) √ 1 + κ2x2 , (A3) and the derivative of κ-logarithm is expressed as d dx lnκ(x) = uκ(x) x , (A4) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The inverse function of uκ(x) is 11 of 13 u−1 κ (x) = exp[arcosh(κx)], (A5) that is the co-function of expκ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The κ-entropy Sκ [2,3] is a κ-generalization of the Gibbs-Shannon entropy SGS = −kB ∑i pi ln pi by replacing the standard logarithm with the κ-logarithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=', Sκ = −kB ∑ i pi lnκ pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (A6) Appendix B Replicator equations and generalized Lotka-Volterra equations We here summarize some known important facts in mathematical biology and evolutional game theory according to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Consider a discrete probability distribution described by a set of n positive variables x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' , xn) with the normalization ∑n i xi = 1, where each xi denotes the proportion of the i-th type in the total population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The replicator equation for this distribution is given by d dt xi = xi(fi(x) − ¯f(x)), (A7) where f(x) = (f1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' , fn(x)) is a fitness landscape and ¯f(x) = ∑n i=1 xi fi(x) is the mean fitness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Replicator dynamics can be described as a time evolutional curve on the simplex ∆n ∶= {x ∈ Rn + ∣ xi ≥ 0,∑i xi = 1} with the matrix component gij(x) of Shahshahani metric [15] g as gij(x) = δij xi , (A8) The inverse matrix is gij(x) = xiδij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Note that the n-simplex ∆n is (n − 1)-dimensional and the Shahshahani metric diverges on the boundary of the simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' So this metric is valid only on the interior Sn of ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' There is a natural mapping: (p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' , pn) → (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Fisher metric is induced by the Shahshahani metric under this mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (gF)ij(x) = E[∂ ln x ∂xi ∂ ln x ∂xj ] = n ∑ k=1 xk δik xi δik xi = δij xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (A9) It is known that the Shahshahani manifolds yields an interpretation of the replicator equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Theorem 1 in [14]: if the differential equation dxi/dt = fi(x) is a Euclidean gradient with fi = ∂V/∂xi, the replicator equation (A7) is a gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Shahshahani metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' A brief explanation is as the followings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Shahshahani metric is (∇gV)i = ∑ j gij ∂V ∂xj = ∑ j xiδij fj = xi fi, (A10) which is the first term in LHS of the replicator equation (A7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The variable xi in the replicator equation has to satisfy the normalization constraint (∑i xi = 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=', the dynamics of each xi is 12 of 13 restricted on the simplex ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Recall that Shahshahni metric is valid only on the interior Sn of ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Indeed, the normalization constraint is satisfied during an time evolution as follows d dt ∑ i xi = ∑ i dxi dt = ∑ i xi(fi − ¯f) = ∑ i xi fi − ¯f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (A11) The state ˆx is said to be evolutionarily stable state (ESS) if for all x ≠ ˆx in some neighborhood of ˆx, x ⋅ f(x) < ˆx ⋅f (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (A12) Let the potential V(x) = D(ˆx∥x) = ∑i ˆxi ln ˆxi − ∑i ˆxi ln xi, then we have d dtV(x) = −∑ i ˆxi 1 xi dxi dt = −∑ i ˆxi(fi − ¯f) = −∑ i ˆxi fi + ¯f = −(ˆx ⋅f −x ⋅ f) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (A13) Hence the Kullback-Leibler (KL) divergence D(ˆx∥x) is a local Lyapunov function for RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Next, if xi = exp(vi(x) − ψ) with dvi(x)/dt = fi(x) and ψ(x) a normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' From the normalization ∑i xi = 1, we have 0 = ∑ i d dt xi = ∑ i ( d dtvi(x) − d dtψ(x))xi = ∑ i xi fi(x) − d dtψ(x) = ¯f(x) − d dtψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (A14) As a result we see that dψ(x)/dt = ¯f(x), and xi satisfies d dt xi = xi( d dtvi(x) − d dtψ(x)) = xi(fi(x) − ¯f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (A15) Consequently the exponential families xi = exp(vi(x) − ψ) are solutions of REs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' If there is no constraint the corresponding dynamics is described by the generalized LV equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' The generalized LV equations and REs are related as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Let each yi satisfies the generalized LV equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Changing the variable yi to xi as xi = yi ∑n j=1 yj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (A16) which lead to the new normalized variables {xi}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=', ∑j xj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Then, we see that dxi dt = dyi dt ∑j yj − yi ∑k dyk dt (∑j yj) 2 = yi fi ∑j yj − yi (∑j yj) ∑k yk fk (∑j yj) = xi(fi − ¯f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' (A17) Thus, the transformed variable xi in (A16) satisfies the RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Kaniadakis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Scarfone, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Matsuzoe H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' On the canonical distributions of a thermal particle in a generalized velocity-dependent potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Physica A, 2020, 541, 123273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' 9.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Kurzhanski and Manfred Peschel, Berlin, Boston: De Gruyter, 1987, 186-195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' Gompertz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} +page_content=' On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQf_vqB/content/2301.00914v1.pdf'} 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sha256:671b3050272fefbf159e73e0f533be7dc4590c09e532fd758df066db9003c0f0 +size 3670061 diff --git a/otE5T4oBgHgl3EQfkQ83/content/tmp_files/2301.05661v1.pdf.txt b/otE5T4oBgHgl3EQfkQ83/content/tmp_files/2301.05661v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a7cbe0a237ad1cab7e1a86ce1bc6c19782755852 --- /dev/null +++ b/otE5T4oBgHgl3EQfkQ83/content/tmp_files/2301.05661v1.pdf.txt @@ -0,0 +1,1400 @@ +arXiv:2301.05661v1 [math.LO] 13 Jan 2023 +Choice-free Dualities for Lattice Expansions: +Application to Logics with a Negation Operator +Chrysafis (Takis) Hartonas +Department of Digital Systems +University of Thessaly, Greece +hartonas@uth.gr +January 13, 2023 +Abstract +Constructive dualities have been recently proposed for some lattice +based algebras and a related project has been outlined by Holliday and +Bezhanishvili, aiming at obtaining “choice-free spatial dualities for other +classes of algebras [...], giving rise to choice-free completeness proofs for +non-classical logics”. +We present in this article a way to complete the Holliday-Bezhanishvili +project (uniformly, for any normal lattice expansion) by recasting re- +cent relational representation and duality results in a choice-free man- +ner. These results have some affinity with the Moshier and Jipsen duality +for bounded lattices with quasi-operators, except for aiming at repre- +senting operators by relations, extending the J´onsson-Tarski approach for +BAOs, and Dunn’s follow up approach for distributive gaggles, to con- +texts where distribution may not be assumed. +To illustrate, we apply +the framework to lattices (and their logics) with some form or other of +a (quasi)complementation operator, obtaining canonical extensions in re- +lational frames and choice-free dualities for lattices with a minimal, or +a Galois quasi-complement, or involutive lattices, including De Morgan +algebras, as well as Ortholattices and Boolean algebras, as special cases. +1 +Introduction +Choice-free dualities have been lately proposed for Boolean algebras, by Holliday +and Bezhanishvili [2], for Ortholattices, by MacDonald and Yamamoto [22], for +modal lattices by Bezhanishvili, Dmitrieva, de Groot and Moraschini [1] and +for De Vries algebras by Massas [21]. These are part of a project, outlined by +Holliday and Bezhanishvili and aiming at obtaining “choice-free spatial dualities +for other classes of algebras [...], giving rise to choice-free completeness proofs +for non-classical logics” [2, page 45]. The project has its origins in Holliday’s +‘possibility frames’ for modal logic [17], as noted in [2]. +1 + +A choice-free representation and duality for bounded lattices with quasiop- +erators, by Moshier and Jipsen [23,24], had already appeared in print, influenc- +ing at least Dmitrieva’s research, with Bezhanishvili, de Groot and Morachini, +on modal lattices. +The Moshier-Jipsen duality is related to results by this +author [10], and with Dunn [14], some detail on these relations is presented +in [13, Remark 4.2, Remark 4.8] and we revisit the issue in Proposition 4.7 in +this article. The duality of [23, 24] is not primarily intended to provide logic +related, relational semantics applications. This becomes clear by their choice +to represent lattice quasi-operators as strongly continuous and meet preserving +point operators on the dual topological spaces, whereas for semantic purposes +one typically aims for first-order definable classes of relational frames. +In the recent few years, this author has pursued a project of extending +J´onsson and Tarski’s approach for Boolean algebras with operators (BAOs) +[19,20] and Dunn’s follow up approach for distributive generalized Galois logics +(gaggles) [4–6] to the case of general lattices with quasi-operators (normal lattice +operators, in our preferred terminology), building on older work by the author +(with Dunn) in [14], while working within the framework of canonical exten- +sions [8] of lattice expansions. This project developed in parallel with Gehrke’s +(with co-workers) generalized Kripke frames approach (RS frames) [7] and the +relations between the two approaches have been detailed in [12]. We note that, +as far as the objectives of the current article are concerned, Gehrke’s approach +of RS-frames in [7] builds on Hartung’s lattice representation [16], which inher- +its from Urquhart’s lattice representation [25] an essential use of the axiom of +choice. Choice was also assumed in this author’s [13,14] (Alexander’s subbasis +lemma, whose proof uses Zorn’s lemma, was used to prove compactness of the +space), but we show in this article that the use of choice is inessential and we +can easily recast the duality in a choice-free manner, switching to a spectral +topology. +In Section 2 we present the algebras of interest, bounded lattices with a quasi- +complementation operator. We restrict attention to some distinguished cases, +allowing for both a distributive (notably De Morgan and Boolean algebras) and +a non-distributive lattice base (such as involutive, or orthocomplement lattices). +Section 3 starts with a review subsection (Section 3.1) for sorted frames with +relations and generated operators, drawing on [13], and concludes with Section +3.2 where frames for quasi-complemented lattices are discussed. A first-order +axiomatic specification of the classes of frames with respect to which the logics +of the algebraic structures of section 2 can be shown to be sound is provided in +Table 1. +Section 4 presents choice-free representations of semilattices (Section 4.1) +and bounded lattices (Section 4.2) and concludes with Section 4.3 detailing the +representation of arbitrary normal lattice operators, drawing on [13]. +In Section 5 we apply the representation framework of section 4.3 to the +particular case of quasi-complementation operators on bounded lattices. The +results of this section establish that the varieties of quasi-complemented lattices +we consider are closed under canonical extensions. Thereby, completeness the- +orems via a canonical model construction can be proven for the logics of the +2 + +algebraic structures considered. +Spectral duality theorems are proven in Section 6. The main result in this +section is Theorem 3.1, where we detail the duality between the categories M +of bounded lattices with a minimal quasi-complementation operator and the +category SRF∗ +νM of sorted residuated frames whose first-order axiomatization is +given in Table 2. The remaining dualities are then easily obtained. In particular, +Proposition 3.7, relying on Theorem 3.5, provides a first-order frame condition +for the lattice of Galois stable sets to be completely distributive, which is then +used for the cases of representation and duality for De Morgan algebras and +Boolean algebras. +Some concluding remarks are made in Section 7. +2 +Quasi-complemented Lattices +Let {1,∂} be a 2-element set, L1 = L and L∂ = Lop (the opposite lattice). +Extending established terminology [19], a function f ∶ L1 × ⋯ × Ln �→ Ln+1 will +be called additive and normal, or a normal operator, if it distributes over finite +joins of the lattice Li, for each i = 1,... n, delivering a join in Ln+1. +Definition 2.1. An n-ary operation f on a bounded lattice L is a normal lattice +operator of distribution type δ(f) = (i1,... ,in;in+1) ∈ {1,∂}n+1 if it is a normal +additive function f ∶ Li1 ×⋯×Lin �→ Lin+1 (distributing over finite joins in each +argument place), where each ij, for j = 1,... ,n + 1, is in the set {1,∂}, hence +Lij is either L, or L∂. +If τ is a tuple (sequence) of distribution types, a normal lattice expansion +of (similarity) type τ is a lattice with a normal lattice operator of distribution +type δ for each δ in τ. +The category NLEτ, for a fixed similarity type τ, has normal lattice expan- +sions of type τ as objects. Its morphisms are the usual algebraic homomor- +phisms. +In this article we focus on the class of lattices L = (L,≤,∧,∨,0,1,ν) with +a quasi-complementation operator ν, of increasing axiomatization strength, in- +cluding at least the following: +(antitonicity) +a ≤ b �→ νb ≤ νa +(normality) +ν0 = 1 +(∨∧) +ν(a ∨ b) ≤ νa ∧ νb. +Given antitonicity, the operation ν satisfies the identity ν(a∨b) = νa∧νb, hence +it is a normal lattice operator of distribution type δ(ν) = (1;∂). We list some +basic facts in Lemma 2.2. +Lemma 2.2. Let L = (L,≤,∧,∨,0,1,ν) be a bounded lattice with an antitone +operation ν. +(1) ν forms a Galois connection on L (a ≤ νb iff b ≤ νa) iff it satisfies the +inequation a ≤ ννa +3 + +(2) if a ≤ ννa holds in the lattice, then the normality axiom ν0 = 1 and the +identity ν(a ∨ b) = νa ∧ νb are derivable +(3) if ννa ≤ a holds in the lattice, then the identity ν(a ∧ b) = νa ∨ νb is +derivable +(4) if either of the De Morgan identities ν(a∨b) = νa∧νb, or ν(a∧b) = νa∨νb, +holds in the lattice, then antitonicity of ν is a derivable property +(5) if ν is an involution (a = ννa) and, in addition, the lattice is distributive, +then it is a De Morgan algebra +(6) if ν is an involution and, in addition, it satisfies the intuitionistic explosion +principle (ex falso quidlibet) a ∧ νa ≤ 0, then the lattice is an Ortholattice +(orthocomplemented lattice) +(7) if ν is an involution satisfying the antilogism rule (a∧b ≤ c �→ a∧νc ≤ νb), +then the lattice is a Boolean algebra +Proof. Each of the claims (1) to (6) has a straightforward proof, left to the +interested reader. For (7), the hypothesis implies that a ∧ b ≤ c iff a ∧ νc ≤ νb. +This means that ∧ is self-conjugate with respect to the involution ν. To see +that this implies distributivity, define a → c = ν(a ∧ νc) and observe that the +conjugacy condition is equivalent to residuation of ∧ and →, i.e. a ∧ b ≤ c iff +a∧νc ≤ νb iff b ≤ a → c. Distribution then follows from residuation. In addition, +by part (2), ν0 = 1 holds and then also ν1 = νν0 = 0. Hence the intuitionistic +principle a ∧ νa ≤ 0 = ν1 follows, since we can infer it from a ∧ 1 ≤ a using +antilogism. By the hypothesis that ν is an involution, the explosion principle +a ∧ νa ≤ 0 is equivalent to excluded middle a ∨ νa = 1. Hence the lattice is a +Boolean algebra. +Figure 1 summarizes the above results, where DMA,O,INV designate the +equational classes (varieties) of De Morgan algebras, Ortholattices and lattices +with an involution, respectively, BA designates the variety of Boolean algebras, +the remaining two labels M,G designate the varieties of lattices with a minimal, +or a Galois connected quasi-complementation operator, respectively, and the +arrow label (dist) indicates addition of the distribution law a∧(b∨c) ≤ (a∧b)∨ +(a ∧ c). +3 +Sorted residuated frames (SRFs) +3.1 +Frames, Relations and (Sorted) Image Operators +We review in this section definitions, notational conventions and results from +[12,13], to the extent needed for our current purposes. +Regard {1,∂} as a set of sorts and let Z = (Z1,Z∂) be a sorted set. Sorted +residuated frames F = (Z1,⍊,Z∂) are triples consisting of nonempty sets Z1 = +X,Z∂ = Y and a binary relation ⍊ ⊆ X × Y . +4 + +Figure 1: (Quasi)Complemented Lattices +BA +DMA +a∧νa=0 +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +a∧b≤c +a∧νc≤νb +O +(dist) +◆◆◆◆◆◆◆◆◆◆◆◆◆◆ +INV +a∧νa=0 +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +(dist) +PPPPPPPPPPPPPP +G +a ≤ ννa +ννa≤a +M +ν0 = 1 +ν(a ∨ b) = νa ∧ νb +The relation ⍊ will be referred to as the Galois relation of the frame. It +generates a Galois connection ( )⍊ ∶ ℘(X) ⇆ ℘(Y )∂ ∶ ⍊( ) (V ⊆ U ⍊ iff U ⊆ ⍊V ) +U ⍊ = {y ∈ Y ∣ ∀x ∈ U x ⍊ y} = {y ∈ Y ∣ U ⍊ y} +⍊V = {x ∈ X ∣ ∀y ∈ V x ⍊ y} = {x ∈ X ∣ x ⍊ V }. +We will also have use for the complement I of the Galois relation ⍊ and we will +designate frames using either the Galois relation ⍊, or its complement I. +A subset A ⊆ X will be called stable if A = ⍊(A⍊). Similarly, a subset B ⊆ Y +will be called co-stable if B = (⍊B)⍊. Stable and co-stable sets will be referred to +as Galois sets, disambiguating to Galois stable or Galois co-stable when needed +and as appropriate. The following quasi-seriality condition will be assumed for +sorted frames +∀x ∈ X∃y ∈ Y xIy +∀y ∈ Y ∃x ∈ X xIy +(1) +Note that assuming (1), the empty set is (co)stable and we have ∅⍊ = Y , ⍊Y = ∅ +and similarly ⍊∅ = X and X⍊ = ∅. +By G(X),G(Y ) we designate the complete lattices of stable and co-stable +sets, respectively. Note that the Galois connection restricts to a dual isomor- +phism ( )⍊ ∶ G(X) ⋍ G(Y )∂ ∶ ⍊( ). +Preorder relations are induced on each of the sorts, by setting for x,z ∈ X, +x ⪯ z iff {x}⍊ ⊆ {z}⍊ and, similarly, for y,v ∈ Y , y ⪯ v iff ⍊{y} ⊆ ⍊{v}. A (sorted) +frame is called separated if the preorders ⪯ (on X and on Y ) are in fact partial +orders ≤. +5 + +Our notational conventions are these of [13, Remark 3.2]. We repeat them +below, for the reader’s convenience. +Remark 3.1 (Notational conventions). For a sorted relation R ⊆ ∏j=n+1 +j=1 +Zij, +where ij ∈ {1,∂} for each j (and thus Zij = X if ij = 1 and Zij = Y when ij = ∂), +we make the convention to regard it as a relation R ⊆ Zin+1 ×∏j=n +j=1 Zij, we agree +to write its sort type as σ(R) = (in+1;i1⋯in) and for a tuple of points of suitable +sort we write uRu1⋯un for (u,u1,... ,un) ∈ R. +We use Γ to designate upper closure ΓU = {z ∈ X ∣ ∃x ∈ U x ⪯ z}, for U ⊆ X, +and similarly for U ⊆ Y . The set U is increasing (an upset) iff U = ΓU. For a +singleton set {x} ⊆ X we write Γx, rather than Γ({x}) and similarly for {y} ⊆ Y . +We typically use the standard Formal Concept Aanalysis priming notation +for each of the two Galois maps ⍊( ),( )⍊. This allows for stating and proving +results for each of G(X),G(Y ) without either repeating definitions and proofs, +or making constant appeals to duality. Thus for a Galois set G, G′ = G⍊, if +G ∈ G(X) (G is a Galois stable set), and otherwise G′ = ⍊G, if G ∈ G(Y ) (G is +a Galois co-stable set). +For an element u in either X or Y and a subset W, respectively of Y or +X, we write u∣W, under a well-sorting assumption, to stand for either u ⍊ W +(which stands for u ⍊ w, for all w ∈ W), or W ⍊ u (which stands for w ⍊ u, for +all w ∈ W), where well-sorting means that either u ∈ X,W ⊆ Y , or W ⊆ X and +u ∈ Y , respectively. Similarly for the notation u∣v, where u,v are elements of +different sort. +We designate n-tuples (of sets, or elements) using a vectorial notation, set- +ting (G1,... ,Gn) = ⃗G ∈ ∏j=n +j=1 G(Zij), ⃗U ∈ ∏j=n +j=1 ℘(Zij), ⃗u ∈ ∏j=n +j=1 Zij (where +ij ∈ {1,∂}). Most of the time we are interested in some particular argument +place 1 ≤ k ≤ n and we write ⃗G[F]k for the tuple ⃗G where Gk = F (or Gk is +replaced by F). Similarly ⃗u[x]k is (u1,... ,uk−1,x,uk+1,... ,un). +For brevity, we write ⃗u ⪯ ⃗v for the pointwise ordering statements u1 ⪯ +v1,... ,un ⪯ vn. We also let ⃗u ∈ ⃗ +W stand for the conjunction of component- +wise membership uj ∈ Wj, for all j = 1,... ,n. +To simplify notation, we write Γ⃗u for the n-tuple (Γu1,... ,Γun). +For a +unary map f and a tuple ⃗u we write f[⃗u] for the tuple (f(u1),... ,f(un)). +Note that the same notation is used for the image f[S] = {f(x) ∣ x ∈ S} of a set +under a function f, but context will make it clear what the intended meaning +is. The convention can be nested, so that if S is a set (or sequence) of tuples +⃗ui, then f[S] is the set (or sequence) consisting of the elements f[⃗ui]. +To refer to sections of relations (the sets obtained by leaving one argument +place unfilled) we make use of the notation ⃗u[ ]k which stands for the (n − 1)- +tuple (u1,... ,uk−1,[ ] ,uk+1,... ,un) and similarly for tuples of sets, extending +the membership convention for tuples to cases such as ⃗u[ ]k ∈ ⃗F[ ]k and similarly +for ordering relations ⃗u[ ]k ⪯ ⃗v[ ]k. +We also quantify over tuples (with, or +without a hole in them), instead of resorting to an iterated quantification over +the elements of the tuple, as for example in ∃⃗u[ ]k ∈ ⃗F[ ]k∃v,w ∈ G wR⃗u[v]k. +We extend the vectorial notation to distribution types, summarily writing +δ = (⃗ij;in+1) for (i1,... ,in;in+1). Then, for example, ⃗ij[∂]k is the tuple with +6 + +ik = ∂. Furthermore, we let ij = ∂, if ij = 1 and ij = 1, when ij = ∂. +Lemma 3.2 ( [13, Lemma 3.3] ). Let F = (X,⍊,Y ) be a polarity and u a point +in Z = X ∪ Y . +1. ⍊ is increasing in each argument place (and thereby its complement I is +decreasing in each argument place). +2. (Γu)′ = {u}′ and Γu = {u}′′ is a Galois set. +3. Galois sets are increasing, i.e. u ∈ G implies Γu ⊆ G. +4. For a Galois set G, G = ⋃u∈G Γu. +5. For a Galois set G, G = ⋁u∈G Γu = ⋂v∣G{v}′. +6. For a Galois set G and any set W, W ′′ ⊆ G iff W ⊆ G. +◻ +It is typical in the context of canonical extensions of lattices to refer to prin- +cipal upper sets Γx ∈ G(X)(x ∈ X = Filt(L)), as closed, or filter elements of +G(X) and to sets ⍊{y} ∈ G(X)(y ∈ Y = Idl(L)) as open, or ideal elements of +G(X), and similarly for sets Γy,{x}⍊ with x ∈ X,y ∈ Y . This creates an unfortu- +nate clash of terminology and we shall have to rely on context to disambiguate. +Furthermore, a closed element Γx is said to be clopen if Γx = ⍊{y} for some +y ∈ Y , which is unique when the frame is separated. +By Lemma 3.2, the closed elements of G(X) join-generate G(X), while the +open elements meet-generate G(X) (similarly for G(Y )). +Definition 3.3 (Galois dual relation). For a relation R, of sort type σ, its +Galois dual relation R′ is the relation defined by uR′⃗v iff ∀w (wR⃗v �→ w∣u). +In other words, R′⃗v = (R⃗v)′. +Definition 3.4 (Sections of relations). For an (n + 1)-ary relation Rσ (of sort +σ) and an n-tuple ⃗u, Rσ⃗u = {w ∣ wRσ⃗u} is the section of Rσ determined by +⃗u. +To designate a section of the relation at the k-th argument place we let +⃗u[ ]k be the tuple with a hole at the k-th argument place. Then wRσ⃗u[ ]k = +{v ∣ wRσ⃗u[v]k} ⊆ Zik is the k-th section of Rσ. +If R is a relation on a sorted residuated frame F = (X,I,Y ), of some sort type +σ = σ(R) = (in+1;i1⋯in), then as in the unsorted case, R generates a (sorted) +image operator αR, defined by (2), of sort σ(αR) = (i1,... ,in;in+1), defined by +the obvious generalization of the J´onsson–Tarski image operators [19], +αR( ⃗W) = {w ∈ Zin+1 ∣ ∃ ⃗w (wR ⃗w ∧ +j=n +⋀ +j=1 +(wj ∈ Wj))} += +⋃ +⃗w∈ ⃗ +W +R ⃗w, +(2) +where for each j, Wj ⊆ Zij (and recall that Zij = X when ij = 1 and Zij = Y , if +ij = ∂). Let αR be the closure of the restriction of αR to Galois sets ⃗F, +αR( ⃗F) = (αR( ⃗F))′′ = ⎛ +⎝ +wj∈Fj +⋃ +j=1,...,n +R ⃗w⎞ +⎠ +′′ += ⋁ +⃗w∈ ⃗ +F +(R ⃗w)′′, +(3) +where Fj ∈ G(Zij), for each j ∈ {1,...,n}. +7 + +Theorem 3.5 ( [13, Theorem 3.12]). Let F = (X,⍊,Y,R) be a frame with an +(n+1)-ary sorted relation, of some sort σ(R) = (in+1; ⃗ij) and assume that for any +w ∈ Zin+1 and any (n − 1)-tuple ⃗p[ ]k with pj ∈ Zij, for each j ∈ {1,...,n} ∖ {k}, +the sections wR′⃗p[ ]k of the Galois dual relation R′ of R are Galois sets. Then +αR distributes at the k-th argument place over arbitrary joins in G(Zik). +◻ +The Galois set operator αR is sorted. Single-sorted operators α1 +R on G(X) +and α∂ +R on G(Y ) are obtained by composition with the Galois connection, which +is a duality of G(X) and G(Y ). +Definition 3.6 (Full complex algebra). Let F = (X,⍊,Y,R) be a polarity with +a relation R of some sort σ(R) = (in+1;i1⋯in). The full complex algebra of F is +the structure F+ = (G(X),α1 +R) and its dual full complex algebra is the structure +F∂ = (G(Y ),α∂ +R). Subalgebras of full complex algebras will be referred to as +complex algebras of a frame. +Proposition 3.7. Let F = (X,⍊,Y ) be a sorted frame (a polarity) and G(X) +the complete lattice of stable sets. Let R be the ternary upper bound relation +on X defined by xRuz iff both u ⪯ x and z ⪯ x. If all sections of the Galois dual +relation R′ of R are Galois sets, then G(X) is completely distributive. +Proof. Let αR be the image operator generated by R, αR(U,W) = ⋃w∈W +u∈U Ruw. +Notice that, for stable sets A,C (more generally, for increasing sets), αR(A,C) = +A ∩ C. Hence αR(A,C) = αR(A,C) = A ∩ C, since Galois sets are closed under +intersection. Given the section stability hypothesis for the Galois dual relation +R′ of R, Theorem 3.5 applies, from which distribution of αR (i.e. of intersection) +over arbitrary joins of stable sets is concluded. +The image operator αR ∶ ∏j=n +j=1 ℘(Zij) �→ ℘(Zin+1) is a sorted normal and +completely additive function in each argument place, therefore it is residuated, +i.e. for each k there is a set-operator βk +R satisfying the condition +αR( ⃗W[V ]k) ⊆ U iff V ⊆ βk +R( ⃗W[U]k). +(4) +Hence βk +R( ⃗W[U]k) is the largest set V s.t. αR( ⃗W[V ]k) ⊆ U and it is thereby +definable by +βk +R( ⃗W[U]k) = ⋃{V ∣ αR( ⃗W[V ]k) ⊆ U}. +(5) +We let βk +R/ be the restriction of βk +R of equation (5) to Galois sets, according to +its sort type, explicitly defined by (6) +βk +R/( ⃗E[G]k) = ⋃{F ∈ G(Zik) ∣ αR( ⃗E[F]k) ⊆ G}. +(6) +Theorem 3.8 ( [13, Theorem 3.14, Lemma 3.15]). If αR is residuated in the k- +th argument place, then βk +R/ is its residual and βk +R/( ⃗E[G]k) is a Galois set, i.e. +the union in equation (6) is actually a join in G(Zik). Furthermore, equations +(7) and (8) +βk +R/( ⃗E[G]k) = ⋃{Γu ∈ G(Zik) ∣ αR( ⃗E[Γu]k) ⊆ G}, +(7) +βk +R/( ⃗E[G]k) = {u ∈ Zik ∣ αR( ⃗E[Γu]k) ⊆ G} +(8) +8 + +give an equivalent definition of βk +R/. +◻ +Thus, under the assumption of section stability for the Galois dual relation +R′, the operation of closure of restriction to Galois sets preserves residuation +and we obtain αR ⊣ βk +R/. +Frame morphisms are the weak bounded morphisms whose definition in [13, +Definition 3.20] is repeated below, where we let I be the complement of the +Galois relation ⍊ of a frame. +Definition 3.9. If π = (p,q) ∶ (X2,I2,Y2) �→ (X1,I1,Y1) is a pair of maps +p ∶ X2 �→ X1, q ∶ Y2 �→ Y1, then π will be called a (sorted) weak bounded +morphism iff +1. ∀x′ ∈ X2∀y′ ∈ Y2 (x′I2y′ �→ π(x′)I1π(y′)) +2. ∀x ∈ X1∀y′ ∈ Y2(xI1π(y′) �→ ∃x′ ∈ X2(x ≤ π(x′) ∧ x′I2y′)) +3. ∀x′ ∈ X2∀y ∈ Y1(π(x′)I1y �→ ∃y′ ∈ Y2(y ≤ π(y′) ∧ x′I2y′)). +Definition 3.10. If (p,q) ∶ (X2,I2,Y2) → (X1,I1,Y1), with p ∶ X2 → X1 and +q ∶ Y2 → Y1, then we let π = (p,q) and we define π−1 by setting +π−1(W) = { p−1(W) ∈ ℘(X2) +if W ⊆ X1 +q−1(W) ∈ ℘(Y2) +if W ⊆ Y1. +Similarly, we let +π(w) = { p(w) ∈ X1 +if w ∈ X2 +q(w) ∈ Y1 +if w ∈ Y2. +Proposition 3.11 ( [13, Corollary 3.21]). The inverse π−1 = (p,q)−1 of a weak +bounded morphism is a complete lattice homomorphism of the lattices of Galois +stable sets of sorted residuated frames. +◻ +For frames with relations, let π be a weak bounded morphism, π = (p,q) ∶ +(X2,I2,Y2,(Sσ)σ∈τ) �→ (X1,I1,Y1,(Rσ)σ∈τ), and let Rσ,Sσ be corresponding +relations in the two frames, of the same sort type. For simplicity, we omit the +subscript σ in the sequel. We recall the following from [13]. +Proposition 3.12 ( [13, Proposition 3.24, Lemma 3.25]). If for any ⃗u it holds +that π−1αR(Γ⃗u) = αS(π−1[Γ⃗u]), then for any tuple ⃗F of Galois sets of the +required sort π−1αR( ⃗F) = αS(π−1[ ⃗F]). Furthermore, equations +π−1αR(Γ⃗u) = αS(π−1[Γ⃗u]), +(9) +π(v)R⃗u iff +∃ ⃗w(⃗u ≤ π[ ⃗w] ∧ vS ⃗w). +(10) +provide equivalents to the above assumption that π−1αR(Γ⃗u) be identical to +αS(π−1[Γ⃗u]). +◻ +9 + +Table 1: Axioms for Sorted Residuated Frames of similarity type τ +(F0) The complement I of the Galois relation ⍊ of the frame is quasi-serial, in +the sense of condition (1) +(F1) The frame is separated +(F2) For each σ = (⃗ij;in+1) in the similarity type τ, each ⃗u ∈ ∏j=n +j=1 Zij, Rσ⃗u is +a closed element of G(Zin+1) +(F3) For each σ = (⃗ij;in+1) in the similarity type τ, each w ∈ Zin+1, the n-ary +relation wRσ is decreasing in every argument place +(F4) All sections of the Galois dual relations R′ +σ of Rσ, for each σ in τ, are +Galois sets +For π = (p,q) ∶ (X2,I2,Y2,S2ν) �→ (X1,I1,Y1,S1ν), where +p ∶ X2 �→ X1 and q ∶ Y2 �→ Y1 +(M1) ∀x′ ∈ X2∀y′ ∈ Y2 (x′I2y′ �→ p(x′)I1q(y′)) +(M2) ∀x ∈ X1∀y′ ∈ Y2(xI1q(y′) �→ ∃x′ ∈ X2(x ≤ p(x′) ∧ x′I2y′)) +(M3) ∀x′ ∈ X2∀y ∈ Y1(p(x′)I1y �→ ∃y′ ∈ Y2(y ≤ q(y′) ∧ x′I2y′)) +(M4) ∀z ∈ X1∀v ∈ Y2(q(v)S1νz �→ ∃x ∈ X2(z ≤ p(x) and vS2νx)) +We list in Table 1, after [13], the minimal axiomatization we shall assume +for a sorted residuated frame with relations F = (X,I,Y,(Rσ)σ∈τ). The axiom- +atization will be strengthened in the sequel imposing, among others, a spectral +topology on each of X,Y . +Note that axioms (F1) and (F2) imply that there is a (sorted) function ̂fR +on the points of the frame such that ̂fR(⃗u) = w iff R⃗u = Γw. The following +immediate observation will be useful in the sequel. +Lemma 3.13 ( [13, Lemma 3.16]). Let F be a frame of similarity type τ and +assume that axioms (F1)–(F3) in Table 1 hold. Then for a frame relation R of +type σ in τ, αR(Γ⃗u) = R⃗u = αR(Γ⃗u) = Γ( ̂fR(⃗u)). +◻ +3.2 +Frames for Quasi-Complemented Lattices +We now consider sorted frames F = (X,⍊,Y,Sν) with σ(Sν) = (∂;1), i.e. Sν ⊆ +Y × X, and we assume that axioms (F0)–(F4) of Table 1 hold. +Let S′ +ν be the Galois dual relation of Sν, defined by S′ +νz = (Sνz)′ for each +z ∈ X, let ηS ∶ ℘(X) �→ ℘(Y ) be the sorted image operator generated by Sν, +10 + +defined on U ⊆ X by +ηS(U) = {y ∈ Y ∣ ∃x ∈ X(ySνx and x ∈ U)} = ⋃ +x∈U +Sνx, +(11) +and ηS ∶ G(X) �→ G(Y ) be the closure of its restriction to Galois sets. Further- +more, let ̂ν ∶ X �→ Y be the point operator of Lemma 3.13, so that for a closed +element Γx ∈ G(X) we have ηS(Γx) = Sνx = ηS(Γx) = Γ(̂ν(x)). By axiom (F2), +Sνx is a Galois set. Hence for a stable set A ∈ G(X), +ηS(A) = ( ⋃ +x∈A +Sνx) +′′ += ⋁ +x∈A +Sνx = ⋁ +x∈A +Γ(̂ν(x)) = ⋁ +x∈A +ηS(Γx) ∈ G(Y ). +(12) +Let also ην(A) = (ηS(A))′ = ⋂z∈A S′ +νz, so that ην is a single-sorted operation +(on G(X)) derived from ηS by composition with the Galois connection. +Remark 3.14 (Switching Notation). Hereafter, we simplify notation, switching +to the more familiar � for the incompatibility relation S′ +ν and letting ην(A) be +designated by A∗. +For subsequent use, we make a note of the fact that +x ∈ A∗ iff ∀z(z ∈ A �→ x�z) iff x�A. +(13) +Since ηS ∶ G(X) �→ G(Y ) distributes over arbitrary joins, by the axioms +in Table 1 and Theorem 3.5, it is residuated with a map ζS ∶ G(Y ) �→ G(X), +which maps meets of G(Y ) (hence joins of G(X)) to meets of G(X), defined on +B ∈ G(Y ) (using also Theorem 3.8) by +ζS(B) = ⋁{A ∈ G(X) ∣ ηS(A) ⊆ B} = ⋃{A ∈ G(X) ∣ ηS(A) ⊆ B}. +By [13, Lemma 3.15], ζS(B) is equivalently defined by equation (14), specializing +equation (8), +ζS(B) = {x ∈ X ∣ ηS(Γx) ⊆ B}. +(14) +By duality of G(X) and G(Y ), every B ∈ G(Y ) is B = C′ for some C ∈ G(X). +Hence we obtain that ηS(A) ⊆ C′ iff A ⊆ ζS(C′). From this, setting ζν(C) = +ζS(C′), we obtain the Galois connection condition C ⊆ ην(A) iff A ⊆ ζν(C). +Recalling that we have switched notation to A∗ for ην(A) and setting A☆ = +ζν(A), we can rewrite the Galois condition as A ⊆ C☆ iff C ⊆ A∗. +Lemma 3.15. Let F = (X,⍊,Y,Sν) be a frame subject to the axioms of Table +1 and let A ∈ G(X) be any stable set. +1. The following are equivalent +(a) � is symmetric +(b) A ⊆ A∗∗ +(c) A∗ = A☆ +11 + +2. A∗☆ = ⋁x∈A ζSηS(Γx) +3. The following are equivalent +(a) A∗☆ ⊆ A, for any A ∈ G(X) +(b) ζSηS(Γx) ⊆ Γx, for any x ∈ X +(c) ∀x,z ∈ X[∀v ∈ Y (vSνz �→ vSνx) �→ x ≤ z] +4. The following are equivalent +(a) � is irreflexive +(b) A ∩ A∗ = ∅ +Proof. For (1) and the case (a)⇒(b), suppose, for a contradiction, that x ∈ A, +but x /∈ A∗∗. Let z ∈ A∗ such that x�z fails. But z ∈ A∗ = {z ∣ ∀u(u ∈ A �→ z�u)} +and x ∈ A, so z�x holds. By symmetry of � it follows x�z, contradiction. Hence +A ⊆ A∗∗, for any A ∈ G(X). +For (b)⇒(c), by Lemma 2.2, ( )∗ forms a Galois connection on G(X). By +uniqueness of adjoints it then follows that A∗ = A☆. +For (c)⇒(b), by definition ( )☆ is Galois connected with ( )∗ and hence if +the two are equal, then the result follows by using Lemma 2.2. +For (b)⇒(a), recall that � = S′ +ν, both sections of which are stable sets, which +are increasing sets by Lemma 3.2, and that by Lemma 2.2 the hypothesis means +that ( )∗ is a Galois connection on the lattice of stable sets. Assuming x�z, we +then have x�Γz, which means that x ∈ (Γz)∗ = {x ∣ ∀u(z ≤ u �→ x�u)}. If x ≤ w +and z ≤ u, then by x�z we also have w�u and this means that Γx ⊆ (Γz)∗. Since +( )∗ is antitone, (Γz)∗∗ ⊆ (Γx)∗. But then Γz ⊆ (Γz)∗∗ ⊆ (Γx)∗ from which we +obtain z ∈ (Γx)∗ and so z�x follows. +For claim (2), using definitions we obtain that +(A∗)☆ += +ζS ((A∗)′) += +ζs ((⋂x∈A{̂ν(x)}′)′) += +ζS (⋁x∈A Γ(̂ν(x))) = +⋁x∈A ζS(Γ(̂ν(x))) += +⋁x∈A ζSηS(Γx). +For claim (3) and the case (a)⇒(b), assuming that A∗☆ ⊆ A, for any A ∈ G(X) +and choosing A = Γx, for arbitrary x ∈ X, it is immediate, given the identity +proven in claim (2), that (Γx)∗☆ = ⋁x≤z ζSηS(Γz) = ζSηS(Γx) ⊆ Γx. +The converse, (b)⇒(a), is immediate: A∗☆ = ⋁x∈A ζSηS(Γx) ⊆ ⋁x∈A Γx = A. +To prove (b)⇔(c), we have +ζSηS(Γx) += +ζS(Sνx) += +⋃{z ∈ X ∣ ηS(Γz) ⊆ Sνx} += +⋃{z ∈ X ∣ Sνz ⊆ Sνx} +Hence, for any x ∈ X +ζSηS(Γx) ⊆ Γx +iff ∀z(Sνz ⊆ Sνx �→ x ≤ z) +iff ∀z[∀v(vSνz �→ vSνx) �→ x ≤ z] +12 + +For claim (4), if x ∈ A ∩ A∗ ≠ ∅, then it must be that x�x and thus � is not +irreflexive. Conversely, assume A ∩ A∗ = ∅, for any A ∈ G(X), but suppose that +x�x for some x ∈ X. Then x ∈ (Γx)∗ and thus Γx∩(Γx)∗ ≠ ∅, contradiction. +Corollary 3.16. Let F = (X,⍊,Y,Sν) be a frame satisfying axioms (F0)–(F4) +of Table 1, where we set � = S′ +ν, and let F+ be its full complex algebra, F+ = +(G(X),⊆,⋂,⋁,∅,X,( )∗). +1. F+ is a complete lattice with a minimal quasi-complementation operator +( )∗ on stable sets +2. F+ is a complete lattice with a quasi-complementation operator ( )∗ which +is a Galois connection on stable sets iff � is symmetric +3. F+ is a complete lattice with an involution ( )∗ iff � is symmetric and the +condition ∀x,z ∈ X[∀v ∈ Y (vSνz �→ vSνx) �→ x ≤ z] holds in the frame +4. F+ is a complete ortholattice iff � is symmetric and irreflexive and the +condition ∀x,z ∈ X[∀v ∈ Y (vSνz �→ vSνx) �→ x ≤ z] holds in the frame +5. F+ is a complete De Morgan algebra if (a) � is symmetric, (b) the condition +∀x,z ∈ X[∀v ∈ Y (vSνz �→ vSνx) �→ x ≤ z] holds in the frame and (c) +the sections of the Galois dual relation R′ of the upper bound relation R +of Proposition 3.7 are stable +6. F+ is a complete Boolean algebra if the conditions (a)–(c) of the previous +case hold in the frame and, in addition, (d) � is irreflexive. +Proof. Immediate, given Lemma 3.15 and Proposition 3.7. +4 +Choice-Free Representation of NLEs +4.1 +Semilattice Representation +Let M = (M,≤,∧,1) be a meet semilattice with a unit (top) element 1 and +X = Filt(M) its set of proper filters (we assume filters are nonempty, i.e. 1 ∈ x for +any x ∈ X). For each a ∈ M, let Xa = {x ∈ X ∣ a ∈ x} and B = {Xa ⊆ X ∣ a ∈ M} +and notice that Xa ∩ Xb = Xa∧b ∈ B, so that B itself is a meet semilattice with +unit element X = X1. +Let X = (X,B) be the topological space with carrier set X and topology Ω +generated by taking B as a basis. +Remark 4.1 (Notation). Principal filters of meet semilattices and lattices +are typically designated with the notation xa (= a↑), for an element a of the +(semi)lattice. Join semilattice and lattice principal ideals are similarly desig- +nated by ya = a↓. We typically use x,z for filters, y,v for ideals and u,w for +either case. +Lemma 4.2. Given a filter F in the lattice Ω(X) of open sets of X, define +F = {a ∈ M ∣ Xa ∈ F} and xF to be the filter of M generated by the set F. Then +13 + +1. for any basic open set Xa ∈ B, xF ∈ Xa iff Xa ∈ F +2. for any open set U of X, if xF ∈ U, then U ∈ F +3. if F is a completely prime filter in the lattice Ω(X) of open sets of X, then +xF ∈ U iff U ∈ F. +Proof. For (1), if xF ∈ Xa, then by definition of Xa = {x ∈ X ∣ a ∈ x} we have +a ∈ xF . By definition of xF , let e1,... ,en ∈ M be such that e1∧⋯∧en ≤ a and for +each i = 1,...,n, Xei ∈ F. Then ⋂n +i=1 Xei ∈ F. Also, ⋂n +i=1 Xei = Xe1∧⋯∧en ⊆ Xa +and so Xa ∈ F as well. Conversely, if Xa ∈ F, then a ∈ xF , which is to say that +xF ∈ Xa. +For (2), if U is open, let E ⊆ M be such that U = ⋃e∈E Xe. If xF ∈ U, then +let e ∈ E be an element such that xF ∈ Xe. By (1), Xe ∈ F and then since +Xe ⊆ U and F is a filter, U ∈ F follows. +For (3), it suffices to show, given (2) above, that if F is completely prime and +U ∈ F, then xF ∈ U. Now U = ⋃e∈E Xe for some E ⊆ M and then by complete +primeness of F we get Xe ∈ F, for some e ∈ E. It then follows by part (1) that +xF ∈ Xe ⊆ U. +Given any space X, a filter F of X is a non-empty upper set (with respect to +the specialization order ⊑ on X) such that for any x,z ∈ F a lower-bound u ∈ X +of {x,z} is in F. We let KOF(X) designate the family of compact-open filters of +X, following the notation of [23]. +Proposition 4.3. Let M be a meet semilattice, X = (X,B) its dual topological +space (where X = Filt(M) and B = {Xa ∣ a ∈ M} is a basis for the topology Ω +on X). Then +1. the space X is a spectral space +2. B = {Xa ∣ a ∈ M} = KOF(X). +Proof. Recall that a topological space is spectral if it is T0, coherent, compact +and sober, which we prove in turn in order to establish part (1). +For the T0 property, if x ≠ z are distinct filters, without loss of generality we +may assume that a ∈ x, but a /∈ z, for some semilattice element a ∈ M. Then the +open set Xa separates x,z, since x ∈ Xa but z /∈ Xa. +For the coherence property we verify that the basis B of the topology consists +of compact-open sets and that it is closed under finite intersections. For the +latter requirement, B is easily seen to be closed under finite intersections, since +Xa∩Xb = Xa∧b and the intersection of the empty family of X′ +as is X itself, which +is identical to X1 = {x ∈ X ∣ 1 ∈ x}. For the first requirement of coherence, the +X′ +as are certainly open, by definition of the topology. +For compactness, let +C ⊆ M and suppose that {Xe ∣ e ∈ C} covers Xa, i.e. Xa ⊆ ⋃e∈C Xe. Then the +principal filter xa = a ↑ ∈ Xa is in Xe, for some e ∈ C, hence e ∈ xa, i.e. a ≤ e. +Then e ∈ x, for any x ∈ Xa and this shows that Xa ⊆ Xe = {z ∈ X ∣ e ∈ z}. Hence +{Xe}, for this e, is the needed finite subcover of Xa. +14 + +Since X = X1 = {x ∈ X ∣ 1 ∈ x}, compactness of X follows from the previous +argument. +Sobriety of the space is equivalent to the requirement that every completely +prime filter F in the lattice Ω(X) of open sets of X is generated by a single point +xF , in other words that F = {U ∈ Ω(X) ∣ xF ∈ U}. This was shown to hold in +Lemma 4.2, part (3). +For part (2), left to right, Xa is compact-open, by the proof of coherence +for X in part (1) of this proposition. Furthermore, x ∈ Xa iff a ∈ x iff xa ⊆ x iff +x ∈ Γxa. Hence Xa is a (principal) filter. +Conversely, let F ⊆ X be a compact-open filter of X. Being open, let F = +⋃a∈E⊆M Xa, so that by compactness, F = Xa1 ∪ ⋯ ∪ Xan for some n. By ai↑ = +xai ∈ Xai ⊆ F, all the xai’s, for i = 1,... ,n, are in F, hence so is their meet +(intersection), since F is a filter. Letting u = ⋂n +i=1 xai, we show that F = Γu = u↑. +Right-to-left is obvious since u ∈ F, which is a filter, so Γu ⊆ F. For left-to-right, +let z ∈ F, so that z ∈ Xak = Γxak for some k ∈ {1,...,n}, hence xak ⊆ z. By +definition of u we obtain u ⊆ xak ⊆ z and then z ∈ Γu. Hence F ⊆ Γu follows, +too. Thus F = Γu = ⋃n +i=1 Xai and thus u ∈ Xai = Γxai for some i. But then +xai ⊆ u = ⋂n +i=1 xai ⊆ xai so that u = xai, for this i, and so F = Γxai = Xai ∈ B. +It should be pointed out that, except for the phrasing, notation and detail, +the arguments in the proofs of Lemma 4.2 and Proposition 4.3 are the same as +these involved in showing that the space of proper filters of a Boolean algebra is +spectral [2, Proposition 3.12], or that the space of proper filters of an ortholattice +is spectral [22, Proposition 3.4.1]. It is really only the semilattice-structure that +is relevant in the argument, which is one of the reasons that we included a proof +of Proposition 4.3, the other reason relating to the observation made in [13,14] +that a lattice can be always regarded as a diagram of dually isomorphic meet +semilattices. For the case of Boolean algebras and ortholattices, this duality +may be taken to be Boolean complementation, or ortho-complementation, re- +spectively. For general bounded lattices the semilattice duality can be taken to +be the identity map ı ∶ L∧ ⇆ (L∨)∂, where L∨ = (L∧)∂, as in [13,14], and as we +explain in more detail in the sequel. +Note that the topology Ω on X is the Scott topology, with respect to the +specialization order ⊑ on X [18, chapter II.1], which is inclusion of filters (of +M). To see that specialization coincides with filter inclusion note that if x ⊑ z, +i.e. N o(x) ⊆ N o(z) and a ∈ x, then x ∈ Xa ∈ N o(x) ⊆ N o(z), hence also z ∈ Xa, +i.e. a ∈ z and so x ⊆ z. Conversely, if x ⊆ z and U is an open neighborhood of +X, let Xa be a basic open such that x ∈ Xa ⊆ U. From x ∈ Xa we get a ∈ x ⊆ z, +so also z ∈ Xa ⊆ U, hence U ∈ N o(z), i.e. N o(x) ⊆ N o(z) which by definition +means that x ⊑ z. +It follows from Proposition 4.3 that the space X is an HMS space (named +so in [23], in honour of Hofmann, Mislove and Stralka), defined by a set of +equivalent conditions in [23, Theorem 2.5]). +The following representation result is an immediate consequence of Propo- +sition 4.3. +15 + +Corollary 4.4. Given a meet semilattice M, the map a ↦ Xa is a semilattice +isomorphism M ⋍ KOF(Filt(M)). +◻ +4.2 +The Canonical Dual Space of a Lattice +If N is a join semilattice with unit (bottom) element 0, then N∂ (the opposite +semilattice, order reversed) is a meet semilattice with unit (top) and Filt(N∂) = +Y = Idl(N). The topology generated by the basis of sets Y a = {y ∈ Y ∣ a ∈ y}, for +a ∈ N, is a spectral topology by Proposition 4.3, observing that Y a ∩Y b = Y a∨b, +which ensures that the basis C = {Y a ∣ a ∈ N} is closed under finite intersections +(with the empty intersection being Y0 = Y itself). +For the lattice case, just as orthonegation is represented in [9] by the binary +relation � ⊆ X × X defined by x ⊥ z iff ∃a(a ∈ x ∧ a� ∈ z), the identity trivial +duality ı ∶ L ⋍ (L∂)∂ is similarly represented [13,14] by the sorted binary relation +⍊ ⊆ X × Y defined by x ⍊ y iff ∃a(a ∈ x ∧ ı(a) ∈ y) iff x ∩ y ≠ ∅. +Note that the quasi-seriality condition (1) holds for the complement of the +canonical Galois relation ⍊. +As in [14], we represent lattices and normal lattice expansions, more gen- +erally, in topologized sorted frames (polarities) F = (X,⍊,Y,(Rσ)σ∈τ), where +for each normal lattice operator f of distribution type σ = (i1,... ,in;in+1) +the frame is equipped with a sorted relation Rσ of sort σ = (in+1;i1⋯in), i.e. +Rσ ⊆ Zin+1 × ∏n +j=1 Zij and where Zij = X when ij = 1 and Zij = Y when ij = ∂. +Proposition 4.5. For a bounded lattice L, the bases B = {Xa ∣ a ∈ L} and +C = {Y a ∣ a ∈ L} of the spaces X = (X,B),Y = (Y,C), where X = Filt(L) +and Y = Idl(L), are the families of compact-open Galois stable and co-stable, +respectively, sets. Furthermore, B and C are dually isomorphic bounded lattices, +with B a sublattice of G(X) and C a sublattice of G(Y ). +Proof. The two cases for B and C are similar. +The proof follows from [14, +Lemma 2.7]. That lemma is phrased in terms of clopen sets in a Hausdorff +space, which are then compact-open and compactness and (co)stability are the +only properties needed and used in the argument. Other than that, the claim +in [14, Lemma 2.7] is phrased in terms of an arbitrary duality ℓ ∶ S ⋍ K∂ ∶ r +between meet semilattices. +The case for lattices follows by specializing the +argument to the trivial duality ℓ = r = ı ∶ L∧ ⇆ (L∨)∂, which we do below. +First, stability of the sets Xa,Y a follows by Lemma 3.2, observing that +Xa = Γxa and Y a = Γya. It remains to show that every stable compact-open +subset A of X is of the form Xa, for some lattice element a ∈ L. Assume A = A′′ +is compact-open and let x /∈ A′′ = A. Let y ∈ A′ such that x /⍊ y, i.e. x ∩ y = ∅. +By y ∈ A′ we have A ⍊ y, i.e. for any z ∈ A we have z ⍊ y, i.e. az ∈ z ∩ y, for +some lattice element az. Thus A ⊆ ⋃z∈A Xaz and, by compactness, it follows +that A ⊆ Xaz1 ∪ ⋯ ∪ Xazn, for some n. Letting ax = az1 ∨ ⋯ ∨ azn it follows that +for all i = 1,... ,n, Xazi ⊆ Xax, hence A ⊆ Xax. Notice that ax /∈ x, since ax ∈ y. +Hence x /∈ Xax. This shows that −A ⊆ −Xax and given we also obtained A ⊆ Xax +it follows that A = Xax. +16 + +That both B,C are (dually isomorphic) lattices follows from the fact that +(Xa)′ = Y a and (Y a)′ = Xa. Joins in B,C are defined by taking closures of +unions: A ∨ C = (A ∪ C)′′, as in G(X) and G(Y ). +Let KOG(X),KOG(Y ) be the families of Galois compact-open subsets of X +and of Y , respectively. By Propositions 4.3 and 4.5, KOF(X) = KOG(X) and +KOF(Y) = KOG(Y ). +The following choice-free lattice representation theorem is an immediate con- +sequence of our so far results in this article. +Theorem 4.6 (Choice-free Lattice Representation). Let L = (L,≤,∧,∨,0,1) be +a bounded lattice and (X,⍊,Y ) its dual filter-ideal frame (X = Filt(L),Y = +Idl(L)), with ⍊ ⊆ X × Y defined by x ⍊ y iff x ∩ y ≠ ∅. Let X = (X,B) and +Y = (Y,C) be the spectral spaces generated by the bases B = {Xa ∣ a ∈ L} and +C = {Y a ∣ a ∈ L}, respectively. +Then the map a ↦ Xa is a lattice isomorphism L ⋍ KOG(Filt(L)) and the +map a ↦ Y a is a dual isomorphism L∂ ⋍ KOG(Idl(L)). +◻ +The representation of (semi)lattices detailed above is essentially the same +as that in [14], by this author and Dunn, the difference lying in the choice of +the topology to be imposed on the filter space. +The lattice G(X) of Galois +stable sets is a canonical extension of the lattice, see [8, Proposition 2.6], which +is unique up to an isomorphism that commutes with the lattice embeddings, +by [8, Proposition 2.7]. +Moshier and Jipsen [23] provide a topological construction of the canonical +extension of a lattice. Proposition 4.5, together with uniqueness of canonical +extensions up to isomorphism, entails that the filter space of a lattice is what +Moshier and Jipsen call a BL-space, defined by a number of equivalent conditions +in [23, Theorem 3.2]. +It can be easily verified that the canonical extension +FSat(X) defined in [23] is literally identical to G(X). +We substantiate this +claim below. Recall first that OF(X) designates in [23] the family of open filters +of X = Filt(L). +Proposition 4.7. The following hold +1. A Galois stable set A = A′′ is a filter of X = Filt(L) and, similarly, a Galois +co-stable set B = B′′ is a filter of Y = Idl(L). +2. Every open filter F ∈ OF(X) is of the form ⍊{y} for a unique, by separation +of the frame (cf Lemma 3.2), ideal y ∈ Y = Idl(L) +3. For any subset U ⊆ X, U ′′ = fsat(U). Therefore, G(X) = FSat(X). +Proof. For part (1), the proof is straightforward and we only discuss the case of +stable sets. First, Galois sets are upsets, by Lemma 3.2. Now let x,z ∈ A = A′′ +and suppose that x ∩ z /∈ A′′. +Then there exists an ideal y ∈ A′ such that +(x ∩ z) /⍊ y, i.e. (x ∩ z) ∩ y = ∅. By x,z ∈ A′′, let a ∈ x ∩ y ≠ ∅ and b ∈ z ∩ y ≠ ∅. +Then both a,b ∈ y, hence a ∨ b ∈ y. But x,z are filters, hence a ∨ b ∈ x ∩ z, which +contradicts the assumption that x ∩ z /∈ A′′. +17 + +For part (2), for any y ∈ Y , ⍊{y} is Galois stable, hence a filter of X, by part +(1). To see that it is an open set, let x ∈ ⍊{y}, i.e. x ⍊ y, so that a ∈ x ∩ y ≠ ∅. +Thus x ∈ Xa. Since a ∈ y, Xa ⍊ y, so that we obtain x ∈ Xa ⊆ ⍊{y}. Thus +⍊{y} ∈ OF(X), for any y ∈ Y . +Conversely, let F ∈ OF(X) and let E ⊆ L be such that F = ⋃a∈E Xa. Let y +be the ideal generated by E. Thus e ∈ y iff there exist e1,... ,en ∈ E, for some +n, such that e ≤ e1 ∨ ⋯ ∨ en. We show that F = ⍊{y}, for this y. +If x ∈ F = ⋃a∈E Xa, then x ∈ Xa, for some a ∈ E. +By definition of y, +we get a ∈ y, so x ⍊ y, i.e. +x ∈ ⍊{y}. +Hence F ⊆ ⍊{y}. +Conversely, let +x ⍊ y so that e ∈ x ∩ y. +Then e ≤ e1 ∨ ⋯ ∨ en, where {e1,... ,en} ⊆ E. +It +follows that xe1 ∩ ⋯ ∩ xen = xe1∨⋯∨en ⊆ xe ⊆ x. Since ei ∈ E, we have Xei ⊆ F. +Because xei ∈ Xei, all principal filters xe1,... ,xen ∈ F. Since F is a filter, their +intersection is in F and then also x ∈ F. Hence ⍊{y} ⊆ F. +For part (3), by Lemma 3.2 the set of open elements ⍊{y} is meet-dense in +G(X). By part (2) above, OF(X) = {⍊{y} ∣ y ∈ Y }. Using also the definition of +F-saturation in [23] we obtain +fsat(U) = ⋂{F ∈ OF(X) ∣ U ⊆ F} = ⋂{⍊{y} ∣ U ⍊ y} = U ′′ +and so FSat(X) = {A ⊆ X ∣ A = fsat(A)} = {A ⊆ X ∣ A = A′′} = G(X). +4.3 +Representing Normal Lattice Operators +Let L = (L,≤,∧,∨,0,1,f) be a bounded lattice with a normal operator f. Then +f extends to a completely normal operator F, of the same distribution type as +f, on the canonical extension G(Filt(L)) of L. +For the proof, we refer the reader to [13, Sections 3.1,4.1,4.2]. The represen- +tation of the operator is the same as that given in [11], the difference lying in +the axiomatization of the dual frame of the lattice expansion, in particular on +the axioms for the relations corresponding to the lattice operator. Particular +instances of the representation were also given in [15]. +To keep this article as self-contained as possible, we sketch the representation +steps, drawing on [13, Section 4.1]. +The base polarity F = (Filt(L),⍊,Idl(L)) consists of the sets X = Filt(L) of +filters and Y = Idl(L) of ideals of the lattice and the relation ⍊ ⊆ Filt(L)×Idl(L), +defined by x ⍊ y iff x ∩ y ≠ ∅, while the representation map ζ1 sends a lattice +element a ∈ L to the set of filters that contain it, ζ1(a) = {x ∈ X ∣ a ∈ x} = +{x ∈ X ∣ xa ⊆ x} = Γxa. Similarly, a co-represenation map ζ∂ is defined by +ζ∂(a) = {y ∈ Y ∣ a ∈ y} = {y ∈ Y ∣ ya ⊆ y} = Γya. +For each normal lattice operator a relation is defined, such that if δ = +(i1,... ,in;in+1) is the distribution type of the operator, then σ = (in+1;i1⋯in) is +the sort type of the relation. Without loss of generality, we may restrict to just +two normal operators f, of output type 1, and h, of output type ∂. We then de- +fine two corresponding relations R,S of respective sort types σ(R) = (1;i1⋯in) +and σ(S) = (∂;t1⋯tn), where for each j, ij and tj are in {1,∂}. In other words +R ⊆ X × ∏j=n +j=1 Zij and S ⊆ Y × ∏j=n +j=1 Ztj. +18 + +To define the relations, we use the point operators introduced in [10] (see +also [11]). In the generic case we examine, we need to define two sorted operators +̂f ∶ +j=n +∏ +j=1 +Zij �→ Z1 +̂h ∶ +j=n +∏ +j=1 +Ztj �→ Z∂ +(recall that Z1 = X,Z∂ = Y ). +Assuming for the moment that the point operators have been defined, the canon- +ical relations R,S are defined by +xR⃗u +iff +̂f(⃗u) ⊆ x (for x ∈ X and ⃗u ∈ +j=n +∏ +j=1 +Zij), +yS⃗v +iff ̂h(⃗v) ⊆ y (for y ∈ Y +and ⃗v ∈ +j=n +∏ +j=1 +Ztj). +(15) +Returning to the point operators and letting xe,ye be the principal filter and +principal ideal, respectively, generated by a lattice element e, these are uniformly +defined as follows, for ⃗u ∈ ∏j=n +j=1 Zij and ⃗v ∈ ∏j=n +j=1 Ztj +̂f(⃗u) = ⋁{xf(⃗a) ∣ ⃗a ∈ ⃗u} +̂h(⃗v) = ⋁{yh(⃗a) ∣ ⃗a ∈ ⃗v}. +(16) +In other words, ̂f(⃗u) is the filter generated by the set {f(⃗a) ∣ ⃗a ∈ ⃗u}. Similarly +̂h(⃗v) is the ideal generated by the set {h(⃗a) ∣ ⃗a ∈ ⃗v}. +Proposition 4.8. In the canonical lattice frame all axioms of Table 1 hold. +In particular, all sections of the Galois dual relations R′,S′ of the canonical +relations R,S, defined by equations (15), are Galois sets. +Proof. The proof for axioms (F1)–(F3) is given in [13, Lemma 4.3]. For axiom +(F4), the claim was first stated as [12, Lemma 25] and a proof of one of the +subcases was detailed, the other one being sufficiently similar. The omitted proof +of the other subcase was provided in [13, Lemma 4.6]. Axiom (F0) obviously +holds in the canonical frame since every proper filter x does not intersect the +principal ideal ya, for any a /∈ x. Similarly for ideals. +The following results will be useful in the sequel. +Lemma 4.9 ( [12, Lemma 23]). In the canonical frame, xR⃗u holds iff ∀⃗a ∈ Ln +(⃗a ∈ ⃗u �→ f(⃗a) ∈ x). Similarly, yS⃗v holds iff ∀⃗a ∈ Ln (⃗a ∈ ⃗v �→ h(⃗a) ∈ y). +◻ +Lemma 4.10 ( [12, Lemma 24]). Where R′,S′ are the Galois dual relations of +the canonical relations R,S, yR′⃗u holds iff ̂f(⃗u) ⍊ y iff ∃⃗b(⃗b ∈ ⃗u ∧ f(⃗b) ∈ y). +Similarly, xS′⃗v holds iff x ⍊ ̂h(⃗v) iff ∃⃗e(⃗e ∈ ⃗v ∧ h(⃗e) ∈ x). +◻ +Each of the relations R,S generates a classical, though sorted, completely +additive image operator αR,ηS, respectively, and we designate by αR,ηS the +closure of their restriction to Galois sets (stable, or co-stable, according to the +distribution types of f,h). By Theorem 3.5 and Proposition 4.8, αR,ηS dis- +tribute over arbitrary joins of Galois sets. Composing with the Galois connec- +tion, which is a duality of the complete lattices of Galois stable and co-stable +19 + +sets, completely normal operators are obtained, αf,ηh ∶ ∏n +i=1 G(X) �→ G(X), +of the same distribution type as f,h, respectively, explicitly defined by +αf(A1,... ,An) = +αR(... , Aj +� +ij=1 +,... , A′ +r +� +ir=∂ +,...) +(A1,... ,An ∈ G(X)), +(17) +ηh(B1,... ,Bn) = +ηS(... , Br +� +ir=∂ +,... , B′ +j +� +ij=1 +,...) +(B1,... ,Bn ∈ G(Y )). +(18) +By [13, Proposition 5.2], αf,ηh restrict to normal operators of the respective +distribution type on the lattice KOG(X) (which, in [13], is identified as the +lattice of clopen (in the lattice-theoretic sense) elements of G(X)). We can then +conclude with the representation theorem below. +Theorem 4.11. Given a similarity type τ, let τ1,τ∂ be the subtypes consisting +of all distribution types in τ of output type 1 and ∂, respectively. If L is a +normal lattice expansion of type τ, L = (L,≤,∧,∨,0,1,(fδ)δ∈τ1,(hδ′)δ′∈τ∂ ), then +the representation map ζ(a) = {x ∈ X ∣ a ∈ x} = Xa, where X = Filt(L), is an +isomorphism ζ ∶ L ⋍ (KOG(X),⊆,∩,∨,∅,X,(αf)δ(f)∈τ1,(ηh)δ(h)∈τ∂ ) of normal +lattice expansions. +◻ +The next Proposition identifies the canonical extension of normal operators +we have defined as their σ/π-extension. +Proposition 4.12. The operations αf, ηh are the σ and π extensions of f,h, +respectively, as these are defined in [8]. +Proof. The argument has been detailed in [12, Section 3.3]. +Roughly, given +Proposition 4.8, Lemma 3.13 applies so that if R is the relation constructed +from a normal lattice operator f, then αR(Γ⃗u) = R⃗u = αR(Γ⃗u) = Γ( ̂fR(⃗u)). As- +suming f is of distribution type δ = (⃗ij;1), f σ as defined in [8] is a sorted map +and it is defined on a tuple ⃗F of Galois sets by extending its definition on closed +elements, by f σ( ⃗F) = ⋁⃗u∈ ⃗ +F fσ(Γ⃗u), where fσ is defined on closed elements. It +is shown in [12, Section 3.3] that fσ(Γ⃗u), as defined in [8], satisfies the identity +fσ(Γ⃗u) = Γ( ̂fR⃗u) ∈ G(X). Thereby, the σ-extension of f coincides with the +operation αR that we defined. For an operator h of distribution type (⃗tj;∂), +with corresponding canonical frame relation S, its dual σ-extension on closed +elements satisfies, respectively, the identity h∂ +σ(Γ⃗u) = Γ(̂hS⃗u) ∈ G(Y ). Extend- +ing to Galois sets we similarly have h∂ +σ( ⃗F) = ηS( ⃗F). The single-sorted σ and +π-extensions of f,h, respectively, are then obtained by composing appropriately +with the Galois connection, resulting in the maps αf = f σ,ηh = hπ, as defined +in equations (17) and (18), respectively. +5 +Representing Quasi-Complemented Lattices +In this section we extend the lattice representation of section 4.2 to the case +of a lattice with an additional quasi-complementation operator, assuming the +20 + +axiomatization of at least the minimal system of Figure 1 and specializing the +constructions of Section 4.3. +The canonical dual frame is the structure (X,⍊,Y,Sν), where X = Filt(L), +Y = Idl(L), ⍊ ⊆ X × Y is defined by x ⍊ y iff x ∩ y ≠ ∅ and Sν ⊆ Y × X is the +canonical relation defined using the point operator ̂ν ∶ X �→ Y , by equations +(15) and (16). For the case at hand, the definitions are given by equation (19) +̂ν(x) = ⋁{yνa ∣ a ∈ x} +ySνx iff ν(x) ⊆ y +(x ∈ X,y ∈ Y ). +(19) +Observe that Sνx = Γ(̂ν(x)). +By Lemma 4.9, Sν ⊆ Y × X is equivalently defined by the condition +ySνx iff ∀a ∈ L(a ∈ x �→ νa ∈ y). +By Lemma 4.10 its Galois dual relation S′ +ν ⊆ X × X is defined by +zS′ +νx iff ∀y ∈ Y (ySνx �→ z ⍊ y) iff z ⍊ ̂ν(x) iff ∃a ∈ L(a ∈ x and νa ∈ z). (20) +Observe also that S′ +νx = {̂ν(x)}′ = ⍊{̂ν(x)}. +A sorted image operator ηS ∶ ℘(X) �→ ℘(Y ) is generated by the relation +Sν, defined by equation (11). +The closure of the restriction of ηS to stable +sets is designated by ηS, defined by equation (12). +By Theorem 3.5, given +also Proposition 4.8, ηS distributes over arbitrary joins of stable sets in G(X), +returning a join of co-stable sets in G(Y ). +The canonical extension ην ∶ G(X) �→ G(X) of the normal lattice operator +ν is then obtained by composing with the Galois connection, setting ην(A) = +(ηS(A))′ ∈ G(X), hence ην(A) = ⋂x∈A S′ +νx, where S′ +ν is the Galois dual relation +of Sν. Given (12), we obtain ην(A) = ⋂x∈A +⍊{̂ν(x)}. +In the terminology of [8] ην = νπ is the π-extension of the lattice operator ν, +defined by +νπ(A) = ⋂ +x∈A +⍊{̂ν(x)} = ⋂ +x∈A +S′ +νx = ην(A) = ⍊(ηS(A)) = ⍊(ηS(A)). +(21) +Remark 5.1 (Switching Notation). Hereafter, we simplify notation as in Re- +mark 3.14, switching to the more familiar � for the incompatibility relation S′ +ν +and letting νπ(A) (i.e. ην(A)) be designated by A∗. +Theorem 5.2. Let L = (L,≤,∧,∨,0,1,ν) be a bounded lattice with an antitone +map ν. If L belongs to one of the lattice varieties of Figure 1, then so does, +respectively, the full complex algebra (G(X),⊆,⋂,⋁,∅,X,( )∗) of its canonical +frame. In other words, each of the varieties of Figure 1 is closed under canonical +extensions. +Proof. Assume first that ν in L is a minimal quasi-complement. By the re- +sults of sections 3 and 4.3, ( )∗ ∶ G(X) �→ G(X) co-distributes over arbitrary +joins in G(X), returning a meet, i.e. +(⋁i∈I Ai)∗ = ⋂i∈I A∗ +i . +Hence the full +complex algebra of the canonical frame (G(X),⊆,⋂,⋁,∅,X,( )∗) is a complete +lattice with a minimal quasi-complementation operator, given that we also have +21 + +∅∗ = (ηS(∅))′ = ∅′ = X, using normality of the classical, though sorted, image +operator ηS. +If ν satisfies the Galois condition a ≤ ννa, then it is immediate that the +canonical relation � = S′ +ν, defined by equation (20), is symmetric. By Lemma +3.15, this implies that A ⊆ A∗∗, for any A ∈ G(X). +Assume now that ν is an involution. To show that A∗∗ ⊆ A, it suffices to +verify that the equivalent condition (3)c of Lemma 3.15 holds in the canonical +frame. Given that Sνz = Γ(̂ν(z)), where for a filter z, ̂ν(z) is the ideal generated +by the set {νe ∣ e ∈ z}, the inclusion Sνz ⊆ Sνx is equivalent to the inclusion +Γ(̂ν(z)) ⊆ Γ(̂ν(x)), hence to ̂ν(x) ⊆ ̂ν(z). +To see that this implies x ⊆ z, +let a ∈ x. +Then νa ∈ ̂ν(x), so νa ∈ ̂ν(z). Let then e1,... ,en ∈ z such that +νa ≤ νe1 ∨ ⋯ ∨ νen. This is equivalent to ν(νe1 ∨ ⋯ ∨ νen) ≤ ννa ≤ a, in turn +equivalent to e1 ∧ ⋯ ∧ en ≤ a. Since z is a filter, this implies a ∈ z and this +completes the proof that x ⊆ z under the given assumption. +If the lattice is an ortholattice, then by the argument for the case of lattices +with an involution previously given and by Lemma 3.15 it suffices to verify that +the canonical relation � = S′ +ν is irreflexive. By Lemma 4.10, x�z holds iff there +exists a lattice element e such that e ∈ z and νe ∈ x. Reflexivity, x�x, would +then imply that e ∧ νe = 0 ∈ x, contradicting the fact that x is a proper filter. +For the case where the lattice is a De Morgan algebra, i.e. a distributive +lattice with an involution, it suffices to prove that G(X) is distributive. An +algebraic proof of this has been given in [8, Lemma 5.1], but we provide here a +new proof based on the constructions we have presented. +Note first that both lattice join ∨ and meet ∧ are trivially normal lattice +operators in the sense of Definition 2.1, but meet is an operator (in the J´onsson- +Tarski sense) only when it distributes over joins. When this is the case, meet +also has the distribution type (1,1;1). Its σ-extension ∧σ, is constructed as +outlined in section 4.3. Specifically, letting ∧ = f, the point operator ̂f on filters +is defined by ̂f(x,z) = ⋁{xa∧b ∣ a ∈ x and b ∈ z} and the canonical relation R∧ is +then defined by xR∧uz iff ∀a,b(a ∈ u and b ∈ z �→ a ∧ b ∈ x), using Lemma 4.9. +Note that R∧ is the upper bound relation of Proposition 3.7. Considering the +image operator αR ∶ ℘(X)×℘(X) �→ ℘(X) defined by αR(U,W) = {x ∈ X ∣ ∃u ∈ +U∃z ∈ W xR∧uz}, we obtain that αR(A,C) = A ∩ C, for A,C ∈ G(X). By +Proposition 4.8, all sections of the Galois dual relation of R∧ are stable. It then +follows by Proposition 3.7 that intersection distributes over arbitrary joins, in +other words, G(X) is a completely distributive lattice. +Finally, for the case of Boolean algebras, combine the arguments given for +ortholattices and De Morgan algebras. +6 +Spectral Duality +Let M,G,INV,DMA,O and BA be the categories of algebras in the respec- +tive varieties M,G,INV,DMA,O and BA of Figure 1 with the usual algebraic +homomorphisms. +As in [13], SRFτ designates the category of sorted residuated frames with +22 + +Table 2: Axioms for SRF∗ +νM +(F0) The complement I of the Galois relation ⍊ of the frame is quasi-serial, in +other words ∀x ∈ X∃y ∈ Y xIy and ∀y ∈ Y ∃x ∈ X xIy +(F1) The frame is separated +(F2) For each z ∈ X, Sνz is a closed element of G(Y ) and if z is a clopen element +(i.e. Γz = ⍊{v} for a (unique, by separation) point v in Y ), then Sνz is a +clopen element of G(Y ) +(F3) For each y ∈ Y the set ySν is decreasing (a down set) +(F4) Both sections of the Galois dual relation S′ +ν of Sν are Galois sets +(F5) Clopen elements are closed under finite intersections in each of G(X),G(Y ) +(F6) The family of closed elements, for each of G(X),G(Y ), is the intersection +closure of the respective family of clopens +(F7) Each of X,Y carries a spectral topology generated by the basis of their +respective families of clopen elements +For a sorted map π = (p,q) ∶ (X2,I2,Y2,S2ν) �→ (X1,I1,Y1,S1ν), where +p ∶ X2 �→ X1 and q ∶ Y2 �→ Y1 +(M1) ∀x′ ∈ X2∀y′ ∈ Y2 (x′I2y′ �→ p(x′)I1q(y′)) +(M2) ∀x ∈ X1∀y′ ∈ Y2(xI1q(y′) �→ ∃x′ ∈ X2(x ≤ p(x′) ∧ x′I2y′)) +(M3) ∀x′ ∈ X2∀y ∈ Y1(p(x′)I1y �→ ∃y′ ∈ Y2(y ≤ q(y′) ∧ x′I2y′)) +(M4) ∀z ∈ X1∀v ∈ Y2(q(v)S1νz �→ ∃x ∈ X2(z ≤ p(x) and vS2νx)) +(M5) for all points u, π−1(Γu) = Γv, for some (unique, by separation) v +a relation Rσ for each σ ∈ τ. For our present purposes we only consider frames +F = (X,⍊,Y,Sν), with σ(Sν) = (∂;1), given that the distribution type of the +normal lattice operator ν under study is δ(ν) = (1;∂). In particular then we let +SRFν = SRF{(1;∂)} designate the category with objects the sorted residuated +frames with a relation Sν as above, subject to the axioms of Table 1. SRFν is +too large a category for duality purposes and we specify full subcategories for +each of the cases of interest. In [13], the notation SRF∗ +τ was used to designate +the intended subcategory and we keep with this notation, while also subscripting +appropriately to distinguish between the different categories of interest in this +article. For a frame F in SRF∗ +τ, we let L(F) be the full complex algebra F+ of +stable sets and L∗(F) the complex algebra of clopen elements. +23 + +Theorem 6.1. Let SRF∗ +νM be the full subcategory of SRFν axiomatized by +the axioms in Table 2. There exist functors F,L∗ forming a categorical duality +F ∶ M ⋍ (SRF∗ +νM)op ∶ L∗. +Proof. Let L = (L,≤,∧,∨,0,1,ν) be a lattice with a minimal quasi complementa- +tion operation. Define F(L) = (Filt(L),⍊,Idl(L),Sν) to be the canonical frame +of the lattice constructed in Section 5. Axioms (F0)–(F4) hold for the canonical +frame, by Proposition 4.8. Note that axiom (F2) of Table 2 is a strengthening +of the corresponding axiom in Table 1. +To verify the stronger version of (F2), suppose Γz = ⍊{v} is a clopen ele- +ment. Clopen elements of G(X) in the canonical frame are precisely the stable +compact-open sets Xa = Γxa = xa↑ = ⍊{ya}. By definition of the point operator +̂ν and of the canonical relation Sν in equation (19), Sνxa = Γ(̂ν(xa)). It is +straightforward to see that ̂ν(xa) = yνa, hence Sνxa = Γyνa = Y νa is a clopen el- +ement of G(Y ), where yνa = (νa)↓ is the principal ideal generated by the lattice +element νa. +Axiom (F5) holds, since Xa ∩ Xb = Xa∧b, while also Y a ∩ Y b = Y a∨b. For +(F6), by join-density of principal filters, any filter x is the join x = ⋁a∈x xa, hence +every closed element Γx of G(X) is an intersection Γx = ⋂a∈x Γxa = Γ(⋁a∈x xa) +and similarly for closed elements Γy ∈ G(Y ). Finally, axiom (F7) was verified +in Proposition 4.3 for meet semilattices and the same proof applies to establish +that the topology on each of X = Filt(L) and Y = Idl(L) is a spectral topology. +By the above argument, F(L) is an object in the category SRF∗ +νM. +By +Proposition 4.5, {Xa ∣ a ∈ L} = KOG(X) is the complex algebra of clopen el- +ements L∗F(L) and we then verified in Theorem 4.6 that the representation +map a ↦ Xa = {x ∈ X ∣ a ∈ x} is a lattice isomorphism L ⋍ KOG(X). Since +Xνa = Γxνa = (Γ(̂ν(xa)))′ = (Γxa)∗ = (Xa)∗, the representation map is an +isomorphism L ⋍ L∗F(L) of lattices with a (minimal) quasi-complementation +operation ν. +For morphisms h ∶ L1 �→ L2, the argument that F(h) ∶ F(L2) �→ F(L1) +is a frame morphism satisfying axioms (M1)–(M4) is a special instance of the +argument given in the proof of [13, Proposition 4.9], handling the general case +of arbitrary normal lattice expansions. The proofs regarding axioms (M5) and +(M6) were given in [13, Proposition 5.6, Proposition 5.7]. This establishes that +F ∶ M �→ (SRF∗ +νM)op is a contravariant functor satisfying L ⋍ L∗F(L). +Now let F be a sorted residuated frame in the category SRF∗ +νM. We have +let L(F) = F+ = (G(X),⊆,⋂,⋀,∅,X,( )∗) be its full complex algebra (Defini- +tion 3.6) and L∗(F) = (KOG(X),⊆,∩,∨,∅,X,( )∗) be its subalgebra of clopen +elements. That the operation ( )∗ restricts to clopen elements is clear, since +(Xa)∗ = Xνa. +By Corollary 3.16, L∗(F) is an object in the category M of +lattices with a minimal quasi-complementation operation. +If π = (p,q) ∶ F2 �→ F1 is a frame morphism in SRF∗ +νM, then it was verified +in [13, Corollary 3.21] that L∗(π) = π−1 ∶ G(X1) �→ G(X2) is a complete lattice +homomorphism of the complete lattices of stable sets of the frames. +Given +axiom (M4), it was established in [13, Proposition 3.24, Lemma 3.25] that π−1 ∶ +F+ +1 �→ F+ +2 is in fact a homorphism of the full complex algebras of the frames. +24 + +By axiom (M5), π−1 preserves closed elements, hence by [13, Lemma 3.23] it +also preserves clopen elements (from which continuity of π follows, since clopen +stable elements are precisely the basic open sets in the topology). +The above argument has established that L∗ is a contravariant functor from +the category SRF∗ +νM to the category M of lattices with a minimal quasi- +complementation operation. +We have already also established that for any +object L in M we have an isomorphism L ⋍ L∗F(L) and it remains to argue +that for any sorted residuated frame F in the category SRF∗ +νM we also have +that F ⋍ FL∗(F). To avoid repetitions, we refer the reader to the proof of the +general duality theorem for any normal lattice expansion [13, Theorem 5.8]. +Definition 6.2. Categories of frames F = (X,⍊,Y,Sν), where we set � = S′ +ν, +corresponding to lattices with a quasi complementation operation are axioma- +tized by the axioms of Table 2 as well as one or more of the additional axioms +below, as specified for each category. +(G) +� is symmetric +(INV) +∀x,z ∈ X[∀v ∈ Y (vSνz �→ vSνx) �→ x ≤ z] +(O) +� is irreflexive +(D) +All sections of the Galois dual relation R′ of the upper bound relation +R of Proposition 3.7 are stable +SRF∗ +νM +Table 2 axioms +SRF∗ +νG +Table 2 axioms + Axiom (G) +SRF∗ +νINV +Table 2 axioms + Axiom (G) + Axiom (INV) +SRF∗ +νO +Table 2 axioms + Axiom (G) + Axiom (INV) + Axiom (O) +SRF∗ +νDMA +Table 2 axioms + Axiom (G) + Axiom (INV) + Axiom (D) +SRF∗ +νBA +Table 2 axioms + Axiom (G) + Axiom (INV) + Axiom (O) + ++ Axiom (D). +Theorem 6.3. The spectral duality of Theorem 6.1 specializes to dualities for +each of the frame categories of Definition 6.2 and their respective categories of +bounded lattices with a (quasi) complementation operation. +Proof. That the double dual of a lattice in one of the varieties of Figure 1 is +in the variety in question was verified in Theorem 5.2. That the (full) complex +algebra of a frame in one of the frame categories which satisfies one or more +axioms from the list in Definition 6.2 is an algebra in the respective variety +corresponding to the frame category was verified in Corollary 3.16. The rest of +the duality argument for each of the cases is the same as in Theorem 6.1. +7 +Concluding Remarks +In this article, we have provided alternative constructions for the choice-free +representation and duality for Boolean algebras and Ortholattices, first given +25 + +in [2,22]. A Stone duality result for De Morgan algebras, using choice, was pub- +lished by Bimb´o in [3] and we have given here a choice-free version of the dual- +ity. Our background motivation has been the J´onsson-Tarski [19,20] approach, +constructing set-operators from relations to represent operators on Boolean al- +gebras, a project that has been extended with Dunn’s research on generalized +Galois logics (gaggles). In [13], we presented a generalization of this project of +relational representation to cases where distribution may not be assumed. The +framework of [13] was applied in this article to the case of bounded lattices with +a quasi-complementation operator, recasting the duality of [13] in a choice-free +manner. By treating both De Morgan (and Boolean) algebras, as well as Or- +tholattices, it has been shown that the presence or not of distribution does not +create any significant obstacle, as distribution in the complete lattice of stable +sets of a sorted frame has been shown to be first-order definable. For the dis- +tributive case, the resulting semantics from the approach presented appears to +have strong affinities to Holliday’s possibility semantics [17]. +The approach presented can be extended to any normal lattice expansion, +including the case of modal lattices studied in [1], based on the framework +developed in [13]. +References +[1] Nick Bezhanishvili, Anna Dmitrieva, Jim de Groot, and Tommaso Moras- +chini. Positive (modal) logic beyond distributivity, 2022. +[2] Nick Bezhanishvili and Wesley Holliday. Choice-free Stone duality. The +Journal of Symbolic Logic, 85(1):109–148, 2020. +[3] Katalin Bimb´o. Functorial duality for ortholattices and De Morgan lattices. +Logica Universalis, 1:311–333, 2007. +[4] Katalin Bimb´o and J. Michael Dunn. Generalized Galois Logics. Relational +Semantics of Nonclassical Logical Calculi, volume 188. CSLI Lecture Notes, +CSLI, Stanford, CA, 2008. +[5] J. Michael Dunn. Gaggle theory: An abstraction of Galois coonections and +resuduation with applications to negations and various logical operations. +In Logics in AI, Proceedings of European Workshop JELIA 1990, LNCS +478, pages 31–51, 1990. +[6] J. Michael Dunn. Partial gaggles applied to logics with restricted structural +rules. In K. Doˇsen and P. Schroeder-Heister, editors, Substructural Logics, +pages 63–108. Clarenton and Oxford University Press, Oxford, UK, 1993. +[7] Mai Gehrke. +Generalized Kripke frames. +Studia Logica, 84(2):241–275, +2006. +[8] Mai Gehrke and John Harding. Bounded lattice expansions. Journal of +Algebra, 238:345–371, 2001. +26 + +[9] Robert Goldblatt. Semantic analysis of orthologic. Journal of Philosophical +Logic, 3:19–35, 1974. +[10] Chrysafis Hartonas. Duality for lattice-ordered algebras and for normal +algebraizable logics. Studia Logica, 58:403–450, 1997. +[11] Chrysafis Hartonas. +Stone duality for lattice expansions. +Oxford Logic +Journal of the IGPL, 26(5):475–504, 2018. +[12] Chrysafis Hartonas. Reconcilliation of approaches to the semantics of logics +without distribution. In Katalin Bimb´o, editor, Relevance Logics and other +Tools for Reasoning: Essays in Honor of J. Michael Dunn, number 46 in +Tributes, pages 215–236. College Publications, 2022. +[13] Chrysafis Hartonas. Duality for normal lattice expansions and sorted resid- +uated frames with relations. Algebra Universalis, (to appear, 2023). +[14] Chrysafis Hartonas and J. Michael Dunn. Stone duality for lattices. Algebra +Universalis, 37:391–401, 1997. +[15] Chrysafis Hartonas and Ewa Or�lowska. +Representation of lattices with +modal operators in two-sorted frames. Fundamenta Informatica, 166(1):29– +56, 2019. +[16] Gerd Hartung. A topological representation for lattices. Algebra Univer- +salis, 29:273–299, 1992. +[17] Wesley H. Holliday. Possibility frames and forcing for modal logic, 2018. +https://escholarship.org/uc/item/0tm6b30q, 2018. +[18] Peter Johnstone. Stone Spaces. Cambridge studies in advanced mathemat- +ics. Cambridge University Press, 1986. +[19] Bjarni J´onsson and Alfred Tarski. Boolean algebras with operators I. Amer- +ican Journal of Mathematics, 73:891–939, 1951. +[20] Bjarni J´onsson and Alfred Tarski. +Boolean algebras with operators II. +American Journal of Mathematics, 74:8127–162, 1952. +[21] Guillaume Massas. Choice-Free de Vries Duality, 2022. +[22] Joseph McDonald and Kentarˆo Yamamoto. Choice-free duality for ortho- +complemented lattices by means of spectral spaces. Algebra Universalis, +83, 2022. +[23] M. Andrew Moshier and Peter Jipsen. Topological duality and lattice ex- +pansions, I: A topological construction of canonical extensions. Algebra +Universalis, 71(2):109–126, Apr 2014. +[24] M. Andrew Moshier and Peter Jipsen. Topological duality and lattice ex- +pansions, II: Lattice expansions with quasioperators. Algebra Universalis, +71(3):221–234, May 2014. +27 + +[25] Alasdair Urquhart. A topological representation of lattices. Algebra Uni- +versalis, 8:45–58, 1978. +28 + diff --git a/otE5T4oBgHgl3EQfkQ83/content/tmp_files/load_file.txt b/otE5T4oBgHgl3EQfkQ83/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9f804a78783a2247c252ba8431b4c30593f8fdcd --- /dev/null +++ b/otE5T4oBgHgl3EQfkQ83/content/tmp_files/load_file.txt @@ -0,0 +1,976 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf,len=975 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='05661v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='LO] 13 Jan 2023 Choice-free Dualities for Lattice Expansions: Application to Logics with a Negation Operator Chrysafis (Takis) Hartonas Department of Digital Systems University of Thessaly, Greece hartonas@uth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='gr January 13, 2023 Abstract Constructive dualities have been recently proposed for some lattice based algebras and a related project has been outlined by Holliday and Bezhanishvili, aiming at obtaining “choice-free spatial dualities for other classes of algebras [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='], giving rise to choice-free completeness proofs for non-classical logics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We present in this article a way to complete the Holliday-Bezhanishvili project (uniformly, for any normal lattice expansion) by recasting re- cent relational representation and duality results in a choice-free man- ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' These results have some affinity with the Moshier and Jipsen duality for bounded lattices with quasi-operators, except for aiming at repre- senting operators by relations, extending the J´onsson-Tarski approach for BAOs, and Dunn’s follow up approach for distributive gaggles, to con- texts where distribution may not be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To illustrate, we apply the framework to lattices (and their logics) with some form or other of a (quasi)complementation operator, obtaining canonical extensions in re- lational frames and choice-free dualities for lattices with a minimal, or a Galois quasi-complement, or involutive lattices, including De Morgan algebras, as well as Ortholattices and Boolean algebras, as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 1 Introduction Choice-free dualities have been lately proposed for Boolean algebras, by Holliday and Bezhanishvili [2], for Ortholattices, by MacDonald and Yamamoto [22], for modal lattices by Bezhanishvili, Dmitrieva, de Groot and Moraschini [1] and for De Vries algebras by Massas [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' These are part of a project, outlined by Holliday and Bezhanishvili and aiming at obtaining “choice-free spatial dualities for other classes of algebras [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='], giving rise to choice-free completeness proofs for non-classical logics” [2, page 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The project has its origins in Holliday’s ‘possibility frames’ for modal logic [17], as noted in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 1 A choice-free representation and duality for bounded lattices with quasiop- erators, by Moshier and Jipsen [23,24], had already appeared in print, influenc- ing at least Dmitrieva’s research, with Bezhanishvili, de Groot and Morachini, on modal lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The Moshier-Jipsen duality is related to results by this author [10], and with Dunn [14], some detail on these relations is presented in [13, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='8] and we revisit the issue in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7 in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The duality of [23, 24] is not primarily intended to provide logic related, relational semantics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' This becomes clear by their choice to represent lattice quasi-operators as strongly continuous and meet preserving point operators on the dual topological spaces, whereas for semantic purposes one typically aims for first-order definable classes of relational frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In the recent few years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' this author has pursued a project of extending J´onsson and Tarski’s approach for Boolean algebras with operators (BAOs) [19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='20] and Dunn’s follow up approach for distributive generalized Galois logics (gaggles) [4–6] to the case of general lattices with quasi-operators (normal lattice operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' in our preferred terminology),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' building on older work by the author (with Dunn) in [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' while working within the framework of canonical exten- sions [8] of lattice expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' This project developed in parallel with Gehrke’s (with co-workers) generalized Kripke frames approach (RS frames) [7] and the relations between the two approaches have been detailed in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We note that, as far as the objectives of the current article are concerned, Gehrke’s approach of RS-frames in [7] builds on Hartung’s lattice representation [16], which inher- its from Urquhart’s lattice representation [25] an essential use of the axiom of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Choice was also assumed in this author’s [13,14] (Alexander’s subbasis lemma, whose proof uses Zorn’s lemma, was used to prove compactness of the space), but we show in this article that the use of choice is inessential and we can easily recast the duality in a choice-free manner, switching to a spectral topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In Section 2 we present the algebras of interest, bounded lattices with a quasi- complementation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We restrict attention to some distinguished cases, allowing for both a distributive (notably De Morgan and Boolean algebras) and a non-distributive lattice base (such as involutive, or orthocomplement lattices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Section 3 starts with a review subsection (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1) for sorted frames with relations and generated operators, drawing on [13], and concludes with Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2 where frames for quasi-complemented lattices are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' A first-order axiomatic specification of the classes of frames with respect to which the logics of the algebraic structures of section 2 can be shown to be sound is provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Section 4 presents choice-free representations of semilattices (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1) and bounded lattices (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2) and concludes with Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3 detailing the representation of arbitrary normal lattice operators, drawing on [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In Section 5 we apply the representation framework of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3 to the particular case of quasi-complementation operators on bounded lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The results of this section establish that the varieties of quasi-complemented lattices we consider are closed under canonical extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Thereby, completeness the- orems via a canonical model construction can be proven for the logics of the 2 algebraic structures considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Spectral duality theorems are proven in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The main result in this section is Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1, where we detail the duality between the categories M of bounded lattices with a minimal quasi-complementation operator and the category SRF∗ νM of sorted residuated frames whose first-order axiomatization is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The remaining dualities are then easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In particular, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7, relying on Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='5, provides a first-order frame condition for the lattice of Galois stable sets to be completely distributive, which is then used for the cases of representation and duality for De Morgan algebras and Boolean algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Some concluding remarks are made in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 2 Quasi-complemented Lattices Let {1,∂} be a 2-element set, L1 = L and L∂ = Lop (the opposite lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Extending established terminology [19], a function f ∶ L1 × ⋯ × Ln �→ Ln+1 will be called additive and normal, or a normal operator, if it distributes over finite joins of the lattice Li, for each i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' n, delivering a join in Ln+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' An n-ary operation f on a bounded lattice L is a normal lattice operator of distribution type δ(f) = (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='in+1) ∈ {1,∂}n+1 if it is a normal additive function f ∶ Li1 ×⋯×Lin �→ Lin+1 (distributing over finite joins in each argument place), where each ij, for j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,n + 1, is in the set {1,∂}, hence Lij is either L, or L∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If τ is a tuple (sequence) of distribution types, a normal lattice expansion of (similarity) type τ is a lattice with a normal lattice operator of distribution type δ for each δ in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The category NLEτ, for a fixed similarity type τ, has normal lattice expan- sions of type τ as objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Its morphisms are the usual algebraic homomor- phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In this article we focus on the class of lattices L = (L,≤,∧,∨,0,1,ν) with a quasi-complementation operator ν, of increasing axiomatization strength, in- cluding at least the following: (antitonicity) a ≤ b �→ νb ≤ νa (normality) ν0 = 1 (∨∧) ν(a ∨ b) ≤ νa ∧ νb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Given antitonicity, the operation ν satisfies the identity ν(a∨b) = νa∧νb, hence it is a normal lattice operator of distribution type δ(ν) = (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We list some basic facts in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let L = (L,≤,∧,∨,0,1,ν) be a bounded lattice with an antitone operation ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (1) ν forms a Galois connection on L (a ≤ νb iff b ≤ νa) iff it satisfies the inequation a ≤ ννa 3 (2) if a ≤ ννa holds in the lattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' then the normality axiom ν0 = 1 and the identity ν(a ∨ b) = νa ∧ νb are derivable (3) if ννa ≤ a holds in the lattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' then the identity ν(a ∧ b) = νa ∨ νb is derivable (4) if either of the De Morgan identities ν(a∨b) = νa∧νb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' or ν(a∧b) = νa∨νb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' holds in the lattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' then antitonicity of ν is a derivable property (5) if ν is an involution (a = ννa) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' in addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' the lattice is distributive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' then it is a De Morgan algebra (6) if ν is an involution and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' in addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' it satisfies the intuitionistic explosion principle (ex falso quidlibet) a ∧ νa ≤ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' then the lattice is an Ortholattice (orthocomplemented lattice) (7) if ν is an involution satisfying the antilogism rule (a∧b ≤ c �→ a∧νc ≤ νb),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' then the lattice is a Boolean algebra Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Each of the claims (1) to (6) has a straightforward proof, left to the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For (7), the hypothesis implies that a ∧ b ≤ c iff a ∧ νc ≤ νb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' This means that ∧ is self-conjugate with respect to the involution ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To see that this implies distributivity, define a → c = ν(a ∧ νc) and observe that the conjugacy condition is equivalent to residuation of ∧ and →, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' a ∧ b ≤ c iff a∧νc ≤ νb iff b ≤ a → c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Distribution then follows from residuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In addition, by part (2), ν0 = 1 holds and then also ν1 = νν0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence the intuitionistic principle a ∧ νa ≤ 0 = ν1 follows, since we can infer it from a ∧ 1 ≤ a using antilogism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By the hypothesis that ν is an involution, the explosion principle a ∧ νa ≤ 0 is equivalent to excluded middle a ∨ νa = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence the lattice is a Boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Figure 1 summarizes the above results, where DMA,O,INV designate the equational classes (varieties) of De Morgan algebras, Ortholattices and lattices with an involution, respectively, BA designates the variety of Boolean algebras, the remaining two labels M,G designate the varieties of lattices with a minimal, or a Galois connected quasi-complementation operator, respectively, and the arrow label (dist) indicates addition of the distribution law a∧(b∨c) ≤ (a∧b)∨ (a ∧ c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 3 Sorted residuated frames (SRFs) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1 Frames, Relations and (Sorted) Image Operators We review in this section definitions, notational conventions and results from [12,13], to the extent needed for our current purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Regard {1,∂} as a set of sorts and let Z = (Z1,Z∂) be a sorted set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Sorted residuated frames F = (Z1,⍊,Z∂) are triples consisting of nonempty sets Z1 = X,Z∂ = Y and a binary relation ⍊ ⊆ X × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 4 Figure 1: (Quasi)Complemented Lattices BA DMA a∧νa=0 ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ a∧b≤c a∧νc≤νb O (dist) ◆◆◆◆◆◆◆◆◆◆◆◆◆◆ INV a∧νa=0 ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ (dist) PPPPPPPPPPPPPP G a ≤ ννa ννa≤a M ν0 = 1 ν(a ∨ b) = νa ∧ νb The relation ⍊ will be referred to as the Galois relation of the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' It generates a Galois connection ( )⍊ ∶ ℘(X) ⇆ ℘(Y )∂ ∶ ⍊( ) (V ⊆ U ⍊ iff U ⊆ ⍊V ) U ⍊ = {y ∈ Y ∣ ∀x ∈ U x ⍊ y} = {y ∈ Y ∣ U ⍊ y} ⍊V = {x ∈ X ∣ ∀y ∈ V x ⍊ y} = {x ∈ X ∣ x ⍊ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We will also have use for the complement I of the Galois relation ⍊ and we will designate frames using either the Galois relation ⍊, or its complement I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' A subset A ⊆ X will be called stable if A = ⍊(A⍊).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Similarly, a subset B ⊆ Y will be called co-stable if B = (⍊B)⍊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Stable and co-stable sets will be referred to as Galois sets, disambiguating to Galois stable or Galois co-stable when needed and as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The following quasi-seriality condition will be assumed for sorted frames ∀x ∈ X∃y ∈ Y xIy ∀y ∈ Y ∃x ∈ X xIy (1) Note that assuming (1), the empty set is (co)stable and we have ∅⍊ = Y , ⍊Y = ∅ and similarly ⍊∅ = X and X⍊ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By G(X),G(Y ) we designate the complete lattices of stable and co-stable sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Note that the Galois connection restricts to a dual isomor- phism ( )⍊ ∶ G(X) ⋍ G(Y )∂ ∶ ⍊( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Preorder relations are induced on each of the sorts, by setting for x,z ∈ X, x ⪯ z iff {x}⍊ ⊆ {z}⍊ and, similarly, for y,v ∈ Y , y ⪯ v iff ⍊{y} ⊆ ⍊{v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' A (sorted) frame is called separated if the preorders ⪯ (on X and on Y ) are in fact partial orders ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 5 Our notational conventions are these of [13, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We repeat them below, for the reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1 (Notational conventions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For a sorted relation R ⊆ ∏j=n+1 j=1 Zij, where ij ∈ {1,∂} for each j (and thus Zij = X if ij = 1 and Zij = Y when ij = ∂), we make the convention to regard it as a relation R ⊆ Zin+1 ×∏j=n j=1 Zij, we agree to write its sort type as σ(R) = (in+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='i1⋯in) and for a tuple of points of suitable sort we write uRu1⋯un for (u,u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,un) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We use Γ to designate upper closure ΓU = {z ∈ X ∣ ∃x ∈ U x ⪯ z}, for U ⊆ X, and similarly for U ⊆ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The set U is increasing (an upset) iff U = ΓU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For a singleton set {x} ⊆ X we write Γx, rather than Γ({x}) and similarly for {y} ⊆ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We typically use the standard Formal Concept Aanalysis priming notation for each of the two Galois maps ⍊( ),( )⍊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' This allows for stating and proving results for each of G(X),G(Y ) without either repeating definitions and proofs, or making constant appeals to duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Thus for a Galois set G, G′ = G⍊, if G ∈ G(X) (G is a Galois stable set), and otherwise G′ = ⍊G, if G ∈ G(Y ) (G is a Galois co-stable set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For an element u in either X or Y and a subset W, respectively of Y or X, we write u∣W, under a well-sorting assumption, to stand for either u ⍊ W (which stands for u ⍊ w, for all w ∈ W), or W ⍊ u (which stands for w ⍊ u, for all w ∈ W), where well-sorting means that either u ∈ X,W ⊆ Y , or W ⊆ X and u ∈ Y , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Similarly for the notation u∣v, where u,v are elements of different sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We designate n-tuples (of sets, or elements) using a vectorial notation, set- ting (G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,Gn) = ⃗G ∈ ∏j=n j=1 G(Zij), ⃗U ∈ ∏j=n j=1 ℘(Zij), ⃗u ∈ ∏j=n j=1 Zij (where ij ∈ {1,∂}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Most of the time we are interested in some particular argument place 1 ≤ k ≤ n and we write ⃗G[F]k for the tuple ⃗G where Gk = F (or Gk is replaced by F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Similarly ⃗u[x]k is (u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,uk−1,x,uk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For brevity, we write ⃗u ⪯ ⃗v for the pointwise ordering statements u1 ⪯ v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,un ⪯ vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We also let ⃗u ∈ ⃗ W stand for the conjunction of component- wise membership uj ∈ Wj, for all j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To simplify notation, we write Γ⃗u for the n-tuple (Γu1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,Γun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For a unary map f and a tuple ⃗u we write f[⃗u] for the tuple (f(u1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,f(un)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Note that the same notation is used for the image f[S] = {f(x) ∣ x ∈ S} of a set under a function f, but context will make it clear what the intended meaning is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The convention can be nested, so that if S is a set (or sequence) of tuples ⃗ui, then f[S] is the set (or sequence) consisting of the elements f[⃗ui].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To refer to sections of relations (the sets obtained by leaving one argument place unfilled) we make use of the notation ⃗u[ ]k which stands for the (n − 1)- tuple (u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,uk−1,[ ] ,uk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,un) and similarly for tuples of sets, extending the membership convention for tuples to cases such as ⃗u[ ]k ∈ ⃗F[ ]k and similarly for ordering relations ⃗u[ ]k ⪯ ⃗v[ ]k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We also quantify over tuples (with, or without a hole in them), instead of resorting to an iterated quantification over the elements of the tuple, as for example in ∃⃗u[ ]k ∈ ⃗F[ ]k∃v,w ∈ G wR⃗u[v]k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We extend the vectorial notation to distribution types, summarily writing δ = (⃗ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='in+1) for (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='in+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then, for example, ⃗ij[∂]k is the tuple with 6 ik = ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Furthermore, we let ij = ∂, if ij = 1 and ij = 1, when ij = ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2 ( [13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let F = (X,⍊,Y ) be a polarity and u a point in Z = X ∪ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ⍊ is increasing in each argument place (and thereby its complement I is decreasing in each argument place).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (Γu)′ = {u}′ and Γu = {u}′′ is a Galois set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Galois sets are increasing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' u ∈ G implies Γu ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For a Galois set G, G = ⋃u∈G Γu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For a Galois set G, G = ⋁u∈G Γu = ⋂v∣G{v}′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For a Galois set G and any set W, W ′′ ⊆ G iff W ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ◻ It is typical in the context of canonical extensions of lattices to refer to prin- cipal upper sets Γx ∈ G(X)(x ∈ X = Filt(L)), as closed, or filter elements of G(X) and to sets ⍊{y} ∈ G(X)(y ∈ Y = Idl(L)) as open, or ideal elements of G(X), and similarly for sets Γy,{x}⍊ with x ∈ X,y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' This creates an unfortu- nate clash of terminology and we shall have to rely on context to disambiguate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Furthermore, a closed element Γx is said to be clopen if Γx = ⍊{y} for some y ∈ Y , which is unique when the frame is separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2, the closed elements of G(X) join-generate G(X), while the open elements meet-generate G(X) (similarly for G(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3 (Galois dual relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For a relation R, of sort type σ, its Galois dual relation R′ is the relation defined by uR′⃗v iff ∀w (wR⃗v �→ w∣u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In other words, R′⃗v = (R⃗v)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='4 (Sections of relations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For an (n + 1)-ary relation Rσ (of sort σ) and an n-tuple ⃗u, Rσ⃗u = {w ∣ wRσ⃗u} is the section of Rσ determined by ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To designate a section of the relation at the k-th argument place we let ⃗u[ ]k be the tuple with a hole at the k-th argument place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then wRσ⃗u[ ]k = {v ∣ wRσ⃗u[v]k} ⊆ Zik is the k-th section of Rσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If R is a relation on a sorted residuated frame F = (X,I,Y ), of some sort type σ = σ(R) = (in+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='i1⋯in), then as in the unsorted case, R generates a (sorted) image operator αR, defined by (2), of sort σ(αR) = (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='in+1), defined by the obvious generalization of the J´onsson–Tarski image operators [19], αR( ⃗W) = {w ∈ Zin+1 ∣ ∃ ⃗w (wR ⃗w ∧ j=n ⋀ j=1 (wj ∈ Wj))} = ⋃ ⃗w∈ ⃗ W R ⃗w, (2) where for each j, Wj ⊆ Zij (and recall that Zij = X when ij = 1 and Zij = Y , if ij = ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let αR be the closure of the restriction of αR to Galois sets ⃗F, αR( ⃗F) = (αR( ⃗F))′′ = ⎛ ⎝ wj∈Fj ⋃ j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=',n R ⃗w⎞ ⎠ ′′ = ⋁ ⃗w∈ ⃗ F (R ⃗w)′′, (3) where Fj ∈ G(Zij), for each j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=',n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 7 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='5 ( [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let F = (X,⍊,Y,R) be a frame with an (n+1)-ary sorted relation, of some sort σ(R) = (in+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ⃗ij) and assume that for any w ∈ Zin+1 and any (n − 1)-tuple ⃗p[ ]k with pj ∈ Zij, for each j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=',n} ∖ {k}, the sections wR′⃗p[ ]k of the Galois dual relation R′ of R are Galois sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then αR distributes at the k-th argument place over arbitrary joins in G(Zik).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ◻ The Galois set operator αR is sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Single-sorted operators α1 R on G(X) and α∂ R on G(Y ) are obtained by composition with the Galois connection, which is a duality of G(X) and G(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='6 (Full complex algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let F = (X,⍊,Y,R) be a polarity with a relation R of some sort σ(R) = (in+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='i1⋯in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The full complex algebra of F is the structure F+ = (G(X),α1 R) and its dual full complex algebra is the structure F∂ = (G(Y ),α∂ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Subalgebras of full complex algebras will be referred to as complex algebras of a frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let F = (X,⍊,Y ) be a sorted frame (a polarity) and G(X) the complete lattice of stable sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let R be the ternary upper bound relation on X defined by xRuz iff both u ⪯ x and z ⪯ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If all sections of the Galois dual relation R′ of R are Galois sets, then G(X) is completely distributive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let αR be the image operator generated by R, αR(U,W) = ⋃w∈W u∈U Ruw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Notice that, for stable sets A,C (more generally, for increasing sets), αR(A,C) = A ∩ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence αR(A,C) = αR(A,C) = A ∩ C, since Galois sets are closed under intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Given the section stability hypothesis for the Galois dual relation R′ of R, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='5 applies, from which distribution of αR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' of intersection) over arbitrary joins of stable sets is concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The image operator αR ∶ ∏j=n j=1 ℘(Zij) �→ ℘(Zin+1) is a sorted normal and completely additive function in each argument place, therefore it is residuated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' for each k there is a set-operator βk R satisfying the condition αR( ⃗W[V ]k) ⊆ U iff V ⊆ βk R( ⃗W[U]k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (4) Hence βk R( ⃗W[U]k) is the largest set V s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' αR( ⃗W[V ]k) ⊆ U and it is thereby definable by βk R( ⃗W[U]k) = ⋃{V ∣ αR( ⃗W[V ]k) ⊆ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (5) We let βk R/ be the restriction of βk R of equation (5) to Galois sets, according to its sort type, explicitly defined by (6) βk R/( ⃗E[G]k) = ⋃{F ∈ G(Zik) ∣ αR( ⃗E[F]k) ⊆ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (6) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='8 ( [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='14, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If αR is residuated in the k- th argument place, then βk R/ is its residual and βk R/( ⃗E[G]k) is a Galois set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' the union in equation (6) is actually a join in G(Zik).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Furthermore, equations (7) and (8) βk R/( ⃗E[G]k) = ⋃{Γu ∈ G(Zik) ∣ αR( ⃗E[Γu]k) ⊆ G}, (7) βk R/( ⃗E[G]k) = {u ∈ Zik ∣ αR( ⃗E[Γu]k) ⊆ G} (8) 8 give an equivalent definition of βk R/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ◻ Thus, under the assumption of section stability for the Galois dual relation R′, the operation of closure of restriction to Galois sets preserves residuation and we obtain αR ⊣ βk R/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Frame morphisms are the weak bounded morphisms whose definition in [13, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='20] is repeated below, where we let I be the complement of the Galois relation ⍊ of a frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If π = (p,q) ∶ (X2,I2,Y2) �→ (X1,I1,Y1) is a pair of maps p ∶ X2 �→ X1, q ∶ Y2 �→ Y1, then π will be called a (sorted) weak bounded morphism iff 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ∀x′ ∈ X2∀y′ ∈ Y2 (x′I2y′ �→ π(x′)I1π(y′)) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ∀x ∈ X1∀y′ ∈ Y2(xI1π(y′) �→ ∃x′ ∈ X2(x ≤ π(x′) ∧ x′I2y′)) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ∀x′ ∈ X2∀y ∈ Y1(π(x′)I1y �→ ∃y′ ∈ Y2(y ≤ π(y′) ∧ x′I2y′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If (p,q) ∶ (X2,I2,Y2) → (X1,I1,Y1), with p ∶ X2 → X1 and q ∶ Y2 → Y1, then we let π = (p,q) and we define π−1 by setting π−1(W) = { p−1(W) ∈ ℘(X2) if W ⊆ X1 q−1(W) ∈ ℘(Y2) if W ⊆ Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Similarly, we let π(w) = { p(w) ∈ X1 if w ∈ X2 q(w) ∈ Y1 if w ∈ Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='11 ( [13, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The inverse π−1 = (p,q)−1 of a weak bounded morphism is a complete lattice homomorphism of the lattices of Galois stable sets of sorted residuated frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ◻ For frames with relations, let π be a weak bounded morphism, π = (p,q) ∶ (X2,I2,Y2,(Sσ)σ∈τ) �→ (X1,I1,Y1,(Rσ)σ∈τ), and let Rσ,Sσ be corresponding relations in the two frames, of the same sort type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For simplicity, we omit the subscript σ in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We recall the following from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='12 ( [13, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='24, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If for any ⃗u it holds that π−1αR(Γ⃗u) = αS(π−1[Γ⃗u]), then for any tuple ⃗F of Galois sets of the required sort π−1αR( ⃗F) = αS(π−1[ ⃗F]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Furthermore, equations π−1αR(Γ⃗u) = αS(π−1[Γ⃗u]), (9) π(v)R⃗u iff ∃ ⃗w(⃗u ≤ π[ ⃗w] ∧ vS ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (10) provide equivalents to the above assumption that π−1αR(Γ⃗u) be identical to αS(π−1[Γ⃗u]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ◻ 9 Table 1: Axioms for Sorted Residuated Frames of similarity type τ (F0) The complement I of the Galois relation ⍊ of the frame is quasi-serial, in the sense of condition (1) (F1) The frame is separated (F2) For each σ = (⃗ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='in+1) in the similarity type τ, each ⃗u ∈ ∏j=n j=1 Zij, Rσ⃗u is a closed element of G(Zin+1) (F3) For each σ = (⃗ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='in+1) in the similarity type τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' each w ∈ Zin+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' the n-ary relation wRσ is decreasing in every argument place (F4) All sections of the Galois dual relations R′ σ of Rσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' for each σ in τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' are Galois sets For π = (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='q) ∶ (X2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='I2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='Y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='S2ν) �→ (X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='I1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='Y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='S1ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' where p ∶ X2 �→ X1 and q ∶ Y2 �→ Y1 (M1) ∀x′ ∈ X2∀y′ ∈ Y2 (x′I2y′ �→ p(x′)I1q(y′)) (M2) ∀x ∈ X1∀y′ ∈ Y2(xI1q(y′) �→ ∃x′ ∈ X2(x ≤ p(x′) ∧ x′I2y′)) (M3) ∀x′ ∈ X2∀y ∈ Y1(p(x′)I1y �→ ∃y′ ∈ Y2(y ≤ q(y′) ∧ x′I2y′)) (M4) ∀z ∈ X1∀v ∈ Y2(q(v)S1νz �→ ∃x ∈ X2(z ≤ p(x) and vS2νx)) We list in Table 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' after [13],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' the minimal axiomatization we shall assume for a sorted residuated frame with relations F = (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='(Rσ)σ∈τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The axiom- atization will be strengthened in the sequel imposing, among others, a spectral topology on each of X,Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Note that axioms (F1) and (F2) imply that there is a (sorted) function ̂fR on the points of the frame such that ̂fR(⃗u) = w iff R⃗u = Γw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The following immediate observation will be useful in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='13 ( [13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let F be a frame of similarity type τ and assume that axioms (F1)–(F3) in Table 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then for a frame relation R of type σ in τ, αR(Γ⃗u) = R⃗u = αR(Γ⃗u) = Γ( ̂fR(⃗u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ◻ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2 Frames for Quasi-Complemented Lattices We now consider sorted frames F = (X,⍊,Y,Sν) with σ(Sν) = (∂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Sν ⊆ Y × X, and we assume that axioms (F0)–(F4) of Table 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let S′ ν be the Galois dual relation of Sν, defined by S′ νz = (Sνz)′ for each z ∈ X, let ηS ∶ ℘(X) �→ ℘(Y ) be the sorted image operator generated by Sν, 10 defined on U ⊆ X by ηS(U) = {y ∈ Y ∣ ∃x ∈ X(ySνx and x ∈ U)} = ⋃ x∈U Sνx, (11) and ηS ∶ G(X) �→ G(Y ) be the closure of its restriction to Galois sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Further- more, let ̂ν ∶ X �→ Y be the point operator of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='13, so that for a closed element Γx ∈ G(X) we have ηS(Γx) = Sνx = ηS(Γx) = Γ(̂ν(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By axiom (F2), Sνx is a Galois set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence for a stable set A ∈ G(X), ηS(A) = ( ⋃ x∈A Sνx) ′′ = ⋁ x∈A Sνx = ⋁ x∈A Γ(̂ν(x)) = ⋁ x∈A ηS(Γx) ∈ G(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (12) Let also ην(A) = (ηS(A))′ = ⋂z∈A S′ νz, so that ην is a single-sorted operation (on G(X)) derived from ηS by composition with the Galois connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='14 (Switching Notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hereafter, we simplify notation, switching to the more familiar � for the incompatibility relation S′ ν and letting ην(A) be designated by A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For subsequent use, we make a note of the fact that x ∈ A∗ iff ∀z(z ∈ A �→ x�z) iff x�A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (13) Since ηS ∶ G(X) �→ G(Y ) distributes over arbitrary joins, by the axioms in Table 1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='5, it is residuated with a map ζS ∶ G(Y ) �→ G(X), which maps meets of G(Y ) (hence joins of G(X)) to meets of G(X), defined on B ∈ G(Y ) (using also Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='8) by ζS(B) = ⋁{A ∈ G(X) ∣ ηS(A) ⊆ B} = ⋃{A ∈ G(X) ∣ ηS(A) ⊆ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By [13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='15], ζS(B) is equivalently defined by equation (14), specializing equation (8), ζS(B) = {x ∈ X ∣ ηS(Γx) ⊆ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (14) By duality of G(X) and G(Y ), every B ∈ G(Y ) is B = C′ for some C ∈ G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence we obtain that ηS(A) ⊆ C′ iff A ⊆ ζS(C′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' From this, setting ζν(C) = ζS(C′), we obtain the Galois connection condition C ⊆ ην(A) iff A ⊆ ζν(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Recalling that we have switched notation to A∗ for ην(A) and setting A☆ = ζν(A), we can rewrite the Galois condition as A ⊆ C☆ iff C ⊆ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let F = (X,⍊,Y,Sν) be a frame subject to the axioms of Table 1 and let A ∈ G(X) be any stable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The following are equivalent (a) � is symmetric (b) A ⊆ A∗∗ (c) A∗ = A☆ 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' A∗☆ = ⋁x∈A ζSηS(Γx) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The following are equivalent (a) A∗☆ ⊆ A, for any A ∈ G(X) (b) ζSηS(Γx) ⊆ Γx, for any x ∈ X (c) ∀x,z ∈ X[∀v ∈ Y (vSνz �→ vSνx) �→ x ≤ z] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The following are equivalent (a) � is irreflexive (b) A ∩ A∗ = ∅ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For (1) and the case (a)⇒(b), suppose, for a contradiction, that x ∈ A, but x /∈ A∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let z ∈ A∗ such that x�z fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' But z ∈ A∗ = {z ∣ ∀u(u ∈ A �→ z�u)} and x ∈ A, so z�x holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By symmetry of � it follows x�z, contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence A ⊆ A∗∗, for any A ∈ G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For (b)⇒(c), by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2, ( )∗ forms a Galois connection on G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By uniqueness of adjoints it then follows that A∗ = A☆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For (c)⇒(b), by definition ( )☆ is Galois connected with ( )∗ and hence if the two are equal, then the result follows by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For (b)⇒(a), recall that � = S′ ν, both sections of which are stable sets, which are increasing sets by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2, and that by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2 the hypothesis means that ( )∗ is a Galois connection on the lattice of stable sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Assuming x�z, we then have x�Γz, which means that x ∈ (Γz)∗ = {x ∣ ∀u(z ≤ u �→ x�u)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If x ≤ w and z ≤ u, then by x�z we also have w�u and this means that Γx ⊆ (Γz)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Since ( )∗ is antitone, (Γz)∗∗ ⊆ (Γx)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' But then Γz ⊆ (Γz)∗∗ ⊆ (Γx)∗ from which we obtain z ∈ (Γx)∗ and so z�x follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For claim (2), using definitions we obtain that (A∗)☆ = ζS ((A∗)′) = ζs ((⋂x∈A{̂ν(x)}′)′) = ζS (⋁x∈A Γ(̂ν(x))) = ⋁x∈A ζS(Γ(̂ν(x))) = ⋁x∈A ζSηS(Γx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For claim (3) and the case (a)⇒(b), assuming that A∗☆ ⊆ A, for any A ∈ G(X) and choosing A = Γx, for arbitrary x ∈ X, it is immediate, given the identity proven in claim (2), that (Γx)∗☆ = ⋁x≤z ζSηS(Γz) = ζSηS(Γx) ⊆ Γx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The converse, (b)⇒(a), is immediate: A∗☆ = ⋁x∈A ζSηS(Γx) ⊆ ⋁x∈A Γx = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To prove (b)⇔(c), we have ζSηS(Γx) = ζS(Sνx) = ⋃{z ∈ X ∣ ηS(Γz) ⊆ Sνx} = ⋃{z ∈ X ∣ Sνz ⊆ Sνx} Hence, for any x ∈ X ζSηS(Γx) ⊆ Γx iff ∀z(Sνz ⊆ Sνx �→ x ≤ z) iff ∀z[∀v(vSνz �→ vSνx) �→ x ≤ z] 12 For claim (4), if x ∈ A ∩ A∗ ≠ ∅, then it must be that x�x and thus � is not irreflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Conversely, assume A ∩ A∗ = ∅, for any A ∈ G(X), but suppose that x�x for some x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then x ∈ (Γx)∗ and thus Γx∩(Γx)∗ ≠ ∅, contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let F = (X,⍊,Y,Sν) be a frame satisfying axioms (F0)–(F4) of Table 1, where we set � = S′ ν, and let F+ be its full complex algebra, F+ = (G(X),⊆,⋂,⋁,∅,X,( )∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' F+ is a complete lattice with a minimal quasi-complementation operator ( )∗ on stable sets 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' F+ is a complete lattice with a quasi-complementation operator ( )∗ which is a Galois connection on stable sets iff � is symmetric 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' F+ is a complete lattice with an involution ( )∗ iff � is symmetric and the condition ∀x,z ∈ X[∀v ∈ Y (vSνz �→ vSνx) �→ x ≤ z] holds in the frame 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' F+ is a complete ortholattice iff � is symmetric and irreflexive and the condition ∀x,z ∈ X[∀v ∈ Y (vSνz �→ vSνx) �→ x ≤ z] holds in the frame 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' F+ is a complete De Morgan algebra if (a) � is symmetric, (b) the condition ∀x,z ∈ X[∀v ∈ Y (vSνz �→ vSνx) �→ x ≤ z] holds in the frame and (c) the sections of the Galois dual relation R′ of the upper bound relation R of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7 are stable 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' F+ is a complete Boolean algebra if the conditions (a)–(c) of the previous case hold in the frame and, in addition, (d) � is irreflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Immediate, given Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='15 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 4 Choice-Free Representation of NLEs 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1 Semilattice Representation Let M = (M,≤,∧,1) be a meet semilattice with a unit (top) element 1 and X = Filt(M) its set of proper filters (we assume filters are nonempty, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 1 ∈ x for any x ∈ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For each a ∈ M, let Xa = {x ∈ X ∣ a ∈ x} and B = {Xa ⊆ X ∣ a ∈ M} and notice that Xa ∩ Xb = Xa∧b ∈ B, so that B itself is a meet semilattice with unit element X = X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let X = (X,B) be the topological space with carrier set X and topology Ω generated by taking B as a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1 (Notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Principal filters of meet semilattices and lattices are typically designated with the notation xa (= a↑), for an element a of the (semi)lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Join semilattice and lattice principal ideals are similarly desig- nated by ya = a↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We typically use x,z for filters, y,v for ideals and u,w for either case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Given a filter F in the lattice Ω(X) of open sets of X, define F = {a ∈ M ∣ Xa ∈ F} and xF to be the filter of M generated by the set F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' for any basic open set Xa ∈ B, xF ∈ Xa iff Xa ∈ F 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' for any open set U of X, if xF ∈ U, then U ∈ F 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' if F is a completely prime filter in the lattice Ω(X) of open sets of X, then xF ∈ U iff U ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For (1), if xF ∈ Xa, then by definition of Xa = {x ∈ X ∣ a ∈ x} we have a ∈ xF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By definition of xF , let e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,en ∈ M be such that e1∧⋯∧en ≤ a and for each i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=',n, Xei ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then ⋂n i=1 Xei ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Also, ⋂n i=1 Xei = Xe1∧⋯∧en ⊆ Xa and so Xa ∈ F as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Conversely, if Xa ∈ F, then a ∈ xF , which is to say that xF ∈ Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For (2), if U is open, let E ⊆ M be such that U = ⋃e∈E Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If xF ∈ U, then let e ∈ E be an element such that xF ∈ Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By (1), Xe ∈ F and then since Xe ⊆ U and F is a filter, U ∈ F follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For (3), it suffices to show, given (2) above, that if F is completely prime and U ∈ F, then xF ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Now U = ⋃e∈E Xe for some E ⊆ M and then by complete primeness of F we get Xe ∈ F, for some e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' It then follows by part (1) that xF ∈ Xe ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Given any space X, a filter F of X is a non-empty upper set (with respect to the specialization order ⊑ on X) such that for any x,z ∈ F a lower-bound u ∈ X of {x,z} is in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We let KOF(X) designate the family of compact-open filters of X, following the notation of [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let M be a meet semilattice, X = (X,B) its dual topological space (where X = Filt(M) and B = {Xa ∣ a ∈ M} is a basis for the topology Ω on X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' the space X is a spectral space 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' B = {Xa ∣ a ∈ M} = KOF(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Recall that a topological space is spectral if it is T0, coherent, compact and sober, which we prove in turn in order to establish part (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For the T0 property, if x ≠ z are distinct filters, without loss of generality we may assume that a ∈ x, but a /∈ z, for some semilattice element a ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then the open set Xa separates x,z, since x ∈ Xa but z /∈ Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For the coherence property we verify that the basis B of the topology consists of compact-open sets and that it is closed under finite intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For the latter requirement, B is easily seen to be closed under finite intersections, since Xa∩Xb = Xa∧b and the intersection of the empty family of X′ as is X itself, which is identical to X1 = {x ∈ X ∣ 1 ∈ x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For the first requirement of coherence, the X′ as are certainly open, by definition of the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For compactness, let C ⊆ M and suppose that {Xe ∣ e ∈ C} covers Xa, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Xa ⊆ ⋃e∈C Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then the principal filter xa = a ↑ ∈ Xa is in Xe, for some e ∈ C, hence e ∈ xa, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' a ≤ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then e ∈ x, for any x ∈ Xa and this shows that Xa ⊆ Xe = {z ∈ X ∣ e ∈ z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence {Xe}, for this e, is the needed finite subcover of Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 14 Since X = X1 = {x ∈ X ∣ 1 ∈ x}, compactness of X follows from the previous argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Sobriety of the space is equivalent to the requirement that every completely prime filter F in the lattice Ω(X) of open sets of X is generated by a single point xF , in other words that F = {U ∈ Ω(X) ∣ xF ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' This was shown to hold in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2, part (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For part (2), left to right, Xa is compact-open, by the proof of coherence for X in part (1) of this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Furthermore, x ∈ Xa iff a ∈ x iff xa ⊆ x iff x ∈ Γxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence Xa is a (principal) filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Conversely, let F ⊆ X be a compact-open filter of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Being open, let F = ⋃a∈E⊆M Xa, so that by compactness, F = Xa1 ∪ ⋯ ∪ Xan for some n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By ai↑ = xai ∈ Xai ⊆ F, all the xai’s, for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,n, are in F, hence so is their meet (intersection), since F is a filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Letting u = ⋂n i=1 xai, we show that F = Γu = u↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Right-to-left is obvious since u ∈ F, which is a filter, so Γu ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For left-to-right, let z ∈ F, so that z ∈ Xak = Γxak for some k ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=',n}, hence xak ⊆ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By definition of u we obtain u ⊆ xak ⊆ z and then z ∈ Γu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence F ⊆ Γu follows, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Thus F = Γu = ⋃n i=1 Xai and thus u ∈ Xai = Γxai for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' But then xai ⊆ u = ⋂n i=1 xai ⊆ xai so that u = xai, for this i, and so F = Γxai = Xai ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' It should be pointed out that, except for the phrasing, notation and detail, the arguments in the proofs of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3 are the same as these involved in showing that the space of proper filters of a Boolean algebra is spectral [2, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='12], or that the space of proper filters of an ortholattice is spectral [22, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' It is really only the semilattice-structure that is relevant in the argument, which is one of the reasons that we included a proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3, the other reason relating to the observation made in [13,14] that a lattice can be always regarded as a diagram of dually isomorphic meet semilattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For the case of Boolean algebras and ortholattices, this duality may be taken to be Boolean complementation, or ortho-complementation, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For general bounded lattices the semilattice duality can be taken to be the identity map ı ∶ L∧ ⇆ (L∨)∂, where L∨ = (L∧)∂, as in [13,14], and as we explain in more detail in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Note that the topology Ω on X is the Scott topology, with respect to the specialization order ⊑ on X [18, chapter II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1], which is inclusion of filters (of M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To see that specialization coincides with filter inclusion note that if x ⊑ z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' N o(x) ⊆ N o(z) and a ∈ x, then x ∈ Xa ∈ N o(x) ⊆ N o(z), hence also z ∈ Xa, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' a ∈ z and so x ⊆ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Conversely, if x ⊆ z and U is an open neighborhood of X, let Xa be a basic open such that x ∈ Xa ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' From x ∈ Xa we get a ∈ x ⊆ z, so also z ∈ Xa ⊆ U, hence U ∈ N o(z), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' N o(x) ⊆ N o(z) which by definition means that x ⊑ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' It follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3 that the space X is an HMS space (named so in [23], in honour of Hofmann, Mislove and Stralka), defined by a set of equivalent conditions in [23, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The following representation result is an immediate consequence of Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 15 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Given a meet semilattice M, the map a ↦ Xa is a semilattice isomorphism M ⋍ KOF(Filt(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ◻ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2 The Canonical Dual Space of a Lattice If N is a join semilattice with unit (bottom) element 0, then N∂ (the opposite semilattice, order reversed) is a meet semilattice with unit (top) and Filt(N∂) = Y = Idl(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The topology generated by the basis of sets Y a = {y ∈ Y ∣ a ∈ y}, for a ∈ N, is a spectral topology by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3, observing that Y a ∩Y b = Y a∨b, which ensures that the basis C = {Y a ∣ a ∈ N} is closed under finite intersections (with the empty intersection being Y0 = Y itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For the lattice case, just as orthonegation is represented in [9] by the binary relation � ⊆ X × X defined by x ⊥ z iff ∃a(a ∈ x ∧ a� ∈ z), the identity trivial duality ı ∶ L ⋍ (L∂)∂ is similarly represented [13,14] by the sorted binary relation ⍊ ⊆ X × Y defined by x ⍊ y iff ∃a(a ∈ x ∧ ı(a) ∈ y) iff x ∩ y ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Note that the quasi-seriality condition (1) holds for the complement of the canonical Galois relation ⍊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' As in [14], we represent lattices and normal lattice expansions, more gen- erally, in topologized sorted frames (polarities) F = (X,⍊,Y,(Rσ)σ∈τ), where for each normal lattice operator f of distribution type σ = (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='in+1) the frame is equipped with a sorted relation Rσ of sort σ = (in+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='i1⋯in), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Rσ ⊆ Zin+1 × ∏n j=1 Zij and where Zij = X when ij = 1 and Zij = Y when ij = ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For a bounded lattice L, the bases B = {Xa ∣ a ∈ L} and C = {Y a ∣ a ∈ L} of the spaces X = (X,B),Y = (Y,C), where X = Filt(L) and Y = Idl(L), are the families of compact-open Galois stable and co-stable, respectively, sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Furthermore, B and C are dually isomorphic bounded lattices, with B a sublattice of G(X) and C a sublattice of G(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The two cases for B and C are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The proof follows from [14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' That lemma is phrased in terms of clopen sets in a Hausdorff space, which are then compact-open and compactness and (co)stability are the only properties needed and used in the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Other than that, the claim in [14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7] is phrased in terms of an arbitrary duality ℓ ∶ S ⋍ K∂ ∶ r between meet semilattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The case for lattices follows by specializing the argument to the trivial duality ℓ = r = ı ∶ L∧ ⇆ (L∨)∂, which we do below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' First, stability of the sets Xa,Y a follows by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2, observing that Xa = Γxa and Y a = Γya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' It remains to show that every stable compact-open subset A of X is of the form Xa, for some lattice element a ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Assume A = A′′ is compact-open and let x /∈ A′′ = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let y ∈ A′ such that x /⍊ y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' x ∩ y = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By y ∈ A′ we have A ⍊ y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' for any z ∈ A we have z ⍊ y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' az ∈ z ∩ y, for some lattice element az.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Thus A ⊆ ⋃z∈A Xaz and, by compactness, it follows that A ⊆ Xaz1 ∪ ⋯ ∪ Xazn, for some n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Letting ax = az1 ∨ ⋯ ∨ azn it follows that for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,n, Xazi ⊆ Xax, hence A ⊆ Xax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Notice that ax /∈ x, since ax ∈ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence x /∈ Xax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' This shows that −A ⊆ −Xax and given we also obtained A ⊆ Xax it follows that A = Xax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 16 That both B,C are (dually isomorphic) lattices follows from the fact that (Xa)′ = Y a and (Y a)′ = Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Joins in B,C are defined by taking closures of unions: A ∨ C = (A ∪ C)′′, as in G(X) and G(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let KOG(X),KOG(Y ) be the families of Galois compact-open subsets of X and of Y , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='5, KOF(X) = KOG(X) and KOF(Y) = KOG(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The following choice-free lattice representation theorem is an immediate con- sequence of our so far results in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='6 (Choice-free Lattice Representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let L = (L,≤,∧,∨,0,1) be a bounded lattice and (X,⍊,Y ) its dual filter-ideal frame (X = Filt(L),Y = Idl(L)), with ⍊ ⊆ X × Y defined by x ⍊ y iff x ∩ y ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let X = (X,B) and Y = (Y,C) be the spectral spaces generated by the bases B = {Xa ∣ a ∈ L} and C = {Y a ∣ a ∈ L}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then the map a ↦ Xa is a lattice isomorphism L ⋍ KOG(Filt(L)) and the map a ↦ Y a is a dual isomorphism L∂ ⋍ KOG(Idl(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ◻ The representation of (semi)lattices detailed above is essentially the same as that in [14], by this author and Dunn, the difference lying in the choice of the topology to be imposed on the filter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The lattice G(X) of Galois stable sets is a canonical extension of the lattice, see [8, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='6], which is unique up to an isomorphism that commutes with the lattice embeddings, by [8, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Moshier and Jipsen [23] provide a topological construction of the canonical extension of a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='5, together with uniqueness of canonical extensions up to isomorphism, entails that the filter space of a lattice is what Moshier and Jipsen call a BL-space, defined by a number of equivalent conditions in [23, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' It can be easily verified that the canonical extension FSat(X) defined in [23] is literally identical to G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We substantiate this claim below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Recall first that OF(X) designates in [23] the family of open filters of X = Filt(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The following hold 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' A Galois stable set A = A′′ is a filter of X = Filt(L) and, similarly, a Galois co-stable set B = B′′ is a filter of Y = Idl(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Every open filter F ∈ OF(X) is of the form ⍊{y} for a unique, by separation of the frame (cf Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2), ideal y ∈ Y = Idl(L) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For any subset U ⊆ X, U ′′ = fsat(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Therefore, G(X) = FSat(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For part (1), the proof is straightforward and we only discuss the case of stable sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' First, Galois sets are upsets, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Now let x,z ∈ A = A′′ and suppose that x ∩ z /∈ A′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then there exists an ideal y ∈ A′ such that (x ∩ z) /⍊ y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (x ∩ z) ∩ y = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By x,z ∈ A′′, let a ∈ x ∩ y ≠ ∅ and b ∈ z ∩ y ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then both a,b ∈ y, hence a ∨ b ∈ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' But x,z are filters, hence a ∨ b ∈ x ∩ z, which contradicts the assumption that x ∩ z /∈ A′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 17 For part (2), for any y ∈ Y , ⍊{y} is Galois stable, hence a filter of X, by part (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To see that it is an open set, let x ∈ ⍊{y}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' x ⍊ y, so that a ∈ x ∩ y ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Thus x ∈ Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Since a ∈ y, Xa ⍊ y, so that we obtain x ∈ Xa ⊆ ⍊{y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Thus ⍊{y} ∈ OF(X), for any y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Conversely, let F ∈ OF(X) and let E ⊆ L be such that F = ⋃a∈E Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let y be the ideal generated by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Thus e ∈ y iff there exist e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,en ∈ E, for some n, such that e ≤ e1 ∨ ⋯ ∨ en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We show that F = ⍊{y}, for this y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If x ∈ F = ⋃a∈E Xa, then x ∈ Xa, for some a ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By definition of y, we get a ∈ y, so x ⍊ y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' x ∈ ⍊{y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence F ⊆ ⍊{y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Conversely, let x ⍊ y so that e ∈ x ∩ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then e ≤ e1 ∨ ⋯ ∨ en, where {e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,en} ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' It follows that xe1 ∩ ⋯ ∩ xen = xe1∨⋯∨en ⊆ xe ⊆ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Since ei ∈ E, we have Xei ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Because xei ∈ Xei, all principal filters xe1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,xen ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Since F is a filter, their intersection is in F and then also x ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence ⍊{y} ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For part (3), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2 the set of open elements ⍊{y} is meet-dense in G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By part (2) above, OF(X) = {⍊{y} ∣ y ∈ Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Using also the definition of F-saturation in [23] we obtain fsat(U) = ⋂{F ∈ OF(X) ∣ U ⊆ F} = ⋂{⍊{y} ∣ U ⍊ y} = U ′′ and so FSat(X) = {A ⊆ X ∣ A = fsat(A)} = {A ⊆ X ∣ A = A′′} = G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3 Representing Normal Lattice Operators Let L = (L,≤,∧,∨,0,1,f) be a bounded lattice with a normal operator f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then f extends to a completely normal operator F, of the same distribution type as f, on the canonical extension G(Filt(L)) of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For the proof, we refer the reader to [13, Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The represen- tation of the operator is the same as that given in [11], the difference lying in the axiomatization of the dual frame of the lattice expansion, in particular on the axioms for the relations corresponding to the lattice operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Particular instances of the representation were also given in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To keep this article as self-contained as possible, we sketch the representation steps, drawing on [13, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The base polarity F = (Filt(L),⍊,Idl(L)) consists of the sets X = Filt(L) of filters and Y = Idl(L) of ideals of the lattice and the relation ⍊ ⊆ Filt(L)×Idl(L), defined by x ⍊ y iff x ∩ y ≠ ∅, while the representation map ζ1 sends a lattice element a ∈ L to the set of filters that contain it, ζ1(a) = {x ∈ X ∣ a ∈ x} = {x ∈ X ∣ xa ⊆ x} = Γxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Similarly, a co-represenation map ζ∂ is defined by ζ∂(a) = {y ∈ Y ∣ a ∈ y} = {y ∈ Y ∣ ya ⊆ y} = Γya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For each normal lattice operator a relation is defined, such that if δ = (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='in+1) is the distribution type of the operator, then σ = (in+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='i1⋯in) is the sort type of the relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Without loss of generality, we may restrict to just two normal operators f, of output type 1, and h, of output type ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We then de- fine two corresponding relations R,S of respective sort types σ(R) = (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='i1⋯in) and σ(S) = (∂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='t1⋯tn), where for each j, ij and tj are in {1,∂}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In other words R ⊆ X × ∏j=n j=1 Zij and S ⊆ Y × ∏j=n j=1 Ztj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 18 To define the relations, we use the point operators introduced in [10] (see also [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In the generic case we examine, we need to define two sorted operators ̂f ∶ j=n ∏ j=1 Zij �→ Z1 ̂h ∶ j=n ∏ j=1 Ztj �→ Z∂ (recall that Z1 = X,Z∂ = Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Assuming for the moment that the point operators have been defined, the canon- ical relations R,S are defined by xR⃗u iff ̂f(⃗u) ⊆ x (for x ∈ X and ⃗u ∈ j=n ∏ j=1 Zij), yS⃗v iff ̂h(⃗v) ⊆ y (for y ∈ Y and ⃗v ∈ j=n ∏ j=1 Ztj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (15) Returning to the point operators and letting xe,ye be the principal filter and principal ideal, respectively, generated by a lattice element e, these are uniformly defined as follows, for ⃗u ∈ ∏j=n j=1 Zij and ⃗v ∈ ∏j=n j=1 Ztj ̂f(⃗u) = ⋁{xf(⃗a) ∣ ⃗a ∈ ⃗u} ̂h(⃗v) = ⋁{yh(⃗a) ∣ ⃗a ∈ ⃗v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (16) In other words, ̂f(⃗u) is the filter generated by the set {f(⃗a) ∣ ⃗a ∈ ⃗u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Similarly ̂h(⃗v) is the ideal generated by the set {h(⃗a) ∣ ⃗a ∈ ⃗v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In the canonical lattice frame all axioms of Table 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In particular, all sections of the Galois dual relations R′,S′ of the canonical relations R,S, defined by equations (15), are Galois sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The proof for axioms (F1)–(F3) is given in [13, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For axiom (F4), the claim was first stated as [12, Lemma 25] and a proof of one of the subcases was detailed, the other one being sufficiently similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The omitted proof of the other subcase was provided in [13, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Axiom (F0) obviously holds in the canonical frame since every proper filter x does not intersect the principal ideal ya, for any a /∈ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Similarly for ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The following results will be useful in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='9 ( [12, Lemma 23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In the canonical frame, xR⃗u holds iff ∀⃗a ∈ Ln (⃗a ∈ ⃗u �→ f(⃗a) ∈ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Similarly, yS⃗v holds iff ∀⃗a ∈ Ln (⃗a ∈ ⃗v �→ h(⃗a) ∈ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ◻ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='10 ( [12, Lemma 24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Where R′,S′ are the Galois dual relations of the canonical relations R,S, yR′⃗u holds iff ̂f(⃗u) ⍊ y iff ∃⃗b(⃗b ∈ ⃗u ∧ f(⃗b) ∈ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Similarly, xS′⃗v holds iff x ⍊ ̂h(⃗v) iff ∃⃗e(⃗e ∈ ⃗v ∧ h(⃗e) ∈ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ◻ Each of the relations R,S generates a classical, though sorted, completely additive image operator αR,ηS, respectively, and we designate by αR,ηS the closure of their restriction to Galois sets (stable, or co-stable, according to the distribution types of f,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='5 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='8, αR,ηS dis- tribute over arbitrary joins of Galois sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Composing with the Galois connec- tion, which is a duality of the complete lattices of Galois stable and co-stable 19 sets, completely normal operators are obtained, αf,ηh ∶ ∏n i=1 G(X) �→ G(X), of the same distribution type as f,h, respectively, explicitly defined by αf(A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,An) = αR(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' , Aj � ij=1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' , A′ r � ir=∂ ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=') (A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,An ∈ G(X)), (17) ηh(B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,Bn) = ηS(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' , Br � ir=∂ ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' , B′ j � ij=1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=') (B1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,Bn ∈ G(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (18) By [13, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2], αf,ηh restrict to normal operators of the respective distribution type on the lattice KOG(X) (which, in [13], is identified as the lattice of clopen (in the lattice-theoretic sense) elements of G(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We can then conclude with the representation theorem below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Given a similarity type τ, let τ1,τ∂ be the subtypes consisting of all distribution types in τ of output type 1 and ∂, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If L is a normal lattice expansion of type τ, L = (L,≤,∧,∨,0,1,(fδ)δ∈τ1,(hδ′)δ′∈τ∂ ), then the representation map ζ(a) = {x ∈ X ∣ a ∈ x} = Xa, where X = Filt(L), is an isomorphism ζ ∶ L ⋍ (KOG(X),⊆,∩,∨,∅,X,(αf)δ(f)∈τ1,(ηh)δ(h)∈τ∂ ) of normal lattice expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ◻ The next Proposition identifies the canonical extension of normal operators we have defined as their σ/π-extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The operations αf, ηh are the σ and π extensions of f,h, respectively, as these are defined in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The argument has been detailed in [12, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Roughly, given Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='8, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='13 applies so that if R is the relation constructed from a normal lattice operator f, then αR(Γ⃗u) = R⃗u = αR(Γ⃗u) = Γ( ̂fR(⃗u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' As- suming f is of distribution type δ = (⃗ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1), f σ as defined in [8] is a sorted map and it is defined on a tuple ⃗F of Galois sets by extending its definition on closed elements, by f σ( ⃗F) = ⋁⃗u∈ ⃗ F fσ(Γ⃗u), where fσ is defined on closed elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' It is shown in [12, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3] that fσ(Γ⃗u), as defined in [8], satisfies the identity fσ(Γ⃗u) = Γ( ̂fR⃗u) ∈ G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Thereby, the σ-extension of f coincides with the operation αR that we defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For an operator h of distribution type (⃗tj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='∂), with corresponding canonical frame relation S, its dual σ-extension on closed elements satisfies, respectively, the identity h∂ σ(Γ⃗u) = Γ(̂hS⃗u) ∈ G(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Extend- ing to Galois sets we similarly have h∂ σ( ⃗F) = ηS( ⃗F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The single-sorted σ and π-extensions of f,h, respectively, are then obtained by composing appropriately with the Galois connection, resulting in the maps αf = f σ,ηh = hπ, as defined in equations (17) and (18), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 5 Representing Quasi-Complemented Lattices In this section we extend the lattice representation of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2 to the case of a lattice with an additional quasi-complementation operator, assuming the 20 axiomatization of at least the minimal system of Figure 1 and specializing the constructions of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The canonical dual frame is the structure (X,⍊,Y,Sν), where X = Filt(L), Y = Idl(L), ⍊ ⊆ X × Y is defined by x ⍊ y iff x ∩ y ≠ ∅ and Sν ⊆ Y × X is the canonical relation defined using the point operator ̂ν ∶ X �→ Y , by equations (15) and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For the case at hand, the definitions are given by equation (19) ̂ν(x) = ⋁{yνa ∣ a ∈ x} ySνx iff ν(x) ⊆ y (x ∈ X,y ∈ Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (19) Observe that Sνx = Γ(̂ν(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='9, Sν ⊆ Y × X is equivalently defined by the condition ySνx iff ∀a ∈ L(a ∈ x �→ νa ∈ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='10 its Galois dual relation S′ ν ⊆ X × X is defined by zS′ νx iff ∀y ∈ Y (ySνx �→ z ⍊ y) iff z ⍊ ̂ν(x) iff ∃a ∈ L(a ∈ x and νa ∈ z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (20) Observe also that S′ νx = {̂ν(x)}′ = ⍊{̂ν(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' A sorted image operator ηS ∶ ℘(X) �→ ℘(Y ) is generated by the relation Sν, defined by equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The closure of the restriction of ηS to stable sets is designated by ηS, defined by equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='5, given also Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='8, ηS distributes over arbitrary joins of stable sets in G(X), returning a join of co-stable sets in G(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The canonical extension ην ∶ G(X) �→ G(X) of the normal lattice operator ν is then obtained by composing with the Galois connection, setting ην(A) = (ηS(A))′ ∈ G(X), hence ην(A) = ⋂x∈A S′ νx, where S′ ν is the Galois dual relation of Sν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Given (12), we obtain ην(A) = ⋂x∈A ⍊{̂ν(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In the terminology of [8] ην = νπ is the π-extension of the lattice operator ν, defined by νπ(A) = ⋂ x∈A ⍊{̂ν(x)} = ⋂ x∈A S′ νx = ην(A) = ⍊(ηS(A)) = ⍊(ηS(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (21) Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1 (Switching Notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hereafter, we simplify notation as in Re- mark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='14, switching to the more familiar � for the incompatibility relation S′ ν and letting νπ(A) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ην(A)) be designated by A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let L = (L,≤,∧,∨,0,1,ν) be a bounded lattice with an antitone map ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If L belongs to one of the lattice varieties of Figure 1, then so does, respectively, the full complex algebra (G(X),⊆,⋂,⋁,∅,X,( )∗) of its canonical frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In other words, each of the varieties of Figure 1 is closed under canonical extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Assume first that ν in L is a minimal quasi-complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By the re- sults of sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3, ( )∗ ∶ G(X) �→ G(X) co-distributes over arbitrary joins in G(X), returning a meet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (⋁i∈I Ai)∗ = ⋂i∈I A∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Hence the full complex algebra of the canonical frame (G(X),⊆,⋂,⋁,∅,X,( )∗) is a complete lattice with a minimal quasi-complementation operator, given that we also have 21 ∅∗ = (ηS(∅))′ = ∅′ = X, using normality of the classical, though sorted, image operator ηS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If ν satisfies the Galois condition a ≤ ννa, then it is immediate that the canonical relation � = S′ ν, defined by equation (20), is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='15, this implies that A ⊆ A∗∗, for any A ∈ G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Assume now that ν is an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To show that A∗∗ ⊆ A, it suffices to verify that the equivalent condition (3)c of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='15 holds in the canonical frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Given that Sνz = Γ(̂ν(z)), where for a filter z, ̂ν(z) is the ideal generated by the set {νe ∣ e ∈ z}, the inclusion Sνz ⊆ Sνx is equivalent to the inclusion Γ(̂ν(z)) ⊆ Γ(̂ν(x)), hence to ̂ν(x) ⊆ ̂ν(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To see that this implies x ⊆ z, let a ∈ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Then νa ∈ ̂ν(x), so νa ∈ ̂ν(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let then e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' ,en ∈ z such that νa ≤ νe1 ∨ ⋯ ∨ νen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' This is equivalent to ν(νe1 ∨ ⋯ ∨ νen) ≤ ννa ≤ a, in turn equivalent to e1 ∧ ⋯ ∧ en ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Since z is a filter, this implies a ∈ z and this completes the proof that x ⊆ z under the given assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If the lattice is an ortholattice, then by the argument for the case of lattices with an involution previously given and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='15 it suffices to verify that the canonical relation � = S′ ν is irreflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='10, x�z holds iff there exists a lattice element e such that e ∈ z and νe ∈ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Reflexivity, x�x, would then imply that e ∧ νe = 0 ∈ x, contradicting the fact that x is a proper filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For the case where the lattice is a De Morgan algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' a distributive lattice with an involution, it suffices to prove that G(X) is distributive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' An algebraic proof of this has been given in [8, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1], but we provide here a new proof based on the constructions we have presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Note first that both lattice join ∨ and meet ∧ are trivially normal lattice operators in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1, but meet is an operator (in the J´onsson- Tarski sense) only when it distributes over joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' When this is the case, meet also has the distribution type (1,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Its σ-extension ∧σ, is constructed as outlined in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Specifically, letting ∧ = f, the point operator ̂f on filters is defined by ̂f(x,z) = ⋁{xa∧b ∣ a ∈ x and b ∈ z} and the canonical relation R∧ is then defined by xR∧uz iff ∀a,b(a ∈ u and b ∈ z �→ a ∧ b ∈ x), using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Note that R∧ is the upper bound relation of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Considering the image operator αR ∶ ℘(X)×℘(X) �→ ℘(X) defined by αR(U,W) = {x ∈ X ∣ ∃u ∈ U∃z ∈ W xR∧uz}, we obtain that αR(A,C) = A ∩ C, for A,C ∈ G(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='8, all sections of the Galois dual relation of R∧ are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' It then follows by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7 that intersection distributes over arbitrary joins, in other words, G(X) is a completely distributive lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Finally, for the case of Boolean algebras, combine the arguments given for ortholattices and De Morgan algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 6 Spectral Duality Let M,G,INV,DMA,O and BA be the categories of algebras in the respec- tive varieties M,G,INV,DMA,O and BA of Figure 1 with the usual algebraic homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' As in [13], SRFτ designates the category of sorted residuated frames with 22 Table 2: Axioms for SRF∗ νM (F0) The complement I of the Galois relation ⍊ of the frame is quasi-serial, in other words ∀x ∈ X∃y ∈ Y xIy and ∀y ∈ Y ∃x ∈ X xIy (F1) The frame is separated (F2) For each z ∈ X, Sνz is a closed element of G(Y ) and if z is a clopen element (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Γz = ⍊{v} for a (unique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' by separation) point v in Y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' then Sνz is a clopen element of G(Y ) (F3) For each y ∈ Y the set ySν is decreasing (a down set) (F4) Both sections of the Galois dual relation S′ ν of Sν are Galois sets (F5) Clopen elements are closed under finite intersections in each of G(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='G(Y ) (F6) The family of closed elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' for each of G(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='G(Y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' is the intersection closure of the respective family of clopens (F7) Each of X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='Y carries a spectral topology generated by the basis of their respective families of clopen elements For a sorted map π = (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='q) ∶ (X2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='I2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='Y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='S2ν) �→ (X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='I1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='Y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='S1ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' where p ∶ X2 �→ X1 and q ∶ Y2 �→ Y1 (M1) ∀x′ ∈ X2∀y′ ∈ Y2 (x′I2y′ �→ p(x′)I1q(y′)) (M2) ∀x ∈ X1∀y′ ∈ Y2(xI1q(y′) �→ ∃x′ ∈ X2(x ≤ p(x′) ∧ x′I2y′)) (M3) ∀x′ ∈ X2∀y ∈ Y1(p(x′)I1y �→ ∃y′ ∈ Y2(y ≤ q(y′) ∧ x′I2y′)) (M4) ∀z ∈ X1∀v ∈ Y2(q(v)S1νz �→ ∃x ∈ X2(z ≤ p(x) and vS2νx)) (M5) for all points u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' π−1(Γu) = Γv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' for some (unique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' by separation) v a relation Rσ for each σ ∈ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For our present purposes we only consider frames F = (X,⍊,Y,Sν), with σ(Sν) = (∂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1), given that the distribution type of the normal lattice operator ν under study is δ(ν) = (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In particular then we let SRFν = SRF{(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='∂)} designate the category with objects the sorted residuated frames with a relation Sν as above, subject to the axioms of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' SRFν is too large a category for duality purposes and we specify full subcategories for each of the cases of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In [13], the notation SRF∗ τ was used to designate the intended subcategory and we keep with this notation, while also subscripting appropriately to distinguish between the different categories of interest in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For a frame F in SRF∗ τ, we let L(F) be the full complex algebra F+ of stable sets and L∗(F) the complex algebra of clopen elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 23 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let SRF∗ νM be the full subcategory of SRFν axiomatized by the axioms in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' There exist functors F,L∗ forming a categorical duality F ∶ M ⋍ (SRF∗ νM)op ∶ L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Let L = (L,≤,∧,∨,0,1,ν) be a lattice with a minimal quasi complementa- tion operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Define F(L) = (Filt(L),⍊,Idl(L),Sν) to be the canonical frame of the lattice constructed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Axioms (F0)–(F4) hold for the canonical frame, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Note that axiom (F2) of Table 2 is a strengthening of the corresponding axiom in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To verify the stronger version of (F2), suppose Γz = ⍊{v} is a clopen ele- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Clopen elements of G(X) in the canonical frame are precisely the stable compact-open sets Xa = Γxa = xa↑ = ⍊{ya}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By definition of the point operator ̂ν and of the canonical relation Sν in equation (19), Sνxa = Γ(̂ν(xa)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' It is straightforward to see that ̂ν(xa) = yνa, hence Sνxa = Γyνa = Y νa is a clopen el- ement of G(Y ), where yνa = (νa)↓ is the principal ideal generated by the lattice element νa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Axiom (F5) holds, since Xa ∩ Xb = Xa∧b, while also Y a ∩ Y b = Y a∨b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For (F6), by join-density of principal filters, any filter x is the join x = ⋁a∈x xa, hence every closed element Γx of G(X) is an intersection Γx = ⋂a∈x Γxa = Γ(⋁a∈x xa) and similarly for closed elements Γy ∈ G(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Finally, axiom (F7) was verified in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3 for meet semilattices and the same proof applies to establish that the topology on each of X = Filt(L) and Y = Idl(L) is a spectral topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By the above argument, F(L) is an object in the category SRF∗ νM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='5, {Xa ∣ a ∈ L} = KOG(X) is the complex algebra of clopen el- ements L∗F(L) and we then verified in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='6 that the representation map a ↦ Xa = {x ∈ X ∣ a ∈ x} is a lattice isomorphism L ⋍ KOG(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Since Xνa = Γxνa = (Γ(̂ν(xa)))′ = (Γxa)∗ = (Xa)∗, the representation map is an isomorphism L ⋍ L∗F(L) of lattices with a (minimal) quasi-complementation operation ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For morphisms h ∶ L1 �→ L2, the argument that F(h) ∶ F(L2) �→ F(L1) is a frame morphism satisfying axioms (M1)–(M4) is a special instance of the argument given in the proof of [13, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='9], handling the general case of arbitrary normal lattice expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The proofs regarding axioms (M5) and (M6) were given in [13, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='6, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' This establishes that F ∶ M �→ (SRF∗ νM)op is a contravariant functor satisfying L ⋍ L∗F(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Now let F be a sorted residuated frame in the category SRF∗ νM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We have let L(F) = F+ = (G(X),⊆,⋂,⋀,∅,X,( )∗) be its full complex algebra (Defini- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='6) and L∗(F) = (KOG(X),⊆,∩,∨,∅,X,( )∗) be its subalgebra of clopen elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' That the operation ( )∗ restricts to clopen elements is clear, since (Xa)∗ = Xνa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='16, L∗(F) is an object in the category M of lattices with a minimal quasi-complementation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' If π = (p,q) ∶ F2 �→ F1 is a frame morphism in SRF∗ νM, then it was verified in [13, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='21] that L∗(π) = π−1 ∶ G(X1) �→ G(X2) is a complete lattice homomorphism of the complete lattices of stable sets of the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Given axiom (M4), it was established in [13, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='24, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='25] that π−1 ∶ F+ 1 �→ F+ 2 is in fact a homorphism of the full complex algebras of the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 24 By axiom (M5), π−1 preserves closed elements, hence by [13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='23] it also preserves clopen elements (from which continuity of π follows, since clopen stable elements are precisely the basic open sets in the topology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The above argument has established that L∗ is a contravariant functor from the category SRF∗ νM to the category M of lattices with a minimal quasi- complementation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' We have already also established that for any object L in M we have an isomorphism L ⋍ L∗F(L) and it remains to argue that for any sorted residuated frame F in the category SRF∗ νM we also have that F ⋍ FL∗(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' To avoid repetitions, we refer the reader to the proof of the general duality theorem for any normal lattice expansion [13, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Categories of frames F = (X,⍊,Y,Sν), where we set � = S′ ν, corresponding to lattices with a quasi complementation operation are axioma- tized by the axioms of Table 2 as well as one or more of the additional axioms below, as specified for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' (G) � is symmetric (INV) ∀x,z ∈ X[∀v ∈ Y (vSνz �→ vSνx) �→ x ≤ z] (O) � is irreflexive (D) All sections of the Galois dual relation R′ of the upper bound relation R of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='7 are stable SRF∗ νM Table 2 axioms SRF∗ νG Table 2 axioms + Axiom (G) SRF∗ νINV Table 2 axioms + Axiom (G) + Axiom (INV) SRF∗ νO Table 2 axioms + Axiom (G) + Axiom (INV) + Axiom (O) SRF∗ νDMA Table 2 axioms + Axiom (G) + Axiom (INV) + Axiom (D) SRF∗ νBA Table 2 axioms + Axiom (G) + Axiom (INV) + Axiom (O) + + Axiom (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The spectral duality of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1 specializes to dualities for each of the frame categories of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2 and their respective categories of bounded lattices with a (quasi) complementation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' That the double dual of a lattice in one of the varieties of Figure 1 is in the variety in question was verified in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' That the (full) complex algebra of a frame in one of the frame categories which satisfies one or more axioms from the list in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='2 is an algebra in the respective variety corresponding to the frame category was verified in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The rest of the duality argument for each of the cases is the same as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 7 Concluding Remarks In this article, we have provided alternative constructions for the choice-free representation and duality for Boolean algebras and Ortholattices, first given 25 in [2,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' A Stone duality result for De Morgan algebras, using choice, was pub- lished by Bimb´o in [3] and we have given here a choice-free version of the dual- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Our background motivation has been the J´onsson-Tarski [19,20] approach, constructing set-operators from relations to represent operators on Boolean al- gebras, a project that has been extended with Dunn’s research on generalized Galois logics (gaggles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' In [13], we presented a generalization of this project of relational representation to cases where distribution may not be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The framework of [13] was applied in this article to the case of bounded lattices with a quasi-complementation operator, recasting the duality of [13] in a choice-free manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' By treating both De Morgan (and Boolean) algebras, as well as Or- tholattices, it has been shown that the presence or not of distribution does not create any significant obstacle, as distribution in the complete lattice of stable sets of a sorted frame has been shown to be first-order definable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' For the dis- tributive case, the resulting semantics from the approach presented appears to have strong affinities to Holliday’s possibility semantics [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The approach presented can be extended to any normal lattice expansion, including the case of modal lattices studied in [1], based on the framework developed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' References [1] Nick Bezhanishvili, Anna Dmitrieva, Jim de Groot, and Tommaso Moras- chini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Positive (modal) logic beyond distributivity, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' [2] Nick Bezhanishvili and Wesley Holliday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Choice-free Stone duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' The Journal of Symbolic Logic, 85(1):109–148, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' [3] Katalin Bimb´o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Functorial duality for ortholattices and De Morgan lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Logica Universalis, 1:311–333, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' [4] Katalin Bimb´o and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Michael Dunn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 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+page_content=' Topological duality and lattice ex- pansions, II: Lattice expansions with quasioperators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Algebra Universalis, 71(3):221–234, May 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 27 [25] Alasdair Urquhart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' A topological representation of lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' Algebra Uni- versalis, 8:45–58, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE5T4oBgHgl3EQfkQ83/content/2301.05661v1.pdf'} +page_content=' 28' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 98160,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Zacatecas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Mexico cSorbonne Universit`es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Universit´e de Technologie de Compi`egne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Heudiasyc UMR 7253,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Compi`egne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 60200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' France Abstract In this paper we address the control problem of aerial cable suspended load trans- portation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' using multiple Unmanned Aerial Vehicles (UAVs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' First, the dynamical model of the coupled system is obtained using the Newton-Euler formalism, for n UAVs transporting a load, where the cables are supposed to be rigid and mass- less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The control problem is stated as a trajectory tracking directly on the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' To do so, a hierarchical control scheme is proposed based on the attractive ellip- soid method, where a virtual controller is calculated for tracking the position of the load, with this, the desired position for each vehicle along with their desired cable tensions are estimated, and used to compute the virtual controller for the position of each vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' This results in an underdetermined system, where an infinite number of drones’ configurations comply with the desired load position, thus additional constrains can be imposed to obtain an unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Further- more, this information is used to compute the attitude reference for the vehicles, which are feed to a quaternion based attitude control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The stability analysis, using an energy-like function, demonstrated the practical stability of the system, it is that all the error signals are attracted and contained in an invariant set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Hence, the proposed scheme assures that, given well posed initial conditions, the closed-loop ∗This work was supported by the Mexican National Council of Science and Technology CONACYT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' ∗∗corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Email address: diego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='mercado@cimat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='mx (Diego Mercado-Mercado) Preprint submitted to Arxiv January 30, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='11350v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='RO] 26 Jan 2023 system guarantees the trajectory tracking of the desired position on the load with bounded errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The proposed control strategy was evaluated in numerical simu- lations for three agents following a smooth desired trajectory on the load, showing good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Keywords: Suspended Payload Control, UAVs, Multi-Agent Systems, Aerial Transportation, Attractive Ellipsoid Method, Practical Stability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Introduction Small scale Unmanned Aerial Vehicles (UAVs) have received increased atten- tion in the last decades, thanks to their great potential in civilian applications such as surveillance, monitoring, photography, structures inspection, fast deployment of food and medical equipment, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' A clear example of this, is the use of UAVs to transport packages quickly, specially in urban environments [2], opti- mizing the distance and time required by avoiding and alleviating traffic conges- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Such potential has not passed unnoticed, to the point that the most important transnational delivery companies have shown great interest, and invested time and money to develop aerial transportation systems using drones capable to deliver packages quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Aerial transport using UAVs presents important design challenges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' given that drones normally have reduced payload capabilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' but it also appears as an inter- esting control problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' provided that the payload mass and inertia matrix may be unknown to the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' also,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' it may be required to optimize the energy employed in the mission,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' since the autonomy of the system is considerably reduced with larger payloads,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' the payload itself introduces uncertainty and distur- bances to the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' for instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' the center of mass of the system is shifted once a payload is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' More in particular, we center our attention to the problem o cable suspended payloads, a problem that arises from missions where the UAV hooks the payload on flight, without the need of landing, a useful feature which allows to avoid the risk of landing in hazardous terrains and situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Cable sus- pended aerial transportation (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 1) has attracted recently the attention from the research community, specially from the automatic control perspective, since the resulting system has more degrees of under-actuation, and the suspended pay- load introduces oscillations and disturbances [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Accordingly, a few works are to be found in the literature [4], where the former studies were centered in the study of a single vehicle under idealized conditions, considering a punctual mass suspended by a rigid mass-less cable [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' From 2 W Y X Z … … Figure 1: Aerial transportation system with multiple UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The cables are assumed rigid and mass-less, while the payload is considered as a punctual mass carried by n aerial vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' there, some works have popped up considering different variations to the problem, assuming conditions closer to a real application, such as assuming the load as a rigid rod [7, 8], or a rigid body [9, 10], or considering that the cables possess elasticity [8, 11] or are flexible [12], or that more than one vehicle contributes to carry the load [13, 14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Nevertheless, small aerial vehicles, particularly multirotor rotor-crafts, suffer from their flight time autonomy and their payload capacity, while increasing their size is undesirable for safety reasons, since the risk of severe accidents scales up with size, to the point of endangering human lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' In that sense, aerial transporta- tion using multiple drones appears as an appealing and challenging alternative, with great potential for applications in the real world, allowing to distribute the payload weight among a team of small UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Furthermore, such multi-agents scheme adds redundancy and enables fault tolerance capabilities to the system, such that the mission can be accomplished even in the case of a problem with some of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Related Works Besides its applicability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' the problem of aerial transportation with multiple drones and cable suspended loads turns out to be a very interesting and challeng- ing problem from the automatic control point of view,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' given that it deals with a nonlinear system of high order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' with multiple degrees of freedom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' and an impor- tant number of degrees of under-actuation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' not to mention the parametric uncer- tainties and disturbances induced by the other agents or the payload itself,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' or even external phenomena affecting the system in real applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' such as wind gusts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Only a handful of works have studied the problem of aerial transportation of cable suspended payloads with multiple agents [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' For instance, in [17], a pas- sivity based control is proposed for two UAVs, but only in the XZ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The proposed control strategy was validated only in numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Also, the authors in [7] present an adaptive dynamic compensator, for the case of two UAVs carrying a cable suspended rigid rod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The cables are considered rigid, and exper- imental validation is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Moreover, in [18], the authors propose a vision- based cable suspended load transport with two quadrotors, in a leader follower approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The load is a rigid rod, but it is modeled as a point mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' An LQR (Least Quadratic Regulator) controller is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The performance of the trans- portation system is validate in real experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Nevertheless, these works only consider a fixed small number of agents, ranging from two to four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' On the other hand, there are some works that study the more general trans- portation problem for n ∈ Z+ | n > 2 UAVs [19, 20, 21], but they focus mainly on the motion planning problem, using simple open-loop controllers, it is, without considering any feedback on the pose of the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' This is somehow similar to the flight formation problem, with the load acting as an external disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Such is the case of the work at [22], where the dynamic model of the transportation system is presented using the hybrid systems formalism, where the payload is modeled either as a point mass or as a rigid body, and the cables are considered rigid and mass-less, with strictly positive tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The hybrid systems are used to model the cases when an agent stops contributing with the payload transportation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=', when for some reason its cable tension equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Then, a differential flatness approach is employed to design a control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' However, such controller lacks of feedback in the payload, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' it acts in open-loop with respect to the payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Experimental results with n = 3 UAVs validate the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Probably the most studied controller schemes in the literature for the cable suspended payload transportation control problem with n UAVs is the use of Ge- ometric controllers that work directly in the Special Euclidean group in three di- mensions SE(3), considering the payload as a rigid body, along with rigid and 4 mass-less cables [10, 9, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Also, in [12] a geometric control is proposed, but the cables are assumed as flexible instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The Euler-Lagrange formalism is nor- mally used to obtain the dynamic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' These works propose interesting solu- tions to the control problem under consideration, with formal stability analysis, nonetheless, such controllers are not straightforward to implement in real-time experiments, hence their validation remains only in the simulation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Another interesting recent related work is the one presented in [24], where a transportation system with multiple UAVs is presented, but instead of a cable suspended payload, multi-links legs are used to hold a rigid payload up, something like a flying parallel robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Experimental results demonstrate the feasibility of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Contributions In this work, we deal with the control problem of load trajectory tracking in an aerial transportation system conformed by n ≤ 2 UAVs and a cable sus- pended payload, considered as a punctual mass, and rigid mass-less cables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' In contrast to former multi-agent aerial systems such as formation flight or consen- sus based, here the control problem is defined directly on the load position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The proposed closed-loop hierarchical controller is designed using the attractive ellip- soid method, where the resultant of the cable tensions is used as a virtual controller to assure trajectory tracking on the load, then, the desired tension on each UAV cable is obtained from the virtual load controller, resulting in an under-determined system, from where additional constrains can be imposed to obtain an unique so- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Afterwards, the desired UAV position is feed to a position controller for each agent, using its desired attitude as a virtual controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Finally, a quaternion based controller is employed to command the attitude of each drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The stability of the closed-loop system was analyzed using the attractive el- lipsoid method, guaranteeing the practical stability of the system as long as the initial conditions are well posed, and the desired load trajectories are smooth, with bounded relative accelerations between the agent and the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' In other words, the load trajectory tracking is assured with bounded errors, which depend mainly on the disturbances caused by the virtual controllers transient response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Numerical simulations validate the good performance of the proposed strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Focusing in the general problem of cable suspended payload aerial transporta- tion with n ≤ 2 agents, in this paper we proposed a closed-loop controller, con- sidering feedback from the load position, which is the main goal in the control formulation, this is in contrast with other works on the literature which focus mainly in the motion planning problem and propose open loop solutions without 5 feedback from the load [19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' This work also offers an alternative solution to the geometric control approach proposed in the works [10, 9, 23, 12], in a more intuitive framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Moreover, the use of the attractive ellipsoid method allows to study the robust- ness of the system by providing the size of the stability region using the attractive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Furthermore, the method defines a procedure to obtain the gains to as- sure the smallest invariant set contemplating the bounds of the disturbances of the coupled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Additionally, taking into consideration the disturbances due to the virtual controllers, an analysis of the necessary conditions for stability is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The contribution of the present work can be summarized as follows: A hierarchical controller based in the attractive ellipsoid method is proposed for the load trajectory tracking problem in cable suspended payload aerial transportation systems with n agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The proposed closed-loop system is continuous and assures practical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The stability analysis was carried out using a Lyapunov-like function and considering the disturbances due to the virtual controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Also, the op- erating conditions were discussed, including the initial conditions and the relative accelerations between the load and the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Numerical simula- tions validated the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The reminder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' In Section 2 the control objective is established and the dynamic model of the aerial transportation sys- tem with multiple agents is presented, while.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Then, in Section 3 we introduce the control strategy, and analyze the closed-loop stability by means of an energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Thereafter, in Section 4, the performance of the closed-loop system is studied in numerical simulations, showing good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Finally, in Section 5, some conclusions and future works are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Problem Statement 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Dynamic model Let us consider an aerial transportation system composed of n ∈ Z+ Un- manned Aerial Vehicles (UAVs) carrying a cable suspended payload, under the following assumptions The load can be considered as a punctual mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 6 All the cables are mass-less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The cables are rigid with positive tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The aerodynamic effects are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The air friction is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The desired trajectories are smooth, with bounded accelerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Each of the agents (UAVs) has a mass mi, and an inertial matrix Ji, then, the dynamics of motion for the i-th agent can be modeled as mi ¨xi = fiRi( ¯qi)e3 −mige3 +Tiαi (1) ˙¯qi = 1 2 ¯qi ⊗ ¯Ωi (2) Ji ˙Ωi +Ωi ×JiΩi = τi (3) where xi ∈ R3 stands for the position of the i-th UAV’s center of mass, with respect to an inertial reference frame W, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='1, Ri ∈ SO(3) is a rotation matrix providing the attitude of each UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' fi ∈ R represents the total thrust force produced by the rotors, e3 ≜ [0 0 1]T, g is the gravity constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Ti ∈ R are the cable tensions, and αi ∈ S2 is a unit vector representing the orientation of each cable connecting the i-th UAV with the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Moreover, Ωi ∈ R3 is the angular velocity in the body fixed frame Bi, and τi ∈ R3 are the input torques, ¯Ωi = [0,ΩT i ]T is a pure quaternion of the vector Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The attitude is parameterized by the unit quaternion [25] ¯qi = [qi0,qT i ]T ∈ Q, where qi0 ∈ R and qi = [qi1,qi2,qi3]T ∈ R3, while its inverse quaternion is given by ¯q∗ i = [qi0,−qT i ]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The quaternion product ⊗, between two unit quaternions p,q ∈ Q is given by [26] p⊗q = � p0 −pT p Ip0 +[p×] ��q0 q � (4) where p× maps the vector p ∈ R3 to a skew symmetric matrix so(3), and the iden- tity matrix is I ∈ R3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Furthermore, the rotation matrices Ri are parameterized with unit quaternions as follows Ri(q) = � � 1−2q2 i2 −2q2 i3 2qi1qi2 −2qi0qi3 2qi1qi3 +2qi0qi2 2qi1qi2 +2qi0qi3 1−2q2 i1 −2q2 i3 2qi2qi3 −2qi0qi1 2qi1qi3 −2qi0qi2 2qi2qi3 +2qi0qi1 1−2q2 i1 −2q2 i2 � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' (5) 7 The cables are assumed to be mass-less and rigid, with length Li, hence, the payload is subject to the following restriction with respect to each aerial drone xi = xL −Liαi (6) The payload is considered as a punctual mass with mass mL, with position xL ∈ R3, then, the dynamics of the load are subject to the following equation mL ¨xL = −mLge3 −ΣTiαi (7) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Control problem We consider the trajectory tracking problem on the load, where the main goal is to design a closed-loop controller that assures that the position of the load xL(t), and its time derivatives, tend to a desired time varying reference xLd(t), it is {xL, ˙xL} → {xLd, ˙xLd}, considering the dynamic of the system (7), subject to the constrains (6) and the coupled dynamics of (1), guaranteeing that the error trajectories {xe, ˙xe} are confined to an attractive invariant set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The controller is continuous and the algorithm considers the underactuation of the aerial vehicles and the coupled dynamics of the other drones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Note that the control objective is defined on the payload’s position, not on the UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' In contrast with formation flight controllers [27, 28], here the formation of the agents is irrelevant as long as they respect the system restrictions imposed by the cables (6), accordingly, there exist infinity configurations (for- mations) where the agents are able to transport the load to its desired reference trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Additional constrains can be imposed to the system in order to obtain a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Control Strategy As presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 2, where the block diagram of the control strategy is depicted, a hierarchical controller is proposed, where a virtual controller is em- ployed to control the load uL, which represents the desired resultant tension, it is the sum of the desired tensions in each cable ∑n i=1 (Tidαid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' From the Load’s control law uL, the desired resultant force on the load is distributed among the vehicles and the desired position of each agent is computed using the desired di- rection of the cable αid and the desired tension Tid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Note that the resultant system is under-determined, and there exist an infinite number of valid solutions, which means that there exists an infinite number of valid configurations for the agents to 8 position control 1 attitude control 1 UAV 1 position control n attitude control n UAV n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' solution selector load control Load Figure 2: Blocks diagram of the closed-loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' A hierarchical control scheme is proposed, where the load control uses the resultant tensions as a virtual control input, then the target tensions are distributed to the agents and a position controller computes the desired thrust force vector in order to guarantee that each agent moves to its desired position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Finally, an attitude controller extracts the desired orientation of each drone and computes the required input torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' get the load to its desired position, however, it is enough to impose additional con- strains on the desired tensions to obtain only one valid solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Then, a virtual controller for the position of the vehicles is computed, providing the desired input force for each quadrotor, this results in a desired thrust force fid and a desired attitude quaternion ¯qid, which is parsed to the attitude control for each vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Therefore, assuring the desired attitude is reached fast enough ¯qi → ¯qid, implies that each of the vehicles tends to their desired position xi → xid, and the tension in the cables tends to the desired tension, resulting in the tracking of the desired position by the load, it is xL → xLd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' In this section, first the attractive ellipsoid method is described, afterwards, the attitude and virtual position controller are obtained, then, the position and load control laws, used in the virtual controllers, are developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Attractive Ellipsoid Method The attractive ellipsoid method assures that the system will be bounded by an attractive set which form is defined by a matrix P, as is defined by 9 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Attractive Ellipsoid [29]: An ellipsoid, represented by ε ⊂ Rn with center in ξc is given by: ε = � ξ ∈ Rn| (ξ −ξc)TP(ξ −ξc) ≤ 1 � (8) is said to be attractive if for all the trajectories of the vector state ξ, it is satisfied that limsup t→∞ (ξ −ξc)TP(ξ −ξc) ≤ 1 (9) where the ellipsoidal matrix P ∈ Rn×n is a symmetric positive definite matrix 0 < P, P = PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' If the set (8) is attractive, it is if it satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' (9), the practical stability [30] of the system is guaranteed, meaning that the system states converge and are bound to a neighborhood around the chosen center of the ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Attitude Controller Considering the control strategy for the attitude of an aerial vehicle in [31], and a desired attitude quaternion ¯qid, let us define the attitude quaternion error ¯qie ≜ ¯q∗ id ⊗ ¯qi = � qid0 qi0 +qT idqi qid0 qi −qi0qid −qid ×qi � (10) with Ωie ≜ Ωi−Ωid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' From now on, subindex d denotes a desired reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Using this variables, the error manifolds si can be defined as si ≜ Ωie +ρiqie (11) where ρi is a positive diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Then, the attitude controller for each vehi- cle τi is given by τi = −Kdisi −βisat(γisi) (12) where Kdi is a positive defined matrix, and βi and γi are positive diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' This controller guarantees the convergence of the error signals to an invariant set [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Position Controller Now, the next step is to estimate the desired attitude signals that will assure the tracking of the position of each vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Following the position control strategy for a UAV in [26], let us define a virtual controller uid = [uid1 uid2 uid3]T ∈ R3 for each one of the vehicles dynamics as uid ≜ fidRid( ¯qid)e3 (13) where the desired thrust fid, and the desired attitude quaternion ¯qid can be com- puted as fid = ∥uid∥ and qid = � ������ 1 2 �2 ˆuid3 +2 − ˆuid2 √2 ˆuid3+2 ˆuid1 √2 ˆuid3+2 0 � ������ (14) with ˆuid = uid/∥uid∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Therefore the corresponding desired velocity can be calcu- lated using the time derivative of the desired direction of the thrust ˙ˆuid as follows Ωid = � ������� − ˙ˆui2+ ˙ˆui3 ˆui2 ˆui3 +1 ˙ˆui1+ ˙ˆu3 (− ˆu1) ˆui3 +1 ˆuid1 ˙ˆuid2 − ˆuid2 ˙ˆuid1 ˆuid3 +1 � ������� (15) Note that the derivative of the controller uid is needed to compute the desired Ωid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Load Control and Error Dynamics Let’s define the error variables for the position of the load xe and the i-th vehicle xei as xe ≜ xL −xLd (16) xei ≜ xi −xid (17) from the system constrains (6), the desired position of each vehicle is xid ≜ xLd −Liαid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' (18) 11 using the 2nd time derivative of (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' and substituting (7) we obtain ¨xe = ¨xL − ¨xLd (19) ¨xe = − 1 mL n ∑ i=1 (Tiαi)−ge3 − ¨xLd (20) now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' consider a virtual controller for the load defined as uL ≜ − n ∑ i=1 (Tidαid) (21) adding and subtracting (21) in (20) results in the open-loop error dynamic of the load ¨xe = − 1 mL � uL − n ∑ i=1 (ζLi) � −ge3 − ¨xLd (22) where ζLi is the error between the actual and desired tension defined as ζLi ≜ Tiαi −Tidαid (23) Note that the disturbance on the load corresponds to the sum of the error in the tensions on each vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Now, for the error dynamics in each vehicle, let us consider the second time derivative of the error position from each UAV dynamics (17) and (18), also, adding and subtracting the virtual position controller (13) results in ¨xei = 1 mi ui + 1 mi ζi −ge3 + 1 mi Tiαi − ¨xLd +Li ¨αid (24) where ζi ≜ fiRie3 − fidRide3 (25) Note that the disturbance ζi is the difference between the desired and the actual input force on each vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Assuming that the desired thrust fid is commanded instantaneously, then, ζi depends on the attitude tracking error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Therefore, as ¯qie → [1 0 0 0]T this disturbance ζi → 0 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Closed-loop Error Dynamics Let us propose the control laws for the virtual controllers uL and ui as uL = −mL (ge3 + ¨xLd)−νL (26) 12 ui = mi (ge3 + ¨xLd)−Tidαid +νi (27) where νL ≜ −kpL(xL −xLd)−kdL(˙xL − ˙xLd)−kiL � (xL −xLd) and νi ≜ −kpi(xi −xLd +Liαid)−kdi(˙xi − ˙xLd +Li ˙αid)−kii � (xi −xLd +Liαid), with the control gains matrices kpL,kdL,kiL,kpi,kdi,kii > 0 ∈ R3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Please note that the desired UAVs positions are given by the desired load’s position xLd and the desired cable orientation αid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Then, the closed-loop error dynamics of the systems result in ¨xe = 1 mL � n ∑ i=1 (ζLi)+νL � (28) ¨xei = 1 mi (νi +ζi +ζLi)+Li ¨αid (29) Now let’s define a state space based on the error dynamics as χ ≜ �� xT e dt,xT e , ˙xT e , � xT e1dt,xT e1, ˙xT e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=', � xT endt,xT en, ˙xT en �T ∈ R9(n+1) (30) and its time derivative ˙χ = ˜Aχ + ˜Bζ (31) where ˜A ≜ � ���� A+ 1 mLKL 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 0 0 A+ 1 m1K1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' A+ 1 mnKn � ����, ˜B ≜ � ������� 1 mLB 0 0 1 mLB 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 1 mLB 0 0 1 m1B 1 m1B L1B 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 0 0 0 0 0 0 1 m2B 1 m2B L2B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 0 0 0 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 1 mnB 1 mnB LnB � ������� ,ζ ≜ � ���������� ζL1 ζ1 ¨α1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' ζLn ζn ¨αnd � ���������� ∈ R9n, 13 A ≜ � � 0 I 0 0 0 I 0 0 0 � �,B ≜ � � 0 0 I � �,KL ≜ � � 0 0 0 0 0 0 −kiL −kpL −kdL � �,Ki ≜ � � 0 0 0 0 0 0 −kii −kpi −kdi � � with the identity matrix I ∈ R3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Assumptions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The attitude controllers of each UAV (12) assure the errors to be attracted and confined into a small invariant set [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Therefore, the position virtual controller is induced, with a small disturbance ζi, with a bound c1i, it is ∥ζi∥2 ≤ c1i (32) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The desired position of the vehicles xid is calculated using αid which is estimated from the payload controller uL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Also, considering well-posed initial conditions in compliance with the constrains (6), along with small initial errors, and assuming that the desired trajectory of the load xLd is smooth and slow varying, then it is assumed that the controller of the load have a bounded transient response, implying that the second time derivative of αid is bounded ∥ ¨αid∥2 ≤ c2i (33) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The difference between the desired and actual tensions on the cables are bounded ∥ζLi∥2 ≤ c3i (34) with {c1i,c2i,c3i} > 0 ∈ R as constant positive bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Stability Analysis Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' For the system (1) and (7) with the restriction (6) and the virtual controls (26) and (27), under assumptions 1 − 3, and considering some matrices ˜A and ˜B, if there exists a symmetric positive definite matrix P ∈ R9(n+1)×9(n+1), P = PT > 0, and some positive constants α and ε that satisfy the inequality WL(P,KL,K1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=',Kn,α,ε) < 0 where WL = �P ˜A+ ˜ATP+αP P ˜B ˜BTP −εI9n×9n � , (35) 14 then, there exists an attractive stability region around the origin of the vector state χ defined by the ellipsoid E = � χ ∈ Rn| χTPχ ≤ β α � (36) where β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Henceforth, the practical stability of the system is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' In order to analyze the system, the following Lyapunov-like energy func- tion is proposed V = χTPχ (37) whose derivative, adding and subtracting αV and ε∥ζ∥2, using assumptions 1-3 is equal to ˙V = χTP ˙χ + ˙χTPχ ±αV ±ε∥ζ∥2 ≤ �χ ζ �T WL �χ ζ � −αV +β (38) where β = ε ∑n i=1 (c1i +c2i +c3i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' As long as it is assured that WL ≤ 0, there exists an attractive stability region around the origin of xe and xei defined by the ellipsoid E = � χ ∈ R9(n+1)| χTPχ ≤ β α � (39) and the solution to the optimization problem is tr � β α P−1� → min α,β,ε,KL,K1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=',Kn,P subject to the restrictions α > 0, ε > 0, 0 < P, WL = WL (KL,K1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=',Kn,P,α,ε,) ≤ 0 (40) Therefore, if there exist the solution to the problem (40) the practical stability is guaranteed that correspond to the smallest attractive region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Note the vector ζ is composed with all the disturbances present in the closed- loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' ζLi corresponds to the difference between the actual and the desired tension in the cable, and appears in the error dynamics of the load and each ve- hicle, this tension increases as the error in position xei increases, meaning that as long as the position controller assures small errors this disturbance would be small enough to assure practical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 15 ¨αid corresponds to the variation in the direction of the desired tension and only appears in the vehicles error dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' This implies that the transient response of the control load directly affects the bound of this disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Therefore, non continuous desired position trajectories should not be used, also, the load virtual controller should be as damped as possible, this results in reducing the upper bound of the ¨αid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' ζi is the disturbance that represents the difference of the desired and actual input forces in each vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Given that the attitude dynamics of the vehicles are faster than the position dynamics, initial small errors in the attitude controller are desirable, in order to assure that this disturbance is small enough and the position will converge to the desired one, even if there are small disturbances of ζLi and ¨αid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The attitude controller has to be highly reactive in order to assure the tracking of the desired trajectory, however, if the position controller is highly reactive, it will induce aggressive attitude trajectories that will result in greater attitude er- rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Therefore there exists a compromise between the controllers, as the attitude controller must be highly reactive and assure the tracking for different trajectories, then the position controller must be reactive enough to assure the convergence of the position compensating the disturbances with smooth transient response, and finally the load controller must have a damped response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Numerical Simulations In order to validate the control strategy proposed in Section 3, and to study the behavior of the closed-loop system, numerical simulations were carried out with the help of Matlab/Simulink®.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' A video showing the simulation is available at https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='be/uzAYqm1-H_U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' As a case of study, let us consider three Table 1: Simulation parameters in International System units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Index i ∈ {1,2,3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Function diag(∗) is a square diagonal matrix whose diagonal values are given by the argument, and βi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' mi mL Ji Li 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='01diag(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='32,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='32,4) 1 kpi kdi kii ρi 40[1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5]T 10[1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='2]T 2[1 1 2]T 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5I kpL kdL kiL Kdi 9[1 1 1]T 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5[1 1 1]T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='2[1 1 1]T 16I 16 Figure 3: Load position xL (top) and load transportation error xe = xL −xLd (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' UAVs carrying the payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The main parameters used in the simulation are pre- sented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The objective is for the load to track a desired trajectory, in this case an as- cending spiral (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 3) of the form xLd = � � 1−cos(2 5πt) sin(2 5πt) t/10 � � (41) 17 Load position [m] 2 real x, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 1 Load 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 1 0 2 3 4 5 6 7 8 9 10 Load trasportation error [m] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='1 ey 0 ez 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='2 0 2 3 4 5 6 8 9 10 time [s]Figure 4: UAV’s positions xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' where t is the time variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The initial positions of the UAVs and the load are xL = [0 0 0]T (42) x1 = R(z,−π/4)R(y,−π/6)[0 0 L1]T (43) x2 = R(z,π/4)R(y,−π/6)[0 0 L2]T (44) x3 = R(y,π/6)[0 0 L3]T (45) with R(a,b) ∈ SO(3) as the basic rotation matrices representing a rotation of an angle b ∈ S1 around one of the main axis a ∈ {x,y,z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Note that from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' (21), 18 UAV Position [m] 2 JAV 0 3 5 6 9 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 2 UAV 0 2 0 6 9 10 3 2 UAV 3 0 0 2 3 4 5 6 9 10 time [s]Figure 5: UAVs’ attitude quaternion ¯qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' (26) we can obtain the desired tension on the cables, but this results in an under- constrained system with infinite solutions, then we can impose additional con- strains to remain with an unique solution, in this case, the desired tensions are selected as follows T1α1d = −mL 3 R(z,−π/4)R(y,−π/6)(ge3) (46) T2α2d = −mL 3 R(z,π/4)R(y,−π/6)(ge3) (47) T3α3d = −uL −T1α1d −T2α2d (48) 19 attitude (o) *b q, (1) quaternion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 q, (2) q, (3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0 3 4 5 6 8 9 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='8 (0) °b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='6 q,(1) quaternion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='4 q, (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='2 0 3 4 5 6 7 8 9 10 9, (0) 93 (1) quaternion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 3 (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0 2 3 4 5 6 7 8 9 10 time [s]Figure 6: Tensions Tiαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The solid lines represent the actual signals, while the dashed lines repre- sent the desired references in the three axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' any other valid solution can be used instead, for instance, it would be interesting to select the solution that minimizes the total energy among the n agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Figures 3-8 demonstrate the good performance of the control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' At the top of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 3 the load position is presented, where we can appreciate how the load follows the desired reference, while the payload error xe is depicted at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' We can observe that the errors converge to the invariant set, hence, they remain small and bounded, which is in compliance with practical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 20 Tensions T,α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' real desired M 3 0 2 3 4 5 6 7 8 9 10 real 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 desired 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0 2 3 6 8 9 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 real desired 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 1 0 2 3 6 8 9 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 real 0 desired 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0 2 3 4 5 6 8 9 10 time [s]Figure 7: Control inputs ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The UAVs’ positions along the mission are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 4, where we can see how the third UAV has to correct its initial formation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Also, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 5 the UAVs’ attitude quaternions are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Moreover, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 6 the desired (dotted lines) and real tensions (solid lines), as well as the total tension (top) are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' We can note that the tension signals are smooth and remain bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The drones’ control inputs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Here we can observe that due to the choice of the desired tensions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' (47)-(48), the third UAV must make a bigger control effort to compensate for the errors in the load positioning, besides 21 Control inputs u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 2 0 2 0 3 5 6 8 9 10 4 2 y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 0 2 0 5 6 8 9 10 10 5 0 2 3 4 5 6 7 8 9 10 time [s]its larger initial error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Nevertheless, all the UAVs converge quickly to their desired trajectories, and remain within small bounded errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Finally, we can appreciate the good overall transportation system performance in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 8, where the top view (top) and the 3D view (bottom) of the UAVs and load trajectories are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The numerical results in this case of study support the validity of the approach, and that the closed-loop coupled system is stable and the position of the load con- verges to the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Also, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='6, the system disturbances are larger in magnitude at the beginning, during the transient state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Even in this scenario, the error signals remain small and bounded, corroborating practical sta- bility of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Conclusions In this work a hierarchical controller for the transportation of a cable sus- pended load by multiple UAVs is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The proposed algorithm considers the collaboration of n vehicles, using a continuous controller that assures that the error signals are confined to an attractive invariant set, assuring practical stability for the coupled system, resulting in the tracking of a time varying trajectory by the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' To do so,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' the desired sum of the cable tensions is used as a virtual controller for the load trajectory tracking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' the desired tensions and direction for each cable can be obtained,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' resulting in an under-determined system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' where additional constrains can be added to obtain an unique solution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' thereafter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' from the cables’ direction we compute the desired position for each UAV agent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' which is in turn fed to a position controller using the desired thrust vector as a new virtual con- troller,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' from which the desired attitude of each drone are drawn and parsed to a quaternion based attitude controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The proposed algorithm, based in the attractive ellipsoid method, considers two disturbances due to the virtual controllers, one that depends on the attitude er- ror, and the other one that depends on the position error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' By means of a Lyapunov- like energy function, we have studied the stability of the closed-loop system, demonstrating its practical stability, and establishing some sufficient conditions in order to guarantee small bounds in the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Numerical simulation were presented in order to demonstrate the viability of the proposed scheme, and study its performance, showing the tracking of a as- cending spiral trajectory by the load carried by three UAVs, obtaining good per- formance with small bounded errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' From the stability analysis and the simula- tions study, we have encounter that the moment when the disturbances are more 22 Figure 8: Trajectories of the cable suspended transportation system with 3 UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Aerial view (x−y plane) on top, 3D view at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' The control strategy demonstrated good performance, being able to accurately track the desired load’s trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' significant is during their initial state, given that both the attitude and position controllers had not converged jet to their invariant sets during the transient state, thus, small initial errors are required in order to assure that this disturbances are bounded and small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' As a future work, we are working on the validation of the proposed control strategy in real-time experiments with three or more drones transporting a load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Also, it would be interesting to propose other suitable approaches to obtain the 23 aerial view 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 x [m] 3D view X, X2 +?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 ~ N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 1 0 y [m] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content='5 x [m] desired cable tensions from the resultant tension, for instance to minimize the total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Finally, we would like to study different control approaches applied to this kind of multi-agent transportation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Kumar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFIT4oBgHgl3EQfwCuC/content/2301.11350v1.pdf'} +page_content=' Michael, Opportunities and challenges with autonomous mi- cro aerial vehicles, The International Journal of Robotics Research 31 (11) (2012) 1279–1291.' metadata={'source': 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0000000000000000000000000000000000000000..fcfac5e60301abb35d28dffc9379ab619fcffba6 --- /dev/null +++ b/qNFRT4oBgHgl3EQfeDf8/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:86844631c7e923c60c15ff20b2dad7708915694ab53349e463e56cde23b13d5b +size 9175085 diff --git a/qdAzT4oBgHgl3EQfcPxh/content/tmp_files/2301.01399v1.pdf.txt b/qdAzT4oBgHgl3EQfcPxh/content/tmp_files/2301.01399v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..57f6d0988cd3c74cdf21191a85c4ab26746ca37b --- /dev/null +++ b/qdAzT4oBgHgl3EQfcPxh/content/tmp_files/2301.01399v1.pdf.txt @@ -0,0 +1,3429 @@ +arXiv:2301.01399v1 [math.AP] 4 Jan 2023 +GLOBAL WELL-POSEDNESS OF WEAK SOLUTIONS TO THE +INCOMPRESSIBLE EULER EQUATIONS WITH HELICAL +SYMMETRY IN R3. +DENGJUN GUO AND LIFENG ZHAO +Abstract. We use a Lagrangian method to prove the global well-posedness +of weak solutions to the three-dimensional Euler equations with helical sym- +metry and without helical swirl in the whole space for initial vorticity in +L1 +1 +� L∞ +1 (R3). The vortex transport formula is also obtained in our article. +1. Introduction +We consider the three-dimensional incompressible Euler equation in R3, + + + + + +∂tU + U · ∇U + ∇P = 0 +∇ · U = 0 +U(x, 0) = U0(x). +(1.1) +The equation describes the motion of an ideal incompressible fluid in R3 with initial +velocity U0(x). Here U = (U 1, U 2, U 3) : R3 × R → R3 represents the velocity and +P : R3 × R → R represents the scalar pressure which can be determined by the +incompressibility condition. +The vorticity of the fluid is defined by Ω(x, t) := ∇ ∧ U(x, t), where +(a1, a2, a3) ∧ (b1, b2, b3) := (a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1). +Moreover, the vorticity satisfies +� +∂tΩ + U · ∇Ω + Ω · ∇U = 0, +Ω(x, 0) = ∇ ∧ U0(x) := Ω0(x), +(1.2) +which is called the vorticity-stream formulation of the Euler equations. Here the +velocity U can be recovered from Ω by the well-known Biot-Savart law +U(x) = 1 +4π +� +R3 +x − y +|x − y|3 ∧ Ω(y) dy. +The local well-posedness of the three-dimensional Euler equation has been studied +in [19] for initial data U0 ∈ Hm and m ≥ 4. However, the global existence of +smooth solutions to the three-dimensional Euler equation remains open due to the +strong nonlinearity of the vorticity stretching term Ω · ∇U, see [1] or [19] for more +references. +Unlike the three-dimensional case, the two-dimensional incompressible Euler equa- +tion is global well-posed in L1 � L∞ due to the absence of the vorticity stretching +term, see [19] and [26]. Moreover, the two-dimensional incompressible Euler equa- +tion can sometimes be viewed as a transport equation and the vortex transport +formula holds for solutions with bounded L1 � L∞ norm, see [19] for references. +L. Zhao is supported by NSFC Grant of China No. 12271497 and the National Key Research +and Development Program of China No. 2020YFA0713100. +Data Availability Statements: Data sharing not applicable to this article as no datasets +were generated or analysed during the current study. +1 + +2 +D. GUO AND L. ZHAO +Yudovich [27] extended the uniqueness results to the solutions whose Lp norm +grows slowly as p → ∞ and Vishik [23] extended the results to Besov type spaces. +The existence of global solutions to the two-dimensional Euler equation with Lp +initial vorticity has been established by Majda and Diperna in [10]. Later, De- +lort proved the global existence for measure valued initial data [9]. However, the +uniqueness might not hold for Lp initial data, we refer to the works by Vishik [24] +and [25]. +Some three-dimensional flows with special symmetries can be reduced to two- +dimensional flows. Among the most important examples are axisymmetric flows and +helical flows without swirl. The global existence and uniqueness of axisymmetric +solutions was obtained by Yudovich in [19]. For helical solutions, in [12] Dutrifoy +proved the global well-posedness for smooth solutions in bounded domains, an key +observation is that the third component Ωz of the vorticity Ω satisfies +∂tΩz + U · ∇Ωz = 0, +(1.3) +which shows that the vorticity is transported by the flow. +Setting w(x1, x2) = +Ωz(x1, x2, 0), Ettinger and Titi reduced the three-dimensional Euler equation to +the following two-dimensional problem: +∂tw + ∇⊥L−1 +H w · ∇w = 0, +where LH is a specific elliptic operator. They also establish the global existence +and uniqueness for this two-dimensional problem in bounded domains [13]. For +the whole space R3, the global existence of weak solutions has been obtained by +Bronzi–Lopes–Lopes [3] for L1 � Lp(R3) initial vorticity with compact support. +The assumption of the compact support was then removed by Jiu, Li and Niu +[16]. However, for Lp solutions, the uniqueness of the solutions and the continuous +dependence on initial data is still an open problem in the whole space R3. Moreover, +the vortex transport formula is still unknown even in bounded domains. +In this paper, we consider the three-dimensional incompressible Euler equation +with helical symmetry and without swirl. +Let Ωz be a solution to (1.3) in R3. +Setting w(x1, x2, t) = Ωz(x1, x2, 0, t), then w satisfies the two-dimensional helical +Euler equation +∂tw + Hw · ∇w = 0, +(1.4) +where +Hw(x) = +� +R2 H(x, y)w(y) dy +and H(x, y) is the modified Biot-Savart kernel defined in Proposition 2.12. The +main result are stated as follows (for the precise definition of Lagrangian weak +solutions and the function spaces, we refer the reader to section 2). +Theorem 1.1. The equation (1.4) is globally well-posed in L1 +1 +� L∞ +1 (R2) : +(i) (Existence). +For all T > 0 and w0 ∈ L1 � L∞(R2), there exists a La- +grangian weak solution w(x, t) ∈ L∞ � +[0, T ], L1 � L∞(R2) +� +to (1.4). More- +over, the velocity Hw(·, t) is locally Log-Lipschitz continuous. +(ii) (Uniqueness). For any w0 ∈ L1 +1 +� L∞ +1 (R2), there exists at most one la- +grangian weak solution to (1.4) in L∞ � +[0, T ], L1 +1 +� L∞ +1 (R2) +� +with initial +vorticity w0. +(iii) (Continuous dependence on initial data). Let w0,n be a sequence of initial +data such that +sup +n ∥w0,n∥L∞ +1 (R2) < ∞ + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +3 +and +∥w0,n − w0∥L1 +1(R2) → 0. +Let w(t) and wn(t) be the solution to (1.4) with initial data w0 and w0,n, +respectively. Then for any T > 0, there holds +sup +t≤T +∥wn(t) − w(t)∥L1 +1(R2) → 0. +. +Remark 1.2 (Vortex transport formula). The velocity field Hw is divergence free. +So in view of Lemma A.4, all weak solutions to (1.4) in L∞ � +[0, T ], L1 +1 +� L∞ +1 (R2) +� +are +indeed Lagrangian and hence the assumption of Lagrangian in (ii) can be removed. +In other words, all weak solutions to (1.4) satisfies the vortex transport formula. +Remark 1.3. With the help of the vortex transport formula, the solution w to +(1.4) with initial data in L1 +1 +� L∞ +1 (R2) belongs to C +� +[0, T ], L1 +1(R2) +� +Remark 1.4. By a similar argument, one can also obtain the global well-posedness +for (1.4) in L1 +m +� L∞ +m(R2) for any m ≥ 1. When m = 0, we only prove the existence +of a global solution to (1.4). Indeed, the continuous dependence on initial data +holds by the same argument in section 6 once we obtain the uniqueness. However, +our method can not be applied to prove the uniqueness when m = 0. The main +difficulty is that Hw(x) is Log-Lipschitz continuous when and m = 1 while only +locally Log-Lipschitz continuous when m = 0. For locally Log-Lipschitz continuous +velocity field, Clop, Jylh¨a, Mateu, and Orobitg proved the uniqueness for continuity +equations and some techniques of optimal transport is needed here. We refer the +reader to [7] for references. +In the proof of Theorem 1.1, we also obtain the well-posedness for (1.3), which is +much easier than (1.4) since the velocity remains bounded there while the velocity +may grows linearly in two-dimensional cases. +Corollary 1.5 (Global well-posedness for (1.3)). The three-dimensional helical +Euler equation (1.3) is globally well-posed in L1 +1 +� L∞ +1 (R3). +Next we state our strategy of the proof. The key ingredients of our proof is to +give a detailed estimates for the modified Biot-Savart kernel H(x, y). Once the +estimates for the difference |H(x, y)−H(z, y)| be established, the existence of weak +Lagrangian solutions to (1.4) then follows by a similar argument as in [19]. We can +not obtain the uniqueness for (1.4) directly since the velocity field Hw(x) might +growth linearly. Instead, we show that (1.3) and (1.4) are equivalent and then it +suffices to prove the uniqueness for (1.3). Finally, the continuous dependence on +initial data follows by the vortex transport formula and a compactness argument. +Our article is organized as follows: In section 2, we give a detailed description +of our problems and reduce the three-dimensional incompressible Euler equation +(1.2)(for initial velocity with helical symmetry and without helical swirl) to the +two-dimensional helical Euler equation (1.4). In section 3, we obtain a detailed +estimates for the modified Biot-Savart kernel H(x, y). In section 4, we show the +uniqueness of the helical Euler equation (1.3) and (1.4) in L1 +1 +� L∞ +1 . In section 5, +we obtain the global existence and the vortex transport formula for (1.4). In section +6, we prove the continuous dependence on initial data to equation (1.3) and (1.4). +2. Mathematical preliminaries and main result. +Our main purpose of this section is to fix notations and state our main results. For +simplicity of presentation, we usually refer a point x = (x1, x2, x3) ∈ R3 or R2 × T +to x = (x′, x3). + +4 +D. GUO AND L. ZHAO +2.1. Helical functions and vector fields. +Definition 2.1 (helical function). A function f : R3 → R is called helical if for all +θ ∈ R, +f(Sθx) = f(x) +for almost every x ∈ R3, where +Sθx := Rθx + + + +0 +0 +θ + + +and +Rθ := + + +cos θ +sin θ +0 +− sin θ +cos θ +0 +0 +0 +1 + + . +(2.1) +Definition 2.2 (helical vector field). A vector field u : R3 → R3 is called helical if +for all θ ∈ R, +R−θu(Sθ(x)) = u(x) +for almost every x ∈ R3. +Assume f is a continuous helical function and u is a continuous helical vector field, +then by the definition above we see that +f(x′, x3) = f(R−x3x′, 0) +(2.2) +and +u(x′, x3) = Rx3u(R−x3x′, 0). +(2.3) +Now if f and u are only measurable function (vector field), then f(x′, 0) and u(x′, 0) +are not well-defined, thus (2.2) and (2.3) do not make sense. However, we have the +following: +Lemma 2.3. Let f be a locally bounded helical function and u be a locally bounded +helical vector field. Define +g(x1, x2) = 1 +2π +� 2π +0 +f(Rax′, a) da +and +v(x1, x2) = 1 +2π +� 2π +0 +R−au(Rax′, a) da. +Then for almost every x ∈ R3, there hold +f(x′, x3) = g(R−x3x′) +(2.4) +and +u(x′, x3) = Rx3v(R−x3x′). +(2.5) +Proof. We only prove (2.4) since the proof of (2.5) is similarly. Setting +Ar = +� +Br +���f(x′, x3) − g(R−x3x′) +��� dx′dx3, +then it suffice to show that Ar = 0 for all r > 0. By the definition of g, we obtain +Ar = +� +Br +����f(x′, x3) − 1 +2π +� 2π +0 +f(Ra−x3x′, a) da +���� dx +≤ 1 +2π +� +Br +� 2π +0 +|f(x′, x3) − f(Ra−x3x′, a)| da dx. + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +5 +Next we make a change of variable a = b + x3, then it follows that +Ar ≤ 1 +2π +� +Br +� 2π−x3 +−x3 +|f(x′, x3) − f(Rbx′, b + x3)| db dx +≤ 1 +2π +� +Br +� 2π+r +−r +|f(x′, x3) − f(Rax′, a + x3)| da dx += 1 +2π +� 2π+r +−r +� +Br +|f(x′, x3) − f(Sa(x′, x3))| dx da. +Thus, by Definition 2.1 we get Ar ≡ 0 and hence (2.4) holds. +□ +Motivated by (2.2), (2.3), by slight abuse of notation, we may refer to g(x1, x2) as +f(x′, 0), and refer to v(x1, x2) as u(x′, 0). Next we derive some useful properties +for helical functions and vector fields. +Lemma 2.4. Let f ∈ L∞ +loc(R3) be a helical function, then +∂3f = x1∂2f − x2∂1f +(2.6) +in the sense of distribution. +Proof. In [13], (2.6) has been shown when f is smooth. For general f ∈ L∞ +loc(R3), +we define f0(x1, x2) := f(x1, x2, 0). Let f0,n ∈ C∞ +c (R2) be a sequence of smooth +function converges to f0 in L1 +loc(R2) and set fn(x′, x3) := f0,n(R−x3x′), then for +any φ ∈ C∞ +c (R3), it follows from (2.1) and Definition 2.1 that +� +R3 f(x)∂3φ(x) dx = +� +R3 f0(R−x3x′)∂3φ(x) dx += +� +R3 f0(x1 cos x3 − x2 sin x3, x1 sin x3 + x2 cos x3)∂3φ(x) dx. +Since φ has compact support, the dominating convergence theorem then gives +� +R3 f(x)∂3φ(x) dx = lim +n +� +R3 f0,n(x1 cos x3 − x2 sin x3, x1 sin x3 + x2 cos x3)∂3φ(x) dx += lim +n +� +R3 fn(x)∂3φ(x) dx. +Note that fn is a smooth helical function, so (2.6) and integrating by parts yields +� +R3 f(x)∂3φ(x) dx = lim +n +� +R3 −∂3fn(x)φ(x) dx += lim +n +� +R3(x2∂1fn − x1∂2fn)φ(x) dx += lim +n +� +R3 fn(x1∂2φ(x) − x2∂1φ(x)) dx += lim +n +� +R3(x1∂2φ(x) − x2∂1φ(x))f0,n(x1 cos x3 − x2 sin x3, x1 sin x3 + x2 cos x3) dx. +Again by dominating convergence theorem, we finally get +� +R3 f(x)∂3φ(x) dx = +� +R3(x1∂2φ(x) − x2∂1φ(x))f0(x1 cos x3 − x2 sin x3, x1 sin x3 + x2 cos x3) dx += +� +R3 f(x) (x1∂2φ(x) − x2∂1φ(x)) dx, +which proves (2.6). +□ +For helical vector fields, by [13] and a similar argument as above, we obtain + +6 +D. GUO AND L. ZHAO +Lemma 2.5. Let u = (u1, u2, u3) ∈ L1 +loc be a helical vector field, then we have +∂3u1 = x1∂2u1 − x2∂1u1 + u2, +∂3u2 = x1∂2u2 − x2∂1u2 − u1, +and +∂3u3 = x1∂2u3 − x2∂1u3. +(2.7) +Proof. Assume u is smooth, then above equations have already been proved in [13]. +For general u ∈ L1 +loc, the proof is only a matter of smoothness as in Lemma 2.4 so +we omit it. +□ +Next we will describe the particle trajectory map associated with a Log-Lipschitz +continuous helical vector fields. +Definition 2.6. A function (vector field) u is called Log-Lipschitz continuous if +sup +x̸=y +|u(x) − u(y)| +F(|x − y|) +< ∞, +where F is the Log-Lipschitz function +F(r) = +� +r(1 − log r) +r ≤ 1 +e +r + 1 +e +r > 1 +e. +(2.8) +Note that F ′(r) = − log r if r ≤ 1 +e and F ′(r) = 1 if r > 1 +e, so F ′(r) is continuous and +decreasing. Thus, F(r) is a concave function which satisfies F(r) ≈ r(1 − log− r), +where +log− r = +� +log r +r ≤ 1 +0 +r > 1. +Definition 2.7. Let u : Rd × [0, T ) → Rd be a locally Log-Lipschitz continuous +velocity field, we say X(α, t) is a particle trajectory map associated with u if it +satisfies + + + +dX(α, t) +dt += u(X(α, t), t) +X(α, 0) = α +for any α ∈ Rd and t ∈ [0, T ). +Assume that the velocity field is helical, then we have +Lemma 2.8. Let u be a locally Log-Lipschitz continuous helical vector field and +X(α, t) be its associated particle trajectory map, then for any α ∈ R3, t ∈ R+ and +θ ∈ R, it holds that +S−θX(Sθα, t) = X(α, t). +(2.9) +Proof. A direct calculation shows that +d (S−θX(Sθα, t)) +dt += d (R−θX(Sθα, t)) +dt += R−θu(X(Sθα, t), t) += u(S−θX(Sθα, t), t). +Thus, both S−θX(Sθα, t) and X(α, t) are the solutions of + + + +dX(α, t) +dt += u(X(α, t), t) +X(α, 0) = α. + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +7 +So the uniqueness of the particle trajectory map implies +SθX(S−θα, t) = X(α, t). +□ +Next we summarize the following results for helical vector fields without helical +swirl. +Definition 2.9. Let ξ(x) = (x2, −x1, 1), then the helical swirl uξ of a vector field +u is defined by +uξ := x2u1 − x1u2 + u3. +Lemma 2.10 ([13]). Let U0 be a smooth helical vector field without helical swirl +and U(x, t) be the smooth helical solution of the three-dimensional Euler equation +(1.2) with initial data U0. Then we have +Uξ(x, t) ≡ 0. +Lemma 2.11 ([13]). Let Ω = (Ωx, Ωy, Ωz) be a smooth solution of the three- +dimensional Euler equation (1.2) without swirl, then +Ω = (x2Ωz, −x1Ωz, Ωz) = Ωzξ. +Moreover, Ωz is a helical function which satisfies the three-dimensional helical Euler +equation (1.3). +2.2. Reduce the three-dimensional Euler equation to a two-dimensional +problem. In this subsection, we will reduce(formally) the three-dimensional Euler +equation (1.2) to the two-dimensional helical equation (1.4). More precisely, we +have the following: +Proposition 2.12 (Two-dimensional helical Euler equation). Let Ω = (Ωx, Ωy, Ωz) +be a smooth solution to the three-dimensional helical Euler equation without swirl. +Set w(x1, x2) = Ωz(x1, x2, 0), +K(x, y) = 1 +4π +� +R +(x1, x2, 0) − (Ra(y), a) +|(x1, x2, 0) − (Ra(y), a) |3 ∧ ξ((Ra(y), a)) da, +(2.10) +U(x1, x2) = +� +R2 K(x, y)w(y) dy +(2.11) +and +Hw := H1w + H2w = (U 1, U 2) + (−x2, x1)U 3, +Then w satisfies the two-dimensional helical Euler equation +∂tw + Hw · ∇w = 0 +with ∇ · Hw = 0. +Proof. Let Ω = (Ωx, Ωy, Ωz) be a smooth helical solution to the equation (1.2) +without swirl, then Lemma 2.11 implies that Ωz is a helical function and satisfies +∂tΩz + U 1∂1Ωz + U 2∂2Ωz + U 3∂3Ωz = 0. +Thus, it follow from (2.7) that +∂tΩz + (U 1 − x2U 3)∂1Ωz + (U2 + x1U 3)∂2Ωz = 0. +Let x3 = 0 in the above equation and set w(x1, x2) := Ωz(x′, 0), then w(x1, x2) +satisfies the two-dimensional helical Euler equation +∂tw + Hw · ∇w = 0, +where +Hw(x1, x2) = +� +U 1(x′, 0) − x2U 3(x′, 0), U 2(x′, 0) + x1U 3(x′, 0) +� +. +(2.12) + +8 +D. GUO AND L. ZHAO +Next we will show that this equation is incompressible. Indeed, the divergence of +Hw is +∇ · Hw = ∂1U 1 − x2∂1U 3 + ∂2U 2 + x1∂2U 3. +Since U is a helical vector field and ∇ · U = 0, then (2.7) gives +∇ · Hw = ∂1U 1 + ∂2U 2 + ∂3U 3 = ∇ · U = 0. +Next we prove (2.10) and (2.11). By Lemma 2.11 and the three-dimensional Biot- +Savart law, it follows that +U(x) = ∇ × ∆−1Ω(x) += 1 +4π +� +R3 +x − y +|x − y|3 ∧ (y2Ωz(y), −y1Ωz(y), Ωz(y)) dy += 1 +4π +� +R3 +x − y +|x − y|3 ∧ ξ(y)w (R−y3(y1, y2)) dy. +Then change of variables (z1, z2) = R−y3(y1, y2) yields +U(x′, 0) = 1 +4π +� +R3 +(x′, 0) − (Ry3(z′), y3) +|(x′, 0) − (Ry3(z′), y3) |3 ∧ ξ((Ry3z′, y3))w(z1, z2) dz1dz2dy3 += 1 +4π +� +R2 +�� +R +(x′, 0) − (Ry3(z′), y3) +|(x′, 0) − (Ry3(z′), y3) |3 ∧ ξ((Ry3z′, y3)) dy3 +� +w(z1, z2) dz′, +which proves (2.10) and (2.11). +□ +By the proof above, the velocity Hw can be written in the form +Hw(x) = +� +R2 H(x, y)w(y) dy := +� +R2 H1(x, y)w(y) dy + +� +R2 H2(x, y)w(y) dy (2.13) +for +H1(x, y) := +� +K1(x, y), K2(x, y) +� +(2.14) +and +H2(x, y) := (−x2, x1)K3(x, y). +(2.15) +We refer to the two-dimensional problem (1.4) as the two-dimensional helical Euler +equation and we refer to the equation (1.3) as the three-dimensional helical Euler +equation. +2.3. Function spaces and the definition of weak solution. In this subsection, +we will introduce the appropriate functional spaces and some definition of the weak +solutions. The three-dimensional weighted Lp +m(R2 × T) norm is defined by +∥f∥Lp +m(R2×T) := +� +R2×T +⟨y′⟩pm|f(y1, y2, y3)|p dy +1 +p . +Similarly, the two-dimensional weighted Lp +m(R2) norm is defined by +∥g∥Lp +m(R2) := +� +R2⟨y⟩pm|g(y1, y2)|p dy +1 +p . +Here and throughout the paper we use the notation ⟨a⟩ = (1 + |a|2) +1 +2 . The norms +Lp +m(R2 × T) and Lp +m(R2) are related as follows: +Lemma 2.13. Let f ∈ Lp +m(R2) and define F : R3 → R as +F(x′, x3) := f(R−x3x′). +Then for any p ∈ [1, +∞], +∥f∥Lp +m(R2) = (2π)−1/p∥F∥Lp +m(R2×T). + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +9 +Proof. The proof follows directly by definition when p = ∞, so it suffices to consider +the case when 1 ≤ p < ∞. In fact, +∥F∥p +Lp +m(R2×T) = +� +T +� +R2(1 + y2 +1 + y2 +2) +mp +2 |f(R−y3(y1, y2))|p dy1dy2 dy3 += +� +T +� +R2(1 + z2 +1 + z2 +2) +mp +2 |f(z1, z2)|p dz1dz2 dy3 += 2π∥f∥p +Lp +m(R2). +□ +We will now formulate the weak problem of the two-dimensional helical Euler equa- +tion (1.4) and the three-dimensional helical Euler equation (1.3). +Definition 2.14. (Weak solutions to the three-dimensional helical Euler equation.) +We say that Ωz(x, t) is a weak solution to (1.3) if +� +R3 Ωz(x, t)φ(x, t) dx − +� +R3 Ωz(x, 0)φ(x, 0) dx = +� t +0 +� +R3 Ω(∂tφ + U · ∇φ) dxds. +for all t ∈ [0, T ] and for all test function φ ∈ C∞ +c (R3 × [0, +∞)). +Similarly, we define +Definition 2.15. (Weak solutions to the two-dimensional helical Euler equation.) +We say that w(x, t) is a weak solution to (1.4) if +� +R2 w(x, t)φ(x, t) dx − +� +R2 w(x, 0)φ(x, 0) dx = +� t +0 +� +R2 w(∂tφ + Hw · ∇φ) dxds. +for all t ∈ [0, T ] and for all test function φ ∈ C∞ +c (R2 × [0, +∞)). +Definition 2.16. (Lagrangian solution) We say w is a Lagrangian solution if it is +a weak solution to (1.4) and it also satisfies the vortex transport formula +w(X(α, t), t) = w(α, 0), +where X(α, t) is the particle trajectory map associated with Hw. +3. Key estimates for the Biot-Savart kernel. +The purpose of this section is to establish the necessary estimates for the modified +Biot-Savart kernel H(x, y), K(x, y) and G(x, y) given in (2.13), (2.10) and (3.21), +respectively. The estimates for the two-dimensional kernel H(x, y) and K(x, y) is +given in subsection 3.1 and 3.2. The estimates for the three-dimensional kernel +G(x, y) is given in subsection 3.3. +3.1. Estimates for the modified Biot-Savart kernel H and K. First we derive +some estimates for the modified Biot-Savart kernel K(x, y) which is given in (2.10). +Proposition 3.1 (Estimates for K(x, y)). Let x, y be two distinct points in R2, +then there holds +|K(x, y)| ≲ min +� +⟨y⟩(1 + +1 +|x − y|), ⟨x⟩(1 + +1 +|x − y|) +� +. +(3.1) +Proof. Note that |ξ(y1, y2, a)| ≲ ⟨y⟩, so we have +|K(x, y)| ≲ +� +R +⟨y⟩ +|x − Ra(y)|2 + a2 da +(3.2) +and hence it suffices to show that +� +R +⟨y⟩ +|x − Ra(y)|2 + a2 da ≲ min +� +⟨y⟩(1 + +1 +|x − y|), ⟨x⟩(1 + +1 +|x − y|) +� +. +(3.3) + +10 +D. GUO AND L. ZHAO +First we consider the case when |y| ≥ 2|x|. In this case +|x − y| +3 +≤ |Ra(x) − y| ≤ 3|x − y|, +so we have +� +R +⟨y⟩ +|x − Ra(y)|2 + a2 da ≲ ⟨y⟩ +� +R +1 +|x − y|2 + a2 da ≈ +⟨y⟩ +|x − y| ≲ 1 + +⟨x⟩ +|x − y|. +Next we consider the case when |y| ≤ 2|x|. Note that ⟨y⟩ ≲ ⟨x⟩, so it suffices to +show +� ∞ +0 +⟨y⟩ +|x − Ra(y)|2 + a2 da ≲ ⟨y⟩ +� +1 + +1 +|x − y| +� +, +which is equivalent to +� ∞ +0 +1 +|x − Ra(y)|2 + a2 da ≲ 1 + +1 +|x − y|. +(3.4) +To prove (3.4), we will assume without loss of generality that 2|x| ≥ |y| ≥ |x|. On +one hand, it follows from Triangle inequality that +|x − y| ≤ +����x − y +|y||x| +���� + |y| − |x|. +On the other hand, the fact |y| ≥ |x| implies +����x − y +|y||x| +���� ≤ |x − y| +and +|y| − |x| ≤ |x − y|, +which yield +����x − y +|y||x| +���� + |y| − |x| ≲ |x − y|. +Gathering the estimates above, we finally obtain +|x − y| ≈ +����x − y +|y||x| +���� + |y| − |x|. +(3.5) +Next we set θx,y = ∠xoy and assume without loss of generality that θx,y ∈ [0, π], +then +θx,y ≈ +���x − +y +|y||x| +��� +|x| +. +(3.6) +Case 1: θx,y ≥ θ0 := 10−4. In view of (3.5) and (3.6), there holds +|x − Ray| ≈ |x − y| +when a ≤ θ0 +2 . Thus, +� ∞ +0 +1 +|x − Ray|2 + a2 da ≲ +� +θ0 +2 +0 +1 +|x − y|2 + a2 da + +� ∞ +θ0 +2 +1 +a2 da +≲ +1 +|x − y| + 1. + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +11 +Case 2: θx,y ≤ θ0. In this case, we have +� ∞ +0 +1 +|x − Ray|2 + a2 da ≲ +� ∞ +π +3 +1 +|x − Ray|2 + a2 da +(3.7) ++ +� +π +3 +2θx,y +1 +|x − Ray|2 + a2 da +(3.8) ++ +� 2θx,y +0 +1 +|x − Ray|2 + a2 da. +(3.9) +For (3.7), a direct calculation shows that +� ∞ +π +3 +1 +|x − Ray|2 + a2 da ≤ +� ∞ +π +3 +1 +a2 da ≲ 1. +For (3.8), π +3 ≥ a ≥ 2θx,y implies |x − Ray| ≥ |x − y| and hence +� +π +3 +2θx,y +1 +|x − Ray|2 + a2 da ≲ +� ∞ +0 +1 +|x − y|2 + a2 da = +1 +|x − y|. +For (3.9), a direct calculation shows that +� 2θx,y +0 +1 +|x − Ray|2 + a2 da ≤ +� θx,y +0 +1 +|x − Ray|2 + a2 da + +� 2θx,y +θx,y +1 +|x − Ray|2 + a2 da +≤ 2 +� θx,y +0 +1 +|x − Ray|2 + a2 da +since |x − Ray| = |x − R(2θx,y−a)y|. To estimate the right hand side, first we use +(3.5) and (3.6) to conclude that +|y − x| ≈ |x − y +|y||x|| + |y| − |x| ≈ |x||θx,y| + |y| − |x|, +which implies +|x − Ray| ≈ |x||θx,Ray| + |R − ay| − |x| +≈ (θx,y − a)|x| + |y| − |x| +for 0 ≤ a ≤ θx,y. Thus, +� 2θx,y +0 +1 +|x − Ray|2 + a2 da ≲ +� θx,y +0 +1 +(θx,y − a)2|x|2 + (|y| − |x|)2 + a2 da += θx,y +� 1 +0 +1 +(1 − a)2|θx,yx|2 + (|y| − |x|)2 + |θx,ya|2 da. +Case 2.1, |y| − |x| ≤ θx,y|x|. From (3.5) and (3.6), we see that |y − x| ≈ θx,y|x| and +hence +� 2θx,y +0 +1 +|x − Ray|2 + a2 da ≲ θx,y +� 1 +0 +1 +(1 − a)2θ2x,y|x|2 + θ2x,ya2 da += +1 +θx,y +� 1 +0 +1 +(1 − a)2|x|2 + a2 da. +Then a direct calculation shows that +� 1 +0 +1 +(1 − a)2|x|2 + a2 da = +� 1 +0 +1 +a2|x|2 + (a − 1)2 da +≤ +� ∞ +−∞ +1 +a2|x|2 + (a − 1)2 da += +π +� +1 + |x|2 , + +12 +D. GUO AND L. ZHAO +which implies (recalling that |y − x| ≈ θx,y|x| ) +� 2θx,y +0 +1 +|x − Ray|2 + a2 da ≲ +1 +θx,y|x| ≲ +1 +|x − y|. +Case 2.2, |y| − |x| ≥ θx,y|x|. Using the estimates (3.5) and (3.6), it follows that +|y − x| ≈ |y| − |x|. Therefore, +� 2θx,y +0 +1 +|x − Ray|2 + a2 da ≲ θx,y +� 1 +0 +1 +(|y| − |x|)2 + |θx,ya|2 da +≲ +� ∞ +0 +1 +(|y| − |x|)2 + a2 da +≈ +1 +|y| − |x| ≈ +1 +|y − x|. +Gathering these estimates, we get (3.3) and (3.1). +□ +Next we derive some estimates for the modified Biot-Savart kernel H(x, y). +Proposition 3.2 (Estimates for H(x, y)). Let x, y be two distinct points in R2 , +then there holds +|H1(x, y)| ≲ min +� +⟨y⟩(1 + +1 +|x − y|), ⟨x⟩(1 + +1 +|x − y|) +� +(3.10) +and +|H2(x, y)| ≲ min +� +⟨x⟩2(1 + +1 +|x − y|), ⟨y⟩2(1 + +1 +|x − y|), ⟨x⟩⟨y⟩(1 + +1 +|x − y|) +� +. +(3.11) +Proof. We only prove (3.11) since (3.10) is a direct consequence of (2.10), (2.14) +and (3.3). From (2.15) and (3.1), we have that +|H2(x, y)| ≲ ⟨x⟩2(1 + +1 +|x − y|) +and +|H2(x, y)| ≲ ⟨y⟩⟨x⟩(1 + +1 +|x − y|). +Therefore, it remains to show +|H2(x, y)| ≲ ⟨y⟩2(1 + +1 +|x − y|). +In fact, it follows from (2.15) that +|H2(x, y)| ≤ ⟨x⟩|K(x, y)| ≤ ⟨y⟩ +� +R +⟨x⟩ +|y − Rax|2 + a2 da. +Observe that |y − Rax| = |x − R−ay|, so inequality (3.1) yields +� +R +⟨x⟩ +|y − Rax|2 + a2 da ≲ ⟨y⟩(1 + +1 +|x − y|) +and hence (3.11) follows. +□ +Next we derive some useful estimates for the velocity field Hw = H1w + H2w. +Proposition 3.3 (Estimates for Hw(x)). For any x ∈ R2, it holds that +|H1w(x)| ≲ min +� +∥w∥L1 +1 +� L∞ +1 , ⟨x⟩∥w∥L1 � L∞ +� +(3.12) +and +|H2w(x)| ≲ min +� +⟨x⟩∥w∥L1 +1 +� L∞ +1 , ⟨x⟩2∥w∥L1 � L∞, ∥w∥L1 +2 +� L∞ +2 +� +. + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +13 +Proof. We only prove (3.12) since the other one is similar. For H1w, recall that +|H1w(x)| = +���� +� +R2 H1(x, y)w(y) dy +���� . +So by (3.10), on one hand, +|H1w(x)| ≲ ⟨x⟩ +� +R2 +� +1 + +1 +|x − y| +� +|w(y)| dy +≤ ⟨x⟩ +� +∥w∥L1 + +� +R2 +|w(y)| +|x − y| dy +� +. +(3.13) +On the other hand, +|H1w(x)| ≲ +� +R2 +� +1 + +1 +|x − y| +� +⟨y⟩|w(y)| dy +≤ ∥w∥L1 +1 + +� +R2 +|⟨y⟩w(y)| +|x − y| dy. +(3.14) +Observe that for a scalar function f ∈ L1 � L∞(R2), there holds +� +R2 +|f(y)| +|x − y| dy ≲ ∥f∥L1 � L∞. +(3.15) +So using (3.13), (3.14) and (3.15), we finally get the desired estimates for H1w. +□ +3.2. Osgood property of Hw. In this subsection, we will show that the particle +trajectory map X(α, t) given by + + + +dX(α, t) +dt += Hw(X(α, t), t) +X(α, 0) = α +is well-defined. To this end, it suffices to show the velocity Hw satisfies the Osgood’s +condition. More precisely, we will show that +sup +x,y∈BR(0) +|Hw(x) − Hw(z)| +F(|x − z|) +≲R 1 +for F the Log-Lipschitz function given in (2.8). Recalling that Hw(x) = (U 1, U 2)+ +(−x2, x1)U 3, so it follow that +|Hw(x) − Hw(z)| ≲ |x − z||U(x)| + ⟨z⟩|U(x) − U(z)|. +(3.16) +Then by the definition of U, +U(x) − U(z) = +� +R2 (K(x, y) − K(z, y)) w(y) dy, +which implies that (using (2.10)) +|Hw(x) − Hw(z)| +≲ +� +R2 +� +R +���� +(x1, x2, 0) − (Ra(y), a) +|(x1, x2, 0) − (Ra(y), a) |3 − +(z1, z2, 0) − (Ra(y), a) +|(z1, z2, 0) − (Ra(y), a) |3 +���� da⟨z⟩⟨y⟩|w(y)| dy ++ |x − z||U(x)|. +(3.17) +For the last term, we use Proposition 3.3 to conclude that +|x − z||U(x)| ≲ F(x − z)⟨x⟩∥w∥L1 � L∞ +(3.18) +and +|x − z||U(x)| ≲ F(x − z)∥w∥L1 +1 +� L∞ +1 . +(3.19) + +14 +D. GUO AND L. ZHAO +The first term can be bounded by +� +R3 +���� +(x′, 0) − (y′, y3) +|(x′, 0) − (y′, y3)|3 − +(z′, 0) − (y′, y3) +|(z′, 0) − (y′, y3)|3 +���� ⟨z′⟩⟨y′⟩|Ωz(y)| dy +(3.20) +after a change of variables Ray = ˜y. To estimate this integral, we have the following: +Lemma 3.4. For any x, z ∈ R3, it holds that +� +R3 +���� +x − y +|x − y|3 − +z − y +|z − y|3 +���� ⟨y′⟩|Ωz(y)| dy ≲ (⟨x⟩ + ⟨z⟩)∥Ωz∥L1 � L∞(R2×T). +Together with (3.17) and (3.20), we see that Hw is locally Osgood continuous and +hence the particle trajectory map X(α, t) associated with Hw is well-defined. The +proof of this lemma will need some estimates of the three-dimensional Biot-Savart +kernel, which will be postponed to next subsection. +3.3. Estimates for the three-dimensional modified Biot-Savart kernel. In +R3, the velocity field with helical symmetry can be recovered by the third compo- +nent of the vorticity: +U(x) = +� +R3 G(x, y)Ωz(y) dy, +where +G(x, y) := +x − y +|x − y|3 ∧ ξ(y). +(3.21) +The main purpose of this subsection is to give a detailed estimates of G(x, y). We +begin with the following auxiliary lemma: +Lemma 3.5. Let Ω(x1, x2, x3) be a 2π-periodic function in x3, then one has +� +R3 +1 +|x − y|2 |Ω(y)| dy ≲ ∥Ω∥L1 � L∞(R2×T). +(3.22) +Proof. First, observe that +� +R3 +1 +|x − y|2 |Ω(y)| dy = +� +R3 +1 +|y|2 |Ω(y + x)| dy +and for any x ∈ R3 +∥Ω(· + x)∥L1 � L∞(R2×T) = ∥Ω∥L1 � L∞(R2×T). +So we may assume without loss of generality that x = 0 and Ω ≥ 0. Then we +estimate +� +R3 +Ω(y) +|y|2 dy = +� +R2 +� 2π +−2π +Ω(y) +|y|2 dy3 dy1dy2 + +� +R2 +� +|y3|≥2π +Ω(y) +|y|2 dy. +(3.23) +For the first term in (3.23), denote +˜Ω(y) := Ω(y)χ|y3|≤2π(y). +Recall that +� +R3 +|f(y)| +|x − y|2 dy ≲ ∥f∥L1 � L∞(R3), +(3.24) +so we have +� +R2 +� 2π +−2π +Ω(y) +|y|2 dy3 dy1dy2 ≲ ∥˜Ω∥L1 � L∞(R3) ≤ 2∥Ω∥L1 � L∞(R2×T). + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +15 +For the second term in (3.23), a direct calculation shows that +� +R2 +� +|y3|≥2π +Ω(y) +|y|2 dy = +� +n̸=−1,0 +� +R2 +� 2π(n+1) +2πn +Ω(y) +|y|2 dy3dy1dy2 +≲ +� +n̸=−1,0 +� +R2 +� 2π(n+1) +2πn +Ω(y) +|y3|2 dy3dy1dy2 +≲ +� +n̸=−1,0 +� +R2 +� 2π(n+1) +2πn +Ω(y) +n2 +dy3dy1dy2 += +� +n̸=−1,0 +1 +n2 ∥Ω∥L1(R2×T) +≲ ∥Ω∥L1(R2×T), +which completes the proof. +□ +As a consequence of Lemma 3.5, we obtain +Corollary 3.6 (Estimates for U(x)). For any x ∈ R3, it holds that +|U(x)| ≲ ∥Ωz∥L1 +1 +� L∞ +1 (R2×T). +(3.25) +Remark 3.7. If we assume that Ωz is helical, then for any x ∈ R3, there holds +|U(x)| ≲ ⟨x′⟩∥Ωz∥L1 � L∞(R2×T). +Indeed, setting w(x1, x2) = Ωz(x1, x2, 0), then by definition of U, we have that +|U(x1, x2, 0)| ≤ +���� +� +R3 +⟨y′⟩ +|(x′, 0) − (y′, y3)|2 |Ωz(y)| dy +���� += +� +R2 |w(y1, y2)| +� +R +⟨y′⟩ +|x′ − Ry3y′|2 + y2 +3 +dy3dy′. +Then (3.3) and Lemma 2.13 gives +|U(x1, x2, 0)| ≲ ⟨x′⟩∥w∥L1 � L∞(R2) +≲ ⟨x′⟩∥Ωz∥L1 � L∞(R2×T), +which implies +|U(x′, x3)| = |U(R−x3x′, 0)| +≲ ⟨x′⟩∥Ωz∥L1 � L∞(R2×T) +since U is a helical vector field. +Next we estimate the difference +|U(x) − U(z)| = +���� +� +R3 +� x − y +|x − y|3 − +z − y +|z − y|3 +� +∧ ξ(y)Ωz(y) dy +���� +≲ +� +R3 +���� +x − y +|x − y|3 − +z − y +|z − y|3 +���� ⟨y′⟩|Ωz(y)| dy. +Lemma 3.8. Let Ω(x1, x2, x3) be a helical function, then for any x, z ∈ R3, it holds +that +� +R3 +���� +x − y +|x − y|3 − +z − y +|z − y|3 +���� |Ω(y)| dy ≲ ∥Ω∥L1 � L∞(R2×T)F(|x − z|), +(3.26) +where F is the Log-Lipschitz function given in (2.8). + +16 +D. GUO AND L. ZHAO +Proof. After changing the variable ˜y = y − z, we obtain +� +R3 +���� +x − y +|x − y|3 − +z − y +|z − y|3 +���� |Ω(y)| dy = +� +R3 +���� +x − z − ˜y +|x − z − ˜y|3 − +−˜y +| − ˜y|3 +���� |Ω(˜y + z)| d˜y. +Observe that for any z ∈ R3, +∥Ω(· + z)∥L1 � L∞(R2×T) = ∥Ω∥L1 � L∞(R2×T). +So we may assume without loss of generality that z = 0 and Ω ≥ 0. To complete +the proof, it suffices to show that +� +|K(x − y) − K(−y)| Ω(y) dy ≲ ∥Ω∥L1 � L∞(R2×T)F(|x|), +where K(x) := +x +|x|3 ∈ C∞(R3 \ 0). To this end, we divide the above integral into +three parts: +� +|K(x − y) − K(−y)| Ω(y) dy = +� +|y−x|≥2 +|K(x − y) − K(−y)| Ω(y) dy +(3.27) ++ +� +2≥|y−x|≥2|x| +|K(x − y) − K(−y)| Ω(y) dy +(3.28) ++ +� +|y−x|≤2|x| +|K(x − y) − K(−y)| Ω(y) dy. +(3.29) +For (3.27), a direct calculation shows that +���� +a +|a|3 − +b +|b|3 +���� +2 += +1 +|a|4 + +1 +|b|4 − 2a · b +|a|3|b|3 += |a|4 + |b|4 − 2|a|a · |b|b +|a|4|b|4 += +���|a|a − |b|b +��� +2 +|a|4|b|4 +, +which implies +���� +a +|a|3 − +b +|b|3 +���� = +���|a|a − |b|b +��� +|a|2|b|2 +≲ +���a|a| − a|b| + a|b| − b|b| +��� +|a|2|b|2 +≲ +���|a| − |b| +��� +|a||b|2 ++ |a − b| +|a|2|b| +≲ +� +1 +|a||b|2 + +1 +|a|2|b| +� +|a − b|, +(3.30) +and hence +|K(x − y) − K(−y)| ≲ |x| +� +1 +|x − y|2|y| + +1 +|x − y||y|2 +� +. +Therefore, +� +|y−x|≥2 +|K(x − y) − K(−y)| Ω(y) dy +≲|x| +� +|x−y|≥2 +� +1 +|x − y|2|y| + +1 +|x − y||y|2 +� +Ω(y) dy. +(3.31) + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +17 +To bound the right-hand side of (3.27), it remains to show that +� +|y−x|≥2 +1 +|x − y||y|2 Ω(y) dy + +� +|y−x|≥2 +1 +|x − y|2|y|Ω(y) dy ≲ ∥Ω∥L1 � L∞(R2×T). +(3.32) +For the first integral above, (3.22) gives +� +|y−x|≥2 +1 +|x − y||y|2 Ω(y) dy ≲ +� +R3 +1 +|y|2 Ω(y) dy ≲ ∥Ω∥L1 � L∞(R2×T). +(3.33) +For the second integral in (3.32), let N = N(x) be the integer such that x ∈ +[2πN, 2π(N + 1)). Then we get +� +|y−x|≥2 +1 +|x − y|2|y|Ω(y) dy = +∞ +� +−∞ +� +R2 +� 2π(n+1) +2πn +Ω(y)χ{|y−x|≥2}(y) +|x − y|2|y| +dy3 dy′ += +� +|n−N|≥2 +� +R2 +� 2π(n+1) +2πn +Ω(y)χ{|y−x|≥2}(y) +|x − y|2|y| +dy3 dy′ +(3.34) ++ +� +|n−N|≤1 +� +R2 +� 2π(n+1) +2πn +Ω(y)χ{|y−x|≥2}(y) +|x − y|2|y| +dy3 dy′. +(3.35) +For (3.34), we only consider the case when n ≥ N +2 since the other part is similar. +Note that |x − y| ≥ |x3 − y3| ≈ |N + 1 − n| for y3 ∈ [2πn, 2π(n + 1)), so it follows +that +∞ +� +n=N+2 +� +R2 +� 2π(n+1) +2πn +Ω(y)χ{|y−x|≥2}(y) +|x − y|2|y| +dy3dy1dy2 +≲ +∞ +� +n=N+2 +� +R2 +� 2π(n+1) +2πn +Ω(y) +|N + 1 − n|2|y| dy3 dy1dy2 += +∞ +� +n=N+2 +1 +|N + 1 − n|2 +� +R2 +� 2π(n+1) +2πn +Ω(y) +|y| dy3 dy1dy2. +Denote +˜Ωn(y) := Ω(y)χ2πn≤y3≤2π(n+1), +then (3.24) yields +∞ +� +n=N+2 +� +R2 +� 2π(n+1) +2πn +Ω(y)χ{|y−x|≥2}(y) +|x − y|2|y| +dy3dy1dy2 +≲ +∞ +� +n=N+2 +1 +|N + 1 − n|2 ∥˜Ωn∥L1 � L∞(R3) +≲ +∞ +� +n=N+2 +1 +|N + 1 − n|2 ∥Ω∥L1 � L∞(R2×T) +≲∥Ω∥L1 � L∞(R2×T). +(3.36) +For (3.35), denote +˜Ω(y) = Ω(y)χ{2π(N−1)≤y3≤2π(N+2)} + +18 +D. GUO AND L. ZHAO +then in view of (3.24), we have +� +|n−N|≤1 +� +R2 +� 2π(n+1) +2πn +Ω(y)χ{|y−x|≥2}(y) +|x − y|2|y| +dy3 dy1dy2 +≲ +� +R3 +˜Ω(y) +|x − y|2|y|χ{|y−x|≥2} dy = +� +|y−x|≥2 +˜Ω(y) +|x − y|2|y| dy +≲ +� +R3 +˜Ω(y) +|y| dy ≲ ∥˜Ω∥L1 � L∞(R3) ≲ ∥Ω∥L1 � L∞(R2×T). +Together with (3.33) and (3.36), we finally obtain (3.32) and hence by (3.31) we get +the desired estimates for (3.27). For (3.28), we only consider the case |x| ≤ 1 since +otherwise the integral vanishes identically. Note that when |y| ≥ 2|x|, the segment +between x − y and −y does not contain the origin. So the mean value theorem +yields +|K(x − y) − K(−y)| = |x∇K(−y + θx)| +≲ |x| +1 +| − y + θx|3 ≲ |x| +|y|3 ≲ +|x| +|y − x|3 . +Therefore, +� +2≥|y−x|≥2|x| +|K(x − y) − K(−y)| Ω(y) dy +≲|x| +� +2≥|y|≥2|x| +Ω(y) +|y − x|3 dy ≲ ∥Ω∥L∞(R2×T)F(|x|). +For (3.3), we estimate +� +|y|≤2|x| +���� +(x − y) +|x − y|3 − (−y) +| − y|3 +���� Ω(y) dy ≲ +� +|y|≤2|x| +Ω(y) +|x − y|2 dy + +� +|y|≤2|x| +Ω(y) +|y|2 dy. +A direct calculation shows that +� +|y|≤2|x| +Ω(y) +|x − y|2 dy ≲ ∥Ω∥L∞ +� +|y|≤2|x| +1 +|x − y|2 dy +≲ ∥Ω∥L∞ +� +|y|≤2|x| +1 +|y|2 dy +≲ ∥Ω∥L∞|x| +and similarly, +� +|y|≤2|x| +Ω(y) +|y|2 dy ≲ ∥Ω∥L∞ +� +|y|≤2|x| +1 +|y|2 dy +≲ ∥Ω∥L∞ +� +|y|≤2|x| +1 +|y|2 dy +≲ ∥Ω∥L∞|x|. +Gathering the estimates above we obtain +� +|y−x|≤2|x| +|K(x − y) − K(−y)| Ω(y) dy ≲ |x| + F(|x|) + |x| ≲ F(|x|), +which completes the proof. +□ +Corollary 3.9. For any x, z ∈ R3 and G(x, y) defined in (3.21), the following +holds. � +R3 |G(x, y) − G(z, y)| |Ω(y)| dy ≲ ∥Ω∥L1 +1 +� L∞ +1 (R2×T)F(|x − z|), +(3.37) + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +19 +� +R3 |G(x, y) − G(z, y)| |Ω(y)| dy ≲ (⟨x′⟩ + ⟨z′⟩)∥Ω∥L1 � L∞(R2×T)F(|x − z|) (3.38) +and +� +R3 |G(y, x) − G(y, z)| |Ω(y)| dy ≲ (⟨x′⟩ + ⟨z′⟩)∥Ω∥L1 � L∞(R2×T)F(|x − z|). (3.39) +Proof. The inequality (3.37) follows directly from Lemma 3.8. In order to prove +(3.38), observe that +� +R3 |G(x, y) − G(z, y)| Ω(y) dy ≲ +� +R3 |K(x − y) − K(z − y)| ⟨y′⟩Ω(y) dy, +so by (3.26) it suffice to consider the integral for max {5|x|, 5|z|} ≤ |y|. +When +|y| ≤ 1, +� +|y|≤1 +|G(x, y) − G(z, y)| Ω(y) dy ≲ +� +|y|≤1 +|K(x − y) − K(z − y)| ⟨y′⟩Ω(y) dy +≲ +� +R3 |K(x − y) − K(z − y)| Ω(y) dy. +Thus (3.26) implies that +� +|y|≤1 +|G(x, y) − G(z, y)| Ω(y) dy ≲ F(|x − z|)∥Ω∥L1 � L∞(R2×T). +When |y| ≥ 1, we use mean value theorem to conclude that +|K(x − y) − K(z − y)| ≲ |x − z| +|y|3 +, +which implies (using (3.22)) +� +max {1,5|x|,5|z|}≤|y| +|G(x, y) − G(z, y)| |Ω(y)| dy +≲|x − z| +� +max{1,5|x|,5|z|}≤|y| +⟨y′⟩ +|y|3 |Ω(y)| dy +≲|x − z| +� +R3 +|Ω(y)| +|y|2 +dy +≲|x − z|∥Ω∥L1 � L∞(R2×T). +Next we consider (3.39). Observe that +G(y, x) − G(y, z) = +y − x +|y − x|3 ∧ ξ(x) − +y − z +|y − z|3 ξ(z) += +� y − x +|y − x|3 − +y − z +|y − z|3 +� +ξ(x) + +y − z +|y − z|3 (ξ(x) − ξ(z)), +so we have +|G(y, x) − G(y, z)| ≲ |ξ(x)| +���� +y − x +|y − x|3 − +y − z +|y − z|3 +���� + |x − z| +1 +|y − z|2 . +Therefore, (3.22) and (3.26) yields +� +|G(y, x) − G(y, z)| |Ω(y)| dy +≲⟨x′⟩ +� ���� +x − y +|x − y|3 − +z − y +|z − y|3 +���� |Ω(y)| dy + |x − z| +� +|Ω(y)| +|y − z|2 dy +≲⟨x′⟩F(x − z)∥Ω∥L1 � L∞(R2×T). +□ + +20 +D. GUO AND L. ZHAO +4. Proof of the uniqueness. +In this section, we prove the uniqueness of the weak solutions to (1.4) and (1.3) in +L1 +1 +� L∞ +1 (R2) and L1 +1 +� L∞ +1 (R3), respectively. +4.1. Uniqueness of the three-dimensional helical Euler equation (1.3). Our +main theorem is: +Theorem 4.1. Assume Ωz, ˜Ωz ∈ L∞([0, T ], L1 +1 +� L∞ +1 (R2 × T)) are two lagrangian +solutions to (1.3) with the same initial data Ωz +0, then Ωz = ˜Ωz. +Before the proof, we first make some useful observations. Recall that Ωz satisfies +∂tΩz + U · ∇Ωz = 0 +and when Ωz ∈ L1 +1 +� L∞ +1 (R2 × T), Corollary 3.9 implies that the velocity field +U(x, t) = +� +G(x, y)Ωz(y) dy +with +G(x, y) = +x − y +|x − y|3 ∧ ξ(y) = +x − y +|x − y|3 ∧ (y2, −y1, 1) +is Log-Lipschitz continuous. Thus, the particle trajectory map + + + +dX(α, t) +dt += U(X(α, t), t) +X(α, 0) = α +is well-defined. Let X(α, t), ˜X(α, t) be the particle trajectory map associated with +U and ˜U, respectively. Then we have +Lemma 4.2. Assume X(α, t) = ˜X(α, t) for all α ∈ supp(Ωz +0) and t ∈ [0, T ], then +Ωz = ˜Ωz. +Proof. We fix x ∈ R3 and t ∈ [0, T ], then there exists unique α and β such that +x = X(α, t) = ˜X(β, t) +since the map X(·, t) and ˜X(·, t) are one-to-one. +Case 1: α, β /∈ supp Ωz +0. In this case, one has +Ωz(x, t) = Ωz(X(α, t), t) = Ωz +0(α) = 0 +and +˜Ωz(x, t) = ˜Ωz( ˜X(β, t), t) = Ωz +0(β) = 0, +which implies Ωz(x, t) = ˜Ωz(x, t) = 0. +Case 2: α ∈ supp Ωz +0. Observe that by assumption in our lemma, we have +˜X(β, t) = x = X(α, t) = ˜X(α, t) +since α ∈ supp Ωz +0. Thus, α = β since ˜X(·, t) is one-to-one and hence +Ωz(x, t) = Ωz +0(α, t) = Ωz +0(β, t) = ˜Ωz(x, t). +The case when β ∈ supp Ωz +0 is similar so we omit it. +□ +Remark 4.3. From Lemma 2.8, we see that X − ˜X is period in α3. So in order to +prove Ωz = ˜Ωz, it suffices to show +X(α, t) = ˜X(α, t) +for all α ∈ supp(Ωz +0) +� +R2 × T. +Now we are ready to prove Theorem 4.1. + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +21 +Proof. Motivated by the remark above, we define the distance +D(t) := +� +R2×T +���X(α, t) − ˜X(α, t) +��� ⟨α⟩|Ωz +0(α)| dα +and it remains to show that E(t) ≡ 0. First we will show that D(t) is continuous +and bounded in [0, T ]. To this end, we estimate +|X(α, t) − α| = +� t +0 +X(α, s) − α +|X(α, s) − α|U(X(α, s), s) ds +≲ T sup +t ∥U(·, t)∥L∞, +which together with (3.25) gives +|X(α, t) − α| ≲ sup +t ∥Ωz∥L1 +1 +� L∞ +1 (R2×T) ≲ 1. +Similarly, +| ˜X(α, t) − α| ≲ 1, +which implies +⟨α⟩ ≈ ⟨X(α, t)⟩ ≈ ⟨ ˜X(α, t)⟩ +(4.1) +and +|X(α, t) − ˜X(α, t)| ≤ |X(α, t) − α| + | ˜X(α, t) − α| ≲ 1. +Therefore, +D(t) ≲ +� +R2×T +⟨α⟩|Ωz +0(α)| dα = ∥Ωz∥L1 +1(R2×T). +Furthermore, the energy D(t) is indeed continuous in [0, T ] by Lebesgue dominating +convergence Theorem. +Next we show D(t) ≡ 0. Observe that if Ωz +0 ≡ 0, then Ωz = ˜Ωz ≡ 0, so we may +assume without loss of generality that Ωz +0 ̸= 0. Set +D∗(t) := +D(t) +∥Ωz +0∥L1 +1(R2×T) +. +We claim that for all t ∈ [0, T ], one has +D∗(t) ≲ +� t +0 +F(D∗(s)) ds. +Recalling that D∗(0) = 0, so it follows from Osgood’s Lemma that D∗(t) ≡ 0 and +hence X(α, t) = ˜X(α, t) for all α ∈ supp (Ωz +0) � R2 × T. We now prove the claim. +A direct calculation shows that +���X(α, t) − ˜X(α, t) +��� ≲ +� t +0 +���U(X(α, s), s) − ˜U( ˜X(α, s), s) +��� ds +≲ +� t +0 +��� ˜U(X(α, s), s) − ˜U( ˜X(α, s), s) +��� ds +(4.2) ++ +� t +0 +���U(X(α, s), s) − ˜U(X(α, s), s) +��� ds. +(4.3) +First we use (3.37) to conclude that +| ˜U(x) − ˜U(z)| ≲ ∥˜Ωz∥L1 +1 +� L∞ +1 (R2×T)F(|x − z|), +which implies +� t +0 +��� ˜U(X, s) − ˜U( ˜X, s) +��� ds ≲ +� t +0 +F +� +|X(α, s) − ˜X(α, s)| +� +ds. + +22 +D. GUO AND L. ZHAO +Recall that F is a concave function, so Jensen’s inequality yields +1 +∥Ωz +0∥L1 +1 +� +R2×T +� t +0 +���U(X, s) − ˜U(X, s) +��� ds⟨α⟩|Ωz +0(α)| dα +≲ +� t +0 +� +R2×T +F +� +|X(α, s) − ˜X(α, s)| +� ⟨α⟩|Ωz +0(α)| +∥Ωz +0∥L1 +1 +dα ds +≲ +� t +0 +F +�� +R2×T +|X(α, s) − ˜X(α, s)| ⟨α⟩|Ωz +0(α)| +∥Ωz +0∥L1 +1 +dα +� +ds += +� t +0 +F(D∗(s)) ds. +Next we estimate the second term in (4.3), +|U(x, s) − ˜U(x, s)| = +���� +� +G(x, y)Ωz(y, s) dy − +� +G(x, y)˜Ωz(y, s) dy +���� += +���� +� +G(x, X(β, s))Ωz +0(β) dβ − +� +G(x, ˜X(β, s))Ωz +0(β) dβ +���� +≲ +� ���G(x, X(β, s)) − G(x, ˜ +X(β, s)) +��� |Ωz +0(β)| dβ. +Then (4.1) gives +� +R2×T +���U(X(α, s), s) − ˜U(X(α, s), s) +��� ⟨α⟩Ωz +0(α) dα +≲ +� +R3 +� +R2×T +���G(X(α, s), X(β, s)) − G(X(α, s), ˜X(β, s)) +��� ⟨α⟩|Ωz +0(β)Ωz +0(α)| dα dβ +≲ +� +R3 +� +R2×T +���G(X(α, s), X(β, s)) − G(X(α, s), ˜X(β, s)) +��� ⟨X(α, s)⟩|Ωz +0(β)Ωz +0(α)| dα dβ +:=A. +Note that |X(α, t)−α| ≲ 1, so there exists N > 0 such that X(·, t) maps R2×[0, 2π] +into R2 × [−2πN, 2πN]. Thus a change of variables x = X(α, s) yields +A ≲ +� +R3 +� +R2×[−2πN,2πN] +���G(x, X(β, s)) − G(x, ˜ +X(β, s)) +��� ⟨x′⟩|Ωz(x, s)| dx|Ωz +0(β)| dβ. +Fix M ≫ N such that for all |β3| ≥ 2πM and all t ∈ [0, T ], |X3(β, t)|, | ˜X3(β, t)| ≥ +4πN. Then we can divide the integral into two parts according to the value of β3. +Let XM = R2 × [−2πM, 2πM], then +A ≲ +� +XM +� +XN +���G(x, X(β, s)) − G(x, ˜X(β, s)) +��� ⟨x′⟩|Ωz(x, s)| dx|Ωz +0(β)| dβ ++ +� +XC +M +� +XN +���G(x, X(β, s)) − G(x, ˜ +X(β, s)) +��� ⟨x′⟩|Ωz(x, s)| dx|Ωz +0(β)| dβ +:=A1 + A2. +(4.4) +For A1, we see directly from (3.38) and (4.1) that +A1 ≲ +� +XM +� +⟨X(β, s)⟩ + ⟨ ˜X(β, s)⟩ +� +F(|X(β, s) − ˜X(β, s)|)|Ωz +0(β)| dβ +≲ +� +XM +F(|X(β, s) − ˜X(β, s)|)|⟨β⟩|Ωz +0(β)| dβ +≲ +� +R2×T +F(|X(β, s) − ˜X(β, s)|)|⟨β⟩|Ωz +0(β)| dβ + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +23 +since Ωz ∈ L∞([0, T ], L1 +1 +� L∞ +1 (R2 × T)). Thus, Jensen’s inequality yields +A1 ≲ F +�� +R2×T +|X(β, s) − ˜X(β, s)| |⟨β⟩Ωz +0(β)| +∥Ωz +0∥L1 +1(R2×T) +dβ +� +. +(4.5) +For A2, first we estimate +���G(x, X(β, s)) − G(x, ˜ +X(β, s)) +��� +(4.6) += +����� +x − X(β, s) +|x − X(β, s)|3 ∧ ξ(X(β, s)) − +x − ˜X(β, s) +|x − ˜X(β, s)|3 ∧ ξ( ˜X(β, s)) +����� +≲ +����� +� +x − X(β, s) +|x − X(β, s)|3 − +x − ˜X(β, s) +|x − ˜X(β, s)|3 +� +∧ ξ(X(β, s)) +����� +(4.7) ++ +����� +x − ˜X(β, s) +|x − ˜X(β, s)|3 ∧ +� +ξ(X(β, s)) − ξ( ˜ +X(β, s)) +������ . +(4.8) +For (4.7), we see from (3.30) that for all |x| ≤ 2πN and |β3| ≥ 2πM, +����� +� +x − X(β, s) +|x − X(β, s)|3 − +x − ˜X(β, s) +|x − ˜X(β, s)|3 +� +∧ ξ(X(β, s)) +����� +≲ +���X(β, s) − ˜X(β, s) +��� ⟨β′⟩ +|β3|3 +≲ +���X(β, s) − ˜X(β, s) +��� ⟨β′⟩ +|β3|2 . +For (4.8), since +���x3 − ˜X3 +��� ≳ 1, it is easy to check that +����� +x − ˜X(β, s) +|x − ˜X(β, s)|3 ∧ +� +ξ(X(β, s)) − ξ( ˜X(β, s)) +������ ≲ +���X(β, s) − ˜X(β, s) +��� +1 +|β3|2 . +Thus, +���G(x, X(β, s)) − G(x, ˜ +X(β, s)) +��� ≲ +���X(β, s) − ˜X(β, s) +��� ⟨β′⟩ +|β3|2 , +which implies that +A2 ≲ ∥Ωz(·, s)∥L1 +1(R2×T) +� +XC +M +���X(β, s) − ˜X(β, s) +��� ⟨β′⟩ +|β3|2 |Ωz +0(β)| dβ +≲ +� +|n|≥M +� +R2 +� 2π(n+1) +2πn +���X(β, s) − ˜X(β, s) +��� ⟨β′⟩ +|β3|2 |Ωz +0(β)| dβ3 dβ1dβ2 +≲ +� +|n|≥M +� +R2 +� 2π(n+1) +2πn +���X(β, s) − ˜X(β, s) +��� ⟨β′⟩ +n2 |Ωz +0(β)| dβ3 dβ1dβ2. +Note that |X(β, t) − ˜X(β, t)|, ⟨β′⟩ and Ωz +0(β) are periodic functions in β3, so it +follows that +A2 ≲ +� +|n|≥M +1 +n2 +� +R2 +� 2π +0 +���X(β, s) − ˜X(β, s) +��� ⟨β′⟩|Ωz +0(β)| dβ3 dβ1dβ2 +≲ F +�� +R2×T +���X(β, s) − ˜X(β, s) +��� +⟨β⟩|Ωz +0(β)| +∥Ωz +0∥L1 +1(R2×T) +dβ +� +, +(4.9) + +24 +D. GUO AND L. ZHAO +where we have used the fact that F(r) ≳ r for r ≥ 0. Thus, (4.4), (4.5) and (4.9) +gives +A ≲ F +�� +R2×T +���X(β, s) − ˜X(β, s) +��� +⟨β⟩|Ωz +0(β)| +∥Ωz +0∥L1 +1(R2×T) +dβ +� +. +Integrating the above inequality from 0 to t, we arrive at +D∗(t) ≲ +� t +0 +F +�� +R2×T +���X(α, s) − ˜X(α, s) +��� ⟨α⟩|Ωz +0(α)| +∥Ωz +0∥L1 +1 +dα +� +ds += +� t +0 +F(D∗(s)) ds. +This completes the proof. +□ +4.2. Uniqueness of the two-dimensional helical Euler equation (1.4) in +L1 +1 +� L∞ +1 (R2). We will show that every weak solution of (1.4) can be lifted to a +Lagrangian weak solution of (1.3), thus the uniqueness of (1.4) follows directly +from Theorem 4.1. +Lemma 4.4. Let w(x, t) ∈ L∞([0, T ], L1 � L∞(R2)) be a weak solution of the two- +dimensional helical Euler equation (1.4), set +Ω(x1, x2, x3, t) = w(R−x3(x1, x2), t) +(4.10) +and +U(x, t) = +� +R3 G(x, y)Ω(y, t) dy. +Then Ω(x, t) satisfies the three-dimensional helical Euler equation (1.3). +Proof. By definition of the weak solution, it suffices to show that for any φ ∈ +C∞ +c (R3 × R), +� +R3 Ω(x, t)φ(x, t) dx − +� +R3 Ω(x, 0)φ(x, 0) dx = +� t +0 +� +R3 Ω(∂tφ + U · ∇φ) dxdt. +First we observe that (using (4.10)) +� +R3 Ω(x, t)φ(x, t) dx − +� +R3 Ω(x, 0)φ(x, 0) dx += +� +R3 w(R−x3x′, t)φ(x, t) dx − +� +R3 w(R−x3x′, 0)φ(x, 0) dx += +� +R +�� +R2 w(x1, x2, t)φ(Rx3x′, x3, t) dx′ − +� +R2 w(x1, x2, 0)φ(Rx3x′, x3, 0) dx′ +� +dx3, +which implies +� +R3 Ω(x, t)φ(x, t) dx − +� +R3 Ω(x, 0)φ(x, 0) dx += +� +R +�� t +0 +� +R2 w(x1, x2, s) [∂s + Hw(x1, x2, s) · ∇] (φ(Rx3x′, x3, s)) dx′ds +� +dx3 += +� t +0 +� +R +�� +R2 w(x1, x2, s) [∂s + Hw(x1, x2, s) · ∇] (φ(Rx3x′, x3, s)) dx′ +� +dx3 ds + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +25 +since w is a weak solution of (1.4). Therefore, to complete the proof of the lemma, +it remains to show that +� +R3 Ω(x, s) [∂s + U · ∇] φ(x, s) dx += +� +R +�� +R2 w(x1, x2, s) [∂s + Hw(x1, x2, s) · ∇] (φ(Rx3x′, x3, s)) dx′ +� +dx3. +(4.11) +Since Ω is a helical function, we have +� +R3 Ω(x, s)∂sφ(x, s) dx = +� +R3 w(R−x3x′, s)∂sφ(x, s) dx += +� +R3 w(x1, x2, s)∂sφ(Rx3x′, x3, s) dx. +(4.12) +Recall that +Hw(x1, x2) = (U 1(x′, 0), U 2(x′, 0)) + (−x2, x1)U 3(x′, 0), +(4.13) +where U 3(·, t) are helical functions on R3. So we use (2.7), (2.6), (4.13) and the +fact that ΩU 3 is a helical function to get the following: +� +R3 Ω(x, s)(U 1∂1φ + U 2∂2φ) dx + +� +R3(ΩU 3)∂3φ dx += +� +R3 Ω(x, s)(U 1∂1φ + U 2∂2φ) dx + +� +R3 Ω(x1U 3∂2φ − x2U 3∂1φ) dx. +(4.14) +Note that for a smooth function φ = φ(x1, x2), +∇ (φ(Rx3x′)) = R−x3[∇φ(Rx3x′)]. +So in view of (4.14), there holds +� +R3 Ω(x, s)(U 1∂1φ + U 2∂2φ) dx + +� +R3(ΩU 3)∂3φ dx += +� +R +� +R2 w(x1, x2, s)Hw(x1, x2, s) · ∇ (φ(Rx3x′, x3, s)) dx1dx2 dx3. +(4.15) +Together with (4.12) we finally get (4.11) and hence completes the proof. +□ +Lemma 4.5. Assume w(x, t) ∈ L∞([0, T ], L1 � L∞(R2)) is Lagrangian weak solu- +tion of (1.4), then +Ω(x′, x3, t) = w(R−x3x′, t) +(4.16) +is a Lagrangian weak solution of (1.3). +Proof. Let X(α1, α2, t) = (X1(α1, α2, t), X2(α1, α2, t)) be the particle trajectory +map associated to Hw. Motivated by (2.9), define +X3(α1, α2, t) = +� t +0 +U3(X1(α1, α2, s), X2(α1, α2, s), 0) ds, +and +Y (α1, α2, 0, t) = (RX3(X1, X2), X3). +(4.17) +We claim that +Y (α1, α2, α3, t) := Sα3Y (R−α3(α1, α2), 0, t) +(4.18) +is the particle trajectory map associated with U in R3. That is, + + + +dY (α, t) +dt += U(Y (α, t), t) +Y (α, 0) = α, + +26 +D. GUO AND L. ZHAO +Indeed, a direct calculation shows that +dY1(α1, α2, 0, t) +dt +=(U1 − X2U3) cos(X3) − X1U3 sin(X3) + (U2 + X1U3) sin(X3) + X2U3 cos(X3) +=U1 cos(X3) + U2 sin(X3), +dY2(α1, α2, 0, t) +dt += −U1 sin(X3) + U2 cos(X3) +and +dY3(α1, α2, 0, t) +dt += U3. +Hence, +dY (α1, α2, 0, t) +dt += RX3U(X1, X2, 0) = U(RX3(X1, X2), X3) +since U is a helical vector field. Then (4.17) yields +dY (α1, α2, 0, t) +dt += U(Y (α1, α2, 0, t), t), +which combined with (4.18), gives +dY (α1, α2, α3, t) +dt += dSα3Y (R−α3(α1, α2), 0, t) +dt += dRα3Y (R−α3(α1, α2), 0, t) +dt += Rα3U(Y (R−α3(α1, α2), 0, t), t). +Finally, using (2.3) and (4.18), we have that +dY (α1, α2, α3, t) +dt += Rα3U(Y (S−α3α, t), t) += U(Sα3Y (S−α3α, t), t) += U(Y (α1, α2, α3, t), t) +which implies that Y is the particle trajectory map of U in R3 and hence completes +the proof of the claim. +Now it remains to show that +Ω(Y (α, t), t) = Ω0(α). +To this end, we set β = R−α3(α1, α2). Then it follows from (2.2) and (2.9) that +Ω(Y (α, t), t) = Ω(Sα3Y (β, 0, t), t) += Ω(Y (β, 0, t), t) += Ω(S−Y3(β,0,t)Y (β, 0, t), t), +which yields (using (4.17) and (4.10)) +Ω(Y (α, t), t) = Ω(S−X3(β,t)(RX3(X1, X2), X3)(β, t), t) += Ω(X1(β, t), X2(β, t), 0, t) += w(X1(β, t), X2(β, t), t) += w0(β) +since w is a Lagrangian solution. Thus in view of (4.16), we get +Ω(Y (α, t), t) = Ω0(α), +which completes the proof. +□ +An immediate consequence is: + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +27 +Corollary 4.6. Every weak solution w(x, t) of (1.4) can be lifted to a weak La- +grangian solution of (1.3) with Ωz(x, t) = w(R−x3(x1, x2), t). Moreover, such lifting +is injective. +Proof. It follows from Lemma A.4 that every weak solution w(·, t) to (1.4) is indeed +a Lagrangian weak solution, so Lemma 4.4 implies that every weak solution of (1.4) +can be lifted to a weak Lagrangian solution of (1.3). Thus it remains to show that +such lifting is injective. Let w and ˜w be two different weak solution of (1.4), then +∥Ωz(·, t) − ˜Ωz(·, t)∥L1 +1 +� L∞ +1 (R2×T) ≈ ∥w(·, t) − ˜w(·, t)∥L1 +1 +� L∞ +1 (R2) ̸= 0 +as a consequence of Lemma 2.13. +□ +Together with Theorem 4.1, we obtain the uniqueness of weak solutions to the +two-dimensional helical Euler equation. +Corollary 4.7. Assume w, ˜w ∈ L∞([0, T ], L1 +1 +� L∞ +1 (R2)) are two weak solutions to +(1.4) with the same initial data w0, then w = ˜w. +5. Global existence of weak solutions to the two-dimensional +helical Euler equation +The main purpose of this section is to show that all weak solutions to the two- +dimensional helical Euler equation (1.4) are global in L1 +1 ∩ L∞ +1 (R2). +5.1. A formal argument. We will assume in this subsection that all functions are +smooth enough and exhibit sufficient decay at infinity. The well-known Beale-Kato- +Majda criterion suggests that a solution Ω of the three-dimensional Euler equation +(1.2) blows-up at time T if and only if +� T +0 +∥Ω∥L∞(R3) dt = ∞. +For a helical solution without swirl, we have +∥Ω∥L∞(R3) = ∥Ωz∥L∞ +1 (R2×T) +with Ω = (x2Ωz, −x1Ωz, Ωz) and Ωz(x1, x2, x3) = w(R−x3(x1, x2)). So it follows +that +∥Ω∥L∞(R3) = ∥Ωz∥L∞ +1 (R3) = ∥w∥L∞ +1 (R2). +Recall that w satisfies the (transport) equation +∂tw + Hw · ∇w = 0 +with ∇ · Hw = 0. Multiply both sides by ⟨x⟩, we obtain +∂t(⟨x⟩w) + Hw · ∇(⟨x⟩w) = wHw · ∇⟨x⟩ += wH1w · ∇⟨x⟩ +since Hw = H1w + H2w and x · H2w=0. Therefore, +∥⟨x⟩w(x, t)∥L1 � L∞ ≤ ∥⟨x⟩w0(x)∥L1 � L∞ + +� t +0 +∥wH1w(x, s)∥L1 � L∞ ds. +To bound the second term in the right hand side, we use (3.12) to estimate +∥wH1w∥L1 � L∞ ≤ ∥H1w∥L∞∥w∥L1 � L∞ +≲ ∥w∥L1 +1 +� L∞ +1 ∥w∥L1 � L∞, +which implies (recall that ∥w(·, t)∥Lp = ∥w0∥Lp since ∇ · Hw = 0) +∥w(·, t)∥L1 +1 +� L∞ +1 ≤ ∥w0∥L1 +1 +� L∞ +1 + C∥w0∥L1 � L∞ +� t +0 +∥w(·, s)∥L1 +1 +� L∞ +1 ds. + +28 +D. GUO AND L. ZHAO +So it follows from Gronwall’s inequality that ∥w(·, t)∥L1 +1 +� L∞ +1 ≲ CeCt for all t ≥ 0 +and hence the Beale-Kato-Majda criterion implies at least formally that solutions +to (1.4) are global. More precisely, we have +Theorem 5.1. For any w0 ∈ C∞ +c (R2), there exist a global Lagrangian weak solution +w(x, t) to (1.4) with initial vorticity w0. Furthermore, for any t ∈ [0, T ], there exists +a constant C depending only on T and ∥w0∥L1 +1 +� L∞ +1 (R2) such that +sup +t∈[0,T ] +∥w(·, t)∥L1 +1 +� L∞ +1 (R2) ≤ CT,w0. +Remark 5.2. By the same argument, one can show that for all T > 0, N ∈ N∗ and +t ∈ [0, T ], there exist some constant C depending only on N, T and ∥w0∥L1 +N +� L∞ +N (R2) +such that +sup +t∈[0,T ] +∥w(·, t)∥L1 +N +� L∞ +N (R2) ≤ CN,T,w0. +The proof of Theorem 5.1 is standard, which is sketched in the appendix. +5.2. Proof of the existence. In this subsection, we will show that for every +w0 ∈ L1 � L∞(R2), there exists a global Lagrangian weak solution to (1.4) with +initial data w0. +Proof of Theorem 1.1 (i). Let {w0,n} ∈ C∞ +c (R2) such that w0,n → w0 ∈ L1(R2) +and ∥w0,n∥L∞(R2) ≲ ∥w0∥L∞(R2). Let wn(·, t) be a sequence of Lagrangian solutions +to (1.4) with initial data w0,n. Then the fact ∇ · Hwn = 0 yields +∥wn(·, t)∥L1 � L∞(R2) = ∥w0,n∥L1 � L∞(R2) ≲ 1, +which implies that (using Proposition 3.3) +|H1wn(x, t)| ≲ ⟨x⟩ +and +|H2wn(x, t)| ≲ ⟨x⟩2. +Then we use (3.17), (3.18) and (3.38) to conclude that +|Hwn(x, t) − Hwn(z, t)| ≲ (⟨x⟩3 + ⟨z⟩3)F(|x − z|). +Thus, it follows from Lemma A.2 that for any R, T > 0, Xn(α, t) and X−t +n (α) are +uniform bounded and equicontinuous on BR × [0, T ]. So there exist X(α, t) and its +inverse map X−t(α) and a subsequence Xnk such that +Xnk(α, t) → X(α, t) +and +X−t +nk (α) → X−t(α) +uniformly in every compact set. Furthermore, for ant t ≥ 0, the map X−t(·) and +X(·, t) preserves Lebesgue measure. Now setting w(x, t) = w0(X−t(x)), we will +show that w(x, t) is the desired solution. To this end, first we write the subsequence +nk as n for simplicity and we claim that for every t ∈ [0, T ], +∥w(·, t) − wn(·, t)∥L1 → 0. +(5.1) +Indeed, since wn(·) is a Lagrangian weak solution, we have +∥w(·, t) − wn(·, t)∥L1 = +� +R2 |w0(X−t(x)) − w0(X−t +n (x)) dx. +Now for any ǫ > 0, let wc ∈ C∞ +c (R2) such that +∥w0 − wc∥L1 ≤ +ǫ +100. + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +29 +Then it follows that +∥w(·, t) − wn(·, t)∥L1 ≤ +� +R2 |w0(X−t(x)) − wc(X−t(x))| dx + +� +R2 |wc(X−t(x)) − wc(X−t +n (x))| dx ++ +� +R2 |wc(X−t +n (x)) − w0(X−t +n (x))| dx. +Observe that X−t(·) and X(·, t) preserves Lebesgue measure, so there holds +� +R2 |w0(X−t(x)) − wc(X−t(x))| dx = ∥w0 − wc∥L1 ≤ +ǫ +100. +Similarly, +� +R2 |w0(X−t +n (x)) − wc(X−t +n (x))| dx = ∥w0 − wc∥L1 ≤ +ǫ +100. +Then the dominating convergence theorem gives +� +R2 |wc(X−t(x)) − wc(X−t +n (x))| dx ≤ 97ǫ +100 +for n large enough since wc ∈ C∞ +c (R2) and X−t +n (x) converges to X−t(x) in compact +set. This completes the proof of the claim. Furthermore, it follows from (A.1) that +Hwn(x, t) → Hw(x, t) +(5.2) +for all x ∈ R2 and t ∈ R. Note that for any φ ∈ C∞ +c (R2), we have +� +wn(x, t)φ(x, t) dx − +� +wn(x, 0)φ(x, 0) dx = +� t +0 +� +R2 wn(∂tφ + Hwn · ∇φ) dxdt +since each wn is a weak solution. So with the help of (5.1) and (5.2), letting n → ∞ +we get the desired conclusion. +□ +6. Continuous dependence on initial data. +In this section we will prove the continuous dependence on initial data. First, we +prove the following simplified version. +Theorem 6.1. Let w0,n be a sequence of initial data such that +sup +n ∥w0,n∥L∞ +1 (R2) < ∞ +and +∥w0,n − w0∥L1 +1(R2) → 0. +Then for any t ≥ 0, one has +∥wn(t) − w(t)∥L1 +1(R2) → 0 +as n → ∞. +Proof. First we claim that for any R, T > 0, Xn(·, ·) → X(·, ·) uniformly in BR × +[0, T ]. Assume this is not true, then there exist R, T, δ > 0, (αk, tk) ∈ BR × [0, T ] +and a subsequence Xnk such that +|Xnk(αk, tk) − X(αk, tk)| ≥ δ > 0. +Observe that BR × [0, T ] is compact, so we may assume without loss of generality +that (αk, tk) → (α0, t0) ∈ BR × [0, T ]. Then by a similar argument as in section +6, there exists ˜X(α, t) such that Xnk → ˜X uniformly in compact set. Thus, if we +set w( ˜ +X(α, t), t) := w0(α), then ˜w is also a weak solution to (1.4) with initial data +w0. Meanwhile, from Corollary 4.7 we know that solutions to (1.4) is unique in +L1 +1 +� L∞ +1 (R2), so w = ˜w and hence X = ˜X. Thus, +X(α0, t0) − ˜X(α0, t0) = 0, + +30 +D. GUO AND L. ZHAO +which leads to a contradiction since +|X(α0, t0) − ˜X(α0, t0)| = lim +k→∞ |X(αk, tk) − Xnk(αk, tk)| ≥ δ > 0. +Therefore, Xn → X uniformly in compact set. +Next we show wn → w in L1 +1(R2). A direct calculation shows that +∥w(·, t) − wn(·, t)∥L1 +1(R2) = +� +R2⟨x⟩|w0,n(X−t +n (x)) − w0(X−t(x))| dx +≤ +� +R2⟨x⟩|w0,n(X−t +n (x)) − w0(X−t +n (x))| dx +(6.1) ++ +� +R2⟨x⟩|w0(X−t +n (x)) − w0(X−t(x))| dx. +(6.2) +For (6.1), we use (4.1) to conclude that for n large enough +� +R2⟨x⟩|w0,n(X−t +n (x)) − w0(X−t +n (x))| dx = +� +R2⟨Xn(α, t)⟩|w0,n(α) − w0(α)| dα +≲ +� +R2⟨α⟩|w0,n(α) − w0(α)| dα +≤ +ǫ +100 +since w0,n → w0 in L1 +1(R2). For (6.2), we choose w0,ǫ ∈ C∞ +c (R2) such that ∥w0,ǫ − +w0∥L1 +1(R2) ≤ δǫ for some δ small enough. Then it follows that +� +R2⟨x⟩|w0(X−t +n (x)) − w0(X−t(x))| dx ≤ +� +R2⟨x⟩|w0(X−t +n (x)) − w0,ǫ(X−t +n (x))| dx +(6.3) ++ +� +R2⟨x⟩|w0,ǫ(X−t +n (x)) − w0,ǫ(X−t(x))| dx +(6.4) ++ +� +R2⟨x⟩|w0,ǫ(X−t(x)) − w0(X−t(x))| dx. +(6.5) +Recall that X−t(·) and X−t +n (·) preserves Lebesgue measure, so in view of (6.1), the +integral in (6.3) and (6.5) can be bounded by +ǫ +100 once δ is taken small enough. For +(6.4), dominating convergence theorem then gives +� +R2⟨x⟩|w0,ǫ(X−t +n (x)) − w0,ǫ(X−t(x))| dx ≤ +ǫ +100 +for n large enough and hence the proof of the theorem is complete. +□ +Next we prove Theorem 1.1 (iii). That is, for any T > 0, there holds +sup +t≤T +∥wn(t) − w(t)∥L1 +1(R2) → 0 +as n → ∞. +Proof. Assume this is not true, then there exists a subsequence of wn(which we still +denote it by wn), tn ∈ [0, T ] and δ > 0 such that +∥wn(tn) − w(tn)∥L1 +1(R2) > δ. + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +31 +We may assume without loss of generality that tn → t0 ∈ [0, T ] as n → ∞ since +[0, T ] is compact. Then we estimate +∥wn(tn) − w(tn)∥L1 +1(R2) = +� +R2⟨x⟩|wn(x, tn) − w(x, tn)| dx += +� +R2⟨x⟩|w0,n(X−tn +n +(x)) − w0(X−tn(x))| dx +≤ +� +R2⟨x⟩|w0,n(X−tn +n +(x)) − w0(X−tn +n +(x))| dx +(6.6) ++ +� +R2⟨x⟩|w0(X−tn +n +(x)) − w0(X−tn(x))| dx. +(6.7) +For (6.6), in view of (4.1), we obtain +� +R2⟨x⟩|w0,n(X−tn +n +(x)) − w0(X−tn +n +(x))| dx = +� +R2⟨Xn(α, tn)⟩|w0,n(α) − w0(α)| dα +≲ +� +R2⟨α⟩|w0,n(α) − w0(α)| dα → 0. +To bound the term in (6.7), we choose wǫ ∈ C∞ +c (R2) such that ∥w0 − wǫ∥L1 +1(R2) ≤ +δ +100. Then by a similar argument as in (6.3), (6.4) and (6.5), there holds +� +R2⟨x⟩|w0(X−tn +n +(x)) − w0(X−tn(x))| dx ≤ +� +R2⟨x⟩|w0(X−tn +n +(x)) − wǫ(X−tn +n +(x))| dx ++ +� +R2⟨x⟩|wǫ(X−tn +n +(x)) − wǫ(X−tn(x))| dx ++ +� +R2⟨x⟩|wǫ(X−tn(x)) − w0(X−tn(x))| dx +≤ 3δ +100 +for n large enough, which contradicts the fact that +∥wn(tn) − w(tn)∥L1 +1(R2) > δ. +Thus the proof of the whole theorem is complete. +□ +Appendix A. +Proposition A.1. For any w ∈ L1 � L∞(R2), it holds that ∇ · Hw = 0 in the +sense of distribution, where +Hw(x) := +� +H(x, y)w(y) dy +and the kernel H(x, y) is given in (2.13). +Proof. Assume w ∈ C∞ +c (R2), then ∇ · Hw = 0 by a direct calculation as in Section +2. Now for general w ∈ L1 � L∞(R2), we choose a sequence of wǫ ∈ C∞ +c (R2) such +that wǫ → w in L1 and |wǫ(x)| ≲ 1. Then for any R > 0, it follows from Corollary +3.2 that +|Hw(x)| ≲ ⟨x⟩2 +� � +1 + +1 +|x − y| +� +|w(y)| dy +≲ ⟨x⟩2 +� +∥w∥L1 + +� +|x−y|>R +|w(y)| +|x − y| dy + +� +|x−y|≤R +|w(y)| +|x − y| dy +� +≲ ⟨x⟩2 +� +∥w∥L1 + ∥w∥L1 +R ++ R∥w∥L∞ +� +, + +32 +D. GUO AND L. ZHAO +which implies +|Hw(x)| ≲ ⟨x⟩2 � +∥w∥L1 + ∥w∥1/2 +L1 ∥w∥1/2 +L∞ +� +(A.1) +once we take R = +∥w∥1/2 +L1 +∥w∥1/2 +L∞ . Therefore, Hwǫ(x) → Hw(x) uniformly in every compact +set since H is a linear operator. Hence for any φ ∈ C∞ +c (R2), we have +� +Hw(x) · ∇φ(x) dx = lim +ǫ→0 +� +Hwǫ(x) · ∇φ(x) dx = 0. +□ +Next we will show that the particle trajectory map of Hw preserves Lebesgue +measure. Before the proof, we need the following technical lemma. +Lemma A.2. Let {Un(x, t)}∞ +n=1 be a sequence of vector fields in Rd satisfying +(i) +���Un(x, t) · +x +|x| +��� ≲ ⟨x⟩. +(ii) supx∈BR |Un(x, t)| ≲R 1. +(iii) sup|x|,|z|≤R |Un(x, t) − Un(z, t)| ≲R F(|x − z|). +Let Xn be the particle trajectory map of Un, then for any R, T > 0, Xn is uniformly +bounded and equicontinuous in BR ×[0, T ]. Similarly, let X−t +n (·) be the inverse map +of Xn(·, t), then X−t +n (x) is uniformly bounded and equicontinuous in BR ×[0, T ] for +any R, T > 0. +Proof. For any |α| ≤ R and t ≤ T , it is easy to check that +|Xn(α, t)| = |α| + +� t +0 +Xn(α, s) +|Xn(α, s)| · Un(Xn(α, s), s) ds +≲ |α| + +� t +0 +|Xn(α, s)| + 1 ds +≲ R + T + +� t +0 +|Xn(α, s)| ds. +So Gronwall’s implies that Xn is uniformly bounded in BR × [0, T ]. In order to +show the equicontinuity of Xn, we use assumption (iii) to estimate +|Xn(α, t) − Xn(β, t)| = +� t +0 +Xn(α, s) − Xn(β, s) +|Xn(α, s) − Xn(β, s)| (Un(Xn(α, s), s) − Un(Xn(β, s), s)) ds +≲ +� t +0 +F (|X(α, s) − X(β, s)|) ds. +Next we set z(t) = |Xn(α, t)−Xn(β, t)| and assume that z(t) ≤ +1 +100 for all t ∈ [0, T ], +then it follows that +z(t) ≤ C +� t +0 +−z(s)(log z(s)) ds. +Thus by comparison theorem we have +z(t) ≤ ee−Ct log(z(0)) +≤ ee−CT log(z(0)) += ee−CT log |α−β|. +So if we take |α−β| ≤ e−10e−CT , then z(t) ≤ e−10 ≤ +1 +100 for all t ∈ [0, T ] and hence +|Xn(α, t) − Xn(β, t)| ≤ ee−CT log |α−β|. + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +33 +Next we estimate +|Xn(α, t1) − Xn(α, t2)| = +���� +� t2 +t1 +Un(Xn(α, s), s) ds +���� . +Recall that Xn is uniformly bounded, so there exists M > 0 such that |Xn(α, t)| ≤ +M for all α ∈ BR(0) and t ∈ [0, T ]. Together with the fact that Un(·, t) is uniformly +bounded in BM × [0, T ], we finally obtain +|Xn(α, t1) − Xn(α, t2)| ≲ +� t2 +t1 +1 ds ≲ |t2 − t1|. +Therefore, for any δ > 0 and n ∈ N ∗, there exists ǫ0 = ǫ0(R, T, δ) such that for all +α, β ∈ BR, t1, t2 ∈ [0, T ] with |α − β| + |t2 − t1| ≤ ǫ0, there holds +|Xn(α, t1) − Xn(β, t2)| ≤ δ. +The estimates for the inverse map X−t +n (x) is similar, see [19] for references. +□ +Now we can show that the particle trajectory map of Hw preserves Lebesgue mea- +sure. More generally, we prove the following. +Lemma A.3. Let U(x, t) be a velocity field in Rd × [0, T ] with +(i) |U(x, t)| ≤ C⟨x⟩2 for some M > 0. +(ii) +���U(x, t) · +x +⟨x⟩ +��� ≤ C⟨x⟩. +(iii) U is locally Log-Lipschitz continuous. +(iv) ∇ · U = 0 in the sense of distribution. +Let X(α, t) be the particle trajectory map of U, then the map X(·, t) : Rd → Rd is +bijective and preserves Lebesgue measure. +Proof. Let η be a smooth positive function supported in B1 and +� +Rd η(x) dx = 1. +Define +ηǫ(x) = 1 +ǫd η(x +ǫ ) +and +Uǫ(x, t) := ηǫ ∗ U(x, t) = +� +Rd ηǫ(x − y)U(y, t) dy. +Then it is easy to check that for all ǫ ≤ 1, +|Uǫ(x, t)| ≲ ⟨x⟩2. +Moreover, +����Uǫ(x) · x +⟨x⟩ +���� = +� +Rd ηǫ(x − y)U(y) · y +⟨y⟩ dy + +� +Rd ηǫ(x − y)U(y) · ( x +⟨x⟩ − y +⟨y⟩) dy +≲ ⟨x⟩ + ⟨x⟩2 +� +Rd |x − y|ηǫ(x − y) dy +≲ ⟨x⟩ + ǫ⟨x⟩2 +(A.2) + +34 +D. GUO AND L. ZHAO +and +sup +|x|,|z|≤R +|Uǫ(x, t) − Uǫ(z, t)| ≲ +� +Rd ηǫ(y)|Uǫ(x − y, t) − Uǫ(z − y, t)| dy +≲ +� +Rd ηǫ(y) +sup +|x|,|z|≤R+1 +|U(x, t) − U(z, t)| dy +≲R +� +Rd ηǫ(y)F(|x − z|) dy +≲R F(|x − z|). +Next we estimate the particle trajectory map Xǫ(α, t) associated with Uǫ(x, t): +⟨Xǫ(α, t)⟩ = ⟨α⟩ + +� t +0 +Xǫ(α, s) +⟨Xǫ(α, s)⟩ · Uǫ(Xǫ(α, s), s) ds. +Setting zǫ(t) = ⟨Xǫ(α, t)⟩, then it follows from (A.2) that +zǫ(t) ≤ ⟨α⟩ + C +� t +0 +zǫ(s) + ǫzǫ(s)2 ds. +Assume zǫ(t) ≤ D1eD2t for all t ∈ [0, T ] and ⟨α⟩ ≤ R, then the above inequality +implies that +zǫ(t) ≤ ⟨α⟩ + CD1 +D2 +(eD2t − 1) + ǫCD2 +1 +2D2 +(e2D2t − 1) +≤ eD2t +� +R + 1 + CD1 +D2 ++ ǫCD2 +1 +2D2 +eD2T +� +. +Thus, if we take D2 = 100(⟨C⟩ + ⟨R⟩) and D1 = 100R + 100, then there exists +ǫ0 = ǫ0(R) such that +z(t) ≤ D1eD2t +2 +for all ǫ ≤ ǫ0. Therefore, a bootstrap argument then gives +⟨Xǫ(α, t)⟩ ≲ ⟨α⟩ +for all ǫ ≤ ǫ0(R) and ⟨α⟩ ≤ R. Furthermore, by a similar argument as in Lemma +A.2, we see that for all R > 0, there exists a positive constant CR such that for +all ǫ ≤ ǫ0(R), Xǫ(α, t) and X−t +ǫ (α) are uniformly bounded and equicontinuous in +BR × [0, T ] with |X(α, t)|, |X−t(α)| ≤ CR. Now we take ǫ ≤ ǫ0(CR) and choose +subsequence Xn(α, t) and X−t +n (α) converges uniformly to X(α, t) and X−t(α) in +BCR × [0, T ] respectively. +Note that +Xn(α, t) = +� t +0 +Un(Xn(α, s), s) ds, +Taking n → ∞ we get +X(α, t) = +� t +0 +U(X(α, s), s) ds, +which implies that X is the particle trajectory map of U for ⟨α⟩ ≤ R. Moreover, +X(X−t(α), t) = lim +n Xn(X−t +n (α), t) = α +for all ⟨α⟩ ≤ R. Thus, X−t is the inverse map of X(·, t) at least for ⟨α⟩ ≤ R. Now +it remains to show that X(·, t) and X−t(·) preserves Lebesgue measure. Recall that +∇·U = 0 in the sense of distribution, so Uǫ is a smooth velocity field with ∇·Uǫ = 0 +and hence Xǫ(·, t) and X−t +ǫ (·) preserves Lebesgue measure. Thus the dominating +convergence theorem yields +m(X(O, t)) = m(O) = m(X−t(O)) + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +35 +for all measurable set O ⊂ BR. Since R is arbitrary, X(·, t) and X−t(·) preserves +Lebesgue measure. In other words, +� +Rd f(X(α, t)) dα = +� +Rd f(x) dx +for all f ∈ L1(Rd). This completes the proof. +□ +Lemma A.4. Every weak solution to (1.4) in L∞([0, T ], L1 +1 +� L∞ +1 ) is indeed la- +grangian. +Proof. Assume w ∈ L∞([0, T ], L1 +1 +� L∞ +1 ), then the argument in Section 3 implies +that |Hw(x, t)| ≲ ⟨x⟩2 and Hw(·, t) is locally log-lip continuous. Set U = Hw and +consider the continuity equation +∂tψ + ∇ · (ψU) = 0. +We say that ψ is a weak solution of the continuity equation on [0,T) with initial +data w0 if for all test function φ ∈ C∞ +c ([0, T ) × R2), there holds +− +� +ψ(x, 0)φ(x, 0) dx = +� T +0 +� +R2 ψ(∂tφ + U · ∇φ) dxdt. +Therefore, one can check directly that the solution w(x, t) to the two-dimensional +helical equation (1.4) is also a weak solution to the continuity equation. Mean- +while, Lemma A.3 implies that ψ(x, t) = w0(X−t(x)) is also a weak solution of the +continuity equation. Note that the weak solutions to the continuity equation in +L1([0, T ), L1) is unique, so w(x, t) = w0(X−t(x)) and hence all weak solutions to +(1.4) is Lagrangian. +□ +Appendix B. +We now give the proof of Theorem 5.1. +Proof of the existence. Recall that solutions to (1.4) is unique in L1 +1 +� L∞ +1 (R2), so +it suffices to prove the existence of weak solutions in [0, T ]. Now we fix T > 0 +and let w0 ∈ C∞ +c (R2). Assume without loss of generality that w0 ̸= 0 and set +w0(x, t) := w0(x). Now we can define wn(x, t) inductively: +� +∂twn+1 + Hwn · ∇wn+1 =0 +wn+1(x, 0) =w0(x). +Multiply both side by ⟨x⟩, then we obtain +∂t⟨x⟩wn+1 + Hwn · ∇(⟨x⟩wn+1) =wn+1Hwn · ∇⟨x⟩ +=wn+1H1wn · x +⟨x⟩ +since Hw = H1w + H2w and H2w · x = 0. Note that ∇ · Hwn = 0, so a direct +calculation shows that +∥wn(·, t)∥Lp(R2) = ∥w0∥Lp(R2) +for any p ∈ [0, ∞]. Moreover, +∥⟨x⟩wn+1(·, t)∥L1 � L∞(R2) +≤∥⟨x⟩w0∥L1 � L∞(R2) + +� t +0 +∥wn+1Hwn∥L1 � L∞(R2) ds +≤∥⟨x⟩w0∥L1 � L∞(R2) + +� t +0 +∥wn+1∥L∞(R2)∥Hwn∥L1 � L∞(R2) ds +≤∥⟨x⟩w0∥L1 � L∞(R2) + ∥w0∥L∞(R2) +� t +0 +∥Hwn∥L1 � L∞(R2) ds. + +36 +D. GUO AND L. ZHAO +Thus, +∥wn(t)∥L1 +1 +� L∞ +1 (R2) ≲ 1 +for all t ∈ [0, T ]. Together with (3.12) we get +|Hwn(x)| ≲ ⟨x⟩ +and +����Hwn(x) · x +⟨x⟩ +���� ≲ 1, +□ +which implies that ⟨Xn(α, t)⟩ ≈ ⟨α⟩ and hence there exists M > 0 such that wn(·, t) +supported in BM(0) for all n > 0 and t ∈ [0, T ]. Furthermore, Lemma 3.4 yields +|Hwn(x, t) − Hwn(z, t)| ≲ (⟨x⟩ + ⟨z⟩)F(|x − z|) +for any R > 0 and t ∈ [0, T ]. So in view of Lemma A.3, there exist X(α, t), ˜X(α, t) +and a subsequence nk such that Xnk and Xnk−1 converges uniformly to X and +˜X, respectively in compact subset of R2 × [0, T ]. Note that wnk+1 satisfies the +transport equation +� +∂twnk+1 + Hwnk · ∇wnk+1 =0 +wnk+1(x, 0) =w0(x). +So letting k → ∞, by a similar argument as in section 5, we find that w satisfies +� +∂tw + H ˜w · ∇w =0 +w(x, 0) =w0(x). +Here w(x, t) := w0(X−t(x)) and ˜w(x, t) := w0( ˜X−t(x)). To complete the proof, it +remains to show that w ≡ ˜w, which is equivalent to X(α, t) = ˜X(α, t) for every α ∈ +supp w0. Therefore, we define the distance +Dn(t) := +� +|Xn+1(α, t) − Xn(α, t)||w0(α)| dα. +Observe that Dn(t) is uniformly bounded since w0 has compact support. +So +by dominating convergence theorem, it suffices to show that for any t ∈ [0, T ], +limn→∞ Dn(t) = 0. First we use Newton-Leibniz formula to conclude that +Dn(t) ≤ +� t +0 +� ��Hwn+1(Xn+1(α, s), s) − Hwn(Xn(α, s), s) +�� |w0(α)| dα ds +≤ +� t +0 +� ��Hwn+1(Xn+1(α, s), s) − Hwn+1(Xn(α, s), s) +�� |w0(α)| dα ds (B.1) ++ +� t +0 +� ��Hwn+1(Xn(α, s), s) − Hwn(Xn(α, s), s) +�� |w0(α)| dα ds. +(B.2) +For (B.1), inequality (3.17), (3.19) and Lemma 3.4 yields +� t +0 +� ��Hwn+1(Xn+1(α, s), s) − Hwn+1(Xn(α, s), s) +�� |w0(α)| dα ds +≲ +� t +0 +F(Dn+1(s)) ds, + +THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY +37 +where we have used the Jensen’s inequality and the fact that w0 ̸= 0. For (B.2), +note that Xn(·, t) preserves Lebesgue measure, so it follows that +� t +0 +� ��Hwn+1(Xn(α, s), s) − Hwn(Xn(α, s), s) +�� |w0(α)| dα ds += +� t +0 +� ��Hwn+1(x, s) − Hwn(x, s) +�� |wn+1(x, s)| dx ds += +� t +0 +� ���� +� +H(x, y)wn+1(y, s) dy − +� +H(x, y)wn(y, s) dy +���� |wn+1(x, s)|dxds. +Recall that wn(Xn−1(α, t), t) = w0(α), so we have +� t +0 +� ��Hwn+1(Xn(α, s), s) − Hwn(Xn(α, s), s) +�� |w0(α)| dα ds += +� t +0 +� ���� +� � +H(x, Xn(β, s)) − H(x, Xn−1(β, s)) +� +|w0(β)| dβ +���� |wn+1(x, s)|dxds. +Thus, (2.10), (2.13), (2.14), (2.15), (3.17), (3.20) and Lemma 3.4 yields +� t +0 +� ��Hwn+1(Xn(α, s), s) − Hwn(Xn(α, s), s) +�� |w0(α)| dα ds +≲ +� t +0 +F(Dn(s)) ds +since w(·, t) supported in BM(0) for all t ∈ [0, T ]. Gathering the estimates above, +we arrive at +Dn(t) ≲ +� t +0 +F(Dn(s)) ds + +� t +0 +F(Dn−1(s)) ds. +Let ˜Dn(t) := supm≥n Dm(t), then it follows that +˜Dn(t) ≲ +� t +0 +F( ˜Dn−1(s)) ds. +Setting D∗(t) := limn→∞ ˜Dn(t) and letting n → +∞, we have +D∗(t) ≲ +� t +0 +F(D∗(s)) ds, +which implies D∗ ≡ 0 by Osgood’s Lemma. Observe that Dn(t) ≤ ˜Dn(t), so we get +lim +n→∞ Dn(t) = 0 +and hence completes the proof. +References +[1] C. Bardos and E. S. Titi, Euler equations of incompressible ideal fluids, Russian Math. +Surveys, 62 (2007), pp. 409–451. +[2] J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the +3-D Euler equations, Comm. Math. Phys., 94 (1984), pp. 61–66. +[3] A. Bronzi, M. Lopes and H. Nussenzveig Lopes, Global existence of a weak solution of the +incompressible Euler equations with helical symmetry and Lp vorticity, Indiana Univ. Math. +J. 64 (1) (2015) 309–341. +[4] G. Cavallaro and C. Marchioro, Time evolution of vortex rings with large radius and very +concentrated vorticity. J. Math. Phys. 62, 053102, 20 pp. (2021). +[5] D. Chae, N. Kim, Axisymmetric weak solutions of the 3-D Euler equations for incompressible +fluid flows, Nonlinear Anal. 29 (1997) 1393–1404. +[6] D. Chiron, Vortex helices for the Gross-Pitaevskii equation. J. Math. Pures Appl. 84(11), +1555–1647 (2005) +[7] A. Clop, H. Jylh¨a, J. Mateu, and J. Orobitg, Well-posedness for the continuity equation for +vector fields with suitable modulus of continuity, J. Funct. Anal. 276 (2019), no. 1, 45–77 + +38 +D. GUO AND L. ZHAO +[8] G. Crippa and G. Stefani, An elementary proof of existence and uniqueness for the Euler +flow in localized Yudovich spaces. arXiv:2110.15648, 2021. +[9] J.-M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc., +4(3):553–586, 1991. +[10] R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the +incompressible fluid equations, Comm. Math. Phys., 108 (1987), pp. 667–689. +[11] R. J. Diperna and A. J. Majda, Concentrations in regularizations for 2-D incompressible +flow, Comm. Pure Appl. Math. 40 (3) (1987) 301–345. +[12] A. Dutrifoy, Existence globale en temps de solutions h´elico¨ıdales des ´equations d’Euler, C. +R. Acad. Sci. Paris S´er. I Math., 329 (1999), pp. 653–656. +[13] B. Ettinger and E. Titi, Global existence and uniqueness of weak solutions of three- +dimensional Euler equations with helical symmetry in the absence of vorticity stretching. +SIAM J. Math. Anal., 41(1):269–96, 2009. +[14] S. Gang and X. Zhu, Axisymmetric solutions to the 3D Euler equations, Nonlinear Anal., +66(9):1938–1948, 2007. +[15] T.Y. Hou, C. Li, Dynamic stability of the three-dimensional axisymmetric Navier–Stokes +equations with swirl, Comm. Pure Appl. Math. 61 (2008) 661–697 +[16] Q. Jiu, J. Li, D. Niu, Global existence of weak solutions to the three-dimensional Euler +equations with helical symmetry. J. Differ. Equ. 262, 5179–5205 (2017) +[17] Q. Jiu and Z. Xin, On strong convergence to 3D axisymmetric vortex sheets, J. Differential +Equations, 233(1):33–50, 2006. +[18] T. Kato, Nonstationay flows of viscous and ideal fluids in R3, J. Funct. Anal., 9:296–305, +1972. +[19] A. J. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure +Appl. Math., 39(S):S187–S220, 1986. +[20] X. Saint Raymond, Remarks on axisymmetric solutions of the incompressible Euler system, +Comm. Partial Differential Equations, 19(1–2):321–334, 1994. +[21] M. R. Ukhovskii and V. Yudovich, Axially symmetric flows of ideal and viscous fluids filling +the whole space, J. Appl. Math. Mech., 32 (1968), pp. 52–61. +[22] I. Vecchi and S. Wu, On L1-vorticity for 2-D incompressible flow, Manuscripta Math., +78(4):403–412, 1993. +[23] M. Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov +type, Ann. Sci. Ecole Norm. Sup. (4), 32 (1999), pp. 769–812. +[24] M. Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of +an ideal incompressible fluid. Part I, arXiv:1805.09426 (2018) +[25] M. Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of +an ideal incompressible fluid. Part II, arXiv:1805.09440 (2018) +[26] V. Yudovich, Nonstationary flow of an ideal incompressible liquid. Zh. Vych. Mat. 3 (1963), +1032–1066. +[27] V. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an +ideal incompressible fluid, Math. Res. Lett., 2 (1995), pp. 27–38. +School of Mathematical Sciences, University of Science and Technology of China, +Hefei 230026, Anhui, China +Email address: guodeng@mail.ustc.edu.cn +School of Mathematical Sciences, University of Science and Technology of China, +Hefei 230026, Anhui, China +Email address: zhaolf@ustc.edu.cn + diff --git a/qdAzT4oBgHgl3EQfcPxh/content/tmp_files/load_file.txt b/qdAzT4oBgHgl3EQfcPxh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a53df02d61fda4cea7e2b5005aafd2bceaab48a --- /dev/null +++ b/qdAzT4oBgHgl3EQfcPxh/content/tmp_files/load_file.txt @@ -0,0 +1,1217 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf,len=1216 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='01399v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='AP] 4 Jan 2023 GLOBAL WELL-POSEDNESS OF WEAK SOLUTIONS TO THE INCOMPRESSIBLE EULER EQUATIONS WITH HELICAL SYMMETRY IN R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' DENGJUN GUO AND LIFENG ZHAO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We use a Lagrangian method to prove the global well-posedness of weak solutions to the three-dimensional Euler equations with helical sym- metry and without helical swirl in the whole space for initial vorticity in L1 1 � L∞ 1 (R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The vortex transport formula is also obtained in our article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Introduction We consider the three-dimensional incompressible Euler equation in R3, \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∂tU + U · ∇U + ∇P = 0 ∇ · U = 0 U(x, 0) = U0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) The equation describes the motion of an ideal incompressible fluid in R3 with initial velocity U0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Here U = (U 1, U 2, U 3) : R3 × R → R3 represents the velocity and P : R3 × R → R represents the scalar pressure which can be determined by the incompressibility condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The vorticity of the fluid is defined by Ω(x, t) := ∇ ∧ U(x, t), where (a1, a2, a3) ∧ (b1, b2, b3) := (a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Moreover, the vorticity satisfies � ∂tΩ + U · ∇Ω + Ω · ∇U = 0, Ω(x, 0) = ∇ ∧ U0(x) := Ω0(x), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) which is called the vorticity-stream formulation of the Euler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Here the velocity U can be recovered from Ω by the well-known Biot-Savart law U(x) = 1 4π � R3 x − y |x − y|3 ∧ Ω(y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The local well-posedness of the three-dimensional Euler equation has been studied in [19] for initial data U0 ∈ Hm and m ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' However, the global existence of smooth solutions to the three-dimensional Euler equation remains open due to the strong nonlinearity of the vorticity stretching term Ω · ∇U, see [1] or [19] for more references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Unlike the three-dimensional case, the two-dimensional incompressible Euler equa- tion is global well-posed in L1 � L∞ due to the absence of the vorticity stretching term, see [19] and [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Moreover, the two-dimensional incompressible Euler equa- tion can sometimes be viewed as a transport equation and the vortex transport formula holds for solutions with bounded L1 � L∞ norm, see [19] for references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Zhao is supported by NSFC Grant of China No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 12271497 and the National Key Research and Development Program of China No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 2020YFA0713100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Data Availability Statements: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 1 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO Yudovich [27] extended the uniqueness results to the solutions whose Lp norm grows slowly as p → ∞ and Vishik [23] extended the results to Besov type spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The existence of global solutions to the two-dimensional Euler equation with Lp initial vorticity has been established by Majda and Diperna in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Later, De- lort proved the global existence for measure valued initial data [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' However, the uniqueness might not hold for Lp initial data, we refer to the works by Vishik [24] and [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Some three-dimensional flows with special symmetries can be reduced to two- dimensional flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Among the most important examples are axisymmetric flows and helical flows without swirl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The global existence and uniqueness of axisymmetric solutions was obtained by Yudovich in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For helical solutions, in [12] Dutrifoy proved the global well-posedness for smooth solutions in bounded domains, an key observation is that the third component Ωz of the vorticity Ω satisfies ∂tΩz + U · ∇Ωz = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) which shows that the vorticity is transported by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Setting w(x1, x2) = Ωz(x1, x2, 0), Ettinger and Titi reduced the three-dimensional Euler equation to the following two-dimensional problem: ∂tw + ∇⊥L−1 H w · ∇w = 0, where LH is a specific elliptic operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' They also establish the global existence and uniqueness for this two-dimensional problem in bounded domains [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For the whole space R3, the global existence of weak solutions has been obtained by Bronzi–Lopes–Lopes [3] for L1 � Lp(R3) initial vorticity with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The assumption of the compact support was then removed by Jiu, Li and Niu [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' However, for Lp solutions, the uniqueness of the solutions and the continuous dependence on initial data is still an open problem in the whole space R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Moreover, the vortex transport formula is still unknown even in bounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In this paper, we consider the three-dimensional incompressible Euler equation with helical symmetry and without swirl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let Ωz be a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Setting w(x1, x2, t) = Ωz(x1, x2, 0, t), then w satisfies the two-dimensional helical Euler equation ∂tw + Hw · ∇w = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) where Hw(x) = � R2 H(x, y)w(y) dy and H(x, y) is the modified Biot-Savart kernel defined in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The main result are stated as follows (for the precise definition of Lagrangian weak solutions and the function spaces, we refer the reader to section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) is globally well-posed in L1 1 � L∞ 1 (R2) : (i) (Existence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For all T > 0 and w0 ∈ L1 � L∞(R2), there exists a La- grangian weak solution w(x, t) ∈ L∞ � [0, T ], L1 � L∞(R2) � to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' More- over, the velocity Hw(·, t) is locally Log-Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (ii) (Uniqueness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For any w0 ∈ L1 1 � L∞ 1 (R2), there exists at most one la- grangian weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) in L∞ � [0, T ], L1 1 � L∞ 1 (R2) � with initial vorticity w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (iii) (Continuous dependence on initial data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let w0,n be a sequence of initial data such that sup n ∥w0,n∥L∞ 1 (R2) < ∞ THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 3 and ∥w0,n − w0∥L1 1(R2) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let w(t) and wn(t) be the solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) with initial data w0 and w0,n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then for any T > 0, there holds sup t≤T ∥wn(t) − w(t)∥L1 1(R2) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2 (Vortex transport formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The velocity field Hw is divergence free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So in view of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4, all weak solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) in L∞ � [0, T ], L1 1 � L∞ 1 (R2) � are indeed Lagrangian and hence the assumption of Lagrangian in (ii) can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In other words, all weak solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) satisfies the vortex transport formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' With the help of the vortex transport formula, the solution w to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) with initial data in L1 1 � L∞ 1 (R2) belongs to C � [0, T ], L1 1(R2) � Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' By a similar argument, one can also obtain the global well-posedness for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) in L1 m � L∞ m(R2) for any m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' When m = 0, we only prove the existence of a global solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Indeed, the continuous dependence on initial data holds by the same argument in section 6 once we obtain the uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' However, our method can not be applied to prove the uniqueness when m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The main difficulty is that Hw(x) is Log-Lipschitz continuous when and m = 1 while only locally Log-Lipschitz continuous when m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For locally Log-Lipschitz continuous velocity field, Clop, Jylh¨a, Mateu, and Orobitg proved the uniqueness for continuity equations and some techniques of optimal transport is needed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We refer the reader to [7] for references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1, we also obtain the well-posedness for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3), which is much easier than (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) since the velocity remains bounded there while the velocity may grows linearly in two-dimensional cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5 (Global well-posedness for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The three-dimensional helical Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) is globally well-posed in L1 1 � L∞ 1 (R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Next we state our strategy of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The key ingredients of our proof is to give a detailed estimates for the modified Biot-Savart kernel H(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Once the estimates for the difference |H(x, y)−H(z, y)| be established, the existence of weak Lagrangian solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) then follows by a similar argument as in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We can not obtain the uniqueness for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) directly since the velocity field Hw(x) might growth linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Instead, we show that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) are equivalent and then it suffices to prove the uniqueness for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Finally, the continuous dependence on initial data follows by the vortex transport formula and a compactness argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Our article is organized as follows: In section 2, we give a detailed description of our problems and reduce the three-dimensional incompressible Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2)(for initial velocity with helical symmetry and without helical swirl) to the two-dimensional helical Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In section 3, we obtain a detailed estimates for the modified Biot-Savart kernel H(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In section 4, we show the uniqueness of the helical Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) in L1 1 � L∞ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In section 5, we obtain the global existence and the vortex transport formula for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In section 6, we prove the continuous dependence on initial data to equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Mathematical preliminaries and main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Our main purpose of this section is to fix notations and state our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For simplicity of presentation, we usually refer a point x = (x1, x2, x3) ∈ R3 or R2 × T to x = (x′, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Helical functions and vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1 (helical function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' A function f : R3 → R is called helical if for all θ ∈ R, f(Sθx) = f(x) for almost every x ∈ R3, where Sθx := Rθx + \uf8eb \uf8ed 0 0 θ \uf8f6 \uf8f8 and Rθ := \uf8eb \uf8ed cos θ sin θ 0 − sin θ cos θ 0 0 0 1 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2 (helical vector field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' A vector field u : R3 → R3 is called helical if for all θ ∈ R, R−θu(Sθ(x)) = u(x) for almost every x ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume f is a continuous helical function and u is a continuous helical vector field, then by the definition above we see that f(x′, x3) = f(R−x3x′, 0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) and u(x′, x3) = Rx3u(R−x3x′, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) Now if f and u are only measurable function (vector field), then f(x′, 0) and u(x′, 0) are not well-defined, thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) do not make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' However, we have the following: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let f be a locally bounded helical function and u be a locally bounded helical vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Define g(x1, x2) = 1 2π � 2π 0 f(Rax′, a) da and v(x1, x2) = 1 2π � 2π 0 R−au(Rax′, a) da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then for almost every x ∈ R3, there hold f(x′, x3) = g(R−x3x′) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) and u(x′, x3) = Rx3v(R−x3x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We only prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) since the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5) is similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Setting Ar = � Br ���f(x′, x3) − g(R−x3x′) ��� dx′dx3, then it suffice to show that Ar = 0 for all r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' By the definition of g, we obtain Ar = � Br ����f(x′, x3) − 1 2π � 2π 0 f(Ra−x3x′, a) da ���� dx ≤ 1 2π � Br � 2π 0 |f(x′, x3) − f(Ra−x3x′, a)| da dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 5 Next we make a change of variable a = b + x3, then it follows that Ar ≤ 1 2π � Br � 2π−x3 −x3 |f(x′, x3) − f(Rbx′, b + x3)| db dx ≤ 1 2π � Br � 2π+r −r |f(x′, x3) − f(Rax′, a + x3)| da dx = 1 2π � 2π+r −r � Br |f(x′, x3) − f(Sa(x′, x3))| dx da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1 we get Ar ≡ 0 and hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Motivated by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3), by slight abuse of notation, we may refer to g(x1, x2) as f(x′, 0), and refer to v(x1, x2) as u(x′, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Next we derive some useful properties for helical functions and vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let f ∈ L∞ loc(R3) be a helical function, then ∂3f = x1∂2f − x2∂1f (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6) in the sense of distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In [13], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6) has been shown when f is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For general f ∈ L∞ loc(R3), we define f0(x1, x2) := f(x1, x2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let f0,n ∈ C∞ c (R2) be a sequence of smooth function converges to f0 in L1 loc(R2) and set fn(x′, x3) := f0,n(R−x3x′), then for any φ ∈ C∞ c (R3), it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) and Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1 that � R3 f(x)∂3φ(x) dx = � R3 f0(R−x3x′)∂3φ(x) dx = � R3 f0(x1 cos x3 − x2 sin x3, x1 sin x3 + x2 cos x3)∂3φ(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Since φ has compact support, the dominating convergence theorem then gives � R3 f(x)∂3φ(x) dx = lim n � R3 f0,n(x1 cos x3 − x2 sin x3, x1 sin x3 + x2 cos x3)∂3φ(x) dx = lim n � R3 fn(x)∂3φ(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Note that fn is a smooth helical function, so (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6) and integrating by parts yields � R3 f(x)∂3φ(x) dx = lim n � R3 −∂3fn(x)φ(x) dx = lim n � R3(x2∂1fn − x1∂2fn)φ(x) dx = lim n � R3 fn(x1∂2φ(x) − x2∂1φ(x)) dx = lim n � R3(x1∂2φ(x) − x2∂1φ(x))f0,n(x1 cos x3 − x2 sin x3, x1 sin x3 + x2 cos x3) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Again by dominating convergence theorem, we finally get � R3 f(x)∂3φ(x) dx = � R3(x1∂2φ(x) − x2∂1φ(x))f0(x1 cos x3 − x2 sin x3, x1 sin x3 + x2 cos x3) dx = � R3 f(x) (x1∂2φ(x) − x2∂1φ(x)) dx, which proves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ For helical vector fields, by [13] and a similar argument as above, we obtain 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let u = (u1, u2, u3) ∈ L1 loc be a helical vector field, then we have ∂3u1 = x1∂2u1 − x2∂1u1 + u2, ∂3u2 = x1∂2u2 − x2∂1u2 − u1, and ∂3u3 = x1∂2u3 − x2∂1u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume u is smooth, then above equations have already been proved in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For general u ∈ L1 loc, the proof is only a matter of smoothness as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4 so we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Next we will describe the particle trajectory map associated with a Log-Lipschitz continuous helical vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' A function (vector field) u is called Log-Lipschitz continuous if sup x̸=y |u(x) − u(y)| F(|x − y|) < ∞, where F is the Log-Lipschitz function F(r) = � r(1 − log r) r ≤ 1 e r + 1 e r > 1 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='8) Note that F ′(r) = − log r if r ≤ 1 e and F ′(r) = 1 if r > 1 e, so F ′(r) is continuous and decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, F(r) is a concave function which satisfies F(r) ≈ r(1 − log− r), where log− r = � log r r ≤ 1 0 r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let u : Rd × [0, T ) → Rd be a locally Log-Lipschitz continuous velocity field, we say X(α, t) is a particle trajectory map associated with u if it satisfies \uf8f1 \uf8f2 \uf8f3 dX(α, t) dt = u(X(α, t), t) X(α, 0) = α for any α ∈ Rd and t ∈ [0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume that the velocity field is helical, then we have Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let u be a locally Log-Lipschitz continuous helical vector field and X(α, t) be its associated particle trajectory map, then for any α ∈ R3, t ∈ R+ and θ ∈ R, it holds that S−θX(Sθα, t) = X(α, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' A direct calculation shows that d (S−θX(Sθα, t)) dt = d (R−θX(Sθα, t)) dt = R−θu(X(Sθα, t), t) = u(S−θX(Sθα, t), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, both S−θX(Sθα, t) and X(α, t) are the solutions of \uf8f1 \uf8f2 \uf8f3 dX(α, t) dt = u(X(α, t), t) X(α, 0) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 7 So the uniqueness of the particle trajectory map implies SθX(S−θα, t) = X(α, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Next we summarize the following results for helical vector fields without helical swirl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let ξ(x) = (x2, −x1, 1), then the helical swirl uξ of a vector field u is defined by uξ := x2u1 − x1u2 + u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10 ([13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let U0 be a smooth helical vector field without helical swirl and U(x, t) be the smooth helical solution of the three-dimensional Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) with initial data U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then we have Uξ(x, t) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='11 ([13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let Ω = (Ωx, Ωy, Ωz) be a smooth solution of the three- dimensional Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) without swirl, then Ω = (x2Ωz, −x1Ωz, Ωz) = Ωzξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Moreover, Ωz is a helical function which satisfies the three-dimensional helical Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Reduce the three-dimensional Euler equation to a two-dimensional problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In this subsection, we will reduce(formally) the three-dimensional Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) to the two-dimensional helical equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' More precisely, we have the following: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='12 (Two-dimensional helical Euler equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let Ω = (Ωx, Ωy, Ωz) be a smooth solution to the three-dimensional helical Euler equation without swirl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Set w(x1, x2) = Ωz(x1, x2, 0), K(x, y) = 1 4π � R (x1, x2, 0) − (Ra(y), a) |(x1, x2, 0) − (Ra(y), a) |3 ∧ ξ((Ra(y), a)) da, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10) U(x1, x2) = � R2 K(x, y)w(y) dy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='11) and Hw := H1w + H2w = (U 1, U 2) + (−x2, x1)U 3, Then w satisfies the two-dimensional helical Euler equation ∂tw + Hw · ∇w = 0 with ∇ · Hw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let Ω = (Ωx, Ωy, Ωz) be a smooth helical solution to the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) without swirl, then Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='11 implies that Ωz is a helical function and satisfies ∂tΩz + U 1∂1Ωz + U 2∂2Ωz + U 3∂3Ωz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, it follow from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7) that ∂tΩz + (U 1 − x2U 3)∂1Ωz + (U2 + x1U 3)∂2Ωz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let x3 = 0 in the above equation and set w(x1, x2) := Ωz(x′, 0), then w(x1, x2) satisfies the two-dimensional helical Euler equation ∂tw + Hw · ∇w = 0, where Hw(x1, x2) = � U 1(x′, 0) − x2U 3(x′, 0), U 2(x′, 0) + x1U 3(x′, 0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='12) 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO Next we will show that this equation is incompressible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Indeed, the divergence of Hw is ∇ · Hw = ∂1U 1 − x2∂1U 3 + ∂2U 2 + x1∂2U 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Since U is a helical vector field and ∇ · U = 0, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7) gives ∇ · Hw = ∂1U 1 + ∂2U 2 + ∂3U 3 = ∇ · U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Next we prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='11 and the three-dimensional Biot- Savart law, it follows that U(x) = ∇ × ∆−1Ω(x) = 1 4π � R3 x − y |x − y|3 ∧ (y2Ωz(y), −y1Ωz(y), Ωz(y)) dy = 1 4π � R3 x − y |x − y|3 ∧ ξ(y)w (R−y3(y1, y2)) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then change of variables (z1, z2) = R−y3(y1, y2) yields U(x′, 0) = 1 4π � R3 (x′, 0) − (Ry3(z′), y3) |(x′, 0) − (Ry3(z′), y3) |3 ∧ ξ((Ry3z′, y3))w(z1, z2) dz1dz2dy3 = 1 4π � R2 �� R (x′, 0) − (Ry3(z′), y3) |(x′, 0) − (Ry3(z′), y3) |3 ∧ ξ((Ry3z′, y3)) dy3 � w(z1, z2) dz′, which proves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ By the proof above, the velocity Hw can be written in the form Hw(x) = � R2 H(x, y)w(y) dy := � R2 H1(x, y)w(y) dy + � R2 H2(x, y)w(y) dy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='13) for H1(x, y) := � K1(x, y), K2(x, y) � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='14) and H2(x, y) := (−x2, x1)K3(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='15) We refer to the two-dimensional problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) as the two-dimensional helical Euler equation and we refer to the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) as the three-dimensional helical Euler equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Function spaces and the definition of weak solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In this subsection, we will introduce the appropriate functional spaces and some definition of the weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The three-dimensional weighted Lp m(R2 × T) norm is defined by ∥f∥Lp m(R2×T) := � R2×T ⟨y′⟩pm|f(y1, y2, y3)|p dy 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Similarly, the two-dimensional weighted Lp m(R2) norm is defined by ∥g∥Lp m(R2) := � R2⟨y⟩pm|g(y1, y2)|p dy 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Here and throughout the paper we use the notation ⟨a⟩ = (1 + |a|2) 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The norms Lp m(R2 × T) and Lp m(R2) are related as follows: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let f ∈ Lp m(R2) and define F : R3 → R as F(x′, x3) := f(R−x3x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then for any p ∈ [1, +∞], ∥f∥Lp m(R2) = (2π)−1/p∥F∥Lp m(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The proof follows directly by definition when p = ∞, so it suffices to consider the case when 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In fact, ∥F∥p Lp m(R2×T) = � T � R2(1 + y2 1 + y2 2) mp 2 |f(R−y3(y1, y2))|p dy1dy2 dy3 = � T � R2(1 + z2 1 + z2 2) mp 2 |f(z1, z2)|p dz1dz2 dy3 = 2π∥f∥p Lp m(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ We will now formulate the weak problem of the two-dimensional helical Euler equa- tion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) and the three-dimensional helical Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (Weak solutions to the three-dimensional helical Euler equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=') We say that Ωz(x, t) is a weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) if � R3 Ωz(x, t)φ(x, t) dx − � R3 Ωz(x, 0)φ(x, 0) dx = � t 0 � R3 Ω(∂tφ + U · ∇φ) dxds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' for all t ∈ [0, T ] and for all test function φ ∈ C∞ c (R3 × [0, +∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Similarly, we define Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (Weak solutions to the two-dimensional helical Euler equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=') We say that w(x, t) is a weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) if � R2 w(x, t)φ(x, t) dx − � R2 w(x, 0)φ(x, 0) dx = � t 0 � R2 w(∂tφ + Hw · ∇φ) dxds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' for all t ∈ [0, T ] and for all test function φ ∈ C∞ c (R2 × [0, +∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (Lagrangian solution) We say w is a Lagrangian solution if it is a weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) and it also satisfies the vortex transport formula w(X(α, t), t) = w(α, 0), where X(α, t) is the particle trajectory map associated with Hw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Key estimates for the Biot-Savart kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The purpose of this section is to establish the necessary estimates for the modified Biot-Savart kernel H(x, y), K(x, y) and G(x, y) given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='13), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='21), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The estimates for the two-dimensional kernel H(x, y) and K(x, y) is given in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The estimates for the three-dimensional kernel G(x, y) is given in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Estimates for the modified Biot-Savart kernel H and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' First we derive some estimates for the modified Biot-Savart kernel K(x, y) which is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1 (Estimates for K(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let x, y be two distinct points in R2, then there holds |K(x, y)| ≲ min � ⟨y⟩(1 + 1 |x − y|), ⟨x⟩(1 + 1 |x − y|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Note that |ξ(y1, y2, a)| ≲ ⟨y⟩, so we have |K(x, y)| ≲ � R ⟨y⟩ |x − Ra(y)|2 + a2 da (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) and hence it suffices to show that � R ⟨y⟩ |x − Ra(y)|2 + a2 da ≲ min � ⟨y⟩(1 + 1 |x − y|), ⟨x⟩(1 + 1 |x − y|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) 10 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO First we consider the case when |y| ≥ 2|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In this case |x − y| 3 ≤ |Ra(x) − y| ≤ 3|x − y|, so we have � R ⟨y⟩ |x − Ra(y)|2 + a2 da ≲ ⟨y⟩ � R 1 |x − y|2 + a2 da ≈ ⟨y⟩ |x − y| ≲ 1 + ⟨x⟩ |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Next we consider the case when |y| ≤ 2|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Note that ⟨y⟩ ≲ ⟨x⟩, so it suffices to show � ∞ 0 ⟨y⟩ |x − Ra(y)|2 + a2 da ≲ ⟨y⟩ � 1 + 1 |x − y| � , which is equivalent to � ∞ 0 1 |x − Ra(y)|2 + a2 da ≲ 1 + 1 |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4), we will assume without loss of generality that 2|x| ≥ |y| ≥ |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' On one hand, it follows from Triangle inequality that |x − y| ≤ ����x − y |y||x| ���� + |y| − |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' On the other hand, the fact |y| ≥ |x| implies ����x − y |y||x| ���� ≤ |x − y| and |y| − |x| ≤ |x − y|, which yield ����x − y |y||x| ���� + |y| − |x| ≲ |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Gathering the estimates above, we finally obtain |x − y| ≈ ����x − y |y||x| ���� + |y| − |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5) Next we set θx,y = ∠xoy and assume without loss of generality that θx,y ∈ [0, π], then θx,y ≈ ���x − y |y||x| ��� |x| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6) Case 1: θx,y ≥ θ0 := 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6), there holds |x − Ray| ≈ |x − y| when a ≤ θ0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, � ∞ 0 1 |x − Ray|2 + a2 da ≲ � θ0 2 0 1 |x − y|2 + a2 da + � ∞ θ0 2 1 a2 da ≲ 1 |x − y| + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 11 Case 2: θx,y ≤ θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In this case, we have � ∞ 0 1 |x − Ray|2 + a2 da ≲ � ∞ π 3 1 |x − Ray|2 + a2 da (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7) + � π 3 2θx,y 1 |x − Ray|2 + a2 da (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='8) + � 2θx,y 0 1 |x − Ray|2 + a2 da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='9) For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7), a direct calculation shows that � ∞ π 3 1 |x − Ray|2 + a2 da ≤ � ∞ π 3 1 a2 da ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='8), π 3 ≥ a ≥ 2θx,y implies |x − Ray| ≥ |x − y| and hence � π 3 2θx,y 1 |x − Ray|2 + a2 da ≲ � ∞ 0 1 |x − y|2 + a2 da = 1 |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='9), a direct calculation shows that � 2θx,y 0 1 |x − Ray|2 + a2 da ≤ � θx,y 0 1 |x − Ray|2 + a2 da + � 2θx,y θx,y 1 |x − Ray|2 + a2 da ≤ 2 � θx,y 0 1 |x − Ray|2 + a2 da since |x − Ray| = |x − R(2θx,y−a)y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' To estimate the right hand side, first we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6) to conclude that |y − x| ≈ |x − y |y||x|| + |y| − |x| ≈ |x||θx,y| + |y| − |x|, which implies |x − Ray| ≈ |x||θx,Ray| + |R − ay| − |x| ≈ (θx,y − a)|x| + |y| − |x| for 0 ≤ a ≤ θx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, � 2θx,y 0 1 |x − Ray|2 + a2 da ≲ � θx,y 0 1 (θx,y − a)2|x|2 + (|y| − |x|)2 + a2 da = θx,y � 1 0 1 (1 − a)2|θx,yx|2 + (|y| − |x|)2 + |θx,ya|2 da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1, |y| − |x| ≤ θx,y|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6), we see that |y − x| ≈ θx,y|x| and hence � 2θx,y 0 1 |x − Ray|2 + a2 da ≲ θx,y � 1 0 1 (1 − a)2θ2x,y|x|2 + θ2x,ya2 da = 1 θx,y � 1 0 1 (1 − a)2|x|2 + a2 da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then a direct calculation shows that � 1 0 1 (1 − a)2|x|2 + a2 da = � 1 0 1 a2|x|2 + (a − 1)2 da ≤ � ∞ −∞ 1 a2|x|2 + (a − 1)2 da = π � 1 + |x|2 , 12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO which implies (recalling that |y − x| ≈ θx,y|x| ) � 2θx,y 0 1 |x − Ray|2 + a2 da ≲ 1 θx,y|x| ≲ 1 |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2, |y| − |x| ≥ θx,y|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Using the estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6), it follows that |y − x| ≈ |y| − |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, � 2θx,y 0 1 |x − Ray|2 + a2 da ≲ θx,y � 1 0 1 (|y| − |x|)2 + |θx,ya|2 da ≲ � ∞ 0 1 (|y| − |x|)2 + a2 da ≈ 1 |y| − |x| ≈ 1 |y − x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Gathering these estimates, we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Next we derive some estimates for the modified Biot-Savart kernel H(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2 (Estimates for H(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let x, y be two distinct points in R2 , then there holds |H1(x, y)| ≲ min � ⟨y⟩(1 + 1 |x − y|), ⟨x⟩(1 + 1 |x − y|) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10) and |H2(x, y)| ≲ min � ⟨x⟩2(1 + 1 |x − y|), ⟨y⟩2(1 + 1 |x − y|), ⟨x⟩⟨y⟩(1 + 1 |x − y|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We only prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='11) since (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10) is a direct consequence of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1), we have that |H2(x, y)| ≲ ⟨x⟩2(1 + 1 |x − y|) and |H2(x, y)| ≲ ⟨y⟩⟨x⟩(1 + 1 |x − y|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, it remains to show |H2(x, y)| ≲ ⟨y⟩2(1 + 1 |x − y|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In fact, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='15) that |H2(x, y)| ≤ ⟨x⟩|K(x, y)| ≤ ⟨y⟩ � R ⟨x⟩ |y − Rax|2 + a2 da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Observe that |y − Rax| = |x − R−ay|, so inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) yields � R ⟨x⟩ |y − Rax|2 + a2 da ≲ ⟨y⟩(1 + 1 |x − y|) and hence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='11) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Next we derive some useful estimates for the velocity field Hw = H1w + H2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3 (Estimates for Hw(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For any x ∈ R2, it holds that |H1w(x)| ≲ min � ∥w∥L1 1 � L∞ 1 , ⟨x⟩∥w∥L1 � L∞ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='12) and |H2w(x)| ≲ min � ⟨x⟩∥w∥L1 1 � L∞ 1 , ⟨x⟩2∥w∥L1 � L∞, ∥w∥L1 2 � L∞ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We only prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='12) since the other one is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For H1w, recall that |H1w(x)| = ���� � R2 H1(x, y)w(y) dy ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10), on one hand, |H1w(x)| ≲ ⟨x⟩ � R2 � 1 + 1 |x − y| � |w(y)| dy ≤ ⟨x⟩ � ∥w∥L1 + � R2 |w(y)| |x − y| dy � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='13) On the other hand, |H1w(x)| ≲ � R2 � 1 + 1 |x − y| � ⟨y⟩|w(y)| dy ≤ ∥w∥L1 1 + � R2 |⟨y⟩w(y)| |x − y| dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='14) Observe that for a scalar function f ∈ L1 � L∞(R2), there holds � R2 |f(y)| |x − y| dy ≲ ∥f∥L1 � L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='15) So using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='15), we finally get the desired estimates for H1w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Osgood property of Hw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In this subsection, we will show that the particle trajectory map X(α, t) given by \uf8f1 \uf8f2 \uf8f3 dX(α, t) dt = Hw(X(α, t), t) X(α, 0) = α is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' To this end, it suffices to show the velocity Hw satisfies the Osgood’s condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' More precisely, we will show that sup x,y∈BR(0) |Hw(x) − Hw(z)| F(|x − z|) ≲R 1 for F the Log-Lipschitz function given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Recalling that Hw(x) = (U 1, U 2)+ (−x2, x1)U 3, so it follow that |Hw(x) − Hw(z)| ≲ |x − z||U(x)| + ⟨z⟩|U(x) − U(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='16) Then by the definition of U, U(x) − U(z) = � R2 (K(x, y) − K(z, y)) w(y) dy, which implies that (using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10)) |Hw(x) − Hw(z)| ≲ � R2 � R ���� (x1, x2, 0) − (Ra(y), a) |(x1, x2, 0) − (Ra(y), a) |3 − (z1, z2, 0) − (Ra(y), a) |(z1, z2, 0) − (Ra(y), a) |3 ���� da⟨z⟩⟨y⟩|w(y)| dy + |x − z||U(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='17) For the last term, we use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3 to conclude that |x − z||U(x)| ≲ F(x − z)⟨x⟩∥w∥L1 � L∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='18) and |x − z||U(x)| ≲ F(x − z)∥w∥L1 1 � L∞ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='19) 14 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO The first term can be bounded by � R3 ���� (x′, 0) − (y′, y3) |(x′, 0) − (y′, y3)|3 − (z′, 0) − (y′, y3) |(z′, 0) − (y′, y3)|3 ���� ⟨z′⟩⟨y′⟩|Ωz(y)| dy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='20) after a change of variables Ray = ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' To estimate this integral, we have the following: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For any x, z ∈ R3, it holds that � R3 ���� x − y |x − y|3 − z − y |z − y|3 ���� ⟨y′⟩|Ωz(y)| dy ≲ (⟨x⟩ + ⟨z⟩)∥Ωz∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='20), we see that Hw is locally Osgood continuous and hence the particle trajectory map X(α, t) associated with Hw is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The proof of this lemma will need some estimates of the three-dimensional Biot-Savart kernel, which will be postponed to next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Estimates for the three-dimensional modified Biot-Savart kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In R3, the velocity field with helical symmetry can be recovered by the third compo- nent of the vorticity: U(x) = � R3 G(x, y)Ωz(y) dy, where G(x, y) := x − y |x − y|3 ∧ ξ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='21) The main purpose of this subsection is to give a detailed estimates of G(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We begin with the following auxiliary lemma: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let Ω(x1, x2, x3) be a 2π-periodic function in x3, then one has � R3 1 |x − y|2 |Ω(y)| dy ≲ ∥Ω∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='22) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' First, observe that � R3 1 |x − y|2 |Ω(y)| dy = � R3 1 |y|2 |Ω(y + x)| dy and for any x ∈ R3 ∥Ω(· + x)∥L1 � L∞(R2×T) = ∥Ω∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So we may assume without loss of generality that x = 0 and Ω ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then we estimate � R3 Ω(y) |y|2 dy = � R2 � 2π −2π Ω(y) |y|2 dy3 dy1dy2 + � R2 � |y3|≥2π Ω(y) |y|2 dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='23) For the first term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='23), denote ˜Ω(y) := Ω(y)χ|y3|≤2π(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Recall that � R3 |f(y)| |x − y|2 dy ≲ ∥f∥L1 � L∞(R3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='24) so we have � R2 � 2π −2π Ω(y) |y|2 dy3 dy1dy2 ≲ ∥˜Ω∥L1 � L∞(R3) ≤ 2∥Ω∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 15 For the second term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='23), a direct calculation shows that � R2 � |y3|≥2π Ω(y) |y|2 dy = � n̸=−1,0 � R2 � 2π(n+1) 2πn Ω(y) |y|2 dy3dy1dy2 ≲ � n̸=−1,0 � R2 � 2π(n+1) 2πn Ω(y) |y3|2 dy3dy1dy2 ≲ � n̸=−1,0 � R2 � 2π(n+1) 2πn Ω(y) n2 dy3dy1dy2 = � n̸=−1,0 1 n2 ∥Ω∥L1(R2×T) ≲ ∥Ω∥L1(R2×T), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ As a consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5, we obtain Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6 (Estimates for U(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For any x ∈ R3, it holds that |U(x)| ≲ ∥Ωz∥L1 1 � L∞ 1 (R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='25) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' If we assume that Ωz is helical, then for any x ∈ R3, there holds |U(x)| ≲ ⟨x′⟩∥Ωz∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Indeed, setting w(x1, x2) = Ωz(x1, x2, 0), then by definition of U, we have that |U(x1, x2, 0)| ≤ ���� � R3 ⟨y′⟩ |(x′, 0) − (y′, y3)|2 |Ωz(y)| dy ���� = � R2 |w(y1, y2)| � R ⟨y′⟩ |x′ − Ry3y′|2 + y2 3 dy3dy′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='13 gives |U(x1, x2, 0)| ≲ ⟨x′⟩∥w∥L1 � L∞(R2) ≲ ⟨x′⟩∥Ωz∥L1 � L∞(R2×T), which implies |U(x′, x3)| = |U(R−x3x′, 0)| ≲ ⟨x′⟩∥Ωz∥L1 � L∞(R2×T) since U is a helical vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Next we estimate the difference |U(x) − U(z)| = ���� � R3 � x − y |x − y|3 − z − y |z − y|3 � ∧ ξ(y)Ωz(y) dy ���� ≲ � R3 ���� x − y |x − y|3 − z − y |z − y|3 ���� ⟨y′⟩|Ωz(y)| dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let Ω(x1, x2, x3) be a helical function, then for any x, z ∈ R3, it holds that � R3 ���� x − y |x − y|3 − z − y |z − y|3 ���� |Ω(y)| dy ≲ ∥Ω∥L1 � L∞(R2×T)F(|x − z|), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='26) where F is the Log-Lipschitz function given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 16 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' After changing the variable ˜y = y − z, we obtain � R3 ���� x − y |x − y|3 − z − y |z − y|3 ���� |Ω(y)| dy = � R3 ���� x − z − ˜y |x − z − ˜y|3 − −˜y | − ˜y|3 ���� |Ω(˜y + z)| d˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Observe that for any z ∈ R3, ∥Ω(· + z)∥L1 � L∞(R2×T) = ∥Ω∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So we may assume without loss of generality that z = 0 and Ω ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' To complete the proof, it suffices to show that � |K(x − y) − K(−y)| Ω(y) dy ≲ ∥Ω∥L1 � L∞(R2×T)F(|x|), where K(x) := x |x|3 ∈ C∞(R3 \\ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' To this end, we divide the above integral into three parts: � |K(x − y) − K(−y)| Ω(y) dy = � |y−x|≥2 |K(x − y) − K(−y)| Ω(y) dy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='27) + � 2≥|y−x|≥2|x| |K(x − y) − K(−y)| Ω(y) dy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='28) + � |y−x|≤2|x| |K(x − y) − K(−y)| Ω(y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='29) For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='27), a direct calculation shows that ���� a |a|3 − b |b|3 ���� 2 = 1 |a|4 + 1 |b|4 − 2a · b |a|3|b|3 = |a|4 + |b|4 − 2|a|a · |b|b |a|4|b|4 = ���|a|a − |b|b ��� 2 |a|4|b|4 , which implies ���� a |a|3 − b |b|3 ���� = ���|a|a − |b|b ��� |a|2|b|2 ≲ ���a|a| − a|b| + a|b| − b|b| ��� |a|2|b|2 ≲ ���|a| − |b| ��� |a||b|2 + |a − b| |a|2|b| ≲ � 1 |a||b|2 + 1 |a|2|b| � |a − b|, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='30) and hence |K(x − y) − K(−y)| ≲ |x| � 1 |x − y|2|y| + 1 |x − y||y|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, � |y−x|≥2 |K(x − y) − K(−y)| Ω(y) dy ≲|x| � |x−y|≥2 � 1 |x − y|2|y| + 1 |x − y||y|2 � Ω(y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='31) THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 17 To bound the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='27), it remains to show that � |y−x|≥2 1 |x − y||y|2 Ω(y) dy + � |y−x|≥2 1 |x − y|2|y|Ω(y) dy ≲ ∥Ω∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='32) For the first integral above, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='22) gives � |y−x|≥2 1 |x − y||y|2 Ω(y) dy ≲ � R3 1 |y|2 Ω(y) dy ≲ ∥Ω∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='33) For the second integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='32), let N = N(x) be the integer such that x ∈ [2πN, 2π(N + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then we get � |y−x|≥2 1 |x − y|2|y|Ω(y) dy = ∞ � −∞ � R2 � 2π(n+1) 2πn Ω(y)χ{|y−x|≥2}(y) |x − y|2|y| dy3 dy′ = � |n−N|≥2 � R2 � 2π(n+1) 2πn Ω(y)χ{|y−x|≥2}(y) |x − y|2|y| dy3 dy′ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='34) + � |n−N|≤1 � R2 � 2π(n+1) 2πn Ω(y)χ{|y−x|≥2}(y) |x − y|2|y| dy3 dy′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='35) For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='34), we only consider the case when n ≥ N +2 since the other part is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Note that |x − y| ≥ |x3 − y3| ≈ |N + 1 − n| for y3 ∈ [2πn, 2π(n + 1)), so it follows that ∞ � n=N+2 � R2 � 2π(n+1) 2πn Ω(y)χ{|y−x|≥2}(y) |x − y|2|y| dy3dy1dy2 ≲ ∞ � n=N+2 � R2 � 2π(n+1) 2πn Ω(y) |N + 1 − n|2|y| dy3 dy1dy2 = ∞ � n=N+2 1 |N + 1 − n|2 � R2 � 2π(n+1) 2πn Ω(y) |y| dy3 dy1dy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Denote ˜Ωn(y) := Ω(y)χ2πn≤y3≤2π(n+1), then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='24) yields ∞ � n=N+2 � R2 � 2π(n+1) 2πn Ω(y)χ{|y−x|≥2}(y) |x − y|2|y| dy3dy1dy2 ≲ ∞ � n=N+2 1 |N + 1 − n|2 ∥˜Ωn∥L1 � L∞(R3) ≲ ∞ � n=N+2 1 |N + 1 − n|2 ∥Ω∥L1 � L∞(R2×T) ≲∥Ω∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='36) For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='35), denote ˜Ω(y) = Ω(y)χ{2π(N−1)≤y3≤2π(N+2)} 18 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO then in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='24), we have � |n−N|≤1 � R2 � 2π(n+1) 2πn Ω(y)χ{|y−x|≥2}(y) |x − y|2|y| dy3 dy1dy2 ≲ � R3 ˜Ω(y) |x − y|2|y|χ{|y−x|≥2} dy = � |y−x|≥2 ˜Ω(y) |x − y|2|y| dy ≲ � R3 ˜Ω(y) |y| dy ≲ ∥˜Ω∥L1 � L∞(R3) ≲ ∥Ω∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='33) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='36), we finally obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='32) and hence by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='31) we get the desired estimates for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='28), we only consider the case |x| ≤ 1 since otherwise the integral vanishes identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Note that when |y| ≥ 2|x|, the segment between x − y and −y does not contain the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So the mean value theorem yields |K(x − y) − K(−y)| = |x∇K(−y + θx)| ≲ |x| 1 | − y + θx|3 ≲ |x| |y|3 ≲ |x| |y − x|3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, � 2≥|y−x|≥2|x| |K(x − y) − K(−y)| Ω(y) dy ≲|x| � 2≥|y|≥2|x| Ω(y) |y − x|3 dy ≲ ∥Ω∥L∞(R2×T)F(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3), we estimate � |y|≤2|x| ���� (x − y) |x − y|3 − (−y) | − y|3 ���� Ω(y) dy ≲ � |y|≤2|x| Ω(y) |x − y|2 dy + � |y|≤2|x| Ω(y) |y|2 dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' A direct calculation shows that � |y|≤2|x| Ω(y) |x − y|2 dy ≲ ∥Ω∥L∞ � |y|≤2|x| 1 |x − y|2 dy ≲ ∥Ω∥L∞ � |y|≤2|x| 1 |y|2 dy ≲ ∥Ω∥L∞|x| and similarly, � |y|≤2|x| Ω(y) |y|2 dy ≲ ∥Ω∥L∞ � |y|≤2|x| 1 |y|2 dy ≲ ∥Ω∥L∞ � |y|≤2|x| 1 |y|2 dy ≲ ∥Ω∥L∞|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Gathering the estimates above we obtain � |y−x|≤2|x| |K(x − y) − K(−y)| Ω(y) dy ≲ |x| + F(|x|) + |x| ≲ F(|x|), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For any x, z ∈ R3 and G(x, y) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='21), the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' � R3 |G(x, y) − G(z, y)| |Ω(y)| dy ≲ ∥Ω∥L1 1 � L∞ 1 (R2×T)F(|x − z|), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='37) THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 19 � R3 |G(x, y) − G(z, y)| |Ω(y)| dy ≲ (⟨x′⟩ + ⟨z′⟩)∥Ω∥L1 � L∞(R2×T)F(|x − z|) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='38) and � R3 |G(y, x) − G(y, z)| |Ω(y)| dy ≲ (⟨x′⟩ + ⟨z′⟩)∥Ω∥L1 � L∞(R2×T)F(|x − z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='39) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='37) follows directly from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In order to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='38), observe that � R3 |G(x, y) − G(z, y)| Ω(y) dy ≲ � R3 |K(x − y) − K(z − y)| ⟨y′⟩Ω(y) dy, so by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='26) it suffice to consider the integral for max {5|x|, 5|z|} ≤ |y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' When |y| ≤ 1, � |y|≤1 |G(x, y) − G(z, y)| Ω(y) dy ≲ � |y|≤1 |K(x − y) − K(z − y)| ⟨y′⟩Ω(y) dy ≲ � R3 |K(x − y) − K(z − y)| Ω(y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='26) implies that � |y|≤1 |G(x, y) − G(z, y)| Ω(y) dy ≲ F(|x − z|)∥Ω∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' When |y| ≥ 1, we use mean value theorem to conclude that |K(x − y) − K(z − y)| ≲ |x − z| |y|3 , which implies (using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='22)) � max {1,5|x|,5|z|}≤|y| |G(x, y) − G(z, y)| |Ω(y)| dy ≲|x − z| � max{1,5|x|,5|z|}≤|y| ⟨y′⟩ |y|3 |Ω(y)| dy ≲|x − z| � R3 |Ω(y)| |y|2 dy ≲|x − z|∥Ω∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Next we consider (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Observe that G(y, x) − G(y, z) = y − x |y − x|3 ∧ ξ(x) − y − z |y − z|3 ξ(z) = � y − x |y − x|3 − y − z |y − z|3 � ξ(x) + y − z |y − z|3 (ξ(x) − ξ(z)), so we have |G(y, x) − G(y, z)| ≲ |ξ(x)| ���� y − x |y − x|3 − y − z |y − z|3 ���� + |x − z| 1 |y − z|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='26) yields � |G(y, x) − G(y, z)| |Ω(y)| dy ≲⟨x′⟩ � ���� x − y |x − y|3 − z − y |z − y|3 ���� |Ω(y)| dy + |x − z| � |Ω(y)| |y − z|2 dy ≲⟨x′⟩F(x − z)∥Ω∥L1 � L∞(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ 20 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof of the uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In this section, we prove the uniqueness of the weak solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) in L1 1 � L∞ 1 (R2) and L1 1 � L∞ 1 (R3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Uniqueness of the three-dimensional helical Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Our main theorem is: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume Ωz, ˜Ωz ∈ L∞([0, T ], L1 1 � L∞ 1 (R2 × T)) are two lagrangian solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) with the same initial data Ωz 0, then Ωz = ˜Ωz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Before the proof, we first make some useful observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Recall that Ωz satisfies ∂tΩz + U · ∇Ωz = 0 and when Ωz ∈ L1 1 � L∞ 1 (R2 × T), Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='9 implies that the velocity field U(x, t) = � G(x, y)Ωz(y) dy with G(x, y) = x − y |x − y|3 ∧ ξ(y) = x − y |x − y|3 ∧ (y2, −y1, 1) is Log-Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, the particle trajectory map \uf8f1 \uf8f2 \uf8f3 dX(α, t) dt = U(X(α, t), t) X(α, 0) = α is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let X(α, t), ˜X(α, t) be the particle trajectory map associated with U and ˜U, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then we have Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume X(α, t) = ˜X(α, t) for all α ∈ supp(Ωz 0) and t ∈ [0, T ], then Ωz = ˜Ωz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We fix x ∈ R3 and t ∈ [0, T ], then there exists unique α and β such that x = X(α, t) = ˜X(β, t) since the map X(·, t) and ˜X(·, t) are one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Case 1: α, β /∈ supp Ωz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In this case, one has Ωz(x, t) = Ωz(X(α, t), t) = Ωz 0(α) = 0 and ˜Ωz(x, t) = ˜Ωz( ˜X(β, t), t) = Ωz 0(β) = 0, which implies Ωz(x, t) = ˜Ωz(x, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Case 2: α ∈ supp Ωz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Observe that by assumption in our lemma, we have ˜X(β, t) = x = X(α, t) = ˜X(α, t) since α ∈ supp Ωz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, α = β since ˜X(·, t) is one-to-one and hence Ωz(x, t) = Ωz 0(α, t) = Ωz 0(β, t) = ˜Ωz(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The case when β ∈ supp Ωz 0 is similar so we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='8, we see that X − ˜X is period in α3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So in order to prove Ωz = ˜Ωz, it suffices to show X(α, t) = ˜X(α, t) for all α ∈ supp(Ωz 0) � R2 × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Now we are ready to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Motivated by the remark above, we define the distance D(t) := � R2×T ���X(α, t) − ˜X(α, t) ��� ⟨α⟩|Ωz 0(α)| dα and it remains to show that E(t) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' First we will show that D(t) is continuous and bounded in [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' To this end, we estimate |X(α, t) − α| = � t 0 X(α, s) − α |X(α, s) − α|U(X(α, s), s) ds ≲ T sup t ∥U(·, t)∥L∞, which together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='25) gives |X(α, t) − α| ≲ sup t ∥Ωz∥L1 1 � L∞ 1 (R2×T) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Similarly, | ˜X(α, t) − α| ≲ 1, which implies ⟨α⟩ ≈ ⟨X(α, t)⟩ ≈ ⟨ ˜X(α, t)⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) and |X(α, t) − ˜X(α, t)| ≤ |X(α, t) − α| + | ˜X(α, t) − α| ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, D(t) ≲ � R2×T ⟨α⟩|Ωz 0(α)| dα = ∥Ωz∥L1 1(R2×T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Furthermore, the energy D(t) is indeed continuous in [0, T ] by Lebesgue dominating convergence Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Next we show D(t) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Observe that if Ωz 0 ≡ 0, then Ωz = ˜Ωz ≡ 0, so we may assume without loss of generality that Ωz 0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Set D∗(t) := D(t) ∥Ωz 0∥L1 1(R2×T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We claim that for all t ∈ [0, T ], one has D∗(t) ≲ � t 0 F(D∗(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Recalling that D∗(0) = 0, so it follows from Osgood’s Lemma that D∗(t) ≡ 0 and hence X(α, t) = ˜X(α, t) for all α ∈ supp (Ωz 0) � R2 × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We now prove the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' A direct calculation shows that ���X(α, t) − ˜X(α, t) ��� ≲ � t 0 ���U(X(α, s), s) − ˜U( ˜X(α, s), s) ��� ds ≲ � t 0 ��� ˜U(X(α, s), s) − ˜U( ˜X(α, s), s) ��� ds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) + � t 0 ���U(X(α, s), s) − ˜U(X(α, s), s) ��� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) First we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='37) to conclude that | ˜U(x) − ˜U(z)| ≲ ∥˜Ωz∥L1 1 � L∞ 1 (R2×T)F(|x − z|), which implies � t 0 ��� ˜U(X, s) − ˜U( ˜X, s) ��� ds ≲ � t 0 F � |X(α, s) − ˜X(α, s)| � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 22 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO Recall that F is a concave function, so Jensen’s inequality yields 1 ∥Ωz 0∥L1 1 � R2×T � t 0 ���U(X, s) − ˜U(X, s) ��� ds⟨α⟩|Ωz 0(α)| dα ≲ � t 0 � R2×T F � |X(α, s) − ˜X(α, s)| � ⟨α⟩|Ωz 0(α)| ∥Ωz 0∥L1 1 dα ds ≲ � t 0 F �� R2×T |X(α, s) − ˜X(α, s)| ⟨α⟩|Ωz 0(α)| ∥Ωz 0∥L1 1 dα � ds = � t 0 F(D∗(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Next we estimate the second term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3), |U(x, s) − ˜U(x, s)| = ���� � G(x, y)Ωz(y, s) dy − � G(x, y)˜Ωz(y, s) dy ���� = ���� � G(x, X(β, s))Ωz 0(β) dβ − � G(x, ˜X(β, s))Ωz 0(β) dβ ���� ≲ � ���G(x, X(β, s)) − G(x, ˜ X(β, s)) ��� |Ωz 0(β)| dβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) gives � R2×T ���U(X(α, s), s) − ˜U(X(α, s), s) ��� ⟨α⟩Ωz 0(α) dα ≲ � R3 � R2×T ���G(X(α, s), X(β, s)) − G(X(α, s), ˜X(β, s)) ��� ⟨α⟩|Ωz 0(β)Ωz 0(α)| dα dβ ≲ � R3 � R2×T ���G(X(α, s), X(β, s)) − G(X(α, s), ˜X(β, s)) ��� ⟨X(α, s)⟩|Ωz 0(β)Ωz 0(α)| dα dβ :=A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Note that |X(α, t)−α| ≲ 1, so there exists N > 0 such that X(·, t) maps R2×[0, 2π] into R2 × [−2πN, 2πN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus a change of variables x = X(α, s) yields A ≲ � R3 � R2×[−2πN,2πN] ���G(x, X(β, s)) − G(x, ˜ X(β, s)) ��� ⟨x′⟩|Ωz(x, s)| dx|Ωz 0(β)| dβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Fix M ≫ N such that for all |β3| ≥ 2πM and all t ∈ [0, T ], |X3(β, t)|, | ˜X3(β, t)| ≥ 4πN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then we can divide the integral into two parts according to the value of β3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let XM = R2 × [−2πM, 2πM], then A ≲ � XM � XN ���G(x, X(β, s)) − G(x, ˜X(β, s)) ��� ⟨x′⟩|Ωz(x, s)| dx|Ωz 0(β)| dβ + � XC M � XN ���G(x, X(β, s)) − G(x, ˜ X(β, s)) ��� ⟨x′⟩|Ωz(x, s)| dx|Ωz 0(β)| dβ :=A1 + A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) For A1, we see directly from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='38) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) that A1 ≲ � XM � ⟨X(β, s)⟩ + ⟨ ˜X(β, s)⟩ � F(|X(β, s) − ˜X(β, s)|)|Ωz 0(β)| dβ ≲ � XM F(|X(β, s) − ˜X(β, s)|)|⟨β⟩|Ωz 0(β)| dβ ≲ � R2×T F(|X(β, s) − ˜X(β, s)|)|⟨β⟩|Ωz 0(β)| dβ THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 23 since Ωz ∈ L∞([0, T ], L1 1 � L∞ 1 (R2 × T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, Jensen’s inequality yields A1 ≲ F �� R2×T |X(β, s) − ˜X(β, s)| |⟨β⟩Ωz 0(β)| ∥Ωz 0∥L1 1(R2×T) dβ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5) For A2, first we estimate ���G(x, X(β, s)) − G(x, ˜ X(β, s)) ��� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6) = ����� x − X(β, s) |x − X(β, s)|3 ∧ ξ(X(β, s)) − x − ˜X(β, s) |x − ˜X(β, s)|3 ∧ ξ( ˜X(β, s)) ����� ≲ ����� � x − X(β, s) |x − X(β, s)|3 − x − ˜X(β, s) |x − ˜X(β, s)|3 � ∧ ξ(X(β, s)) ����� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7) + ����� x − ˜X(β, s) |x − ˜X(β, s)|3 ∧ � ξ(X(β, s)) − ξ( ˜ X(β, s)) ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='8) For (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7), we see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='30) that for all |x| ≤ 2πN and |β3| ≥ 2πM, ����� � x − X(β, s) |x − X(β, s)|3 − x − ˜X(β, s) |x − ˜X(β, s)|3 � ∧ ξ(X(β, s)) ����� ≲ ���X(β, s) − ˜X(β, s) ��� ⟨β′⟩ |β3|3 ≲ ���X(β, s) − ˜X(β, s) ��� ⟨β′⟩ |β3|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='8), since ���x3 − ˜X3 ��� ≳ 1, it is easy to check that ����� x − ˜X(β, s) |x − ˜X(β, s)|3 ∧ � ξ(X(β, s)) − ξ( ˜X(β, s)) ������ ≲ ���X(β, s) − ˜X(β, s) ��� 1 |β3|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, ���G(x, X(β, s)) − G(x, ˜ X(β, s)) ��� ≲ ���X(β, s) − ˜X(β, s) ��� ⟨β′⟩ |β3|2 , which implies that A2 ≲ ∥Ωz(·, s)∥L1 1(R2×T) � XC M ���X(β, s) − ˜X(β, s) ��� ⟨β′⟩ |β3|2 |Ωz 0(β)| dβ ≲ � |n|≥M � R2 � 2π(n+1) 2πn ���X(β, s) − ˜X(β, s) ��� ⟨β′⟩ |β3|2 |Ωz 0(β)| dβ3 dβ1dβ2 ≲ � |n|≥M � R2 � 2π(n+1) 2πn ���X(β, s) − ˜X(β, s) ��� ⟨β′⟩ n2 |Ωz 0(β)| dβ3 dβ1dβ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Note that |X(β, t) − ˜X(β, t)|, ⟨β′⟩ and Ωz 0(β) are periodic functions in β3, so it follows that A2 ≲ � |n|≥M 1 n2 � R2 � 2π 0 ���X(β, s) − ˜X(β, s) ��� ⟨β′⟩|Ωz 0(β)| dβ3 dβ1dβ2 ≲ F �� R2×T ���X(β, s) − ˜X(β, s) ��� ⟨β⟩|Ωz 0(β)| ∥Ωz 0∥L1 1(R2×T) dβ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='9) 24 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO where we have used the fact that F(r) ≳ r for r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='9) gives A ≲ F �� R2×T ���X(β, s) − ˜X(β, s) ��� ⟨β⟩|Ωz 0(β)| ∥Ωz 0∥L1 1(R2×T) dβ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Integrating the above inequality from 0 to t, we arrive at D∗(t) ≲ � t 0 F �� R2×T ���X(α, s) − ˜X(α, s) ��� ⟨α⟩|Ωz 0(α)| ∥Ωz 0∥L1 1 dα � ds = � t 0 F(D∗(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Uniqueness of the two-dimensional helical Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) in L1 1 � L∞ 1 (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We will show that every weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) can be lifted to a Lagrangian weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3), thus the uniqueness of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) follows directly from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let w(x, t) ∈ L∞([0, T ], L1 � L∞(R2)) be a weak solution of the two- dimensional helical Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4), set Ω(x1, x2, x3, t) = w(R−x3(x1, x2), t) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10) and U(x, t) = � R3 G(x, y)Ω(y, t) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then Ω(x, t) satisfies the three-dimensional helical Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' By definition of the weak solution, it suffices to show that for any φ ∈ C∞ c (R3 × R), � R3 Ω(x, t)φ(x, t) dx − � R3 Ω(x, 0)φ(x, 0) dx = � t 0 � R3 Ω(∂tφ + U · ∇φ) dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' First we observe that (using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10)) � R3 Ω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' t)φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' t) dx − � R3 Ω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 0)φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 0) dx = � R3 w(R−x3x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' t)φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' t) dx − � R3 w(R−x3x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 0)φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 0) dx = � R �� R2 w(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' t)φ(Rx3x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' t) dx′ − � R2 w(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 0)φ(Rx3x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 0) dx′ � dx3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' which implies � R3 Ω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' t)φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' t) dx − � R3 Ω(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 0)φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 0) dx = � R �� t 0 � R2 w(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' s) [∂s + Hw(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' s) · ∇] (φ(Rx3x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' s)) dx′ds � dx3 = � t 0 � R �� R2 w(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' s) [∂s + Hw(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' s) · ∇] (φ(Rx3x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' s)) dx′ � dx3 ds THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 25 since w is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, to complete the proof of the lemma, it remains to show that � R3 Ω(x, s) [∂s + U · ∇] φ(x, s) dx = � R �� R2 w(x1, x2, s) [∂s + Hw(x1, x2, s) · ∇] (φ(Rx3x′, x3, s)) dx′ � dx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='11) Since Ω is a helical function, we have � R3 Ω(x, s)∂sφ(x, s) dx = � R3 w(R−x3x′, s)∂sφ(x, s) dx = � R3 w(x1, x2, s)∂sφ(Rx3x′, x3, s) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='12) Recall that Hw(x1, x2) = (U 1(x′, 0), U 2(x′, 0)) + (−x2, x1)U 3(x′, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='13) where U 3(·, t) are helical functions on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='13) and the fact that ΩU 3 is a helical function to get the following: � R3 Ω(x, s)(U 1∂1φ + U 2∂2φ) dx + � R3(ΩU 3)∂3φ dx = � R3 Ω(x, s)(U 1∂1φ + U 2∂2φ) dx + � R3 Ω(x1U 3∂2φ − x2U 3∂1φ) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='14) Note that for a smooth function φ = φ(x1, x2), ∇ (φ(Rx3x′)) = R−x3[∇φ(Rx3x′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='14), there holds � R3 Ω(x, s)(U 1∂1φ + U 2∂2φ) dx + � R3(ΩU 3)∂3φ dx = � R � R2 w(x1, x2, s)Hw(x1, x2, s) · ∇ (φ(Rx3x′, x3, s)) dx1dx2 dx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='15) Together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='12) we finally get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='11) and hence completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume w(x, t) ∈ L∞([0, T ], L1 � L∞(R2)) is Lagrangian weak solu- tion of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4), then Ω(x′, x3, t) = w(R−x3x′, t) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='16) is a Lagrangian weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let X(α1, α2, t) = (X1(α1, α2, t), X2(α1, α2, t)) be the particle trajectory map associated to Hw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Motivated by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='9), define X3(α1, α2, t) = � t 0 U3(X1(α1, α2, s), X2(α1, α2, s), 0) ds, and Y (α1, α2, 0, t) = (RX3(X1, X2), X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='17) We claim that Y (α1, α2, α3, t) := Sα3Y (R−α3(α1, α2), 0, t) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='18) is the particle trajectory map associated with U in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' That is, \uf8f1 \uf8f2 \uf8f3 dY (α, t) dt = U(Y (α, t), t) Y (α, 0) = α, 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO Indeed, a direct calculation shows that dY1(α1, α2, 0, t) dt =(U1 − X2U3) cos(X3) − X1U3 sin(X3) + (U2 + X1U3) sin(X3) + X2U3 cos(X3) =U1 cos(X3) + U2 sin(X3), dY2(α1, α2, 0, t) dt = −U1 sin(X3) + U2 cos(X3) and dY3(α1, α2, 0, t) dt = U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Hence, dY (α1, α2, 0, t) dt = RX3U(X1, X2, 0) = U(RX3(X1, X2), X3) since U is a helical vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='17) yields dY (α1, α2, 0, t) dt = U(Y (α1, α2, 0, t), t), which combined with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='18), gives dY (α1, α2, α3, t) dt = dSα3Y (R−α3(α1, α2), 0, t) dt = dRα3Y (R−α3(α1, α2), 0, t) dt = Rα3U(Y (R−α3(α1, α2), 0, t), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Finally, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='18), we have that dY (α1, α2, α3, t) dt = Rα3U(Y (S−α3α, t), t) = U(Sα3Y (S−α3α, t), t) = U(Y (α1, α2, α3, t), t) which implies that Y is the particle trajectory map of U in R3 and hence completes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Now it remains to show that Ω(Y (α, t), t) = Ω0(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' To this end, we set β = R−α3(α1, α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='9) that Ω(Y (α, t), t) = Ω(Sα3Y (β, 0, t), t) = Ω(Y (β, 0, t), t) = Ω(S−Y3(β,0,t)Y (β, 0, t), t), which yields (using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10)) Ω(Y (α, t), t) = Ω(S−X3(β,t)(RX3(X1, X2), X3)(β, t), t) = Ω(X1(β, t), X2(β, t), 0, t) = w(X1(β, t), X2(β, t), t) = w0(β) since w is a Lagrangian solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='16), we get Ω(Y (α, t), t) = Ω0(α), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ An immediate consequence is: THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 27 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Every weak solution w(x, t) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) can be lifted to a weak La- grangian solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) with Ωz(x, t) = w(R−x3(x1, x2), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Moreover, such lifting is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' It follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4 that every weak solution w(·, t) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) is indeed a Lagrangian weak solution, so Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4 implies that every weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) can be lifted to a weak Lagrangian solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus it remains to show that such lifting is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let w and ˜w be two different weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4), then ∥Ωz(·, t) − ˜Ωz(·, t)∥L1 1 � L∞ 1 (R2×T) ≈ ∥w(·, t) − ˜w(·, t)∥L1 1 � L∞ 1 (R2) ̸= 0 as a consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Together with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1, we obtain the uniqueness of weak solutions to the two-dimensional helical Euler equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume w, ˜w ∈ L∞([0, T ], L1 1 � L∞ 1 (R2)) are two weak solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) with the same initial data w0, then w = ˜w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Global existence of weak solutions to the two-dimensional helical Euler equation The main purpose of this section is to show that all weak solutions to the two- dimensional helical Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) are global in L1 1 ∩ L∞ 1 (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' A formal argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We will assume in this subsection that all functions are smooth enough and exhibit sufficient decay at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The well-known Beale-Kato- Majda criterion suggests that a solution Ω of the three-dimensional Euler equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) blows-up at time T if and only if � T 0 ∥Ω∥L∞(R3) dt = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For a helical solution without swirl, we have ∥Ω∥L∞(R3) = ∥Ωz∥L∞ 1 (R2×T) with Ω = (x2Ωz, −x1Ωz, Ωz) and Ωz(x1, x2, x3) = w(R−x3(x1, x2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So it follows that ∥Ω∥L∞(R3) = ∥Ωz∥L∞ 1 (R3) = ∥w∥L∞ 1 (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Recall that w satisfies the (transport) equation ∂tw + Hw · ∇w = 0 with ∇ · Hw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Multiply both sides by ⟨x⟩, we obtain ∂t(⟨x⟩w) + Hw · ∇(⟨x⟩w) = wHw · ∇⟨x⟩ = wH1w · ∇⟨x⟩ since Hw = H1w + H2w and x · H2w=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, ∥⟨x⟩w(x, t)∥L1 � L∞ ≤ ∥⟨x⟩w0(x)∥L1 � L∞ + � t 0 ∥wH1w(x, s)∥L1 � L∞ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' To bound the second term in the right hand side, we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='12) to estimate ∥wH1w∥L1 � L∞ ≤ ∥H1w∥L∞∥w∥L1 � L∞ ≲ ∥w∥L1 1 � L∞ 1 ∥w∥L1 � L∞, which implies (recall that ∥w(·, t)∥Lp = ∥w0∥Lp since ∇ · Hw = 0) ∥w(·, t)∥L1 1 � L∞ 1 ≤ ∥w0∥L1 1 � L∞ 1 + C∥w0∥L1 � L∞ � t 0 ∥w(·, s)∥L1 1 � L∞ 1 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 28 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO So it follows from Gronwall’s inequality that ∥w(·, t)∥L1 1 � L∞ 1 ≲ CeCt for all t ≥ 0 and hence the Beale-Kato-Majda criterion implies at least formally that solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) are global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' More precisely, we have Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For any w0 ∈ C∞ c (R2), there exist a global Lagrangian weak solution w(x, t) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) with initial vorticity w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Furthermore, for any t ∈ [0, T ], there exists a constant C depending only on T and ∥w0∥L1 1 � L∞ 1 (R2) such that sup t∈[0,T ] ∥w(·, t)∥L1 1 � L∞ 1 (R2) ≤ CT,w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' By the same argument, one can show that for all T > 0, N ∈ N∗ and t ∈ [0, T ], there exist some constant C depending only on N, T and ∥w0∥L1 N � L∞ N (R2) such that sup t∈[0,T ] ∥w(·, t)∥L1 N � L∞ N (R2) ≤ CN,T,w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1 is standard, which is sketched in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof of the existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In this subsection, we will show that for every w0 ∈ L1 � L∞(R2), there exists a global Lagrangian weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) with initial data w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let {w0,n} ∈ C∞ c (R2) such that w0,n → w0 ∈ L1(R2) and ∥w0,n∥L∞(R2) ≲ ∥w0∥L∞(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let wn(·, t) be a sequence of Lagrangian solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) with initial data w0,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then the fact ∇ · Hwn = 0 yields ∥wn(·, t)∥L1 � L∞(R2) = ∥w0,n∥L1 � L∞(R2) ≲ 1, which implies that (using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) |H1wn(x, t)| ≲ ⟨x⟩ and |H2wn(x, t)| ≲ ⟨x⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='17), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='38) to conclude that |Hwn(x, t) − Hwn(z, t)| ≲ (⟨x⟩3 + ⟨z⟩3)F(|x − z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, it follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2 that for any R, T > 0, Xn(α, t) and X−t n (α) are uniform bounded and equicontinuous on BR × [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So there exist X(α, t) and its inverse map X−t(α) and a subsequence Xnk such that Xnk(α, t) → X(α, t) and X−t nk (α) → X−t(α) uniformly in every compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Furthermore, for ant t ≥ 0, the map X−t(·) and X(·, t) preserves Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Now setting w(x, t) = w0(X−t(x)), we will show that w(x, t) is the desired solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' To this end, first we write the subsequence nk as n for simplicity and we claim that for every t ∈ [0, T ], ∥w(·, t) − wn(·, t)∥L1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) Indeed, since wn(·) is a Lagrangian weak solution, we have ∥w(·, t) − wn(·, t)∥L1 = � R2 |w0(X−t(x)) − w0(X−t n (x)) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Now for any ǫ > 0, let wc ∈ C∞ c (R2) such that ∥w0 − wc∥L1 ≤ ǫ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 29 Then it follows that ∥w(·, t) − wn(·, t)∥L1 ≤ � R2 |w0(X−t(x)) − wc(X−t(x))| dx + � R2 |wc(X−t(x)) − wc(X−t n (x))| dx + � R2 |wc(X−t n (x)) − w0(X−t n (x))| dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Observe that X−t(·) and X(·, t) preserves Lebesgue measure, so there holds � R2 |w0(X−t(x)) − wc(X−t(x))| dx = ∥w0 − wc∥L1 ≤ ǫ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Similarly, � R2 |w0(X−t n (x)) − wc(X−t n (x))| dx = ∥w0 − wc∥L1 ≤ ǫ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then the dominating convergence theorem gives � R2 |wc(X−t(x)) − wc(X−t n (x))| dx ≤ 97ǫ 100 for n large enough since wc ∈ C∞ c (R2) and X−t n (x) converges to X−t(x) in compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' This completes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Furthermore, it follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) that Hwn(x, t) → Hw(x, t) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) for all x ∈ R2 and t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Note that for any φ ∈ C∞ c (R2), we have � wn(x, t)φ(x, t) dx − � wn(x, 0)φ(x, 0) dx = � t 0 � R2 wn(∂tφ + Hwn · ∇φ) dxdt since each wn is a weak solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So with the help of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2), letting n → ∞ we get the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Continuous dependence on initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In this section we will prove the continuous dependence on initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' First, we prove the following simplified version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let w0,n be a sequence of initial data such that sup n ∥w0,n∥L∞ 1 (R2) < ∞ and ∥w0,n − w0∥L1 1(R2) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then for any t ≥ 0, one has ∥wn(t) − w(t)∥L1 1(R2) → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' First we claim that for any R, T > 0, Xn(·, ·) → X(·, ·) uniformly in BR × [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume this is not true, then there exist R, T, δ > 0, (αk, tk) ∈ BR × [0, T ] and a subsequence Xnk such that |Xnk(αk, tk) − X(αk, tk)| ≥ δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Observe that BR × [0, T ] is compact, so we may assume without loss of generality that (αk, tk) → (α0, t0) ∈ BR × [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then by a similar argument as in section 6, there exists ˜X(α, t) such that Xnk → ˜X uniformly in compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, if we set w( ˜ X(α, t), t) := w0(α), then ˜w is also a weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) with initial data w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Meanwhile, from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7 we know that solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) is unique in L1 1 � L∞ 1 (R2), so w = ˜w and hence X = ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, X(α0, t0) − ˜X(α0, t0) = 0, 30 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO which leads to a contradiction since |X(α0, t0) − ˜X(α0, t0)| = lim k→∞ |X(αk, tk) − Xnk(αk, tk)| ≥ δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, Xn → X uniformly in compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Next we show wn → w in L1 1(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' A direct calculation shows that ∥w(·, t) − wn(·, t)∥L1 1(R2) = � R2⟨x⟩|w0,n(X−t n (x)) − w0(X−t(x))| dx ≤ � R2⟨x⟩|w0,n(X−t n (x)) − w0(X−t n (x))| dx (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) + � R2⟨x⟩|w0(X−t n (x)) − w0(X−t(x))| dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) For (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1), we use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) to conclude that for n large enough � R2⟨x⟩|w0,n(X−t n (x)) − w0(X−t n (x))| dx = � R2⟨Xn(α, t)⟩|w0,n(α) − w0(α)| dα ≲ � R2⟨α⟩|w0,n(α) − w0(α)| dα ≤ ǫ 100 since w0,n → w0 in L1 1(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2), we choose w0,ǫ ∈ C∞ c (R2) such that ∥w0,ǫ − w0∥L1 1(R2) ≤ δǫ for some δ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then it follows that � R2⟨x⟩|w0(X−t n (x)) − w0(X−t(x))| dx ≤ � R2⟨x⟩|w0(X−t n (x)) − w0,ǫ(X−t n (x))| dx (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) + � R2⟨x⟩|w0,ǫ(X−t n (x)) − w0,ǫ(X−t(x))| dx (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) + � R2⟨x⟩|w0,ǫ(X−t(x)) − w0(X−t(x))| dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5) Recall that X−t(·) and X−t n (·) preserves Lebesgue measure, so in view of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1), the integral in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5) can be bounded by ǫ 100 once δ is taken small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4), dominating convergence theorem then gives � R2⟨x⟩|w0,ǫ(X−t n (x)) − w0,ǫ(X−t(x))| dx ≤ ǫ 100 for n large enough and hence the proof of the theorem is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Next we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' That is, for any T > 0, there holds sup t≤T ∥wn(t) − w(t)∥L1 1(R2) → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume this is not true, then there exists a subsequence of wn(which we still denote it by wn), tn ∈ [0, T ] and δ > 0 such that ∥wn(tn) − w(tn)∥L1 1(R2) > δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 31 We may assume without loss of generality that tn → t0 ∈ [0, T ] as n → ∞ since [0, T ] is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then we estimate ∥wn(tn) − w(tn)∥L1 1(R2) = � R2⟨x⟩|wn(x, tn) − w(x, tn)| dx = � R2⟨x⟩|w0,n(X−tn n (x)) − w0(X−tn(x))| dx ≤ � R2⟨x⟩|w0,n(X−tn n (x)) − w0(X−tn n (x))| dx (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6) + � R2⟨x⟩|w0(X−tn n (x)) − w0(X−tn(x))| dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7) For (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='6), in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1), we obtain � R2⟨x⟩|w0,n(X−tn n (x)) − w0(X−tn n (x))| dx = � R2⟨Xn(α, tn)⟩|w0,n(α) − w0(α)| dα ≲ � R2⟨α⟩|w0,n(α) − w0(α)| dα → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' To bound the term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='7), we choose wǫ ∈ C∞ c (R2) such that ∥w0 − wǫ∥L1 1(R2) ≤ δ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then by a similar argument as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='5), there holds � R2⟨x⟩|w0(X−tn n (x)) − w0(X−tn(x))| dx ≤ � R2⟨x⟩|w0(X−tn n (x)) − wǫ(X−tn n (x))| dx + � R2⟨x⟩|wǫ(X−tn n (x)) − wǫ(X−tn(x))| dx + � R2⟨x⟩|wǫ(X−tn(x)) − w0(X−tn(x))| dx ≤ 3δ 100 for n large enough, which contradicts the fact that ∥wn(tn) − w(tn)∥L1 1(R2) > δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus the proof of the whole theorem is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For any w ∈ L1 � L∞(R2), it holds that ∇ · Hw = 0 in the sense of distribution, where Hw(x) := � H(x, y)w(y) dy and the kernel H(x, y) is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume w ∈ C∞ c (R2), then ∇ · Hw = 0 by a direct calculation as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Now for general w ∈ L1 � L∞(R2), we choose a sequence of wǫ ∈ C∞ c (R2) such that wǫ → w in L1 and |wǫ(x)| ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then for any R > 0, it follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2 that |Hw(x)| ≲ ⟨x⟩2 � � 1 + 1 |x − y| � |w(y)| dy ≲ ⟨x⟩2 � ∥w∥L1 + � |x−y|>R |w(y)| |x − y| dy + � |x−y|≤R |w(y)| |x − y| dy � ≲ ⟨x⟩2 � ∥w∥L1 + ∥w∥L1 R + R∥w∥L∞ � , 32 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO which implies |Hw(x)| ≲ ⟨x⟩2 � ∥w∥L1 + ∥w∥1/2 L1 ∥w∥1/2 L∞ � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) once we take R = ∥w∥1/2 L1 ∥w∥1/2 L∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, Hwǫ(x) → Hw(x) uniformly in every compact set since H is a linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Hence for any φ ∈ C∞ c (R2), we have � Hw(x) · ∇φ(x) dx = lim ǫ→0 � Hwǫ(x) · ∇φ(x) dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Next we will show that the particle trajectory map of Hw preserves Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Before the proof, we need the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let {Un(x, t)}∞ n=1 be a sequence of vector fields in Rd satisfying (i) ���Un(x, t) · x |x| ��� ≲ ⟨x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (ii) supx∈BR |Un(x, t)| ≲R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (iii) sup|x|,|z|≤R |Un(x, t) − Un(z, t)| ≲R F(|x − z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let Xn be the particle trajectory map of Un, then for any R, T > 0, Xn is uniformly bounded and equicontinuous in BR ×[0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Similarly, let X−t n (·) be the inverse map of Xn(·, t), then X−t n (x) is uniformly bounded and equicontinuous in BR ×[0, T ] for any R, T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For any |α| ≤ R and t ≤ T , it is easy to check that |Xn(α, t)| = |α| + � t 0 Xn(α, s) |Xn(α, s)| · Un(Xn(α, s), s) ds ≲ |α| + � t 0 |Xn(α, s)| + 1 ds ≲ R + T + � t 0 |Xn(α, s)| ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So Gronwall’s implies that Xn is uniformly bounded in BR × [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In order to show the equicontinuity of Xn, we use assumption (iii) to estimate |Xn(α, t) − Xn(β, t)| = � t 0 Xn(α, s) − Xn(β, s) |Xn(α, s) − Xn(β, s)| (Un(Xn(α, s), s) − Un(Xn(β, s), s)) ds ≲ � t 0 F (|X(α, s) − X(β, s)|) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Next we set z(t) = |Xn(α, t)−Xn(β, t)| and assume that z(t) ≤ 1 100 for all t ∈ [0, T ], then it follows that z(t) ≤ C � t 0 −z(s)(log z(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus by comparison theorem we have z(t) ≤ ee−Ct log(z(0)) ≤ ee−CT log(z(0)) = ee−CT log |α−β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So if we take |α−β| ≤ e−10e−CT , then z(t) ≤ e−10 ≤ 1 100 for all t ∈ [0, T ] and hence |Xn(α, t) − Xn(β, t)| ≤ ee−CT log |α−β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 33 Next we estimate |Xn(α, t1) − Xn(α, t2)| = ���� � t2 t1 Un(Xn(α, s), s) ds ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Recall that Xn is uniformly bounded, so there exists M > 0 such that |Xn(α, t)| ≤ M for all α ∈ BR(0) and t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Together with the fact that Un(·, t) is uniformly bounded in BM × [0, T ], we finally obtain |Xn(α, t1) − Xn(α, t2)| ≲ � t2 t1 1 ds ≲ |t2 − t1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, for any δ > 0 and n ∈ N ∗, there exists ǫ0 = ǫ0(R, T, δ) such that for all α, β ∈ BR, t1, t2 ∈ [0, T ] with |α − β| + |t2 − t1| ≤ ǫ0, there holds |Xn(α, t1) − Xn(β, t2)| ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' The estimates for the inverse map X−t n (x) is similar, see [19] for references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Now we can show that the particle trajectory map of Hw preserves Lebesgue mea- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' More generally, we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let U(x, t) be a velocity field in Rd × [0, T ] with (i) |U(x, t)| ≤ C⟨x⟩2 for some M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (ii) ���U(x, t) · x ⟨x⟩ ��� ≤ C⟨x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (iii) U is locally Log-Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (iv) ∇ · U = 0 in the sense of distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let X(α, t) be the particle trajectory map of U, then the map X(·, t) : Rd → Rd is bijective and preserves Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let η be a smooth positive function supported in B1 and � Rd η(x) dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Define ηǫ(x) = 1 ǫd η(x ǫ ) and Uǫ(x, t) := ηǫ ∗ U(x, t) = � Rd ηǫ(x − y)U(y, t) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Then it is easy to check that for all ǫ ≤ 1, |Uǫ(x, t)| ≲ ⟨x⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Moreover, ����Uǫ(x) · x ⟨x⟩ ���� = � Rd ηǫ(x − y)U(y) · y ⟨y⟩ dy + � Rd ηǫ(x − y)U(y) · ( x ⟨x⟩ − y ⟨y⟩) dy ≲ ⟨x⟩ + ⟨x⟩2 � Rd |x − y|ηǫ(x − y) dy ≲ ⟨x⟩ + ǫ⟨x⟩2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) 34 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO and sup |x|,|z|≤R |Uǫ(x, t) − Uǫ(z, t)| ≲ � Rd ηǫ(y)|Uǫ(x − y, t) − Uǫ(z − y, t)| dy ≲ � Rd ηǫ(y) sup |x|,|z|≤R+1 |U(x, t) − U(z, t)| dy ≲R � Rd ηǫ(y)F(|x − z|) dy ≲R F(|x − z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Next we estimate the particle trajectory map Xǫ(α, t) associated with Uǫ(x, t): ⟨Xǫ(α, t)⟩ = ⟨α⟩ + � t 0 Xǫ(α, s) ⟨Xǫ(α, s)⟩ · Uǫ(Xǫ(α, s), s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Setting zǫ(t) = ⟨Xǫ(α, t)⟩, then it follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) that zǫ(t) ≤ ⟨α⟩ + C � t 0 zǫ(s) + ǫzǫ(s)2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume zǫ(t) ≤ D1eD2t for all t ∈ [0, T ] and ⟨α⟩ ≤ R, then the above inequality implies that zǫ(t) ≤ ⟨α⟩ + CD1 D2 (eD2t − 1) + ǫCD2 1 2D2 (e2D2t − 1) ≤ eD2t � R + 1 + CD1 D2 + ǫCD2 1 2D2 eD2T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, if we take D2 = 100(⟨C⟩ + ⟨R⟩) and D1 = 100R + 100, then there exists ǫ0 = ǫ0(R) such that z(t) ≤ D1eD2t 2 for all ǫ ≤ ǫ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, a bootstrap argument then gives ⟨Xǫ(α, t)⟩ ≲ ⟨α⟩ for all ǫ ≤ ǫ0(R) and ⟨α⟩ ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Furthermore, by a similar argument as in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2, we see that for all R > 0, there exists a positive constant CR such that for all ǫ ≤ ǫ0(R), Xǫ(α, t) and X−t ǫ (α) are uniformly bounded and equicontinuous in BR × [0, T ] with |X(α, t)|, |X−t(α)| ≤ CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Now we take ǫ ≤ ǫ0(CR) and choose subsequence Xn(α, t) and X−t n (α) converges uniformly to X(α, t) and X−t(α) in BCR × [0, T ] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Note that Xn(α, t) = � t 0 Un(Xn(α, s), s) ds, Taking n → ∞ we get X(α, t) = � t 0 U(X(α, s), s) ds, which implies that X is the particle trajectory map of U for ⟨α⟩ ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Moreover, X(X−t(α), t) = lim n Xn(X−t n (α), t) = α for all ⟨α⟩ ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, X−t is the inverse map of X(·, t) at least for ⟨α⟩ ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Now it remains to show that X(·, t) and X−t(·) preserves Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Recall that ∇·U = 0 in the sense of distribution, so Uǫ is a smooth velocity field with ∇·Uǫ = 0 and hence Xǫ(·, t) and X−t ǫ (·) preserves Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus the dominating convergence theorem yields m(X(O, t)) = m(O) = m(X−t(O)) THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 35 for all measurable set O ⊂ BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Since R is arbitrary, X(·, t) and X−t(·) preserves Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' In other words, � Rd f(X(α, t)) dα = � Rd f(x) dx for all f ∈ L1(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Every weak solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) in L∞([0, T ], L1 1 � L∞ 1 ) is indeed la- grangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume w ∈ L∞([0, T ], L1 1 � L∞ 1 ), then the argument in Section 3 implies that |Hw(x, t)| ≲ ⟨x⟩2 and Hw(·, t) is locally log-lip continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Set U = Hw and consider the continuity equation ∂tψ + ∇ · (ψU) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We say that ψ is a weak solution of the continuity equation on [0,T) with initial data w0 if for all test function φ ∈ C∞ c ([0, T ) × R2), there holds − � ψ(x, 0)φ(x, 0) dx = � T 0 � R2 ψ(∂tφ + U · ∇φ) dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, one can check directly that the solution w(x, t) to the two-dimensional helical equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) is also a weak solution to the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Mean- while, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3 implies that ψ(x, t) = w0(X−t(x)) is also a weak solution of the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Note that the weak solutions to the continuity equation in L1([0, T ), L1) is unique, so w(x, t) = w0(X−t(x)) and hence all weak solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) is Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' □ Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' We now give the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Proof of the existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Recall that solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4) is unique in L1 1 � L∞ 1 (R2), so it suffices to prove the existence of weak solutions in [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Now we fix T > 0 and let w0 ∈ C∞ c (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Assume without loss of generality that w0 ̸= 0 and set w0(x, t) := w0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Now we can define wn(x, t) inductively: � ∂twn+1 + Hwn · ∇wn+1 =0 wn+1(x, 0) =w0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Multiply both side by ⟨x⟩, then we obtain ∂t⟨x⟩wn+1 + Hwn · ∇(⟨x⟩wn+1) =wn+1Hwn · ∇⟨x⟩ =wn+1H1wn · x ⟨x⟩ since Hw = H1w + H2w and H2w · x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Note that ∇ · Hwn = 0, so a direct calculation shows that ∥wn(·, t)∥Lp(R2) = ∥w0∥Lp(R2) for any p ∈ [0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Moreover, ∥⟨x⟩wn+1(·, t)∥L1 � L∞(R2) ≤∥⟨x⟩w0∥L1 � L∞(R2) + � t 0 ∥wn+1Hwn∥L1 � L∞(R2) ds ≤∥⟨x⟩w0∥L1 � L∞(R2) + � t 0 ∥wn+1∥L∞(R2)∥Hwn∥L1 � L∞(R2) ds ≤∥⟨x⟩w0∥L1 � L∞(R2) + ∥w0∥L∞(R2) � t 0 ∥Hwn∥L1 � L∞(R2) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' 36 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' GUO AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' ZHAO Thus, ∥wn(t)∥L1 1 � L∞ 1 (R2) ≲ 1 for all t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='12) we get |Hwn(x)| ≲ ⟨x⟩ and ����Hwn(x) · x ⟨x⟩ ���� ≲ 1, □ which implies that ⟨Xn(α, t)⟩ ≈ ⟨α⟩ and hence there exists M > 0 such that wn(·, t) supported in BM(0) for all n > 0 and t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Furthermore, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4 yields |Hwn(x, t) − Hwn(z, t)| ≲ (⟨x⟩ + ⟨z⟩)F(|x − z|) for any R > 0 and t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So in view of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='3, there exist X(α, t), ˜X(α, t) and a subsequence nk such that Xnk and Xnk−1 converges uniformly to X and ˜X, respectively in compact subset of R2 × [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Note that wnk+1 satisfies the transport equation � ∂twnk+1 + Hwnk · ∇wnk+1 =0 wnk+1(x, 0) =w0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So letting k → ∞, by a similar argument as in section 5, we find that w satisfies � ∂tw + H ˜w · ∇w =0 w(x, 0) =w0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Here w(x, t) := w0(X−t(x)) and ˜w(x, t) := w0( ˜X−t(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' To complete the proof, it remains to show that w ≡ ˜w, which is equivalent to X(α, t) = ˜X(α, t) for every α ∈ supp w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Therefore, we define the distance Dn(t) := � |Xn+1(α, t) − Xn(α, t)||w0(α)| dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Observe that Dn(t) is uniformly bounded since w0 has compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' So by dominating convergence theorem, it suffices to show that for any t ∈ [0, T ], limn→∞ Dn(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' First we use Newton-Leibniz formula to conclude that Dn(t) ≤ � t 0 � ��Hwn+1(Xn+1(α, s), s) − Hwn(Xn(α, s), s) �� |w0(α)| dα ds ≤ � t 0 � ��Hwn+1(Xn+1(α, s), s) − Hwn+1(Xn(α, s), s) �� |w0(α)| dα ds (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1) + � t 0 � ��Hwn+1(Xn(α, s), s) − Hwn(Xn(α, s), s) �� |w0(α)| dα ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2) For (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='1), inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='17), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='19) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4 yields � t 0 � ��Hwn+1(Xn+1(α, s), s) − Hwn+1(Xn(α, s), s) �� |w0(α)| dα ds ≲ � t 0 F(Dn+1(s)) ds, THREE-DIMENSIONAL EULER EQUATION WITH HELICAL SYMMETRY 37 where we have used the Jensen’s inequality and the fact that w0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' For (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='2), note that Xn(·, t) preserves Lebesgue measure, so it follows that � t 0 � ��Hwn+1(Xn(α, s), s) − Hwn(Xn(α, s), s) �� |w0(α)| dα ds = � t 0 � ��Hwn+1(x, s) − Hwn(x, s) �� |wn+1(x, s)| dx ds = � t 0 � ���� � H(x, y)wn+1(y, s) dy − � H(x, y)wn(y, s) dy ���� |wn+1(x, s)|dxds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Recall that wn(Xn−1(α, t), t) = w0(α), so we have � t 0 � ��Hwn+1(Xn(α, s), s) − Hwn(Xn(α, s), s) �� |w0(α)| dα ds = � t 0 � ���� � � H(x, Xn(β, s)) − H(x, Xn−1(β, s)) � |w0(β)| dβ ���� |wn+1(x, s)|dxds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Thus, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='13), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='14), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='15), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='17), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='20) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='4 yields � t 0 � ��Hwn+1(Xn(α, s), s) − Hwn(Xn(α, s), s) �� |w0(α)| dα ds ≲ � t 0 F(Dn(s)) ds since w(·, t) supported in BM(0) for all t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Gathering the estimates above, we arrive at Dn(t) ≲ � t 0 F(Dn(s)) ds + � t 0 F(Dn−1(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Let ˜Dn(t) := supm≥n Dm(t), then it follows that ˜Dn(t) ≲ � t 0 F( ˜Dn−1(s)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Setting D∗(t) := limn→∞ ˜Dn(t) and letting n → +∞, we have D∗(t) ≲ � t 0 F(D∗(s)) ds, which implies D∗ ≡ 0 by Osgood’s Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content=' Observe that Dn(t) ≤ ˜Dn(t), so we get lim n→∞ Dn(t) = 0 and hence completes the proof.' metadata={'source': 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Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China Email address: zhaolf@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfcPxh/content/2301.01399v1.pdf'} diff --git a/sNE1T4oBgHgl3EQfQANY/content/tmp_files/2301.03034v1.pdf.txt b/sNE1T4oBgHgl3EQfQANY/content/tmp_files/2301.03034v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..02c8ecaf2aea8cc3ce83862c00512046d7385f2c --- /dev/null +++ b/sNE1T4oBgHgl3EQfQANY/content/tmp_files/2301.03034v1.pdf.txt @@ -0,0 +1,1014 @@ +Hunter: Using Change Point Detection to Hunt for Performance +Regressions +Matt Fleming∗ +matt@codeblueprint.co.uk +Piotr Kołaczkowski∗ +pkolaczk@datastax.com +Ishita Kumar∗ +ishitakumar@umass.edu +Shaunak Das∗ +shaunak.das@datastax.com +Sean McCarthy∗ +sean.mccarthy@datastax.com +Pushkala Pattabhiraman∗ +Pushkala.Pattabhiraman@datastax.com +Henrik Ingo∗ +henrik.ingo@avoinelama.fi +ABSTRACT +Change point detection has recently gained popularity as a method +of detecting performance changes in software due to its ability to +cope with noisy data. In this paper we present Hunter, an open +source tool that automatically detects performance regressions and +improvements in time-series data. Hunter uses a modified E-divisive +means algorithm to identify statistically significant changes in +normally-distributed performance metrics. We describe the changes +we made to the E-divisive means algorithm along with their motiva- +tion. The main change we adopted was to replace the significance +test using randomized permutations with a Student’s t-test, as we +discovered that the randomized approach did not produce deter- +ministic results, at least not with a reasonable number of iterations. +In addition we’ve made tweaks that allow us to find change points +the original algorithm would not, such as two nearby changes. For +evaluation, we developed a method to generate real timeseries, but +with artificially injected changes in latency. We used these data +sets to compare Hunter against two other well known algorithms, +PELT and DYNP. Finally, we conclude with lessons we’ve learned +supporting Hunter across teams with individual responsibility for +the performance of their project. +KEYWORDS +change point detection, performance, benchmarking, continuous +integration +ACM Reference Format: +Matt Fleming, Piotr Kołaczkowski, Ishita Kumar, Shaunak Das, Sean Mc- +Carthy, Pushkala Pattabhiraman, and Henrik Ingo. 2023. Hunter: Using +Change Point Detection to Hunt for Performance Regressions. In Proceedings +∗DataStax, Inc +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +ICPE2023, April 15-19, 2023, Coimbra, Portugal +© 2023 Association for Computing Machinery. +ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 +https://doi.org/10.1145/1122445.1122456 +of ICPE 2023: International Conference on Performance Engineering (ICPE2023). +ACM, New York, NY, USA, 8 pages. https://doi.org/10.1145/1122445.1122456 +1 +INTRODUCTION +Testing the performance of distributed databases, such as Apache +Casandra, is an integral part of the development process and is +often incorporated into Continuous Integration pipelines where +performance tests and benchmarks can be run periodically or in +response to pushing changes to source code repositories. But given +the complex nature of distributed systems, their performance is +often unstable and performance test results can fluctuate from run +to run even on the same hardware. This result instability is due to a +number of factors including variability of the underlying hardware +[7], background processes and CPU frequency scaling at the OS +level, and application-level request scheduling and prioritisation +[1]. All of this makes the job of identifying whether the change in +performance is the result of a software change or simply noise from +the test extremely difficult to do automatically. Threshold-based +techniques are covered in the literature, but these methods do not +handle noise in benchmark data well and require that threshold +values be set per-test [5]. Additionally, thresholds need to be peri- +odically tuned as performance improvements are added and new +baselines are established. +In the past, we have relied heavily on experienced engineers to +visually inspect graphs and benchmark data to identify changes in +performance. However, this suffers from a number of drawbacks +including: +• Expert knowledge for identifying changes is difficult to teach +other engineers +• Small teams of experts have a limit on the number of tests +they can inspect +• Even experienced engineers can miss changes +Because of these drawbacks, we have recently created Hunter[15], +an open source tool that uses change point detection to find statisti- +cally significant changes in time-series data. Change point detection +has recently gained favour as a method of coping with the inherent +instability, or noise, in performance test and benchmark data [5] +and can identify both performance regressions and improvements. +Hunter was designed with the goal of eliminating the need for +a dedicated group of engineers to sift through performance test +arXiv:2301.03034v1 [cs.DB] 8 Jan 2023 + +ICPE2023, April 15-19, 2023, Coimbra, Portugal +Fleming and Kołaczkowski, et al. +results. Instead, individual teams can feed their benchmark data to +a central datastore which Hunter pulls from and analyses. We use +Hunter for validating multiple releases across various distributed +database and streaming products which has required that we make +Hunter intuitive and user-friendly for engineers that are experts in +their particular area but not performance experts. +The contributions in this paper are: +• We present an open-source tool that can run change point +detection on any time-series data containing multiple metrics +in either a .CSV file or stored on a graphite server. +• We discuss the modifications we have made to the E-divisive +means algorithm to improve its performance and predictabil- +ity of results. +• We develop a method for generating timeseries of real bench- +marking results, with artificially injected changes to latency +at discrete points in time. This allows us to evaluate the ac- +curacy of an algorithm objectively, against a known set of +correct change points. +• We compare Hunter (modified E-divisive means) against two +other change point detection algorithms, DYNP and PELT. +• We share the lessons that we have learned from running +Hunter in a multi-team environment where each team is re- +sponsible for a different product and favors different bench- +marks. +2 +HIGH-LEVEL OVERVIEW +Hunter is a command-line tool, written in Python, that detects sta- +tistically significant changes in time-series data stored either in a +CSV file or on a graphite server. It is designed to be easily integrated +into build pipelines [10] and provide automated performance analy- +sis that can decide whether code should be deployed to production. +As well as printing change point data on the command-line, Hunter +also includes support for Slack and can be configured to send results +to a Slack channel. +2.1 +Data Source +Hunter can run analysis on data pulled from a graphite server or +from data contained in an CSV file. Graphite support was necessary +to integrate Hunter into our testing and deployment workflow. If +developers are not using graphite as their central repository of +benchmark data, the CSV support provides a common denominator +for feeding data to Hunter. +2.2 +Configuration +The data sources that Hunter uses are specified in a YAML configu- +ration file. This configuration file has sections for graphite servers, +Slack tokens, and data definitions. Hunter even supports templat- +ing which allows common definitions to be reused and avoids test +definition duplication. Since we routinely use Hunter on hundreds +of tests and metrics, the template feature helps to keep our con- +figuration file small. For example, we use graphite metric prefixes +to group related metrics together so that all metrics for a specific +Apache Cassandra version are linked by a common string. +One example of this is the test db.20k-rw-ts.fixed, a benchmark +running on Datastax Enterprise that performs read and write op- +erations at a fixed rate of throughput. We run this test in both a +configuration with replication factor 1 and with replication factor +3 and yet despite this difference we can reuse around 95% of the +Hunter configuration because the metric types are the same. +Below is an example configuration file. +graphite: +url: http://graphite.local +suffixes: +- ebdse_read.result +templates: +common_metrics: +metrics: +throughput: +scale: 1 +direction: 1 +p99: +scale: 1.0e-6 +direction: -1 +tests: +db.20k-rw-ts.fixed: +inherit: +- common_metrics +tags: +- db.20k-rw-ts.fixed.1-bm2small-rf-1 +prefix: performance_regressions.db.20k-rw-ts.fixed +For test data in CSV files, Hunter allows users to specify at- +tributes of the file such as file path, which columns contain times- +tamps, which contains metrics, and the delimiter character used to +separate fields on each line. +2.3 +Continuous Integration +Since Hunter is a simple Python application, it has proven trivial to +connect with different teams’ CI pipelines. We use a docker image +to run Hunter against daily performance test results which are +stored on a central graphite server. The Docker image is launched +from a Jenkins job that runs once a day. +2.4 +Sending Results to Slack +After running change point detection on a given time-series, Hunter +can submit the results of its analysis to a Slack channel. We have +found that this is the perfect location to notify developers of changes +in performance mainly because each channel is already categorised +by team or project. Developers usually triage the results of Hunter +by investigating any unexpected changes in performance to identify +whether there is a genuine change in performance for the product +or the result was caused by noise in the workload or the platform. +Having Hunter’s results displayed in such a prominent location +as Slack channels has resulted in improvements to the underlying +infrastructure used to run performance tests. When test results +show frequent fluctuations because of noise from the platform, one +of our teams improved the stability of those platforms so that they +are provided with more actionable results from Hunter. + +Hunter: Using Change Point Detection to Hunt for Performance Regressions +ICPE2023, April 15-19, 2023, Coimbra, Portugal +3 +IMPLEMENTATION +Hunter is built on top of the E-divisive Means algorithm available +in the signal_processing_algorithms library from MongoDB [14] +but we have extended it in two ways to improve its efficiency (so +that we can generate results faster) and to get repeatable results +when performing multiple iterations on the same data set. +3.1 +E-divisive Means Algorithm +The E-divisive means [12] is an offline change point detection al- +gorithm that uses hierarchical estimation to estimate the number +and locations of change points in a distribution. Since it’s a non- +parametric method, it makes no assumptions about the underlying +data distribution or the change in distribution and is well suited for +use with benchmarking data that is often non-normal. The hierar- +chical aspect comes into play when deciding which collection of +data points to search for change points. E-divisive means divides +the time-series into two parts and recursively searches for change +points. +Individual points are tested using a test statistic from previous +change points which the literature calls ˆq, and the p-value of ˆq +is determined using random permutation testing which requires +multiple calculations. Using random permutations comes with a +performance cost and we found that detecting change points took +an unreasonably long time for our data set. Additionally, because the +permutations are random we found that the results of Hunter were +non-deterministic and varied from run to run. It is possible to reduce +the non-determinism in the results by increasing the number of +permutations but this has the negative effect of increasing Hunter’s +runtime. In our case running Hunter using the standard E-divisive +means algorithm on hundreds of data points for a single test and +single metric took 2-3 seconds. But to validate a nightly build or +release, developers need the ability to run change point detection +on tens of tests where each test recorded tens of metrics. This would +push the runtime to several minutes, which was no longer ideal. +3.2 +Significance Testing +When initially developing Hunter we profiled the code to under- +stand which parts were taking the longest to detect change points +in our data. We discovered that the vast majority of the time was +spent performing significance testing. This wasn’t entirely sur- +prising given the use of the ˆq statistic and its reliance on random +permutations. We switched to using Student’s t-test and saw the +runtime of Hunter reduce by an order of magnitude as well as +providing consistent results when run multiple times on the same +data set. While Student’s t-test is not a robust measure of statistical +significance for arbitrary data sets, it turned out it works extremely +well for our scenario. +We also tested using the Mann-Whitney U test. This would +have been appealing since, unlike the Student’s t-test, it is a non- +parametric test that doesn’t assume the input data is normally +distributed. But it turned out to not behave very well on small +amounts of data, as it requires 30 points to be conclusive. In con- +trast both the original E-Divisive, and our Student’s t-version, are +able to find changes in extremely short time series with only 4-7 +points. Since E-Divisive is a hierarchical algorithm that splits the +Figure 1: Temporary anomaly example +original time series into ever smaller windows, this is a significant +difference. +3.3 +Fixed-Sized Windows +As we began using Hunter on larger and larger data series, we +discovered that change points identified in previous runs would +suddenly disappear from Hunter’s results. This issue turned out to +be caused by performance regressions that were fixed shortly after +being introduced. This is a known issue with E-divisive means and +is discussed in [5]. Because E-divisive means divides the time series +into two parts, most of the data points on either side of the split +showed similar values. The algorithm therefore, by design, would +treat the two nearby changes as a temporary anomaly, rather than +a persistent change, and therefore filter it out. +Figure 1 illustrates this issue. +Our solution to this problem was to split the entire time series +into fixed-sized windows and run the E-divisive means algorithm +on each window individually. Change points that exist at window +boundaries require special attention since change point detection +algorithms in general are unable to identify whether the most recent +point in a data series is a change point. To address this problem +Hunter allows the windows to overlap and care is taken so that +a change point isn’t reported multiple times because it exists in +multiple windows. +3.4 +Weak Change Points +Splitting the data series into windows partially addresses the prob- +lem of missing change points in large data sets, but we also needed +a method of forcing the E-divisive means algorithm to continue re- +cursively analysing the data series. The E-divisive means algorithm +terminates when the significance test, Student’s t-test in Hunter, +fails. If the algorithm first selects a change point with a p-value +above the threshold set by the user (usually 0.05), it will terminate +immediately, even if it would have detected change points below +the p-value had it continued. We refer to change points that fail the +significance test but would lead to other points below the p-value +as weak change points. +The process of handling weak change points has two steps. First, +we use a larger p-value threshold when splitting so that it allows +detection of weak change points. Second, we reevaluate the p-values +and merge the split data series in a bottom-up way by removing +change points that have a p-value above the smaller, user-specified +threshold. We found that without forcing recursion to continue +Hunter would miss some change points. Our modification results +in much more accurate p-values. + +Throughput (ops/s) +450 K +400 K +350 K +300 K +250 K +200 K +150 K +100 K + 50 K +9/25 +9/28 +10/1 +10/4 +10/7 +10/10 +10/13 +10/16 +10/19 +10/22 +10/25 +10/27 +10/30ICPE2023, April 15-19, 2023, Coimbra, Portugal +Fleming and Kołaczkowski, et al. +Additionally, we filter out change points that show a small rela- +tive change, e.g. change points where the difference in metric value +is below 5%. This relative threshold acts as a filter to discard change +points that are not actionable, i.e. change points that are too small +for developers to reproduce or verify a fix. +4 +EVALUATION +We evaluated our algorithms using benchmark data taken from +a daily Gatling [9] performance test on Datastax Enterprise. The +benchmark data was saved to a CSV file and passed to Hunter using +the following command-line: +poetry run hunter analyse db-gatling.csv +. +The data in db-gatling.csv contains 175 entries and covers 15 +months’ worth of data. There are multiple performance changes +contained within, both improvements (higher throughput or lower +latency) and regressions (lower throughput or higher latency). +We opted for reading the data from a CSV file to avoid network +communication delays with the graphite server influencing the +duration of each run. Every algorithm was run 30 times on the +same CSV file and the mean value, along with 95% confidence +intervals, are reported in Table 1. +Since we found that the permutation algorithm produced unsta- +ble results, we have also included the average number of change +points detected for each of the algorithms in Table 1 as well as the +standard deviations. +4.1 +Quickly Reverted Regressions +Around 2020-10-10 on the graph in Figure 1 we can see a drop +in the throughput. This performance regression was caused by a +change to the way network packet decoding and processing was +done in Datastax Enterprise. This problematic change was reverted +on 2020-10-21 which explains why the throughput metric returns +to previous values shortly after. Two red lines demarcate the data +range where the regression is present. This is a known problem +with change point detection and is explicitly mentioned in [5]. +Both the Student’s t-test and weak change points algorithms +detected this regression and revert in each of the 30 runs through +the data. The permutation based algorithm, only detected these +changes for 15 of the 30 runs, or 50% of the time. +4.2 +MongoDB Performance Test Result Dataset +We also used the publicly available MongoDB Performance Test +Result Dataset [13] to compare the performance of E-divisive means +with random permutations, and Student’s t-test with weak change +points filtering as the statistical significance test. This is the same +data set used in [5], so this analysis should be comparable and +familiar to the emerging change point detection community. +We arbitrarily selected 5 tests from the microbenchmark suite, +and only focus on the max_ops_per_sec result. All results are from +task misc_read_commands and variant linux-wt-standalone. The 5 +timeseries are shown in Figure 2. To avoid clutter, only one time- +series was decorated with the change points found, but the results +for the other 4 are similar. +As discussed above, the original E-divisive algorithm is not de- +terministic. Table 2 shows how many change points were found for +Figure 2: 5 tests from the public MongoDB microbench- +marks dataset. +100 iterations of each timeseries. The results are alarming. For ex- +ample for Remove.IntNonIdNoIndex (row 5) it finds 4 change points +43% of the time, 3 points 50% of the time, but 7% of the time it finds +zero change points! +Figure 2 also shows the other main issue that we have addressed. +When a regresion is quickly followed by a fix or rebound, then the +original algorithm tends to ignore one or both changes as noise. +The 9th change point in the graph is such an example, it is only +found due to the approach with fixed sized windows in this work. +Finally we can clearly see that our implementation finds many +more change points. (The red diamonds are found only with the +Student’s t-test configuration.) This is an expected result as most +of the modifications are motivated by making the algorithm as +sensitive as possible. Whether all 16 change points are meaningful +is ultimately a subjective judgement, but looking closely at the +graph one can at least understand why the algorithm would have +chosen each point. The implementation offers to filter out changes +that are too small to be actionable, but this feature was not used in +these tests, as we wanted to show the full output of our modified +E-divisive algorithm, without post filtering. +An obvious question we can already anticipate is that if the +original implementation from MongoDB performs thís poorly, how +come MongoDB itself has used it so successfully? The answer is +that a a higher level in MongoDB’s performance CI was designed +such that if a point was once flagged as a change point, the system +will forever remember it. This way change points would not ran- +domly disappear while a developer is already working to fix it, and +likewise a point marked as a false positive will remain muted and +not re-appear the next day. But the effect relative to the problems +highlighted in this paper is that the overall MongoDB CI system +will eventually find and remember all change points, even if on a +given day the E-divisive algorithm may stop early and only return +a subset. This higher level system was documented in [11] and the +recorded change points are also part of [13]. +4.3 +Evaluation with a Dataset with Known +Change Points +One weakness in the above analysis, and to our knowledge all +previous literature published on this topic so far, is that judging the +accuracy of the algorithm or its implementation is always subjective. +Ultimately it’s the human evaluator who decides whether a reported + +70000 +60000 +50000 +40000 +Mixed.FindOneUpdatelntld-50-50 + Mixed.FindThenUpdate-50-50 +Remove.Intld +30000 +Remove.IntNonldindex +Remove.IntNonldNolndex +Permutation +20000 +Student's T +Both +10000Hunter: Using Change Point Detection to Hunt for Performance Regressions +ICPE2023, April 15-19, 2023, Coimbra, Portugal +Table 1: Performance and result accuracy for different significance tests in Hunter’s e.divisive implementation. +Algorithm +Mean Duration +Mean 95% CI +Change points +Change points stddev +Permutation +2.221 +2.209, 2.233 +16 +1.174 +Student’s t-test +1.863 +1.853, 1.873 +20 +0 +Student’s + Weak Change Points +1.594 +1.584, 1.603 +16 +0 +Table 2: Distribution of nr of change points found with different statistical tests. (MongoDB data set, 100 iterations) +Algorithm +Test name +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +12 +14 +16 +18 +Permutation +Mixed.FindOneUpdateIntId-50-50 +1 +1 +98 +Permutation +Mixed.FindThenUpdate-50-50 +1 +99 +Permutation +Remove.IntId +64 +32 +4 +Permutation +Remove.IntNonIdIndex +61 +34 +5 +Permutation +Remove.IntNonIdNoIndex +7 +50 +43 +Student’s + Weak +Mixed.FindOneUpdateIntId-50-50 +100 +Student’s + Weak +Mixed.FindThenUpdate-50-50 +100 +Student’s + Weak +Remove.IntId +100 +Student’s + Weak +Remove.IntNonIdIndex +100 +Student’s + Weak +Remove.IntNonIdNoIndex +100 +change point is a true positive, or "useful", or "actionable". And note +that those may not be the same! This is because an objective truth +about the correct set of change points is not available. If we had +that knowledge, we would not have needed this system to begin +with. +It’s of course possible to generate synthetic timeseries with +changes injected at known steps, such as a sine wave or even white +noise, where the mean or amplitude is changing at discrete points. +However these tests tend to feel naive and E-divisive performs quite +well against them. +To obtain a real data set, we employed Chaos Mesh[3] to artifi- +cially generate network latency in the system under test. In other +words we artificially injected real changes, at known points in time, +into a real benchmark producing otherwise realistic results. The +benchmark used was to test Cassandra with the same toolchain +used for CI[6]. +In order to create different time series, we decided on a group of +variables that we varied to generate varying scenarios. We created 9 +different scenarios by altering the values of the following variables: +number of changepoints, magnitude of change of variance between +groups, magnitude of change between groups and the length of +groups. A group is defined as the set of points occurring between +two changepoints. Each scenario contains 5 test series, each with a +minor variation. The scenarios themselves can be grouped into three +categories. The first scenario, change in mean, creates change points +by changing the mean of the groups. The variance remains relative +constant. Similarly, change in variance has constant mean and +varying variance. This case is great to replicate noisy environment’s. +Change in both mean and variance realistically replicates noisy +environment’s with random latencies. +Note that the timeseries used for this evaluation is different +from those in previous sections. Whereas previous timeseries have +been a sequence of (nightly) builds, and the data points represent +values like average throughput during a test, in this evaluation the +timeseries is from a single benchmark, and the values are snapshots +each second. This is because waiting for a year to create a time +series of true nightly builds was not practical. +4.3.1 +Evaluation Metrics. Having obtained data sets with known +change points, we can now employ objective statistical tests to +measure the accuracy of Hunter. Essentially we have recast the +evaluation task as a machine learning problem, where an algorithm +is expected to produce a known output from a given input training +dataset. +We will evaluate hunter using two metrics, F1 score and Rand +Index. In particular we will be evaluating the p99 metric. p99 in- +dicates that 1 in every 100 users will encounter latency. It is a +common industry standard and also used for performance targets +when developing Cassandra. +4.3.2 +True Positives. We will be using the following two variables +to represent the sets of ground truth and predicted points: +𝑋 ∗ : set of ground truth +𝑋 : set of predicted points +True positives are defined as the set of change-points in the +detected class that are real change points i.e they are present in the +set. M represents the scope of error. If the difference between the +predicted points and the true changepoint in less that or equal to +M we will consider it as a correcty predicted changepoint. +𝑇𝑃(𝑋,𝑋 ∗) := {𝑥 ∈ 𝑋 |∃𝑥∗ ∈ 𝑋 ∗𝑠.𝑡.|𝑥 − 𝑥∗| ≤ 𝑀} +It was important to ensure that there were no duplication’s ie. if +two points in the predicted set were in the margin of error of the +same point in 𝑋 ∗. A changepoint in 𝑋 ∗ was marked as visited once +a point in 𝑋 was within 𝑀 and added to 𝑇𝑃(𝑋,𝑋 ∗) set and can not +be considered again. +4.3.3 +F1 Score. The reasons for using the F1 Score to calculate +the accuracy of hunter are that it is unaffected by the size and +the density of data, it penalizes false positives and credits correct +detections. +F1 score is defined as + +ICPE2023, April 15-19, 2023, Coimbra, Portugal +Fleming and Kołaczkowski, et al. +𝐹1 = 2 ∗ 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 ∗ 𝑟𝑒𝑐𝑎𝑙𝑙 +𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 + 𝑟𝑒𝑐𝑎𝑙𝑙 +Precision is the proportion of predicted change points that are +true change points:[17] +𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = |𝑇𝑃(𝑋,𝑋 ∗)| +|𝑋 |∗ +Recall is the proportion of true change points that are well +predicted:[17] +𝑅𝑒𝑐𝑎𝑙𝑙 = |𝑇𝑃(𝑋,𝑋 ∗)| +|𝑋 | +4.3.4 +Rand Index. Another metric we evaluated on was the rand +index. +𝑅𝑎𝑛𝑑𝐼𝑛𝑑𝑒𝑥(𝑋,𝑋 ∗) = +𝑇𝑃 +𝑇𝑁 +𝑇𝑃 +𝑇𝑁 + 𝐹𝑃 + 𝐹𝑁 +• TP : correctly predicts the positive class : True change points +calculated +• TN : correctly predicts the negative class : None in this case +• FP : model incorrectly predicts positive class : |𝑋 ∗|−|𝑇𝑃(𝑋,𝑋 ∗)| +• FN :model incorrectly predicts negative class: |𝑋 |−|𝑇𝑃(𝑋,𝑋 ∗)| +4.3.5 +Benchmarking against PELT and DYNP. With an objective +truth to benchmark against, it also becomes possible to compare +Hunter against alternative change point detection algorithms. We +therefore also present results from two other well-known offline +algorithms PELT and DYNP. These algorithms were used using the +ruptures package in python.[17] +4.3.6 +Results. We ran hunter over 45 test runs. The test runs were +ran on a GKE cluster using Kubernetes. The cluster was ran on 4 x +n2-standard-4 nodes in zone us-central1-a. +All the algorithms were evaluated on two metrics. It can be seen +that hunter has consistently outperformed both pelt and dynp on +both the metrics. In all the experiments we had an margin of error +as 10 seconds. +4.3.7 +Correlation to the number of points. There is a positive corre- +lation between the number of points and accuracy. As the number +of points increase so does hunter’s performance. For Figure 3 it +Figure 3: Correlation between F1 and number of points +can be seen that hunter outperformed both PELT and DYNP with a +huge margine. +4.3.8 +Correlation between delta error and algorithms. Hunter’s per- +formance increases if we allow a larger margin of error. Hunter is +able to get an F1 score of 0.1481 with an delta error of one second, +where PELT and DYNP need a margin of error of at least 3 seconds +to get a non-zero score. This is a key characteristic why E-divisive +has served us well for detecting regressions in CI. Preferably we +like to know the exact commit that caused a regression, not just the +general area whereabouts a regression is suspected. E-divisive is +superior in satisfying especially this requirement! The performance +of all algorithms increases drastically as we give them slightly more +flexibility in terms of margin of error. With a margin of error of 4 +seconds we see that the performance increases to 0.612 for Hunter. +DYNP starts to catch up with Hunters accuracy at 15 seconds. +Figure 4: Correlation between F1 and delta error +5 +LESSONS LEARNED +We have now been operating Hunter for multiple teams for close +to 2 years. In that time we’ve made a number of improvements +in addition to the algorithmic changes covered in Section 3. The +lessons we have learned, and the changes made in response, helped +Hunter to become the de facto choice for statistical significance +detection inside of DataStax. +5.1 +More Data Points Are Better +We originally started off with 2 weeks worth of data points. Given +that performance tests were run once a day this gave us 14 data +points. This decision was primarily because we wanted to avoid the +delay in collecting lots of data from our graphite server. This proved +to be far too few data points to get meaningful results from Hunter +and we increased it to a month (around 30) by default. This wider +time range has allowed Hunter to deal with noise in the results +much better and now we see fewer false positive change points. We +plan to experiment with data sets covering a longer period of time +in the future to see whether we can reduce the false positive rate +even further. + +F1 Scares grauped by algorithm +0.B3 +0.B +Hunter +PELT +0.7 +DYNP +89'0 +0.6 +adi +0.5 +0.40 +0.4 - +6 +0.30 +6 0.3 +0.19 +0.10 +0.12 +0.02 +0.05 +0.D +1 +2 +4F1 score w.r.t vs delta emor +0.B +0.6 +0.2 +Hunter +PELT +0. +DYNP +5 +15 +21 +25 +31 +F1 ScoreHunter: Using Change Point Detection to Hunt for Performance Regressions +ICPE2023, April 15-19, 2023, Coimbra, Portugal +Table 3: Evaluation results +Algorithm +Hunter +Pelt +Dynp +Metric +𝐹1 +𝑅𝑎𝑛𝑑 +𝐹1 +𝑅𝑎𝑛𝑑 +𝐹1 +𝑅𝑎𝑛𝑑 +Single Change Point +Scenario1 +0.261904 +0.166667 +0.0 +0.0 +0.0 +0.0 +Scenario2 +0.466667 +0.305556 +0.027027 +0.013698 +0.285714 +0.166667 +Scenario3 +0.466667 +0.305556 +0.040556 +0.020698 +0.285714 +0.166667 +Two Change Points +Scenario4 +0.666667 +0.666667 +0.060606 +0.031746 +0.190476 +0.111112 +Scenario5 +0.888889 +0.833334 +0.090909 +0.047619 +0.095238 +0.055556 +Scenario6 +0.490476 +0.327777 +0.0 +0.0 +0.0 +0.0 +Four Change Points +Scenario7 +0.925926 +0.866667 +0.249999 +0.142857 +0.444445 +0.285714 +Scenario8 +0.731313 +0.579365 +0.085713 +0.045073 +0.095238 +0.055556 +Scenario9 +0.818182 +0.714286 +0.016460 +0.00843 +0.0 +0.0 +5.2 +New Change Points Matter Most +Once a change point has been reported to a developer it does not +make sense to keep reporting it. When Hunter discovers many +change points, reporting them via Slack can make the results over- +whelming and make it difficult for developers to analyse. Things +are made worse if a change point signals a performance regression +that has since been fixed because Hunter will report both the old +regression and more recent improvement as separate changes. +To quieten the output of Hunter’s Slack feature, we capped re- +sults to only show change points from the last 7 days. While this +does ignore valuable data because the magnitude of the change +point can be updated as new data is processed, those changes are +not important enough to spam everyone on the Slack channel. In +the case where developers need to see the full list of results they +can run Hunter manually on the data series. +5.3 +Change Point Detection Cannot Fix Noisy +Data +One of the teams using Hunter was afflicted with frequent change +point messages via the Slack bot. After investigating these change +points they discovered that the performance of the application +hadn’t changed, rather the change in benchmark results was caused +by unstable hardware performance in a private data center. Changes +of +- 10% for the median latency were typical. +While Hunter can detect statistically significant changes in time +series data, it is still not impervious to data that contains wildly +fluctuating points such as that produced by running benchmarks +on untuned hardware. +However, the fact that the team was unable to fully take advan- +tage of Hunter motivated them to investigate the underlying issue +and then migrate their benchmarks and tests to the cloud, which +was shown to produce more repeatable results than the internal +Figure 5: Unstable performance example +benchmarking lab hardware. After the migration, the benchmark +results were much more stable and Hunter produced far fewer false +positives. Figure 5 shows benchmark data for a single Paxos-based +performance test. Before running the test on the public cloud on +2021-09-18 Hunter detected 3 change points per month, on average. +All of these were false positives, that is changes in results that were +not caused by software or configuration modifications. After the +migration Hunter hasn’t detected a single false positive. +6 +RELATED WORK +We used the work in [5] directly when creating Hunter and the +novel contributions in this paper address some of the open questions +posed there. Specifically, the authors of [5] noted the bias inherent +in the E-divisive means algorithm which favours detecting change +points in the center of clusters. They make up for this bias by +combining change point detection with anomaly detection which +can identify large changes in performance as soon as the first data +point in the new series is seen. Our use of windows for analysing +data series addresses this same bias without resorting to anomaly +detection which lacks the same sensitivity to changes as change +point detection. Additionally, we are able to detect changes sooner, +usually within 1-2 days. + +Paxos Performance Test +1.0 s +35 K +30 K +800 ms +25 K +600 ms +20 K +400 ms +15 K +200 ms +10K +O ns +5K +5/16 +6/1 +6/16 +7/1 +7/16 +8/1 +8/16 +9/1 +9/16 +10/1 +p50 +p75 +p90 +p95 +p99 +- p999 +throughputICPE2023, April 15-19, 2023, Coimbra, Portugal +Fleming and Kołaczkowski, et al. +Continuous Benchmarking [10] is a common technique for en- +suring the performance of a product is maintained or improved +as new code is merged into the source code repository and the +literature includes examples of using change point detection [4] +and threshold-based methods to identify changes in software per- +formance [16] as part of a continuous integration pipeline. Multiple +change point detection algorithms can also be combined into an en- +semble which can outperform the individual algorithms [19] when +identifying performance changes. +The change point detection literature is vast and [2] and [17] +provide excellent overviews and taxonomies of online and offline, +supervised vs unsupervised, change point detection algorithms. In +[2] in particular, online sliding window algorithms are covered in +detail. +Online change point detection has also been applied to identify- +ing changes in performance. [20] combines change point detection +with probabilistic model checking of interval Markov chains to +promptly detect changes in the parameters of software systems and +verify the system’s correctness, reliability, and performance. +Running performance tests in the cloud is known to be suscepti- +ble to performance variability [18] even when running the same +software on the same hardware at different times. Historical per- +formance data can be used to predict the future performance in +cloud environments and [21] explores two change point detection +algorithms, robseg and breakout, to predict variability in the cloud +which enables users to plan repeatable experiments. [8] uses the E- +divisive means algorithm to answer the question: does performance +stability of serverless applications vary over time? +7 +CONCLUSION +Detecting performance regressions across a range of product ver- +sions requires automation to be able to identify them quickly and +without needing expert developers to manually detect them. Change +point detection has emerged as a solution to this problem because +of its ability to cope with noise in the data that is inherent to per- +formance testing. +Hunter is an open source[15] tool that uses change point detec- +tion to automatically identify changes in time-series data, taken +from either a graphite server or CSV file, and report the presence +of change points. Hunter extends the E-divisive Means algorithm +to incorporate a Student’s t-test which removes the indeterminism +present in the original version and provides reproducible results +every time it is run on a single data series. We also introduced a +sliding window technique to detect change points that are tempo- +rally close to each other. In addition to outperforming the original +E-divisive means implementation, Hunter seems to also outperform +two other well known algorithms, PELT and DYNP. +8 +ACKNOWLEDGMENTS +We are grateful to Guy Bolton King for his contributions to Hunter. +REFERENCES +[1] 2014. Tales of the Tail: Hardware, OS, and Application-Level Sources of Tail Latency +(Seattle, WA, USA). Association for Computing Machinery, New York, NY, USA. +https://doi.org/10.1145/2670979.2670988 +[2] Samaneh Aminikhanghahi and Diane Cook. 2017. A Survey of Methods for Time +Series Change Point Detection. Knowledge and Information Systems 51 (05 2017). +https://doi.org/10.1007/s10115-016-0987-z +[3] Cloud Native Computing Foundation. [n.d.]. Chaos Mesh: A Powerful Chaos +Engineering Platform for Kubernetes. https://chaos-mesh.org/ Accessed: 2022- +10-20. +[4] David Daly. 2021. Creating a Virtuous Cycle in Performance Testing at MongoDB. +In Proceedings of the ACM/SPEC International Conference on Performance Engi- +neering (Virtual Event, France) (ICPE ’21). Association for Computing Machinery, +New York, NY, USA, 33–41. https://doi.org/10.1145/3427921.3450234 +[5] David Daly, William Brown, Henrik Ingo, Jim O’Leary, and David Bradford. +2020. The Use of Change Point Detection to Identify Software Performance +Regressions in a Continuous Integration System. In Proceedings of the 2020 +ACM/SPEC International Conference on Performance Engineering(ICPE ’20) (2020). +https://doi.org/10.1145/3358960.3375791 +[6] Datastax. [n.d.]. Fallout: Distributed System Testing as a Service. https://github. +com/datastax/fallout Accessed: 2022-10-20. +[7] Dmitry Duplyakin, Alexandru Uta, Aleksander Maricq, and Robert Ricci. 2020. In +Datacenter Performance, The Only Constant Is Change. In 2020 20th IEEE/ACM +International Symposium on Cluster, Cloud and Internet Computing (CCGRID). +370–379. https://doi.org/10.1109/CCGrid49817.2020.00-56 +[8] Simon Eismann, Diego Elias Costa, Lizhi Liao, Cor-Paul Bezemer, Weiyi Shang, +André van Hoorn, and Samuel Kounev. 2021. A Case Study on the Stability of +Performance Tests for Serverless Applications. arXiv:cs.DC/2107.13320 +[9] Gatling. [n.d.]. Gatling Open-Source Load Testing - For DevOps and CI/CD. +https://gatling.io/ Accessed: 2021-10-07. +[10] Martin Grambow, Fabian Lehmann, and David Bermbach. 2019. Continuous +Benchmarking: Using System Benchmarking in Build Pipelines. In 2019 IEEE +International Conference on Cloud Engineering (IC2E). 241–246. https://doi.org/ +10.1109/IC2E.2019.00039 +[11] Henrik Ingo and David Daly. 2020. Automated System Performance Testing at +MongoDB. In Workshop on Testing Database Systems (DBTest’20) (2020). https: +//doi.org/10.1145/3395032.3395323 +[12] David S. Matteson and Nicholas A. James. 2014. A Nonparametric Approach for +Multiple Change Point Analysis of Multivariate Data. J. Amer. Statist. Assoc. 109, +505 (2014), 334–345. http://www.jstor.org/stable/24247158 +[13] MongoDB. [n.d.]. MongoDB Performance Test Result Dataset. https://zenodo. +org/record/5138516#.YW6T-erMIUH Accessed: 2021-10-13. +[14] MongoDB. [n.d.]. Signal Processing Algorithms. https://github.com/mongodb/ +signal-processing-algorithms Accessed: 2021-10-13. +[15] Piotr Kołaczkowski. [n.d.]. Hunter – Hunts Performance Regressions. +https: +//github.com/datastax-labs/hunter Accessed: 2022-12-12. +[16] Kim-Thomas Rehmann, Changyun Seo, Dongwon Hwang, Binh Than Truong, +Alexander Böhm, and Dong Hun Lee. 2016. Performance Monitoring in SAP +HANA’s Continuous Integration Process. ACM SIGMETRICS Performance Evalu- +ation Review. 43 (2016), 43–52. https://doi.org/10.1145/2897356.2897362 +[17] Charles Truong, Laurent Oudre, and Nicolas Vayatis. 2019. Selective review of +offline change point detection methods. Signal Processing 167 (09 2019), 107299. +https://doi.org/10.1016/j.sigpro.2019.107299 +[18] Alexandru Uta, Alexandru Custura, Dmitry Duplyakin, Ivo Jimenez, Jan Reller- +meyer, Carlos Maltzahn, Robert Ricci, and Alexandru Iosup. 2020. Is Big Data +Performance Reproducible in Modern Cloud Networks?. In 17th USENIX Sym- +posium on Networked Systems Design and Implementation (NSDI 20). USENIX +Association, Santa Clara, CA, 513–527. +https://www.usenix.org/conference/ +nsdi20/presentation/uta +[19] Tim van der Horst. 2021. Change Point Detection In Continuous Integration +Performance Tests. (2021). https://repository.tudelft.nl/islandora/object/uuid: +b9ef4b8e-a18e-40cb-b222-a4221cb22431 MSc Thesis. +[20] Xingyu Zhao, Radu Calinescu, Simos Gerasimou, Valentin Robu, and David Flynn. +2020. Interval Change-Point Detection for Runtime Probabilistic Model Checking. +In 35th IEEE/ACM International Conference on Automated Software Engineering. +[21] Yuxuan Zhao, Dmitry Duplyakin, Robert Ricci, and Alexandru Uta. 2021. Cloud +Performance Variability Prediction. In Companion of the ACM/SPEC Interna- +tional Conference on Performance Engineering (Virtual Event, France) (ICPE ’21). +Association for Computing Machinery, New York, NY, USA, 35–40. +https: +//doi.org/10.1145/3447545.3451182 + diff --git a/sNE1T4oBgHgl3EQfQANY/content/tmp_files/load_file.txt b/sNE1T4oBgHgl3EQfQANY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28149d9fe2cd71c79a10de4e750ec572b65b4a47 --- /dev/null +++ b/sNE1T4oBgHgl3EQfQANY/content/tmp_files/load_file.txt @@ -0,0 +1,598 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf,len=597 +page_content='Hunter: Using Change Point Detection to Hunt for Performance Regressions Matt Fleming∗ matt@codeblueprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='uk Piotr Kołaczkowski∗ pkolaczk@datastax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='com Ishita Kumar∗ ishitakumar@umass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='edu Shaunak Das∗ shaunak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='das@datastax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='com Sean McCarthy∗ sean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='mccarthy@datastax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='com Pushkala Pattabhiraman∗ Pushkala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='Pattabhiraman@datastax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='com Henrik Ingo∗ henrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='ingo@avoinelama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='fi ABSTRACT Change point detection has recently gained popularity as a method of detecting performance changes in software due to its ability to cope with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In this paper we present Hunter, an open source tool that automatically detects performance regressions and improvements in time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Hunter uses a modified E-divisive means algorithm to identify statistically significant changes in normally-distributed performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We describe the changes we made to the E-divisive means algorithm along with their motiva- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The main change we adopted was to replace the significance test using randomized permutations with a Student’s t-test, as we discovered that the randomized approach did not produce deter- ministic results, at least not with a reasonable number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In addition we’ve made tweaks that allow us to find change points the original algorithm would not, such as two nearby changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' For evaluation, we developed a method to generate real timeseries, but with artificially injected changes in latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We used these data sets to compare Hunter against two other well known algorithms, PELT and DYNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Finally, we conclude with lessons we’ve learned supporting Hunter across teams with individual responsibility for the performance of their project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' KEYWORDS change point detection, performance, benchmarking, continuous integration ACM Reference Format: Matt Fleming, Piotr Kołaczkowski, Ishita Kumar, Shaunak Das, Sean Mc- Carthy, Pushkala Pattabhiraman, and Henrik Ingo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Hunter: Using Change Point Detection to Hunt for Performance Regressions.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1145/1122445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1122456 of ICPE 2023: International Conference on Performance Engineering (ICPE2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' ACM, New York, NY, USA, 8 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1145/1122445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1122456 1 INTRODUCTION Testing the performance of distributed databases, such as Apache Casandra, is an integral part of the development process and is often incorporated into Continuous Integration pipelines where performance tests and benchmarks can be run periodically or in response to pushing changes to source code repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' But given the complex nature of distributed systems, their performance is often unstable and performance test results can fluctuate from run to run even on the same hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This result instability is due to a number of factors including variability of the underlying hardware [7], background processes and CPU frequency scaling at the OS level, and application-level request scheduling and prioritisation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' All of this makes the job of identifying whether the change in performance is the result of a software change or simply noise from the test extremely difficult to do automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Threshold-based techniques are covered in the literature, but these methods do not handle noise in benchmark data well and require that threshold values be set per-test [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Additionally, thresholds need to be peri- odically tuned as performance improvements are added and new baselines are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In the past, we have relied heavily on experienced engineers to visually inspect graphs and benchmark data to identify changes in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' However, this suffers from a number of drawbacks including: Expert knowledge for identifying changes is difficult to teach other engineers Small teams of experts have a limit on the number of tests they can inspect Even experienced engineers can miss changes Because of these drawbacks, we have recently created Hunter[15], an open source tool that uses change point detection to find statisti- cally significant changes in time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Change point detection has recently gained favour as a method of coping with the inherent instability, or noise, in performance test and benchmark data [5] and can identify both performance regressions and improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Hunter was designed with the goal of eliminating the need for a dedicated group of engineers to sift through performance test arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='03034v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='DB] 8 Jan 2023 ICPE2023, April 15-19, 2023, Coimbra, Portugal Fleming and Kołaczkowski, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Instead, individual teams can feed their benchmark data to a central datastore which Hunter pulls from and analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We use Hunter for validating multiple releases across various distributed database and streaming products which has required that we make Hunter intuitive and user-friendly for engineers that are experts in their particular area but not performance experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The contributions in this paper are: We present an open-source tool that can run change point detection on any time-series data containing multiple metrics in either a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='CSV file or stored on a graphite server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We discuss the modifications we have made to the E-divisive means algorithm to improve its performance and predictabil- ity of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We develop a method for generating timeseries of real bench- marking results, with artificially injected changes to latency at discrete points in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This allows us to evaluate the ac- curacy of an algorithm objectively, against a known set of correct change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We compare Hunter (modified E-divisive means) against two other change point detection algorithms, DYNP and PELT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We share the lessons that we have learned from running Hunter in a multi-team environment where each team is re- sponsible for a different product and favors different bench- marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 2 HIGH-LEVEL OVERVIEW Hunter is a command-line tool, written in Python, that detects sta- tistically significant changes in time-series data stored either in a CSV file or on a graphite server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' It is designed to be easily integrated into build pipelines [10] and provide automated performance analy- sis that can decide whether code should be deployed to production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' As well as printing change point data on the command-line, Hunter also includes support for Slack and can be configured to send results to a Slack channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1 Data Source Hunter can run analysis on data pulled from a graphite server or from data contained in an CSV file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Graphite support was necessary to integrate Hunter into our testing and deployment workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' If developers are not using graphite as their central repository of benchmark data, the CSV support provides a common denominator for feeding data to Hunter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='2 Configuration The data sources that Hunter uses are specified in a YAML configu- ration file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This configuration file has sections for graphite servers, Slack tokens, and data definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Hunter even supports templat- ing which allows common definitions to be reused and avoids test definition duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Since we routinely use Hunter on hundreds of tests and metrics, the template feature helps to keep our con- figuration file small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' For example, we use graphite metric prefixes to group related metrics together so that all metrics for a specific Apache Cassandra version are linked by a common string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' One example of this is the test db.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='20k-rw-ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='fixed, a benchmark running on Datastax Enterprise that performs read and write op- erations at a fixed rate of throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We run this test in both a configuration with replication factor 1 and with replication factor 3 and yet despite this difference we can reuse around 95% of the Hunter configuration because the metric types are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Below is an example configuration file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' graphite: url: http://graphite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='local suffixes: ebdse_read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='result templates: common_metrics: metrics: throughput: scale: 1 direction: 1 p99: scale: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='0e-6 direction: -1 tests: db.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='20k-rw-ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='fixed: inherit: common_metrics tags: db.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='20k-rw-ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1-bm2small-rf-1 prefix: performance_regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='db.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='20k-rw-ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='fixed For test data in CSV files, Hunter allows users to specify at- tributes of the file such as file path, which columns contain times- tamps, which contains metrics, and the delimiter character used to separate fields on each line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3 Continuous Integration Since Hunter is a simple Python application, it has proven trivial to connect with different teams’ CI pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We use a docker image to run Hunter against daily performance test results which are stored on a central graphite server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The Docker image is launched from a Jenkins job that runs once a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='4 Sending Results to Slack After running change point detection on a given time-series, Hunter can submit the results of its analysis to a Slack channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We have found that this is the perfect location to notify developers of changes in performance mainly because each channel is already categorised by team or project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Developers usually triage the results of Hunter by investigating any unexpected changes in performance to identify whether there is a genuine change in performance for the product or the result was caused by noise in the workload or the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Having Hunter’s results displayed in such a prominent location as Slack channels has resulted in improvements to the underlying infrastructure used to run performance tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' When test results show frequent fluctuations because of noise from the platform, one of our teams improved the stability of those platforms so that they are provided with more actionable results from Hunter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Hunter: Using Change Point Detection to Hunt for Performance Regressions ICPE2023, April 15-19, 2023, Coimbra, Portugal 3 IMPLEMENTATION Hunter is built on top of the E-divisive Means algorithm available in the signal_processing_algorithms library from MongoDB [14] but we have extended it in two ways to improve its efficiency (so that we can generate results faster) and to get repeatable results when performing multiple iterations on the same data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1 E-divisive Means Algorithm The E-divisive means [12] is an offline change point detection al- gorithm that uses hierarchical estimation to estimate the number and locations of change points in a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Since it’s a non- parametric method, it makes no assumptions about the underlying data distribution or the change in distribution and is well suited for use with benchmarking data that is often non-normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The hierar- chical aspect comes into play when deciding which collection of data points to search for change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' E-divisive means divides the time-series into two parts and recursively searches for change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Individual points are tested using a test statistic from previous change points which the literature calls ˆq, and the p-value of ˆq is determined using random permutation testing which requires multiple calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Using random permutations comes with a performance cost and we found that detecting change points took an unreasonably long time for our data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Additionally, because the permutations are random we found that the results of Hunter were non-deterministic and varied from run to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' It is possible to reduce the non-determinism in the results by increasing the number of permutations but this has the negative effect of increasing Hunter’s runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In our case running Hunter using the standard E-divisive means algorithm on hundreds of data points for a single test and single metric took 2-3 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' But to validate a nightly build or release, developers need the ability to run change point detection on tens of tests where each test recorded tens of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This would push the runtime to several minutes, which was no longer ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='2 Significance Testing When initially developing Hunter we profiled the code to under- stand which parts were taking the longest to detect change points in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We discovered that the vast majority of the time was spent performing significance testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This wasn’t entirely sur- prising given the use of the ˆq statistic and its reliance on random permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We switched to using Student’s t-test and saw the runtime of Hunter reduce by an order of magnitude as well as providing consistent results when run multiple times on the same data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' While Student’s t-test is not a robust measure of statistical significance for arbitrary data sets, it turned out it works extremely well for our scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We also tested using the Mann-Whitney U test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This would have been appealing since, unlike the Student’s t-test, it is a non- parametric test that doesn’t assume the input data is normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' But it turned out to not behave very well on small amounts of data, as it requires 30 points to be conclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In con- trast both the original E-Divisive, and our Student’s t-version, are able to find changes in extremely short time series with only 4-7 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Since E-Divisive is a hierarchical algorithm that splits the Figure 1: Temporary anomaly example original time series into ever smaller windows, this is a significant difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3 Fixed-Sized Windows As we began using Hunter on larger and larger data series, we discovered that change points identified in previous runs would suddenly disappear from Hunter’s results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This issue turned out to be caused by performance regressions that were fixed shortly after being introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This is a known issue with E-divisive means and is discussed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Because E-divisive means divides the time series into two parts, most of the data points on either side of the split showed similar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The algorithm therefore, by design, would treat the two nearby changes as a temporary anomaly, rather than a persistent change, and therefore filter it out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Figure 1 illustrates this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Our solution to this problem was to split the entire time series into fixed-sized windows and run the E-divisive means algorithm on each window individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Change points that exist at window boundaries require special attention since change point detection algorithms in general are unable to identify whether the most recent point in a data series is a change point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' To address this problem Hunter allows the windows to overlap and care is taken so that a change point isn’t reported multiple times because it exists in multiple windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='4 Weak Change Points Splitting the data series into windows partially addresses the prob- lem of missing change points in large data sets, but we also needed a method of forcing the E-divisive means algorithm to continue re- cursively analysing the data series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The E-divisive means algorithm terminates when the significance test, Student’s t-test in Hunter, fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' If the algorithm first selects a change point with a p-value above the threshold set by the user (usually 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='05), it will terminate immediately, even if it would have detected change points below the p-value had it continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We refer to change points that fail the significance test but would lead to other points below the p-value as weak change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The process of handling weak change points has two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' First, we use a larger p-value threshold when splitting so that it allows detection of weak change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Second, we reevaluate the p-values and merge the split data series in a bottom-up way by removing change points that have a p-value above the smaller, user-specified threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We found that without forcing recursion to continue Hunter would miss some change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Our modification results in much more accurate p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Throughput (ops/s) 450 K 400 K 350 K 300 K 250 K 200 K 150 K 100 K 50 K 9/25 9/28 10/1 10/4 10/7 10/10 10/13 10/16 10/19 10/22 10/25 10/27 10/30ICPE2023, April 15-19, 2023, Coimbra, Portugal Fleming and Kołaczkowski, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Additionally, we filter out change points that show a small rela- tive change, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' change points where the difference in metric value is below 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This relative threshold acts as a filter to discard change points that are not actionable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' change points that are too small for developers to reproduce or verify a fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 4 EVALUATION We evaluated our algorithms using benchmark data taken from a daily Gatling [9] performance test on Datastax Enterprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The benchmark data was saved to a CSV file and passed to Hunter using the following command-line: poetry run hunter analyse db-gatling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='csv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The data in db-gatling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='csv contains 175 entries and covers 15 months’ worth of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' There are multiple performance changes contained within, both improvements (higher throughput or lower latency) and regressions (lower throughput or higher latency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We opted for reading the data from a CSV file to avoid network communication delays with the graphite server influencing the duration of each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Every algorithm was run 30 times on the same CSV file and the mean value, along with 95% confidence intervals, are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Since we found that the permutation algorithm produced unsta- ble results, we have also included the average number of change points detected for each of the algorithms in Table 1 as well as the standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1 Quickly Reverted Regressions Around 2020-10-10 on the graph in Figure 1 we can see a drop in the throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This performance regression was caused by a change to the way network packet decoding and processing was done in Datastax Enterprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This problematic change was reverted on 2020-10-21 which explains why the throughput metric returns to previous values shortly after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Two red lines demarcate the data range where the regression is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This is a known problem with change point detection and is explicitly mentioned in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Both the Student’s t-test and weak change points algorithms detected this regression and revert in each of the 30 runs through the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The permutation based algorithm, only detected these changes for 15 of the 30 runs, or 50% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='2 MongoDB Performance Test Result Dataset We also used the publicly available MongoDB Performance Test Result Dataset [13] to compare the performance of E-divisive means with random permutations, and Student’s t-test with weak change points filtering as the statistical significance test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This is the same data set used in [5], so this analysis should be comparable and familiar to the emerging change point detection community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We arbitrarily selected 5 tests from the microbenchmark suite, and only focus on the max_ops_per_sec result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' All results are from task misc_read_commands and variant linux-wt-standalone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The 5 timeseries are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' To avoid clutter, only one time- series was decorated with the change points found, but the results for the other 4 are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' As discussed above, the original E-divisive algorithm is not de- terministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Table 2 shows how many change points were found for Figure 2: 5 tests from the public MongoDB microbench- marks dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 100 iterations of each timeseries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The results are alarming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' For ex- ample for Remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='IntNonIdNoIndex (row 5) it finds 4 change points 43% of the time, 3 points 50% of the time, but 7% of the time it finds zero change points!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Figure 2 also shows the other main issue that we have addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' When a regresion is quickly followed by a fix or rebound, then the original algorithm tends to ignore one or both changes as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The 9th change point in the graph is such an example, it is only found due to the approach with fixed sized windows in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Finally we can clearly see that our implementation finds many more change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' (The red diamonds are found only with the Student’s t-test configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=') This is an expected result as most of the modifications are motivated by making the algorithm as sensitive as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Whether all 16 change points are meaningful is ultimately a subjective judgement, but looking closely at the graph one can at least understand why the algorithm would have chosen each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The implementation offers to filter out changes that are too small to be actionable, but this feature was not used in these tests, as we wanted to show the full output of our modified E-divisive algorithm, without post filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' An obvious question we can already anticipate is that if the original implementation from MongoDB performs thís poorly, how come MongoDB itself has used it so successfully?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The answer is that a a higher level in MongoDB’s performance CI was designed such that if a point was once flagged as a change point, the system will forever remember it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This way change points would not ran- domly disappear while a developer is already working to fix it, and likewise a point marked as a false positive will remain muted and not re-appear the next day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' But the effect relative to the problems highlighted in this paper is that the overall MongoDB CI system will eventually find and remember all change points, even if on a given day the E-divisive algorithm may stop early and only return a subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This higher level system was documented in [11] and the recorded change points are also part of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3 Evaluation with a Dataset with Known Change Points One weakness in the above analysis, and to our knowledge all previous literature published on this topic so far, is that judging the accuracy of the algorithm or its implementation is always subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Ultimately it’s the human evaluator who decides whether a reported 70000 60000 50000 40000 Mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='FindOneUpdatelntld-50-50 Mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='FindThenUpdate-50-50 Remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='Intld 30000 Remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='IntNonldindex Remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content="IntNonldNolndex Permutation 20000 Student's T Both 10000Hunter: Using Change Point Detection to Hunt for Performance Regressions ICPE2023, April 15-19, 2023, Coimbra, Portugal Table 1: Performance and result accuracy for different significance tests in Hunter’s e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='divisive implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Algorithm Mean Duration Mean 95% CI Change points Change points stddev Permutation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='221 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='209, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='233 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='174 Student’s t-test 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='863 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='853, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='873 20 0 Student’s + Weak Change Points 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='594 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='584, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='603 16 0 Table 2: Distribution of nr of change points found with different statistical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' (MongoDB data set, 100 iterations) Algorithm Test name 0 1 2 3 4 5 6 7 8 9 12 14 16 18 Permutation Mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='FindOneUpdateIntId-50-50 1 1 98 Permutation Mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='FindThenUpdate-50-50 1 99 Permutation Remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='IntId 64 32 4 Permutation Remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='IntNonIdIndex 61 34 5 Permutation Remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='IntNonIdNoIndex 7 50 43 Student’s + Weak Mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='FindOneUpdateIntId-50-50 100 Student’s + Weak Mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='FindThenUpdate-50-50 100 Student’s + Weak Remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='IntId 100 Student’s + Weak Remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='IntNonIdIndex 100 Student’s + Weak Remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='IntNonIdNoIndex 100 change point is a true positive, or "useful", or "actionable".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' And note that those may not be the same!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This is because an objective truth about the correct set of change points is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' If we had that knowledge, we would not have needed this system to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' It’s of course possible to generate synthetic timeseries with changes injected at known steps, such as a sine wave or even white noise, where the mean or amplitude is changing at discrete points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' However these tests tend to feel naive and E-divisive performs quite well against them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' To obtain a real data set, we employed Chaos Mesh[3] to artifi- cially generate network latency in the system under test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In other words we artificially injected real changes, at known points in time, into a real benchmark producing otherwise realistic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The benchmark used was to test Cassandra with the same toolchain used for CI[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In order to create different time series, we decided on a group of variables that we varied to generate varying scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We created 9 different scenarios by altering the values of the following variables: number of changepoints, magnitude of change of variance between groups, magnitude of change between groups and the length of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' A group is defined as the set of points occurring between two changepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Each scenario contains 5 test series, each with a minor variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The scenarios themselves can be grouped into three categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The first scenario, change in mean, creates change points by changing the mean of the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The variance remains relative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Similarly, change in variance has constant mean and varying variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This case is great to replicate noisy environment’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Change in both mean and variance realistically replicates noisy environment’s with random latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Note that the timeseries used for this evaluation is different from those in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Whereas previous timeseries have been a sequence of (nightly) builds, and the data points represent values like average throughput during a test, in this evaluation the timeseries is from a single benchmark, and the values are snapshots each second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This is because waiting for a year to create a time series of true nightly builds was not practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1 Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Having obtained data sets with known change points, we can now employ objective statistical tests to measure the accuracy of Hunter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Essentially we have recast the evaluation task as a machine learning problem, where an algorithm is expected to produce a known output from a given input training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We will evaluate hunter using two metrics, F1 score and Rand Index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In particular we will be evaluating the p99 metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' p99 in- dicates that 1 in every 100 users will encounter latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' It is a common industry standard and also used for performance targets when developing Cassandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='2 True Positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We will be using the following two variables to represent the sets of ground truth and predicted points: 𝑋 ∗ : set of ground truth 𝑋 : set of predicted points True positives are defined as the set of change-points in the detected class that are real change points i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='e they are present in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' M represents the scope of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' If the difference between the predicted points and the true changepoint in less that or equal to M we will consider it as a correcty predicted changepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 𝑇𝑃(𝑋,𝑋 ∗) := {𝑥 ∈ 𝑋 |∃𝑥∗ ∈ 𝑋 ∗𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='|𝑥 − 𝑥∗| ≤ 𝑀} It was important to ensure that there were no duplication’s ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' if two points in the predicted set were in the margin of error of the same point in 𝑋 ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' A changepoint in 𝑋 ∗ was marked as visited once a point in 𝑋 was within 𝑀 and added to 𝑇𝑃(𝑋,𝑋 ∗) set and can not be considered again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3 F1 Score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The reasons for using the F1 Score to calculate the accuracy of hunter are that it is unaffected by the size and the density of data, it penalizes false positives and credits correct detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' F1 score is defined as ICPE2023, April 15-19, 2023, Coimbra, Portugal Fleming and Kołaczkowski, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 𝐹1 = 2 ∗ 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 ∗ 𝑟𝑒𝑐𝑎𝑙𝑙 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 + 𝑟𝑒𝑐𝑎𝑙𝑙 Precision is the proportion of predicted change points that are true change points:[17] 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = |𝑇𝑃(𝑋,𝑋 ∗)| |𝑋 |∗ Recall is the proportion of true change points that are well predicted:[17] 𝑅𝑒𝑐𝑎𝑙𝑙 = |𝑇𝑃(𝑋,𝑋 ∗)| |𝑋 | 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='4 Rand Index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Another metric we evaluated on was the rand index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 𝑅𝑎𝑛𝑑𝐼𝑛𝑑𝑒𝑥(𝑋,𝑋 ∗) = 𝑇𝑃 +𝑇𝑁 𝑇𝑃 +𝑇𝑁 + 𝐹𝑃 + 𝐹𝑁 TP : correctly predicts the positive class : True change points calculated TN : correctly predicts the negative class : None in this case FP : model incorrectly predicts positive class : |𝑋 ∗|−|𝑇𝑃(𝑋,𝑋 ∗)| FN :model incorrectly predicts negative class: |𝑋 |−|𝑇𝑃(𝑋,𝑋 ∗)| 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='5 Benchmarking against PELT and DYNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' With an objective truth to benchmark against, it also becomes possible to compare Hunter against alternative change point detection algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We therefore also present results from two other well-known offline algorithms PELT and DYNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' These algorithms were used using the ruptures package in python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' [17] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='6 Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We ran hunter over 45 test runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The test runs were ran on a GKE cluster using Kubernetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The cluster was ran on 4 x n2-standard-4 nodes in zone us-central1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' All the algorithms were evaluated on two metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' It can be seen that hunter has consistently outperformed both pelt and dynp on both the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In all the experiments we had an margin of error as 10 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='7 Correlation to the number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' There is a positive corre- lation between the number of points and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' As the number of points increase so does hunter’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' For Figure 3 it Figure 3: Correlation between F1 and number of points can be seen that hunter outperformed both PELT and DYNP with a huge margine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='8 Correlation between delta error and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Hunter’s per- formance increases if we allow a larger margin of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Hunter is able to get an F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1481 with an delta error of one second, where PELT and DYNP need a margin of error of at least 3 seconds to get a non-zero score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This is a key characteristic why E-divisive has served us well for detecting regressions in CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Preferably we like to know the exact commit that caused a regression, not just the general area whereabouts a regression is suspected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' E-divisive is superior in satisfying especially this requirement!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The performance of all algorithms increases drastically as we give them slightly more flexibility in terms of margin of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' With a margin of error of 4 seconds we see that the performance increases to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='612 for Hunter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' DYNP starts to catch up with Hunters accuracy at 15 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Figure 4: Correlation between F1 and delta error 5 LESSONS LEARNED We have now been operating Hunter for multiple teams for close to 2 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In that time we’ve made a number of improvements in addition to the algorithmic changes covered in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The lessons we have learned, and the changes made in response, helped Hunter to become the de facto choice for statistical significance detection inside of DataStax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1 More Data Points Are Better We originally started off with 2 weeks worth of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Given that performance tests were run once a day this gave us 14 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This decision was primarily because we wanted to avoid the delay in collecting lots of data from our graphite server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This proved to be far too few data points to get meaningful results from Hunter and we increased it to a month (around 30) by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' This wider time range has allowed Hunter to deal with noise in the results much better and now we see fewer false positive change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We plan to experiment with data sets covering a longer period of time in the future to see whether we can reduce the false positive rate even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' F1 Scares grauped by algorithm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='B3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='B Hunter PELT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content="7 DYNP 89'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='6 adi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='4 - 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='30 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='D 1 2 4F1 score w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='t vs delta emor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='2 Hunter PELT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' DYNP 5 15 21 25 31 F1 ScoreHunter: Using Change Point Detection to Hunt for Performance Regressions ICPE2023, April 15-19, 2023, Coimbra, Portugal Table 3: Evaluation results Algorithm Hunter Pelt Dynp Metric 𝐹1 𝑅𝑎𝑛𝑑 𝐹1 𝑅𝑎𝑛𝑑 𝐹1 𝑅𝑎𝑛𝑑 Single Change Point Scenario1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='261904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='166667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='0 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='490476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='327777 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='0 Four Change Points Scenario7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='925926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='866667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='249999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='142857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='444445 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='285714 Scenario8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='731313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='579365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='085713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='045073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='095238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='055556 Scenario9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='818182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='714286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='016460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='00843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='2 New Change Points Matter Most Once a change point has been reported to a developer it does not make sense to keep reporting it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' When Hunter discovers many change points, reporting them via Slack can make the results over- whelming and make it difficult for developers to analyse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Things are made worse if a change point signals a performance regression that has since been fixed because Hunter will report both the old regression and more recent improvement as separate changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' To quieten the output of Hunter’s Slack feature, we capped re- sults to only show change points from the last 7 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' While this does ignore valuable data because the magnitude of the change point can be updated as new data is processed, those changes are not important enough to spam everyone on the Slack channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In the case where developers need to see the full list of results they can run Hunter manually on the data series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3 Change Point Detection Cannot Fix Noisy Data One of the teams using Hunter was afflicted with frequent change point messages via the Slack bot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' After investigating these change points they discovered that the performance of the application hadn’t changed, rather the change in benchmark results was caused by unstable hardware performance in a private data center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Changes of +- 10% for the median latency were typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' While Hunter can detect statistically significant changes in time series data, it is still not impervious to data that contains wildly fluctuating points such as that produced by running benchmarks on untuned hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' However, the fact that the team was unable to fully take advan- tage of Hunter motivated them to investigate the underlying issue and then migrate their benchmarks and tests to the cloud, which was shown to produce more repeatable results than the internal Figure 5: Unstable performance example benchmarking lab hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' After the migration, the benchmark results were much more stable and Hunter produced far fewer false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Figure 5 shows benchmark data for a single Paxos-based performance test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Before running the test on the public cloud on 2021-09-18 Hunter detected 3 change points per month, on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' All of these were false positives, that is changes in results that were not caused by software or configuration modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' After the migration Hunter hasn’t detected a single false positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 6 RELATED WORK We used the work in [5] directly when creating Hunter and the novel contributions in this paper address some of the open questions posed there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Specifically, the authors of [5] noted the bias inherent in the E-divisive means algorithm which favours detecting change points in the center of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' They make up for this bias by combining change point detection with anomaly detection which can identify large changes in performance as soon as the first data point in the new series is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Our use of windows for analysing data series addresses this same bias without resorting to anomaly detection which lacks the same sensitivity to changes as change point detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Additionally, we are able to detect changes sooner, usually within 1-2 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Paxos Performance Test 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='0 s 35 K 30 K 800 ms 25 K 600 ms 20 K 400 ms 15 K 200 ms 10K O ns 5K 5/16 6/1 6/16 7/1 7/16 8/1 8/16 9/1 9/16 10/1 p50 p75 p90 p95 p99 p999 throughputICPE2023, April 15-19, 2023, Coimbra, Portugal Fleming and Kołaczkowski, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Continuous Benchmarking [10] is a common technique for en- suring the performance of a product is maintained or improved as new code is merged into the source code repository and the literature includes examples of using change point detection [4] and threshold-based methods to identify changes in software per- formance [16] as part of a continuous integration pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Multiple change point detection algorithms can also be combined into an en- semble which can outperform the individual algorithms [19] when identifying performance changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' The change point detection literature is vast and [2] and [17] provide excellent overviews and taxonomies of online and offline, supervised vs unsupervised, change point detection algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In [2] in particular, online sliding window algorithms are covered in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Online change point detection has also been applied to identify- ing changes in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' [20] combines change point detection with probabilistic model checking of interval Markov chains to promptly detect changes in the parameters of software systems and verify the system’s correctness, reliability, and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Running performance tests in the cloud is known to be suscepti- ble to performance variability [18] even when running the same software on the same hardware at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Historical per- formance data can be used to predict the future performance in cloud environments and [21] explores two change point detection algorithms, robseg and breakout, to predict variability in the cloud which enables users to plan repeatable experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' [8] uses the E- divisive means algorithm to answer the question: does performance stability of serverless applications vary over time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 7 CONCLUSION Detecting performance regressions across a range of product ver- sions requires automation to be able to identify them quickly and without needing expert developers to manually detect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Change point detection has emerged as a solution to this problem because of its ability to cope with noise in the data that is inherent to per- formance testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Hunter is an open source[15] tool that uses change point detec- tion to automatically identify changes in time-series data, taken from either a graphite server or CSV file, and report the presence of change points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Hunter extends the E-divisive Means algorithm to incorporate a Student’s t-test which removes the indeterminism present in the original version and provides reproducible results every time it is run on a single data series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' We also introduced a sliding window technique to detect change points that are tempo- rally close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In addition to outperforming the original E-divisive means implementation, Hunter seems to also outperform two other well known algorithms, PELT and DYNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 8 ACKNOWLEDGMENTS We are grateful to Guy Bolton King for his contributions to Hunter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' REFERENCES [1] 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Tales of the Tail: Hardware, OS, and Application-Level Sources of Tail Latency (Seattle, WA, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1145/2670979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='2670988 [2] Samaneh Aminikhanghahi and Diane Cook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' A Survey of Methods for Time Series Change Point Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Knowledge and Information Systems 51 (05 2017).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Performance Monitoring in SAP HANA’s Continuous Integration Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' ACM SIGMETRICS Performance Evalu- ation Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 43 (2016), 43–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='org/10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='sigpro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='107299 [18] Alexandru Uta, Alexandru Custura, Dmitry Duplyakin, Ivo Jimenez, Jan Reller- meyer, Carlos Maltzahn, Robert Ricci, and Alexandru Iosup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 2020.' metadata={'source': 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In 17th USENIX Sym- posium on Networked Systems Design and Implementation (NSDI 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' USENIX Association, Santa Clara, CA, 513–527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='usenix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='org/conference/ nsdi20/presentation/uta [19] Tim van der Horst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Change Point Detection In Continuous Integration Performance Tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' https://repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='nl/islandora/object/uuid: b9ef4b8e-a18e-40cb-b222-a4221cb22431 MSc Thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' [20] Xingyu Zhao, Radu Calinescu, Simos Gerasimou, Valentin Robu, and David Flynn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Interval Change-Point Detection for Runtime Probabilistic Model Checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In 35th IEEE/ACM International Conference on Automated Software Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' [21] Yuxuan Zhao, Dmitry Duplyakin, Robert Ricci, and Alexandru Uta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Cloud Performance Variability Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' In Companion of the ACM/SPEC Interna- tional Conference on Performance Engineering (Virtual Event, France) (ICPE ’21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 35–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='1145/3447545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} +page_content='3451182' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfQANY/content/2301.03034v1.pdf'} diff --git a/tdAzT4oBgHgl3EQfr_1k/content/tmp_files/2301.01652v1.pdf.txt b/tdAzT4oBgHgl3EQfr_1k/content/tmp_files/2301.01652v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f13e662f9a59a1a1e805996781bd8431428cf6ce --- /dev/null +++ b/tdAzT4oBgHgl3EQfr_1k/content/tmp_files/2301.01652v1.pdf.txt @@ -0,0 +1,1214 @@ +Role of ionic surfactant in magnetic dynamics of +self-assembled dispersions of nanoplatelets +Hajnalka Nádasia,∗, Melvin Küsterb, Alenka Merteljc, Nerea Sebastiánc, Patricija +Hribar Bo˘stjan˘ci˘cc,d, Darja Lisjake, Thilo Viereckb, Margaret Rosenbergf, Alexey +O. Ivanovh,i, Sofia S. Kantorovichf,g, Alexey Eremina, Frank Ludwigb,∗ +aInstitute of Physics, Otto von Guericke University, Universitätsplatz +2, Magdeburg, 39106, Germany +bInstitute of Electrical Measurement Science and Fundamental Electrical Engineering (EMG) +and Laboratory for Emerging Nanometrology (LENA), TU Braunschweig, Hans-Sommer-Str. +66, Braunschweig, 38106, Germany +cDepartment of Complex Matter, Jožef Stefan Institute, Jamova cesta +39, Ljubljana, SI-1000, Slovenia +dJožef Stefan International Postgraduate School, Jamova cesta 39, Ljubljana, SI-1000, Slovenia +eDepartment for Materials Synthesis, Jožef Stefan Institute, Jamova cesta +39, Ljubljana, SI-1000, Slovenia +fFaculty of Physics, University of Vienna, Boltzmanngasse 5, Vienna, 1090, , Austria +gResearch Platform “Mathematics-Magnetism-Materials”, University of +Vienna, Oskar-Morgenstern-Platz 1, Vienna, 1090, Austria +hUral Federal University, 51 Lenin Ave., Ekaterinburg, Russian Federation +i M.N. Mikheev Institute of Metal Physics UB RAS, 18 S. Kovalevskaya +str., Ekaterinburg, Russian Federation +Abstract +In complex colloidal systems, interparticle interactions strongly affect the dynam- +ics of the constituting particles. A study of the dynamical response also provides +invaluable information on the character of those interactions. Here we demon- +strate how tuning the electrostatic interactions by an ionic surfactant in dispersions +of magnetic nanoplatelets leads to developing new dynamic modes in magnetic +response spectra. The collective modes can be induced or suppressed by either +varying the concentration ratio of the magnetic nanoplatelets (MP) to the surfac- +tant or increasing the MP concentration reflecting the nanoscale characteristics of +∗Corresponding authors. E-mail adresses: +hajnalka.nadasi@ovgu.de (H. Nádasi) +f.ludwig@tu-braunschweig.de (F. Ludwig). +Preprint submitted to Journal of Molecular Liquids +January 5, 2023 +arXiv:2301.01652v1 [cond-mat.soft] 4 Jan 2023 + +this fluid magnet. +Keywords: soft matter, magnetic nanoplatelets, barium hexaferrite, +ferromagnetic nematics, magnetic dynamics, AC susceptometry +1. Introduction +Ferrofluids are one of the most spectacular examples of complex fluids where +nanoscale properties can be transformed to the macroscopic level [1, 2, 3]. Un- +derstanding the nanoscale dynamics in such systems allows accurate manipulation +and fine-tuning of their macroscopic properties [4, 5, 6, 7]. These dynamics are +governed by two key attributes of the system: the electrostatic interactions, which +ensure the colloidal stability of the magnetic nanoparticles in the liquid dispersion +medium, and the magnetic dipole moment of the nanoparticles. Understand the in- +terplay between these two properties is critical to the tailoring of the ferrofluids’s +microstructure, which can then give rise to the desired macroscopic properties. +Surfactants stabilise colloidal suspensions by physical or chemical adsorption +on the suspended particles’ surfaces. Depending on the solvent affinity of the sur- +factant’s head either they form a single layer around the particles adsorbed by their +solvophobic head(s) while dispersing them in the medium by the solvophilic tail, +or they organise into a double layer, whereby the overlapping solvophobic tails are +enclosed by the solvophilic heads. The key to successfully integrating the parti- +cles in the dispersing medium is to match the dielectric properties of the surfactant +and the solvent. However, the suspension’s longevity depends on the effective pre- +vention of aggregation [8, 9]. In the case of magnetic fluids, it is also essential to +counteract the colloidal self-assembly caused by dipolar interactions, which can +be significantly enhanced by the magnetic field. While reversible agglomeration +is desirable in specific use cases, such as magnetooptical effects [10, 11, 12, 13], +extensive irreversible aggregation of dipolar nanoparticles is generally undesirable +due to the loss of colloidal stability. +Ferrimagnetic scandium-doped barium hexaferrite platelets (Sc-BaHF) are (shape- +) anisotropic particles with their magnetisation perpendicular to the basal plane +[14, 15]. +When such platelets are dispersed into a nematic liquid crystal matrix (5CB +(4-Cyano-4’-pentylbiphenyl), E7), the anchoring of the nematic director at the par- +ticle surfaces may stabilise the colloidal ferromagnetic nematic order as demon- +strated in [16, 17, 18]. It is the first example of the long-anticipated ferromagnetic +nematic, as predicted by de Gennes [19]. Ferromagnetic nematic order can also be +2 + +realised in the pure colloidal liquid crystal, where the nanoplatelets are dispersed +in an isotropic fluid, e.g. 1-butanol [20, 21, 22]. +The high magnetic moment of ferrimagnetic Sc-BaHF platelets requires elec- +trostatic stabilisation against aggregation, which can be achieved by an ionic sur- +factant such as dodecylbenzenesulfonic acid (DBSA) [23]. +In our previous paper [24] we investigated the dynamic magnetic response of +a dilution series of Sc-BaHF ferrofluids. +The dispersion medium 1-butanol contained 4.5 mM surfactant (DBSA) in the +stock suspension. To ensure the stability of the ferrofluid, it was diluted by the +stock solution with the same concentration of DBSA during preparation. In an +oscillating magnetic field, we observed collective modes of the as-prepared fer- +rofluid series. The collective modes were well resolved in the AC susceptibility +(ACS) spectra of the ferrofluids with low magnetic particle concentration. A pos- +sible explanation for the appearance of those modes is that as we dilute the stock +suspension by 4.5 mM DBSA solution, the degree of concentration-change is dif- +ferent for the MPs compared to that of the DBSA. +The adsorption-desorption equilibrium of the surfactant at the MP surface de- +pends not only on the DBSA but also on the MP concentration (푐MP). Boštjančič et +al. [25] investigated suspensions of Sc-BaHF platelets in the concentration range +of 푐MP = 5 g L−1 − 30 g L−1. The ratio of the adsorbed DBSA to the total amount of +DBSA in the suspension increases with increasing platelet concentration. The dis- +solved molecules in the dispersing medium are partly dissociated. With increasing +concentration of this ionic form, the ionic strength increases, whereby the Debye +screening length of the platelets decreases. The repulsive electrostatic interactions +also depend on the effective charge of the platelets, which was found to almost +independent of the DBSA concentration (푐DBSA). +To illustrate the effects of altering the DBSA concentration on degree of self- +assembly in the colloidal suspension, Figure 1 shows how even a simple approxi- +mation of the interparticle interactions is strongly affected by changes in the screen- +ing length. The dipolar interaction between two magnetic nanoplatelets with mag- +netic moment 흁푖 and separated by the vector 퐫 can be written as: +푈dd(퐫) = 휇0 +4휋 +((흁푖흁푗) +|푟|3 +− +3(흁푖퐫)(흁푗퐫) +|푟|5 +) +(1) +and the Coulomb potential with added Yukawa screening term can written as: +푈C−DH(퐫) = +1 +4휋휖0 +푞푖 ⋅ 푞푗 +|푟| +exp (−휅|푟|) +(2) +3 + +where 푞푖 and 푞푗 are the particle charges and 휅 is the Debye screening constant. For +this approximation, we assume that the two platelets are of the same size, and have +the same magnetic moment and effective charge, which reduces the total potential +to: +푈tot(퐫) = 푝m ⋅ −2휇2 +푟3 ++ 푝c ⋅ 푞2 +푟 exp (−휅푟) +(3) +where 푝m and 푝c are scaling factors which encompass the relative strength of the +magnetic and electrostatic interactions. Based on experimental observations, we +fix 푝c > 푝m and vary 휅 to approximate the effect of added DBSA. +Figure 1: Total interaction potential of a pair of charged magnetic discs, oriented head to tail in a +simplified model of the effects of increased Debye screening. The 푥-axis represents the distance +between disc centers, while the 푦-axis shows the potential energy, normalised by the magnetic +dipole-dipole interaction prefactor. The three curves show that as the screening is increased, the +potential goes from repulsive to mildly attractive. +Although the simplifying assumptions in Figure 1 are too strong to claim that +this potential quantitatively represents the experimental system - both the high +degree of polydispersity and the more complex electrostatics would need to be +considered in greater detail - it qualitatively shows that gradually increasing the +screening length will destabilise the system, leading to self-assembly. +4 + +2.0 +No screening +1.5 +Low screening +High screening +1.0 +(y)"d/on +0.5 +0.0 +-0.5 +-1.0 +1.5 +-2.0 +2 +3 +4 +5To understand the influence of the ratio 푐MP∕푐DBSA on the dynamic magnetic re- +sponse, in this work, we explored magnetic dynamics in two series of suspensions: +(i) a series with low but constant magnetic particle concentration (푐MP = 8 g L−1) +and varying 푐DBSA (Set 1) and (ii) a dilution series where we kept the 푐MP∕푐DBSA +ratio constant (Set 2). The latter was chosen so that the ferrofluid with the low- +est platelet concentration (푐MP = 8 g L−1) does not give rise to collective modes +in the ACS spectra. The charge and the ionic strength determine the electrostatic +environment of the ferrofluid: the desorption of the surfactant molecules from the +positively charged platelets and the dissociation of the dissolved DBSA. On the +first approach, we presume that keeping the 푐MP∕푐DBSA constant, the electrostatic +environment does not change significantly (the adsorption-desorption equilibrium +of the surfactant is not shifted) and the response would only depend on the mag- +netic particle concentration i.e the magnetostatic interactions, hence the study of +Set 2. +In the following we discuss collective modes of magnetic dynamics in Sc- +BaHF suspensions. We show, that even at low magnetic particle concentration, +as in Set 1 (푐MP = 8 g L−1), collective modes emerge on increasing surfactant con- +centration (푐DBSA). Besides, investigating Set 2, we also demonstrate the role of +the magnetic particle concentration (푐MP) in the collective behaviour. +2. Materials and methods +2.1. Ferrofluid preparation +The stock ferrofluid was prepared as described in [24]. The sets of ferroflu- +ids were prepared by diluting the stock suspension (푐MP = 304 g L−1 and 푐DBSA = +43 g L−1) to the necessary concentration of MPs and DBSA with appropriate DBSA +solution in 1-butanol and/or by 1-butanol. The resulting suspensions were soni- +cated for a minute. Set 1 marks the suspensions with constant 푐MP = 8 g L−1 and +varying 푐DBSA. They are designated as MP8 and the corresponding weight per- +cent of DBSA as in Table 1. Set 2 identifies the dilution series with varied 푐MP +but constant 푐MP∕푐DBSA. They are designated with the abbreviation MP and the +corresponding magnetic particle concentration as in Table 1. +2.2. AC susceptometry +Measurements of the AC susceptibility were carried out with a custom-made +setup, which was originally built for measurements of the dynamics of magnetic +nanoparticles in a rotating magnetic field [26]. For the ACS measurements just one +set of Helmholtz coils was used to generate the sinusoidal excitation field. ACS +5 + +Set 1 푐MP=8 g L−1 +Set 2 푐MP∕푐DBSA=6.7 +Designation +%DBSA +휙MP/휙DBSA +Designation +푐MP∕gL−1 +휙MP +MP8_12 +12 +1.44 +MP8 +8 +0.0015 +MP8_13 +13 +1.37 +MP12 +12 +0.0024 +MP8_16 +16 +1.04 +MP32 +32 +0.0063 +MP8_21 +21 +0.73 +MP40 +40 +0.0077 +MP8_25 +25 +0.61 +MP92 +92 +0.0170 +MP8_26 +26 +0.57 +MP126 +126 +0.0229 +MP8_50 +50 +0.20 +MP158 +158 +0.0285 +Table 1: In Set 1 the magnetic particle concentration 푐MP = 푚MP∕푉1−BuOH is fixed to 8 g L−1. The +DBSA concentration is given as %DBSA = 푚DBSA∕(푚MP+푚DBSA). The volume percents apply to +the total volume of the ferrofluid. In the dilution series Set 2 the concentration ratio 푐MP∕푐DBSA is +fixed to 6.7. The concentration 푐MP = 푚MP∕푉1−BuOH is utilized to designate the suspensions. The +volume fraction 휙MP = 푉MP∕푉FF. MP8_13 of Set 1 and MP8 of Set 2 are identical suspensions. +spectra were recorded at 298 K in a frequency range between 0.1 Hz and 2.2 kHz +and at field amplitudes between 0.5 mT and 5 mT. +The exposure of a ferrofluid to an alternating magnetic field results in aligned +magnetisation, which will oscillate with a phase lag. The dynamic response can +be described by the real 휒′ and imaginary 휒′′ parts of the magnetic susceptibility. +The ACS susceptibility spectra of MP suspensions are generally analysed with the +Debye model where the complex susceptibility is given by +휒(휔) = +휒0 +1 + 푖휔휏 +(4) +with the static susceptibility 휒0, the angular frequency 휔 = 2휋푓 and the char- +acteristic relaxation time of the MP 휏. Since the Sc-BaHF platelets are thermally +blocked [24], only Brownian relaxation occurs. The Brownian relaxation time [27] +휏B is determined by the viscosity of the medium 휂, the hydrodynamic volume of +the MP 푉H and the temperature 푇 : +휏B = 3휂푉H +푘B푇 +(5) +The field dependence of the Brownian relaxation time is well described by the +empirical model by Yoshida and Enpuku [28] in the frame of the Fokker-Planck +formalism. For non-interacting thermally blocked MPs, their model is valid for +magnetic field amplitudes with a Langevin parameter 휉 = 푚휇0퐻∕ (푘B푇 ) of up to +300. The relaxation time is given by: +6 + +휏B,H = +휏B,0 +√ +1 + 0.126휉1.72, +(6) +where 휏B,0 is the relaxation time in the limit 퐻 → 0. +To account for a distribution of relaxation times due to the polydispersity of +MPs and interparticle interactions, we apply the Cole-Cole equation [29]. +휒(푓) = 휒∞ + +Δ휒 +1 + (푖휔휏B,H +)1−훼 +(7) +considering symmetric broadening only. Here 휒∞ is the susceptibility in the high- +frequency limit, Δ휒 is proportional to the amplitude of the susceptibility, and 훼 is +the broadening coefficient (훼 = 0 corresponds to the Debye model). In order to +analyse ACS spectra with multiple relaxation modes, the experimental data were +fitted with a sum of several Cole-Cole equations: +휒(푓) = 휒∞ + +∑ +푖 +Δ휒푖 +1 + (푖휔휏B,H,푖 +)1−훼푖 +(8) +In our study, up to three modes 푖 could be resolved. +To match these experiment to further theoretical work, we will later fit the +calculated Langevin susceptibility 휒퐿 to an adaption of the susceptibility which +includes chain formation [11]. However, this model only was dervied for dipolar +hard spheres, the shape of which (and thus the self-assembly process) significantly +differs from that of spheres. Therefore, we have calculated a new expression for the +partition function, the calculation of which is described in the SI). The resulting +expression was then fitted using a least-squares fit to determine 휆. +3. Results +3.1. Analysis of ACS spectra of Set 1 +To explore the influence of the surfactant concentration on the magnetic re- +sponse, we measured the set of ferrofluids 1 containing a fixed low concentration +of MPs (8 g L−1) and varied 푐DBSA. The ACS spectra of the suspensions where +휙MP∕휙DBSA > 1 are characterised by a single peak in the high-frequency range +attributed to the relaxation of single platelets (Fig. 2). In the following, we desig- +nate this relaxation mode as a high-frequency (HF) mode. With increasing field +7 + +amplitude the peaks’ maximum shifts to higher frequencies and its amplitude de- +creases. Similar behaviour was observed in aqueous suspensions of spherical [30] +and rod-like nanoparticles [31]. +Figure 2: The ACS spectra of MP8_13 recorded for different amplitudes of the probe field consist +of a single symmetric high-frequency peak related to the single-platelet relaxation mode. The +maximum shifts to higher frequencies with increasing probe field. +A further increase of 푐DBSA results in the emergence of low-frequency modes +(Fig. 3). The onset of the slowest mode (LF) can only be distinguished at low probe +fields. The spectral maximum of this mode shifts to higher frequencies with in- +creasing probe field while its amplitude decreases and the peak apparently flattens. +This contribution acts effectively as an offset of the spectrum. +An additional peak in the intermediate range, denoted as middle-frequency +mode (MF), emerges in this concentration range. This mode merges into the HF +mode with increasing probe field while the amplitude seemingly decreases. How- +ever, since the modes overlap, it is difficult to estimate the relaxation times accu- +rately. +The ACS spectra of MP8_50 have very distinctive features (Fig. 4). They con- +sist of a very pronounced HF peak and especially prominent low-frequency collec- +8 + +0.07 +μoH / mT +0.5 +0.06 +1.0 +1.5 +0.05 +2.0 +2.5 +0.04 +a +3.0 +0.03 +3.5 +4.0 +0.02 +4.5 +5.0 +0.01 +0.00 +0.1 +10 +1 +100 +1000 +f/Hz(a) +(b) +(c) +Figure 3: Multiple peaks of MP8_26 at (a) 0.5 mT, (b) 2.5 mT and (c) 5 mT. With increasing probe +field, the middle- and high-frequency peaks collapse as a result of progressing mode-overlapping. +The slowest mode (LF mode) shifts to higher frequencies and broadens, transforming into a dimin- +ishing flattened peak. +Figure 4: The ACS spectra of MP8_50 recorded for different amplitudes of the probe field are +dominated by collective modes. The nearly linear in log-scale slope in the frequency range below +100 Hz +for AC fields between 3 mT - 5 mT implies the complex spectral structure of mul- +tiple overlapping peaks. +9 + +0.08 +LF-mode +MF-mode +0.06- +HF-mode +--Eq. (5) fit +1 +Exp. data +.U +0.04 +a +0.02- +0.00- +0.1 +1 +10 +100 +1000 +10000 +f /Hz0.08 +LF-mode +MF-mode +0.06- +HF-mode +--Eq. (5) fit +Exp. data +n' +0.04 +a +X +0.02 +10:00888 +0.00. +0.1 +1 +10 +100 +1000 +10000 +f /Hz0.08 +LF-mode +MF-mode +0.06- +HF-mode +--- Eq. (5) fit +0 +0 +0 +Exp. data +n' +0.04 +0. +a +0 +0 +0 +9 +0 +0.02 +0 +0 +00.000 +0.00 +0.1 +1 +10 +100 +1000 +10000 +f /Hz0.08 +μoH/ mT +0.5 +3.0 +0.07 +1.0 +3.5 +1.5 +4.0 +0.06 +2.0 +4.5 +2.5 +5.0 +一 +0.05 +a +0.04 +0.03 +0.02 +0.01 +0.1 +1 +10 +100 +1000 +f/Hztive modes. Already in 0.5 mT probe field, the remaining LF peak is significantly +stronger than the HF peak maximum. In 0.5 mT and 1 mT fields, one can fit three +peaks that can be assigned to the LF, MF and HF modes (Fig. 5). In higher fields, +we observe a nearly linear increase of the imaginary part of the magnetic suscep- +tibility as a function of log 푓 and an HF peak. Its mode structure could only be +implicitly determined using Eq. (8) over a reduced frequency interval. Multiple +overlapping in the ACS spectra of MP8_50 in high probe fields does not allow +accurate Cole-Cole fits. As a result, we cannot accurately determine the position +of the HF peak and its dependence on the probe field amplitude from the fits. +The ACS measurement results described above are in qualitative agreement +with the findings in [25]. At a sufficiently low DBSA concentration (12-16%; cf. +Table 1), we observe just an HF mode in the ACS spectra, which we attribute to +the Brownian rotation of single platelets, while at higher DBSA concentrations, +collective modes appear, which indicate that the electrostatic repulsions are less +effective to prevent the platelets from clustering. +(a) +(b) +Figure 5: Multiple peaks in spectra of MP8_50 recorded in probe fields with amplitude 0.5 mT (a), +and and 1 mT (b) with dominating but well distinguishable low- and middle-frequency modes. +3.1.1. Field dependence of the Brownian relaxation time of the HF mode +The Brownian relaxation times 휏B,H estimated from the position of the HF peak +in the ACS spectra as a function of the applied field amplitude are depicted in +Fig. 6(a) along with the fits with Eq. (6). Since 휏B,0 and 휉 are treated as free pa- +rameters, the fitting allows one to determine the field-free Brownian relaxation +10 + +0.09 +LF-mode +0.08 +Q +MF-mode +0.07 +HF-mode +0.06 +---Eq. (5) fit +0 +0.05 +Exp. data +a.u. +Q +0.04 +0 +0 +88 +X +0.03 +0.02 +0.01- +0.00- +-0.01 +0.1 +1 +10 +100 +1000 +10000 +f/Hz0.09 +LF-mode +0.08 +0 +MF-mode +0.07 +:HF-mode +0.06 +---- Eq. (5) fit +0.05 +Exp. data +0 +a.u. +1 +0.04 +0 +X +0.03 +0 +0.02 +0 +0 +0 +0.01- +0.00- +-0.01 +0.1 +1 +10 +100 +1000 +10000 +f/Hztime 휏B,0 and the magnetic moment 푚 for the individual samples. The results are +depicted in Figs. 6(b, c). +(a) +(b) +(c) +Figure 6: (a) Fit of the field dependence of the relaxation time of the HF mode using the model +by Yoshida and Enpuku (Eq. (6)) for Set 1. The extracted magnetic moment (b) and field-free +relaxation time values (c) as a function of the concentration ratio of the magnetic particles to the +surfactant DBSA. +As shown in Fig. 6(b), the effective magnetic coupling as measured via 푚 +slightly decreases with increasing amount of DBSA. Since the magnetic moment +of the particles directly corresponds to their size, the gradual decrease indicates +that with increasing surfactant concentration means the successively smaller par- +ticles are responding to the field. As Figs. 3-5 show, additional low-frequency +relaxation modes appear at higher DBSA concentrations, which can be attributed +to MP clusters. +Due to the polydispersity of the platelets, these observations suggest that smaller +platelets can still freely rotate contributing to the HF peak, while larger platelets +form clusters. The remaining platelets contributing to the high-frequency range of +the spectrum are, on average, smaller and thus have smaller magnetic moments 푚, +resulting in shorter relaxation times. +3.2. Analysis of ACS spectra of Set 2 +As the results of Set 1 indicate, a ratio 푐MP∕푐DBSA = 6.7 prevents the Sc-BaHF +platelets - at least at comparably low 푐MP - from clustering. The second set of +magnetic fluids (Set 2) was prepared on the simplified assumption that keeping the +ratio of magnetic particles to the surfactant constant at 6.7 results in an unaltered +electrostatic environment in the ferrofluids. The spectra of the ferrofluids with +the lowest MP concentrations, 휙MP = 0.0015 and 0.0024 (MP8 and MP12) have a +single high-frequency peak corresponding to the single platelet relaxation (Fig. 2). +Upon further increasing the MP concentration to 휙MP = 0.0063 and 0.0077 (MP32 +11 + +4.65 - +4.60 - +4.55 - +4.50- +4.45 +4.40- +/ w +4.35 +4.30 - +4.25- +4.20- +4.15- +0.6 +0.8 +1.0 +1.2 +1.4 +UMP / ΦDBSA0.29- +0.28. +0.27 +S +m +0.26 +0.25 +0.24 - +0.23 +0.6 +0.8 +1.0 +1.2 +1.4 +UMP / DBSAMP8 +26 +0.27 +MP8 25 +MP8 +21 +0.24 +MP8 +16 +ms +MP8 +13 +0.21. +MP8 +12 +0.18- +0.15. +0.5 +1.0 +1.5 +2025 +5 3.0 3.5 +4.0 +4.5 +5.0 +μoH/ mTand MP40), a weak LF peak emerges at 0.5 mT, which is suppressed in stronger +fields. First, at 휙MP = 0.0170 (MP92), we could detect three modes in low probe +fields (0.5 mT - 1.5 mT) and two modes in higher fields (2 mT - 5 mT). The LF +mode again shows a very broad spectrum, as described above. At higher concen- +tration 휙MP = 0.0229 (MP126), the fits reveal two modes (Fig. 7). In low probe +fields (0.5 mT - 1.5 mT), there is an HF and an LF mode. With further increas- +ing the probe field, the LF mode is not detectable anymore; the remaining peak is +symmetric in shape. +However, in even higher fields (4.5 mT - 5 mT), the peak again becomes slightly +asymmetric, and a new collective mode emerges. It has, nonetheless much lower +amplitude than the low-frequency mode in low magnetic fields. +At the highest MP concentration 휙MP = 0.0285 (MP158), the character of the +ACS spectra is similar, apart from the fact that the high probe field collective mode +occurs already at 4 mT. The effect of dipolar interactions on the ACS spectra at +moderate MP concentrations has theoretically been studied by Ivanov and Camp. +With increasing MP concentration, which affects the amount of self-assembly in +the system and considering self-assembled clusters of MP, the position of the max- +imum in the ACS imaginary part shifts to smaller frequencies [32]. Extending the +work to higher dipolar coupling constants 휆 ≥ 4, but still low MP concentrations, +the authors found the appearance of additional peaks in the ACS imaginary part, +which they attribute to the response of chains and rings [33]. While at low dipolar +interaction parameters 휆, the Brownian rotation of single MP dominate, at inter- +mediate 휆, the formation of particle chains and rings occurs with a peak frequency +in the ACS imaginary part well below that of single MP. This seems to be the same +underlying phenomenon as we find in our study, as dipolar magnetic nanoplatelets +also form chains for at high values of 휆 (or low electrostatic repulsion). However, +our study does not show a high-frequency peak (at about 24휔휏퐵), which they at- +tribute to the motion of particles inside chains or rings. We exclude the latter mode +for our case since no indication of such a mode was found in the measured ACS +spectra- which makes sense given that platelets do not form rings and are sterically +precluded from rotation inside chains. +3.2.1. Field dependence of the Brownian relaxation time of HF mode +The field dependence of the Brownian relaxation time of individual platelets, +as determined from the peak frequency of the HF mode, is depicted in Fig. 8(a) +in dependence of the applied field amplitude. The field-free Brownian relaxation +time and its decay with increasing field amplitude strongly increase with 휙MP. The +latter would reflect an increase in the effective coupling, which would be measured +12 + +(a) +(b) +(c) +(d) +Figure 7: Cole-Cole fits of the ACS spectra of MP126 in AC fields of (a) 0.5 mT, (b) 1.5 mT, (c) +2 mT and (d) 4.5 mT. The low-frequency collective mode shifts with increasing field amplitude +much stronger to higher frequencies than the HF peak does, so that at 2 mT the ACS spectrum +consists of one symmetric peak. At an even higher field amplitude of 4.5 mT, the nematic mode +emerges as a new low-frequency mode. +as an increase in the magnetic moment 푚. +Recently, Rusanov et al. [34] extended the model for the field-dependent Brow- +nian relaxation time by Yoshida and Enpuku by additionally considering the effect +of dipolar interactions. For the Brownian relaxations, they obtained the following +empirical formula: +13 + +1.6 +LF-mode +1.4 + HF-mode +0 +00 +---·Eq. (5) fit +1.2 +00 +Exp. data +0 +1.0- +0 +0 +0 +0 +0.8 +a +0 +0.6 +1 +0 +0.4- +0 +0.2- +0.0. +0.1 +1 +10 +100 +1000 +10000 +f/Hz1.6 +LF-mode +1.4 +HF-mode +-- Eq. (5) fit +1.2 +0 +0 +Exp. data +0 +0 +1.0- +0 +0.8 +0 +a +0.6 +0.4- +0 +0 +0.2. +0 +0.0. +0.1 +1 +10 +100 +1000 +10000 +f/Hz1.6 +--·Eq. (5) fit +1.4 +Exp. data +1.2 +0 +0 +0 +0 +0 +1.0- +0 +0 +Q +0 +0.8 +0 +a +0 +0.6 +Z +0.4 - +0 +0 +0.2- +0.0- +0.1 +1 +10 +100 +1000 +10000 +f/Hz1.6 +LF-mode +1.4 +HF-mode +--·Eq. (5) fit +1.2 +Exp. data +1.0- +0.8 +a +0 +17 +0.6 +0 +0 +0 +0.4- +0.2- +0.0- +0.1 +1 +10 +100 +1000 +10000 +f/Hz휏B,H = +휏B,0 +√ +(1 − 휒eff +3 )2 + 0.076휉2 +(9) +(a) +(b) +(c) +Figure 8: (a) Field dependence of the Brownian relaxation time 휏B,H for Set 2. Lines show the fits +with Eq. (9). (b) shows the extracted field-free Brownian relaxation time 휏B,0 vs MP concentration +휙MP, with lines as guides to the eye. Figure (c) shows the effective static susceptibility 휒eff vs the +volume fraction 휙MP (points) fitted to the chain-corrected model of Langevin susceptibility using +the correct platelet partition function (line). +Here the effective susceptibility 휒eff describes the effect of magnetic interac- +tions of the single platelets with the surrounding medium on the Brownian relax- +ation time 휏B,H. To fit the data in Fig. 8(a), we determined the mean magnetic +moment 푚 of the platelets and the field-free Brownian relaxation time 휏B,0 of non- +interacting particles by fitting the data points of sample MP8 while setting 휒eff = 0 +14 + +1.6 +0.0015 +1.4 +0.0024 +1.2 +0.0063 +0.0077 +ms +1.0 +0.0170 +0.0229 +0.8 +0.0285 +0.6 +0.4 +0.2 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +μoH/mT0.65 +0.60 +0.55 +0.50 +S +sw +0.45 +0.40 +0.35. +0.30 +0.25 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.03010 +0.1 +0.01 +Model fit with 2=4.26536 +Experimental +0.001 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030(negligible dipolar interactions between particles). For the fits of the other data +sets, 푚 was fixed at this value (푚 = 4.58 × 10−18 A m2) while both 휒eff and 휏B,0 +were taken as free parameters (Fig. 8). Although both the mean magnetic mo- +ment 푚 and the mean hydrodynamic size of the platelets are independent of MP +concentration, the use of 휏B,0 as a free parameter is due to the fact that the vis- +cosity is expected to increase with increasing 휙MP. The results for 휒eff and 휏B,0 +in dependence of 휙MP are depicted in Fig. 8(b) and 8(c). The viscosity increases +by about a factor of 2, i.e., assuming that the viscosity of pure 1-butanol at room +temperature amounts to 2.54 mPa s, it increases to about 6.2 mPa s for the sample +with a Sc-BaHF concentration of 158 g L−1. The rise of 휒eff with 휙MP is weaker +than linear. The effective susceptibility 휒eff in Eq. 9 is dominated by clusters due +to their high magnetic moments. The static susceptibility of clusters was theo- +retically described by Mendelev and Ivanov [11]. Adapting the partition function +used in this theory to platelets (see SI), the expression for 휒eff was fitted to the data +points in Fig. 8(c). As a measure of magnetic dipolar interactions, the adjusted +dipolar coupling constant 휆 = 휇0푚2∕(4휋퐷3푘B푇 ) is used, where 푚 is the magnetic +moment and 퐷 is the diameter of the platelets. In the curve in Fig. 8(c), the aspect +ratio of the platelet was taken to be 1 ∶ 10, which resulted in a fitted parameter +휆 = 4.3. This aspect ratio corresponds to a mean thickness of 5 nm and diameter +of 50 nm of the platelets as reported in [24]. Note that a value 휆 = 4.1 is calculated +for 푚 = 4.58 × 10−18 A m2, 퐷 = 50 nm and 푇 = 296 K, in excellent agreement +with the fitted value. Deviations of the fitted line from the experimental data in +Fig. 8(c) may be caused by the fact that the ratio between single and clustered +platelets, which differently enter 휒eff, may vary when changing the concentration +of Sc-BaHF, 휙MP. In addition, no electrostatic interactions but only dipolar ones +are included, point-like dipoles rather than distributed and asymmetric ones are +considered and distributions of parameters are not accounted for. +3.3. Effect of a DC-Bias Field +Studying the effect of a DC bias field on the magnetic response allows us to se- +lectively suppress the modes and explore their individual behaviour in dependence +on the orientation of the bias field. +One example containing pronounced LF and MF modes is MP8_25. Apply- +ing a DC bias field parallel to the AC probe field for sample MP8_25 results in a +suppression of the collective modes in the low-frequency range and, depending on +the DC field strength, in a shift and repression of the HF peak maxima (Fig. 9(a)). +The suppression of the low-frequency mode indicates, that the larger assemblies of +magnetic nanoparticles are aligned with the field, and Brownian relaxation of the +15 + +(a) +(b) +Figure 9: Influence of DC bias field applied parallel to the AC probe field on the ACS spectra of +MP8_25 (a) and MP158 (b). (a) In a low DC field (0.5 mT), the low-frequency collective modes +are suppressed as the clusters with high effective magnetic moment align first along the field. By +a further increase of the DC field (2.5 mT), the relaxation of single platelets with bigger size i.e. +higher magnetic moment gets constrained, too. Hence, on (b), the single peak of the ACS spectrum +belongs two more than one mode since the low DC bias field (0.5 mT) mitigates it. +smaller particles now dominates the spectrum. Indeed, LF modes result from the +collective behaviour of large ensembles (clusters) of MPs which are most suscep- +tible to aligning along the DC field, hence as a primary response, a low (0.5 mT) +DC field suppresses the LF modes. +Non-correlated single platelets couple individually to the magnetic field hav- +ing much smaller coupling energy than the clusters do, and their relaxation can be +quenched in significantly higher fields. As a result, the suppression of the HF peak +occurs at a much higher DC bias field proving, that single platelets with compara- +bly small 푚 contribute to the HF peak. The shift of the peak frequency to higher +values and the decrease of its amplitude with increasing DC bias field strength are +in agreement with theoretical models [35, 36]. +This approach allows us to also characterise the magnetic dynamics in MP158 +(Fig. 9(b)) having a single broad spectral feature in the ACS spectrum. With in- +creasing DC bias field, the peak frequency shifts from about 50 Hz at zero bias +field to 782 Hz at 2.5 mT. As seen in Fig. 9(a), a DC bias field suppresses the +low-frequency part of the spectrum already at 퐻bias = 0.5 mT so that the peak +at 2.5 mT is expected to be dominated by the HF mode. The fact that the peak +frequency of the HF mode in Fig. 9(b) is slightly lower than that in Fig. 9(a) can +be attributed to the effect of the Zeeman and dipolar interactions (cf. Sect. 3.2.1). +16 + +0.08 +Bias field in mT +0.07 +0 +0.06 +0.5 +0.05 +一 2.5 +0.04 +a +0.03 +X +0.02 +0.01 +0.00 +-0.01 +0.1 +10 +100 +1000 +1 +f/Hz2.0 +Bias field in mT +1.8 +0 +1.6 +0.5 +1.4 +2.5 +1.2- +u +1.0- +a +0.8- +0.6 - +0.4 +0.2 - +0.0- +-0.2 +0.1 +10 +100 +1000 +1 +f/HzFor comparison, the peak position in the absence of bias field in Fig. 9(b) lies at 50 +Hz for MP158, at a more than a magnitude lower frequency than that in Fig. 9(a) +for MP8_25 indicating, that the whole spectrum in Fig. 9(b) is the superposition +of collective low-frequency and single-platelet high-frequency relaxation modes. +Figure 10: Relaxation times in a 0.5 mT DC bias field of the single-platelet (▪) and the collective (▪) +modes as a function of the magnetic particle concentration for Set 2. The two modes reveal opposite +tendencies with increasing 푐MP: while the single-platelet mode slows down, the collective mode +speeds up. +4. Conclusions and outlook +In our experiments we demonstrated that - tuning the electrostatic interactions +by adjusting the DBSA concentration in the dispersions of Sc-BaHF nanoplatelets +- strongly affects the structure of the low-frequency magnetic response (Set 1). +17 + +LF-mode +HF-mode +0.1 +S +0.01 +T +0.001 +1E-4 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +CMP / gL-1An increase in the DBSA concentration reduces the repulsive interactions and en- +hances interparticle correlations resulting in the growth of collective behaviour. +Collective modes developing in the system are very sensitive to the magnetic field +amplitude becoming faster in stronger fields, which suggests a high magnetic mo- +ment is associated with the relaxation modes. Another feature of these modes +is their shift to higher frequencies with growing concentration of MPs (Set 2). +At the same time, the single-platelet relaxation mode becomes slower with in- +creasing 푐MP, because of increasing dipolar interactions and increasing viscosity +as it is expected (Fig. 10). As the MP concentration increases, the motion of +the particles becomes strongly correlated leading to the development of orienta- +tional order and a restoring force, which accelerates the relaxation processes. This +restoring force also contributes to the nematic order above a critical concentra- +tion (푐MP = 126 g L−1). At somewhat lower concentration, highly correlated clus- +ters appear in the isotropic suspensions and are responsible for a strong magnetic +and magnetooptical response. A similar situation of the so-called para-nematic +state has been observed in dispersions of rod- and plate-shaped pigment parti- +cles [37, 38, 39, 40]. +As a next step to determine the significance of viscosity change with increasing +magnetic particle concentration, we are planning to measure viscosity of varied +concentration of ferrofluids in and without magnetic field. We will also elaborate +on the ACS measurements in DC bias fields - parallel and perpendicular to the AC +field - to comprehend more facets of the complexity of the spectra. +CRediT authorship contribution statement +Hajnalka Nádasi: Conceptualisation, Investigation, Analysis, Writing - orig- +inal draft, Writing - Review & Editing. Melvin Küster: Conceptualisation, Inves- +tigation, Analysis, Writing - original draft, Writing - Review & Editing, Visual- +ization. Alenka Mertelj: Investigation, Resources, Writing - Review & Editing. +Nerea Sebastián: Investigation, Resources, Writing - original draft, Writing - Re- +view & Editing. Patricija Hribar Boštjančič: Investigation, Resources. Darja +Lisjak: Investigation, Resources. Thilo Viereck: Conceptualisation, Resources, +Review & Editing. Margaret Rosenberg: Computation, Simulations, Writing - +Review & Editing. Alexey O. Ivanov: Conceptualisation, Computation, Writing - +Review & Editing. Sofia Kantorovich: Conceptualisation, Computation, Writing +- Review & Editing. Alexey Eremin: Conceptualisation, Writing - original draft, +Writing - Review & Editing. Frank Ludwig: Conceptualisation, Investigation, +Resources, Writing - original draft, Writing - Review & Editing. +18 + +Conflicts of interest +There are no conflicts to declare. +Acknowledgements +F.L., M.K., A.E. and H.N. acknowledge the support of the Deutsche Forschungs- +gemeinschaft (Projects NA 1668/1-1 and LU 800/7-1). 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Eremin, R. Stannarius, S. D. Peroukidis, S. H. L. Klapp, S. Klein, +Colloidal Suspensions of Rodlike Nanocrystals and Magnetic Spheres under +an External Magnetic Stimulus: Experiment and Molecular Dynamics Sim- +ulation, Langmuir 32 (20) (2016) 5085. +[39] K. May, A. Eremin, R. Stannarius, B. Szabó, T. Börzsönyi, I. Appel, +S. Behrens, S. Klein, Exceptionally large magneto-optical response in dis- +persions of plate-like nanocrystallites and magnetic nanoparticles, Journal +of Magnetism and Magnetic Materials 431 (2016) 79. +[40] K. May, R. Stannarius, K. Kang, P. K. Challa, S. Sprunt, A. Jákli, S. Klein, +A. Eremin, Collective dynamics in dispersions of anisometric pigment parti- +cles, Journal of Molecular Liquids 267 (2018) 322. +22 + diff --git a/tdAzT4oBgHgl3EQfr_1k/content/tmp_files/load_file.txt b/tdAzT4oBgHgl3EQfr_1k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..15e87c73341061a95fef00c0cb67992aa7ca7bef --- /dev/null +++ b/tdAzT4oBgHgl3EQfr_1k/content/tmp_files/load_file.txt @@ -0,0 +1,964 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf,len=963 +page_content='Role of ionic surfactant in magnetic dynamics of self-assembled dispersions of nanoplatelets Hajnalka Nádasia,∗, Melvin Küsterb, Alenka Merteljc, Nerea Sebastiánc, Patricija Hribar Bo˘stjan˘ci˘cc,d, Darja Lisjake, Thilo Viereckb, Margaret Rosenbergf, Alexey O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Ivanovh,i, Sofia S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Kantorovichf,g, Alexey Eremina, Frank Ludwigb,∗ aInstitute of Physics, Otto von Guericke University, Universitätsplatz 2, Magdeburg, 39106, Germany bInstitute of Electrical Measurement Science and Fundamental Electrical Engineering (EMG) and Laboratory for Emerging Nanometrology (LENA), TU Braunschweig, Hans-Sommer-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 66,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Braunschweig,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 38106,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Germany cDepartment of Complex Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Jožef Stefan Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Jamova cesta 39,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Ljubljana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' SI-1000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Slovenia dJožef Stefan International Postgraduate School,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Jamova cesta 39,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Ljubljana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' SI-1000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Slovenia eDepartment for Materials Synthesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Jožef Stefan Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Jamova cesta 39,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Ljubljana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' SI-1000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Slovenia fFaculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' University of Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Boltzmanngasse 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 1090,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Austria gResearch Platform “Mathematics-Magnetism-Materials”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' University of Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Oskar-Morgenstern-Platz 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 1090,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Austria hUral Federal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 51 Lenin Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=', Ekaterinburg, Russian Federation i M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Mikheev Institute of Metal Physics UB RAS, 18 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Kovalevskaya str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=', Ekaterinburg, Russian Federation Abstract In complex colloidal systems, interparticle interactions strongly affect the dynam- ics of the constituting particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' A study of the dynamical response also provides invaluable information on the character of those interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Here we demon- strate how tuning the electrostatic interactions by an ionic surfactant in dispersions of magnetic nanoplatelets leads to developing new dynamic modes in magnetic response spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The collective modes can be induced or suppressed by either varying the concentration ratio of the magnetic nanoplatelets (MP) to the surfac- tant or increasing the MP concentration reflecting the nanoscale characteristics of ∗Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' E-mail adresses: hajnalka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='nadasi@ovgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='de (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Nádasi) f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='ludwig@tu-braunschweig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='de (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Ludwig).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Preprint submitted to Journal of Molecular Liquids January 5, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='01652v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='soft] 4 Jan 2023 this fluid magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Keywords: soft matter, magnetic nanoplatelets, barium hexaferrite, ferromagnetic nematics, magnetic dynamics, AC susceptometry 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Introduction Ferrofluids are one of the most spectacular examples of complex fluids where nanoscale properties can be transformed to the macroscopic level [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Un- derstanding the nanoscale dynamics in such systems allows accurate manipulation and fine-tuning of their macroscopic properties [4, 5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' These dynamics are governed by two key attributes of the system: the electrostatic interactions, which ensure the colloidal stability of the magnetic nanoparticles in the liquid dispersion medium, and the magnetic dipole moment of the nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Understand the in- terplay between these two properties is critical to the tailoring of the ferrofluids’s microstructure, which can then give rise to the desired macroscopic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Surfactants stabilise colloidal suspensions by physical or chemical adsorption on the suspended particles’ surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Depending on the solvent affinity of the sur- factant’s head either they form a single layer around the particles adsorbed by their solvophobic head(s) while dispersing them in the medium by the solvophilic tail, or they organise into a double layer, whereby the overlapping solvophobic tails are enclosed by the solvophilic heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The key to successfully integrating the parti- cles in the dispersing medium is to match the dielectric properties of the surfactant and the solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' However, the suspension’s longevity depends on the effective pre- vention of aggregation [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' In the case of magnetic fluids, it is also essential to counteract the colloidal self-assembly caused by dipolar interactions, which can be significantly enhanced by the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' While reversible agglomeration is desirable in specific use cases, such as magnetooptical effects [10, 11, 12, 13], extensive irreversible aggregation of dipolar nanoparticles is generally undesirable due to the loss of colloidal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Ferrimagnetic scandium-doped barium hexaferrite platelets (Sc-BaHF) are (shape- ) anisotropic particles with their magnetisation perpendicular to the basal plane [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' When such platelets are dispersed into a nematic liquid crystal matrix (5CB (4-Cyano-4’-pentylbiphenyl), E7), the anchoring of the nematic director at the par- ticle surfaces may stabilise the colloidal ferromagnetic nematic order as demon- strated in [16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' It is the first example of the long-anticipated ferromagnetic nematic, as predicted by de Gennes [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Ferromagnetic nematic order can also be 2 realised in the pure colloidal liquid crystal, where the nanoplatelets are dispersed in an isotropic fluid, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 1-butanol [20, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The high magnetic moment of ferrimagnetic Sc-BaHF platelets requires elec- trostatic stabilisation against aggregation, which can be achieved by an ionic sur- factant such as dodecylbenzenesulfonic acid (DBSA) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' In our previous paper [24] we investigated the dynamic magnetic response of a dilution series of Sc-BaHF ferrofluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The dispersion medium 1-butanol contained 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mM surfactant (DBSA) in the stock suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' To ensure the stability of the ferrofluid, it was diluted by the stock solution with the same concentration of DBSA during preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' In an oscillating magnetic field, we observed collective modes of the as-prepared fer- rofluid series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The collective modes were well resolved in the AC susceptibility (ACS) spectra of the ferrofluids with low magnetic particle concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' A pos- sible explanation for the appearance of those modes is that as we dilute the stock suspension by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mM DBSA solution, the degree of concentration-change is dif- ferent for the MPs compared to that of the DBSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The adsorption-desorption equilibrium of the surfactant at the MP surface de- pends not only on the DBSA but also on the MP concentration (푐MP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Boštjančič et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' [25] investigated suspensions of Sc-BaHF platelets in the concentration range of 푐MP = 5 g L−1 − 30 g L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The ratio of the adsorbed DBSA to the total amount of DBSA in the suspension increases with increasing platelet concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The dis- solved molecules in the dispersing medium are partly dissociated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' With increasing concentration of this ionic form, the ionic strength increases, whereby the Debye screening length of the platelets decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The repulsive electrostatic interactions also depend on the effective charge of the platelets, which was found to almost independent of the DBSA concentration (푐DBSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' To illustrate the effects of altering the DBSA concentration on degree of self- assembly in the colloidal suspension, Figure 1 shows how even a simple approxi- mation of the interparticle interactions is strongly affected by changes in the screen- ing length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The dipolar interaction between two magnetic nanoplatelets with mag- netic moment 흁푖 and separated by the vector 퐫 can be written as: 푈dd(퐫) = 휇0 4휋 ((흁푖흁푗) |푟|3 − 3(흁푖퐫)(흁푗퐫) |푟|5 ) (1) and the Coulomb potential with added Yukawa screening term can written as: 푈C−DH(퐫) = 1 4휋휖0 푞푖 ⋅ 푞푗 |푟| exp (−휅|푟|) (2) 3 where 푞푖 and 푞푗 are the particle charges and 휅 is the Debye screening constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' For this approximation, we assume that the two platelets are of the same size, and have the same magnetic moment and effective charge, which reduces the total potential to: 푈tot(퐫) = 푝m ⋅ −2휇2 푟3 + 푝c ⋅ 푞2 푟 exp (−휅푟) (3) where 푝m and 푝c are scaling factors which encompass the relative strength of the magnetic and electrostatic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Based on experimental observations, we fix 푝c > 푝m and vary 휅 to approximate the effect of added DBSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Figure 1: Total interaction potential of a pair of charged magnetic discs, oriented head to tail in a simplified model of the effects of increased Debye screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The 푥-axis represents the distance between disc centers, while the 푦-axis shows the potential energy, normalised by the magnetic dipole-dipole interaction prefactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The three curves show that as the screening is increased, the potential goes from repulsive to mildly attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Although the simplifying assumptions in Figure 1 are too strong to claim that this potential quantitatively represents the experimental system - both the high degree of polydispersity and the more complex electrostatics would need to be considered in greater detail - it qualitatively shows that gradually increasing the screening length will destabilise the system, leading to self-assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 No screening 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 Low screening High screening 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 (y)"d/on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 2 3 4 5To understand the influence of the ratio 푐MP∕푐DBSA on the dynamic magnetic re- sponse, in this work, we explored magnetic dynamics in two series of suspensions: (i) a series with low but constant magnetic particle concentration (푐MP = 8 g L−1) and varying 푐DBSA (Set 1) and (ii) a dilution series where we kept the 푐MP∕푐DBSA ratio constant (Set 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The latter was chosen so that the ferrofluid with the low- est platelet concentration (푐MP = 8 g L−1) does not give rise to collective modes in the ACS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The charge and the ionic strength determine the electrostatic environment of the ferrofluid: the desorption of the surfactant molecules from the positively charged platelets and the dissociation of the dissolved DBSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' On the first approach, we presume that keeping the 푐MP∕푐DBSA constant, the electrostatic environment does not change significantly (the adsorption-desorption equilibrium of the surfactant is not shifted) and the response would only depend on the mag- netic particle concentration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='e the magnetostatic interactions, hence the study of Set 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' In the following we discuss collective modes of magnetic dynamics in Sc- BaHF suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' We show, that even at low magnetic particle concentration, as in Set 1 (푐MP = 8 g L−1), collective modes emerge on increasing surfactant con- centration (푐DBSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Besides, investigating Set 2, we also demonstrate the role of the magnetic particle concentration (푐MP) in the collective behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Materials and methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Ferrofluid preparation The stock ferrofluid was prepared as described in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The sets of ferroflu- ids were prepared by diluting the stock suspension (푐MP = 304 g L−1 and 푐DBSA = 43 g L−1) to the necessary concentration of MPs and DBSA with appropriate DBSA solution in 1-butanol and/or by 1-butanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The resulting suspensions were soni- cated for a minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Set 1 marks the suspensions with constant 푐MP = 8 g L−1 and varying 푐DBSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' They are designated as MP8 and the corresponding weight per- cent of DBSA as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Set 2 identifies the dilution series with varied 푐MP but constant 푐MP∕푐DBSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' They are designated with the abbreviation MP and the corresponding magnetic particle concentration as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' AC susceptometry Measurements of the AC susceptibility were carried out with a custom-made setup, which was originally built for measurements of the dynamics of magnetic nanoparticles in a rotating magnetic field [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' For the ACS measurements just one set of Helmholtz coils was used to generate the sinusoidal excitation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' ACS 5 Set 1 푐MP=8 g L−1 Set 2 푐MP∕푐DBSA=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='7 Designation %DBSA 휙MP/휙DBSA Designation 푐MP∕gL−1 휙MP MP8_12 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='44 MP8 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0015 MP8_13 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='37 MP12 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0024 MP8_16 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='04 MP32 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0063 MP8_21 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='73 MP40 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0077 MP8_25 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='61 MP92 92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0170 MP8_26 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='57 MP126 126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0229 MP8_50 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='20 MP158 158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0285 Table 1: In Set 1 the magnetic particle concentration 푐MP = 푚MP∕푉1−BuOH is fixed to 8 g L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The DBSA concentration is given as %DBSA = 푚DBSA∕(푚MP+푚DBSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The volume percents apply to the total volume of the ferrofluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' In the dilution series Set 2 the concentration ratio 푐MP∕푐DBSA is fixed to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The concentration 푐MP = 푚MP∕푉1−BuOH is utilized to designate the suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The volume fraction 휙MP = 푉MP∕푉FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' MP8_13 of Set 1 and MP8 of Set 2 are identical suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' spectra were recorded at 298 K in a frequency range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 Hz and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2 kHz and at field amplitudes between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT and 5 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The exposure of a ferrofluid to an alternating magnetic field results in aligned magnetisation, which will oscillate with a phase lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The dynamic response can be described by the real 휒′ and imaginary 휒′′ parts of the magnetic susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The ACS susceptibility spectra of MP suspensions are generally analysed with the Debye model where the complex susceptibility is given by 휒(휔) = 휒0 1 + 푖휔휏 (4) with the static susceptibility 휒0, the angular frequency 휔 = 2휋푓 and the char- acteristic relaxation time of the MP 휏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Since the Sc-BaHF platelets are thermally blocked [24], only Brownian relaxation occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The Brownian relaxation time [27] 휏B is determined by the viscosity of the medium 휂, the hydrodynamic volume of the MP 푉H and the temperature 푇 : 휏B = 3휂푉H 푘B푇 (5) The field dependence of the Brownian relaxation time is well described by the empirical model by Yoshida and Enpuku [28] in the frame of the Fokker-Planck formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' For non-interacting thermally blocked MPs, their model is valid for magnetic field amplitudes with a Langevin parameter 휉 = 푚휇0퐻∕ (푘B푇 ) of up to 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The relaxation time is given by: 6 휏B,H = 휏B,0 √ 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='126휉1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='72, (6) where 휏B,0 is the relaxation time in the limit 퐻 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' To account for a distribution of relaxation times due to the polydispersity of MPs and interparticle interactions, we apply the Cole-Cole equation [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 휒(푓) = 휒∞ + Δ휒 1 + (푖휔휏B,H )1−훼 (7) considering symmetric broadening only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Here 휒∞ is the susceptibility in the high- frequency limit, Δ휒 is proportional to the amplitude of the susceptibility, and 훼 is the broadening coefficient (훼 = 0 corresponds to the Debye model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' In order to analyse ACS spectra with multiple relaxation modes, the experimental data were fitted with a sum of several Cole-Cole equations: 휒(푓) = 휒∞ + ∑ 푖 Δ휒푖 1 + (푖휔휏B,H,푖 )1−훼푖 (8) In our study, up to three modes 푖 could be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' To match these experiment to further theoretical work, we will later fit the calculated Langevin susceptibility 휒퐿 to an adaption of the susceptibility which includes chain formation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' However, this model only was dervied for dipolar hard spheres, the shape of which (and thus the self-assembly process) significantly differs from that of spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Therefore, we have calculated a new expression for the partition function, the calculation of which is described in the SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The resulting expression was then fitted using a least-squares fit to determine 휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Analysis of ACS spectra of Set 1 To explore the influence of the surfactant concentration on the magnetic re- sponse, we measured the set of ferrofluids 1 containing a fixed low concentration of MPs (8 g L−1) and varied 푐DBSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The ACS spectra of the suspensions where 휙MP∕휙DBSA > 1 are characterised by a single peak in the high-frequency range attributed to the relaxation of single platelets (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' In the following, we desig- nate this relaxation mode as a high-frequency (HF) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' With increasing field 7 amplitude the peaks’ maximum shifts to higher frequencies and its amplitude de- creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Similar behaviour was observed in aqueous suspensions of spherical [30] and rod-like nanoparticles [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Figure 2: The ACS spectra of MP8_13 recorded for different amplitudes of the probe field consist of a single symmetric high-frequency peak related to the single-platelet relaxation mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The maximum shifts to higher frequencies with increasing probe field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' A further increase of 푐DBSA results in the emergence of low-frequency modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The onset of the slowest mode (LF) can only be distinguished at low probe fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The spectral maximum of this mode shifts to higher frequencies with in- creasing probe field while its amplitude decreases and the peak apparently flattens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' This contribution acts effectively as an offset of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' An additional peak in the intermediate range, denoted as middle-frequency mode (MF), emerges in this concentration range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' This mode merges into the HF mode with increasing probe field while the amplitude seemingly decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' How- ever, since the modes overlap, it is difficult to estimate the relaxation times accu- rately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The ACS spectra of MP8_50 have very distinctive features (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' They con- sist of a very pronounced HF peak and especially prominent low-frequency collec- 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='07 μoH / mT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='04 a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 10 1 100 1000 f/Hz(a) (b) (c) Figure 3: Multiple peaks of MP8_26 at (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT, (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT and (c) 5 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' With increasing probe field, the middle- and high-frequency peaks collapse as a result of progressing mode-overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The slowest mode (LF mode) shifts to higher frequencies and broadens, transforming into a dimin- ishing flattened peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Figure 4: The ACS spectra of MP8_50 recorded for different amplitudes of the probe field are dominated by collective modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The nearly linear in log-scale slope in the frequency range below 100 Hz for AC fields between 3 mT - 5 mT implies the complex spectral structure of mul- tiple overlapping peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='08 LF-mode MF-mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='06- HF-mode --Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (5) fit 1 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' data .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='04 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='02- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='00- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 1 10 100 1000 10000 f /Hz0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='08 LF-mode MF-mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='06- HF-mode --Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (5) fit Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=" data n' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='04 a X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='02 10:00888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 1 10 100 1000 10000 f /Hz0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='08 LF-mode MF-mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='06- HF-mode --- Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (5) fit 0 0 0 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=" data n' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' a 0 0 0 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='02 0 0 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 1 10 100 1000 10000 f /Hz0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='08 μoH/ mT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 一 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='05 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 1 10 100 1000 f/Hztive modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Already in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT probe field, the remaining LF peak is significantly stronger than the HF peak maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' In 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT and 1 mT fields, one can fit three peaks that can be assigned to the LF, MF and HF modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' In higher fields, we observe a nearly linear increase of the imaginary part of the magnetic suscep- tibility as a function of log 푓 and an HF peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Its mode structure could only be implicitly determined using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (8) over a reduced frequency interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Multiple overlapping in the ACS spectra of MP8_50 in high probe fields does not allow accurate Cole-Cole fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' As a result, we cannot accurately determine the position of the HF peak and its dependence on the probe field amplitude from the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The ACS measurement results described above are in qualitative agreement with the findings in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' At a sufficiently low DBSA concentration (12-16%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Table 1), we observe just an HF mode in the ACS spectra, which we attribute to the Brownian rotation of single platelets, while at higher DBSA concentrations, collective modes appear, which indicate that the electrostatic repulsions are less effective to prevent the platelets from clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (a) (b) Figure 5: Multiple peaks in spectra of MP8_50 recorded in probe fields with amplitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT (a), and and 1 mT (b) with dominating but well distinguishable low- and middle-frequency modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Field dependence of the Brownian relaxation time of the HF mode The Brownian relaxation times 휏B,H estimated from the position of the HF peak in the ACS spectra as a function of the applied field amplitude are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 6(a) along with the fits with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Since 휏B,0 and 휉 are treated as free pa- rameters, the fitting allows one to determine the field-free Brownian relaxation 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='09 LF-mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='08 Q MF-mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='07 HF-mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='06 ---Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (5) fit 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='05 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' data a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='04 0 0 88 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='01- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='00- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 1 10 100 1000 10000 f/Hz0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='09 LF-mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='08 0 MF-mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='07 :HF-mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='06 ---- Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (5) fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='05 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' data 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='04 0 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='03 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='02 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='01- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='00- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 1 10 100 1000 10000 f/Hztime 휏B,0 and the magnetic moment 푚 for the individual samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The results are depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 6(b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (a) (b) (c) Figure 6: (a) Fit of the field dependence of the relaxation time of the HF mode using the model by Yoshida and Enpuku (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (6)) for Set 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The extracted magnetic moment (b) and field-free relaxation time values (c) as a function of the concentration ratio of the magnetic particles to the surfactant DBSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 6(b), the effective magnetic coupling as measured via 푚 slightly decreases with increasing amount of DBSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Since the magnetic moment of the particles directly corresponds to their size, the gradual decrease indicates that with increasing surfactant concentration means the successively smaller par- ticles are responding to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' As Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 3-5 show, additional low-frequency relaxation modes appear at higher DBSA concentrations, which can be attributed to MP clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Due to the polydispersity of the platelets, these observations suggest that smaller platelets can still freely rotate contributing to the HF peak, while larger platelets form clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The remaining platelets contributing to the high-frequency range of the spectrum are, on average, smaller and thus have smaller magnetic moments 푚, resulting in shorter relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Analysis of ACS spectra of Set 2 As the results of Set 1 indicate, a ratio 푐MP∕푐DBSA = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='7 prevents the Sc-BaHF platelets - at least at comparably low 푐MP - from clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The second set of magnetic fluids (Set 2) was prepared on the simplified assumption that keeping the ratio of magnetic particles to the surfactant constant at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='7 results in an unaltered electrostatic environment in the ferrofluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The spectra of the ferrofluids with the lowest MP concentrations, 휙MP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0015 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0024 (MP8 and MP12) have a single high-frequency peak corresponding to the single platelet relaxation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Upon further increasing the MP concentration to 휙MP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0063 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0077 (MP32 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='65 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='60 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='55 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='50- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='40- / w 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='30 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='25- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='20- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='15- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4 UMP / ΦDBSA0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='29- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='27 S m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='24 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4 UMP / DBSAMP8 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='27 MP8 25 MP8 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='24 MP8 16 ms MP8 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' MP8 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='18- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 2025 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 μoH/ mTand MP40), a weak LF peak emerges at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT, which is suppressed in stronger fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' First, at 휙MP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0170 (MP92), we could detect three modes in low probe fields (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT) and two modes in higher fields (2 mT - 5 mT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The LF mode again shows a very broad spectrum, as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' At higher concen- tration 휙MP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0229 (MP126), the fits reveal two modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' In low probe fields (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT), there is an HF and an LF mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' With further increas- ing the probe field, the LF mode is not detectable anymore;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' the remaining peak is symmetric in shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' However, in even higher fields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT - 5 mT), the peak again becomes slightly asymmetric, and a new collective mode emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' It has, nonetheless much lower amplitude than the low-frequency mode in low magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' At the highest MP concentration 휙MP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0285 (MP158), the character of the ACS spectra is similar, apart from the fact that the high probe field collective mode occurs already at 4 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The effect of dipolar interactions on the ACS spectra at moderate MP concentrations has theoretically been studied by Ivanov and Camp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' With increasing MP concentration, which affects the amount of self-assembly in the system and considering self-assembled clusters of MP, the position of the max- imum in the ACS imaginary part shifts to smaller frequencies [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Extending the work to higher dipolar coupling constants 휆 ≥ 4, but still low MP concentrations, the authors found the appearance of additional peaks in the ACS imaginary part, which they attribute to the response of chains and rings [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' While at low dipolar interaction parameters 휆, the Brownian rotation of single MP dominate, at inter- mediate 휆, the formation of particle chains and rings occurs with a peak frequency in the ACS imaginary part well below that of single MP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' This seems to be the same underlying phenomenon as we find in our study, as dipolar magnetic nanoplatelets also form chains for at high values of 휆 (or low electrostatic repulsion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' However, our study does not show a high-frequency peak (at about 24휔휏퐵), which they at- tribute to the motion of particles inside chains or rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' We exclude the latter mode for our case since no indication of such a mode was found in the measured ACS spectra- which makes sense given that platelets do not form rings and are sterically precluded from rotation inside chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Field dependence of the Brownian relaxation time of HF mode The field dependence of the Brownian relaxation time of individual platelets, as determined from the peak frequency of the HF mode, is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 8(a) in dependence of the applied field amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The field-free Brownian relaxation time and its decay with increasing field amplitude strongly increase with 휙MP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The latter would reflect an increase in the effective coupling, which would be measured 12 (a) (b) (c) (d) Figure 7: Cole-Cole fits of the ACS spectra of MP126 in AC fields of (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT, (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT, (c) 2 mT and (d) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The low-frequency collective mode shifts with increasing field amplitude much stronger to higher frequencies than the HF peak does, so that at 2 mT the ACS spectrum consists of one symmetric peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' At an even higher field amplitude of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT, the nematic mode emerges as a new low-frequency mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' as an increase in the magnetic moment 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Recently, Rusanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' [34] extended the model for the field-dependent Brow- nian relaxation time by Yoshida and Enpuku by additionally considering the effect of dipolar interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' For the Brownian relaxations, they obtained the following empirical formula: 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 LF-mode 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4 HF-mode 0 00 ---·Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (5) fit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2 00 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' data 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0- 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='8 a 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4- 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 1 10 100 1000 10000 f/Hz1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 LF-mode 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4 HF-mode -- Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (5) fit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2 0 0 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' data 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0- 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='8 0 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4- 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 1 10 100 1000 10000 f/Hz1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 --·Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (5) fit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2 0 0 0 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0- 0 0 Q 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='8 0 a 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4 - 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 1 10 100 1000 10000 f/Hz1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 LF-mode 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4 HF-mode --·Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (5) fit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='8 a 0 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 1 10 100 1000 10000 f/Hz휏B,H = 휏B,0 √ (1 − 휒eff 3 )2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='076휉2 (9) (a) (b) (c) Figure 8: (a) Field dependence of the Brownian relaxation time 휏B,H for Set 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Lines show the fits with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (b) shows the extracted field-free Brownian relaxation time 휏B,0 vs MP concentration 휙MP, with lines as guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Figure (c) shows the effective static susceptibility 휒eff vs the volume fraction 휙MP (points) fitted to the chain-corrected model of Langevin susceptibility using the correct platelet partition function (line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Here the effective susceptibility 휒eff describes the effect of magnetic interac- tions of the single platelets with the surrounding medium on the Brownian relax- ation time 휏B,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' To fit the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 8(a), we determined the mean magnetic moment 푚 of the platelets and the field-free Brownian relaxation time 휏B,0 of non- interacting particles by fitting the data points of sample MP8 while setting 휒eff = 0 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0024 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0077 ms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='03010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='01 Model fit with 2=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='26536 Experimental 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='030(negligible dipolar interactions between particles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' For the fits of the other data sets, 푚 was fixed at this value (푚 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='58 × 10−18 A m2) while both 휒eff and 휏B,0 were taken as free parameters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Although both the mean magnetic mo- ment 푚 and the mean hydrodynamic size of the platelets are independent of MP concentration, the use of 휏B,0 as a free parameter is due to the fact that the vis- cosity is expected to increase with increasing 휙MP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The results for 휒eff and 휏B,0 in dependence of 휙MP are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 8(b) and 8(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The viscosity increases by about a factor of 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=', assuming that the viscosity of pure 1-butanol at room temperature amounts to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='54 mPa s, it increases to about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2 mPa s for the sample with a Sc-BaHF concentration of 158 g L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The rise of 휒eff with 휙MP is weaker than linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The effective susceptibility 휒eff in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 9 is dominated by clusters due to their high magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The static susceptibility of clusters was theo- retically described by Mendelev and Ivanov [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Adapting the partition function used in this theory to platelets (see SI), the expression for 휒eff was fitted to the data points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 8(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' As a measure of magnetic dipolar interactions, the adjusted dipolar coupling constant 휆 = 휇0푚2∕(4휋퐷3푘B푇 ) is used, where 푚 is the magnetic moment and 퐷 is the diameter of the platelets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' In the curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 8(c), the aspect ratio of the platelet was taken to be 1 ∶ 10, which resulted in a fitted parameter 휆 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' This aspect ratio corresponds to a mean thickness of 5 nm and diameter of 50 nm of the platelets as reported in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Note that a value 휆 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 is calculated for 푚 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='58 × 10−18 A m2, 퐷 = 50 nm and 푇 = 296 K, in excellent agreement with the fitted value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Deviations of the fitted line from the experimental data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 8(c) may be caused by the fact that the ratio between single and clustered platelets, which differently enter 휒eff, may vary when changing the concentration of Sc-BaHF, 휙MP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' In addition, no electrostatic interactions but only dipolar ones are included, point-like dipoles rather than distributed and asymmetric ones are considered and distributions of parameters are not accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Effect of a DC-Bias Field Studying the effect of a DC bias field on the magnetic response allows us to se- lectively suppress the modes and explore their individual behaviour in dependence on the orientation of the bias field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' One example containing pronounced LF and MF modes is MP8_25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Apply- ing a DC bias field parallel to the AC probe field for sample MP8_25 results in a suppression of the collective modes in the low-frequency range and, depending on the DC field strength, in a shift and repression of the HF peak maxima (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 9(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The suppression of the low-frequency mode indicates, that the larger assemblies of magnetic nanoparticles are aligned with the field, and Brownian relaxation of the 15 (a) (b) Figure 9: Influence of DC bias field applied parallel to the AC probe field on the ACS spectra of MP8_25 (a) and MP158 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' (a) In a low DC field (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT), the low-frequency collective modes are suppressed as the clusters with high effective magnetic moment align first along the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' By a further increase of the DC field (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT), the relaxation of single platelets with bigger size i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' higher magnetic moment gets constrained, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Hence, on (b), the single peak of the ACS spectrum belongs two more than one mode since the low DC bias field (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT) mitigates it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' smaller particles now dominates the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Indeed, LF modes result from the collective behaviour of large ensembles (clusters) of MPs which are most suscep- tible to aligning along the DC field, hence as a primary response, a low (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT) DC field suppresses the LF modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Non-correlated single platelets couple individually to the magnetic field hav- ing much smaller coupling energy than the clusters do, and their relaxation can be quenched in significantly higher fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' As a result, the suppression of the HF peak occurs at a much higher DC bias field proving, that single platelets with compara- bly small 푚 contribute to the HF peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The shift of the peak frequency to higher values and the decrease of its amplitude with increasing DC bias field strength are in agreement with theoretical models [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' This approach allows us to also characterise the magnetic dynamics in MP158 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 9(b)) having a single broad spectral feature in the ACS spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' With in- creasing DC bias field, the peak frequency shifts from about 50 Hz at zero bias field to 782 Hz at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 9(a), a DC bias field suppresses the low-frequency part of the spectrum already at 퐻bias = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT so that the peak at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT is expected to be dominated by the HF mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The fact that the peak frequency of the HF mode in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 9(b) is slightly lower than that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 9(a) can be attributed to the effect of the Zeeman and dipolar interactions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='08 Bias field in mT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='07 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='05 一 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='04 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='03 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 10 100 1000 1 f/Hz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0 Bias field in mT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='8 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2- u 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0- a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='8- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 10 100 1000 1 f/HzFor comparison, the peak position in the absence of bias field in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 9(b) lies at 50 Hz for MP158, at a more than a magnitude lower frequency than that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 9(a) for MP8_25 indicating, that the whole spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 9(b) is the superposition of collective low-frequency and single-platelet high-frequency relaxation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Figure 10: Relaxation times in a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='5 mT DC bias field of the single-platelet (▪) and the collective (▪) modes as a function of the magnetic particle concentration for Set 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' The two modes reveal opposite tendencies with increasing 푐MP: while the single-platelet mode slows down, the collective mode speeds up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Conclusions and outlook In our experiments we demonstrated that - tuning the electrostatic interactions by adjusting the DBSA concentration in the dispersions of Sc-BaHF nanoplatelets strongly affects the structure of the low-frequency magnetic response (Set 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 17 LF-mode HF-mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='1 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='01 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='001 1E-4 0 20 40 60 80 100 120 140 160 180 CMP / gL-1An increase in the DBSA concentration reduces the repulsive interactions and en- hances interparticle correlations resulting in the growth of collective behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Collective modes developing in the system are very sensitive to the magnetic field amplitude becoming faster in stronger fields, which suggests a high magnetic mo- ment is associated with the relaxation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Another feature of these modes is their shift to higher frequencies with growing concentration of MPs (Set 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' At the same time, the single-platelet relaxation mode becomes slower with in- creasing 푐MP, because of increasing dipolar interactions and increasing viscosity as it is expected (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' As the MP concentration increases, the motion of the particles becomes strongly correlated leading to the development of orienta- tional order and a restoring force, which accelerates the relaxation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' This restoring force also contributes to the nematic order above a critical concentra- tion (푐MP = 126 g L−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' At somewhat lower concentration, highly correlated clus- ters appear in the isotropic suspensions and are responsible for a strong magnetic and magnetooptical response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' A similar situation of the so-called para-nematic state has been observed in dispersions of rod- and plate-shaped pigment parti- cles [37, 38, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' As a next step to determine the significance of viscosity change with increasing magnetic particle concentration, we are planning to measure viscosity of varied concentration of ferrofluids in and without magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' We will also elaborate on the ACS measurements in DC bias fields - parallel and perpendicular to the AC field - to comprehend more facets of the complexity of the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' CRediT authorship contribution statement Hajnalka Nádasi: Conceptualisation, Investigation, Analysis, Writing - orig- inal draft, Writing - Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Melvin Küster: Conceptualisation, Inves- tigation, Analysis, Writing - original draft, Writing - Review & Editing, Visual- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Alenka Mertelj: Investigation, Resources, Writing - Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Nerea Sebastián: Investigation, Resources, Writing - original draft, Writing - Re- view & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Patricija Hribar Boštjančič: Investigation, Resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Darja Lisjak: Investigation, Resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Thilo Viereck: Conceptualisation, Resources, Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Margaret Rosenberg: Computation, Simulations, Writing - Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Alexey O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Ivanov: Conceptualisation, Computation, Writing - Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Sofia Kantorovich: Conceptualisation, Computation, Writing Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Alexey Eremin: Conceptualisation, Writing - original draft, Writing - Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Frank Ludwig: Conceptualisation, Investigation, Resources, Writing - original draft, Writing - Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' 18 Conflicts of interest There are no conflicts to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' Acknowledgements F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' acknowledge the support of the Deutsche Forschungs- gemeinschaft (Projects NA 1668/1-1 and LU 800/7-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='L, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQfr_1k/content/2301.01652v1.pdf'} +page_content='M.' metadata={'source': 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Barseghyan1 and Nadia Tahiri2 +1 Département d’Informatique, Université du Québec à Montréal, Case postale +8888, Succursale Centre-ville, Montreal, QC, H3C 3P8, Canada +2 Département d’Informatique, Université de Sherbrooke, 2500 Boulevard de +l’Université, Sherbrooke, Québec J1K 2R1, Canada +*Corresponding author: makarenkov.vladimir@uqam.ca +Abstract: Phylogenetic trees (i.e. evolutionary trees, additive trees or X-trees) play +a key role in the processes of modeling and representing species evolution. Genome +evolution of a given group of species is usually modeled by a species phylogenetic +tree that represents the main patterns of vertical descent. However, the evolution of +each gene is unique. It can be represented by its own gene tree which can differ +substantially from a general species tree representation. Consensus trees and super- +trees have been widely used in evolutionary studies to combine phylogenetic infor- +mation contained in individual gene trees. Nevertheless, if the available gene trees +are quite different from each other, then the resulting consensus tree or supertree +can either include many unresolved subtrees corresponding to internal nodes of high +degree or can simply be a star tree. This may happen if the available gene trees have +been affected by different reticulate evolutionary events, such as horizontal gene +transfer, hybridization or genetic recombination. Thus, the problem of inferring +multiple alternative consensus trees or supertrees, using clustering, becomes rele- +vant since it allows one to regroup in different clusters gene trees having similar +evolutionary patterns (e.g. gene trees representing genes that have undergone the +same horizontal gene transfer or recombination events). We critically review recent +advances and methods in the field of phylogenetic tree clustering, discuss the meth- +ods’ mathematical properties, and describe the main advantages and limitations of +multiple consensus tree and supertree approaches. In the application section, we +show how the multiple supertree clustering approach can be used to cluster aaRS +gene trees according to their evolutionary patterns. +Keywords: Clustering, Cluster validity index, Consensus tree, k-means, k-medoids, +Phylogenetic tree, Robinson and Foulds distance, Supertree. + +2 + + + + +Introduction +The term phylogeny (i.e. phylogenetic tree or evolutionary tree) was introduced by +Haeckel in 1866, who defined it as "the history of the paleontological development +of organisms by analogy with ontogeny or the history of individual development". +A phylogenetic tree represents a hypothesis about evolution of a given group of +species which are usually associated with the tree leaves. + +In mathematics, phylogenetic trees are called additive trees or X-trees (as their +leaves are often associated with the set of species X; Barthélemy and Guénoche +1991). Let us now present some necessary mathematical definitions related to phy- +logenetic trees. The distance δ(x,y) between two vertices x and y in a phylogenetic +tree T is defined as the sum of the edge lengths in the unique path linking x and y in +T. Such a path is denoted (x,y). A leaf is a vertex of degree one. Usually, a leaf +represents a contemporary species (or taxa). + +Definition 1. Let X be a finite set of n taxa. A dissimilarity d on X is a non-negative +function on X × X such that for any x, y from X: +d(x,y) = d(y,x) ≥ d(x,x) = 0. + +Definition 2. A dissimilarity d on X satisfies the four-point condition if for any x, y, +z, and w from X: d(x,y) + d(z,w) ≤ max {d(x,z) + d(y,w); d(x,w) + d(y,z)}. + +Definition 3. For a finite set X, a phylogenetic tree (i.e. an additive tree or an X- +tree, i.e. a tree whose leaves are labeled according to a final set of species X) is an +ordered pair (T, φ) consisting of a tree T, with vertex set V, and a map φ: X→ V +with the property that, for all x ∈ X with degree at most two, x ∈ φ(X). A phyloge- +netic tree is binary if φ is a bijection from X into the leaf set of T and every interior +vertex has degree three. + +The theorem relating the four-point condition and a dissimilarity representability by +a phylogenetic tree is as follows: +Theorem 1 (Zarestskii, Buneman, Patrinos & Hakimi, Dobson). Any dissimilarity +satisfying the four-point condition on X × X (where X is a finite set of species) can +be represented by a phylogenetic tree T such that for any x, y from X, d(x,y) is equal +to the length of the path linking the leaves x and y in T. This dissimilarity is called +a tree metric. Furthermore, this tree is unique. + +Figure 1 gives an example of a tree metric on the set X of five taxa and the corre- +sponding phylogenetic tree. + +Unfortunately, real-life evolutionary distances (or dissimilarities) rarely satisfy the +four-point condition. Thus, one need to carry out an approximation algorithm to +infer a tree metric matrix from a given matrix of evolutionary distances (Gascuel +2005). Among the most known distance-based approximation algorithms we can + +3 + + + + +mention Neighbor-Joining (Saitou 1988), UPGMA (Sokal and Michener 1958), +FITCH (Felsenstein 1997), and MW (Makarenkov and Leclerc 1996, 1999). + +Fig 1. An example of a tree metric on the set X of five taxa (on the left) and the +corresponding phylogenetic tree (additive tree or X-tree) on the right. + +Biologists often need to compare phylogenetic trees to each other in order to dis- +cover different evolutionary histories that govern a given set of species. There are +several measures for comparing phylogenetic trees. The most popular of them in- +clude the Robinson and Foulds topological distance (RF) (Robinson and Foulds +1981), the least-squares distance (LS), the bipartition dissimilarity (BD) (Boc et al. +2010), and the quartet distance (QD) (Bryant et al. 2000). In this literature review, +we will mainly explore the methods based on the Robinson and Foulds distance. +The Robinson and Foulds topological distance (Robinson and Foulds 1981) be- +tween two trees is the minimum number of elementary operations (contraction and +expansion) of nodes needed to transform one phylogenetic tree into another. It is +also the number of splits (or bipartitions) that are present in one tree and absent in +the other. The two phylogenetic trees in question must have the same set of taxa. +The closer two phylogenetic trees are topologically, the smaller the value of the RF +distance. It is often relevant to normalize the value of the RF distance by dividing it +by its maximum possible value (equal to 2n-6) for two binary phylogenetic trees +with n leaves. The RF distance calculation between two trees with n leaves can be +carried out in O(n) (Day 1985, Makarenkov 1997, Makarenkov and Leclerc 2000). + +Often phylogenetic tree reconstruction methods do not return a single phylogenetic +tree as output, but a collection of different trees (Gascuel 2005). Moreover, phylo- +genetic trees inferred for different genes often differ from each other. There is no +absolute criterion for determining whether one tree is better than the others (except +for the use of intrinsic criteria, e.g., the use of bootstrap scores). For this reason, it +is preferable to seek a consensus representation of these trees, such that their con- +cordant parts appear clearly in relation to the discordant parts. The resulting repre- +sentation is called a consensus tree. Traditional consensus methods generate a single +phylogenetic tree that is a representative of all of the input trees (Bryant 2003). One +of the first consensus methods was proposed by Adams (Adams 1972). Since then, +a wide variety of methods have been developed. How to use them has been the +subject of much debate (Bryant 2003, Dong et al. 2010). + +The main types of consensus trees are the following: the strict consensus tree (Sokal +and Rohlf 1981, Moon and Eulenstein 2017), the majority-rule consensus tree (Mar- +gush and McMorris 1981), the Nelson consensus tree (Nelson 1979), and the + +x1 +x2 +x2 +x3 +x4 +x5 +1 +x1 +6 +6 +4 +2 +2 +2 +x2 +2 +4 +6 +/1 +1 +x3 +4 +6 +x5 +x3 +x4 +4 +x44 + + + + +extended majority-rule consensus tree (Felsenstein 1985). Let us briefly recall the +main characteristics of each of these consensus trees. + +The strict consensus tree (or Nelson's cladogram) is inferred by considering only +those tree splits (i.e. bipartitions induced by the internal tree edges) that are identical +in all trees compared. Conflicting parts of phylogenetic trees are represented by +multifurcations in a strict consensus tree. + +It is sometimes more convenient to have a less strict criterion than the one used by +the strict consensus tree in order to allow bipartitions that are not necessarily present +in all trees. When comparing a set of phylogenetic trees with different topologies, +it is possible to search for the monophyletic groups that appear most frequently (of- +ten in more than 50% of the trees) among all the trees compared. The resulting tree +is the majority-rule consensus tree. + +The extended majority-rule consensus tree contains all majority bipartitions to +which the remaining compatible bipartitions are added in turn, starting with the most +frequent bipartitions for the given tree set. The process stops when a completely +resolved (i.e. binary) tree is obtained. The extended majority consensus tree is the +most frequently used in molecular biology, as it is always the best resolved among +the three types of consensus trees discussed so far. + +The Nelson consensus tree includes the heaviest set of compatible bipartitions. It +consists in finding a clique of maximum weight in a compatibility graph of the entire +bipartition set, which is NP-hard (Nelson 1979, Bryant 2003). + +Unfortunately, in many practical situations, phylogenetic trees used as input of con- +sensus tree reconstruction methods can be quite divergent. This can happen, for ex- +ample, when the input trees represent the evolution of different genes which have +been affected by multiple reticulate evolutionary events such as horizontal gene +transfer, hybridization or intragenic/intergenic recombination, ancient gene dupli- +cation or gene loss (Makarenkov and Legendre 2000, Mirkin et al. 2003, Bapteste +et al. 2004). These evolutionary events can be unique for a subgroup of the input +gene trees. Thus, it seems to be much more appropriate to represent this subgroup +by its own consensus tree. However, the conventional consensus tree methods pro- +vide only one candidate tree for a given set of input gene phylogenies without con- +sidering their possible subgroups (or clusters) (Maddison et al. 2007). + +Figure 2 shows an example of four seven-leaf phylogenetic trees T1, T2, T3, and T4. +Here, the solution consisting of two majority-rule consensus trees, T12 and T34, +seems to be much more appropriate than the conventional consensus solution con- +sisting of a single majority-rule consensus tree, T1234, i.e., here a star tree (a tree +having no internal edges at all). + + +5 + + + + +1 +2 +3 +4 +6 +Tree T1 +Tree T2 +7 +5 +1 +2 +3 +4 +5 +7 +6 +1 +5 +3 +6 +2 +Tree T3 +Tree T4 +4 +7 +1 +5 +3 +6 +7 +4 +2 +1 +2 +3 +4 +6 +Majority-rule consensus tree T12 +7 +5 +Single (traditional) majority-rule +consensus tree T1234 +7 +5 +1 +6 +4 +3 +2 +1 +5 +3 +6 +4 +Majority-rule consensus tree T34 +7 +2 + +Fig 2. Four phylogenetic trees T1, T2, T3, and T4 defined on the same set of seven +leaves. Their single (traditional) majority-rule consensus tree is a star tree T1234. +The majority-rule consensus trees, T12 and T34, constructed for the pairs of topolog- +ically close trees: T1 and T2, and T3 and T4, respectively. + +In many evolutionary studies gene trees to be combined are defined on different, +but partially overlapping, sets of taxa (e.g. see Tree of Life project; Maddison et al. +2007). It is very unlikely that all the genes considered have been sequenced for the + +6 + + + + +same sets of species. In order to reconcile such trees, supertree reconstruction meth- +ods should be applied (Bininda-Emonds 2004, Wilkinson et al. 2007, McMorris and +Wilkinson 2011, Warnow 2018). Supertrees synthesize a given set of small (i.e. +partial) trees with partial taxon overlap into comprehensive supertrees that include +all taxa present in the given set of trees. + + +Fig 3. Four phylogenetic trees T1, T2, T3, and T4 defined on different, but mutually +overlapping, sets of seven taxa. Their single (traditional) majority-rule supertree is +a star tree T1234. The majority-rule supertrees, T12 and T34, constructed for the pairs +of topologically close trees: T1 and T2, and T3 and T4, respectively. + + 1 +2 +3 +4 +6 +Tree T1 +Tree T2 +7 +1 +2 +3 +4 +5 +1 +5 +3 +6 +2 +Tree T3 +Tree T4 +4 +7 +1 +5 +3 +6 +7 +4 +1 +2 +3 +4 +6 +Majority-rule supertree T12 +7 +5 +Single (traditional) majority-rule +supertree T1234 +7 +5 +1 +6 +4 +3 +2 +1 +5 +3 +6 +4 +Majority-rule supertree T34 +7 +2 + +7 + + + + +The most known supertree inference method is Matrix Representation with Parsi- +mony (MRP) (Baum 1992, Ragan 1992) that carries out matrix-like aggregation of +the given partial trees. The supertree reconstruction methods are commonly used +for phylogenetic analysis of organisms with large genomes (Mank et al. 2005, +Bininda-Emonds et al. 2007, Faurby et al. 2016, Kimbal et al. 2019). For organisms +with small genomes, such as prokaryotes, several approaches to genomic phyloge- +netic analysis have been adopted. In particular, supertree analysis provides new in- +sights into the evolution of prokaryotes that could not be solved by many other ap- +proaches (Daubin et al. 2001). Recently, Makarenkov et al. (2021) and Tahiri et al. +(2022) have used supertree phylogenetic analysis to characterize the evolution of +SARS-CoV-2 genes. + +As in the case of consensus trees, in many practical situations multiple conservative +supertrees should be inferred to best represent the evolution of a given group of +gene trees. Figure 3 shows an example of four phylogenetic trees T1, T2, T3, and T4 +defined on different, but mutually overlapping, sets of seven taxa. Here, the solution +consisting of two majority-rule supertrees, T12 and T34, is more appropriate than that +consisting of a single majority-rule supertree, T1234, i.e., here a star tree, yielded by +the traditional supertree approach. +The idea of building multiple consensus trees was originally formulated by Maddi- +son (Maddison 1991). He discovered that consensus trees for some subsets of input +trees may differ a lot and that they are generally much better resolved than the single +traditional consensus tree characterizing the whole set of the input trees. Many ap- +proaches have been developed to provide solutions for classifying phylogenetic +trees based on the well-known clustering algorithms, such as k-means and k-me- +doids. We discuss their main features in the Methods section. + +Partitioning is a clustering approach used to divide a given set of elements (or taxa) +into a meaningful set of groups of elements (objects or entities) called clusters (or +classes) (Mirkin 1996, Mirkin 2005). The objective of partitioning is to find groups +of similar elements according to a given similarity measure. The four main parti- +tioning approaches that can be used to group the elements based on the set of their +features (or variables) are the following: 1) a center of gravity, i.e., the k-means +algorithm (Lloyd 1957, MacQueen 1967), where k denotes the number of clusters; +2) a geometric median, i.e., k-medians (Bradley et al. 1997); 3) a center containing +the most frequent modes, i.e., k-modes (Huang 1998); 4) a medoid- based approach, +in which a medoid is a cluster element that minimizes the sum of the distances be- +tween it and all other cluster elements, i.e., k-medoids (Kaufman and Rousseeuw +1990). In our literature review, we will mainly focus only on the k-means and k- +medoids algorithms as they have been extensively used in tree clustering (see the +Methods section). Both of them are very fast, as the time complexity of k-means is +𝛰(𝐼 × 𝐾 × 𝑀 × 𝑁), where I is the number of iterations in the internal loop of k- +means, K is the number of clusters, M is the number of features characterizing the +given set of elements, and N is the number of elements, whereas the time complexity +of k-medoids is 𝛰(𝐼 × 𝐾 × 𝑀 × (𝑁 − 𝐾)2). It is worth noting that the k-medoids +algorithm is much less sensitive to outliers than k-means. The Euclidean, Manhattan +and Minkowski metrics are the most frequently used in the objective function of k- + +8 + + + + +means and k-medoids (Mirkin 2005, de Amorim and Mirkin 2012, de Amorim and +Makarenkov 2016). However, in the case of tree clustering the Robison and Foulds +topological distance or another tree distance should be used instead, and phyloge- +netic trees will play the role of cluster elements. +Methods +The Phylogenetic Islands (Maddison 1991) is a method that divides a collection +of trees based on the branch length of the trees and the number of branch rearrange- +ments by which the input trees differ. The author considers the three following types +of branch rearrangement: NNI (nearest neighbor interchange), SPR (subtree prun- +ing-regrafting), and TBR (tree bisection reconnection) (Swofford and Olsen 1990, +Swofford 1991). In NNI rearrangements, a clade (i.e. a subtree) can be moved to a +nearby branch only, in SPR, it can be moved to a nearby or a distant branch, and in +TBR, it can be moved to a nearby or a distant branch, with the clade also being +rerooted. This method was developed to find the most-parsimonious trees using tree +search algorithms, i.e., it starts with multiple starting points to find multiple islands. +Maddison formally defines an island of trees of length L as a collection of n trees +that satisfy three requirements: (1) all trees are of length +𝑁 +2), and Tm is +the tree T restricted to its majority edges. The weight of each bipartition Bi is the +number Ni of X-trees in the profile containing this bipartition. +The author generalizes the score (4), defining it for a partition of trees 𝑃𝜋 in k clas- +ses, as follows: + +𝑊𝑘(𝑃𝜋) = +∑ 𝑝𝑖 × 𝑊𝜋𝑖 +𝑖=1,…,𝑘 +(𝑇𝑖 +𝑚𝑎𝑗), +(5) +where 𝑃𝜋 is a partition of the set of trees 𝜋 in k classes (𝜋1, … , 𝜋𝑘) containing re- +spectively {p1, …, pk} trees, and 𝑇𝑖 +𝑚𝑎𝑗 is the majority consensus trees corresponding +to class i. + +Islands of Trees (Silva and Wilkinson 2021) is the method based on any appropri- +ate pairwise tree-to-tree distance metric that extends the notion of island to any set +or multiset of trees, such as those that can be generated by Bayesian or bootstrap +methods and facilitates finding islands of trees a posteriori. This can be useful when +the strict consensus of most parsimonious trees is relatively unresolved, although it +relies on the analytical program (Silva and Wilkinson used PAUP*) to identify not +only the number of islands, but also the constituents of most parsimonious trees. +Distinct subsets of trees, such as tree islands, are complementary to other means of +data exploration that involve attempts at partitioning sets of trees to obtain better +summaries and promote better understanding of evolution. However, this method is +of limited use for large phylogenetic tree distributions because it replaces the cal- +culation of the distance with a very large number of pairwise comparisons of trees. + +11 + + + + + +Inferring multiple consensus trees using k-medoids (Tahiri et al. 2018) is a fast +method for inferring multiple consensus trees from a given set of phylogenetic trees +defined on the same set of species. This method is based on the k-medoids parti- +tioning algorithm to partition a given set of trees into multiple tree clusters. The +well-known Silhouette and Caliński-Harabasz cluster validity indices have been +adapted for tree clustering with k-medoids to determine the most appropriate num- +ber of clusters. It can be used to identify groups of gene trees that have similar evo- +lutionary histories within the group and different evolutionary histories between the +groups. This method is suitable for the analysis of large genomic and phylogenetic +datasets. +Compared to the objective function used by Stockham et al. (2002) (see Equation +1), Tahiri et al. (2018) used the majority-rule consensus tree instead of the strict +consensus tree, and the unsquared RF distances instead of the squared one. The +straightforward objective function to be minimized is then as follows: + +𝑂𝐹 = ∑ ∑ 𝑅𝐹(𝑇𝑘 +𝑚𝑎𝑗, 𝑇𝑘𝑖) +𝑁𝑘 +𝑖=1 +𝐾 +𝑘=1 +, +(6) +where RF is the Robinson and Foulds distance between the tree Tki (i.e. tree i of +cluster k) and 𝑇𝑘 +𝑚𝑎𝑗 that is the majority-rule consensus tree of cluster k. Neverthe- +less, computing the majority-rule consensus tree or the extended majority-rule con- +sensus tree requires at least O(nN) time, where n is the number of leaves (taxa or +species) in each tree and N is the number of trees. +Thus, Tahiri et al. (2018) used the following objective function in their method +which is based on k-medoids: + +𝑂𝐹𝑚𝑒𝑑 = ∑ ∑ 𝑅𝐹(𝑇𝑘 +𝑚, 𝑇𝑘𝑖) +𝑁𝑘 +𝑖=1 +𝐾 +𝑘=1 +, +(7) +where 𝑇𝑘 +𝑚 is the medoid of cluster k, defined as a tree belonging to cluster k that +minimizes the sum of the RF distances between it and all other trees in k. This ver- +sion of the objective function is much faster than that based on Equation (6) because +it does not require the majority-rule consensus tree recomputation at each basic step +of clustering algorithm. The running time of this method is O(nN2+rK(N-K)2I), +where O(nN2) is the time needed to precalculate the matrix of pairwise RF distances +of size (N×N) between all input trees, K is the number of clusters, I is the number +of iterations in the internal loop of k-medoids, and r is the number of different ran- +dom starts used in k-medoids (usually hundreds of different random starts are +needed to obtain good clustering results; Mirkin 2005). + +Inferring multiple consensus trees and supertrees using k-means (Tahiri et al. +2022) is a new method for inferring multiple alternative consensus trees and super- +trees that best represent the main evolutionary patterns of a given set of gene trees. +This method is based on the use of the popular k-means clustering algorithm and +the Robinson and Foulds topological distance. It partitions a given set of trees into + +12 + + + + +one, for homogeneous data, or multiple, for heterogeneous data, cluster(s) of trees. +The authors show how the popular Caliński-Harabasz, Silhouette, Ball and Hall, +and Gap cluster validity indices can be used in tree clustering with k-means. The +Euclidean property of the square root of the Robinson and Foulds distance is used +to define a fast and efficient objective function that is as follows: +𝑂𝐹𝐸𝐴 = ∑ 1 +𝑁𝑘 +∑ ∑ 𝑅𝐹(𝑇𝑘𝑖, 𝑇𝑘𝑗) +𝑁𝑘 +𝑗=𝑖+1 +𝑁𝑘−1 +𝑖=1 +𝐾 +𝑘=1 +, +(8) +The time complexity of the tree clustering algorithm based on Equation (8) is +O(nN2+rNKI). + +Moreover, the authors establish some interesting properties, and use them in the clus- +tering process, of the general objective function defined in Equation (6). Specifically, +the lower and the upper bounds of this objective function OF are established in The- +orem 2 below: + +Theorem 2 (Tahiri et al. 2022). For a given cluster k containing Nk phylogenetic +trees (i.e. additive trees or X-trees) the following inequalities hold: +1 +𝑁𝑘 − 1 ∑ ∑ 𝑅𝐹(𝑇𝑘𝑖, 𝑇𝑘𝑗) +𝑁𝑘 +𝑗=𝑖+1 +𝑁𝑘−1 +𝑖=1 +≤ ∑ 𝑅𝐹(𝑇𝑘 +𝑚𝑎𝑗, 𝑇𝑘𝑖) +𝑁𝑘 +𝑖=1 +≤ 2 +𝑁𝑘 +∑ ∑ 𝑅𝐹(𝑇𝑘𝑖, 𝑇𝑘𝑗), +𝑁𝑘 +𝑗=𝑖+1 +𝑁𝑘−1 +𝑖=1 + (9) +where Nk is the number of trees in cluster k, Tki and Tkj are, respectively, trees i and j +in cluster k, and 𝑇𝑘 +𝑚𝑎𝑗 is the majority-rule consensus tree of cluster k. + +In the same paper, Tahiri et al. show how their method can be extended to the case +of supertree clustering. In the supertree clustering context, we assume that a given set +of N unrooted phylogenetic trees may contain different, but mutually overlapping, +sets of leaves. In this case, the original objective function OF shown in Equation (6) +can be reformulated as follows: +𝑂𝐹𝑆𝑇 = ∑ ∑ 𝑅𝐹𝑛𝑜𝑟𝑚(𝑆𝑇𝑘, 𝑇𝑘𝑖) +𝑁𝑘 +𝑖=1 +𝐾 +𝑘=1 += ∑ ∑ ( +𝑅𝐹(𝑆𝑇𝑘, 𝑇𝑘𝑖) +2𝑛(𝑆𝑇𝑘, 𝑇𝑘𝑖) − 6) +𝑁𝑘 +𝑖=1 +𝐾 +𝑘=1 +, +(10) +where K is the number of clusters, Nk is the number of trees in cluster k, +RFnorm(STk,Tki) is the normalized Robinson and Foulds topological distance between +tree i of cluster k, denoted Tki, and the majority-rule supertree of this cluster, denoted +STk, reduced to a subtree having all leaves in common with Tki. The RF distance is +normalized here by dividing it by its maximum possible value (i.e. 2n(STk,Tki)-6, +where n(STk,Tki) is the number of common leaves in STk and Tki). The RF distance +normalization is performed here to account equally the contribution of each tree to +clustering. Clearly, Equation (10) can be used only if the number of common leaves +in STk and Tki is larger than 3. + +An analog of Equation (8) can be used in supertree clustering to avoid supertree re- +calculations at each step of k-means. This can be done using the following objective +function: + +13 + + + + +𝑂𝐹𝑆𝑇_𝐸𝐴 = ∑ 1 +𝑁𝑘 +∑ ∑ ( 𝑅𝐹(𝑇𝑘𝑖, 𝑇𝑘𝑗) +2𝑛(𝑇𝑘𝑖, 𝑇𝑘𝑗) − 6 + 𝛼 × 𝑛(𝑇𝑘𝑖) + 𝑛(𝑇𝑘𝑗) − 2𝑛(𝑇𝑘𝑖, 𝑇𝑘𝑗) +𝑛(𝑇𝑘𝑖) + 𝑛(𝑇𝑘𝑗) +) , +𝑁𝑘 +𝑗=𝑖+1 +𝑁𝑘−1 +𝑖=1 +𝐾 +𝑘=1 + (11) +where n(Tki) is the number of leaves in tree Tki, n(Tkj) is the number of leaves in tree +Tkj, n(Tki,Tkj) is the number of common leaves in trees Tki and Tkj, and α is the penal- +ization (tuning) parameter, taking values between 0 and 1, needed to prevent from +putting to the same cluster trees having small percentages of common leaves. + +The simulations conducted by Tahiri at al. (2022) illustrated that their new tree clus- +tering method is faster and generally more efficient than the methods of Stockham +et al. (2002), Tahiri et al. (2018) and Bonnard et al. (2006) discussed earlier in this +section. +Cluster validity indices adapted to tree clustering +In this section, we show how the popular Caliński-Harabasz, Silhouette, Ball and +Hall, and Gap cluster validity indices can be used in tree clustering with k-means. +Caliński-Harabasz cluster validity index adapted for tree clustering +The first cluster validity index we consider here is the Caliński-Harabasz index +(Caliński and Harabasz 1974). This index, sometimes called the variance ratio cri- +terion, is defined as follows: +𝐶𝐻 = 𝑆𝑆𝐵 +𝑆𝑆𝑊 +× 𝑁 − 𝐾 +𝐾 − 1, +(12) +where SSB is the index of intergroup evaluation, SSW is the index of intragroup eval- +uation, K is the number of clusters and N is the number of elements (i.e. trees in our +case). The optimal number of clusters corresponds to the largest value of CH. + +In the traditional version of CH, when the Euclidean distance is considered, the SSB +coefficient is evaluated by using the L2-norm: +𝑆𝑆𝐵 = ∑ 𝑁𝑘 +𝐾 +𝑘=1 +‖𝑚𝑘 − 𝑚‖2, +(13) +where mk (k = 1 ... K) is the centroid of cluster k, m is the overall mean (i.e. centroid) +of all elements in the given dataset X, and Nk is the number of elements in cluster k. +In the context of the Euclidean distance, the SSW index can be calculated using the +two following equivalent expressions: +𝑆𝑆𝑊 = ∑ ∑‖𝑥𝑘𝑖 − 𝑚𝑘‖2 = ∑ 1 +𝑁𝑘 +( ∑ +∑ ‖𝑥𝑘𝑖 − 𝑥𝑘𝑗‖ +2 +𝑁𝑘 +𝑗=𝑖+1 +𝑁𝑘−1 +𝑖=1 +) +𝐾 +𝑘=1 +, +𝑁𝑘 +𝑖=1 +𝐾 +𝑘=1 + +(14) +where xki and xkj are elements i and j of cluster k, respectively (Caliński and Harabasz +1974). + +14 + + + + + +To use the analogues of Equations (13) and (14) in tree clustering, Tahiri et al. +(2022) used the concept of centroid for a given set of trees. The median tree +(Barthélemy and Monjardet 1981; Barthélemy and McMorris 1986) plays the role +of this centroid in a tree clustering algorithm. The median procedure (Barthélemy +and Monjardet 1981) is defined below. The set of median trees, Md(Π), for a given +set of trees Π = {T1, …, TN} having the same set of leaves S, is the set of all trees T +defined on S, such that: ∑ +𝑅𝐹(𝑇, 𝑇𝑖) +𝑁 +𝑖=1 + is minimized. If N is odd, then the majority- +rule consensus tree, Maj(Π) of Π, is the only element of Md(Π). If N is even, then +Md(Π) is composed of Maj(Π) and of some more resolved trees. + +Tahiri et al. (2022) proposed to use some formulas based on the properties of the +Euclidean distance to define SSB and SSW in k-means-like tree clustering. These for- +mulas do not require the computation of the majority (or the extended majority)- +rule consensus trees at each iteration of k-means. Precisely, they replace the term +‖𝑥𝑘𝑖 − 𝑥𝑘𝑗‖ +2 in Equation (14) by 𝑅𝐹(𝑇𝑘𝑖, 𝑇𝑘𝑗) to obtain the formula for SSW: +𝑆𝑆𝑊 = ∑ 1 +𝑁𝑘 +∑ +∑ 𝑅𝐹(𝑇𝑘𝑖, 𝑇𝑘𝑗) +𝑁𝑘 +𝑗=𝑖+1 +𝑁𝑘−1 +𝑖=1 +, +𝐾 +𝑘=1 + +(15) +where Tki and Tkj are trees i and j of cluster k, respectively. + +Also, in the case of the Euclidean distance, the formula is as follows: +𝑆𝑆𝐵 + 𝑆𝑆𝑊 = 1 +𝑁 (∑ ∑ ‖𝑥𝑖 − 𝑥𝑗‖ +2 +𝑁 +𝑗=𝑖+1 +𝑁−1 +𝑖=1 +), +(16) +where 𝑥𝑖 and 𝑥𝑗 are two different elements of X (Caliński and Harabasz 1974). +As a result, the approximation to the global variance between groups, SSB, can be +evaluated as follows: +𝑆𝑆𝐵 = 1 +𝑁 (∑ ∑ 𝑅𝐹(𝑇𝑖, 𝑇𝑗) +𝑁 +𝑗=𝑖+1 +𝑁−1 +𝑖=1 +) − 𝑆𝑆𝑊, +(17) +where Ti and Tj are trees i and j in the set of trees Π, and 𝑆𝑆𝑊 is calculated according +to Equation (15). + +Based on the Euclidean properties of the square root of the Robinson and Foulds +distance, Equations (15) and (17) establish the exact formulas for calculating the +indices SSB and SSW for the objective function 𝑂𝐹𝐸𝐴 defined by Equation (8). Inter- +estingly the objective function 𝑂𝐹𝐸𝐴 can also be used as an approximation of the +objective function defined in Equation (6) (obviously, the centroid of a cluster of +trees is not necessarily a consensus tree of the cluster; furthermore, it is not neces- +sarily a phylogenetic tree). + + +15 + + + + +Ball-Hall index adapted for tree clustering +Another relevant criterion to consider in this review is the Ball-Hall index. In 1965, +Ball and Hall (BH) introduced the ISODATA procedure to measure the average +dispersion of groups of objects with respect to the mean square root distance, i.e. +the intra-group distance. Unlike the CH index, the BH index can be used to find +solutions consisting of a single consensus tree. Tahiri et al. (2022) adapted the BH +criterion for tree clustering with k-means, which led to the following formula: +𝐵𝐻 = 1 +𝐾 ∑ 1 +𝑁𝑘 +𝐾 +𝑘=1 +∑ 𝑅𝐹(𝑇𝑘 +𝑚𝑎𝑗, 𝑇𝑘𝑖) +𝑁𝑘 +𝑖=1 +. +(18) +Furthermore, the following formula can be used to avoid the majority-rule tree cal- +culation: +𝐵𝐻 = 1 +𝐾 ∑ 1 +𝑁𝑘 +2 +𝐾 +𝑘=1 +∑ +∑ 𝑅𝐹(𝑇𝑘𝑖, 𝑇𝑘𝑗) +𝑁𝑘 +𝑗=𝑖+1 +𝑁𝑘−1 +𝑖=1 +. +(19) +Silhouette index adapted for tree clustering +The next popular criterion we consider here is the Silhouette (SH) width index +(Rousseeuw 1987). Traditionally, the Silhouette width of cluster k is defined as fol- +lows: +𝑠(𝑘) = 1 +𝑁𝑘 +[∑ +𝑏(𝑖) − 𝑎(𝑖) +𝑚𝑎𝑥( 𝑎(𝑖), 𝑏(𝑖) +𝑁𝑘 +𝑖=1 +], +(20) +where Nk is the number of elements belonging to cluster k, a(i) is the average dis- +tance between element i and all other elements belonging to cluster k, and b(i) is the +smallest, over-all clusters k’ different from k, of all average distances between i and +all the elements of cluster k’. + +Equations (21) and (22) can be used to calculate a(i) and b(i), respectively, in case +of tree clustering: +𝑎(𝑖) = +∑ +𝑅𝐹(𝑇𝑘𝑖, 𝑇𝑘𝑗) +𝑁𝑘 +𝑗=1 +𝑁𝑘 +, +(21) +𝑏(𝑖) = +𝑚𝑖𝑛 +1≤𝑘′≤𝐾,𝑘′≠𝑘 +∑ +𝑅𝐹(𝑇𝑘𝑖, 𝑇𝑘′𝑗) +𝑁𝑘′ +𝑗=1 +𝑁𝑘′ +, +(22) +where Tk’j is tree j of cluster k’, such that k’ ≠ k, and Nk’ is the number of trees in +cluster k’. + +The optimal number of clusters, K, corresponds to the maximum average value of +SH that is calculated as follows: + +16 + + + + +𝑆𝐻 = 𝑠(𝐾) = ∑ +[𝑠(𝑘)] +𝐾 +𝐾 +𝑘=1 +. +(23) +The value of the SH index defined by Equation (23) is located in the interval be- +tween -1 and +1. +Gap statistic adapted for tree clustering +The last criterion that we are discussing here is the Gap statistic (Tibshirani et al. +2001). As the BH index, Gap allows solutions consisting of a single consensus tree. +The formulas proposed by Tibshirani et al. (2001) are based on the properties of the +Euclidean distance. In the context of tree clustering, Tahiri et al. (2022) adapted the +Gap statistic by defining the total intracluster distance, 𝐷𝑘, characterizing the cohe- +sion between the trees belonging to the same cluster k, as follows: +𝐷𝑘 = ∑ ∑ 𝑅𝐹(𝑇𝑘𝑖 +𝑁𝑘 +𝑗=1 +𝑁𝑘 +𝑖=1 +, 𝑇𝑘𝑗). +(24) +The sum of the average total intracluster distances, 𝑉𝐾, can be calculated using the +following formula: +𝑉𝐾 = ∑ 1 +2𝑁𝑘 +𝐾 +𝑘=1 +𝐷𝑘. +(25) +The Gap statistic, which reflects the quality of a given clustering solution with K +clusters, is traditionally defined as follows: +𝐺𝑎𝑝𝑁(𝐾) = 𝐸𝑁 +∗ {log( 𝑉𝐾)} − log( 𝑉𝐾), +(26) +where 𝐸𝑁 +∗ denotes expectation under a sample of size N from the reference distribu- +tion. The following formula (Tibshirani et al. 2001) for the expectation of 𝑙𝑜𝑔( 𝑉𝐾) +was used in our method: +𝐸𝑁 +∗ {log( 𝑉𝐾)} = log (𝑁𝑛 +12) − (2 +𝑛) log( 𝐾), +(27) +where n is the number of tree leaves. The largest value of the Gap statistic corre- +sponds to the best clustering. + +Example of application to evolutionary data + +Aminoacyl-tRNA synthetases (aaRSs) are enzymes that attach the appropriate +amino acid to their cognate transfer RNA. The structure-function aspect of aaRSs +has long interested biologists (Woese et al. 2000, Godwin et al. 2018). It has been +observed that the central role played by aaRSs in translation suggest that their evo- +lutionary histories and that of the genetic code can be closely related (Woese et al. +2000). This information would make aaRS gene domain analysis a key component +of tree-of-life inference (Bullwinkle and Ibba 2014, Unvert et al. 2017). Woese et +al. examined the evolutionary profiles of each of the 20 standard aaRSs used by + +17 + + + + +living cells to construct the evolutionary history of proteins organized into 5 groups +(nonpolar aliphatic R group, nonpolar, aromatic R group, polar, uncharged R group, +positively charged R group, and negatively charged R group). To conduct their fa- +mous aaRS analysis Woese et al. considered a total of 72 species from 3 main do- +mains (Archaea, Eukarya and Bacteria), which can be represented by leaves of the +related phylogenetic trees. +In our study, we used 36 aaRS phylogenetic trees (i.e. aaRS gene trees) originally +constructed by Woese et al. These trees had different, but mutually overlapping, +sets of leaves (in total 72 different species were considered). They are available on +our GitHub repository along with our program at the following URL address: +https://github.com/TahiriNadia/KMeansSuperTreeClustering. These 36 trees were +used as input for our KMeansSuperTreeClustering algorithm (Tahiri et al. 2022). +Our supertree clustering algorithm was carried out with the following options: the +Caliński-Harabasz (Caliński and Harabasz 1974) cluster validity index was used to +select the best number of clusters (the number of clusters varied from 2 to 10 in our +experiments) and the penalization parameter α was set to 1. + +In these settings, our algorithm found that the best solution for these data corre- +sponds to a 2-cluster partitioning. Each of these clusters of trees can be represented +by its own supertree. The first obtained cluster includes 19 trees for a total of 61 +different species, while the second obtained cluster includes 17 trees for a total of +56 species. The supertrees (see Figures 3 and 4) for the two obtained tree clusters +were inferred using the CLANN program (Creevey and McInerney 2005). In +CLANN, we used the most similar supertree (dfit) method (Creevey et al. 2004) +with the mrp criterion. This criterion involves a matrix representation based on the +parsimony criterion. Next, we inferred the most common (by cluster) horizontal +gene transfers (HGT) that characterize the evolution of phylogenetic trees included +in the two obtained clusters of trees. The HGT detection method by Boc et al. (2010) +was used for this purpose. It proceeds by reconciliation of the species and gene +phylogenetic trees. In our case, the two obtained supertrees played the role of gene +trees, while the species phylogenetic trees followed the NCBI taxonomic classifi- +cation (see https://www.ncbi.nlm.nih.gov/Taxonomy/CommonTree/wwwcmt.cgi); +they are presented by full edges in Figures 4 and 5. These supertrees were not fully +resolved (i.e. the first supertree, see Fig. 4 contains 9 internal nodes with degree +greater than 3, whereas the second supertree, see Fig. 5 contains 10 internal nodes +with degree greater than 3). We used the version of the HGT algorithm available on +the T-Rex website (Boc et al. 2012) and Armadillo 1.1 (Lord et al. 2012) workflow +platform to identify the scenarios of HGT events that reconcile each species tree +with the corresponding supertree. The root of all of these trees was placed on the +edge that splits the clade of Bacteria with those of Eukarya and Archaea. Two fre- +quent horizontal gene transfers were found for the first supertree and four for the +second supertree. Our results indicate that most of aminoacyl-tRNA synthetases un- +derwent a two-way evolution. The obtained results are in line with the results of +Dohm et al. (2006) and Sharaf et al. (2019) that aminoacyl-tRNA synthetases pos- +sess two versions of most tRS, one cytosolic and one mitochondrial. + +18 + + + + + + +Fig 4. Species tree (full edges) corresponding to the NCBI taxonomic classification +constructed for 61 species from the first cluster of 19 aaRS phylogenetic trees. The +two horizontal gene transfers (indicated by arrows) were found using the HGT- +Detection program of Boc et al. (2012). +A. aeolicus +T. maritima +Synechocystis sp. PCC 6803 +C. tepidum +P. gingivalis +C. trachomatis +M. pneumoniae +M. genitalium +D. radiodurans +T. aquaticus +T. pallidum +B. burgdorferi +S. coelicolor +M. tuberculosis +C. glutamicum +T. ferrooxidans +H. pylori +C. crescentus +R. capsulatus +R. prowazekii +B. bacilliformis +B. pertussis +N. gonorrhoeae +H. influenzae +P. aeruginosa +F. tularensis +S. typhimurium +E. coli +C. longisporum +C. acetobutylicum +B. subtilis +S. aureus +E. faecalis +L. delbrueckii +L. casei +S. pyogenes +S. pneumoniae +T. thermophila +P. falciparum +T. vaginalis +H. sapiens +D. melanogaster +N. locustae +S. cerevisiae +O. sativa +N. tabacum +A. thaliana +C. symbiosum +A. pernix +P. aerophilum +S. solfataricus +M. thermautotrophicus +H. salinarum +A. fulgidus +M. barkeri +P. horikoshii +P. furiosus +M. maripaludis +M. jannaschii +G. intestinalis +Eukaryota +Archaea +Bacteria +G. lambia + +19 + + + + + +Fig 5. Species tree (full edges) corresponding to the NCBI taxonomic classification +constructed for 56 species from the first cluster of 17 aaRS phylogenetic trees. The +four horizontal gene transfers (indicated by arrows) were found using the HGT- +Detection program of Boc et al. (2012). +A. aeolicus +T. maritima +Synechocystis sp. PCC 6803 +C. tepidum +P. gingivalis +C. trachomatis +M. pneumoniae +M. genitalium +S. coelicolor +M. tuberculosis +D. radiodurans +T. thermophilus +T. pallidum +B. burgdorferi +C. acetobutylicum +L. bulgaricus +E. faecalis +S. pyogenes +B. subtilis +S. aureus +T. ferrooxidans +B. pertussis +N. gonorrhoeae +H. pylori +C. jejuni +C. crescentus +R. capsulatus +R. prowazekii +Z. mobilis +R. meliloti +A. brasilense +C. burnetii +H. influenzae +E. coli +F. tularensis +A. calcoaceticus +P. fluorescens +P. aeruginosa +P. falciparum +G. intestinalis +S. cerevisiae +H. sapiens +C. elegans +L. luteus +A. thaliana +C. symbiosum +A. pernix +P. aerophilum +S. solfataricus +M. thermautotrophicus +P. horikoshii +H. marismortui +A. fulgidus +M. barkeri +M. maripaludis +M. jannaschii +Archaea +Eukaryota +Bacteria + +20 + + + + +Conclusion + +In this paper, we have reviewed the state-of-the-art systematic methods for inferring +multiple alternative consensus trees and supertrees from a given set of phylogenetic +trees (i.e. additive trees, evolutionary trees or X-trees). Most of the reviewed papers +describe algorithms proceeding by k-means or k-medoids clustering of tree topolo- +gies. In the case of consensus tree clustering problem, all the trees should be defined +on the same set of taxa (i.e. species associated to the tree leaves), whereas in the +case of supertree clustering problem, the trees can be defined on different, but mu- +tually overlapping, sets of taxa. In many instances, multiple consensus trees and +supertrees represent more relevant evolutionary models than traditional single con- +sensus trees and supertrees. The resolution of multiple consensus trees and super- +trees is generally much better than that of single consensus trees or supertrees in- +ferred by conventional methods (Maddison 1991). Thus, multiple consensus trees +and supertrees have the potential of preserving much more plausible information +from a set of given gene trees. Clustering seems to be an intuitive natural solution +for inferring multiple consensus trees and supertrees. Tree clustering has a direct +practical application in evolutionary studies. It allows one to identify sets of genes +that have been affected to the same horizontal gene transfer, hybridization, intra- +genic/intergenic recombination events, or those that have undergone the same an- +cient gene duplications and gene losses during their evolution (Makarenkov et al. +2004, Bapteste et al. 2004, Diallo et al. 2006, Boc and Makarenkov 2011). + +Since the beginning of the Tree of Life inference project (Maddison et al. 2007), the +number of studies dealing with supertree theory has grown considerably. The meth- +ods described in this paper can be used for inferring multiple alternative subtrees of +the Tree of Life as it contains many unresolved clades (i.e. subtrees with high de- +grees of its internal nodes). From the practical point of view the problem of con- +structing multiple alternative supertrees is more relevant than that of constructing +multiple alternative consensus trees because most of currently available gene trees +are not defined on exactly the same sets of taxa. However, to the best of our +knowledge, the only study addressing this relevant problem remains the recent work +of Tahiri et al. (2022). The authors of this work showed how some remarkable prop- +erties of the Robinson and Foulds topological distance (original or normalized) and +the k-means partitioning algorithm can be used to achieve very promising tree clus- +tering performance. Finally, in the application section, we showed how this method +can be applied to cluster phylogenetic trees from the famous aaRS phylogenetic da- +taset originally described by Woese et al. 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Microbiol Mol Biol Rev 64(1):202-236 + + diff --git a/vNAyT4oBgHgl3EQfm_j-/content/tmp_files/load_file.txt b/vNAyT4oBgHgl3EQfm_j-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..feeba1f88cb3126c93b910c2f40e19c6e41deb7f --- /dev/null +++ b/vNAyT4oBgHgl3EQfm_j-/content/tmp_files/load_file.txt @@ -0,0 +1,666 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf,len=665 +page_content='Inferring multiple consensus trees and super- trees using clustering: a review Vladimir Makarenkov1*, Gayane S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Barseghyan1 and Nadia Tahiri2 1 Département d’Informatique, Université du Québec à Montréal, Case postale 8888, Succursale Centre-ville, Montreal, QC, H3C 3P8, Canada 2 Département d’Informatique, Université de Sherbrooke, 2500 Boulevard de l’Université, Sherbrooke, Québec J1K 2R1, Canada *Corresponding author: makarenkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='vladimir@uqam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='ca Abstract: Phylogenetic trees (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' evolutionary trees, additive trees or X-trees) play a key role in the processes of modeling and representing species evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Genome evolution of a given group of species is usually modeled by a species phylogenetic tree that represents the main patterns of vertical descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' However, the evolution of each gene is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' It can be represented by its own gene tree which can differ substantially from a general species tree representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Consensus trees and super- trees have been widely used in evolutionary studies to combine phylogenetic infor- mation contained in individual gene trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Nevertheless, if the available gene trees are quite different from each other, then the resulting consensus tree or supertree can either include many unresolved subtrees corresponding to internal nodes of high degree or can simply be a star tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' This may happen if the available gene trees have been affected by different reticulate evolutionary events, such as horizontal gene transfer, hybridization or genetic recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Thus, the problem of inferring multiple alternative consensus trees or supertrees, using clustering, becomes rele- vant since it allows one to regroup in different clusters gene trees having similar evolutionary patterns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' gene trees representing genes that have undergone the same horizontal gene transfer or recombination events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' We critically review recent advances and methods in the field of phylogenetic tree clustering, discuss the meth- ods’ mathematical properties, and describe the main advantages and limitations of multiple consensus tree and supertree approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' In the application section, we show how the multiple supertree clustering approach can be used to cluster aaRS gene trees according to their evolutionary patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Keywords: Clustering, Cluster validity index, Consensus tree, k-means, k-medoids, Phylogenetic tree, Robinson and Foulds distance, Supertree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2 Introduction The term phylogeny (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' phylogenetic tree or evolutionary tree) was introduced by Haeckel in 1866, who defined it as "the history of the paleontological development of organisms by analogy with ontogeny or the history of individual development".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' A phylogenetic tree represents a hypothesis about evolution of a given group of species which are usually associated with the tree leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' In mathematics, phylogenetic trees are called additive trees or X-trees (as their leaves are often associated with the set of species X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Barthélemy and Guénoche 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Let us now present some necessary mathematical definitions related to phy- logenetic trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The distance δ(x,y) between two vertices x and y in a phylogenetic tree T is defined as the sum of the edge lengths in the unique path linking x and y in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Such a path is denoted (x,y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' A leaf is a vertex of degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Usually, a leaf represents a contemporary species (or taxa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Let X be a finite set of n taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' A dissimilarity d on X is a non-negative function on X × X such that for any x, y from X: d(x,y) = d(y,x) ≥ d(x,x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' A dissimilarity d on X satisfies the four-point condition if for any x, y, z, and w from X: d(x,y) + d(z,w) ≤ max {d(x,z) + d(y,w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' d(x,w) + d(y,z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' For a finite set X, a phylogenetic tree (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' an additive tree or an X- tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' a tree whose leaves are labeled according to a final set of species X) is an ordered pair (T, φ) consisting of a tree T, with vertex set V, and a map φ: X→ V with the property that, for all x ∈ X with degree at most two, x ∈ φ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' A phyloge- netic tree is binary if φ is a bijection from X into the leaf set of T and every interior vertex has degree three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The theorem relating the four-point condition and a dissimilarity representability by a phylogenetic tree is as follows: Theorem 1 (Zarestskii, Buneman, Patrinos & Hakimi, Dobson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Any dissimilarity satisfying the four-point condition on X × X (where X is a finite set of species) can be represented by a phylogenetic tree T such that for any x, y from X, d(x,y) is equal to the length of the path linking the leaves x and y in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' This dissimilarity is called a tree metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Furthermore, this tree is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Figure 1 gives an example of a tree metric on the set X of five taxa and the corre- sponding phylogenetic tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Unfortunately, real-life evolutionary distances (or dissimilarities) rarely satisfy the four-point condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Thus, one need to carry out an approximation algorithm to infer a tree metric matrix from a given matrix of evolutionary distances (Gascuel 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Among the most known distance-based approximation algorithms we can 3 mention Neighbor-Joining (Saitou 1988), UPGMA (Sokal and Michener 1958), FITCH (Felsenstein 1997), and MW (Makarenkov and Leclerc 1996, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' An example of a tree metric on the set X of five taxa (on the left) and the corresponding phylogenetic tree (additive tree or X-tree) on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Biologists often need to compare phylogenetic trees to each other in order to dis- cover different evolutionary histories that govern a given set of species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' There are several measures for comparing phylogenetic trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The most popular of them in- clude the Robinson and Foulds topological distance (RF) (Robinson and Foulds 1981), the least-squares distance (LS), the bipartition dissimilarity (BD) (Boc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2010), and the quartet distance (QD) (Bryant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' In this literature review, we will mainly explore the methods based on the Robinson and Foulds distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The Robinson and Foulds topological distance (Robinson and Foulds 1981) be- tween two trees is the minimum number of elementary operations (contraction and expansion) of nodes needed to transform one phylogenetic tree into another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' It is also the number of splits (or bipartitions) that are present in one tree and absent in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The two phylogenetic trees in question must have the same set of taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The closer two phylogenetic trees are topologically, the smaller the value of the RF distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' It is often relevant to normalize the value of the RF distance by dividing it by its maximum possible value (equal to 2n-6) for two binary phylogenetic trees with n leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The RF distance calculation between two trees with n leaves can be carried out in O(n) (Day 1985, Makarenkov 1997, Makarenkov and Leclerc 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Often phylogenetic tree reconstruction methods do not return a single phylogenetic tree as output, but a collection of different trees (Gascuel 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Moreover, phylo- genetic trees inferred for different genes often differ from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' There is no absolute criterion for determining whether one tree is better than the others (except for the use of intrinsic criteria, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=', the use of bootstrap scores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' For this reason, it is preferable to seek a consensus representation of these trees, such that their con- cordant parts appear clearly in relation to the discordant parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The resulting repre- sentation is called a consensus tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Traditional consensus methods generate a single phylogenetic tree that is a representative of all of the input trees (Bryant 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' One of the first consensus methods was proposed by Adams (Adams 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Since then, a wide variety of methods have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' How to use them has been the subject of much debate (Bryant 2003, Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The main types of consensus trees are the following: the strict consensus tree (Sokal and Rohlf 1981, Moon and Eulenstein 2017), the majority-rule consensus tree (Mar- gush and McMorris 1981), the Nelson consensus tree (Nelson 1979), and the x1 x2 x2 x3 x4 x5 1 x1 6 6 4 2 2 2 x2 2 4 6 /1 1 x3 4 6 x5 x3 x4 4 x44 extended majority-rule consensus tree (Felsenstein 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Let us briefly recall the main characteristics of each of these consensus trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=" The strict consensus tree (or Nelson's cladogram) is inferred by considering only those tree splits (i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' bipartitions induced by the internal tree edges) that are identical in all trees compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Conflicting parts of phylogenetic trees are represented by multifurcations in a strict consensus tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' It is sometimes more convenient to have a less strict criterion than the one used by the strict consensus tree in order to allow bipartitions that are not necessarily present in all trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' When comparing a set of phylogenetic trees with different topologies, it is possible to search for the monophyletic groups that appear most frequently (of- ten in more than 50% of the trees) among all the trees compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The resulting tree is the majority-rule consensus tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The extended majority-rule consensus tree contains all majority bipartitions to which the remaining compatible bipartitions are added in turn, starting with the most frequent bipartitions for the given tree set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The process stops when a completely resolved (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' binary) tree is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The extended majority consensus tree is the most frequently used in molecular biology, as it is always the best resolved among the three types of consensus trees discussed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The Nelson consensus tree includes the heaviest set of compatible bipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' It consists in finding a clique of maximum weight in a compatibility graph of the entire bipartition set, which is NP-hard (Nelson 1979, Bryant 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Unfortunately, in many practical situations, phylogenetic trees used as input of con- sensus tree reconstruction methods can be quite divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' This can happen, for ex- ample, when the input trees represent the evolution of different genes which have been affected by multiple reticulate evolutionary events such as horizontal gene transfer, hybridization or intragenic/intergenic recombination, ancient gene dupli- cation or gene loss (Makarenkov and Legendre 2000, Mirkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2003, Bapteste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' These evolutionary events can be unique for a subgroup of the input gene trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Thus, it seems to be much more appropriate to represent this subgroup by its own consensus tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' However, the conventional consensus tree methods pro- vide only one candidate tree for a given set of input gene phylogenies without con- sidering their possible subgroups (or clusters) (Maddison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Figure 2 shows an example of four seven-leaf phylogenetic trees T1, T2, T3, and T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Here, the solution consisting of two majority-rule consensus trees, T12 and T34, seems to be much more appropriate than the conventional consensus solution con- sisting of a single majority-rule consensus tree, T1234, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=', here a star tree (a tree having no internal edges at all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 5 1 2 3 4 6 Tree T1 Tree T2 7 5 1 2 3 4 5 7 6 1 5 3 6 2 Tree T3 Tree T4 4 7 1 5 3 6 7 4 2 1 2 3 4 6 Majority rule consensus tree T12 7 5 Single (traditional) majority rule consensus tree T1234 7 5 1 6 4 3 2 1 5 3 6 4 Majority rule consensus tree T34 7 2 Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Four phylogenetic trees T1, T2, T3, and T4 defined on the same set of seven leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Their single (traditional) majority-rule consensus tree is a star tree T1234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The majority-rule consensus trees, T12 and T34, constructed for the pairs of topolog- ically close trees: T1 and T2, and T3 and T4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' In many evolutionary studies gene trees to be combined are defined on different, but partially overlapping, sets of taxa (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' see Tree of Life project;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Maddison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' It is very unlikely that all the genes considered have been sequenced for the 6 same sets of species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' In order to reconcile such trees, supertree reconstruction meth- ods should be applied (Bininda-Emonds 2004, Wilkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2007, McMorris and Wilkinson 2011, Warnow 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Supertrees synthesize a given set of small (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' partial) trees with partial taxon overlap into comprehensive supertrees that include all taxa present in the given set of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Four phylogenetic trees T1, T2, T3, and T4 defined on different, but mutually overlapping, sets of seven taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Their single (traditional) majority-rule supertree is a star tree T1234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The majority-rule supertrees, T12 and T34, constructed for the pairs of topologically close trees: T1 and T2, and T3 and T4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 1 2 3 4 6 Tree T1 Tree T2 7 1 2 3 4 5 1 5 3 6 2 Tree T3 Tree T4 4 7 1 5 3 6 7 4 1 2 3 4 6 Majority rule supertree T12 7 5 Single (traditional) majority rule supertree T1234 7 5 1 6 4 3 2 1 5 3 6 4 Majority rule supertree T34 7 2 7 The most known supertree inference method is Matrix Representation with Parsi- mony (MRP) (Baum 1992, Ragan 1992) that carries out matrix-like aggregation of the given partial trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The supertree reconstruction methods are commonly used for phylogenetic analysis of organisms with large genomes (Mank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2005, Bininda-Emonds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2007, Faurby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2016, Kimbal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' For organisms with small genomes, such as prokaryotes, several approaches to genomic phyloge- netic analysis have been adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' In particular, supertree analysis provides new in- sights into the evolution of prokaryotes that could not be solved by many other ap- proaches (Daubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Recently, Makarenkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' (2021) and Tahiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' (2022) have used supertree phylogenetic analysis to characterize the evolution of SARS-CoV-2 genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' As in the case of consensus trees, in many practical situations multiple conservative supertrees should be inferred to best represent the evolution of a given group of gene trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Figure 3 shows an example of four phylogenetic trees T1, T2, T3, and T4 defined on different, but mutually overlapping, sets of seven taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Here, the solution consisting of two majority-rule supertrees, T12 and T34, is more appropriate than that consisting of a single majority-rule supertree, T1234, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=', here a star tree, yielded by the traditional supertree approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The idea of building multiple consensus trees was originally formulated by Maddi- son (Maddison 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' He discovered that consensus trees for some subsets of input trees may differ a lot and that they are generally much better resolved than the single traditional consensus tree characterizing the whole set of the input trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Many ap- proaches have been developed to provide solutions for classifying phylogenetic trees based on the well-known clustering algorithms, such as k-means and k-me- doids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' We discuss their main features in the Methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Partitioning is a clustering approach used to divide a given set of elements (or taxa) into a meaningful set of groups of elements (objects or entities) called clusters (or classes) (Mirkin 1996, Mirkin 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The objective of partitioning is to find groups of similar elements according to a given similarity measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The four main parti- tioning approaches that can be used to group the elements based on the set of their features (or variables) are the following: 1) a center of gravity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=', the k-means algorithm (Lloyd 1957, MacQueen 1967), where k denotes the number of clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 2) a geometric median, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=', k-medians (Bradley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 3) a center containing the most frequent modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=', k-modes (Huang 1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' 4) a medoid- based approach, in which a medoid is a cluster element that minimizes the sum of the distances be- tween it and all other cluster elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=', k-medoids (Kaufman and Rousseeuw 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' In our literature review, we will mainly focus only on the k-means and k- medoids algorithms as they have been extensively used in tree clustering (see the Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Both of them are very fast, as the time complexity of k-means is 𝛰(𝐼 × 𝐾 × 𝑀 × 𝑁), where I is the number of iterations in the internal loop of k- means, K is the number of clusters, M is the number of features characterizing the given set of elements, and N is the number of elements, whereas the time complexity of k-medoids is 𝛰(𝐼 × 𝐾 × 𝑀 × (𝑁 − 𝐾)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' It is worth noting that the k-medoids algorithm is much less sensitive to outliers than k-means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The Euclidean, Manhattan and Minkowski metrics are the most frequently used in the objective function of k- 8 means and k-medoids (Mirkin 2005, de Amorim and Mirkin 2012, de Amorim and Makarenkov 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' However, in the case of tree clustering the Robison and Foulds topological distance or another tree distance should be used instead, and phyloge- netic trees will play the role of cluster elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Methods The Phylogenetic Islands (Maddison 1991) is a method that divides a collection of trees based on the branch length of the trees and the number of branch rearrange- ments by which the input trees differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' The author considers the three following types of branch rearrangement: NNI (nearest neighbor interchange), SPR (subtree prun- ing-regrafting), and TBR (tree bisection reconnection) (Swofford and Olsen 1990, Swofford 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' In NNI rearrangements, a clade (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' a subtree) can be moved to a nearby branch only, in SPR, it can be moved to a nearby or a distant branch, and in TBR, it can be moved to a nearby or a distant branch, with the clade also being rerooted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' This method was developed to find the most-parsimonious trees using tree search algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=', it starts with multiple starting points to find multiple islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfm_j-/content/2301.00483v1.pdf'} +page_content=' Maddison formally defines an island of trees of length L as a collection of n trees that satisfy three requirements: (1) all trees are of length u when H0 is true} +Power: P{∃ a peak in the local domain with height > u when H1 is true} +(1) +Formulas for type I error have been developed for stationary fields in 1D and isotropic fields in 2D and 3D +(Cheng and Schwartzman, 2015, Cheng and Schwartzman, 2017 and Cheng and Schwartzman, 2018). However, +there is no formula to calculate power. In order to get an appropriate estimate of power, we need to know +the peak height distribution for non-centered (the mean function is not 0) random fields. Generally speaking, +it is very difficult to calculate the peak height distribution especially when the random field has non-zero +mean. Durnez et al. (2016) suggests using Gaussian distribution to describe the non-null peaks and truncated +Gaussian distribution to approximate the overshoot distribution. This approach is easy to implement but not +very accurate because the peak height distribution is in reality always skewed and not close to any Gaussian +distribution. +In this article, we propose to approximate the probability of an observed peak exceeding the detection +threshold u by calculating the expected number of peaks above the threshold u. We show that the approximation, +which is also an upper bound, works well under certain scenarios. For the entire domain, we can approximate +the average peakwise power by taking the arithmetic mean of the approximation proposed in this paper over +non-null voxels. +The proposed approximation makes the problem more tractable, but in general, it does not have an explicit +form. In order to make it applicable in practice, we further simplify the formula under the isotropy assumption +and show its explicit form in 1D, 2D, and 3D. The explicit results are validated through 2D and 3D computer +simulations carried out in MATLAB. The simulation also covers multiple scenarios by modifying the parameters +used to generate the data. The performance of power approximation and its conservative adjustment under +these scenarios are discussed. +Finally, to assess the real-data performance of our power approximation method, we apply it to a 3D +simulation induced by a real brain imaging dataset, where the parameters are estimated from the Human +Connectome Project (Van Essen et al., 2012) fMRI data. By testing the method in a realistic setting, we also +demonstrate how effect size and other parameters affect the power. +The paper is organized as follows. We first show in Section 2 the problem setup and theoretical results in +certain scenarios. In Section 3, we derive the explicit formulas under isotropy. Simulation in 2D is conducted +in Section 4. Details regarding how to apply our formula in application setting are discussed in Section 5. The +methodology is applied to a 3D real dataset in Section 6. +2 +Power Approximation +2.1 +Setup +Let Y (s) = σ(s)Z(s) + µ(s) where Z = {Z(s), s ∈ D} representing the noise is a centered (zero-mean) smooth +unit-variance Gaussian random field on an N dimensional non-empty domain D ⊂ RN, σ(s) is the standard +deviation of the noise and µ(s) is the mean function. Let X(s) = Y (s)/σ(s) = Z(s) + θ(s) where the ratio +θ(s) = µ(s)/σ(s) is the standardized mean function, which we assume to be C2. +Here C3 is a sufficient +smoothness condition for Z, and this will be clarified in Assumption 1 below. +Let +Xi(s) = ∂X(s) +∂si +, +∇X(s) = (X1(s), . . . , XN(s)), +Xij(s) = ∂2X(s) +∂sisj +, +∇2X(s) = (Xij(s))1≤i,j≤N, +Zi(s) = ∂Z(s) +∂si +, +∇Z(s) = (Z1(s), . . . , ZN(s)), +Zij(s) = ∂2Z(s) +∂sisj +, +∇2Z(s) = (Zij(s))1≤i,j≤N. +2 + +We will make use of the following assumptions: +Assumption 1. Z ∈ C2(D) almost surely and its second derivatives satisfy the mean-square H¨older condition: +for any s0 ∈ D, there exists positive constants L, η and δ such that +E(Zij(s) − Zij(t))2 ≤ L2∥s − t∥2η, +∀t, s ∈ Us0(δ), i, j = 1, ..., N. +where Us0(δ) = s0 ⊕(−δ/2, δ/2)N is the N dimensional open cube of side length δ centered at s0. This condition +is satisfied, for example, if Z is C3(D). +Assumption 2. For every pair (t, s) ∈ D × D with s ̸= t, the Gaussian random vector +(Z(s), ∇Z(s), Zij(s), Z(t), ∇Z(t), Zij(t), 1 ≤ i ≤ j ≤ N) +is non-degenerate, i.e. its covariance matrix has full rank. +2.2 +Peak detection +Following the notation in the problem setup, the null and alternative hypothesis can be written as: +H0 : µ(s) = 0 for all s ∈ D +vs +H1 : µ(s) > 0, ∇µ(s) = 0, ∇2µ(s) ≺ 0 for some s ∈ D +The mean function µ(s) is not directly observed, so the hypothesis is tested based on the peak height of X(s). +For a peak detection procedure that aims to test this hypothesis, a threshold u for the peak height of X(s) +needs to be set in advance. If a local maximum with height greater than u is observed, we would choose to +reject the null hypothesis due to the strong evidence against it. The probability that a peak of X exceeds u +P +� +∃ s ∈ D s.t. X(s) > u|∇X(s) = 0 and ∇2X(s) ≺ 0 +� +(2) +is the type I error under H0 and power under H1. The threshold u can be obtained based on the peak height +distribution under H0. A formula for peak height distribution of smooth isotropic Gaussian random fields has +been derived in Cheng and Schwartzman (2018) and it can also be derived directly from a special case of the +formulas presented in this paper. Usually, u is set to be some quantile of the null distribution of peak height +to maintain the nominal α type I error. More details about selecting the threshold will be discussed in the real +data example. Selecting u is not the main focus of this paper and our method can be applied to any choice of +u. +2.3 +Power approximation +Let Mu be the number of local maxima of the random field X above u over the local domain D. The power +defined in (2) can be represented as P[Mu ≥ 1]. We call this the power function, seen as a function of the +threshold u. Note that +P[Mu ≥ 1] = +∞ +� +k=1 +P[Mu = k] ≤ +∞ +� +k=1 +kP[Mu = k] = E[Mu]. +(3) +On the other hand, +E[Mu] − P[Mu ≥ 1] = +∞ +� +k=2 +(k − 1)P[Mu = k] ≤ 1 +2 +∞ +� +k=2 +k(k − 1)P[Mu = k] = 1 +2E[Mu(Mu − 1)]. +(4) +Thus, we have +E[Mu] − 1 +2E[Mu(Mu − 1)] ≤ P[Mu ≥ 1] ≤ E[Mu]. +(5) +This inequality tells us that for any fixed u, the power is bounded within an interval of length E[Mu(Mu −1)]/2. +Thus, E[Mu] is a good approximation of power if one of the two conditions below is satisfied: +1. The factorial moment E[Mu(Mu − 1)] converges to 0 and E[Mu] does not. +2. They both converge to 0 and E[Mu(Mu − 1)] converges faster than E[Mu]. +The convergence above refers to conditions on the signal and noise parameters. In the rest of this section, we +introduce four interesting results. The first result can be useful for simplifying the power function and the other +three results give different scenarios where one of the conditions above holds. +3 + +2.4 +Adjusted E[Mu] +We have provided evidence of using E[Mu] to approximate power through (5). However, E[Mu] alone might not +be sufficient for power approximation since it only gives an upper bound. Also, unlike power, E[Mu] sometimes +exceeds 1. To correct for this, we define the adjusted E[Mu] as +E[Mu]adj = E[Mu]/ max(1, E[M−∞]). +(6) +The adjusted E[Mu] is the same as E[Mu] when the expected number of local maxima E[M−∞] is less or equal +to 1. When E[M−∞] is greater than 1, we divide E[Mu] by E[M−∞] to make sure it never exceeds 1. The +adjusted E[Mu] is more conservative, and we conjecture that it is a lower bound of power when there exists at +least one local maximum in the domain D. In applications, people are interested in a conservative estimator +so that the test is guaranteed to have enough power. Combining E[Mu] and E[Mu]/E[M−∞], we can get an +approximate range of the true power. We will compare E[Mu] and adjusted E[Mu] in simulation studies. +2.5 +Height equivariance +Our first result does not concern the approximation (5) yet, but it offers a simplification of the power function +and E[Mu] that will be used later. The proposition below states that the power function and E[Mu] for peak +detection are translation equivariant with respect to peak height. +Proposition 1. Let θ(s) = h(s) + θ0 be a peak signal with height θ0, where h(s) is a unimodal mean function +with maximum equal to 0 at s0 in D. Then the power function for peak detection and E[Mu] can be written in +the form F(u − θ0), where F(u) is the power function or E[Mu] at θ0 = 0. +Proof. Let ˜θ(s) = θ(s) − θ0 = h(s) + 0 and ˜ +Mu be the number of local maxima of the random field ˜X(s) = +Z(s) + ˜θ(s) above u over D. Considering the definition of power, we have +F(u − θ0) = P[ ˜ +Mu−θ0 ≥ 1] = P[Mu ≥ 1]. +Given that E[ ˜ +Mu−θ0] = E[Mu], is is also straightforward to show E[Mu] is translation equivariant with respect +to θ0. +Next, we give three scenarios where the equality in (5) can be achieved asymptotically: small domain size, +large threshold, and sharp signal. +2.6 +Small domain +If the size of the local domain D where a single peak exists is small enough, it can be shown that equality in +(5) can be achieved asymptotically. +Theorem 1. Consider a local domain Dϵ = U(s0, ϵ) for any fixed s0 ∈ D where U(s0, ϵ) = t0 ⊕ (−ϵ/2, ϵ/2)N +is the N-dimensional open cube of side ϵ centered at t0. For sufficiently small ϵ and fixed threshold u, +P[Mu ≥ 1] = E[Mu](1 − o(1)) = E[Mu]adj(1 − o(1)), +(7) +Proof. The proof is based on the proof of Lemma 3 in Piterbarg (1996) and Lemma 4.1 in Cheng and Schwartz- +man (2015). +E[Mu(Mu − 1)] = +� +Dϵ +� +Dϵ +� ∞ +u +� ∞ +u +E +� +|det∇2X(s)||det∇2X(t)| +���� +X(s) = x1, X(t) = x2 +∇X(s) = ∇X(t) = 0 +� +PX(s),X(t),∇X(s),∇X(t)(x1, x2, 0, 0) dx1 dx2 ds dt. +(8) +Let +E1(s, t) = E +� +|det∇2X(s)||det∇2X(t)| +���� +X(s) = x +∇X(s) = ∇X(t) = 0 +� +and replace one of the integration limits in (8) by −∞, we have +E[Mu(Mu − 1)] ≤ +� +Dϵ +� +Dϵ +P∇X(s),∇X(t)(0, 0)dsdt +� ∞ +u +E1(s, t)PX(s) (x|∇X(s) = ∇X(t) = 0) dx. +Then we can take the Taylor expansion +∇X(t) = ∇X(s) + ∇2X(s)(t − s) + ||t − s||1+αYs,t +4 + +where Ys,t = (Y 1 +s,t, ..., Y N +s,t)T is a Gaussian vector field. Note that the determinant of ∇2X(s) is equal to the +determinant of +� +� +� +� +� +1 +−(t1 − s1) +. . . +−(tN − sN) +0 +... +∇2X(s) +0 +� +� +� +� +� +(9) +For i = 2, ..., N + 1, multiply the ith column of this matrix by (ti − si)/||ti − si||2, take the sum of all such +columns and add the result to the first column. Since ∇X(s) = ∇X(t) = 0, we can derive ∇2X(s)(t − s) = +−||t − s||1+αYs,t, and obtain the matrix below with the same determinant as (9) +� +� +� +� +� +0 +−(t1 − s1) +. . . +−(tN − sN) +−||t − s||−1+αY 1 +s,t +... +∇2X(s) +−||s − t||−1+αY N +s,t +� +� +� +� +� +Let r = max1≤i≤N |ti − si|, +As,t = +� +� +� +� +� +0 +−(t1 − s1)/r +. . . +−(tN − sN)/r +Y 1 +s,t +... +∇2X(s) +Y N +s,t +� +� +� +� +� +So we have +E1(s, t) ≤ ||t − s||αE2(s, t) +where +E2(s, t) = E +� +|detAs,t||det∇2X(t)| +���� +X(s) = x, ∇X(s) = 0 +∇2X(s)(t − s) = −||t − s||1+αYs,t +� +. +Using the inequality of arithmetic and geometric means, we can bound the determinant +|det∇2X(t)| ≤ N 2N−2 � +i,j +|Xij(t)|N +|detAs,t| ≤ (N + 1)2N � +i,j +|aij|N+1 +where aij is the i, j entry of As,t. Apply the inequality again +|det∇2X(t)||detAs,t| ≤ 1 +2N 2N−2(N + 1)2N+1 +� +�� +i,j +|Xij(t)|2N + +� +i,j +|aij|2N+2 +� +� . +For any Gaussian variable X and integer N ≥ 0, the following inequality holds +E[X2N] ≤ 22N(E[X]2N + CNVar(X)2N) +where CN is a constant depending on N. Next, we can focus on the conditional expectation and conditional +variance of Xij(t) and Ys,t. +By Assumption 1 and 2 and the fact that the conditional variance of a Gaussian variable is less or equal to +the unconditional variance, we can conclude that the conditional variance of Xij(t) and Ys,t are bounded above +by some constant. +Summarizing the results above, +sup +s,t∈Dϵ,s̸=t +|E2(s, t)| ≤ C1 +for some constant C1 > 0 and +E1(s, t) ≤ ||t − s||αE2(s, t) ≤ C1||t − s||α. +5 + +Combine the results above and with a fixed threshold u +� ∞ +u +E1(s, t)PX(s) (x|∇X(s) = ∇X(t) = 0) dx +≤ C1||t − s||α +� ∞ +u +PX(s) (x|∇X(s) = ∇X(t) = 0) dx += C1||t − s||α +� ∞ +u +exp(−(Ax − B)2)dx +for some constant A, B += C2||t − s||α +for some constant C2 > 0. +Next, by the proof of Lemma 4.1 in Cheng and Schwartzman (2015) +p∇X(s),∇X(t)(0, 0) ≤ C3||t − s||−N +for some constant C3 > 0. +Therefore, there exists C4 such that +E[Mu(Mu − 1)] ≤ C4 +� +Dϵ +� +Dϵ +1 +||t − s||N−α dtds = o(ϵN). +For E[Mu], by Kac-Rice formula in Adler and Taylor (2007) +E[Mu] = +� +Dϵ +p∇X(s)(0)E +� +|det∇2X(s)|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 +� +ds. +Denote the integrand by g(s). The function g(s) is continuous and positive over the compact domain Dϵ. +Thus infs∈Dϵ g(s) ≥ g0 > 0, implying +E[Mu] ≥ g0ϵN. +Then (7) is an immediate consequence of (5). +For E[M−∞], by Kac-Rice formula +E[M−∞] = +� +Dϵ +p∇X(s)(0)E +� +|det∇2X(s)|1{∇2X(s)≺0}|∇X(s) = 0 +� +ds. +The integrand is also continuous and positive over the compact domain Dϵ indicating E[M−∞] = o(1) for small +ϵ. Thus we have +E[Mu]adj = E[Mu]/ max(1, E[M−∞]) = E[Mu]/ max(1, o(1)) = E[Mu] +for sufficiently small ϵ. +2.7 +Large threshold +For large threshold u, the following asymptotic result shows power can be precisely approximated by E[Mu]. +Theorem 2. For any fixed domain D, as u → ∞ +P[Mu ≥ 1] = E[Mu](1 − o(e−αu2)) +(10) +where the error term o(e−αu2) is non-negative and α > 0 is some constant. +Proof. By lemma 3 of Piterbarg (1996), as u → ∞, the factorial moment is super-exponentially small. That +means ∃α > 0 s.t. +E[Mu(Mu − 1)] = o(e− u2 +2 −αu2). +Also +E[Mu] ≥ P[Mu ≥ 1] ≥ P[supX(s) ≥ u] = O(e− u2 +2 ). +Thus, the factorial moment decays exponentially faster than E[Mu]. The result is an immediate consequence of +(5). +Notice that the threshold u does not affect the value of E[M−∞] which is part of the adjusted E[Mu]. By +(10) +P[Mu ≥ 1] = E[Mu]adj(1 − o(e−αu2)) max(1, E[M−∞]). +If E[M−∞] > 1, the adjusted E[Mu] might be overly conservative for large threshold u. Therefore, we only +recommend E[Mu] for this scenario. +6 + +2.8 +Sharp signal +The following theorem provides an asymptotic power approximation when the signal is sharp. Interestingly, +while the power function is generally non-Gaussian, it becomes closer to Gaussian as the signal peaks become +sharper. +Theorem 3. Let θ(s) = ah(s) + θ0 where h(s) is a unimodal mean function with maximum equal to 0 at s0, +a > 0, and θ0 represents the height. For any fixed threshold u, as a → ∞ +P[Mu ≥ 1] = E[Mu] + o(1) = E[Mu]adj + o(1) = Φ(θ0 − u)(1 + o(1)), +(11) +where Φ(x) is CDF of the standard Gaussian distribution. +Proof. By lemma A.1 of Cheng and Schwartzman (2017), as a → ∞ +P(M−∞ = 1) ≥ 1 − O(exp(−ca2)), +where c > 0 is some constant. Therefore M−∞ +p→ 1. +Since Mu ≤ M−∞ and both of them only take non-negative integer values, |Mu(Mu −1)| and |M−∞(M−∞ − +1)| are bounded above by |M(M − 1)| where M is the number of critical points of the random field X. Apply +Kac-Rice formula +E[M(M − 1)] = +� +D +� +D +E +� +|det∇2X(s)||det∇2X(t)| +��∇X(s) = ∇X(t) = 0 +� +P∇X(s),∇X(t)(0, 0)dsdt. +Denote the integrand by g(s, t, a). The function g(s, t, a) is continuous and positive over the compact domain +D and M(M − 1) +p→ 0 as a → ∞. Thus there exists g0 > 0 such that E[M(M − 1)] ≤ g0. Then by dominated +convergence theorem +E[Mu(Mu − 1)] → 0 +as a → ∞. Since M−∞ +p→ 1, the adjusted E[Mu] +E[Mu]adj = E[Mu] max(1, E[M−∞]) = E[Mu](1 + o(1)) = E[Mu] + o(1). +To calculate E[Mu], apply Kac-Rice formula +E[Mu] = +� +D +p∇X(s)(0)E +� +|det∇2X(s)|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 +� +ds += +� +D +p∇X(s)(0)E +� +|det∇2X(s)|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 +� +ds += +� +D +1 +(2π)N/2� +det(Λ) +exp(−a2(∇h(s))T Λ−1∇h(s)/2) +E +� +|det(∇2Z(s) + a∇2h(s))|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 +� +ds +(12) +where Λ is the covariance matrix of ∇h(s). Let f(s) = (∇h(s))T Λ−1∇h(s)/2 which attains its minimum 0 only +at s0. Similar to the proof of A.4 in Cheng and Schwartzman (2017), as a → ∞, (12) can be approximated by +applying Laplace’s method +E[Mu] = det(a∇2h(s0)) +(2π)N/2� +det(Λ) +� +(2π)N det(Λ) +a2N det(∇2h(s0)) +�1/2 +Φ(θ0 − u) + O(a−2) +=Φ(θ0 − u) + O(a−2). +This finishes the proof. +3 +Explicit formulas +We have showed that the power for peak detection can be approximated by the expected number of local +maxima above u, E[Mu], under certain scenarios such as small domain and large threshold. Although we can +apply the Kac-Rice formula to calculate E[Mu], it remains difficult to evaluate it explicitly for N > 1 without +making any further assumptions. In this section, we focus on computing E[Mu] and show a general formula +can be obtained if the noise field is isotropic. Furthermore, explicit formulas when N = 1, 2, 3 are derived for +application purposes. +7 + +3.1 +Isotropic Gaussian fields +Suppose Z is a zero-mean unit-variance isotropic random field. We can write the covariance function of Z as +E{Z(s)Z(t)} = ρ(∥s − t∥2) for an appropriate function ρ(·) : [0, ∞) → R. Denote +ρ′ = ρ′(0), +ρ′′ = ρ′′(0), +κ = −ρ′/ +� +ρ′′. +(13) +where ρ′ and ρ′′ are first and second derivative of function ρ respectively. +The following lemma comes from Cheng and Schwartzman (2018). +Lemma 1. For each s ∈ RN and i, j, k, l ∈ {1, . . . , N}, +E{Zi(s)Z(s)} = E{Zi(s)Zjk(s)} = 0, +E{Zi(s)Zj(s)} = −E{Zij(s)Z(s)} = −2ρ′δij, +E{Zij(s)Zkl(s)} = 4ρ′′(δijδkl + δikδjl + δilδjk) +where ρ′ and ρ′′ are defined in (13) and δij is the Kronecker delta function. +In particular, it follows from Lemma 1 that Var(Zi(s)) = −2ρ′ and Var(Zii(s)) = 12ρ′′ for any i ∈ {1, . . . , N}, +implying ρ′ < 0 and ρ′′ > 0 and hence κ > 0. +We can use theoretical results from Gaussian Orthogonally Invariant (GOI) matrices to make the calculation +of E[Mu] easier. GOI matrices were first introduced in Schwartzman et al. (2008), and used for the first time +in the context of random fields in Cheng and Schwartzman (2018). It is a class of Gaussian random matrices +that are invariant under orthogonal transformations, and can be useful for computing the expected number +of critical points of isotropic Gaussian fields. We call an N × N random matrix G = (Gij)1≤i,j≤N GOI with +covariance parameter c, denoted by GOI(c), if it is symmetric and all entries are centered Gaussian variables +such that +E[GijGkl] = 1 +2(δikδjl + δilδjk) + cδijδkl. +(14) +The following lemma is Lemma 3.4 from Cheng and Schwartzman (2018). +Lemma 2. Let the assumptions in Lemma 1 hold. Let �G and G be GOI(1/2) and GOI((1 − κ2)/2) matrices +respectively. IN denotes N × N identity matrix. +(i) The distribution of ∇2Z(s) is the same as that of √8ρ′′ �G. +(ii) The distribution of (∇2Z(s)|Z(s) = z) is the same as that of √8ρ′′� +G − +� +κz/ +√ +2 +� +IN +� +. +Lemma 2 shows the distribution and conditional distribution of the Hessian matrix of a centered random +field Z(s). Next, we establish the corresponding result for non-centered random field X(s) = Z(s) + θ(s). +Lemma 3. Let �G and G be GOI(1/2) and GOI((1 − κ2)/2) matrices respectively. +(i) The distribution of ∇2X(s) is the same as that of +� +8ρ′′ �G + ∇2θ(s). +(ii) The distribution of (∇2X(s)|X(s) = x) is the same as that of +� +8ρ′′ +� +G − κ(x − θ(s)) +√ +2 +IN +� ++ ∇2θ(s). +Proof. Part (i) is a direct consequence of Lemma 2. For part (ii), note that (∇2X(s)|X(s) = x) is equivalent +to (∇2Z(s)|Z(s) = x − θ(s)) + ∇2θ(s), and the result follows immediately from Lemma 2. +3.2 +General formula under isotropy +Theorem 4. Let X(s) = Z(s) + θ(s), where Z(s) is a smooth zero-mean unit-variance isotropic Gaussian +random field satisfying Assumption 1, 2. Let θ(s) a smooth C3 mean function such that ∇2θ(s) is a non- +singular matrix with ordered eigenvalues θ′′ +1(s)...θ′′ +N(s) at all critical points s. Then for any domain D +E[Mu] = +� 2ρ′′ +−πρ′ +�N/2 � +D +e +∥∇θ(s)∥2 +4ρ′ +� ∞ +u +φ (x − θ(s)) E +� +|det(Matrix(s))|1{Matrix(s)≺0} +� +dx ds +(15) +where φ(x) is the PDF of the standard Gaussian distribution, Matrix(s) = G − κ(x − θ(s))IN/ +√ +2 + +diag{θ′′ +1(s), . . . , θ′′ +N(s)}/√8ρ′′, G as in Lemma 3 represents GOI((1-κ2)/2), and 1{·} denotes the indicator func- +tion. +8 + +Proof. By the Kac-Rice formula +E[Mu] = +� +D +p∇X(s)(0)E +� +|det∇2X(s)|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 +� +ds += +� +D +p∇Z(s)+∇θ(s)(0)E +� +|det∇2X(s)|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 +� +ds += +� +D +1 +(2π)N/2(−2ρ′)N/2 e +∥∇θ(s)∥2 +4ρ′ +E +� +|det∇2X(s)|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 +� +ds += +� +D +(8ρ′′2)N/2 +(2π)N/2(−2ρ′)N/2 e +∥∇θ(s)∥2 +4ρ′ +� ∞ +u +φ (x − θ(s)) E +� +|det(Matrix(s))|1{Matrix(s)≺0} +� +dx ds += +� +D +� 2ρ′′ +−πρ′ +�N/2 +e +∥∇θ(s)∥2 +4ρ′ +� ∞ +u +φ (x − θ(s)) E +� +|det(Matrix(s))|1{Matrix(s)≺0} +� +dx ds += +� 2ρ′′ +−πρ′ +�N/2 � +D +e +∥∇θ(s)∥2 +4ρ′ +� ∞ +u +φ (x − θ(s)) E +� +|det(Matrix(s))|1{Matrix(s)≺0} +� +dx ds +Next, we show the derivation from the third to the fourth line in the equation above. Since we assume ∇2θ(s) is +a non-singular matrix at all critical points, then there exists an orthonormal matrix, denoted by A(s), such that +A(s)T ∇2θ(s)A(s) = diag{θ′′ +1(s), θ′′ +2, . . . , θ′′ +N(s)}, where θ′′ +1 ≤ . . . ≤ θ′′ +N(s) are ordered eigenvalues of ∇2θ(s). On +the other hand, GOI matrices are invariant under orthonormal transformations. By Lemma 3, the conditional +expectation E[|det(∇2X(s))|1{∇2X(s)≺0}|X(s) = x] is therefore += E +�����det +�� +8ρ′′ +� +G − κ(x − θ(s)) +√ +2 +IN +� ++ ∇2θ(s) +����� 1{Matrix(s)≺0} +� += E +�����det +�� +8ρ′′ +� +G − κ(x − θ(s)) +√ +2 +IN +� ++ A(s)T ∇2θ(s)A(s) +����� 1{Matrix(s)≺0} +� += ( +� +8ρ′′)NE +�����det +�� +G − κ(x − θ(s)) +√ +2 +IN +� ++ A(s)T ∇2θ(s)A(s)/ +� +8ρ′′ +����� 1{Matrix(s)≺0} +� += ( +� +8ρ′′)NE +�����det +� +G − κ(x − θ(s)) +√ +2 +IN + diag{θ′′ +1(s), θ′′ +2, . . . , θ′′ +N(s)}/ +� +8ρ′′ +����� 1{Matrix(s)≺0} +� +. +(16) +The expression (15) can be simplified further if we further assume the mean function θ(s) to be a rotational +symmetric paraboloid centered at s0. In this case, the Hessian of θ(s) is the identity matrix multiplied by a +constant, i.e. +θ′′ = θ′′ +1(s) = θ′′ +2(s) = ... = θ′′ +N(s). +Then we can write the mean function as θ(s) = θ0 + θ′′∥s − s0∥2/2. Define +η = +θ′′ +2κ√ρ′′ = +θ′′ +−2ρ′ = +θ′′ +Var(Z1(s)). +(17) +and +H(˜x) = EN +GOI((1−κ2)/2) +� +� +N +� +j=1 +����λj − κ˜x +√ +2 +���� 1{λN< κ˜x +√ +2 } +� +� . +(18) +E[Mu] can be simplified as +E[Mu] = +� 2ρ′′ +−πρ′ +�N/2 � +D +e +θ′′2∥s−s0∥2 +4ρ′ +� ∞ +˜u(s) +φ (˜x + η) H(˜x)d˜xds +(19) +where we make a change of variable ˜x = x − θ(s) − η and ˜u(s) = u − θ(s) − η. Note that the parameter κ +depends on the correlation structure of Z(s). +3.3 +Explicit formulas in 1D, 2D and 3D +In (19), a general formula for E[Mu] under isotropy was derived. +To make the formula easier to apply in +practice, we have the following results for computing it in 1D, 2D, and 3D. When N = 1, the derivation is +simple enough that we do not need additional assumptions on the mean function θ(s) except those in Theorem +9 + +4, and it follows directly from Kac-Rice formula. When N = 2 and 3, we assume the mean function θ(s) is +a rotational symmetric paraboloid centered at s0. E[Mu] is calculated by first obtaining explicit formulas for +H(˜x), and plugging H into (19). +Proposition 2. Let N = 1, X(s) = Z(s) + θ(s), where Z(s) is a smooth zero-mean unit-variance Gaussian +process and θ(s) is a smooth mean function. Assume additionally that Z(s) is stationary, then +E[Mu] = +� +D +� +−2ρ′(3 − κ2) +κ +φ +� θ′(s) +√−2ρ′ +� � ∞ +u +φ(x − θ(s))ψ +�κ[x − θ(s) − η(s)] +√ +3 − κ2 +� +dx ds, +(20) +where the function ψ is defined as +ψ(x) = +� x +−∞ +Φ(y)dy = φ(x) + xΦ(x), +x ∈ R. +Proof. Since we assume that Z(s) is stationary, Z′(s) is independent of Z(s) and Z′′(s), and ρ′ = −Var(Z′(s))/2 = +E[Z(s)Z′′(s)]/2 and ρ′′ = Var(Z′′(s))/12 do not depend on s. Therefore, +Var(X(s)) = 1, +Var(X′(s)) = −Cov[X(s)X′′(s)] = −2ρ′ +and +Var(X′′(s)) = 12ρ′′. +Note that, by the formula of conditional Gaussian distributions, +X′′(s)|X(s) = x ∼ N(θ′′(s) + 2ρ′(x − θ(s)), 12ρ′′ − 4ρ′2). +By the Kac-Rice formula +E[Mu] = +� +D +pX′(s)(0)E[|X′′(s)|1{X(s)>u}1{X′′(s)<0}|X′(s) = 0]ds += +� +D +pX′(s)(0) +� ∞ +u +φ(x − θ(s)) +� 0 +−∞ +(−x′′) +1 +� +12ρ′′ − 4ρ′2 φ +� +x′′ − θ′′(s) − 2ρ′(x − θ(s)) +� +12ρ′′ − 4ρ′2 +� +dx′′ dx ds += +� +D +pX′(s)(0) +� +12ρ′′ − 4ρ′2 +� ∞ +u +φ(x − θ(s))ψ +� +−2ρ′(x − θ(s)) − θ′′(s) +� +12ρ′′ − 4ρ′2 +� +dx ds += +� +D +1 +√−2ρ′ φ +� θ′(s) +√−2ρ′ +� � +12ρ′′ − 4ρ′2 +� ∞ +u +φ(x − θ(s))ψ +� +−2ρ′(x − θ(s)) − θ′′(s) +� +12ρ′′ − 4ρ′2 +� +dx ds += +� +D +� +12ρ′′ − 4ρ′2 +√−2ρ′ +φ +� θ′(s) +√−2ρ′ +� � ∞ +u +φ(x − θ(s))ψ +� +−2ρ′(x − θ(s)) − θ′′(s) +� +12ρ′′ − 4ρ′2 +� +dx ds. +The second to third line is due to the fact that +� 0 +−∞ +(−x)1 +b φ +�x + a +b +� +dx = +� 0 +−∞ +Φ +�x + a +b +� +dx = b +� a/b +−∞ +Φ(y)dy = bψ +�a +b +� +. +Recall the κ (13) and η (17) parameters defined before. We can rewrite E[Mu] as +� +D +� +−2ρ′(3 − κ2) +κ +φ +� θ′(s) +√−2ρ′ +� � ∞ +u +φ(x − θ(s))ψ +�κ[x − θ(s) − η(s)] +√ +3 − κ2 +� +dx ds. +This finishes the proof. +Note that when N = 1, +H(˜x) = φ +� +κ˜x +√ +3 − κ2 +� ++ +κ˜x +√ +3 − κ2 Φ +� +κ˜x +√ +3 − κ2 +� += ψ +� +κ˜x +√ +3 − κ2 +� +(21) +We need the following lemmas to calculate H(˜x) explicitly when N = 2 and N = 3. They are direct calculation +of integral by parts. +Lemma 4. Let N = 2, for constant a > − 1 +2 and b ∈ R +� +R2 exp +� +− 1 +2 +2 +� +i=1 +λ2 +i − a +2 +� +2 +� +i=1 +λi +�2 +� � 2 +� +i=1 +|λi − b| +� +|λ1 − λ2|1{λ1<λ2 0 and b ∈ R +� +R3 exp +� +− 1 +2 +3 +� +i=1 +λ2 +i + +a +2(2 + 3a) +� +3 +� +i=1 +λi +�2 +� � 3 +� +i=1 +|λi − b| +� +� +1≤i u for ith simulated sample), +(28) +ˆE[Mu] = 1 +B +B +� +i=1 +# peaks in D with height > u for ith simulated sample. +(29) +Figure 2 displays the power, E[Mu] and adjusted E[Mu] curves under the four scenarios. The first panel is to +validate scenario 1 (height equivariance). As stipulated by Proposition 1, the power, E[Mu] and adjusted E[Mu] +curves are parallel for different signal height h having other parameters remain the same. In the second panel, +both the E[Mu] and adjusted E[Mu] curve are close to the power curve under scenario 2 (small domain) which +indicates that for a smaller domain (quantified by Rad(D)), using E[Mu] and adjusted E[Mu] to approximate +power becomes more accurate as stipulated by Theorem 1. We can also find in all three panels that when u is +large enough, the E[Mu] curve converges to the power curve as u increases, as stipulated by Theorem 2. The +adjusted E[Mu] curve also converges to the power curve but with a slower rate compared to E[Mu]. The third +panel shows the power, E[Mu] and adjusted E[Mu] curve all converge to the Gaussian CDF for sharp signal +(small ξ), as stipulated by Theorem 3. +4.2 +Constant mean function +When the mean function θ(s) is constant, i.e. it does not depend on location s, X(s) reduces to a centered +isotropic Gaussian random field. Within the context of this paper, θ(s) = 0 can be seen as the null hypothesis +and the power function becomes the probability of Type I error. We use the peak height distribution when +θ(s) = 0 (Cheng and Schwartzman, 2018) to decide the cutoff point such that the test meets the nominal type +I error. The simulation result when θ(s) = 0 is displayed in Figure 3. +The performance of Type I error approximation when the mean function is 0 is similar to what we find when +the mean function is quadratic (scenario 4 is ignored since the shape parameter does not exist when the mean +function is constant). The conclusion is that under large u (which is guaranteed in order to control the Type I +error) or small domain, we have good Type I error approximation. +13 + +0 +-5 +-10 +-15 +60 +40 +60 +40 +20 +20 +0 +02 +1 +0 +-1 +-2 +60 +40 +60 +40 +20 +20 +0 +00 +-5 +-10 +-15 +60 +40 +60 +40 +20 +20 +0 +0(a) Height equivariance +(b) Small domain size +(c) Sharp signal +Figure 2: +2D simulation: Power approximation using E[Mu] under four different scenarios (scenario 3 is +displayed in all three panels) when the mean function is quadratic. +(a) Height equivariance +(b) Small domain size +Figure 3: 2D simulation: Type I error approximation using E[Mu]. +14 + +1.4 +4.5 +1.2 +Y += 0 +0.8 +EFM +0.6 +E[M +adj +0.4 +0.2 +0 +-4 +-2 +0 +2 +4 +6 +8 +u2 +Rad = 20 +Rad = 10 +1.5 +Rad= 5 +Rad = 3 +E[M.] +E[M +u'adi +0.5 +0 +0 +2 +4 +6 +u1.4 +$= 14 +1.2 +L = 3. +$= 3 +1 +=1 +..E[M.] +0.8 +.E[M.], +u'adj +0.6 +GaussianCDF +0.4 +0.2 +0 +0 +2 +4 +6 +u0.7 +4.5 +0 +0.6 +3 +0 +0.5 += 1.5 +0.4 +0 +E[M.] +0.3 +.E[M.. +"adi +0.2 +0.1 +0 +-4 +-2 +0 +2 +4 +6 +8 +u2.5 +Rad = 20 +Rad = 10 +2 +Rad = 5 +Rad = 3 +E[M.] +1.5 +E[M. +u'adi +1 +0.5 +0 +-2 +-1 +0 +1 +2 +3 +4 +u(a) Height equivariance +(b) Small domain size +(c) Sharp signal (Rad(D) = 3) +(d) Sharp signal (Rad(D) = 20) +Figure 4: 2D simulation: Power approximation using E[Mu] when the mean function is Gaussian. +4.3 +Gaussian mean function +The simulation results under Gaussian mean are displayed in Figure 4. +For scenario 2 and 3, the results +are consistent with those under quadratic mean. For scenario 1, since θ0 controls both the signal height and +sharpness, the power, E[Mu] and adjusted E[Mu] are no longer equivariant in terms of θ0. For scenario 4, if +we look at Figure 4c with Figure 4d, it can be seen that as the signal becomes sharper, the power, E[Mu] and +adjusted E[Mu] curve converges to the Gaussian CDF only when the domain size (quantified by Rad(D)) is +small. In this case, the asymptotic curve is a mixture of Gaussian CDF and E[Mu] under constant mean. This +is due to the shape of Gaussian density as it converges to 0 if we expand the domain. +In conclusion, for Gaussian mean function, we recommend applying our method to approximate power only +when the domain size is small. +5 +Estimation from data +To use our power approximation formula in real peak detection problems, we need to estimate the spatial +covariance function of the noise as well as the mean function from the data. In this section, we demonstrate +the 3D application setting and how to estimate the noise spatial covariance function and the mean function. +Consider an imaging dataset with n subjects, and let Yi(s) represent the signal plus noise for subject i +Yi(s) = µ(s) + σ(s)εi(s) +where s = (s1, s2, s3)′ ∈ R3, the signal µ(s), standard deviation σ(s) and noise ε(s) are assumed to be smooth +C3 functions. If we compute the standardized mean of all n subjects, we will get a standardized random field +X(s) = ¯Y (s)/SE( ¯Y (s)) = √nµ(s)/σ(s) + √n¯ε(s) +(30) +This standardized random field X(s) has constant standard deviation of 1. We can treat √nµ(s)/σ(s) as the +new signal and √n¯ε(s) as the new noise of the standardized field. Let θ(s) = √nµ(s)/σ(s) and Z(s) = √n¯ε(s). +We propose using the following method to estimate the new signal and noise. +5.1 +Estimation of the noise spatial covariance function +We consider the noise Z(s) to be constructed by convolving Gaussian white noise with a kernel: +15 + +2 +Rad = 20 +Rad = 10 +1.5 +Rad= 5 +Rad = 3 +E[M.] +.E[M.. +u'adi +0.5 +0 +-2 +0 +2 +4 +6 +u = 14 +$=7 +0.8 +$= 3 += 1 +.. E[M... +0.6 +- - .E[M.,] +u'adj +Gaussian CDF +0.4 +0.2 +0 +0 +2 +4 +6 +u1.5 +$= 14 +L = 3. += 3 += 1 +GaussianCDF +0.5 +0 +-2 +0 +2 +4 +6 +8 +u1.2 +4.5 +0.8 += 0 +E[M.. +0.6 +.E[M +0.4 +0.2 +0 +0 +2 +4 +6 +8 +uZ(s) = +� +RN K(t − s)dB(t) +(31) +where K(·) is a N dimensional kernel function, and dB(s) is Gaussian white noise. Assume that the kernel is +rotationally symmetric so that the noise Z(s) is isotropic. Under model (31), we would be able to simulate the +noise if we were able to estimate the kernel function from the data. +It can be shown that the autocorrelation of Z(s) is the convolution of the kernel with itself: +Cor(Z(s), Z(s′)) = +� +RN K(t − s)K(t − s′)dt = +� +RN K(t − (s − s′))K(t)dt. +By the convolution theorem, convolution in the original domain equals point-wise multiplication in the +Fourier-transformed domain. Thus the kernel function can be estimated empirically using the following method: +1. Determine a location s0 of interest (e.g. +center of the peak), and calculate the empirical correlation +vectors between Y (s0) and Y (s) where s lies on the three orthogonal axes centered at s0, and belongs to +a subdomain of interest. +2. Take the average of the three estimated correlation vectors (forcing the noise to be isotropic) and perform +Fourier transform. +3. Take the square root of the Fourier coefficients, then the estimated kernel function can be obtained by +performing the inverse Fourier transform. +5.2 +Estimation of the mean function +Our explicit formulas are derived assuming the Hessian of the mean function is a constant times the identity +matrix. Therefore, we aim to find a rotational symmetric paraboloid ˆθ(s) that best represents the mean function: +θ(s) = β0 + β1||s||2 + (β2, β3, β4) · s +(32) +where the dot represents the vector inner product in R3. Note that all the quadratic terms share the same +coefficient which is due to the rotational symmetry. To estimate (32), we can fit a linear regression using all +X(s) within the subdomain as outcome. +6 +A 3D real data example +As an application, we illustrate the methods in a group analysis of fMRI data from the Human Connectome +Project (HCP) (Van Essen et al., 2012). The data is used here to get realistic 3D signal and noise parameters +from which to do 3D simulations as well as evaluate the performance of our formulas in power approximation. +The dataset contains fMRI brain scans for 80 subjects. For each subject, the size of the fMRI image is 91×109×91 +voxels. The mean and standard deviation of the data are displayed in Figure 5a and 5b. +6.1 +Data preprocessing +Gaussian kernel smoothing is applied to the dataset to make the mean function unimodal around the peak and +increase the signal-to-noise ratio. The standard deviation of the smoothing kernel we use in this example is 1 +voxel which translates to full width at half maximum (FWHM) being around 2.235. It is obvious from Figure 5b +that the standard deviation of the noise is not a constant for different locations, thus we use the transformation +described in Section 5 to standardize the smoothed data before analyzing it. The standardized mean of the +smoothed data X(s) is displayed in Figure 5c. +6.2 +Estimation of the autocorrelation and mean functions +After standardizing the data, our next step is to estimate the mean and kernel functions using the methods +described in Section 5. Here a 15 × 15 × 15 subdomain (the red box in Figure 5c) around the peak is taken +to estimate the kernel. Since we assume the noise is isotropic, the correlation around the peak is supposed to +be strictly symmetric along any dimension. However, this is not always true in real data. To tackle this, for +each of the three dimensions, we first save the correlation Cor(X(s), X(s0)) around the peak s0 as a vector and +create a new vector by flipping the saved correlation vector. Then we take the mean of the two vectors so that +it is guaranteed to be symmetric. The empirical correlation after such symmetrization and the corresponding +estimated kernel function are displayed in Figure 6. +16 + +(a) Mean of the data +(b) Standard deviation of the data +(c) Standardized mean of the smoothed data +Figure 5: HCP data: Mean, standard deviation of the data, and standardized mean of the smoothed data +(transverse sliced at the peak of the image along the third dimension). The blue box represents the subdomain +of the peak and the red box represents the subdomain we use to estimate the noise spatial covariance function. +Figure 6: The empirical correlation after symmetrization and the estimated kernel from a subdomain of HCP +data. +17 + +60 +20 +40 +40 +20 +60 +0 +80 +-20 +20 +40 +60 +80 +10060 +20 +50 +40 +40 +30 +60 +20 +10 +80 +0 +20 +40 +60 +80 +10010 +20 +5 +40 +0 +60 +-5 +80 +20 +40 +60 +80 +100Empiricalcorrelation +Estimated kernel +0.8 +0.6 +0.4 +0.2 +0 +-5 +0 +5Figure 7: 3D Simulation induced by data: Simulated Peak height distribution under the null (zero mean) with +different levels of effect sizes and threshold u. +We consider two approaches to estimate the mean function, nonparametric and parametric. The nonpara- +metric mean estimation is obtained as a voxelwise average over subjects. +ˆθ(s) = +n +� +i=1 +Xi(s) = ¯X(s) +(33) +The parametric mean estimation is obtained by fitting a linear regression model (32) using all observed data +X(s) within the subdomain of size 6 × 6 × 6 (the blue box in Figure 5c) as outcome and their corresponding +location variables ||s||2, s as covariates. The least square estimate of the mean is +ˆθ(s) = 13.03 − 0.26||s||2 + (0.20, 0.11, 0.39) · s +(34) +We will compare the difference in simulated power and E[Mu] when the mean function is estimated by the +nonparametric approach (33) vs the parametric approach (34). +6.3 +3D Simulation induced by data +We have done several simulation studies under a well-designed 2D setting where the formulas are supposed to +work well, but eventually, we want to apply the formulas to real-life data which is more complicated. Besides, in +terms of fMRI data analysis, the image is always 3D by nature. Considering all these factors, a 3D simulation +study induced by real data is necessary to validate the performance of the formulas under a more realistic +setting, +In the previous two subsections, we have studied the signal and noise of the HCP data. For the simulation, +we would like to generate 3D images using the estimated mean and kernel function. The noise field is generated +by convolving the estimated kernel (displayed in Figure 6) and Gaussian white noise. +For each simulation +setting, 10,000 such noise fields are generated. +The signal from the standardized data is very strong (see Figure 5c). For illustrative purposes, we choose to +weaken the signal by scaling down the estimated mean function (34) while maintaining the same shape. Here +signal strength is measured by effect size, and the amount of scaling is determined by different levels of effect +size. In traditional t-test or z-test, Cohen’s d values of 0.2, 0.5, and 0.8 (corresponding to 0.58 th 0.69 th and +0.79 th quantiles of the standard Gaussian distribution) are considered as small, medium, and large effect sizes +(Cohen, 1988). The peak height distribution under the null hypothesis (zero mean function) is displayed in +Figure 7, and does not follow a Gaussian distribution. Therefore, we take 0.58 th, 0.69 th and 0.79 th quantile +of the null distribution minus the mean as small (0.16), medium (0.40), and large (0.65) effect size (see the +black dash-dot lines in Figure 7). For simplicity, we see the peak height of the mean function as effect size in +this simulation. However, this is not the most accurate way of defining effect size in the peak detection setting. +More details will be discussed in 7.2. The threshold u for peak detection is chosen as the 0.99 th quantile of +the peak height distribution under the null (≈ 3.42) according to Cheng and Schwartzman (2017) (see the red +dashed line in Figure 7). +Similar to the 2D simulation, the search domain D is assumed to be a sphere centered at the true peak, and +we use radius of D to control the domain size. Signal sharpness is fixed since it is estimated from the data. The +empirical power and E[Mu] are computed using (28) and (29). +To derive the explicit formulas for E[Mu], we assume the mean function to be a rotational symmetric +paraboloid, and this assumption might cause some bias in applications. +In Figure 8, we demonstrate the +difference in simulated power and E[Mu] when the mean function is estimated by the nonparametric approach +18 + +-Mean +n effect size +Threshold +0.8 +Medium +0.6 +0.4 +0.2 +0 +-1 +0 +1 +2 +3 +4Figure 8: 3D Simulation induced by data (Rad(D) = 3, medium effect size): Simulated power and E[Mu] +when mean function is obtained from raw data vs quadratic estimation (34). Here E[Mu] and adjusted E[Mu] +are the same since E[M−∞] < 1. +Figure 9: 3D Simulation induced by data (Rad(D) = 6, medium effect size): Simulated vs theoretical E[Mu] +and adjusted E[Mu]. +(33) vs the parametric approach (34). It can be observed that the quadratic approximation only has a small +impact on power and E[Mu] in this example. In Figure 9, we validate our theoretical formula for E[Mu] (19) as +well as the adjusted E[Mu]. As we can see, the theoretical curve for E[Mu] and adjusted E[Mu] closely mirrors +the empirical curve. The figure also shows that the power approximation using E[Mu] is accurate for large u +as stipulated by Theorem 2. Power curves using three different effect sizes, and comparisons between large and +small domain sizes are displayed in Figure 10. We can see from the figure that the approximation works well +for small and large sample sizes, and E[Mu]adj provides a conservative determination of the sample when E[Mu] +exceeds 1. We can also observe that the performance of power approximation using E[Mu] becomes better if +the domain size is smaller as stipulated by Theorem 1. +7 +Discussion +7.1 +Explicit formulas and approximations +Calculating power for peak detection (1) has been a difficult problem in random field theory due to the lack of +formula that can compute it directly. In this paper, we have discussed the rationale of using E[Mu] and E[Mu]adj +to approximate peak detection power under different scenarios and derived formulas to compute E[Mu] assuming +isotropy. Isotropy is assumed so that we are able to use the GOI matrix (Cheng and Schwartzman, 2018) as a +tool to calculate E[Mu] via the Kac-Rice formula. +We also showed explicit formulas for H(˜x) (defined as (18)) when N = 1, 2, 3 assuming the mean function +is a paraboloid. Computing H(˜x) involves applying the probability density function for the eigenvalues of GOI +matrices and details can be found in the proof of Proposition 2, 3 and 4. Then E[Mu] can be calculated by +plugging H(˜x) to (19). The integration in (19), however, can not be evaluated explicitly. In practice, one may +evaluate it numerically. For higher dimensions (N > 3), it remains difficult to get an explicit form of H(˜x) due +to the fact inferred by Proposition 2, 3 and 4 that the integration becomes extremely complicated as N becomes +large. +19 + +0.8 +Powerfittedmean +...E[M. ] fitted mean +0.6 + -E[M. Jod: fitted mean +adi +-Powerrawdata +.. E[M. ] raw data +0.4 +0.2 +0 +-1 +0 +1 +2 +3 +u0.8 +-Power +..Theoretical E[M] +.Empirical E[M,] +0.6 +.Empirical E[M,], +'ad +0.4 +0.2 +0 +1 +2 +3 +4 +5 +u(a) Small effect size (large domain +size) +(b) Medium effect size (large do- +main size) +(c) Large effect size (large domain +size) +(d) Small effect size (small domain +size) +(e) Medium effect size (small do- +main size) +(f) Large effect size (small domain +size) +Figure 10: 3D Simulation induced by data: Power curves when the signal has small, medium, and large effect +size, and comparisons between large and small domain size. +7.2 +Effect size +We want to emphasize that the power depends on both the signal strength parameter θ0 and shape parameter +η. In a traditional z-test or t-test which tests a single null hypothesis that the mean value equal to 0, the +detection power depends only on a single parameter we call effect size. Here the test is conditional on the point +being a local maximum. Applying a simple z-test or t-test, one could reject the null hypothesis as long as the +peak height θ0 exceeds the pre-specified threshold. This approach is not accurate since the peak height does not +follow a Gaussian or t distribution. To address this, the threshold can be determined by the null distribution +of peak height (Cheng and Schwartzman, 2018) to control the type I error at a nominal level. However, power +calculation based on the test over peak height is still biased since the true effect size depends both on the signal +height and curvature. The height of the peak affects the likelihood of exceeding the threshold and the curvature +affects the likelihood of existing such peak in the domain. It follows that a sharp and high peak is easier to +detect compared to a flat and low peak, leading to a larger detection power. +For an interpretation of the parameter η = θ′′/(−2ρ′), we consider two types of mean function: paraboloid +and Gaussian. Suppose the noise is the result of the convolution of white noise with a Gaussian kernel with +spatial std. dev. ν resulting in the covariance function with ρ(r) = exp(−r/(2ν2)) as specified in Section +3.1. This is the same noise as we simulated in Section 5. When the mean function is paraboloid, consider +θ(s) = −∥s∥2/(2ξ2) + θ0 as in (27). Here we obtain θ′′ = −1/ξ2 and ρ′ = −1/(2ν2), yielding η = θ′′/(−2ρ′) = +−ν2/ξ2. Thus, η is a shape parameter representing the relative sharpness of the mean function with respect +to the curvature of the noise. +When the mean function is Gaussian, consider θ(s) = a exp(−∥s∥2/(2τ 2)). +This expression is obtained, for example, if the signal is the result of the convolution of a delta function +with a Gaussian kernel with spatial std. +dev. +τ. +We obtain θ′′ = −a/τ 2 and ρ′ = −1/(2ν2), yielding +η = θ′′/(−2ρ′) = −aν2/τ 2. Thus, η is the height of the signal a, scaled by the ratio of the spatial extent of +the noise and signal filters. In both cases, the parameter η, and thus the power, are invariant under isotropic +scaling of the domain, in a similar fashion to the peak height distribution under the null hypothesis (Cheng and +Schwartzman, 2020). +Figure 11 illustrates how θ0 and η affect power and E[Mu] in the 2D simulation described in Section 4. As +we have explained, θ0 and η together determine the effect size. Although deriving an explicit form of effect +size as a function of θ0 and η is difficult, we are able to roughly show how the two parameters relate to power. +θ0 which can be seen as signal-to-noise ratio (SNR) plays a major role. Having η stay the same, the power +monotonically increases with respect to θ0. On the other hand, power monotonically decreases with respect to +η having θ0 stays the same. In this simulation example, the impact of θ0 on power is about 10 times stronger +than η if we fit a linear model of power using θ0 and η. We can also observe from the figure that the effect of η +20 + +3.5 +Power +3 +EFM +2.5 +2 +1.5 +1 +0.5 +0 +0 +100 +200 +300 +400 +500 +Sample Size3.5 +Power +3 +E[M.. +EFM +2.5 +2 +1.5 +0.5 +0 +0 +50 +100 +150 +Sample Size3.5 +Power +3 +E[M.. +EIM +2.5 +2 +1.5 +0.5 +0 +0 +10 +20 +30 +40 +50 +Sample Size2 +Power +E[M.. +1.5 +EIM +0.5 +0 +0 +10 +20 +30 +40 +50 +Sample Size2 +Power +E[M. +1.5 +E[M +ad +1 +0.5 +0 +50 +100 +150 +Sample Size1.5 +Power +EFM +EFM +0.5 +0 +100 +200 +300 +400 +500 +Sample Size(a) Power +(b) E[Mu] +Figure 11: 2D simulation: Power and E[Mu] for different θ and η (u = 3.92 and Rad(D) = 10). +on power is stronger for large θ0 compared to that for small θ0. +7.3 +Application to data +To use our formula to calculate power in practice, one needs to assume the peak to be a certain type such as +paraboloid or Gaussian. However, sometimes it might not be plausible to make such assumptions, leading to +inaccurate power estimate. +Regarding the conjecture of E[Mu]adj being a lower bound when there exists at least one local maximum in +the domain D, it remains difficult to prove in general, but as we showed in the real data example, it seems to +be correct in practice. When it comes to a real-life problem, we can take both E[Mu] and the E[Mu]adj into +consideration to get a better understanding of the true sample size. We suggest using E[Mu] as an approximation +to power when the sample size is small, considering E[Mu]adj when the sample size is large. E[Mu]adj also gives +a more conservative estimate of power compared to E[Mu] which is useful to guarantee that the test is powerful +enough when we design future studies. Because of its difficulty, we leave further study of E[Mu]adj for future +work. +8 +Acknowledgments +Y.Z., D.C. and A.S. were partially supported by NIH grant R01EB026859. Data were provided in part by +the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil +Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for +Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University. +9 +Appendix +References +R. J. Adler, J. E. Taylor, Random Fields and Geometry., New York: Springer, 2007. +D. Cheng, A. Schwartzman, Distribution of the height of local maxima of gaussian random fields., Extremes 18 +(2015) 213–240. +D. Cheng, A. Schwartzman, Multiple testing of local maxima for detection of peaks in random fields, Annals of +Statistics 45(2) (2017) 529–556. +D. Cheng, A. Schwartzman, Expected number and height distribution of critical points of smooth isotropic +gaussian random fields., Bernoulli 24(4B) (2018) 3422–3446. +D. Cheng, A. Schwartzman, On critical points of gaussian random fields under diffeomorphic transformations, +Statistics & Probability Letters 158 (2020) 108672. +J. Cohen, Statistical Power Analysis for the Behavioral Sciences, Routledge, 1988. +J. Durnez, J. Degryse, B. Moerkerke, R. Seurinck, V. Sochat, R. A. Poldrack, T. E. Nichols, Power and sample +size calculations for fmri studies based on the prevalence of active peaks. (2016). +21 + +Power +0.5 +0 +8 +9- +6 +-4 +4 +2 +-2 +0 +0 +0 +n["'W] +0.5 +0 +-6 +8 +6 +-4 +4 +-2 +2 +0 +0 +0 +nC. R. Genovese, N. A. Lazar, T. Nichols, Thresholding of statistical maps in functional neuroimaging using the +false discovery rate, Neuroimage 15 (2002) 870–878. +R. Heller, D. Stanley, D. Yekutieli, N. Rubin, Y. Benjamini, Cluster-based analysis of FMRI data, Neuroimage +33 (2006) 599–608. +V. I. Piterbarg, Rice’s method for large excursions of gaussian random fields., Technical Report NO 478. Center +for Stochastic Processes, Univ. North Carolina (1996). +A. Schwartzman, W. Mascarenhas, J. Taylor, Inference for eigenvalues and eigenvectors of gaussian symmetric +matrices., Annals of Statistics 36(6) (2008) 2886–2919. +A. Schwartzman, F. Telschow, Peak p-values and false discovery rate inference in neuroimaging., NeuroImage +197 (2019) 402–413. +D. C. Van Essen, K. Ugurbil, E. Auerbach, D. Barch, T. E. J. Behrens, R. Bucholz, A. Chang, L. Chen, +M. Corbetta, S. W. Curtiss, S. Della Penna, D. Feinberg, M. F. Glasser, N. Harel, A. C. Heath, L. Larson- +Prior, D. Marcus, G. Michalareas, S. Moeller, R. Oostenveld, S. E. Petersen, F. Prior, B. L. Schlaggar, S. M. +Smith, A. Z. Snyder, J. Xu, E. Yacoub, WU-Minn HCP Consortium, The human connectome project: a data +acquisition perspective, Neuroimage 62 (2012) 2222–2231. +22 + diff --git a/wdE3T4oBgHgl3EQf_Asa/content/tmp_files/load_file.txt b/wdE3T4oBgHgl3EQf_Asa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6ed3fae39e4865e13baba3d5c0f681c1a3017712 --- /dev/null +++ b/wdE3T4oBgHgl3EQf_Asa/content/tmp_files/load_file.txt @@ -0,0 +1,1074 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf,len=1073 +page_content='An approximation to peak detection power using Gaussian random field theory Yu Zhao1 Dan Cheng3 Armin Schwartzman1,2 1Division of Biostatistics, Herbert Wertheim School of Public Health and Human Longevity Science, University of California San Diego, 9500 Gilman Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=', La Jolla, CA 92093, USA 2Halicioˇglu Data Science Institute, University of California San Diego, 9500 Gilman Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=', La Jolla, CA 92093, USA 3School of Mathematical and Statistical Sciences, Arizona State University, 900 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Palm Walk, Tempe, AZ 85281, USA Abstract We study power approximation formulas for peak detection using Gaussian random field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The approximation, based on the expected number of local maxima above the threshold u, E[Mu], is proved to work well under three asymptotic scenarios: small domain, large threshold, and sharp signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' An adjusted version of E[Mu] is also proposed to improve accuracy when the expected number of local maxima E[M−∞] exceeds 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Cheng and Schwartzman (2018) developed explicit formulas for E[Mu] of smooth isotropic Gaussian random fields with zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' In this paper, these formulas are extended to allow for rotational symmetric mean functions, so that they are suitable for power calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' We also apply our formulas to 2D and 3D simulated datasets, and the 3D data is induced by a group analysis of fMRI data from the Human Connectome Project to measure performance in a realistic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Key words: Power calculations, peak detection, Gaussian random field, image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 1 Introduction Detection of peaks (local maxima) is an important topic in image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For example, a fundamental goal in fMRI analysis is to identify the local hotspots of brain activity (see, for example, Genovese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=', 2002 and Heller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=', 2006), which are typically captured by peaks in the fMRI signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The detection of such peaks can be posed as a statistical testing problem intended to test whether the underlying signal has a peak at a given location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' This is challenging because such tests are conducted only at locations of observed peaks, which depend on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Therefore, the height distribution of the observed peak is conditional on a peak being observed at that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' This is a nonstandard problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Solutions exist using random field theory (RFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' RFT is a statistical framework that can be used to perform topological inference and modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' RFT-based peak detection has been studied in Cheng and Schwartzman (2017) and Schwartzman and Telschow (2019), which provide the peak height distribution for isotropic noise under the complete null hypothesis of no signal anywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' In general, for any statistical testing problem, accurate power calculations help researchers decide the min- imum sample size required for an informative test, and thus reduce cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Power calculation formulas exist for common univariate tests, such as z-tests and t-tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' However, particular challenges arise when we perform power calculations in peak detection settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Due to the nature of imaging data, the number and location of the signal peaks are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Besides, the power is affected by other spatial aspects of the problem, such as the shape of the peak function and the spatial autocorrelation of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Considering these difficulties, it requires some extra effort to derive a power formula for peak detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' A formal definition of power in peak detection is necessary to perform power calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' In Cheng and Schwartzman (2017) and Durnez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (2016), the authors explored approaches to control the false discovery rate (FDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For the entire domain, average peakwise power, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' power averaged over all non-null voxels, is a natural choice for these approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For a local domain where a single peak exists, the power can be defined as the probability of successfully detecting that peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Following this idea, we describe the null and alternative hypothesis and the definition of detection power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' We do so informally here for didactic purposes and present formal definitions in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='04830v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='ME] 12 Jan 2023 Consider a local domain where a single peak may exist, and consider the hypotheses H0 : “the signal is equal to 0 in the local domain.” vs H1 : “the signal has at least one positive peak in the local domain.” Suppose we observe a random field to be used as test statistic at every location, typically as the result of statistical modeling of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For a fixed threshold u, the existence of observed peaks with height greater than u would lead to rejecting the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Therefore, we define the type I error and power as the probability of existing at least one local maximum above u under H0 and H1 respectively: Type I error: P{∃ a peak in the local domain with height > u when H0 is true} Power: P{∃ a peak in the local domain with height > u when H1 is true} (1) Formulas for type I error have been developed for stationary fields in 1D and isotropic fields in 2D and 3D (Cheng and Schwartzman, 2015, Cheng and Schwartzman, 2017 and Cheng and Schwartzman, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' However, there is no formula to calculate power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' In order to get an appropriate estimate of power, we need to know the peak height distribution for non-centered (the mean function is not 0) random fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Generally speaking, it is very difficult to calculate the peak height distribution especially when the random field has non-zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Durnez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (2016) suggests using Gaussian distribution to describe the non-null peaks and truncated Gaussian distribution to approximate the overshoot distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' This approach is easy to implement but not very accurate because the peak height distribution is in reality always skewed and not close to any Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' In this article, we propose to approximate the probability of an observed peak exceeding the detection threshold u by calculating the expected number of peaks above the threshold u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' We show that the approximation, which is also an upper bound, works well under certain scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For the entire domain, we can approximate the average peakwise power by taking the arithmetic mean of the approximation proposed in this paper over non-null voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The proposed approximation makes the problem more tractable, but in general, it does not have an explicit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' In order to make it applicable in practice, we further simplify the formula under the isotropy assumption and show its explicit form in 1D, 2D, and 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The explicit results are validated through 2D and 3D computer simulations carried out in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The simulation also covers multiple scenarios by modifying the parameters used to generate the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The performance of power approximation and its conservative adjustment under these scenarios are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Finally, to assess the real-data performance of our power approximation method, we apply it to a 3D simulation induced by a real brain imaging dataset, where the parameters are estimated from the Human Connectome Project (Van Essen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=', 2012) fMRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' By testing the method in a realistic setting, we also demonstrate how effect size and other parameters affect the power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' We first show in Section 2 the problem setup and theoretical results in certain scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' In Section 3, we derive the explicit formulas under isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Simulation in 2D is conducted in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Details regarding how to apply our formula in application setting are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The methodology is applied to a 3D real dataset in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 2 Power Approximation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='1 Setup Let Y (s) = σ(s)Z(s) + µ(s) where Z = {Z(s), s ∈ D} representing the noise is a centered (zero-mean) smooth unit-variance Gaussian random field on an N dimensional non-empty domain D ⊂ RN, σ(s) is the standard deviation of the noise and µ(s) is the mean function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let X(s) = Y (s)/σ(s) = Z(s) + θ(s) where the ratio θ(s) = µ(s)/σ(s) is the standardized mean function, which we assume to be C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Here C3 is a sufficient smoothness condition for Z, and this will be clarified in Assumption 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let Xi(s) = ∂X(s) ∂si , ∇X(s) = (X1(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' , XN(s)), Xij(s) = ∂2X(s) ∂sisj , ∇2X(s) = (Xij(s))1≤i,j≤N, Zi(s) = ∂Z(s) ∂si , ∇Z(s) = (Z1(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' , ZN(s)), Zij(s) = ∂2Z(s) ∂sisj , ∇2Z(s) = (Zij(s))1≤i,j≤N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 2 We will make use of the following assumptions: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Z ∈ C2(D) almost surely and its second derivatives satisfy the mean-square H¨older condition: for any s0 ∈ D, there exists positive constants L, η and δ such that E(Zij(s) − Zij(t))2 ≤ L2∥s − t∥2η, ∀t, s ∈ Us0(δ), i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' where Us0(δ) = s0 ⊕(−δ/2, δ/2)N is the N dimensional open cube of side length δ centered at s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' This condition is satisfied, for example, if Z is C3(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For every pair (t, s) ∈ D × D with s ̸= t, the Gaussian random vector (Z(s), ∇Z(s), Zij(s), Z(t), ∇Z(t), Zij(t), 1 ≤ i ≤ j ≤ N) is non-degenerate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' its covariance matrix has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='2 Peak detection Following the notation in the problem setup, the null and alternative hypothesis can be written as: H0 : µ(s) = 0 for all s ∈ D vs H1 : µ(s) > 0, ∇µ(s) = 0, ∇2µ(s) ≺ 0 for some s ∈ D The mean function µ(s) is not directly observed, so the hypothesis is tested based on the peak height of X(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For a peak detection procedure that aims to test this hypothesis, a threshold u for the peak height of X(s) needs to be set in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' If a local maximum with height greater than u is observed, we would choose to reject the null hypothesis due to the strong evidence against it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The probability that a peak of X exceeds u P � ∃ s ∈ D s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' X(s) > u|∇X(s) = 0 and ∇2X(s) ≺ 0 � (2) is the type I error under H0 and power under H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The threshold u can be obtained based on the peak height distribution under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' A formula for peak height distribution of smooth isotropic Gaussian random fields has been derived in Cheng and Schwartzman (2018) and it can also be derived directly from a special case of the formulas presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Usually, u is set to be some quantile of the null distribution of peak height to maintain the nominal α type I error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' More details about selecting the threshold will be discussed in the real data example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Selecting u is not the main focus of this paper and our method can be applied to any choice of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='3 Power approximation Let Mu be the number of local maxima of the random field X above u over the local domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The power defined in (2) can be represented as P[Mu ≥ 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' We call this the power function, seen as a function of the threshold u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Note that P[Mu ≥ 1] = ∞ � k=1 P[Mu = k] ≤ ∞ � k=1 kP[Mu = k] = E[Mu].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (3) On the other hand, E[Mu] − P[Mu ≥ 1] = ∞ � k=2 (k − 1)P[Mu = k] ≤ 1 2 ∞ � k=2 k(k − 1)P[Mu = k] = 1 2E[Mu(Mu − 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (4) Thus, we have E[Mu] − 1 2E[Mu(Mu − 1)] ≤ P[Mu ≥ 1] ≤ E[Mu].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (5) This inequality tells us that for any fixed u, the power is bounded within an interval of length E[Mu(Mu −1)]/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Thus, E[Mu] is a good approximation of power if one of the two conditions below is satisfied: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The factorial moment E[Mu(Mu − 1)] converges to 0 and E[Mu] does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' They both converge to 0 and E[Mu(Mu − 1)] converges faster than E[Mu].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The convergence above refers to conditions on the signal and noise parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' In the rest of this section, we introduce four interesting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The first result can be useful for simplifying the power function and the other three results give different scenarios where one of the conditions above holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='4 Adjusted E[Mu] We have provided evidence of using E[Mu] to approximate power through (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' However, E[Mu] alone might not be sufficient for power approximation since it only gives an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Also, unlike power, E[Mu] sometimes exceeds 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' To correct for this, we define the adjusted E[Mu] as E[Mu]adj = E[Mu]/ max(1, E[M−∞]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (6) The adjusted E[Mu] is the same as E[Mu] when the expected number of local maxima E[M−∞] is less or equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' When E[M−∞] is greater than 1, we divide E[Mu] by E[M−∞] to make sure it never exceeds 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The adjusted E[Mu] is more conservative, and we conjecture that it is a lower bound of power when there exists at least one local maximum in the domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' In applications, people are interested in a conservative estimator so that the test is guaranteed to have enough power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Combining E[Mu] and E[Mu]/E[M−∞], we can get an approximate range of the true power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' We will compare E[Mu] and adjusted E[Mu] in simulation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='5 Height equivariance Our first result does not concern the approximation (5) yet, but it offers a simplification of the power function and E[Mu] that will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The proposition below states that the power function and E[Mu] for peak detection are translation equivariant with respect to peak height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let θ(s) = h(s) + θ0 be a peak signal with height θ0, where h(s) is a unimodal mean function with maximum equal to 0 at s0 in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Then the power function for peak detection and E[Mu] can be written in the form F(u − θ0), where F(u) is the power function or E[Mu] at θ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let ˜θ(s) = θ(s) − θ0 = h(s) + 0 and ˜ Mu be the number of local maxima of the random field ˜X(s) = Z(s) + ˜θ(s) above u over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Considering the definition of power, we have F(u − θ0) = P[ ˜ Mu−θ0 ≥ 1] = P[Mu ≥ 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Given that E[ ˜ Mu−θ0] = E[Mu], is is also straightforward to show E[Mu] is translation equivariant with respect to θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Next, we give three scenarios where the equality in (5) can be achieved asymptotically: small domain size, large threshold, and sharp signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='6 Small domain If the size of the local domain D where a single peak exists is small enough, it can be shown that equality in (5) can be achieved asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Consider a local domain Dϵ = U(s0, ϵ) for any fixed s0 ∈ D where U(s0, ϵ) = t0 ⊕ (−ϵ/2, ϵ/2)N is the N-dimensional open cube of side ϵ centered at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For sufficiently small ϵ and fixed threshold u, P[Mu ≥ 1] = E[Mu](1 − o(1)) = E[Mu]adj(1 − o(1)), (7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The proof is based on the proof of Lemma 3 in Piterbarg (1996) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='1 in Cheng and Schwartz- man (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' E[Mu(Mu − 1)] = � Dϵ � Dϵ � ∞ u � ∞ u E � |det∇2X(s)||det∇2X(t)| ���� X(s) = x1, X(t) = x2 ∇X(s) = ∇X(t) = 0 � PX(s),X(t),∇X(s),∇X(t)(x1, x2, 0, 0) dx1 dx2 ds dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (8) Let E1(s, t) = E � |det∇2X(s)||det∇2X(t)| ���� X(s) = x ∇X(s) = ∇X(t) = 0 � and replace one of the integration limits in (8) by −∞, we have E[Mu(Mu − 1)] ≤ � Dϵ � Dϵ P∇X(s),∇X(t)(0, 0)dsdt � ∞ u E1(s, t)PX(s) (x|∇X(s) = ∇X(t) = 0) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Then we can take the Taylor expansion ∇X(t) = ∇X(s) + ∇2X(s)(t − s) + ||t − s||1+αYs,t 4 where Ys,t = (Y 1 s,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=', Y N s,t)T is a Gaussian vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Note that the determinant of ∇2X(s) is equal to the determinant of � � � � � 1 −(t1 − s1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' −(tN − sN) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' ∇2X(s) 0 � � � � � (9) For i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=', N + 1, multiply the ith column of this matrix by (ti − si)/||ti − si||2, take the sum of all such columns and add the result to the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Since ∇X(s) = ∇X(t) = 0, we can derive ∇2X(s)(t − s) = −||t − s||1+αYs,t, and obtain the matrix below with the same determinant as (9) � � � � � 0 −(t1 − s1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' −(tN − sN) −||t − s||−1+αY 1 s,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' ∇2X(s) −||s − t||−1+αY N s,t � � � � � Let r = max1≤i≤N |ti − si|, As,t = � � � � � 0 −(t1 − s1)/r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' −(tN − sN)/r Y 1 s,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' ∇2X(s) Y N s,t � � � � � So we have E1(s, t) ≤ ||t − s||αE2(s, t) where E2(s, t) = E � |detAs,t||det∇2X(t)| ���� X(s) = x, ∇X(s) = 0 ∇2X(s)(t − s) = −||t − s||1+αYs,t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Using the inequality of arithmetic and geometric means, we can bound the determinant |det∇2X(t)| ≤ N 2N−2 � i,j |Xij(t)|N |detAs,t| ≤ (N + 1)2N � i,j |aij|N+1 where aij is the i, j entry of As,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Apply the inequality again |det∇2X(t)||detAs,t| ≤ 1 2N 2N−2(N + 1)2N+1 � �� i,j |Xij(t)|2N + � i,j |aij|2N+2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For any Gaussian variable X and integer N ≥ 0, the following inequality holds E[X2N] ≤ 22N(E[X]2N + CNVar(X)2N) where CN is a constant depending on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Next, we can focus on the conditional expectation and conditional variance of Xij(t) and Ys,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' By Assumption 1 and 2 and the fact that the conditional variance of a Gaussian variable is less or equal to the unconditional variance, we can conclude that the conditional variance of Xij(t) and Ys,t are bounded above by some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Summarizing the results above, sup s,t∈Dϵ,s̸=t |E2(s, t)| ≤ C1 for some constant C1 > 0 and E1(s, t) ≤ ||t − s||αE2(s, t) ≤ C1||t − s||α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 5 Combine the results above and with a fixed threshold u � ∞ u E1(s, t)PX(s) (x|∇X(s) = ∇X(t) = 0) dx ≤ C1||t − s||α � ∞ u PX(s) (x|∇X(s) = ∇X(t) = 0) dx = C1||t − s||α � ∞ u exp(−(Ax − B)2)dx for some constant A, B = C2||t − s||α for some constant C2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Next, by the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='1 in Cheng and Schwartzman (2015) p∇X(s),∇X(t)(0, 0) ≤ C3||t − s||−N for some constant C3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Therefore, there exists C4 such that E[Mu(Mu − 1)] ≤ C4 � Dϵ � Dϵ 1 ||t − s||N−α dtds = o(ϵN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For E[Mu], by Kac-Rice formula in Adler and Taylor (2007) E[Mu] = � Dϵ p∇X(s)(0)E � |det∇2X(s)|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Denote the integrand by g(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The function g(s) is continuous and positive over the compact domain Dϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Thus infs∈Dϵ g(s) ≥ g0 > 0, implying E[Mu] ≥ g0ϵN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Then (7) is an immediate consequence of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For E[M−∞], by Kac-Rice formula E[M−∞] = � Dϵ p∇X(s)(0)E � |det∇2X(s)|1{∇2X(s)≺0}|∇X(s) = 0 � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The integrand is also continuous and positive over the compact domain Dϵ indicating E[M−∞] = o(1) for small ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Thus we have E[Mu]adj = E[Mu]/ max(1, E[M−∞]) = E[Mu]/ max(1, o(1)) = E[Mu] for sufficiently small ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='7 Large threshold For large threshold u, the following asymptotic result shows power can be precisely approximated by E[Mu].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For any fixed domain D, as u → ∞ P[Mu ≥ 1] = E[Mu](1 − o(e−αu2)) (10) where the error term o(e−αu2) is non-negative and α > 0 is some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' By lemma 3 of Piterbarg (1996), as u → ∞, the factorial moment is super-exponentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' That means ∃α > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' E[Mu(Mu − 1)] = o(e− u2 2 −αu2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Also E[Mu] ≥ P[Mu ≥ 1] ≥ P[supX(s) ≥ u] = O(e− u2 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Thus, the factorial moment decays exponentially faster than E[Mu].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The result is an immediate consequence of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Notice that the threshold u does not affect the value of E[M−∞] which is part of the adjusted E[Mu].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' By (10) P[Mu ≥ 1] = E[Mu]adj(1 − o(e−αu2)) max(1, E[M−∞]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' If E[M−∞] > 1, the adjusted E[Mu] might be overly conservative for large threshold u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Therefore, we only recommend E[Mu] for this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='8 Sharp signal The following theorem provides an asymptotic power approximation when the signal is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Interestingly, while the power function is generally non-Gaussian, it becomes closer to Gaussian as the signal peaks become sharper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let θ(s) = ah(s) + θ0 where h(s) is a unimodal mean function with maximum equal to 0 at s0, a > 0, and θ0 represents the height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For any fixed threshold u, as a → ∞ P[Mu ≥ 1] = E[Mu] + o(1) = E[Mu]adj + o(1) = Φ(θ0 − u)(1 + o(1)), (11) where Φ(x) is CDF of the standard Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' By lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='1 of Cheng and Schwartzman (2017), as a → ∞ P(M−∞ = 1) ≥ 1 − O(exp(−ca2)), where c > 0 is some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Therefore M−∞ p→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Since Mu ≤ M−∞ and both of them only take non-negative integer values, |Mu(Mu −1)| and |M−∞(M−∞ − 1)| are bounded above by |M(M − 1)| where M is the number of critical points of the random field X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Apply Kac-Rice formula E[M(M − 1)] = � D � D E � |det∇2X(s)||det∇2X(t)| ��∇X(s) = ∇X(t) = 0 � P∇X(s),∇X(t)(0, 0)dsdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Denote the integrand by g(s, t, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The function g(s, t, a) is continuous and positive over the compact domain D and M(M − 1) p→ 0 as a → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Thus there exists g0 > 0 such that E[M(M − 1)] ≤ g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Then by dominated convergence theorem E[Mu(Mu − 1)] → 0 as a → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Since M−∞ p→ 1, the adjusted E[Mu] E[Mu]adj = E[Mu] max(1, E[M−∞]) = E[Mu](1 + o(1)) = E[Mu] + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' To calculate E[Mu], apply Kac-Rice formula E[Mu] = � D p∇X(s)(0)E � |det∇2X(s)|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 � ds = � D p∇X(s)(0)E � |det∇2X(s)|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 � ds = � D 1 (2π)N/2� det(Λ) exp(−a2(∇h(s))T Λ−1∇h(s)/2) E � |det(∇2Z(s) + a∇2h(s))|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 � ds (12) where Λ is the covariance matrix of ∇h(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let f(s) = (∇h(s))T Λ−1∇h(s)/2 which attains its minimum 0 only at s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Similar to the proof of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='4 in Cheng and Schwartzman (2017), as a → ∞, (12) can be approximated by applying Laplace’s method E[Mu] = det(a∇2h(s0)) (2π)N/2� det(Λ) � (2π)N det(Λ) a2N det(∇2h(s0)) �1/2 Φ(θ0 − u) + O(a−2) =Φ(θ0 − u) + O(a−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 3 Explicit formulas We have showed that the power for peak detection can be approximated by the expected number of local maxima above u, E[Mu], under certain scenarios such as small domain and large threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Although we can apply the Kac-Rice formula to calculate E[Mu], it remains difficult to evaluate it explicitly for N > 1 without making any further assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' In this section, we focus on computing E[Mu] and show a general formula can be obtained if the noise field is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Furthermore, explicit formulas when N = 1, 2, 3 are derived for application purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='1 Isotropic Gaussian fields Suppose Z is a zero-mean unit-variance isotropic random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' We can write the covariance function of Z as E{Z(s)Z(t)} = ρ(∥s − t∥2) for an appropriate function ρ(·) : [0, ∞) → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Denote ρ′ = ρ′(0), ρ′′ = ρ′′(0), κ = −ρ′/ � ρ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (13) where ρ′ and ρ′′ are first and second derivative of function ρ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The following lemma comes from Cheng and Schwartzman (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For each s ∈ RN and i, j, k, l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' , N}, E{Zi(s)Z(s)} = E{Zi(s)Zjk(s)} = 0, E{Zi(s)Zj(s)} = −E{Zij(s)Z(s)} = −2ρ′δij, E{Zij(s)Zkl(s)} = 4ρ′′(δijδkl + δikδjl + δilδjk) where ρ′ and ρ′′ are defined in (13) and δij is the Kronecker delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' In particular, it follows from Lemma 1 that Var(Zi(s)) = −2ρ′ and Var(Zii(s)) = 12ρ′′ for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' , N}, implying ρ′ < 0 and ρ′′ > 0 and hence κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' We can use theoretical results from Gaussian Orthogonally Invariant (GOI) matrices to make the calculation of E[Mu] easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' GOI matrices were first introduced in Schwartzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (2008), and used for the first time in the context of random fields in Cheng and Schwartzman (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' It is a class of Gaussian random matrices that are invariant under orthogonal transformations, and can be useful for computing the expected number of critical points of isotropic Gaussian fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' We call an N × N random matrix G = (Gij)1≤i,j≤N GOI with covariance parameter c, denoted by GOI(c), if it is symmetric and all entries are centered Gaussian variables such that E[GijGkl] = 1 2(δikδjl + δilδjk) + cδijδkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (14) The following lemma is Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='4 from Cheng and Schwartzman (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let the assumptions in Lemma 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let �G and G be GOI(1/2) and GOI((1 − κ2)/2) matrices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' IN denotes N × N identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (i) The distribution of ∇2Z(s) is the same as that of √8ρ′′ �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (ii) The distribution of (∇2Z(s)|Z(s) = z) is the same as that of √8ρ′′� G − � κz/ √ 2 � IN � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Lemma 2 shows the distribution and conditional distribution of the Hessian matrix of a centered random field Z(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Next, we establish the corresponding result for non-centered random field X(s) = Z(s) + θ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let �G and G be GOI(1/2) and GOI((1 − κ2)/2) matrices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (i) The distribution of ∇2X(s) is the same as that of � 8ρ′′ �G + ∇2θ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (ii) The distribution of (∇2X(s)|X(s) = x) is the same as that of � 8ρ′′ � G − κ(x − θ(s)) √ 2 IN � + ∇2θ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Part (i) is a direct consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' For part (ii), note that (∇2X(s)|X(s) = x) is equivalent to (∇2Z(s)|Z(s) = x − θ(s)) + ∇2θ(s), and the result follows immediately from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='2 General formula under isotropy Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let X(s) = Z(s) + θ(s), where Z(s) is a smooth zero-mean unit-variance isotropic Gaussian random field satisfying Assumption 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let θ(s) a smooth C3 mean function such that ∇2θ(s) is a non- singular matrix with ordered eigenvalues θ′′ 1(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='θ′′ N(s) at all critical points s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Then for any domain D E[Mu] = � 2ρ′′ −πρ′ �N/2 � D e ∥∇θ(s)∥2 4ρ′ � ∞ u φ (x − θ(s)) E � |det(Matrix(s))|1{Matrix(s)≺0} � dx ds (15) where φ(x) is the PDF of the standard Gaussian distribution, Matrix(s) = G − κ(x − θ(s))IN/ √ 2 + diag{θ′′ 1(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' , θ′′ N(s)}/√8ρ′′, G as in Lemma 3 represents GOI((1-κ2)/2), and 1{·} denotes the indicator func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' By the Kac-Rice formula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='E[Mu] = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='p∇X(s)(0)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='|det∇2X(s)|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='p∇Z(s)+∇θ(s)(0)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='|det∇2X(s)|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='(2π)N/2(−2ρ′)N/2 e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='∥∇θ(s)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='4ρ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='|det∇2X(s)|1{∇2X(s)≺0}1{X(s)>u}|∇X(s) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='(8ρ′′2)N/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='(2π)N/2(−2ρ′)N/2 e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='∥∇θ(s)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='4ρ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='φ (x − θ(s)) E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='|det(Matrix(s))|1{Matrix(s)≺0} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='dx ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� 2ρ′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='−πρ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='�N/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='∥∇θ(s)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='4ρ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='φ (x − θ(s)) E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='|det(Matrix(s))|1{Matrix(s)≺0} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='dx ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� 2ρ′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='−πρ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='�N/2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='∥∇θ(s)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='4ρ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='φ (x − θ(s)) E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='|det(Matrix(s))|1{Matrix(s)≺0} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='dx ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' we show the derivation from the third to the fourth line in the equation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Since we assume ∇2θ(s) is a non-singular matrix at all critical points, then there exists an orthonormal matrix, denoted by A(s), such that A(s)T ∇2θ(s)A(s) = diag{θ′′ 1(s), θ′′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' , θ′′ N(s)}, where θ′′ 1 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' ≤ θ′′ N(s) are ordered eigenvalues of ∇2θ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' On the other hand, GOI matrices are invariant under orthonormal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' By Lemma 3, the conditional expectation E[|det(∇2X(s))|1{∇2X(s)≺0}|X(s) = x] is therefore = E �����det �� 8ρ′′ � G − κ(x − θ(s)) √ 2 IN � + ∇2θ(s) ����� 1{Matrix(s)≺0} � = E �����det �� 8ρ′′ � G − κ(x − θ(s)) √ 2 IN � + A(s)T ∇2θ(s)A(s) ����� 1{Matrix(s)≺0} � = ( � 8ρ′′)NE �����det �� G − κ(x − θ(s)) √ 2 IN � + A(s)T ∇2θ(s)A(s)/ � 8ρ′′ ����� 1{Matrix(s)≺0} � = ( � 8ρ′′)NE �����det � G − κ(x − θ(s)) √ 2 IN + diag{θ′′ 1(s), θ′′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' , θ′′ N(s)}/ � 8ρ′′ ����� 1{Matrix(s)≺0} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (16) The expression (15) can be simplified further if we further assume the mean function θ(s) to be a rotational symmetric paraboloid centered at s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' In this case, the Hessian of θ(s) is the identity matrix multiplied by a constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' θ′′ = θ′′ 1(s) = θ′′ 2(s) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' = θ′′ N(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Then we can write the mean function as θ(s) = θ0 + θ′′∥s − s0∥2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Define η = θ′′ 2κ√ρ′′ = θ′′ −2ρ′ = θ′′ Var(Z1(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (17) and H(˜x) = EN GOI((1−κ2)/2) � � N � j=1 ����λj − κ˜x √ 2 ���� 1{λN< κ˜x √ 2 } � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' (18) E[Mu] can be simplified as E[Mu] = � 2ρ′′ −πρ′ �N/2 � D e θ′′2∥s−s0∥2 4ρ′ � ∞ ˜u(s) φ (˜x + η) H(˜x)d˜xds (19) where we make a change of variable ˜x = x − θ(s) − η and ˜u(s) = u − θ(s) − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Note that the parameter κ depends on the correlation structure of Z(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='3 Explicit formulas in 1D, 2D and 3D In (19), a general formula for E[Mu] under isotropy was derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' To make the formula easier to apply in practice, we have the following results for computing it in 1D, 2D, and 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' When N = 1, the derivation is simple enough that we do not need additional assumptions on the mean function θ(s) except those in Theorem 9 4, and it follows directly from Kac-Rice formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' When N = 2 and 3, we assume the mean function θ(s) is a rotational symmetric paraboloid centered at s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' E[Mu] is calculated by first obtaining explicit formulas for H(˜x), and plugging H into (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let N = 1, X(s) = Z(s) + θ(s), where Z(s) is a smooth zero-mean unit-variance Gaussian process and θ(s) is a smooth mean function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Assume additionally that Z(s) is stationary, then E[Mu] = � D � −2ρ′(3 − κ2) κ φ � θ′(s) √−2ρ′ � � ∞ u φ(x − θ(s))ψ �κ[x − θ(s) − η(s)] √ 3 − κ2 � dx ds, (20) where the function ψ is defined as ψ(x) = � x −∞ Φ(y)dy = φ(x) + xΦ(x), x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Since we assume that Z(s) is stationary, Z′(s) is independent of Z(s) and Z′′(s), and ρ′ = −Var(Z′(s))/2 = E[Z(s)Z′′(s)]/2 and ρ′′ = Var(Z′′(s))/12 do not depend on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Therefore, Var(X(s)) = 1, Var(X′(s)) = −Cov[X(s)X′′(s)] = −2ρ′ and Var(X′′(s)) = 12ρ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Note that, by the formula of conditional Gaussian distributions, X′′(s)|X(s) = x ∼ N(θ′′(s) + 2ρ′(x − θ(s)), 12ρ′′ − 4ρ′2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='By the Kac-Rice formula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='E[Mu] = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='pX′(s)(0)E[|X′′(s)|1{X(s)>u}1{X′′(s)<0}|X′(s) = 0]ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='pX′(s)(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='φ(x − θ(s)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='(−x′′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='12ρ′′ − 4ρ′2 φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='x′′ − θ′′(s) − 2ρ′(x − θ(s)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='12ρ′′ − 4ρ′2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='dx′′ dx ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='pX′(s)(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='12ρ′′ − 4ρ′2 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='√−2ρ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� � ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='φ(x − θ(s))ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='−2ρ′(x − θ(s)) − θ′′(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='12ρ′′ − 4ρ′2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content='dx ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' The second to third line is due to the fact that � 0 −∞ (−x)1 b φ �x + a b � dx = � 0 −∞ Φ �x + a b � dx = b � a/b −∞ Φ(y)dy = bψ �a b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Recall the κ (13) and η (17) parameters defined before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' We can rewrite E[Mu] as � D � −2ρ′(3 − κ2) κ φ � θ′(s) √−2ρ′ � � ∞ u φ(x − θ(s))ψ �κ[x − θ(s) − η(s)] √ 3 − κ2 � dx ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Note that when N = 1, H(˜x) = φ � κ˜x √ 3 − κ2 � + κ˜x √ 3 − κ2 Φ � κ˜x √ 3 − κ2 � = ψ � κ˜x √ 3 − κ2 � (21) We need the following lemmas to calculate H(˜x) explicitly when N = 2 and N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' They are direct calculation of integral by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQf_Asa/content/2301.04830v1.pdf'} +page_content=' Let N = 2, for constant a > − 1 2 and b ∈ R � R2 exp � − 1 2 2 � i=1 λ2 i − a 2 � 2 � i=1 λi �2 � � 2 � i=1 |λi − b| � |λ1 − λ2|1{λ1<λ2 1) overlaps between 2 +bubbles. This happens much earlier in larger ξ systems +thus producing shorter simulation trajectories overall. +Fluctuations in the system’s total potential energy +change significantly for different simulated viscosities. +Two distinct limits are observed, as shown in Fig. 1. +Low viscosity simulations (ξ ≤ 0.001), produce large fluc- +tuations in ∆U(∆t = 1)/U(t) (see Fig. 1a, indicative +of avalanchey, intermittent dynamics. +These are sug- +gestive of the system following the ‘bumpy’ lower lev- +els of the energy landscape. Conversely, with higher ξ +values, the system no longer moves from minimum to +minimum of the underlying energy landscape but evolves +in a dynamic force balance between the larger interac- +tion forces and viscous stresses. This allows the system +to fly over the barriers and rugged features of the energy +landscape, with a higher time average potential energy +for the system (see Fig. 6a). This change in the fluctua- +tions is shown more clearly in the distribution of energy +drops, Fig. 1b, which becomes more heavy-tailed at lower +ξ. Further, a similar trend can also be seen in Fig. 1c, +where the average coordination number (over a system +configuration) is higher at higher viscosities, indicating + +3 +(a) +(b) +(c) +FIG. 1. (a) Traces of relative energy differences ∆U/U(t) (for +∆t = 1 or simulation points spaced by 1 time unit) are sensi- +tive to intermittent dynamics. For lower damping, ξ ≲ 0.01, +the relative change in energy shows abrupt peaks character- +istic of intermittent motion. (b) Lower ξ simulations show +a heavy-tailed probability distribution of energy fluctuations, +typical of an avalanchey system. As the system becomes more +viscous, the energy fluctuations become more Gaussian. (c) +The average system coordination number ⟨z⟩ remains low for +lower ξ simulations, characteristic of lower energy configura- +tions close to potential energy minima on the landscape. zC +is the critical coordination for jamming, with zC = 6 in 3−D +[21]. +that viscous stresses are shifting the foam structure away +from the minimum energy states, and farther from jam- +ming, defined as coordination with ⟨z⟩ ≃ zc [21]. Thus +foam configurations formed at low damping explore lower +and more tortuous portions of the energy landscape and +ones with higher damping cruise through higher and ap- +parently smoother portions of the energy landscape. +To characterize the system’s high dimensional motion +over the energy landscape, we look at the path traversed +by the system through configuration space for the range +of viscosities studied. +The different time points on a +simulation trajectory in configuration space are analyzed +for end-to-end distances (∆R2) and path contour lengths +(∆s). This serves as a measure of the tortuosity of the 3N +dimensional configurational trajectory taken by the sys- +tem over time. As expected from our conclusions above, +we observe that lower viscosities yield fractal, self-similar +(a) +(b) +FIG. 2. Analysis of bubble motion in 3-N dimensional space +and real space shows a mixture of fractal and ballistic motion. +(a) The different simulation points in high dimensional (3N) +space are analyzed for end-to-end distances ∆R2 and contour +lengths ∆s to study fractal scaling over different length scales. +Simulations with larger ξ values give almost ballistic scaling in +hyperspace. However, lower ξ lead to a more tortuous trajec- +tory characteristic of a fractal path, leading to super-diffusive +scaling. +The grey dashed line is a reference with ballistic +scaling ∆R2 ∼ ∆s throughout. All data points above rep- +resent values pooled over 4 simulations and log-bin-averaged +over contour distances. (b) Time and ensemble (4) averaged +mean-squared displacement for an ensemble of bubbles that +remain finite sized throughout our simulations, plotted for +the different ξ values shows ballistic motion rolling over to a +super-diffusive form for lower ξ simulations. In comparison, +more viscous simulations show more ballistic behavior over a +larger range of τ. +scaling at large lengthscales with a fractal dimension +of Df ∼ 2/1.38 (slope at large distances) ≃ 1.45 (see +Fig. 2a) – capturing the intrinsic fractal physics of the +landscape [13]. Simulations with higher damping show +almost no large lengthscale fractal character, indicative +of their ability to avoid lower energy portions of the en- +ergy landscape. Alternatively, the slight bends on Fig. 2 +may be interpreted as a shift in the lengthscale (as a func- +tion of ξ) over which a fractal slope would be observed. +This, however, is further evidence for the self-similar frac- +tal nature of the landscape and indicates that one would +have to examine considerable lag times (or configuration + +Quasi-static +$ = 0.001 +$=0.1 +ε = 0.0001 +$= 0.01 +$=1.0 +△U(△t = 1)/U(t) +175 +200 +225 +250 +275 +300 +t +1.6 +P(△U(△t = 1))/U(t) +102 +1.4 +101 +1.2 +100 +Z +1.0 +0.8 +10-1 +0.6 +10-3 +10-2 +10-1 +150 +200 +△U(△t = 1)/U(t) +tTTTT +Quasi-static += 0.0001 +102 +E += 0.001 + = 0.01 +101 += 0.1 +1.38 +s += 1.0 +2 +100 +R +LL +2.0 +10-1 +10-2 +3 +10- +100 +101 +10-1 +102 +As +100 +1.28 +10-1 +LLL +一 +2 +10 +3 +10 +10-5 +2.0 +100 +101 +102 +T4 +distances traveled by the system) to observe soft-glassy +mechanics for systems with larger damping. +Here, it +must be noted that when particles shrink to zero size +as a result of ripening, we fix their positions in space, +thus conserving the number of dimensions (3N) used to +calculate ∆R2. In Fig. 2b, we compute the ensemble and +time-averaged mean-squared displacement as a function +of lag time (τ). These curves show a functional form that +is similar to that of the ∆R2 above because the mean- +squared displacement is a projection of those curves to +3-D space; the slight difference in the exponent is due +to the calculation being done on a slightly different en- +semble of bubbles that remain finite sized throughout the +simulation. +Here, one may also identify a dimensionless group of +interest called the Deborah number De, which can be +expressed as the ratio of time scales associated with re- +laxation and the mode of driving—the two relevant dy- +namic processes for this system. +Here, that would be +the damped relaxation time from Eq. 2 (τR = ξ⟨a⟩2/ϵ) +and timescale associated with changing bubble radii (a) +imparted by the ripening process, Eq. 3 (τC = ⟨a⟩2/α1 +when α1 > α2). This gives us a ripening Deborah num- +ber, (Deα = ξα1/ϵ) which is a ratio of the relaxation +(τR) and coarsening (τC) times (typically ranging be- +tween 10−3 − 10−6 for our simulations). This dimension- +less group presumably depends on the system’s volume +fraction φ and its proximity to the jamming volume frac- +tion φJ [11, 18]. +This dimensionless group formalism can be a useful +way to explain many previous experimental and simula- +tion results [12, 16, 18]. We begin by noting that the +avalanchey dynamics and intermittent rearrangements +observed in our simulations resemble previous studies +of similar systems [11] driven by shear strain instead +of coarsening. +Various comprehensive studies [16, 18] +using 2D shear strain point out a similar transition to +avalanchey rearrangement events below a certain shear +strain rate. +Thus, our results can be interpreted as a +transition in landscape physics as a function of Deα while +shear simulation results [16] can be explained using a cor- +responding shear Deborah number, Deγ. +For a foam experiment, we note that the energy scale +and damping factor vary as the system evolves: ϵ ≃ σ⟨a⟩2 +[11] and ξ ∝ ⟨a⟩, while α1 is effectively independent of +⟨a⟩. Thus experimentally, Deα ∝ ⟨a⟩3 changes for dy- +namically aging foam where ⟨a⟩ increases as a function +of time [13, 22](see Fig. 5). This keeps pushing the aging +system away from the landscape-dominated regime, po- +tentially explaining the issue associated with the shifting +cut-off [1], and tending to produce behavior akin to high +ξ simulations. +B. +Rheology of SGMs +The rheology of soft-glassy systems is typically found +to be weakly frequency dependent (solid-like), often with +a power-law form, while different experiments on foams +[8, 10] yield apparently conflicting results. Computation- +ally, capturing low-frequency responses to applied strains +can be very expensive, making the determination of rhe- +ology difficult [13]. Here, we provide a numerical proce- +dure that derives its essentials from a microrheological +approach [19, 24] that computes the power spectra of the +active, fluctuating shear strain and stress from the par- +ticle motions, and computes the dynamic shear modulus +from their ratio. +We begin by noting that one can relate the stress (σ(t)) +and strain (γ(t)) to the creep compliance (J(t)) using the +theory of linear response [25, 26] and the Boltzmann su- +perposition principle, relating them through a convolu- +tion: +J(t) ⊛ ˙σ(t) =γ(t) +� t +−∞ +J(t − t′) ˙σ(t′)dt′ =γ(t) +(4) +While this basic constitutive equation represents the +relation between the macroscopic stress and strain for a +linear material, we extend this formalism to its micro- +rheological version wherein each bubble/ particle can be +treated as a tracer moving in a homogeneous viscoelas- +tic continuum (formed by all the other bubbles) driven +by active fluctuating stresses. Thus, typically one can +use the bubbles’ positional vectors describing their mo- +tion in the effective medium to describe the local, time- +dependent strain in the effective medium [24]. Similarly, +the local fluctuating active stress acting on each bubble +in the system can be computed as follows [27, 28]: +σ(ri) = − +� +� +nn +� +j +rij ⊗ Fij +� +� δ(r − ri) +(5) +where rij and Fij represent the inter-particle displace- +ments and forces between particles i and j. +Applying the above equations directly to the data +would be impractical because the σ(t) and γ(t) signals +for each bubble are random functions of time. Instead, +we transform the equation described in Appendix B 1 to +a relation between the ensemble-averaged mean squared +differences (MSD), the stress, and strain. +The stress +MSD is calculated by considering the squared difference +between the three off-diagonal elements of the bubble- +wise symmetric tensor (see Eq. 5). Further, we consider +the ensemble average over all bubbles in our system and +over similar lag times to get a statistically consistent +MSD. Meanwhile, the strain MSD can be estimated using +the positional MSD or mean-squared displacement intro- +duced earlier (see Fig. 2a). These quantities can further +be related using the modified Fourier transformed (FT) +version of the above equation [13, 19, 20]: + +5 +|G∗(ω)|2 ≃ +� +∆σ2(ω) +3π ⟨a⟩ � +∆r2(ω) +(6) +To avoid assumptions and approximations related to +computing Fourier transforms of these MSDs over a finite +range of lag times, [13, 20], we consider the exact convolu- +tional relation described in Appendix B 1. This equation +can be further modified using the Wiener-Khinchin the- +orem and the relationship between autocorrelation and +MSD for the stress and strain, giving us the following +equation: +2J2(0)⟨σ2⟩ + +� τi +0 +f(τi − t′)(⟨σ2⟩ − ⟨∆σ2⟩(t′)/2)dt′ += (⟨γ2⟩ − ⟨∆γ2⟩(τi)/2) +≃ 3π⟨a⟩(⟨r2⟩ − ⟨∆r2⟩(τi)/2)/⟨a⟩3 +where f(τi) is defined as follows, +f(τi) = +�� τi +0 +˙J(τi − t”) ˙J(t”)dt” + 2J(0) ˙J(τi) +� +(7) +where ∆σ2(τ) and ∆γ2(τ) represent the time-averaged, +mean-squared difference of the bubbles’ stress and +strains, in our analyses. We approximate the strain using +the position vector, r [19, 24] as discussed above. Further, +we ensemble average our MSDs over 4 simulation runs. +Finally, to represent the creep compliance, we use a mod- +ified version of the model suggested by Lavergne and co- +authors in Ref. [8]: J(t) = 1/G∞+kD/G∞[(1+t/τ0)β−1] +(more details in Appendix B 2). Using this as a model +for the viscoelastic rheology for the foam, we undertake a +simultaneous fitting operation for the parameters of the +model, i.e., G∞, kD and β, at various lag times or τi in +the convolutional integral equation shown above (Eq. 7). +Using the optimal parameters from the fit gives us the +creep compliance and, subsequently, the complex modu- +lus G∗(ω) using the relation, G∗(ω)J(ω) = 1/iω. Further +details of the derivation and mathematics of the numer- +ical procedure are provided in Appendix B 1. It may be +further noted that attempts to model the rheology using +a Maxwell model produced inferior solutions to Eq. 7, +with the power-law model cited above providing signifi- +cantly better fits. +The results from the computed creep compliance and +dynamic shear moduli are summarized in Fig. 3. G∗(ω) +exhibits a power-law regime over the ω range of inter- +est and is characteristic of behavior predicted in theory +[1], simulations [13] and observed in experiments [8–10]. +Recent experiments and our simulation results here (see +Fig. 3), evaluated with a robust numerical approach pro- +vide clear evidence in support of the existence of power- +law rheology in SGMs. Fig. 3a, shows the fits for the J(t) +model described above, with a family of curves with simi- +lar power-low exponents. Considering the semi-analytical +FT to obtain G∗(ω) gives us the viscoelastic moduli with +(a) +(b) +FIG. 3. We compute the viscoelastic moduli for the dynamic +viscous simulations considered from the fluctuating stresses +and displacements of bubbles in the simulation, as described +in the text. ξ values with data over a significant τ range were +considered for the calculation. The dotted grey lines indicate +the τ range of the MSD data used for the above calculation. +(a) Fitting the model explained in the text to simulation data +gives us suitable fits with a family of curves with power-law +behavior. The creep compliance scales as J(τ) ∼ tβ in the +lag time range shown above. (b) G∗(ω) obtained from J(τ), +gives us power-law rheology in ω i.e., G∗(ω) ∼ ωβ. +This +behavior is observed at all ξ values calculated above. (inset) +The predicted β values, indicative of the log-slope for the +curves in (b), hover consistently in the range ∼ 0.15 − 0.2, +similar to previously observed values in simulation [13], and +experiments [8, 9]. +a power regime defined by G∗(ω) ∼ ωβ, with weak depen- +dence of the exponent β on damping, showing that this +is a universal feature for foams, regardless of damping. +C. +Memory and recovery in perturbed SGMs +The SGM system shows a significant downhill descent +in energy as the largest bubbles coarsen and grow. As this +downward trend continues, the system reaches a dynam- +ical scaling steady state [13]. While it is unclear whether +configurations in this regime form an ergodic ensemble +over some characteristic time, the bubbles show stable +trends in various structural quantities like average co- + +Quasi-static +$= 0.01 +ε= 0.1 +ε= 0.001 + = 0.04 +3 +102 +101 +I +100 +10-1 +101 +102 +103 +104 +T +0.20 +B +0.15 +ZA +一 +2 +0 +10 +10 +10 +- +cS +I +10- +T +10-3 +10-2 +100 +101 +10-4 +10-1 +36 +(a) +(b) +FIG. 4. +Scrambling a quasi-static system shows an imme- +diate return to trend. (a) We scramble the configurational +positions of a system in steady state (at t = 400) in a quasi- +static (ξ = 0) simulation. +Surprisingly, the system always +finds a ’new’ steady state right away, as indicated by the co- +ordination number (z − zC) measured here, and continues to +evolve with similar dynamic properties. The dark and light +symbols represent the scrambled and unscrambled simulation, +respectively. (b) Running multiple (∼ 100) such scrambles at +t = 400, gives us a Gaussian distribution of ⟨z⟩ as shown +above. This overlaps well with ⟨z⟩ values obtained at t = 400 +for 10 different realizations of the same simulation as indicated +by the mean ± standard deviations. This tells us that the +scrambled simulation returns to the newly found steady state +instantaneously. +(inset) Moreover, the temporal autocorre- +lations for these z ensembles - scrambled and unscrambled - +provide similar decorrelation times. These findings indicate +similar dynamic properties for the scrambled and unscram- +bled simulation. +ordination number, mean bubble radius, normalized ra- +dial distribution, etc. Bubbles initially move around to +reach the steady state, defined by the dynamical scaling +‘attractor’ on the energy landscape, and then continue +to evolve in this steady state ensemble. Any perturba- +tion away from the attractor would thus lead the system +back to a ’new’ steady state as defined by the structural +and dynamical properties of the attractor and the sys- +tem landscape. Experiments have observed [14, 15] that +a strain-perturbed foam relaxed back to its unperturbed +steady state after an unexpectedly long waiting time, and +FIG. 5. Scrambling a quasi-static system shows no change to +ripening evolution. Here, we look at the structure through +the radial distribution formed at steady state for a scrambled +and unscrambled system. As previously in Fig. 4, the system +instantaneously continues in steady state. As can be noticed, +the slope changes for the scrambled simulation at t = 400 (in- +dicated by arrows), indicative of a new foam initiation time +[13]; however, the trend remains linear, consistent with dy- +namic scaling state behavior ⟨a⟩2 ∼ tage. +have described this as a memory phenomenon or measure +of history dependence. The consensus [14, 15] on the ori- +gin of this memory is that coarsening mediated excita- +tions are needed to enable the system to overcome local +minima that the perturbed system relaxes into. Thus, +the long waiting time has been considered a result of +slow coarsening. +To study this phenomenon’s structural and dynamical +significance computationally, we run a set of simulations +using our modified damped model over various ξ values. +We consider the theoretical extreme of a perturbation +by introducing positional scrambles in our system. To +do so, we begin with a typical steady-state system and +randomly scramble the various 3N positions of the bub- +bles. This scramble randomly assigns a point in hyper- +space for the system of soft spheres, providing a ran- +dom structural perturbation. We then continue with the +relaxation-coarsening procedure described previously in +Section II A. It must be noted here that for the quasi- +static case when ξ = 0, we relax the system to its first en- +ergy minimum (i.e. mechanical equilibrium) using FIRE +[29] instead of using Eq. 2. +For the quasi-static case, we see that the system, upon +one (or even multiple) scrambles, returns to the earlier +dynamical scaling steady trend (see Fig. 4a) immediately. +Indicators like ⟨z⟩ and ⟨a⟩2 show no significant change +from steady-state behavior, as can be seen in Fig. 4 which +plots the scrambled (at t = 400) and unscrambled aver- +age coordination number as a function of time. Here the +scrambled system experiences no barriers to reaching this +’new’ steady state with FIRE traversing the large config- +urational distance on a relatively smooth portion of the +energy landscape (at higher energies) to find the nearest +(primary) minima. It may be noted that the scramble + +1.75 +0.75 +0.50 +1.50 +0.25 +0.00 +1.25 +Zc +-0.25 +1.00 +100 +101 +T +0.75 +0.50 +300 +400 +500 +600 +700 +t +8 +6 +0 + 0.01, we stuck with the use of +dt = 0.001, as the system moves further away from me- +chanically stable states and the above approximation in +Eq. A3 fails to strictly hold. +3. +Dimensionless group analysis: Deborah number +Apart from evaluating the Deborah number De as +the ratio of the damped relaxation time from Eq. 2 +(τR = ξ⟨a⟩2/ϵ) and probing time associated with chang- +ing bubble radii (a) imparted by the coarsening process, +Eq. 3 (τC = ⟨a⟩2/α1, we can do a simple Buckingham Pi +analysis to determine the relevant Π group. Below, we +present the analysis to derive the Deborah number as a +Π group. +One can re-model the system through an experimental +lens and pose the problem statement as measuring the +average radii ⟨a⟩ as a function of time. Intuitively, this +might be influenced by system properties like ϵ, ρ, α1, ξ. +These 4 quantities along with ⟨a⟩, are comprised of the +dimensions M, L and T. Thus 2 Π groups can be made +using these variables for every combination of 3 repeating +variables being chosen. Here, we choose ρ, α, and ξ as +are repeating variables. +Π1 = f(ϵ, ρ, α1, ξ) += ϵρxαy +1ξz +(A7) +Solving for x, y, and z so that Π1 is dimensionless, we +get Π1 = ϵ/(α1ξ) or De = ξα1/ϵ. + +10 +Appendix B: Rheology +1. +Analytical Derivation +Here, we provide a derivation for the integral equation +Eq. 7, which we used to compute the viscoelastic moduli +for our simulation. We start by noting that the theory of +viscoelasticity for linear materials [25, 26] shows that the +creep compliance J, can be related to the stress σ and +strain γ as follows: +� t +−∞ +J(t − t′) ˙σ(t′)dt′ = γ(t) +(B1) +Here we may note that J, σ and γ are = 0 ∀t ∈ (−∞, 0) +and ≥ 0 ∀t ∈ [0, ∞). Thus, we can extend the integral +limits by doing the following: +� ∞ +−∞ +J(t − t′) ˙σ(t − t′)dt′ = γ(t) +� ∞ +−∞ +˙J(t − t′)σ(t − t′)dt′ = γ(t) +using the product rule +� t +0 +˙J(t − t′)σ(t − t′)dt′ + J(0)σ(t) = γ(t) +(B2) +Taking the Fourier Transform of the non-decomposed +equation above and applying the convolution theorem +gives us, +�˙J �σ = �γ +(B3) +While one could potentially work with Eq. B2 or +Eq. B3, the numerical inaccuracies associated with an +FT [13, 20] and the statistical noise in a trajectory func- +tion like σ(t) or γ(t), would make the procedure more +difficult. +Thus, we use the Wiener–Khinchin theorem +and further transform the auto-correlation into its mean +squared version as follows: +�˙J �˙J = ||�γ||2 +||�σ||2 += +� +Rγγ +� +Rσσ += +� +⟨γ2⟩ − ⟨∆γ2⟩/2 +� +⟨σ2⟩ − ⟨∆σ2⟩/2 +(B4) +Reshuffling this equation and taking the inverse FT +yields an integral equation. We decompose the limits to +stay between 0 and t, which adds a few boundary terms +for the step function jump in J and σ at t = 0. Further, +we change our notation for t to τ, to be consistent with +the MSDs which are calculated as averages over lag times. +2J2(0)⟨σ2⟩ + +� τ +0 +f(τ − t′)(⟨σ2⟩ − ⟨∆σ2⟩(t′)/2)dt′ += (⟨γ2⟩ − ⟨∆γ2⟩(τ)/2) +where f(τ) is defined as follows, +f(τ) = +�� τ +0 +˙J(τ − t”) ˙J(t”)dt” + 2J(0) ˙J(τ) +� +(B5) +This equation can be approximated, using similar +mathematical approximations as used earlier in Ref. [20]. +� τ +0 +g(τ − t′)⟨∆σ2⟩(t′)dt′ = ⟨∆γ2⟩(τ) +where g(τ) is defined as follows, +g(τ) = +�� τ +0 +˙J(τ − t”) ˙J(t”)dt” +� +(B6) +We approximate the right-hand side of this equa- +tion using the bubble portions r [19, 24], giving: +≃ +3π⟨a⟩(⟨r2⟩−⟨∆r2⟩(τi)/2)/⟨a⟩3. However, it may be noted +that this equation is not well defined at τ = 0. Thus we +evaluate this only for lag times greater than zero. To get +an accurate solution, we consider the above equation at +various finite lag time values or τi and solve a set of simul- +taneous equations to find the appropriate creep compli- +ance, J(t). Specifically, we choose τi ∈ {τ1, τ2, τ3...τmax}. +Here, τ1 can be as small as dt. We report here results +for τ1 = 1. +This choice, however, brings in some nu- +merical error due to the integrals going from 0 → τ. It +may be noted that this equation is mathematically exact +for ∀τ > 0 and that the upper limit of our observation +– τmax, does not affect the numerical procedure, effec- +tively avoiding a source of truncation error present in +many earlier approaches. +2. +Choice of Fitting Model +One may notice that solving Eq. B5 or Eq. B6 requires +a model for J(t). Here we choose a modified version of +the model suggested in Ref. [8]. 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Series A: Mathematical, +Physical and Engineering Sciences 459, 2347 (2003). +[29] E. Bitzek, P. Koskinen, F. G¨ahler, M. Moseler, and +P. Gumbsch, Structural relaxation made simple, Phys. +Rev. Lett. 97, 170201 (2006). + diff --git a/wtFQT4oBgHgl3EQfwDZl/content/tmp_files/load_file.txt b/wtFQT4oBgHgl3EQfwDZl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..535e83e50bb2ecf03a404fb6c811662dff1e9a89 --- /dev/null +++ b/wtFQT4oBgHgl3EQfwDZl/content/tmp_files/load_file.txt @@ -0,0 +1,666 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf,len=665 +page_content='The emergence of soft-glassy mechanics in simulated foams Amruthesh Thirumalaiswamy,1 Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Riggleman,1, ∗ and John C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Crocker1, † 1Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Pennsylvania Several seemingly different soft materials, including foams, cells, and many complex fluids, exhibit remarkably similar rheological properties and microscopic dynamics, termed soft glassy mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Here, we show that such behavior emerges from a simple model of a damped ripening foam, for sufficiently weak damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' In particular, we observe intermittent avalanchey dynamics, bubble super-diffusion, and power-law rheology that vary as the damping factor is changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' In the limit of weak damping, the dynamics are determined by the tortuous low-lying portions of the energy landscape, as described in a recent study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' For strong damping the viscous stresses cause the system configuration to evolve along higher energy paths, washing out small-scale tortuosity and produc- ing motion with an increasingly ballistic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Using a microrheological approach, the linear viscoelastic response of the model can be efficiently calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This resembles the power-law rhe- ology expected for soft glassy mechanics, but unexpectedly, is only weakly sensitive to the damping parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Lastly, we study the reported memory effect in foams after large perturbations and find that the timescale of the memory goes to zero as the damping parameter vanishes, suggesting that the effect is due to viscous stress relaxation rather than slow structural changes stabilized by the energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' INTRODUCTION Soft glassy materials (SGMs) [1–3] such as foams and emulsions exhibit complex physical and rheological prop- erties that continue to defy explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Moreover, the similarity of soft glassy mechanics to that of living cells [4–6] and glassy materials [7] has long been noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Pre- vious experimental and theoretical models have captured different aspects of such systems while falling short of a complete physical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' For foams, rheological exper- iments have shown conflicting results — showing weak [8, 9] or no [10] power-law frequency dependence of the dynamic shear modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Modeling efforts have largely focused on the now canonical ‘bubble model’ [11, 12], but the dynamic shear modulus of this model has not been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' While a more recent study did report power-law rheology [13] it used a simplified system with- out damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Further, experiments have shown memory effects [14, 15] in which a deformed foam shows perturbed mechanics which relaxes back to the unperturbed trend after a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The physical origin of this memory effect remains poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Here, we study the soft glassy mechanics and rheology of foams, as well as their recovery from mechanical per- turbation using a 3-D bubble model [11, 12, 16] with a simple damping law [7, 12], driven by simulated Ostwald ripening [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Previous stress-strain simulations [16, 18] of a 2-D bubble model without ripening have indicated a transition to avalanchey dynamics with reduced ap- plied strain rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' We look for a similar effect in our ripening foam model by changing the damping param- eter ξ, effectively changing the relative rates of ripening ∗ rrig@seas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='edu † jcrocker@seas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='edu and viscous relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This however requires the com- putationally expensive integration of the bubble model’s equation of motion at low ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' We find that for sufficiently low damping (or equivalently slow ripening), the system dynamics are determined by the tortuous character of the energy landscape, as observed in a damping-free model [13], leading to avalanches in energy, super-diffusive bub- ble motion, and fractal configuration-space paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' For stronger damping, this behavior disappears, being re- placed by a more continuous motion having a ballistic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' We use a microrheological approach to deter- mine the dynamic shear modulus of our model from its in- trinsic, non-thermal fluctuations [19, 20], and find that it generically has power-law rheology resembling recent ex- perimental measurements [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The rheology exponent is, unexpectedly, only a weak function of damping, pro- viding new insights into the origin of power-law rheology in SGMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Lastly, we study foam’s recovery from mechan- ical perturbation by randomly scrambling the locations of bubbles in our model, finding that scrambling leads to perturbed mechanics that slowly return to the (average) unperturbed baseline, resembling experimental reports of mechanical memory in foams [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The foam recov- ers to the baseline more quickly as the damping factor is reduced, and does so immediately when damping is removed, indicating that the memory effect is controlled by viscous stress relaxation, and not due to activation between energy minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' DAMPED SGM MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Coarsening bubble dynamics We model a coarsening foam using the bubble model [11, 12] with a simplified damping rule and simulated Ostwald ripening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' While the bubble model has been tra- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='13400v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='soft] 31 Jan 2023 2 ditionally used to simulate foams [11, 18], it also serves as an effective model for many other SGMs [7, 13, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The constituent bubbles of foam in this model are treated as soft-sphere particles that can overlap and interact via a pairwise repulsive potential when overlapping: V (rij) = � � � ϵ 2 � 1 − ∥rij∥ ai+aj �2 , if ∥rij∥ < ai + aj 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' (1) with rij being the distance between two bubbles of radii ai and aj, and V (rij) being the corresponding potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The positions of the bubbles interacting via the pair- wise potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 1 are evolved using an (over- damped) equation of motion [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Notably, we consider a simplified version of the viscous force, Fi = ξvi [7, 12], on each bubble to reduce computational overload while preserving relevant model physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' ξ dri dt = − nn � j ∂V (rij) ∂ri (2) with the right-hand side representing a summation over neighboring bubbles that contribute to the force on bub- ble i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Meanwhile, the left side is the damping force with ξ being the effective viscous damping factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' To simulate the mass exchange between bubbles due to Ostwald ripening [17], the bubble radii are allowed to evolve while keeping total bubble volume constant (pre- serving notional mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Ripening causes larger bubbles to grow and smaller ones to shrink over time via a pair- wise mass flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' We model this process with a flow rate that depends on the degree of overlap between neighbor- ing bubbles (over the overlap cross-section), along with a mean-field flux that flows through the connected phase medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Qi = −α1ρ nn � j � 1 ai − 1 aj � Aoverlap � �� � neighbor-neighbor −α2ρ � 1 ai − 1 ⟨a⟩ � ai � �� � mean-field (3) As indicated, the first term represents the pair-wise mass- flux between neighboring bubbles over the cross-section of overlap and the second represents the mean field con- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The values for α1(= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='05) and α2(= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='002) were chose akin to our previous study [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' When a bub- ble’s volume turns negative over the course of the simula- tion, we remove it from the simulation box, while ensur- ing that the mass of the deleted bubble and its neighbors is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' To realize the overall evolution of the system, we inde- pendently evolve the system using the equation of motion (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 2) and ripening rules (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 3) for all bubbles at every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' We use a simple explicit Euler scheme with small dt values, to numerically integrate the equations of motion (more details in Appendix A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The use of other integrators like a second-order Runge-Kutta numerical discretization led to similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' While we note that the equations of motion can be physically unstable at very high energies (when there is a significant overlap between bubbles), we verify that such large overlaps are not present at the energy levels presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Further, while using small step-sizes within the range of numerical stability (more details in Appendix A 2), we ensure the simulation has converged by cross-validating with smaller step-sizes (dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' It may be noted here that smaller step- sizes (dt) are required for lower values of ξ, and thus are computationally more expensive to simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Bubble model simulations [13] when initialized ran- domly, eventually reach a dynamic steady state with a characteristic bubble size distribution and various sys- tematic trends in properties such as total energy or mean bubbles size, as with experiments [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' We ini- tialize a system of N ∼ 1000 bubbles at a volume frac- tion of φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='75 (just above its jamming volume frac- tion [13, 23]) with a Weibull bubble radius distribution, P(a) ∼ (k/λ)(x/λ)k−1, where k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='75, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='73, and let the system evolve as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This distri- bution is representative of the steady state bubble size distribution that a Gaussian initialized system reaches when evolved in the quasi-static limit (ξ → 0) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Us- ing this as a starting point for all our simulations, we model over a range of damping factors (ξ) and calculate various physical quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' It must be noted that we end our simulations when bubbles grow consid- erably large leading to multiple (> 1) overlaps between 2 bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This happens much earlier in larger ξ systems thus producing shorter simulation trajectories overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Fluctuations in the system’s total potential energy change significantly for different simulated viscosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Two distinct limits are observed, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Low viscosity simulations (ξ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='001), produce large fluc- tuations in ∆U(∆t = 1)/U(t) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 1a, indicative of avalanchey, intermittent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' These are sug- gestive of the system following the ‘bumpy’ lower lev- els of the energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Conversely, with higher ξ values, the system no longer moves from minimum to minimum of the underlying energy landscape but evolves in a dynamic force balance between the larger interac- tion forces and viscous stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This allows the system to fly over the barriers and rugged features of the energy landscape, with a higher time average potential energy for the system (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This change in the fluctua- tions is shown more clearly in the distribution of energy drops, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 1b, which becomes more heavy-tailed at lower ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Further, a similar trend can also be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 1c, where the average coordination number (over a system configuration) is higher at higher viscosities, indicating 3 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' (a) Traces of relative energy differences ∆U/U(t) (for ∆t = 1 or simulation points spaced by 1 time unit) are sensi- tive to intermittent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' For lower damping, ξ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='01, the relative change in energy shows abrupt peaks character- istic of intermittent motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' (b) Lower ξ simulations show a heavy-tailed probability distribution of energy fluctuations, typical of an avalanchey system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' As the system becomes more viscous, the energy fluctuations become more Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' (c) The average system coordination number ⟨z⟩ remains low for lower ξ simulations, characteristic of lower energy configura- tions close to potential energy minima on the landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' zC is the critical coordination for jamming, with zC = 6 in 3−D [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' that viscous stresses are shifting the foam structure away from the minimum energy states, and farther from jam- ming, defined as coordination with ⟨z⟩ ≃ zc [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Thus foam configurations formed at low damping explore lower and more tortuous portions of the energy landscape and ones with higher damping cruise through higher and ap- parently smoother portions of the energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' To characterize the system’s high dimensional motion over the energy landscape, we look at the path traversed by the system through configuration space for the range of viscosities studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The different time points on a simulation trajectory in configuration space are analyzed for end-to-end distances (∆R2) and path contour lengths (∆s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This serves as a measure of the tortuosity of the 3N dimensional configurational trajectory taken by the sys- tem over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' As expected from our conclusions above, we observe that lower viscosities yield fractal, self-similar (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Analysis of bubble motion in 3-N dimensional space and real space shows a mixture of fractal and ballistic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' (a) The different simulation points in high dimensional (3N) space are analyzed for end-to-end distances ∆R2 and contour lengths ∆s to study fractal scaling over different length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Simulations with larger ξ values give almost ballistic scaling in hyperspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' However, lower ξ lead to a more tortuous trajec- tory characteristic of a fractal path, leading to super-diffusive scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The grey dashed line is a reference with ballistic scaling ∆R2 ∼ ∆s throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' All data points above rep- resent values pooled over 4 simulations and log-bin-averaged over contour distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' (b) Time and ensemble (4) averaged mean-squared displacement for an ensemble of bubbles that remain finite sized throughout our simulations, plotted for the different ξ values shows ballistic motion rolling over to a super-diffusive form for lower ξ simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' In comparison, more viscous simulations show more ballistic behavior over a larger range of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' scaling at large lengthscales with a fractal dimension of Df ∼ 2/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='38 (slope at large distances) ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='45 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 2a) – capturing the intrinsic fractal physics of the landscape [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Simulations with higher damping show almost no large lengthscale fractal character, indicative of their ability to avoid lower energy portions of the en- ergy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Alternatively, the slight bends on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 2 may be interpreted as a shift in the lengthscale (as a func- tion of ξ) over which a fractal slope would be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This, however, is further evidence for the self-similar frac- tal nature of the landscape and indicates that one would have to examine considerable lag times (or configuration Quasi-static $ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='001 $=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='1 ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='0001 $= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='01 $=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='0 △U(△t = 1)/U(t) 175 200 225 250 275 300 t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='6 P(△U(△t = 1))/U(t) 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='4 101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='2 100 Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='8 10-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='6 10-3 10-2 10-1 150 200 △U(△t = 1)/U(t) tTTTT Quasi-static = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='0001 102 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='001 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='01 101 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='38 s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='0 2 100 R LL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='0 10-1 10-2 3 10- 100 101 10-1 102 As 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='28 10-1 LLL 一 2 10 3 10 10-5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='0 100 101 102 T4 distances traveled by the system) to observe soft-glassy mechanics for systems with larger damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Here, it must be noted that when particles shrink to zero size as a result of ripening, we fix their positions in space, thus conserving the number of dimensions (3N) used to calculate ∆R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 2b, we compute the ensemble and time-averaged mean-squared displacement as a function of lag time (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' These curves show a functional form that is similar to that of the ∆R2 above because the mean- squared displacement is a projection of those curves to 3-D space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' the slight difference in the exponent is due to the calculation being done on a slightly different en- semble of bubbles that remain finite sized throughout the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Here, one may also identify a dimensionless group of interest called the Deborah number De, which can be expressed as the ratio of time scales associated with re- laxation and the mode of driving—the two relevant dy- namic processes for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Here, that would be the damped relaxation time from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 2 (τR = ξ⟨a⟩2/ϵ) and timescale associated with changing bubble radii (a) imparted by the ripening process, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 3 (τC = ⟨a⟩2/α1 when α1 > α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This gives us a ripening Deborah num- ber, (Deα = ξα1/ϵ) which is a ratio of the relaxation (τR) and coarsening (τC) times (typically ranging be- tween 10−3 − 10−6 for our simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This dimension- less group presumably depends on the system’s volume fraction φ and its proximity to the jamming volume frac- tion φJ [11, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This dimensionless group formalism can be a useful way to explain many previous experimental and simula- tion results [12, 16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' We begin by noting that the avalanchey dynamics and intermittent rearrangements observed in our simulations resemble previous studies of similar systems [11] driven by shear strain instead of coarsening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Various comprehensive studies [16, 18] using 2D shear strain point out a similar transition to avalanchey rearrangement events below a certain shear strain rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Thus, our results can be interpreted as a transition in landscape physics as a function of Deα while shear simulation results [16] can be explained using a cor- responding shear Deborah number, Deγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' For a foam experiment, we note that the energy scale and damping factor vary as the system evolves: ϵ ≃ σ⟨a⟩2 [11] and ξ ∝ ⟨a⟩, while α1 is effectively independent of ⟨a⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Thus experimentally, Deα ∝ ⟨a⟩3 changes for dy- namically aging foam where ⟨a⟩ increases as a function of time [13, 22](see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This keeps pushing the aging system away from the landscape-dominated regime, po- tentially explaining the issue associated with the shifting cut-off [1], and tending to produce behavior akin to high ξ simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Rheology of SGMs The rheology of soft-glassy systems is typically found to be weakly frequency dependent (solid-like), often with a power-law form, while different experiments on foams [8, 10] yield apparently conflicting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Computation- ally, capturing low-frequency responses to applied strains can be very expensive, making the determination of rhe- ology difficult [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Here, we provide a numerical proce- dure that derives its essentials from a microrheological approach [19, 24] that computes the power spectra of the active, fluctuating shear strain and stress from the par- ticle motions, and computes the dynamic shear modulus from their ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' We begin by noting that one can relate the stress (σ(t)) and strain (γ(t)) to the creep compliance (J(t)) using the theory of linear response [25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 26] and the Boltzmann su- perposition principle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' relating them through a convolu- tion: J(t) ⊛ ˙σ(t) =γ(t) � t −∞ J(t − t′) ˙σ(t′)dt′ =γ(t) (4) While this basic constitutive equation represents the relation between the macroscopic stress and strain for a linear material,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' we extend this formalism to its micro- rheological version wherein each bubble/ particle can be treated as a tracer moving in a homogeneous viscoelas- tic continuum (formed by all the other bubbles) driven by active fluctuating stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Thus, typically one can use the bubbles’ positional vectors describing their mo- tion in the effective medium to describe the local, time- dependent strain in the effective medium [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Similarly, the local fluctuating active stress acting on each bubble in the system can be computed as follows [27, 28]: σ(ri) = − � � nn � j rij ⊗ Fij � � δ(r − ri) (5) where rij and Fij represent the inter-particle displace- ments and forces between particles i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Applying the above equations directly to the data would be impractical because the σ(t) and γ(t) signals for each bubble are random functions of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Instead, we transform the equation described in Appendix B 1 to a relation between the ensemble-averaged mean squared differences (MSD), the stress, and strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The stress MSD is calculated by considering the squared difference between the three off-diagonal elements of the bubble- wise symmetric tensor (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Further, we consider the ensemble average over all bubbles in our system and over similar lag times to get a statistically consistent MSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Meanwhile, the strain MSD can be estimated using the positional MSD or mean-squared displacement intro- duced earlier (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' These quantities can further be related using the modified Fourier transformed (FT) version of the above equation [13, 19, 20]: 5 |G∗(ω)|2 ≃ � ∆σ2(ω) 3π ⟨a⟩ � ∆r2(ω) (6) To avoid assumptions and approximations related to computing Fourier transforms of these MSDs over a finite range of lag times, [13, 20], we consider the exact convolu- tional relation described in Appendix B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This equation can be further modified using the Wiener-Khinchin the- orem and the relationship between autocorrelation and MSD for the stress and strain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' giving us the following equation: 2J2(0)⟨σ2⟩ + � τi 0 f(τi − t′)(⟨σ2⟩ − ⟨∆σ2⟩(t′)/2)dt′ = (⟨γ2⟩ − ⟨∆γ2⟩(τi)/2) ≃ 3π⟨a⟩(⟨r2⟩ − ⟨∆r2⟩(τi)/2)/⟨a⟩3 where f(τi) is defined as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' f(τi) = �� τi 0 ˙J(τi − t”) ˙J(t”)dt” + 2J(0) ˙J(τi) � (7) where ∆σ2(τ) and ∆γ2(τ) represent the time-averaged,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' mean-squared difference of the bubbles’ stress and strains,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' in our analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' We approximate the strain using the position vector, r [19, 24] as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Further, we ensemble average our MSDs over 4 simulation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Finally, to represent the creep compliance, we use a mod- ified version of the model suggested by Lavergne and co- authors in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' [8]: J(t) = 1/G∞+kD/G∞[(1+t/τ0)β−1] (more details in Appendix B 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Using this as a model for the viscoelastic rheology for the foam, we undertake a simultaneous fitting operation for the parameters of the model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=', G∞, kD and β, at various lag times or τi in the convolutional integral equation shown above (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Using the optimal parameters from the fit gives us the creep compliance and, subsequently, the complex modu- lus G∗(ω) using the relation, G∗(ω)J(ω) = 1/iω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Further details of the derivation and mathematics of the numer- ical procedure are provided in Appendix B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' It may be further noted that attempts to model the rheology using a Maxwell model produced inferior solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 7, with the power-law model cited above providing signifi- cantly better fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The results from the computed creep compliance and dynamic shear moduli are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' G∗(ω) exhibits a power-law regime over the ω range of inter- est and is characteristic of behavior predicted in theory [1], simulations [13] and observed in experiments [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Recent experiments and our simulation results here (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 3), evaluated with a robust numerical approach pro- vide clear evidence in support of the existence of power- law rheology in SGMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 3a, shows the fits for the J(t) model described above, with a family of curves with simi- lar power-low exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Considering the semi-analytical FT to obtain G∗(ω) gives us the viscoelastic moduli with (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' We compute the viscoelastic moduli for the dynamic viscous simulations considered from the fluctuating stresses and displacements of bubbles in the simulation, as described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' ξ values with data over a significant τ range were considered for the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The dotted grey lines indicate the τ range of the MSD data used for the above calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' (a) Fitting the model explained in the text to simulation data gives us suitable fits with a family of curves with power-law behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The creep compliance scales as J(τ) ∼ tβ in the lag time range shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' (b) G∗(ω) obtained from J(τ), gives us power-law rheology in ω i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=', G∗(ω) ∼ ωβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This behavior is observed at all ξ values calculated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' (inset) The predicted β values, indicative of the log-slope for the curves in (b), hover consistently in the range ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='15 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='2, similar to previously observed values in simulation [13], and experiments [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' a power regime defined by G∗(ω) ∼ ωβ, with weak depen- dence of the exponent β on damping, showing that this is a universal feature for foams, regardless of damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Memory and recovery in perturbed SGMs The SGM system shows a significant downhill descent in energy as the largest bubbles coarsen and grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' As this downward trend continues, the system reaches a dynam- ical scaling steady state [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' While it is unclear whether configurations in this regime form an ergodic ensemble over some characteristic time, the bubbles show stable trends in various structural quantities like average co- Quasi-static $= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='01 ε= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='1 ε= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='001 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='04 3 102 101 I 100 10-1 101 102 103 104 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='20 B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='15 ZA 一 2 0 10 10 10 cS I 10- T 10-3 10-2 100 101 10-4 10-1 36 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Scrambling a quasi-static system shows an imme- diate return to trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' (a) We scramble the configurational positions of a system in steady state (at t = 400) in a quasi- static (ξ = 0) simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Surprisingly, the system always finds a ’new’ steady state right away, as indicated by the co- ordination number (z − zC) measured here, and continues to evolve with similar dynamic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The dark and light symbols represent the scrambled and unscrambled simulation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' (b) Running multiple (∼ 100) such scrambles at t = 400, gives us a Gaussian distribution of ⟨z⟩ as shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This overlaps well with ⟨z⟩ values obtained at t = 400 for 10 different realizations of the same simulation as indicated by the mean ± standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This tells us that the scrambled simulation returns to the newly found steady state instantaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' (inset) Moreover, the temporal autocorre- lations for these z ensembles - scrambled and unscrambled - provide similar decorrelation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' These findings indicate similar dynamic properties for the scrambled and unscram- bled simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' ordination number, mean bubble radius, normalized ra- dial distribution, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Bubbles initially move around to reach the steady state, defined by the dynamical scaling ‘attractor’ on the energy landscape, and then continue to evolve in this steady state ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Any perturba- tion away from the attractor would thus lead the system back to a ’new’ steady state as defined by the structural and dynamical properties of the attractor and the sys- tem landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Experiments have observed [14, 15] that a strain-perturbed foam relaxed back to its unperturbed steady state after an unexpectedly long waiting time, and FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Scrambling a quasi-static system shows no change to ripening evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Here, we look at the structure through the radial distribution formed at steady state for a scrambled and unscrambled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' As previously in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 4, the system instantaneously continues in steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' As can be noticed, the slope changes for the scrambled simulation at t = 400 (in- dicated by arrows), indicative of a new foam initiation time [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' however, the trend remains linear, consistent with dy- namic scaling state behavior ⟨a⟩2 ∼ tage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' have described this as a memory phenomenon or measure of history dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' The consensus [14, 15] on the ori- gin of this memory is that coarsening mediated excita- tions are needed to enable the system to overcome local minima that the perturbed system relaxes into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Thus, the long waiting time has been considered a result of slow coarsening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' To study this phenomenon’s structural and dynamical significance computationally, we run a set of simulations using our modified damped model over various ξ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' We consider the theoretical extreme of a perturbation by introducing positional scrambles in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' To do so, we begin with a typical steady-state system and randomly scramble the various 3N positions of the bub- bles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' This scramble randomly assigns a point in hyper- space for the system of soft spheres, providing a ran- dom structural perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' We then continue with the relaxation-coarsening procedure described previously in Section II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' It must be noted here that for the quasi- static case when ξ = 0, we relax the system to its first en- ergy minimum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' mechanical equilibrium) using FIRE [29] instead of using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' For the quasi-static case, we see that the system, upon one (or even multiple) scrambles, returns to the earlier dynamical scaling steady trend (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 4a) immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Indicators like ⟨z⟩ and ⟨a⟩2 show no significant change from steady-state behavior, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' 4 which plots the scrambled (at t = 400) and unscrambled aver- age coordination number as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' Here the scrambled system experiences no barriers to reaching this ’new’ steady state with FIRE traversing the large config- urational distance on a relatively smooth portion of the energy landscape (at higher energies) to find the nearest (primary) minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content=' It may be noted that the scramble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='25 Zc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='00 100 101 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFQT4oBgHgl3EQfwDZl/content/2301.13400v1.pdf'} +page_content='50 300 400 500 600 700 t 8 6 0