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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nAutomatic speech recognition is a fundamental technology used on a daily basis by millions of end-users and businesses.\nApplications include automated phone systems, video captioning and voice assistants providing an intuitive and seemless interface between users and end systems.\nCurrent ASR approaches rely solely on utilizing audio input to produce transcriptions.\nHowever, the wide availability of cameras in smartphones and home devices acts as motivation to build AV-ASR models that rely on and benefit from multimodal input.\n\nTraditional AV-ASR systems focus on tracking the user's facial movements and performing lipreading to augment the auditory inputs \\cite{Potamianos97speakerindependent,mroueh2015deep,Tao2018AligningAF}.\nThe applicability of such models in real world environments is limited, due to the need for accurate audio-video alignment and careful camera placement.\nInstead, we focus on using video to contextualize the auditory input and perform multimodal grounding. For example, a basketball court is more likely to include the term ``lay-up'' whereas an office place is more likely include the term ``lay-off''.\nThis approach can boost ASR performance, while the requirements for video input are kept relaxed \\cite{how2-baseline, av-grounding-asr}. Additionally we consider a multiresolution loss that takes into account transcriptions at the character and subword level. We show that this scheme regularizes our model showing significant improvements over subword models. Multitask learning on multiple levels has been previously explored in the literature, mainly in the context of CTC~\\cite{sanabria2018hierarchical,krishna2018hierarchical,ueno2018acoustic}. A mix of seq2seq and CTC approaches combine word and character level~\\cite{kremer2018inductive,ueno2018acoustic} or utilize explicit phonetic information \\cite{toshniwal2017multitask,sanabria2018hierarchical}.\n\n\n\n\nModern ASR systems rely on end-to-end, alignment free neural architectures, i.e. CTC  \\cite{ctc} or sequence to sequence models \\cite{rnnt,deep-cnn}.\nThe use of attention mechanisms significantly improve results in \\cite{attention-asr} and \\cite{las}.\nRecently, the success of transformer architectures for NLP tasks \\cite{transformer,bert,transformer-xl} has motivated speech researchers to investigate their efficacy in end-to-end ASR \\cite{karita2019comparative}.\nZhou et. al., apply an end-to-end transformer architecture for Mandarin Chinese ASR \\cite{zhou2018syllable}.\nSpeech-Transformer extends the scaled dot-product attention mechanism to $2$D and achieves competitive results for character level recognition  \\cite{speech-transformer, improving-speech-transformer}.\nPham et. al. introduce the idea of stochastically deactivating layers during training to achieve a very deep model \\cite{very-deep-transformer}.\nA major challenge of the transformer architecture is the quadratic memory complexity as a function of the input sequence length.\nMost architectures employ consecutive feature stacking \\cite{very-deep-transformer} or CNN preprocessing \\cite{speech-transformer,karita2019comparative} to downsample input feature vectors.\n\\citet{vggtransformer} use a VGG-based input network to downsample the input sequence and achieve learnable positional embeddings.\n\n\n\\begin{figure*}[htb]\n\t\\centering\n\t\\includegraphics[width=.8\\linewidth]{figs\/schematic_2.pdf}\n\t\\caption{Overall system architecture. A cross-modal scaled dot-product attention layer is used to project the visual data into the audio feature space followed by an additive fusion.}\n\t\\label{fig:architecture}\n\\end{figure*}\n\nMultimodal grounding for ASR systems has been explored in \\cite{how2-baseline}, where a pretrained RNN-based ASR model is finetuned with visual information through Visual Adaptive Training.\nFurthermore, \\citet{av-grounding-asr} use a weakly supervised semantic alignment criterion to improve ASR results when visual information is present.\nMultimodal extensions of the transformer architecture have also been explored. These extensions mainly fuse visual and language modalities in the fields of Multimodal Translation and Image Captioning.\nMost approaches focus on using the scaled dot-product attention layer for multimodal fusion and cross-modal mapping.\n\\citet{afouras2018deep} present a transformer model for AV-ASR targeted for lip-reading in the wild tasks. It uses a self attention block to encode the audio and visual dimension independently. A decoder individually attends to the audio and video modalities producing character transcriptions. In comparison our study uses the video features to provide contextual information to our ASR.\n\\citet{libovicky-etal-2018-input} employ two encoder networks for the textual and visual modalities and propose four methods of using the decoder attention layer for multimodal fusion, with hierarchical fusion yielding the best results.\n\\citet{yu2019multimodal} propose an encoder variant to fuse deep, multi-view image features and use them to produce image captions in the decoder.\n\\citet{le-etal-2019-multimodal} use cascaded multimodal attention layers to fuse visual information and dialog history for a multimodal dialogue system.\n\\citet{cross-modal-transformer} present Multimodal Transformers, relying on  a deep pairwise cascade of cross-modal attention mechanisms to map between modalities for multimodal sentiment analysis.\n\nIn relation to the previous studies, the main contributions of this study are a) a fusion mechanism for audio and visual modalities based on the crossmodal scaled-dot product attention, b) an end to end training procedure for multimodal grounding in ASR and c) the use of a multiresolution training scheme for character and subword level recognition in a seq2seq setting without relying on explicit phonetic information.\nWe evaluate our system in the $300$ hour subset of the How2 database \\cite{sanabria18how2}, achieving relative gains up to 3.76\\% with the addition of visual information. Further we show relative gains of 18\\% with the multiresolution loss. Our results are comparable to state-of-the-art ASR performance on this database. \n\n\n\n\\section{Proposed Method}\nOur transformer architecture uses two transformer encoders to individually process acoustic and visual information (Fig. \\ref{fig:architecture}).\nAudio frames are fed to the first set of encoder layers.\nWe denote the space of the encoded audio features as the audio space $\\mathbb{A}$.\nSimilarly, video features are projected to the video space $\\mathbb{V}$ using the second encoder network.\nFeatures from audio and visual space are passed through a tied feed forward layer that projects them into a common space before passing them to their individual encoder layers respectively. This tied embedding layer is important for fusion as it helps align the semantic audio and video spaces.\nWe then use a cross-modal attention layer that maps projected video representations to the projected audio space (Section \\ref{ssec:cm_attention}).\nThe outputs of this layer are added to the original audio features using a learnable parameter $\\alpha$ to weigh their contributions.\nThe fused features are then fed into the decoder stack followed by dense layers to generate character and subword outputs. For multiresolution predictions (Section \\ref{ssec:multiresolution}), we use a common decoder for both character and subword level predictions, followed by a dense output layer for each prediction. This reduces the model parameters and enhances the regularization effect of multitask learning.\n\n\n\\subsection{Cross-modal Attention}\n\\label{ssec:cm_attention}\nScaled dot-product attention operates by constructing three matrices, $K$, $V$ and $Q$ from sequences of inputs.\n$K$ and $V$ may be considered keys and values in a ``soft'' dictionary, while $Q$ is a query that contextualizes the attention weights.\nThe attention mechanism is described in Eq.~\\ref{eq:att}, where $\\sigma$ denotes the softmax operation.\n\\begin{equation}\n    Y = \\sigma(KQ^T)V\n\\label{eq:att}\n\\end{equation}\n\nThe case where $K$, $V$ and $Q$ are constructed using the same input sequence consists a self-attention mechanism.\nWe are interested in cross-modal attention, where $K$ and $V$ are constructed using inputs from one modality $\\mathbb{M}_1$, video in our case (Fig.~\\ref{fig:architecture}) and $Q$ using another modality $\\mathbb{M}_2$, audio.\nThis configuration as an effective way to map features from $\\mathbb{M}_1$ to $\\mathbb{M}_2$ \\cite{cross-modal-transformer}.\nNote, that such a configuration is used in the decoder layer of the original transformer architecture \\cite{transformer} where targets are attended based on the encoder outputs.\n\\subsection{Multiresolution training}\n\\label{ssec:multiresolution}\nWe propose the use of a multitask training scheme where the model predicts both character and subword level transcriptions.\nWe jointly optimize the model using the weighted sum of character and subword level loss, as in Eq.~\\ref{eq:loss}:\n\\begin{equation}\n    L = \\gamma * L_{subword} + (1 - \\gamma) * L_{character}\n\\label{eq:loss}\n\\end{equation}\n\\noindent where $\\gamma$ is a hyperparameter that controls the importance of each task.\n\nThe intuition for this stems from the reasoning that character and subword level models perform different kinds of mistakes.\nFor character prediction, the model tends to predict words that sound phonetically similar to the ground truths, but are syntactically disjoint with the rest of the sentence.\nSubword prediction, yields more syntactically correct results, but rare words tend to be broken down to more common words that sound similar but are semantically irrelevant.\nFor example, character level prediction may turn ``\\textit{old-fashioned}'' into ``\\textit{old-fashioning}'', while subword level\nturns the sentence ``\\textit{ukuleles are different}'' to ``\\textit{you go release are different}''.\nWhen combining the losses, subword prediction, which shows superior performance is kept as the preliminary output, while the character prediction is used as an auxiliary task for regularization.\n\n\n\n\n\n\\section{Experimental Setup}\nWe conduct our experiments on the How2 instructional videos database \\cite{sanabria18how2}.\nThe dataset consists of $300$ hours of instructional videos from the YouTube platform.\nThese videos depict people showcasing particular skills and have high variation in video\/audio quality, camera angles and duration.\nThe transcriptions are mined from the YouTube subtitles, which contain a mix of automatically generated and human annotated transcriptions.\nAudio is encoded using $40$ mel-filterbank coefficients and $3$ pitch features with a frame size of $10$ ms, yielding $43$-dimensional feature vectors.\nThe final samples are segments of the original videos, obtained using word-level alignment.\nWe follow the video representation of the original paper \\cite{how2-baseline}, where a $3$D ResNeXt-101 architecture, pretrained on action recognition, is used to extract $2048$D features \\cite{hara2018can}.\nVideo features are average pooled over the video frames yielding a single feature vector. For our experiments, we use the train, development and test splits proposed by \\cite{sanabria18how2}, which have sizes $298.2$ hours, $3.2$ hours and $3.7$ hours respectively.\n\nOur model consists of $6$ encoder layers and $4$ decoder layers.\nWe use transformer dimension $480$, intermediate ReLU layer size $1920$ and $0.2$ dropout.\nAll attention layers have $6$ attention heads.\nThe model is trained using Adam optimizer with learning rate $10^{-3}$ and $8000$ warmup steps.\nWe employ label smoothing of $0.1$.\nWe weigh the multitask loss with $\\gamma=0.5$ which gives the best performance.\nA coarse search was performed for tuning all hyperparameters over the development set.\nFor character-level prediction, we extract $41$ graphemes from the transcripts.\nFor subword-level prediction, we train a SentencePiece tokenizer \\cite{kudo2018sentencepiece} over the train set transcriptions using byte-pair encoding and vocabulary size $1200$.\nFor decoding we use beam search with beam size $5$ and length normalization parameter $0.7$. \nWe train models for up to $200$ epochs and the model achieving the best loss is selected using early stopping. Any tuning of the original architecture is performed on the development split.\nNo language model or ensemble decoding is used in the output.\n\\section{Results and Discussion}\n\\begin{table}[tb]\n\t\n\t\\begin{center}\n\t\t\\begin{tabular}{|c c c |}\n\t\t\t\\hline\n\t\t\tInput handling & Recognition level & WER  \\\\ [0.5ex]\n\t\t\t\\hline\\hline\n\t\t\tFiltering      & Character & $33.0$ \\\\\n\t\t\t\\hline\n\t\t\tFiltering      & Subword   & $29.7$ \\\\\n\t\t\t\\hline\n\t\t\tChunking & Character & $31.3$  \\\\\n\t\t\t\\hline\n\t\t\tChunking & Subword   & $29.9$ \\\\\n\t\t\t\\hline\n\t\t\tStacking & Character & $28.3$  \\\\\n\t\t\t\\hline\n\t\t\tStacking & Subword   & $26.1$ \\\\\n\t\t\t\\hline\n\t\t\tStacking & MR & $\\mathbf{21.3}$ \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Results for different methods of input filtering for different prediction resolutions. \\emph{MR} stands for multiresolution.}\n\t\\label{tab:subword-character}\n\\end{table}\n\nOne of the challenges using scaled dot-product attention is the quadratic increase of layerwise memory complexity as a function of the input sequence length.\nThis issue is particularly prevalent in ASR tasks, with large input sequences. We explore three simple approaches to work around this limitation.\nFirst, we filter out large input sequences ($x>15s$), leading to loss of $100$ hours of data.\nSecond we, chunk the input samples to smaller sequences, using forced-alignment with a conventional DNN-HMM model to find pauses to split the input and the transcriptions.\nFinally, we stack $4$ consecutive input frames into a single feature vector, thus reducing the input length by $4$. Note that this only reshapes the input data as the dimension of our input is increased by the stacking process\n\\footnote{We tried to use the convolutional architecture from \\cite{vggtransformer}, but it failed to converge in our experiments, possibly due to lack of data}.\nResults for the downsampling techniques for character and subword level predictions are summarized in Table~\\ref{tab:subword-character}.\nWe observe that subword-level model performs better than the character level (upto 10\\% relative) in all settings.\nThis can be attributed to the smaller number of decoding steps needed for the subword model, where error accumulation is smaller.\nFurthermore, we see that the naive filtering of large sequences yields to underperforming systems due to the large data loss. Additionally, we see that frame stacking has superior performance to chunking.\nThis is not surprising as splitting the input samples to smaller chunks leads to the loss of contextual information which is preserved with frame stacking.\nWe evaluate the proposed multiresolution training technique with the frame stacking technique, observing a significant improvement(18.3\\%) in the final WER.\nWe thus observe that predicting finer resolutions as an auxiliary task can be used as an effective means of regularization for this sequence to sequence speech recognition task. Furthermore, we have empirically observed that when training in multiple resolutions, models can converge around 50\\% faster than single resolution models.\n\n\\begin{table}[tb]\n\t\\small\n\t\\begin{center}\n\t\t\\begin{tabular}{|c c c c|}\n\t\t\t\\hline\n\t\t\t&  &   & $\\Uparrow$ \\\\\n\t\t\tFeatures       &  Level     &   WER   & over audio \\\\\n\t\t\t[0.5ex]\n\t\t\t\\hline\\hline\n\t\t\tAudio & Subword & $26.1$ & -\\\\\n\t\t\t\\hline\n\t\t\tAudio + ResNeXt & Subword &  $25.0$ & $3.45\\%$ \\\\\n\t\t\n\t\t\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\tAudio & MR & $\\mathbf{21.3}$ & - \\\\\n\t\t\t\\hline\n\t\t\tAudio + ResNeXt & MR & $20.5$ & $3.76\\%$ \\\\\n\t\t\n\t\t\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\tAudio (B) & Subword & $\\mathbf{19.2}$ & - \\\\\n\t\t\t\\hline\n\t\t\tAudio + ResNext (B) & Subword & $\\mathbf{18.4}$ & $3.13\\%$ \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Comparison of audio only ASR models versus AVASR models with ResNeXt image features. \\emph{MR} stands for multiresolution. \\emph{(B)} shows the results for the LAS model \\cite{how2-baseline}}\n\t\\vspace{-0.6cm}\n\t\\label{tab:multimodal}\n\t\n\\end{table}\nNext, we evaluate relative performance improvement obtained from utilizing the visual features (Table~\\ref{tab:multimodal}).\nWe observe that incorporating visual information improves ASR results. Our AV-ASR system yields gains $>3\\%$ over audio only models for both subword and multiresolution predictions. \nFinally, we observe that while the Listen, Attend and Spell-based architecture of \\cite{how2-baseline} is slightly stronger than the transformer model, the gains from adding visual information is consistent across models. It is important to note that our models are trained end-to-end with both audio and video features.\n\n\n\nAn important question for real-world deployment of multimodal ASR systems is their performance when the visual modality is absent.\nIdeally, a robust system satisfactorily performs when the user's camera is off or in low light conditions.\nWe evaluate our AV-ASR systems in the absence of visual data with the following experiments - a) replace visual feature vectors by zeros b) initialize visual features with gaussian noise with standard deviation $0.2$ c) tweak the value $\\alpha$ to 0 on inference, gating the visual features completely. Table~\\ref{tab:audio-only} shows the results for the different experiments. Results indicate gating visual inputs works better than zeroing them out. Adding a gaussian noise performs best which again indicates the limited availability of data. Overall, in the absence of visual information, without retraining, the AV-ASR model relatively worsens by 6\\% compared to audio only models. \n\n\\begin{table}[tb]\n\n    \\begin{center}\n        \\begin{tabular}{|c c |}\n        \\hline\n        Missing input handling & WER  \\\\ [0.5ex]\n        \\hline\\hline\n        Zeros &  23.1 \\\\\n        \\hline\n        Gaussian Noise $\\sigma$=0.2 & 22.6 \\\\\n        \\hline\n        Gating visual input $\\alpha$=0 & 22.8 \\\\\n        \\hline\n       \\end{tabular}\n   \\end{center}\n\t\\vspace{-0.3cm}\n   \\caption{Experimental evaluation of AV-ASR model for handling missing visual input. Here $\\sigma$ denotes the standard deviation of the noise}\n   \\label{tab:audio-only}\n\\end{table}\n\n\\section{Conclusions}\nThis paper explores the applicability of the transformer architecture for multimodal grounding in ASR. Our proposed framework uses a crossmodal dot-product attention to map visual features to audio feature space. Audio and visual features are then combined with a scalar additive fusion and used to predict character as well as subword transcriptions. We employ a novel multitask loss that combines the subword level and character losses. Results on the How2 database show that a) multiresolution losses regularizes our model producing significant gains in WER over character level and subword level losses individually b) Adding visual information results in relative gains of 3.76\\% over audio model's results validating our model.\n\nDue to large memory requirements of the attention mechanism, we apply aggressive preprocessing to shorten the input sequences, which may hurt model performance. In the future, we plan to alleviate this by incorporating ideas from sparse transformer variants \\cite{reformer,sparsetransformer}.\nFurthermore, we will experiment with more ellaborate, attention-based fusion mechanisms.\nFinally, we will evaluate the multiresolution loss on larger datasets to analyze it's regularizing effects.\n\\bibliographystyle{acl_natbib}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and notations}\n\nAn interconnection topology can be represented by a graph $G=(V,E)$, where $V$ denotes the processors and $E$ the communication links.\nThe hypercube $Q_n$ is a popular interconnection network because of its structural properties.\\\\\n\\indent The Fibonacci cube was introduced in \\cite{hsu93} as a new interconnection network.\nThis graph is an isometric subgraph of the hypercube which is inspired in the Fibonacci numbers. It has attractive recurrent structures such as\nits decomposition into two subgraphs which are also Fibonacci cubes by themselves.  Structural properties of these graphs were more extensively \nstudied afterwards. See \\cite{survey} for a survey. \\\\\n\\indent Lucas cubes, introduced in \\cite{mupe2001}, have attracted the attention as well due to the fact that these cubes are \nclosely related to the Fibonacci cubes. They have also been widely studied\n\\cite{dedo2002, ra2013, ca2011, klmope2011, camo2012, km2012}. \\\\\n\n\\indent We will next define some concepts needed in this paper. \nLet $G$ be a connected graph. The \\emph{open neighbourhood} of a vertex $u$ is $N_G(u)$ the set of vertices adjacent to $u$. The \\emph{closed neighbourhood} of $u$ is $N_G[u]=N_G(u)\\cup\\{u\\}$. The \\emph{distance} between two vertices noted $d_G(x,y)$ is the length of a shortest path between $x$ and $y$. We have thus $N_G[u]=\\{v \\in V(G);d_G(u,v)\\leq1\\}$. We will use the notations $d(x,y)$ and $N[u]$ when the graph is unambiguous.\\\\\nA \\emph{dominating set} $D$ of $G$ is a set of vertices such that every vertex of $G$ belongs to the closed neighbourhood of at least one vertex of $D$. \nIn \\cite{Biggs}, Biggs initiated the study of perfect codes in graphs a generalization of classical 1-error perfect correcting codes. A \\emph{code} $C$ in $G$ is a set of vertices $C$ such that for every pair of distinct vertices $c,c'$ of $C$ we have $N_G[c]\\cap N_G[c']=\\emptyset$ or equivalently such that $d_G(c,c')\\geq3$.\n\nA \\emph{perfect code} of a graph $G$ is both a dominating set and a code. It is thus a set of vertices $C$  such that every vertex of $G$ belongs to the closed neighbourhood of exactly one vertex of $C$. A perfect code is also known as an efficient dominating set. The existence or non-existence of perfect codes have been considered for many graphs. See the introduction of \\cite{aabfk2016} for some references.\n\nThe vertex set of the \\emph{$n$-cube} $Q_n$ is the set $\\mathbb{B}_n$ of  binary strings of length $n$, two vertices being adjacent if they differ in precisely one position. Classical 1-error correcting codes and perfect codes are codes and perfect codes in the graph $Q_n$. The \\emph{weight} of a binary string is the number of 1's.\nThe concatenation of strings $\\bm{x}$ and $\\bm{y}$ is denoted $\\bm{x}||\\bm{y}$ or just $\\bm{x}\\bm{y}$ when there is no ambiguity. A string $\\bm{f}$ is a \\emph{substring} of a string $\\bm{s}$ if there exist strings $\\bm{x}$ and $\\bm{y}$, may be empty, such that $\\bm{s}=\\bm{x}\\bm{f}\\bm{y}$.\n\nA {\\em Fibonacci string} of length $n$ is a binary string $\\bm{b}=b_1\\ldots b_n$ with $b_i\\cdot b_{i+1}=0$ for $1\\leq i<n$. In other words a Fibonacci string is a binary string without $11$ as substring.\nThe {\\em Fibonacci cube} $\\Gamma_n$ ($n\\geq 1$) is the subgraph of $Q_n$ induced by the Fibonacci strings of length $n$.\nAdjacent vertices in $\\Gamma_n$ differ in one bit. Because of the empty string, $\\Gamma_0 = K_1$. \n\nA Fibonacci string  of length $n$ is a \\textit{Lucas string} if $b_1 \\, \\cdotp b_n \\neq 1$. \nThat is, a Lucas string has no two consecutive 1's including the first and the last elements of the string. \nThe \\textit{Lucas cube} $\\Lambda_n$ is the subgraph of $Q_n$ induced by the Lucas strings of length $n$. \nWe have $\\Lambda_0 = \\Lambda_1 = K_1$.\\\\\nLet $\\Fib_n$ and $\\Luc_n$ be the set of strings of Fibonacci strings and Lucas strings of length $n$.\\\\\nBy  $\\Gamma_{n,k}$ and $\\Lambda_{n,k}$ we denote the vertices of of weight $k$ in respectively $\\Gamma_n$ and $\\Lambda_n$\\\\\n\nSince $$\\Luc_n=\\{0\\bm{s}; \\bm{s}\\in \\Fib_{n-1}\\}\\cup\\{10\\bm{s}0; \\bm{s}\\in \\Fib_{n-3}\\} $$ and $$|\\Gamma_{n,k}|=\\binom{n-k+1}{k}$$\nit is immediate  to derive  the following classical result.\n\\begin{proposition}\\label{orderrr}\nLet $n \\ge 1$. The number of vertices of weight $k \\leq n$ in $\\Lambda_n$ is  \n$$|\\Lambda_{n,k}|= \\binom{n-k}{k}+ \\binom{n-k-1}{k-1}.$$\n\\end{proposition}\n\\begin{figure}\n  \\centering\n\\setlength{\\unitlength}{1 mm}\n\\begin{picture}(160, 30)\n\n\\newsavebox{\\gtwo}\n\\savebox{\\gtwo}\n  (30,30)[bl]\n  \n  \n\\put(10,10){\\circle*{2}}\n\\put(10,20){\\circle*{2}}\n\\put(10,30){\\circle*{2}}\n\n\\put(10,10){\\line(0,1){10}}\n\\put(10,20){\\line(0,1){10}}\n\n\\put(5,10){$01$}\n\\put(5,20){$00$}\n\\put(5,30){$10$}\n\n\n  \n\n \n \\newsavebox{\\lthree}\n\\savebox{\\lthree}\n  (40,30)[bl]\n  \n  \n\\put(10,10){\\circle*{2}}\n\\put(10,20){\\circle*{2}}\n\\put(10,30){\\circle*{2}}\n\n\\put(20,20){\\circle*{2}}\n\n\\put(10,20){\\line(1,0){10}}\n\\put(10,10){\\line(0,1){10}}\n\\put(10,20){\\line(0,1){10}} \n\n\\put(3,10){$001$}\n\\put(3,20){$000$}\n\\put(3,30){$010$}\n\\put(21,20){$100$}\n\n \n \n\\newsavebox{\\lfour}\n\\savebox{\\lfour}\n  (40,30)[bl]\n  \n\n\n\\put(10,20){\\circle*{2}}\n\\put(10,30){\\circle*{2}}\n\\put(20,10){\\circle*{2}}\n\\put(20,20){\\circle*{2}}\n\\put(20,30){\\circle*{2}}\n\\put(30,10){\\circle*{2}}\n\\put(30,20){\\circle*{2}}\n\n\n\\put(10,20){\\line(1,0){10}}\n\\put(10,30){\\line(1,0){10}}\n\n\\put(10,20){\\line(0,1){10}}\n\n\\put(20,10){\\line(1,0){10}}\n\\put(20,20){\\line(1,0){10}}\n\n\\put(20,10){\\line(0,1){10}}\n\\put(20,20){\\line(0,1){10}}\n\\put(30,10){\\line(0,1){10}} \n\n\\put(20,12){$0001$}\n\\put(20,22){$0000$}\n\\put(20,32){$0010$}\n\\put(31,20){$0100$}\n\\put(1,20){$1000$}\n\\put(1,30){$1010$}\n\\put(31,10){$0101$}\n \n  \\newsavebox{\\gggfour}\n\\savebox{\\gggfour}\n  (40,40)[bl]\n  \n \n\\put(10,10){\\circle*{2}}\n\\put(10,20){\\circle*{2}}\n\\put(10,30){\\circle*{2}}\n\\put(20,10){\\circle*{2}}\n\\put(20,20){\\circle*{2}}\n\\put(20,30){\\circle*{2}}\n\\put(30,10){\\circle*{2}}\n\\put(30,20){\\circle*{2}}\n\n\\put(10,10){\\line(1,0){10}}\n\\put(20,10){\\line(1,0){10}}\n\\put(10,20){\\line(1,0){10}}\n\\put(20,20){\\line(1,0){10}}\n\\put(10,30){\\line(1,0){10}}\n\\put(10,10){\\line(0,1){10}}\n\\put(10,20){\\line(0,1){10}}\n\\put(20,10){\\line(0,1){10}}\n\\put(20,20){\\line(0,1){10}}\n\\put(30,10){\\line(0,1){10}}\n  \n\\newsavebox{\\lfive}\n\\savebox{\\lfive}\n  (40,40)[bl]\n  \n  \n\\put(0,0){\\usebox{\\gggfour}}\n\n\\put(24,24){\\circle*{2}}\n\\put(24,34){\\circle*{2}}\n\\put(34,24){\\circle*{2}}\n\\put(24,24){\\line(1,0){10}}\n\\put(24,24){\\line(0,1){10}}\n\n\\put(20,20){\\line(1,1){4}}\n\\put(20,30){\\line(1,1){4}}\n\\put(30,20){\\line(1,1){4}} \n\\put(14,6){$00001$}\n\\put(14,16){$00000$}\n\\put(14,26){$00010$}\n\\put(25,16){$00100$}\n\\put(-1,10){$01001$}\n\\put(25,25){$10000$}\n\\put(-1,20){$01000$}\n\\put(-1,30){$01010$}\n\\put(25,06){$00101$}\n\\put(35,20){$10100$}\n\\put(25,35){$10010$}\n\n \n  \n\\put(0,-5){\\usebox{\\gtwo}}  \n\\put(25,-5){\\usebox{\\lthree}}  \n\\put(63,-5){\\usebox{\\lfour}}\n\\put(120,-5){\\usebox{\\lfive}}\n\\end{picture}\n\\caption{$\\Gamma_2=\\Lambda_2$,  $\\Lambda_3$, $\\Lambda_4$ and $\\Lambda_5$}\\label{F1}\n\n\\end{figure}\n\n\n\n\nIt will be convenient to consider the binary strings of length $n$ as vectors   of $\\mathbb{F}^n$ the vector space of dimension $n$ over the field $\\mathbb{F}=\\mathbb{Z}_2$ thus to associate to a string $x_1 x_2 \\dots x_n$ the vector $\\theta(x_1 x_2 \\dots x_n)=(x_1,x_2,\\ldots,x_n)$.\nThe \\emph{Hamming distance} between two vectors $\\bm{x},\\bm{y} \\in \\mathbb{F}^n$, $d(\\bm{x},\\bm{y})$ is the number of coordinates in which they differ.\nBy the correspondence $\\theta$ we can define the binary sum $\\bm{x}+\\bm{y}$ and the Hamming distance $d(\\bm{x},\\bm{y})$ of  strings in $\\mathbb{B}_n$. Note that the Hamming distance is the usual graph distance in $Q_n$.\n\nWe will first recall some basic results about perfect codes in $Q_n$.\nSince $Q_n$ is a regular graph of degree $n$ the existence of a perfect code of cardinality $|C|$ implies $|C|(n+1)=2^n$ thus a necessary condition of existence is that $n+1$ is a power of 2 thus that $n=2^p-1$ for some integer $p$.\n\nFor any integer $p$ Hamming \\cite{Ha1950} constructed, a linear subspace of $\\mathbb{F}^{2^p-1}$ which is a perfect code. It is easy to prove that all linear perfect codes are Hamming codes. Notice that $1^n$ belongs to the Hamming code of length $n$. \\\\\nIn 1961  Vasilev \\cite{{Va1962}}, and later many authors, see \\cite{Co1,Sol2008} for a survey, constructed perfect codes which are not linear codes.\\\\\n\nIn a recent work  \\cite{aabfk2016} Ashrafi and his co-authors proved the non-existence of perfect codes in  $\\Gamma_n$ for $n\\geq 4$. As an open problem the authors suggest to consider the existence of perfect codes in generalizations of Fibonacci cubes.\nThe most complete generalization  proposed in \\cite{ikr} is, for a given string $\\bm{f}$, to consider $\\Gamma_n(\\bm{f})$ the subgraph of $Q_n$ induced by strings that do not contain $\\bm{f}$ as substring. Since Fibonacci cubes are $\\Gamma_n(11)$ the most immediate generalization \\cite{hsuliu,Zag} is to consider  $\\Gamma_n({1^s})$ for a given integer $s$. In \\cite {mo2018} we proved the existence of a perfect code in $\\Gamma_n(1^s)$ for $n=2^p-1$ and $s \\geq 3.2^{p-2}$ for any integer $p\\geq 2$.\\\\\n\nIn the next section we will prove the main result of this note.\n\n\\begin{theorem}\\label{thmain}\nThe Lucas cube $\\Lambda_n , n\\geq 0$, admits a perfect code if and only if $n\\leq 3$ .\n \\end{theorem}\n\n\\section{Perfect codes in Lucas cube}\n\nIt can be easily checked by hand that  $\\{0^n\\}$ is a perfect code of $\\Lambda_n$ for $n\\leq 3$ and that $\\Lambda_4$ or $\\Lambda_5$ does not contain a perfect code (Figure~\\ref{F1}).\\\\\n Assume thus $n\\geq 6$.\\\\ \n\nNote first that from Proposition \\ref{orderrr} we have $$|\\Lambda_{n,2}|=\\frac{n(n-3)}{2} \\text{ and }|\\Lambda_{n,3}|=\\frac{n(n-4)(n-5)}{6}$$.\\\\\nTherefore $\\Lambda_{n,2}$ and $\\Lambda_{n,3}$ are none empty.\\\\\nLet  $\\Lambda_{n,k}^1$ be the vertices of $\\Lambda_{n,k}$ that start with $1$. Since  $\\Luc_n=\\{0\\bm{s}; \\bm{s}\\in \\Fib_{n-1}\\}\\cup\\{10\\bm{s}0; \\bm{s}\\in \\Fib_{n-3}\\} $ the number of vertices in $\\Lambda_{n,k}^1$ is $$|\\Lambda_{n,k}^1|=|\\Gamma_{n-3,k-1}|=\\binom{n-1-k}{k-1}$$.\n\n\\begin{lemma}\nIf $n\\geq 6$ and $C$ is a perfect code of $\\Lambda_n$ then $0^n\\in C$.\n\\end{lemma}\n\\begin{proof}\n\nSuppose on the contrary that $0^n\\notin C$. Since $0^n$ must be dominated there exists a vertex in $\\Lambda_{n,1}\\cap C$. This vertex is unique and because of the circular symmetry of $\\Lambda_n$ we can assume $10^{n-1}\\in C$. \n\nSince $0^n\\notin C$ the other vertices of $\\Lambda_{n,1}$ must be dominated by vertices in  $\\Lambda_{n,2}$. But a vertex in  $\\Lambda_{n,2}$ has precisely two neighbors in $\\Lambda_{n,1}$ thus $n$ must be odd and $$|\\Lambda_{n,2}\\cap C|=\\frac{n-1}{2}.$$\\\\\nThe unique vertex $10^{n-1}$ in $\\Lambda_{n,1}\\cap C$ has exactly  $n-3$ neighbors in  $\\Lambda_{n,2}$. Let $D$ be the vertices of $\\Lambda_{n,2}$ not in $C$ and not dominated by $10^{n-1}$. Vertices in  $D$ must be dominated by vertices in $\\Lambda_{n,3}\\cap C$. Each vertex of $\\Lambda_{n,3}\\cap C$ has exactly  exacty three neighbors  in  $\\Lambda_{n,2}$. \nThus $3$ divides the number of vertices in $D$. This number is  $$|D|=|\\Lambda_{n,2}|-(n-3)-\\frac{n-1}{2}=\\frac{n^2-6n+7}{2}.$$\n This is not possible since there exists no odd integer $n$ such that $6$ divides $n^2+1$. Indeed since\\textit{} $n$ is odd,  $6$ does not divide $n$ thus divides $(n+1)(n-1)= n^2-1$ or $(n+2)(n-2)=n^2-4$ or $(n+3)(n-3)=n^2-9$ thus cannot divide $n^2+1$.\n\\end{proof}\n\\qed  \\\\\nLet $n\\geq 6$ and $C$  be a perfect code. Since  $0^n\\in C$   all vertices of  $\\Lambda_{n,1}$ are dominated by  $0^n$ and thus $\\Lambda_{n,2}\\cap C=\\Lambda_{n,1}\\cap C=\\emptyset$.\nConsequently, each vertex of $\\Lambda_{n,2}$ must be  dominated by a vertex in $\\Lambda_{n,3}$.\nSince each vertex in $\\Lambda_{n,3}$ has precisely three neighbors in  $\\Lambda_{n,2}$ we obtain that $$|\\Lambda_{n,3}\\cap C|=\\frac{|\\Lambda_{n,2}|}{3}.$$\nThis number must be an integer thus $3$ divides $|\\Lambda_{n,2}|=\\frac{n(n-3)}{2}$ and therefore $3$ divides $n(n-3)$. This is only possible if $n$ is a multiple of $3$.\\\\\nEach vertex of $\\Lambda_{n,2}^1$ must be  dominated by a vertex in $\\Lambda_{n,3}^1$. Furthermore a vertex in  $\\Lambda_{n,3}^1$  has precisely two neighbors in  $\\Lambda_{n,2}^1$.  Therefore $|\\Lambda_{n,2}^1|= n-3$ must be even and thus $n=6p+3$ for some integer $p\\geq1$.\\\\\nLet $E$ be  the  set of vertices of  $\\Lambda_{n,3}$ not in $C$. Vertices in $E$ must be dominated by a vertex in $\\Lambda_{n,4}$. Furthermore each vertex in $\\Lambda_{n,4}$ has precisely four  neighbors in  $\\Lambda_{n,3}$.\\\\ Therefore $4$ divides $|E|$ with $$|E|=|\\Lambda_{n,3}|-|\\Lambda_{n,3}\\cap C|= \\frac{n(n-4)(n-5)}{6}-\\frac{n(n-3)}{6}=\\frac{n(n^2-10n+23)}{6}. $$\nReplacing $n$ by $6p+3$ we obtain that $4$ divides the odd number $(2p+1)(18p^2 -12p +1)$. This contradiction prove the Theorem.\n\\qed\n \n\\section{Perfect codes in generalized Lucas cube}\nThe analogous of the generalisation of Fibonacci cube $\\Gamma_n(1^s)$ for Lucas cube  is the family $\\Lambda_n(1^s)$ of subgraphs of $Q_n$ induced by strings without $1^s$ as a substring in a circular manner where $s\\geq 2$ is a given integer.\nMore formally \\cite{ikr2012} for any binary strings $b_1b_2\\dots b_n$ and each $1 \\leq i \\leq n $, call $b_i b_{i+1}\\dots b_n b_1 \\dots b_{i-1}$ the $i$\\emph{-th circulation} of $b_1b_2\\dots b_n$. The \\emph{generalized Lucas cube }$\\Lambda_n(1^s)$ is the subgraph of $Q_n$ induced by strings without a circulation containing $1^s$ as a substring.\\\\\nIn \\cite{mo2018} the existence of a perfect code in $\\Gamma_n(1^s)$ is proved for $n=2^p-1$ and $s \\geq 3.2^{p-2}$ for any integer $p\\geq 2$.\\\\\nThe strategy used in this construction is to build a perfect code $C$ in $Q_n$ such that no vertex of $C$ contains $1^s$ as substring. The set $C$ is also a perfect code in $\\Gamma_n(1^s)$ since each vertex of $\\Gamma_n(1^s)$ belongs to the unique closed neighbourhood in $Q_n$ thus in $\\Gamma_n(1^s)$ of a vertex in $C$. Because of the following proposition we cannot use the same idea  for $\\Lambda_n(1^s)$ and $s\\leq n-1$.\n\\begin{proposition}\nLet $n$ an integer and $2\\leq s\\leq n-1$. There exist no perfect code $C$ in $Q_n$ such that the vertices of $C$ are without a circulation containing $1^s$ as a substring.\n\\end{proposition}\n\\begin {proof}\n\nLet $C$ be a such a perfect code in $Q_n$ then $1^n\\notin C$. Thus $1^n$ must be neighbour of a vertex $c$ in $C$. Since $c=1^i01^{n-1-i}$ for some integer $i$ the ${i+1}$th-circulation of $c$ is $1^{n-1}0$. \n\\end {proof}\nWe can complete this proposition by the two following  results\n\\begin{proposition}\nLet $p\\geq2$ and $n=2^p-1$ then there exists a perfect code in $\\Lambda_n(1^n)$ of order $|C|=\\frac{2^n}{n+1}$.\n\\end{proposition}\n\\begin{proof} Let $D$ be a Hamming code of length $n$ and $C=\\{d+(0^{n-1}1);d\\in D\\}$. Since $1^n\\in D$ $C$ is a perfect code of $Q_n$ such that $1^n\\notin C$. Since $\\Lambda_n(1^n)$ is obtained from $Q_n$ by the deletion of $1^n$ every vertex of  $\\Lambda_n(1^n)$ is in the closed neighbourhood of exactly one vertex of $C$.\n\\end{proof}\n\\begin{proposition}\nLet $p\\geq2$ and $n=2^p-1$ then there exists a perfect code in $\\Lambda_n(1^{n-1})$ and in $\\Lambda_n(1^{n-2})$ of order $|C|=\\frac{2^n}{n+1}-1$.\n\\end{proposition}\n\\begin{proof} Let $D$ be a Hamming code of length $n$. Then $D$ is a perfect code of $Q_n$ such that $1^n\\in D$. \nSince $\\Lambda_n(1^{n-1})$ is obtained from $Q_n$ by the deletion of the closed neighbourhood of $1^n$ every vertex of  $\\Lambda_n(1^{n-1})$ is in the closed neighbourhood of exactly one vertex of $C=D-\\{1^n\\}$.\n Furthermore since $1^n\\in D$ there is no vertex of weight $n-2$ in $D$. Let $u$ be a vertex of  $\\Lambda_n(1^{n-2})$ and $f(u)$ be the vertex in $D$ such that $u\\in N_{Q^n}[u]$. Since there is no vertex in $D$ with weight $n-1$ or $n-2$ there is no circulation of $f(u)$ containing $1^{n-2}$ as a substring. \nTherefore $f(u)$ is a vertex of $\\Lambda_n(1^{n-2})$ and $u\\in N_{\\Lambda_n(1^{n-2})}[f(u)]$.\nSince a code in $Q_n$ is a code in each of its subgraph $C$ is a perfect code of $\\Lambda_n(1^{n-2})$.\n\\end{proof}\\textit{}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nThe future LISA space mission \\cite{ESA12,NASA11} will build upon the success\nof current ground-based detectors by observing new gravitational wave sources in the milli-Hertz band\n\\cite{AmarETC13,AmarETC17,BerrETC19}.  Among these new sources are extreme-mass-ratio inspirals (EMRIs): binaries\ncomposed of a stellar-mass compact object ($\\mu \\sim 10 M_\\odot$) in bound orbit about a massive black hole\n($M\\sim 10^6 M_\\odot$).  With their small mass ratios ($\\eps\\equiv\\mu\/M \\ll 1$), EMRIs evolve adiabatically due\nto orbit-averaged gravitational wave fluxes, with the smaller secondary body completing $\\sim 10^5$ orbits around the\nmore massive primary as the binary emits gravitational waves visible to low-frequency detectors.  As a result of their\nlong durations, EMRI signals are expected to have cumulative signal-to-noise ratios (SNRs) of several tens to several\nhundreds, allowing high-precision measurements that exceed the capabilities of current ground-based gravitational\nwave observatories \\cite{BerrETC19,BakeETC19}.  Such high SNR measurements will still require accurate waveform\nmodels to assist in detecting events and in measuring physical source parameters \\cite{BabaETC17}.\n\nEMRIs are naturally modeled within the framework of black hole perturbation theory (BHPT), in which the small compact\nobject is treated as a perturbing body in the stationary background spacetime associated with the primary black hole\n\\cite{Mino03,PoisPounVega11}. On the orbital timescale $T_\\mathrm{orb}\\sim M$, the dynamics of the small body closely\napproximates a geodesic with a trio of fundamental frequencies (the radial frequency $\\O_r$, polar frequency\n$\\O_\\th$, and azimuthal frequency $\\O_\\vp$) \\cite{Schm02,DrasFlanHugh05,FujiHiki09}.  However, as the system evolves\nthe small body is gradually pushed away from this geodesic as it interacts with its own gravitational perturbation.\nThis behavior is typically described in terms of a local gravitational self-force (GSF) \\cite{MinoSasaTana97,QuinWald97}\nthat acts on the secondary and, at leading perturbative-order, makes an $O(\\eps)$ correction to its motion.  The\ndissipative piece of the GSF drives the adiabatic inspiral of the secondary, while the conservative piece provides\nnonsecular perturbations to the motion.  The secular effect of the averaged dissipative GSF dominates the phase\nevolution of the orbit, which accumulates like $\\sim \\eps^{-1}$ at leading adiabatic order.  This defines the inspiral\nor radiation timescale $T_\\mathrm{rr}\\sim M\\eps^{-1}$, over which the orbital frequencies undergo order-unity\nfractional changes. Furthermore, at a more subtle level, perturbations due to the oscillatory part of the dissipative\nand conservative GSF produce small shifts in the orbital frequencies that affect the orbit and cumulative\nphase at a level $O(\\eps)$ relative to the leading order adiabatic inspiral \\cite{HindFlan08}.  This correction is\nthe post-1 adiabatic order effect.\n\nIn almost all astrophysically-relevant EMRIs, the evolution of the orbital frequencies due to the GSF will cause\nthese systems to pass through one or more consequential orbital resonances at some point in their observed inspirals\n\\cite{RuanHugh14,BerrETC16}.  An orbital resonance occurs when at least two frequencies of orbital motion, $\\O_1$\nand $\\O_2$, form a rational ratio, i.e., $\\O_2\/\\O_1 =\\b_2\/\\b_1$ with coprime $\\b_1,\\b_2\\in \\mathbb{Z}$.  The smaller\nthat the integers $\\beta_1$ and $\\beta_2$ are, the stronger the resonance.  In the solar system, orbital resonances\ncommonly occur among satellites sharing the same primary, such as the 2:3 resonance of Neptune and Pluto, and the\n1:2 resonance of the Galilean satellites Io and Europa.\n\nFor EMRIs, resonances can form between any two of the three frequencies of the smaller body's orbital motion, with\ndifferent resonant combinations leading to different physical effects.\\footnote{See\nRefs.~\\cite{GairYuneBend12,Vand14a,BaraPoun19} for comprehensive lists of research focused on orbital resonances in\nKerr spacetime, and \\cite{GairYuneBend12,LukeWitz21} for more general discussions of EMRI resonances.}  For instance,\n$r\\vp$ \\cite{Vand14b} and\n$\\th\\vp$ resonances \\cite{Hira11} can lead to anisotropic radiation of gravitational waves, resulting in resonant\n`kicks' to the velocity of an EMRI's center-of-mass.  Such effects are expected to contribute to an EMRI's phase\nevolution and waveform at $O(\\eps^{3\/2})$ relative to adiabatic order (i.e., post-$\\tfrac{3}{2}$ adiabatic order)\n\\cite{Vand14b}.  Because LISA waveforms require a phase accuracy of $\\sim 0.1$ radians, $r\\vp$- and\n$\\th\\vp$ resonances are presumably safe to neglect at present in EMRI models.  On the other hand, $r\\th$ resonances,\nwhich only arise in EMRIs with Kerr primaries, will enhance or diminish the time-averaged gravitational wave flux,\nthereby influencing the evolution of the frequencies $\\O_r$, $\\O_\\th$, and $\\O_\\vp$ (and similarly the orbital\nenergy, angular momentum, and Carter constant) \\cite{FlanHind12,FlanHughRuan14,BerrETC16}.  The strongest\nresonances will have frequency ratios such as 1:3, 1:2, or 2:3 \\cite{RuanHugh14,MihaGair17} and will ordinarily\npersist for a resonant period $T_\\text{res}\\sim M\\eps^{-1\/2}$ \\cite{FlanHind12}.  Because these orbital resonances\nare expected to be transient,\\footnote{It may be possible for certain EMRIs to be caught in a sustained\n$r\\th$ resonance, but a system must meet stringent (if even possible) conditions for this to occur \\cite{Vand14a}. }\ntheir effect is at post-$\\tfrac{1}{2}$ adiabatic order \\cite{FlanHind12}.\\footnote{While not the focus of this\nwork, tidal resonances, which may occur for EMRIs that are perturbed by one or more external bodies, can experience\nsimilar post-$\\tfrac{1}{2}$ adiabatic corrections to the phase evolution \\cite{YangCasa17,BongYangHugh19, GuptETC21}.}\n\nIncorporating this new resonant timescale into evolutionary models poses difficulties with, for example, near-identity\ntransformations \\cite{VandWarb18} and multiscale expansions \\cite{BaraPoun19,LukeWitz21}.  In effect, on a timescale\njust larger than $T_\\text{res}$, an EMRI's orbital parameters appear to experience $O(\\eps^{1\/2})$ shifts or jumps in\ntheir values, as shown in \\cite{FlanHind12,BerrETC16}.  These jumps, which are sensitive to the orbital phase of the\nEMRI as it enters resonance \\cite{FlanHughRuan14}, lead to an overall $O(\\eps^{-1\/2})$ shift in the cumulative phase\nof the system.  Failing to account for these resonant phase shifts may\nonly slightly degrade EMRI detection rates \\cite{BerrETC16}, but it will introduce significant systematic biases to\nEMRI parameter estimation that will dominate over standard statistical errors \\cite{SperGair21}.\nBecause nearly all EMRIs will encounter either a 1:3, 1:2, or 2:3 $r\\th$ resonance as they emit gravitational wave\nsignals in the LISA passband \\cite{RuanHugh14}, properly modeling transient $r\\th$ resonances is essential to\nproducing subradian phase-accurate waveforms for the detection and characterization of EMRIs by LISA.\n\nTo date, numerical investigations of transient $r\\th$ resonances have not incorporated local strong-field\nconservative perturbations \\cite{FlanHind12,FlanHughRuan14,BerrETC16,SperGair21}.\nWhile conservative perturbations vanish at adiabatic order for nonresonant motion, it is not yet known if they may\ncontribute to the adiabatic evolution of EMRIs during $r\\th$ resonances.\nFlanagan and Hinderer \\cite{FlanHind12} argue that conservative\neffects will not contribute at leading order during resonances based on their \\textit{integrability\nconjecture}.\\footnote{We follow \\cite{FlanHughRuan14} in referring to Flanagan and Hinderer's argument as the\nintegrability conjecture.}  The validity of this conjecture remains unclear.  Isoyama\net al.~ \\cite{IsoyETC13,IsoyETC19} have found the presence of potential conservative contributions to the evolution\nof the orbit-averaged Carter constant through their Hamiltonian formulation of EMRI dynamics.  For the integrability\nconjecture to hold, these terms would have to vanish when integrated over an entire orbit, which has not been\ndemonstrated analytically.\n\nThe integrability conjecture can potentially be tested through numerical calculations of conservative quantities,\nsuch as a computation of the local GSF in Kerr spacetime during an $r\\th$ resonance.  First-order GSF\ncalculations for generic bound orbits about a Schwarzschild black hole are well advanced, to the point of allowing\nlong-term evolutionary computations \\cite{WarbETC12,OsbuWarbEvan16,WarbOsbuEvan17,VandWarb18}.  Indeed, even\nsecond-order GSF calculations for restricted orbits in Schwarzschild are now emerging \\cite{PounETC20}.  Recent work\nby van de Meent provided the first GSF results for generic orbits in a Kerr EMRI \\cite{Vand18}.  However, these\ncalculations are much more computationally demanding than their Schwarzschild counterparts, and long-term, self-forced\nevolutions of Kerr EMRIs have not yet been produced.  Not even snapshots of the GSF in Kerr EMRIs during\n$r\\th$ resonances have been attempted.\n\nThus, as a first step in exploring local radiation-reaction effects driven by resonances, we consider a\nscalar field analogue and calculate the scalar self-force (SSF) that arises due to a bound particle with scalar\ncharge $q$ in an $r\\th$ resonance about a Kerr black hole.  This work builds off of a\nprevious paper \\cite{NasiOsbuEvan19} in which we presented the first calculation of the SSF for nonresonant\ninclined, eccentric orbits in Kerr spacetime.  By treating $q^2\/(\\mu M)$ as a small parameter, this SSF model mimics\nthe GSF problem.  As is common for adiabatic and osculating geodesic evolutions, we assume the motion of the particle\nto be exactly geodesic and calculate the resulting \\textit{geodesic scalar self-force}.  This assumption produces\nan error that is on the order of the time-averaged dissipative part of the second-order self-force.  In this work\nwe are not yet concerned with applying the self-force and calculating a portion of the evolution based on the\nbackreaction.\n\n\\begin{figure*}[th]\n\t\\includegraphics[width=0.35\\textwidth]{Figures\/geoGen1.png}\n\t\\hspace{1cm}\n\t\\includegraphics[width=0.35\\textwidth]{Figures\/geoRes1.png}\n\t\\caption{Inclined eccentric geodesics around a Kerr black hole with spin\n\t$a\/M=0.9$ and mass $M=1$. The plot on the left is a nonresonant geodesic with\n\torbital parameters $(p,e,x)=(4.700,0.5,\\cos\\pi\/4)$. The plot on\n\tthe right, in contrast, is a 1:2 $r\\th$-resonant geodesic with orbital\n\tparameters $(p,e,x)\\simeq (4.607,0.5,\\cos\\pi\/4)$. The\n\tsemilatus rectum $p$ is truncated at four significant digits and\n\tspecifically chosen so that the particle undergoes one radial\n\tlibration for every two polar librations. The solid (blue) curves trace out the\n\tthree-dimensional motion of the two geodesics in Boyer-Lindquist coordinates\n\t$(r,\\th,\\vp)$.  The solid (grey) curves show various two-dimensional projections\n\tof the three-dimensional orbit.}\n\\label{fig:resGeoExample}\n\\end{figure*}\n\nTo set notation, in Sec.~\\ref{sec:geo} we review nonresonant and resonant geodesics in Kerr spacetime, along with\nvarious parametrizations of geodesic functions. In Sec.~\\ref{sec:ssf} we review the scalar self-force model\npresented in \\cite{NasiOsbuEvan19}.  In Sec.~\\ref{sec:constructSSF}, we extend our previous methods for calculating\nthe SSF along nonresonant orbits to the case of $r\\th$ resonances by making use of a simple shifting relation\nbased on symmetries (Killing vectors) of the Kerr spacetime.  In Sec.~\\ref{sec:results}, we numerically calculate\nthe SSF experienced by a scalar-charged particle for six different $r\\th$-resonant geodesics.  We use these results\nin Sec.~\\ref{sec:flux} to compute the secular evolution (orbit-averaged time rate-of-change) of the orbital constants\n(e.g., $\\langle \\dot{\\mathcal{E}}\\rangle$, $\\langle \\dot{\\mathcal{L}}_z\\rangle$, $\\langle \\dot{\\mathcal{Q}}\\rangle$).\nWe demonstrate that the conservative components of the SSF do not contribute to the evolution of the orbital energy and\nangular momentum, $\\langle \\dot{\\mathcal{E}}\\rangle$ and $\\langle \\dot{\\mathcal{L}}_z\\rangle$, respectively,\nas expected from flux-balance arguments.  We do find nonzero contributions from the conservative SSF\nto the secular evolution of the Carter constant $\\langle \\dot{\\mathcal{Q}}\\rangle$, though these\ncontributions are consistent with the level of systematic numerical errors produced by our regularization scheme.\nWe conclude with a discussion of these results in Sec.~\\ref{sec:conclusion}.  Because we only focus on\n$r\\th$ resonances in this work, henceforth we will occasionally refer to these as simply resonances.\nFor this paper we use units such that $c=G=1$, use metric signature $(-+++)$ and sign conventions, where\napplicable, of Misner, Thorne, and Wheeler \\cite{MisnThorWhee73}.\n\n\\section{Geodesics and resonant motion in Kerr spacetime}\n\\label{sec:geo}\n\nThe instantaneous motion of a small mass orbiting a much more massive rotating black hole is approximated by that\nof a bound, timelike geodesic in Kerr spacetime.  This is the zeroth-order motion in BHPT.  Bound Kerr geodesics\nlibrate at different frequencies in the radial and polar directions, as shown in the left panel of\nFig.~\\ref{fig:resGeoExample}.  Generally, the instantaneous periods of a Kerr EMRI's radial and polar motions are\nincommensurate.  In these circumstances, the geodesic is ergodic: given an infinite time, the motion passes through\nevery point in a finite, bounded region in $r$ and $\\theta$.  For certain orbital parameters, however, these radial\nand polar periods will be in a rational-number ratio, giving rise to a resonance.  In these cases the motion is\nnot ergodic. Instead it loops back on itself, and it does not fill the corresponding bounded region in $r$ and $\\theta$\n\\cite{GrosLeviPere12} (see the right panel of Fig.~\\ref{fig:resGeoExample}).\n\nTo better understand these different behaviors, and to establish notation, we review the analytic framework for\nstudying bound geodesics in the Kerr spacetime, primarily following the work of\n\\cite{Cart68,MisnThorWhee73,Schm02,DrasFlanHugh05,DrasHugh06,FlanHughRuan14}, though we have consolidated and\nadapted notation for consistency.  We also discuss various prescriptions for parametrizing geodesics and how those\nparametrizations must be handled when describing resonances.\n\n\\subsection{Separation of the geodesic equations}\n\nConsider a point particle with mass $\\mu$ on a bound geodesic $x_p^\\mu(\\tau)$ with four-velocity $u^\\alpha \\equiv\ndx_p^\\mu\/d\\tau$, where $\\tau$\nis the particle's proper time.  The background spacetime, described by the metric $g_{\\a\\b}$, is parametrized by\nthe black hole spin $a$ and mass $M$.  Adopting Boyer-Lindquist coordinates $(t,r,\\th,\\vp)$, the Kerr line element\nreads\n\\begin{multline}\n\tds^2 = -\\left(1-\\frac{2Mr}{\\Sig}\\right)dt^2 + \\frac{\\Sig}{\\D}dr^2\n\t-\\frac{4Mar\\sin^2\\th}{\\Sig}dtd\\vp\n\t\\\\\n\t+ \\Sig d\\th^2 + \\frac{\\sin^2\\th}{\\Sig}(\\varpi^4-a^2\\D\\sin^2\\th)d\\vp^2,\n\\end{multline}\nwhere $\\Sig \\equiv r^2 + a^2 \\cos^2\\th$, $\\D \\equiv r^2 - 2 M r + a^2$, and $\\varpi^2 \\equiv r^2+a^2$.\n\nGeodesic motion in Kerr spacetime is completely integrable, leading to three constants of motion---the specific\nenergy $\\mathcal{E}$, the $z$ component of the specific angular momentum $\\mathcal{L}_z$, and the (scaled) Carter\nconstant $\\mathcal{Q}$ \\cite{Cart68}---all of which can be related to the Killing symmetries of Kerr.  The specific\nenergy and angular momentum correspond to the Killing vectors $\\xi^\\mu_{(t)}$ and $\\xi^\\mu_{(\\vp)}$,\n\\begin{align}\n\\label{eqn:EAndLzConstants}\n\t\\mathcal{E} &\\equiv -\\xi^{\\mu}_{(t)}u_\\mu = -u_t,\n\t\\\\\n\t\\mathcal{L}_z &\\equiv + \\xi^{\\mu}_{(\\vp)}u_\\mu = + u_\\vp,\n\\end{align}\nwhile the Carter constant is associated with the Killing tensor $K^{\\mu\\nu}$ \\cite{WalkPenr70},\n\\begin{gather}\n\\label{eqn:Q}\n\t\\mathcal{Q} \\equiv K^{\\mu\\nu}u_\\mu u_\\nu -(\\mathcal{L}_z-a\\mathcal{E})^2,\n\\end{gather}\nwhich is discussed further in Appendix \\ref{app:Qdot}.\n\nIntroducing the (Carter-)Mino time parameter $\\la$ \\cite{Cart68,MisnThorWhee73,Mino03,DrasFlanHugh05}, defined by\n$d\\la \\equiv \\Sig_p^{-1}d\\tau$, the geodesic equations separate\n\\begin{align}\n\\label{eqn:geoT}\n\t\\frac{dt_p}{d\\la} &= V_{tr}(r_p) + V_{t\\th}(\\th_p),\n\t\\\\  \\label{eqn:geoR}\n\t\\frac{dr_p}{d\\la} &= \\pm \\sqrt{V_r(r_p)},\n\t\\\\  \\label{eqn:geoTh}\n\t\\frac{d\\th_p}{d\\la} &= \\pm \\sqrt{V_\\th(\\th_p)},\n\t\\\\ \\label{eqn:geoPhi}\n\t\\frac{d\\vp_p}{d\\la} &= V_{\\vp r}(r_p) + V_{\\vp\\th}(\\th_p),\n\\end{align}\nwhere the various $V$ functions are defined in \\cite{DrasFlanHugh05,NasiOsbuEvan19} and the subscript $p$ denotes\nthat a function is evaluated on the particle's worldline, $x_p^\\mu$.\n\nRather than specifying a geodesic by directly choosing values for $\\mathcal{E}$, $\\mathcal{L}_z$, and $\\mathcal{Q}$,\nwe use relativistic definitions of semilatus rectum $p$ and orbital eccentricity $e$ that are analogous to those of\nKeplerian orbits,\n\\begin{gather}\n\tp \\equiv \\frac{2r_\\text{min}r_\\text{max}}{M(r_\\text{max}+r_\\text{min})},\n\t\\qquad\n\te \\equiv \\frac{r_\\text{max}-r_\\text{min}}{r_\\text{max}+r_\\text{min}} .\n\\end{gather}\nWe add to that the projection of the orbital inclination\n\\begin{equation}\n\tx \\equiv \\cos(\\frac{\\pi}{2}-\\th_\\text{min}) ,\n\\end{equation}\nto round out the parametrization of the orbits.  Here, $r_\\text{min}$ and $r_\\text{max}$ are the minimum and maximum\nradii reached by the point mass and $\\th_\\text{min} = \\pi - \\th_\\text{max}$ is its minimum polar angle.  These are\nthe turning points of the geodesic where\n$V_r(r_\\text{min})=V_r(r_\\text{max})=V_\\th(\\th_\\text{min})=V_\\th(\\pi-\\th_\\text{min})=0$.\nOnce $p$, $e$, and $x$ are specified for an orbit, it is straightforward to determine the corresponding $\\mathcal{E}$,\n$\\mathcal{L}_z$, and $\\mathcal{Q}$ \\cite{Schm02,DrasHugh04}.  One can then solve \\eqref{eqn:geoT}-\\eqref{eqn:geoPhi}\nusing spectral integration methods \\cite{HoppETC15,NasiOsbuEvan19} or analytic special functions\n\\cite{DrasHugh04,FujiHiki09}.  In this work we use a hybrid scheme: we sample the analytic geodesics solutions\npresented in \\cite{FujiHiki09}, then construct their discrete Fourier representations, which provide an\nexponentially-convergent numerical approximation of the geodesics.\n\n\\subsection{Frequencies of generic bound motion}\n\\label{sec:freq}\n\nFor inclined, eccentric (bound) geodesics the point mass librates in $r$ and $\\th$ with radial and polar Mino time\nperiods\n\\begin{align}\n\t\\La_r &\\equiv 2 \\int_{r_\\text{min}}^{r_\\text{max}} \\frac{dr}{\\sqrt{V_r(r)}},\n\t\\\\\n\t\\La_\\th &\\equiv 2 \\int_{\\th_\\text{min}}^{\\pi-\\th_\\text{min}}\n\t\\frac{d\\th}{\\sqrt{V_\\th(\\th)}},\n\\end{align}\nand corresponding Mino time frequencies\n\\begin{equation}\n\t\\Up_{r} \\equiv \\frac{2\\pi}{\\La_r} , \\qquad \\Up_{\\th} \\equiv\n\t\\frac{2\\pi}{\\La_\\th}.\n\\end{equation}\nIn Kerr spacetime, $\\Up_\\th$ is always greater than $\\Up_r$ for bound motion.  The time and azimuthal\nlocations, which depend on the radial and polar motions [see \\eqref{eqn:geoT} and \\eqref{eqn:geoPhi}], accumulate\nat average rates in $\\la$ denoted by $\\G$ and $\\Up_\\vp$ and are given by\n\\begin{align}\n\t\\G &=\\frac{1}{\\La_r} \\int_0^{\\La_r} V_{tr}(r_p)d\\la\n\t+ \\frac{1}{\\La_\\th} \\int_0^{\\La_\\th} V_{t\\th} (\\th_p)d\\la ,\n\t\\\\\n\t\\Up_\\vp &=\\frac{1}{\\La_r} \\int_0^{\\La_r} V_{\\vp r} (r_p)d\\la\n\t+ \\frac{1}{\\La_\\th} \\int_0^{\\La_\\th} V_{\\vp\\th}(\\th_p)d\\la,\n\\end{align}\nwhere $r_p$ and $\\th_p$ are understood to be functions of $\\la$.  Together, these form the complete set of Mino\ntime frequencies $\\Up_\\a=(\\G,\\Up_r,\\Up_\\th,\\Up_\\vp)$.  They are related to the fundamental coordinate-time\nfrequencies by\n\\begin{equation}\n\t\\O_r = \\frac{\\Up_r}{\\G},\n\t\\qquad \\O_\\th = \\frac{\\Up_\\th}{\\G},\n\t\\qquad \\O_\\th = \\frac{\\Up_\\vp}{\\G},\n\\end{equation}\nwhich then define the discrete frequencies\n\\begin{equation}\n\\label{eqn:genFreq}\n\t\\o_{mkn} \\equiv m\\O_\\vp +k\\O_\\th + n\\O_r ,\n\\end{equation}\nin the multiperiodic Fourier spectrum of any variable made time dependent by the geodesic motion.  Note that these\ncoordinate-time frequencies $\\O_\\a$ do not uniquely define a geodesic due to the existence of isofrequency\npairings \\cite{WarbBaraSago13}, though it is well argued that (up to initial conditions) geodesics are uniquely\ndefined by their Mino time frequencies $\\Up_\\a$ \\cite{Vand14a}.\n\n\\subsection{Analytic structure of geodesic solutions and their dependence on initial conditions}\n\nWe consider next families of geodesics and their dependence on initial conditions.  Let an inclined eccentric\ngeodesic be parametrized by $\\la$, $\\hat{x}^\\mu = \\hat{x}^\\mu_p(\\la)$, and have the following initial conditions\n\\begin{gather}\n\\label{eqn:fiducialConditions}\n\t\\hat{x}_p^\\mu(0)=(0,r_\\text{min},\\th_\\text{min},0),\n\t\\\\\n\t\\hat{u}^r(0)=\\hat{u}^\\th(0)=0 .\n\\end{gather}\nFollowing the nomenclature of \\cite{DrasFlanHugh05}, we refer to a trajectory with these initial conditions as a\nfiducial geodesic and distinguish it with a hat.  Integrating \\eqref{eqn:geoT}-\\eqref{eqn:geoPhi} and enforcing these\nfiducial initial conditions, the periodicity in the motion gives rise to solutions that have the form\n\\begin{align}\n\\label{eqn:geoTFid}\n\t\\hat{t}_p(\\la) &= \\G\\la\n\t+ \\D\\hat{t}^{(r)}(\\Up_r\\la) + \\D\\hat{t}^{(\\th)}({\\Up_\\th\\la})\n\t\\\\ \\label{eqn:geoRFid}\n\t\\hat{r}_p(\\la)& = r_\\text{min} + \\D\\hat{r}^{(r)}(\\Up_r\\la),\n\t\\\\ \\label{eqn:geoThFid}\n\t\\hat{\\th}_p(\\la)& = \\th_\\text{min} + \\D\\hat{\\th}^{(\\th)}(\\Up_\\th\\la),\n\t\\\\ \\label{eqn:geoPhiFid}\n\t\\hat{\\vp}_p(\\la)&= \\Up_\\vp\\la\n\t+ \\D\\hat{\\vp}^{(r)}(\\Up_r\\la) + \\D\\hat{\\vp}^{(\\th)}({\\Up_\\th\\la}),\n\\end{align}\nwhere the various $\\Delta \\hat{x}$ terms are oscillatory, periodic functions.  Here we use $\\D\\hat{x}$ to represent\n$\\D \\hat{t}$, $\\D \\hat{r}$, $\\D \\hat{\\th}$, and $\\D \\hat{\\vp}$, which have the\nfollowing properties\n\\begin{align} \\label{eqn:periodicRTh}\n\t\\D\\hat{x}^{(r)}(2\\pi+\\Up_{r}\\la) &= \\D\\hat{x}^{(r)}(\\Up_{r}\\la),\n\t& &\\D\\hat{x}^{(r)}(0) = 0,\n\t\\\\\n\t\\D\\hat{x}^{(\\th)}(2\\pi+\\Up_{\\th}\\la) &= \\D\\hat{x}^{(\\th)}(\\Up_{\\th}\\la),\n\t& &\\D\\hat{x}^{(\\th)}(0) = 0 .\n\\end{align}\nFurthermore, from the way a fiducial geodesic is defined, all of these periodic functions are either even or odd with\nrespect to $\\la=0$, with $\\Delta \\hat{t}$ and $\\Delta \\hat{\\vp}$ being odd (antisymmetric) and $\\Delta \\hat{r}^{(r)}$ and\n$\\Delta \\hat{\\th}^{(\\th)}$ being even (symmetric).  Exact definitions of these geodesic functions are provided in\n\\cite{DrasFlanHugh05,FujiHiki09,NasiOsbuEvan19}.\n\nNext, we consider an inclined, eccentric geodesic $x^\\mu_p(\\la)$ with arbitrary initial conditions\n\\begin{gather}\n\tx_p^\\mu(0)=(t_0,r_0,\\th_0,\\vp_0),\n\t\\\\\n\tu^r(0)=u^r_0, \\qquad \\qquad u^\\th(0)=u^\\th_0 .\n\\end{gather}\nAn arbitrary geodesic can be expressed in terms of the fiducial solutions by shifting the arguments of the periodic functions,\n\\begin{align}\n\\label{eqn:geoTGen}\n\tt_p&(\\la;t_0,\\la^{(r)}_0,\\la^{(\\th)}_0) = t_0\n\t- \\D\\hat{t}(\\Up_r\\la^{(r)}_0,\\Up_\\th\\la^{(\\th)}_0) \\notag\n\t\\\\\n\t& \\quad\n\t+ \\G\\la +\n\t\\D\\hat{t}\\left(\\Up_r(\\la+\\la^{(r)}_0),\\Up_\\th(\\la+\\la^{(\\th)}_0)\\right),\n\t\\\\\n\tr_p&(\\la;\\la^{(r)}_0)= r_\\text{min}\n\t+ \\D\\hat{r}^{(r)}(\\Up_r\\la+\\Up_r\\la^{(r)}_0),\n\t\\\\\n\t\\th_p&(\\la; \\la^{(\\th)}_0) = \\th_\\text{min}\n\t+ \\D\\hat{\\th}^{(\\th)}(\\Up_\\th\\la+\\Up_\\th\\la^{(\\th)}_0),\n\t\\\\ \\label{eqn:geoPhiGen}\n\t\\vp_p&(\\la; \\vp_0,\\la^{(r)}_0,\\la^{(\\th)}_0)= \\vp_0\n\t- \\D\\hat{\\vp}(\\Up_r\\la^{(r)}_0,\\Up_\\th\\la^{(\\th)}_0) \\notag\n\t\\\\ & \\;\\;\\;\n\t+ \\Up_\\vp\\la\n\t+ \\D\\hat{\\vp}\\left(\\Up_r(\\la+\\la^{(r)}_0),\\Up_\\th(\\la+\\la^{(\\th)}_0)\\right).\n\\end{align}\nIn the above expressions, we introduced the compact notation,\n\\begin{align}\n\t\\D\\hat{t}(\\Up_r\\la,\\Up_\\th\\ti{\\la})&\\equiv \\D\\hat{t}^{(r)}(\\Up_r\\la) + \\D\\hat{t}^{(\\th)}(\\Up_\\th\\ti{\\la}),\n\t\\\\\n\t\\D\\hat{\\vp}(\\Up_r\\la,\\Up_\\th\\ti{\\la})&\\equiv \\D\\hat{\\vp}^{(r)}(\\Up_r\\la) +\n\t\\D\\hat{\\vp}^{(\\th)}(\\Up_\\th\\ti{\\la}),\n\\end{align}\nfor the sums of the radial and polar dependencies of the time and azimuthal components.  The initial orbital offsets,\n$\\la^{(r)}_0$ and $\\la^{(\\th)}_0$, are defined in terms of the initial radial and polar positions and velocities by\n\\begin{align*}\n \\D\\hat{r}^{(r)}(\\Up_r\\la^{(r)}_0) &= r_0 - r_\\text{min},\n \\;\\; \\sign\\left(\\sin\\Up_r \\la^{(r)}_0\\right) = \\sign u^r_0,\n \\\\\n \\D\\hat{\\th}^{(\\th)}(\\Up_\\th\\la^{(\\th)}_0) &= \\th_0 - \\th_\\text{min},\n \\;\\; \\sign\\left(\\sin\\Up_\\th \\la^{(\\th)}_0\\right) = \\sign u^\\th_0.\n\\end{align*}\nHere, $\\sign$ represents the sign function. The fiducial case is recovered by setting\n$t_0=\\vp_0=\\la^{(r)}_0=\\la^{(\\th)}_0=0$.\n\n\\begin{figure*}[!th]\n\\centering\n\\includegraphics[width=\\textwidth]{Figures\/resOrbitPanel.pdf}\n\\caption{Projected motion for two 1:2 $r\\th$-resonant geodesics around a Kerr black hole with spin $a\/M=0.9$\nand mass $M=1$.  The left plot suppresses the azimuthal motion and depicts the radial and polar motion of both\ngeodesics in the poloidal plane.  Three-dimensional projections of these two geodesics are mapped separately in\nthe center and right plots. Both geodesics share the orbital parameters $(p,e,x)\\approx(4.607,0.5,\\cos\\pi\/4)$ and\nthe same $r\\th$-resonant frequencies.  While they both start at $r_p(0)=r_\\text{min}$, they differ in their initial\npolar positions and polar velocities.  The solid (blue) line in the left plot and the center plot trace out a\ngeodesic with the initial offset $\\la^{(\\th)}_0=0$ ($q_{\\th 0}=\\bar{q}_0=0$), while the dashed (green) line in the left\nplot and the far right plot represent a geodesic with the initial offset $\\la^{(\\th)}_0=-3\\La_\\th\/8$\n($q_{\\th 0}=\\b_\\theta \\bar{q}_0=-3\\pi\/4$). The motion for each plot is shown from $\\la=0$ to $\\la=47$.  }\n\\label{fig:resInitOrbits}\n\\end{figure*}\n\nAll bound geodesics can be described by \\eqref{eqn:geoTGen}-\\eqref{eqn:geoPhiGen}, though any geodesic that passes\nthrough a simultaneous minimum in the radial and polar motion (i.e., $r_p=r_\\text{min}$ and $\\th_p=\\th_\\text{min}$)\ncan be mapped to a fiducial geodesic via trivial offsets in $t_0$ and $\\vp_0$.  As long as there exist integers\n$k'$ and $n'$ such that\n\\begin{equation}\n\\label{eqn:fiducialCondition}\n\t\\la^{(\\th)}_0 - \\la^{(r)}_0 = n'\\La_r-k'\\La_\\th,\n\\end{equation}\nis satisfied, then a simultaneous turning point at these minimum positions will occur.\\footnote{A similar symmetry\nalso exists for other simultaneous turning points [e.g., $(r_\\mathrm{min},\\th_\\mathrm{max})$,\n$(r_\\mathrm{max},\\th_\\mathrm{max})$], which leads to the alternate constraint of Eq.~(4.9) in \\cite{DrasFlanHugh05}.\nFor simplicity we only focus on the case of a simultaneous minimum turning point $(r_\\mathrm{min},\\th_\\mathrm{min})$,\nwhich is sufficient for our discussion.}  Because nonresonant eccentric inclined geodesics are ergodic, this\nsimultaneous turning point always exists (i.e., on a long enough timescale), and, therefore, nonresonant geodesics\ncan be described by the fiducial expressions in \\eqref{eqn:geoTFid}-\\eqref{eqn:geoPhiFid}, without loss of\ngenerality up to a trivial shifting of Mino time $\\la$.\n\n\\subsection{Special case of $r\\th$-resonant geodesics}\n\\label{sec:resGeo}\n\nWe classify $r\\th$ resonances in terms of the (relatively prime) integers $\\b_r$ and $\\b_\\th$ that define the ratio\nbetween the radial and polar frequencies, i.e.,\n\\begin{equation}\n\t\\frac{\\O_r}{\\O_\\th} = \\frac{\\b_r}{\\b_\\th} .\n\\end{equation}\nLow-integer ratios (i.e., ones with small integers like $\\b_r$:$\\b_\\th=$ 1:2, 2:3) are referred to as strong\nresonances, while high-integer ratios (e.g., 10:11, 1:20) are weak resonances.  Because the radial and polar\nfrequencies are commensurate during an $r\\th$ resonance, the discrete frequency spectrum of resonant geodesics\nreduces to\n\\begin{equation}\n\\label{eqn:resFreq}\n\t\\o_{mN}\\equiv m\\O_\\vp + N\\O,\n\\end{equation}\nwhere $\\O\\equiv \\O_r\/\\b_r=\\O_\\th\/\\b_\\th$ and $N\\equiv k\\b_\\th+n\\b_r \\in \\mathbb{Z}$.  In other words, for an\n$r\\th$ resonance the normally separate sets of harmonics of the radial and polar fundamental frequencies merge into\none set of harmonics, $N$, of a new, lower net frequency, $\\Omega$.\n\nResonant geodesics can also be described by\n\\eqref{eqn:geoTGen}-\\eqref{eqn:geoPhiGen}.  However, unlike nonresonant geodesics which are ergodic,\n$r\\theta$ resonances follow restricted paths through the poloidal plane---as shown in\nFig.~\\ref{fig:resInitOrbits}---and these paths are sensitive to the initial conditions $\\la^{(r)}_0$ and\n$\\la^{(\\th)}_0$.  For a resonance, the radial and polar motions oscillate with the shared (net) Mino time frequency\nand period\n\\begin{equation}\n\t\\Up\\equiv \\frac{\\Up_r}{\\b_r}=\\frac{\\Up_\\th}{\\b_\\th}, \\qquad \\qquad \\La\\equiv \\b_r\\La_r=\\b_\\th\\La_\\th.\n\\end{equation}\nFor simultaneous minimum turning points to occur during a resonance, \\eqref{eqn:fiducialCondition} reduces to the\nmore stringent restriction that $\\la^{(\\th)}_0-\\la^{(r)}_0=N'\\La$, for some integer $N'$, which will not hold true\nfor most choices of $\\la^{(r)}_0$ and $\\la^{(\\th)}_0$.  Therefore, most resonant orbits cannot be mapped to the\nfiducial geodesic with the same frequencies.\n\nNevertheless, the symmetries of Kerr spacetime allow the same functions describing the fiducial orbit to be applied\nmore broadly to the general resonant case.  This process starts with defining the initial resonant offset\n$\\la_0\\equiv \\la^{(\\th)}_0-\\la^{(r)}_0$.  Any two $r\\th$ resonances that share the same value of $\\la_0$ (modulo\n$\\La$) can be mapped onto one another.  Taking advantage of this mapping, one can simplify the description of geodesic\norbits by setting $\\la^{(r)}_0$ or $\\la^{(\\th)}_0$ to 0.  In this work, we choose $\\la^{(r)}_0=0$ so that\n$\\la_0=\\la^{(\\th)}_0$, without loss of generality.  The parametrization of the resonant motion in terms of the\noffset $\\la_0$ is then\n\\begin{align}\n\\label{eqn:geoTRes}\n\tt_p&(\\la;\\la_0) =\n\t\\G\\la - \\D\\hat{t}(0,\\b_\\th\\Up\\la_0)\n\t\\\\\n\t& \\qquad \\qquad \\quad \\;\\,\n\t+ \\D\\hat{t}\\left(\\b_r\\Up\\la,\\b_\\th\\Up(\\la+\\la_0)\\right) \\notag\n\t\\\\\n\tr_p&(\\la;\\la_0)= r_\\text{min}\n\t+ \\D\\hat{r}^{(r)}(\\b_r\\Up\\la),\n\t\\\\\n\t\\th_p&(\\la; \\la_0) = \\th_\\text{min}\n\t+ \\D\\hat{\\th}^{(\\th)}(\\b_\\th\\Up(\\la+\\la_0)),\n\t\\\\ \\label{eqn:geoPhiRes}\n\t\\vp_p&(\\la; \\la_0)=\n\t\\Up_\\vp\\la - \\D\\hat{\\vp}(0,\\b_\\th\\Up\\la_0)\n\t\\\\\n\t& \\qquad \\qquad \\qquad \\,\n\t+ \\D\\hat{\\vp}\\left(\\b_r\\Up\\la,\\b_\\th\\Up(\\la+\\la_0)\\right). \\notag\n\\end{align}\nBy varying the value of the offset in the range $0\\leq\\la_0 < \\La$, we can generate all resonant paths through\nthe poloidal plane that are characterized by the same orbital parameters and frequencies but have different\ninitial positions (see Fig.~\\ref{fig:resInitOrbits}).\n\n\\begin{figure*}[!th]\n\t\\centering\n\t\\includegraphics[width=0.37\\textwidth]{Figures\/torusFig.pdf}\n\t\\hspace{6mm}\n\t\\includegraphics[width=0.37\\textwidth]{Figures\/torusFigRes.pdf}\n\t\\caption{Poloidal motion depicted on the torus for orbits with parameters\n\t$(a\/M,p,e,x)=(0.9,6,0.5,\\cos\\pi\/4)$ (left panel) and\n\t$(a\/M,p,e,x)=(0.9,4.607,0.5,\\cos\\pi\/4)$ (right panel).  In the right plot parameters\n\tare chosen to generate a 1:2 $r\\th$ resonance.  The orbit in the left panel is nonresonant.\n\tThere the solid (blue) line follows the path of a geodesic with fiducial initial\n\tconditions $q_{(a)0}=(t_0,q_{r0},q_{\\th 0},\\vp_0)=(0,0,0,0)$, while the dashed (red) line has\n\tinitial conditions $q_{(a)0}=(0,\\pi,\\pi\/2,0)$. (Both paths are plotted over the Mino time interval $\\la\\in [0,4]$,\n\twith $M=1$.)  Given sufficient time, both paths will fill the entire torus and completely overlap.\n\tIn the right panel the solid (blue) line follows\n\tthe path of a geodesic with the initial resonant phase $\\bar{q}_0=0$ on the Mino time\n\tinterval $\\la\\in [0,\\La]$, where $\\La\\simeq 4.494$ is the (net) resonant Mino time period.  The\n\tdashed (red) line follows a geodesic with initial phase\n\t$\\bar{q}_0=\\pi\/2$ ($\\th_p(\\la=0)=\\pi-\\th_\\text{min}$) on the same time interval. Unlike the nonresonant paths,\n\tthe resonant geodesic flows in the right panel will never overlap with one another.}\n\t\\label{fig:twoTorus}\n\\end{figure*}\n\n\\subsection{Parametrizing geodesics with angle variables}\n\\label{sec:angleVar}\n\nThe integrability of geodesic motion in Kerr spacetime also leads to a natural representation of the motion in terms\nof action-angle variables.  This formalism forms the basis of the two-timescale description of EMRI dynamics\n\\cite{HindFlan08} and provides a characteristic parametrization for functions that depend on the librating radial\nand polar motions.  The angle variable parametrization is also used for presenting gravitational self-force results\n\\cite{Vand18}, which we adopt in similar fashion.\n\nThe angle variables are related to $\\la$ by the Mino frequencies,\n\\begin{equation}\n\tq_r =  \\Up_r \\la, \\qquad \\qquad q_\\th = \\Up_\\th \\la .\n\\end{equation}\nWe similarly define the initial orbital phases\n\\begin{align} \\label{eqn:initialPhasesTuple}\n\tq_{(\\a) 0} = \\{ q_{t0},q_{r0},q_{\\th 0},q_{\\vp 0} \\} =\n\t\\{ t_0, \\Up_r\\la^{(r)}_0, \\Up_\\th\\la^{(\\th)}_0, \\vp_0 \\} .\n\\end{align}\n(Technically, for $q_{t0}$ to represent a dimensionless phase it should be rescaled, i.e., $\\Omega t_0$, but the\ndistinction is unimportant for our present purposes.)  Functions that are periodic with respect to the Mino time\nperiods, $\\La_{r}$ and $\\La_{\\th}$, can then be parametrized in terms of the corresponding angle variables,\n$q_{r}$ and $q_{\\th}$, e.g.,\n\\begin{gather*}\n\t\\D\\hat{r}^{(r)}\\left(\\Up_r(\\la+\\la^{(r)}_0)\\right)\n\t\\rightarrow \\D\\hat{r}^{(r)}(q_r+q_{r0}),\n\t\\\\\n\t\\D\\hat{\\th}^{(\\th)}\\left(\\Up_\\th(\\la+\\la^{(\\th)}_0)\\right)\n\t\\rightarrow \\D\\hat{\\th}^{(\\th)}(q_\\th+q_{\\th 0}).\n\\end{gather*}\n\nUpon parametrizing functions in terms of the angle variables, it is straightforward to project function values on\nthe two-torus spanned by the two angle variables, such as the tori depicted in Fig.~\\ref{fig:twoTorus}.  Each\n(invariant) two-torus forms a section of configuration space for the radial and polar motion of the small body.  The\ngeodesic flow on the torus then describes the evolution of the motion through this configuration\nspace, as discussed in \\cite{BrinGeyeHind15b}.  For a nonresonant orbit, starting these paths at different points on\nthe torus is equivalent to choosing different initial conditions.  The effect of initial conditions on possible paths\nis shown in the top panel of Fig.~\\ref{fig:twoTorus}.  Given an infinite amount of Mino time, a system following a\nnonresonant geodesic will sample every point on the torus.\n\nIn the case of a resonance, the system executes a closed, repeating motion through the domain, as seen in the\nbottom panel of Fig.~\\ref{fig:twoTorus}.  Choosing different initial positions on the torus can generate unique\npaths.  The only way to sample all of the points on the torus is to consider an infinite number of resonant orbits\nthat share the same frequencies $\\Up_\\a$ but different initial offsets $\\la_0$.\n\nTo distinguish resonant geodesics, we define a\nsingle angle variable $\\bar{q}$ for resonant motion and then a single angle parameter $\\bar{q}_0$ for the initial resonant\nphase\\footnote{Our resonant angle $\\bar{q}$ differs from the resonant variable $q_\\perp = \\beta_\\theta q_r - \\beta_r q_\\theta$ used by other authors (e.g., \\cite{Vand14a}).},\n\\begin{equation}\n\\label{eqn:resAngleVar}\n\t\\bar{q} \\equiv  \\Up \\la = \\frac{q_r}{\\b_r} = \\frac{q_\\th}{\\b_\\th},\n\t\\qquad\n\t\\bar{q}_0 \\equiv  \\Up\\la_0 = \\frac{q_{\\th 0}}{\\b_\\th} - \\frac{q_{r 0}}{\\b_r} .\n\\end{equation}\nIn this mapping, $\\bar{q}_0$ is a constant as the system evolves.\n\nIn terms of these resonant angles, we denote reparametrized functions with an overbar, such that\n\\begin{align}\n\t\\D\\bar{r}^{(r)}(\\bar{q}) &\\equiv \\D\\hat{r}^{(r)}(\\b_r \\bar{q}) ,\n\t\\\\\n\t\\D\\bar{\\th}^{(\\th)}(\\bar{q};\\bar{q}_{0}) &\\equiv \\D\\hat{\\th}^{(\\th)}(\\b_\\th \\bar{q}+\\b_\\th \\bar{q}_0),\n\t\\\\\n\t\\D\\bar{t}(\\bar{q};\\bar{q}_{0}) &\\equiv \\D\\hat{t}(\\b_r \\bar{q}, \\b_\\th \\bar{q}+\\b_\\th \\bar{q}_0),\n\t\\\\\n\t\\D\\bar{\\vp}(\\bar{q};\\bar{q}_{0}) &\\equiv \\D\\hat{\\vp}(\\b_r \\bar{q}, \\b_\\th \\bar{q}+\\b_\\th \\bar{q}_0).\n\\end{align}\nThis description is particularly useful for calculating and visualizing the SSF in later sections.\n\n\\section{Scalar self-force problem}\n\\label{sec:ssf}\n\n\\subsection{Overview}\n\nWe use the same scalar model and resulting SSF formalism as outlined in our previous paper \\cite{NasiOsbuEvan19}.\nWe give a brief summary here to establish notation.  The small body is treated as a point particle with mass $\\mu$\nfollowing an (arbitrary) geodesic about a Kerr black hole with mass $M$ and spin $a$, but with the particle endowed\nwith a scalar charge $q \\sim \\mu \\ll M$.  We neglect gravitational perturbations and the GSF due to the mass $\\mu$.\nThe motion of the charge sources a radiative scalar field $\\Phi$, which satisfies the curved-space Klein-Gordon\nequation \\cite{Teuk73}\n\\begin{equation}\n\\label{eqn:KG}\n\tg^{\\a\\b}\\nabla_\\a \\nabla_\\b \\Phi = -4\\pi \\rho,\n\\end{equation}\nwhere $\\rho$ is the scalar charge density and the covariant derivative $\\nabla_\\a$ is taken with respect to the\nstationary Kerr background $g_{\\a\\b}$.  The charge density takes the form\n\\begin{equation}\n\\label{eqn:pointSource}\n\\rho =\nq\\frac{\\d(r-r_p(t))\\d(\\cos\\th-\\cos\\th_p(t))\\d(\\vp-\\vp_p(t))}\n{V_{tr}(r)+V_{t\\th}(\\th)} .\n\\end{equation}\n\nThe scalar field produces a backreaction on the scalar charge in the form of a SSF $F_\\a$ (per unit charge) that\ndrives its motion off of a background geodesic \\cite{Quin00},\n\\begin{equation}\n\tu^\\b \\, \\nabla_\\b \\l \\mu u^\\a \\r = q^2 F^\\a.\n\\end{equation}\nUnlike the gravitational case, the SSF is not orthogonal to the four-velocity and contributes to a variation in the\nrest mass\n\\begin{equation}\n\t\\frac{d\\mu}{d\\tau} = -q^2 u^\\a F_\\a ,\n\\end{equation}\nrequiring all four components of the SSF to be computed.\n\nThe contribution of $\\Phi$ to the SSF can be completely encoded in the Detweiler-Whiting regular field\n$\\Phi^\\text{R}$ \\cite{Quin00,DetwWhit03}\n\\begin{equation}\n\tq^2 F^\\a = \\lim_{x^\\mu\\rightarrow x_p^\\mu}\n\t\tq g^{\\a\\b}\\nabla_\\b \\Phi^\\text{R} .\n\\end{equation}\nThe regular field $\\Phi^\\text{R}=\\Phi^\\text{ret}-\\Phi^\\text{S}$ is defined as the difference between the retarded\nfield $\\Phi^\\text{ret}$, which satisfies Eq.~\\eqref{eqn:KG} with causal boundary conditions, and the singular field\n$\\Phi^\\text{S}$, which also satisfies Eq.~\\eqref{eqn:KG} (but with different boundary conditions) and which captures\nthe local, singular behavior.\n\nThis makes both $\\Phi^\\text{ret}$ and $\\Phi^\\text{S}$ divergent along the point-particle worldline, and their\nsubtraction from one another requires a careful regularization procedure.  We make use of mode-sum regularization\n\\cite{BaraOri00,BaraOri03a}\n\\begin{align}\n\\label{eqn:modeSumSSF}\n\tF_\\a &= \\lim_{x^\\mu \\rightarrow \\pm x^\\mu_p} \\sum_{l=0}^\\infty\n\t(F_{\\a\\pm}^{\\text{ret},l}-F_{\\a\\pm}^{\\text{S},l}) ,\n\\end{align}\nwhere $F_{\\a\\pm}^{\\text{ret},l}$ and $F_{\\a\\pm}^{\\text{S},l}$ are the finite, spherical harmonic moments of the\nfull divergent quantities $q^2F_\\a^\\text{ret}\\equiv q\\nabla_\\a\\Phi^\\text{ret}$ and\n$q^2F_\\a^\\text{S}\\equiv q\\nabla_\\a\\Phi^\\text{S}$.  The $\\pm$ notation accounts for the fact that the moments for some\nvector components are discontinuous at the point source, with their value depending upon the direction in $r$ in\nwhich the limit is taken.\n\nIn the following subsections we outline how $F_\\a^{\\text{ret},l}$ and $F_\\a^{\\text{S},l}$ are constructed for a\npoint source following an arbitrary geodesic.  We calculate the geodesic SSF, finding the force along a background\ngeodesic, and not the self-consistent SSF that would result from applying the backreaction continuously.\n\n\\subsection{The retarded field $\\Phi^\\text{ret}$}\n\\label{sec:phi}\n\nIn the Kerr background Eq.~\\eqref{eqn:KG} is amenable to separation of variables if we both decompose\n$\\Phi^\\text{ret}$ in azimuthal $m$ modes and transform to the frequency domain \\cite{Teuk73,BrilETC72}.  The\ndiscrete spectrum of the bound source motion reduces the frequency-domain representation of the field to a set\nof discrete sums\n\\begin{equation} \\label{eqn:fieldDecomposition}\n\t\\Phi^\\text{ret} = q\n\t\\sum_{\\hat{l} mkn} R_{\\hat{l} mkn}(r) S_{\\hat{l} mkn}(\\th)e^{im\\vp}e^{-i\\o_{mkn}t},\n\\end{equation}\nwhere the discrete frequency spectrum $\\o_{mkn}$ is defined in Eq.~\\eqref{eqn:genFreq} and the sum\n\\begin{equation}\n\t\\sum_{\\hat{l} mkn} \\equiv \\sum_{\\hat{l}=0}^{+\\infty} \\sum_{m = -\\hat{l}}^{\\hat{l}}\n\t\\sum_{k=-\\infty}^{+\\infty} \\sum_{n=-\\infty}^{+\\infty},\n\\end{equation}\nis a compact notation for the sums over $\\hat{l}$ and $m$ and the harmonics of the polar and radial motions.\n\nIn the decomposition, $S_{\\hat{l} mkn}(\\th)$ is the scalar spheroidal harmonic (spin weight 0) with spheroidicity\n$\\s^2=-a^2\\o_{mkn}^2$, and $R_{\\hat{l} mkn}(r)$ is the solution to the (scalar) radial inhomogeneous Teukolsky equation\n\\cite{Teuk73}.  We distinguish between spheroidal and spherical harmonic indices using $\\hat{l}$ and $l$, respectively.\nIn our calculations, we construct $S_{\\hat{l} mkn}$ as a sum over spherical harmonics\n\\begin{equation}\n\tS_{\\hat{l} mkn}(\\th)e^{im\\vp} = \\sum_{l=|m|}^\\infty b^l_{\\hat{l} mkn}\n\tY_{lm}(\\th,\\vp),\n\\end{equation}\nwhere the coefficients $b^{l}_{\\hat{l} mkn}$ satisfy a three-term recurrence relation described in\n\\cite{NasiOsbuEvan19}.\n\n\\subsection{Radial mode functions and extended homogeneous solutions}\n\nThe radial mode functions $R_{\\hat{l} mkn}$ are solved in a way that allows us to apply the method of extended\nhomogeneous solutions \\cite{BaraOriSago08,HoppEvan10,WarbBara10,WarbBara11,Warb15}.  This technique circumvents the\nappearance of Gibbs ringing in the time-domain retarded field \\eqref{eqn:fieldDecomposition} when the source terms\nare pointlike distributions.  The method starts with calculating the radial mode functions in the frequency\ndomain.  We first transform to the tortoise coordinate $r_*$ by integrating\n\\begin{equation}\n\t\\frac{dr_*}{dr}=\\frac{\\varpi^2}{\\D},\n\\end{equation}\nand then introduce scaled radial functions\n\\begin{equation} \\label{eqn:gsnTransform}\n\tX_{\\hat{l} mkn}= \\varpi R_{\\hat{l} mkn}.\n\\end{equation}\nThe new radial functions satisfy a radial wave equation\n\\begin{equation} \\label{eqn:gsnODE}\n\t\\left[\\frac{d^2}{dr_*^2}-U_{\\hat{l} mkn}(r)\\right]X_{\\hat{l} mkn}(r)\n\t= Z_{\\hat{l} mkn}(r),\n\\end{equation}\nwhere the radial potential $U_{\\hat{l} mkn}$ and source term $Z_{\\hat{l} mkn}$ are defined in Sec.~II C of\n\\cite{NasiOsbuEvan19} and Appendix \\ref{app:normC} of this paper.\n\nWe then construct (unit-normalized) homogeneous solutions $\\ti{X}^\\pm_{\\hat{l} mkn}$ that satisfy downgoing ($-$) and\noutgoing ($+$) wave boundary conditions at the horizon and infinity, respectively (also referred to as the in-wave\nand up-wave \\cite{Galt82}).  In practice, rather than numerically constructing solutions by solving \\eqref{eqn:gsnODE},\nwe directly compute the homogeneous radial Teukolsky solutions $\\tilde{R}^\\pm_{\\hat{l} mkn}$ using the Mano-Suzuki-Takasugi function\nexpansions \\cite{ManoSuzuTaka96b,SasaTago03}, and then obtain $\\tilde{X}^\\pm_{\\hat{l} mkn}$ from \\eqref{eqn:gsnTransform}.\n\nThrough variation of parameters, we calculate the normalization\ncoefficients (or Teukolsky amplitudes) ${C}^\\pm_{\\hat{l} mkn}$ that relate the homogeneous and inhomogeneous solutions\nin the source-free regions\n\\begin{align}\n\tX_{\\hat{l} mkn}(r<r_\\text{min}) &= {C}^-_{\\hat{l} mkn}\\ti{X}^-_{\\hat{l} mkn}(r),\n\t\\\\\n\tX_{\\hat{l} mkn}(r>r_\\text{max}) &= {C}^+_{\\hat{l} mkn}\\ti{X}^+_{\\hat{l} mkn}(r).\n\\end{align}\nThe calculation of ${C}^\\pm_{\\hat{l} mkn}$ is described in Appendix \\ref{app:normC} and \\cite{NasiOsbuEvan19}.  As\nnoted previously \\cite{DrasFlanHugh05,FlanHughRuan14}, varying the initial conditions of the source changes\n${C}^\\pm_{\\hat{l} mkn}$ by a phase factor \\cite{DrasFlanHugh05}\n\\begin{equation}\n\\label{eqn:fidToGenClmkn}\n\tC^\\pm_{\\hat{l} mkn}(q_{(\\a)0}) =\n\te^{i\\xi_{mkn}(q_{(\\a)0})}\\hat{C}^\\pm_{\\hat{l} mkn},\n\\end{equation}\nwhere the set of initial conditions $q_{(\\a)0}$ is defined in \\eqref{eqn:initialPhasesTuple}. The phase offset takes the form\n\\begin{multline}\n\t\\xi_{mkn}(q_{(\\a)0})\\equiv m(\\D \\hat{\\vp}(q_{r0},q_{\\th0})-\\vp_0)\n\t\\\\\n\t-\\o_{mkn}(\\D \\hat{t}(q_{r0},q_{\\th0})-t_0)\n\t\t\\\\\n\t\t- k q_{\\th0} - n q_{r0} ,\n\\end{multline}\nand hatted normalization constants $\\hat{C}^\\pm_{\\hat{l} mkn}$ are calculated assuming the fiducial orbit.\nWe provide a derivation of this relationship in Appendix \\ref{app:normC}.  Equation \\eqref{eqn:fidToGenClmkn}\nalso holds true for the Teukolsky amplitudes calculated for gravitational perturbations \\cite{FlanHughRuan14}.\n\nWith these normalized homogeneous solutions, we define the following extended homogeneous functions\n\\begin{align}\n\\label{eqn:EHS}\n\t&{\\phi}^\\pm_{lm}(t,r)\\equiv \\sum_{\\hat{l}=|m|}^{+\\infty}\n\t\\sum_{k=-\\infty}^{+\\infty}\\sum_{n=-\\infty}^{+\\infty}\n\t{\\phi}^\\pm_{l\\hat{l} mkn}(r)e^{-i\\o_{mkn}t},\n\t\\\\ \\label{eqn:EHSmodes}\n\t&{\\phi}^\\pm_{l\\hat{l} mkn}(r)\\equiv \\frac{1}{\\varpi}\n\tb^l_{\\hat{l} mkn}{C}^\\pm_{\\hat{l} mkn}\n\t\\ti{X}^\\pm_{\\hat{l} mkn}(r) ,\n\\end{align}\nfor each spherical harmonic $Y_{lm}(\\th,\\vp)$.  We refer to ${\\phi}^\\pm_{lm}(t,r)$ as extended homogeneous\nfunctions, not extended homogeneous solutions, since by construction there are no time domain wave equations that\nthey satisfy (the Teukolsky equation does not separate with ordinary spherical harmonics).  These functions do\nhowever have the advantage that the sums in \\eqref{eqn:EHS}\nconverge exponentially, unlike those in \\eqref{eqn:fieldDecomposition}.  Furthermore, even though the individual\n$l\\hat{l} mkn$ modes of \\eqref{eqn:EHSmodes} are not valid solutions of the inhomogeneous Teukolsky\nequation in the source region $r_\\text{min}\\leq r \\leq r_\\text{max}$, once the full field is reconstructed in the\ntime domain by summing over all modes, we are left with extended homogeneous solutions that provide an accurate and\nconvergent representation of the retarded field up to the location of the point charge\n\\begin{align}\n\\label{eqn:phiRet}\n\t{\\Phi}^\\text{ret}(t,r,\\th,\\vp)&=\n\t{\\Phi}^{-}(t,r,\\th,\\vp)\\Theta\\left(r_p(t)-r\\right)\n \\\\\n &\\qquad + \\hat{\\Phi}^{+}(t,r,\\th,\\vp)\\Theta\\left(r-r_p(t)\\right), \\notag\n \\\\  \\label{eqn:phiPM}\n {\\Phi}^{\\pm}(t,r,\\th,\\vp) &=\n q \\sum_{lm} {\\phi}^\\pm_{lm}(t,r) Y_{lm}(\\th,\\vp).\n\\end{align}\n\n\\subsection{Retarded and singular contributions to the SSF}\n\\label{sec:ssfReg}\n\nUsing \\eqref{eqn:phiRet} and \\eqref{eqn:phiPM}, we construct the $l$ modes of the force ${F}^{\\text{ret},l}_{\\a\\pm}$,\nalong the particle worldline\n\\begin{align}\n\\label{eqn:ssfRetMino}\n\t{F}^{\\text{ret},l}_{\\a\\pm}(\\la) &=\n\t\\sum_{m=-l}^l\n\t(\\mathcal{D}^{lm}_\\a {\\phi}_{lm}^{\\pm})({t}_p,{r}_p)\n\tY_{lm}({\\th}_p,{\\vp}_p) ,\n\\end{align}\nwhere the coordinate positions of the particle are understood to be functions of Mino time [e.g., ${t}_p={t}_p(\\la)$],\nand the operator $\\mathcal{D}^{lm}_\\a$ performs the following operations on the extended homogeneous functions:\n\\begin{align}\n\t\\mathcal{D}_t^{lm}\\phi^\\pm_{lm}&\\equiv\\partial_t\\phi^\\pm_{lm},\n\t\\\\\n\t\\mathcal{D}_r^{lm}\\phi^\\pm_{lm}&\\equiv\\partial_r\\phi^\\pm_{lm},\n\t\\\\ \\label{eqn:delTheta}\n\t\\mathcal{D}_\\th^{lm}\\phi^\\pm_{lm}&\\equiv\\b^{(-3)}_{l+3,m}\\phi^\\pm_{l+3,m}\n\t+\\b^{(-1)}_{l+1,m}\\phi^\\pm_{l+1,m}\n\t\\\\\n\t&\\qquad\n\t+\\b^{(+1)}_{l-1,m}\\phi^\\pm_{l-1,m}\n\t+\\b^{(+3)}_{l-3,m}\\phi^\\pm_{l-3,m}, \\notag\n\t\\\\\n\t\\mathcal{D}_\\vp^{lm}\\phi^\\pm_{lm}&\\equiv im\\phi^\\pm_{lm} .\n\\end{align}\nThe coefficients $\\b_{lm}^{(\\pm i)}$ are defined in Appendix A of \\cite{NasiOsbuEvan19}\\footnote{A minor error exists in\nEq. (A4) of [39]. The coefficients $\\beta^{(\\pm 3)}_{lm}$ are missing a minus sign in front of the\nparentheses on the righthand side of Eq. (A4).} and are\nobtained by first applying the window function proposed by Warburton \\cite{Warb15} and then reprojecting the\nderivatives $\\partial_\\th Y_{lm}$ onto the $Y_{lm}$ basis.  Details of this operation can be found in\n\\cite{Warb15}, with a correction added in \\cite{NasiOsbuEvan19}.\n\nThe singular contribution is obtained through a local analytic expansion in the neighborhood of the source worldline\n\\cite{BaraOri00,BaraOri03a,DetwMessWhit03}\n\\begin{equation}\n\\label{eqn:regParameters}\n\tF^{\\text{S},l}_{\\a\\pm} = A^\\pm_{\\a}L + B_\\a\n\t+\\sum_{n=1}^{+\\infty} \\frac{D_{\\a,2n}}{\\prod_{k=1}^n (2L-2k)(2L+2k)},\n\\end{equation}\nwhere $L \\equiv l+1\/2$, and the regularization parameters $A_{\\a\\pm}$, $B_\\a$, and\n$D_{\\a,2n}$ are independent of $l$ but functions of $r_p$, $\\th_p$, $u^r$, $u^\\th$, $\\mathcal{E}$, $\\mathcal{L}_z$,\nand $\\mathcal{Q}$ (as well as $a$ and $M$).  Only $A_{\\a\\pm}$ and $B_\\a$ are known analytically for generic bound\norbits in Kerr spacetime \\cite{BaraOri03a}, while $D_{\\a,2}$ is known analytically for equatorial orbits in Kerr\n\\cite{HeffOtteWard14}.\n\nThe terms with higher-order parameters $D_{\\a,2n}$ have the useful property that their $l$-dependent weights vanish\nupon summing over all $l$,\n\\begin{equation}\n\t\\sum_{l=0}^{+\\infty} \\left[\\prod_{k=1}^n (2L-2k)(2L+2k)\\right]^{-1} = 0 .\n\\end{equation}\nOnly $A_{\\a\\pm}$ and $B_\\a$ are needed for convergent results, but if we neglect the $D_{\\a,2n}$ terms upon combining\nEqs.~\\eqref{eqn:modeSumSSF} and \\eqref{eqn:regParameters}, $F_\\a$ converges at a rate $\\sim l^{-2}$.  Each\n$D_{\\a,2n}$ term reintroduced to the regularization procedure improves the convergence rate by another factor of\n$l^{-2}$.  Since we truncate the sum over $l$ modes around $l_\\text{max}\\sim 20$, we must numerically fit for the\nhigher-order regularization parameters to improve the convergence of the mode-sum regularization.  Our fitting\nprocedure is described in \\cite{NasiOsbuEvan19}.  The uncertainties associated with this fitting procedure often\ndominate other numerical errors in the calculation.  We use this to obtain uncertainty estimates for our SSF\nresults.\n\n\\section{Constructing the SSF for resonant and nonresonant sources}\n\\label{sec:constructSSF}\n\n\\subsection{Nonresonant sources}\n\\label{sec:genSSF}\n\nWith a generic nonresonant orbit, the SSF is multiply periodic and never repeats over the entire interval\n$-\\infty<\\la<\\infty$.  Rather than sampling the SSF over this infinite domain in $\\la$, we map the SSF\nto the angle variables introduced in Sec.~\\ref{sec:angleVar},\n\\begin{equation} \\label{eqn:angleVarSSF}\n\t\\hat{F}^{\\text{ret},l}_{\\a\\pm}(q_r,q_\\th) =\n\t\\sum_{m=-l}^l (\\mathcal{D}^{lm}_\\a \\hat{\\phi}_{lm}^{\\pm})(q_r,q_\\th)\n\t\\hat{Y}_{lm}(q_r,q_\\th) .\n\\end{equation}\nNote that we have placed hats on all functions that are evaluated using the fiducial geodesic solutions of\n\\eqref{eqn:geoTFid}-\\eqref{eqn:geoPhiFid}.  The evolution of $\\vp$ is dependent on the motion of both $r$ and\n$\\theta$, so in a slight abuse of notation we reparametrize functions to have the following meaning\n\\begin{align}\n\\label{eqn:angleVarYlm}\n\t\\hat{Y}_{lm}(q_r,q_\\th)\\equiv Y_{lm}(\\hat{\\th}_p(q_\\th),\n\t\\D\\hat{\\vp}(q_r,q_\\th)) ,\n\\end{align}\nand\n\\begin{align}\n\t&\\hat{\\phi}^\\pm_{lm}(q_r,q_\\th)\\equiv \\sum_{\\hat{l} kn}\n\t{\\hat{\\phi}^\\pm_{l\\hat{l} mkn}(q_r)}\n\t e^{-i(\\o_{mkn}\\D\\hat{t}(q_r,q_\\th)+kq_\\th+nq_r)}, \\notag\n\t\\\\   \\label{eqn:angleVarEHS}\n\t&\\hat{\\phi}^\\pm_{l\\hat{l} mkn}(q_r)\\equiv \\hat{\\varpi}^{-1}_p(q_r)b^l_{\\hat{l} mkn}\\hat{C}^\\pm_{\\hat{l} mkn}\n\t\\ti{X}^\\pm_{\\hat{l} mkn}(\\hat{r}_p(q_r)) .\n\\end{align}\nHere $\\hat{\\varpi}_p(q_r)=(\\hat{r}_p^2(q_r)+a^2)^{1\/2}$ and the operator $\\mathcal{D}^{lm}_{\\a}$ performs the\nsame function as before.  Because the regularization parameters only vary with respect to $r_p$, $\\th_p$, $u^r$,\nand $u^\\th$ (assuming the orbital constants are fixed), the singular field can also be translated into this angle\nvariable parametrization, ultimately providing a description of the SSF in terms of $q_r$ and $q_\\th$\n\\begin{equation}\n\\label{eqn:ssfGenAngleVar}\n\t\\hat{F}_\\a(q_r,q_\\th) = \\sum_{l=0}^\\infty \\left(\n\t\\hat{F}_{\\a\\pm}^{\\text{ret},l}(q_r,q_\\th)\n\t-\\hat{F}^{\\text{S},l}_{\\a\\pm}(q_r,q_\\th)\n\t\\right) .\n\\end{equation}\n\nThe angle variable parametrization maps the entire self-force history onto the finite domain of the invariant\ntwo-torus visualized in Fig.~\\ref{fig:twoTorus}.  The SSF, projected onto this torus, can then also be represented\nby the (double) Fourier series\n\\begin{align}\n\\label{eqn:ssfFourier}\n\tF_{\\a}&(q_r,q_\\th)= \\sum_{k=-\\infty}^{+\\infty}\\sum_{n=-\\infty}^{+\\infty}\n\tg^{kn}_\\a e^{-i(kq_\\th+nq_r)},\n\t\\\\\n\tg^{kn}_\\a &= \\frac{1}{4\\pi^2}\\int_0^{2\\pi} dq_r\\int_0^{2\\pi} dq_\\th\\;\n\tF_\\a\\left(q_{r}, q_{\\th}\\right)e^{i(kq_{\\th}+nq_{r})}. \\notag\n\\end{align}\nBy densely sampling values of $q_r$ and $q_\\th$ over the torus at evenly-spaced points $q_{r,i}=2\\pi i\/N_r$ and\n$q_{\\th,j}=2\\pi j\/N_\\th$ (where $N_r, N_\\th \\in \\mathbb{Z}$), we can construct a discrete Fourier representation of\nthe SSF\n\\begin{align}\n\\label{eqn:ssfDiscreteFourier}\n\tF_{\\a}&(q_r,q_\\th)\\simeq \\sum_{k=0}^{N_\\th-1}\\sum_{n=0}^{N_r-1}\n\tf^{kn}_\\a e^{-i(kq_\\th+nq_r)},\n\t\\\\\n\tf^{kn}_\\a &= \\frac{1}{N_r N_\\th}\\sum_{i=0}^{N_\\th-1}\\sum_{j=0}^{N_r-1}\n\tF_\\a\\left(q_{r,i}, q_{\\th,j}\\right)e^{i(kq_{\\th,j}+nq_{r,i})} . \\notag\n\\end{align}\nGiven $N_r$ and $N_\\th$ that are large enough such that $\\text{max}|f^{kn}_\\a-g^{kn}_\\a|<\\eps_\\text{FS}$, where\n$\\eps_\\text{FS}$ is some predefined accuracy goal, the discrete representation will provide an accurate\napproximation of Eq.~\\eqref{eqn:ssfFourier} \\cite{HoppETC15,NasiOsbuEvan19}.  We found that sample numbers of\n$N_r=N_\\th=2^8$ were typically sufficient for constructing a discrete representation that was accurate to about\n$\\eps_\\text{FS}\\sim 10^{-8}-10^{-10}$.  The discrete Fourier series provides an efficient method for storing and\ninterpolating SSF data.\n\nWe can easily generalize our results to geodesics with arbitrary initial conditions by applying the following\nshifting relation,\n\\begin{equation}\n\\label{eqn:ssfFid2Gen}\n\tF_\\a(q_r,q_\\th;q_{a0})\n\t= \\hat{F}_\\a(q_r+q_{r 0},q_\\th+q_{\\th 0}).\n\\end{equation}\nA proof of Eq.~\\eqref{eqn:ssfFid2Gen}, which applies for both the SSF and GSF, is provided in Appendix\n\\ref{app:ssfInvariant}.  While this result seems almost trivial for the nonresonant case, it surprisingly plays a\nrole in improving the efficiency of SSF calculations for resonant orbits as well, as discussed in\nSec.~\\ref{sec:resSSF}.\n\n\n\\subsection{Resonant sources}\n\\label{sec:resSSF}\n\nThe SSF experienced by a charge following an $r\\th$-resonant geodesic requires a different treatment.  The worldline\nof the charge is described by \\eqref{eqn:geoTRes}-\\eqref{eqn:geoPhiRes}.  In contrast to the SSF for a nonresonant\norbit [e.g., $\\hat{F}_\\a(q_r,q_\\th)$], we construct the resonant-SSF $\\bar{F}_\\a$ to be a function of the single\nresonant angle variable $\\bar{q}$ and the initial resonant phase $\\bar{q}_0$ [defined in Eq.~\\eqref{eqn:resAngleVar}].\nWe describe here two methods of calculating $\\bar{F}^\\mathrm{res}_\\a(\\bar{q},\\bar{q}_0)$: the first uses the reduced\nmode spectrum $\\o_{mN}$ defined in \\eqref{eqn:resFreq} to construct the SSF on an $l\\hat{l} mN$ basis, while the second\nuses the generic mode spectrum $\\o_{mkn}$ to construct the SSF on the $l\\hat{l} mkn$ basis, just as we outlined in the\nprevious section for nonresonant orbits.  These two approaches are similar to the two approaches for calculating\ngravitational wave fluxes discussed in \\cite{FlanHughRuan14}.\n\n\\subsubsection{Constructing the resonant SSF on an $l\\hat{l} m N$ basis}\n\nThe retarded SSF sourced by an $r\\th$ resonant geodesic, when parametrized in terms of the resonant angle variable\nand resonant phase, takes the form\n\\begin{equation}\n\\label{eqn:fRetRes}\n\t\\bar{F}^{\\text{ret},l}_{\\a\\pm}(\\bar{q};\\bar{q}_0) =\n\t\\sum_{m=-l}^l\n\t(\\mathcal{D}^{lm}_\\a \\bar{\\phi}_{p,lm}^{\\pm})(\\bar{q};\\bar{q}_0)\\bar{Y}_{lm}(\\bar{q};\\bar{q}_0),\n\\end{equation}\nwhere, in contrast to Eqs.~\\eqref{eqn:angleVarYlm} and \\eqref{eqn:angleVarEHS},\n\\begin{multline}\n\t\\bar{Y}_{lm}(\\bar{q};\\bar{q}_0)\\equiv\n\tY_{lm}(\\bar{\\th}_p(\\bar{q};\\bar{q}_0),\\D\\bar{\\vp}(\\bar{q};\\bar{q}_0)-\\D\\bar{\\vp}(0;\\bar{q}_0)),\n\\end{multline}\nand\n\\begin{align}\n\\label{eqn:resEHS}\n\t&\\bar{\\phi}^\\pm_{lm}(\\bar{q};\\bar{q}_0)\\equiv \\sum_{\\hat{l}=|m|}^{+\\infty} \\sum_{N=-\\infty}^{+\\infty}\n\t{\\bar{\\phi}^\\pm_{l\\hat{l} mN}(\\bar{q};\\bar{q}_0)}\n\t\\\\\n\t&\\qquad \\qquad \\qquad\n\t \\times e^{-i\\o_{mN}(\\D\\bar{t}(\\bar{q};\\bar{q}_0)-\\D\\bar{t}(0;\\bar{q}_0)+N\\bar{q})}, \\notag\n\t\\\\\n\t&\\bar{\\phi}^\\pm_{l\\hat{l} mN}(\\bar{q};\\bar{q}_0)\\equiv\n\t\\bar{\\varpi}^{-1}_p(\\bar{q})b^l_{\\hat{l} mN}\\bar{C}^\\pm_{\\hat{l} mN}(\\bar{q}_0)\n\t\\ti{X}^\\pm_{\\hat{l} mN}(\\bar{r}_p(\\bar{q})). \\notag\n\\end{align}\nAll functions and coefficients with an overbar are evaluated using the resonant geodesic solutions described by\n\\eqref{eqn:geoTRes}-\\eqref{eqn:geoPhiRes}.  The $\\bar{C}^\\pm_{\\hat{l} mN}$ are defined in Appendix \\ref{app:normC} and\nvary with the resonant phase parameter $\\bar{q}_0$.  Unlike $C^\\pm_{\\hat{l} mkn}(\\bar{q}_0)$,\n$\\bar{C}^\\pm_{\\hat{l} mN}(\\bar{q}_0)$ is not related to the fiducial case $\\bar{C}^\\pm_{\\hat{l} mN}(0)$ by a simple phase\nfactor.  Each time we calculate the SSF for a new value of $\\bar{q}_0$, the source term must be integrated over a\nnew resonant orbit.  Since source integration is a computationally-intensive aspect of the SSF calculation, needing\nto repeat this operation is not ideal.  Thus, the advantage of reduced dimensionality in the mode spectrum must be\nweighed against the disadvantage of repeated source integration.\n\n\\subsubsection{Constructing the resonant SSF on an $l\\hat{l} m k n$ basis}\n\nAlternatively, we first construct the fiducial SSF $\\hat{F}_\\a(q_r,q_\\th)$ using the methods outlined in\nSec.~\\ref{sec:genSSF}.  Combining \\eqref{eqn:ssfGenAngleVar} and \\eqref{eqn:ssfFid2Gen}, we can then relate the\nresonant SSF $\\bar{F}^\\mathrm{res}_\\a(\\bar{q};\\bar{q}_0)$ to the fiducial result by fixing the relationship between\n$q_r$ and $q_\\theta$\n\\begin{equation} \\label{eqn:ssfqq0}\n\t\\bar{F}^\\mathrm{res}_{\\a}\n\t(\\bar{q};\\bar{q}_0) =\n\t\\hat{F}_{\\a}\n\t(\\b_r \\bar{q}, \\b_\\th \\bar{q} + \\b_\\th \\bar{q}_0) .\n\\end{equation}\nIn this way, we simply construct the fiducial SSF $\\hat{F}_{\\a}(q_r,q_\\th)$ on an $l\\hat{l} m k n$ basis by relating\nthe $\\hat{l} mN$-mode functions and constants to their $\\hat{l} mkn$-mode counterparts\n\\begin{gather}\n\\label{eqn:freqMKNfreqMN}\n\\o_{mN}=\\o_{m(kn)_N} ,\n\\\\\n\\ti{X}_{\\hat{l} mN}=\\ti{X}_{\\hat{l} m(kn)_N}, \\qquad b^l_{\\hat{l} mN}=b^l_{\\hat{l} m(kn)_N} ,\n\\end{gather}\nwhere one must be careful to understand that $(k,n)_N$ represents the set of all $k$ and $n$ values that produce\nthe same value $N$ that satisfies $N=k\\b_\\th+n\\b_r$.  Significant computational time is saved by recycling values\nof the homogeneous radial functions for different values of $k$ and $n$, provided\nthey share the same frequency and spheroidal mode numbers $(\\hat{l}, m)$.\n\nThe normalization coefficients are related by a coherent sum over all $k$ and $n$ modes that share the same frequency\n(given by $N$)\n\\begin{align}\n\\label{eqn:ClmN2Clmkn}\n\t\\bar{C}^\\pm_{\\hat{l} mN}(\\bar{q}_0)&=  \\sum_{(k,n)_N}\n\te^{i\\ti{\\xi}_{mkn}(\\bar{q}_0)}\\hat{C}^\\pm_{\\hat{l} mkn} ,\n\\end{align}\nas demonstrated in Appendix A of \\cite{FlanHughRuan14} and Sec.~III D of \\cite{GrosLeviPere13}.  In this way,\neach $\\bar{C}^\\pm_{\\hat{l} mN}(\\bar{q}_0)$ is a superposition of many amplitudes $\\hat{C}^\\pm_{\\hat{l} mkn}$ that\nwould have been regarded as independent in the nonresonant case.  In a complex square, this superposition leads to\nconstructive or destructive interference terms in the fluxes.  Note that\n$\\ti{\\xi}_{mkn}(\\bar{q}_0) \\equiv{\\xi}_{mkn}(0,0,\\b_\\th q_{\\th 0},0)$.  Substituting\nEqs.~\\eqref{eqn:freqMKNfreqMN}-\\eqref{eqn:ClmN2Clmkn} into Eqs.~\\eqref{eqn:fRetRes}-\\eqref{eqn:resEHS}, brings them\ninto the same form as Eqs.~\\eqref{eqn:angleVarSSF}-\\eqref{eqn:angleVarEHS}.  Unlike $\\ti{X}_{\\hat{l} mkn}$, each\n$C_{\\hat{l} mkn}$ must be calculated independently, even if they share the same frequencies and spheroidal harmonic\nmode numbers.  Essentially, by introducing the more generic mode spectrum $\\o_{mkn}$, we circumvent the need to\nrepeatedly evaluate each $lmN$ mode at different initial phases, but at the expense of summing over an additional\nmode number.  The advantage of this approach is that, once a code has already been built to calculate the fiducial\nSSF for nonresonant orbits, it can be easily modified to produce the SSF for resonant sources and avoids the need\nto construct an entirely separate code.\n\n\\subsubsection{Discrete Fourier representation of the resonant SSF}\n\nThe resonant SSF is periodic with respect to $\\bar{q}$ and $\\bar{q}_0$, and therefore can be expressed as a multiple\nFourier series.  By sampling the resonant SSF on an evenly spaced two-dimensional grid in $\\bar{q}$ and $\\bar{q}_0$,\nthe discrete Fourier representation of $\\bar{F}^\\mathrm{res}_\\a$ is\n\\begin{align}\n\\label{eqn:FresFourier}\n\t&\\bar{F}^\\text{res}_\\a(\\bar{q};\\bar{q}_0) \\simeq \\sum_{K=0}^{N_0-1}\\sum_{N=0}^{N_\\text{res}-1}\n\t\\bar{g}_\\a^{KN} e^{-iN\\bar{q}}e^{-iK \\bar{q}_0},\n\t\\\\\n\t&\\bar{g}_\\a^{KN} = \\frac{1}{N_0 N_\\text{res}}\n\t\\sum_{\\jmath=0}^{N_0-1}\\sum_{\\imath=0}^{N_\\text{res}-1}\n\t\\bar{F}^\\text{res}_\\a(\\bar{q}_\\imath;\\bar{q}_{0\\jmath}) e^{iN\\bar{q}_\\imath}e^{iK \\bar{q}_{0\\jmath}} ,\n\\end{align}\nwhere $\\bar{q}_{i}=2\\pi i\/N_\\mathrm{res}$ and $\\bar{q}_{0,j}=2\\pi j\/N_0$, with $N_\\mathrm{res}, N_0 \\in \\mathbb{Z}$.\nBy comparing \\eqref{eqn:FresFourier} with \\eqref{eqn:ssfDiscreteFourier} and \\eqref{eqn:ssfqq0}, we can relate\n$\\hat{f}_\\a^{kn}$ and $\\bar{g}_\\a^{KN}$ by\n\\begin{equation}\n\t\\bar{g}_\\a^{KN} = \\hat{f}_\\a^{K\/\\b_\\th,(N-K)\/\\b_r} .\n\\end{equation}\nFrom this relation, we see that $\\bar{g}_\\a^{KN}=0$ unless $K$ is a multiple of $\\b_\\th$ and $N-K$ is a multiple of\n$\\b_r$.  Thus, while the resonant angle variable and the initial resonant phase more naturally capture both the coupled nature of\nthe radial and polar motion and the sensitivity of the source to initial conditions, this parametrization is less\nefficient at capturing the behavior of the self-force.  For example, if one wants to calculate $\\hat{f}^{kn}_\\a$\nfor $0\\leq k < N_\\th$, $0\\leq n < N_r$, then one would need to sample $N_r \\times N_\\th$ points in the\n$q_r$-$q_\\th$ domain, but $\\b_rN_r \\times (\\b_\\th N_\\th+\\b_r N_r)$ points in the $\\bar{q}$-$\\bar{q}_0$ domain.  This\noversampling occurs because the resonant parametrization does not take full advantage of the symmetries of the\norbit, which are better captured by the separation of the radial and polar motion in the $q_r$-$q_\\th$ angle\nparametrization.\n\n\\subsection{Dissipative and conservative SSF}\n\nIrrespective of the type of orbit, the self-force can be decomposed into conservative and dissipative parts,\n$F_\\a^\\text{cons}$ and $F_\\a^\\text{diss}$.  These parts impact the evolution of EMRIs in different ways\n\\cite{Bara09,DiazETC04,Mino03,HindFlan08} and computationally converge at different rates in the mode-sum\nregularization procedure.  The dissipative part $F_\\a^\\text{diss}$ does not require regularization and converges\nexponentially.  The conservative part $F_\\a^\\text{cons}$ requires regularization and converges as a power law in\nthe number of $l$ modes.\n\nSummarizing our previous discussion \\cite{NasiOsbuEvan19} of this decomposition, the split depends on both the\nretarded force and the advanced force $F^\\text{adv}_\\a$, which depends on the advanced scalar field solution.\nThe decomposition is made in terms of spherical harmonic elements, e.g., $F^{\\text{adv},l}_\\a$ and is given by\n\\begin{align}\n\tF^\\text{diss}_{\\a} &=\\sum_{l=0}^{+\\infty}\\frac{1}{2}\n\t\\left( F^{\\text{ret},l}_{\\a\\pm}-\n\tF^{\\text{adv},l}_{\\a\\pm}\\right),\n\t\\\\ \\label{eqn:consSSF}\n\tF^\\text{cons}_{\\a} &=\\sum_{l=0}^{+\\infty}\\left\\{\\frac{1}{2}\n\t\\left( F^{\\text{ret},l}_{\\a\\pm}+\n\tF^{\\text{adv},l}_{\\a\\pm}\\right)-F^{\\text{S},l}_{\\a\\pm} \\right\\}.\n\\end{align}\nThe inconvenience of calculating the advanced scalar field solution is avoided by using symmetries of Kerr\ngeodesics \\cite{Mino03,HindFlan08,Bara09} (summarized also in \\cite{NasiOsbuEvan19}), which lead to convenient\nrelationships between spacetime components of $F^{\\text{ret},l}_\\a$ and $F^{\\text{adv},l}_\\a$,\n\\begin{equation}\n\\label{eqn:ssfRet2Adv}\n\tF^{\\text{adv},l}_\\a(q_r,q_\\th) = \\eps_{(\\a)}\n\tF^{\\text{ret},l}_\\a(2\\pi-q_r,2\\pi-q_\\th) ,\n\\end{equation}\nwhere $\\eps_{(\\a)}=(-1,1,1,-1)$.  Thus, $F_t^\\text{diss}$, $F_r^\\text{cons}$, $F_\\th^\\text{cons}$, and\n$F_\\vp^\\text{diss}$ are symmetric (even) functions on the $q_r-q_\\th$ two-torus, while $F_t^\\text{cons}$,\n$F_r^\\text{diss}$, $F_\\th^\\text{diss}$, and $F_\\vp^\\text{cons}$ are antisymmetric (odd).  These relationships\nbetween advanced and retarded solutions have been previously discussed \\cite{WarbBara11,Warb15,ThorWard17} in the\ncontext of restricted orbits but, in fact, Eq.~\\eqref{eqn:ssfRet2Adv} holds for arbitrary geodesic motion.\n\n{\\renewcommand{\\arraystretch}{1.25}\n\\begin{table}[t!]\n\t\\caption{Summary of the resonant orbits considered in Sec.~\\ref{sec:results}.  In all cases the primary\n\tspin is $a=0.9$ (with $M=1$). The real number values are truncated in the table to four significant figures for brevity.}\n\t\\label{tab:orbits}\n\t\\centering\n\t\\begin{tabular*}{\\columnwidth}{c @{\\extracolsep{\\fill}} c c c c c}\n\t\t\\hline\n\t\t\\hline\n\t\t\\multicolumn{2}{c }{\\quad Model} & $p$ & $e$ & $x_\\text{inc}$ & $\\b_r$:$\\b_\\th$\n\t\t\\\\\n\t\t\\hline\n\t\t\\multicolumn{2}{c}{\\quad $e02.13$} & 3.622 & 0.2 & $\\cos(\\pi\/4)$ & 1:3\n\t\t\\\\\n\t\t\\multicolumn{2}{c}{\\quad $e02.12$} & 4.508 & 0.2 & $\\cos(\\pi\/4)$ & 1:2\n\t\t\\\\\n\t\t\\multicolumn{2}{c}{\\quad $e02.23$} & 6.643 & 0.2 & $\\cos(\\pi\/4)$ & 2:3\n\t\t\\\\\n\t\t\\multicolumn{2}{c}{\\quad $e05.13$} & 3.804 & 0.5 & $\\cos(\\pi\/4)$ & 1:3\n\t\t\\\\\n\t\t\\multicolumn{2}{c}{\\quad $e05.12$} & 4.607 & 0.5 & $\\cos(\\pi\/4)$ & 1:2\n\t\t\\\\\n\t\t\\multicolumn{2}{c}{\\quad $e05.23$} & 6.707 & 0.5 & $\\cos(\\pi\/4)$ & 2:3\n\t\t\\\\\n\t\t\\hline\n\t\t\\hline\n\t\\end{tabular*}\n\\end{table}\n}\n\n\\section{Resonant SSF results}\n\\label{sec:results}\n\nUsing the methods outlined in the prior sections, we generated new results for the SSF on six different\nresonant orbits, the orbital parameters of which are listed in Table \\ref{tab:orbits}.  These calculations were made\nwith a \\textsc{Mathematica} code first described in \\cite{NasiOsbuEvan19}.  These calculations also made use of\nsoftware from the Black Hole Perturbation Toolkit \\cite{BHPTK18}, specifically the \\textsc{KerrGeodesics} and\n\\textsc{SpinWeightedSpheroidalHarmonics} packages.\n\nIn generating numerical results we set $M=1$, which is assumed for the remainder of this work.  Each resonant orbit\nhad primary spin $a=0.9$.  We focused on 1:3, 1:2, and 2:3 $r\\th$ resonances, the three resonances an EMRI is most\nlikely to encounter during its final years of evolution when its signal falls within the LISA passband\n\\cite{RuanHugh14,BerrETC16}.  To pick orbital parameters $(p,e,x)$ that produce $r\\th$-resonant frequencies, we\nfollow the approach of Brink, Geyer, and Hinderer \\cite{BrinGeyeHind15a,BrinGeyeHind15b}.  Specified values of $e$\nand $x$ are chosen first, and then $p$ is numerically calculated using the root-finding method described in Sec.~V$\\,$E\nof \\cite{BrinGeyeHind15b}.  In our work all of the orbits share the same inclination, $x=\\cos(\\pi\/4)$, while two\ndifferent eccentricities, $e=0.2$ and $e=0.5$, are considered.  The resulting values of $p$ (to four places) for\neach resonant orbit are listed in Table \\ref{tab:orbits}.\n\n\\begin{figure*}[th!]\n\t\t\\includegraphics[width=0.7\\textwidth]{Figures\/ssf_converge_save.pdf}\n\t\t\\caption{Convergence of the SSF $l$ modes for resonant models listed in Table \\ref{tab:orbits}.\n\t\tThe dashed and dotted lines depict comparative power-law rates of\n\t\tconvergence for $\\bar{F}^\\text{res}_\\vp(\\bar{q}=5\\pi\/16;\\bar{q}_0=5\\pi\/32\/\\b_\\th)$ as more\n\t\tregularization terms are incorporated.  The (black) squares represent individual $l$ modes\n\t\tof the unregularized SSF, the sum of which clearly diverges.  The (red) triangles are the residuals\n\t\tfrom subtracting $A_\\vp$ and $B_\\vp$.  The (blue) diamonds represent the residuals after subtracting\n\t\t$D_{\\vp,2}$, obtained through numerical fitting.  The (purple) circles represent the inclusion of\n\t\t$D_{\\vp,4}$, also approximated via a numerical fit.}\n\t\t\\label{fig:convergenceFph}\n\\end{figure*}\n\nAs discussed in Sec.~\\ref{sec:constructSSF}, for resonant orbits we express the SSF as a function of the\nresonant angle variable $\\bar{q}$ and the resonant phase parameter $\\bar{q}_0$, i.e.,\n$\\bar{F}^\\text{res}_\\a(\\bar{q};\\bar{q}_0)$, or (as convenient) as a function of the more general angle\nvariables $q_r$ and $q_\\th$ and the initial phases $q_{r0}$ and $q_{\\th 0}$, i.e.,\n$\\hat{F}_\\a(\\b_r \\bar{q},\\b_\\th \\bar{q}+\\b_\\th \\bar{q}_{0})=\\hat{F}_\\a(q_r,q_\\th+q_{\\th 0})$.  Plotting the SSF as\na function of $\\bar{q}$, as shown in Sec.~\\ref{sec:FresQQ0}, highlights the periodicity of the SSF during resonances\nand is qualitatively representative of the Mino or coordinate time dependence of the SSF.  On the other hand,\nplotting the SSF as a function of $q_r$ and $q_\\th$, as shown in Sec.~\\ref{sec:FresQrQth}, separates the dependence\nof the SSF on the radial and polar motion of the orbit.  This way of depicting the SSF mirrors the parametrizations\nused for nonresonant orbits, as seen in \\cite{Vand18,NasiOsbuEvan19}.  To better analyze the impact of different\norbital parameters and types of resonances, we present each spacetime component of the self-force separately.\n\n\\subsection{Regularization and convergence of results}\n\\label{sec:regAndConvergeRes}\n\nThe SSF is constructed by mode-sum regularization and the numerical fitting procedures discussed in\nSec.~\\ref{sec:ssfReg}.  The convergence of the mode-sum regularization procedure is well understood: subtracting\nthe analytically known regularization parameters, $A_\\a$ and $B_\\a$, produces residuals that fall off as\n$\\sim l^{-2}$ for large $l$.  There is no fundamental difference when an orbit is on resonance.  In\nFig.~\\ref{fig:convergenceFph} we plot the mode-sum convergence of $\\bar{F}^\\text{res}_\\vp$ at the point\n$(\\bar{q}=5\\pi\/16,\\b_\\th \\bar{q}_0=5\\pi\/32)$ for all six resonant configurations.  Points refer to the $l$-mode\nresiduals that result from subtracting the analytically known and numerically fitted regularization parameters, while\nthe lines depict expected power-law convergence rates for large $l$.  In each resonance that we consider, the\nresiduals approach their expected asymptotic rates of convergence.\n\nWhile all of the models have the same asymptotic behavior at large $l$, Fig.~\\ref{fig:convergenceFph} demonstrates\nthat for low $l$ modes the $e=0.2$ sources converge faster than those with $e=0.5$, the 2:3 resonances converge\nfaster than the 1:2 resonances, and the 1:2 resonances converge faster than the 1:3 ones.  Higher eccentricities\nrequire a broader frequency spectrum to capture the radial motion.  Additionally, sources that orbit farther into\nthe strong field excite larger perturbations and require higher frequency modes to capture the behavior of the\nself-force.  The 1:3 resonances have the smallest pericentric separations, the 1:2 resonances have the next smallest,\nand the 2:3 resonances have the largest, which is reflected in varying rates of convergence at low $l$.\n\nGiven these factors, the $e05.13$ orbit presents the greatest challenge.  For this model it takes thousands of\nadditional modes to capture the behavior of the SSF compared to other resonant configurations.  Because of the slow\nconvergence at low multipoles, truncating mode summations at the same value of $l_\\text{max}$ as the other orbits\nwill introduce larger numerical errors in the retarded SSF contributions.  While these numerical errors are still\nrelatively small, they are significant enough that they make it much more difficult to fit for higher-order\nregularization parameters.  The accuracy of the conservative component of the SSF suffers because of this.  In\nconsequence, the conservative SSF is only known to $\\gtrsim 2$ digits of accuracy for the $e05.13$ orbit, with the\nnumerical error greatest when a component of the SSF is in the vicinity of passing through zero.  Fortunately, the\ndissipative component typically dominates over the conservative contribution in regions of the orbit where the\nconservative contribution is known less accurately.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.95\\textwidth]{Figures\/ssf_r_var_rmin_save}\n\t\\caption{Radial component of the SSF as a function of the resonant angle variable $\\bar{q}$,\n\ti.e., $\\bar{F}^\\text{res}_r(\\bar{q};\\bar{q}_0)$, for the six resonant geodesics listed in Table\n\t\\ref{tab:orbits}.  The SSF is weighted by the cube of the pericentric radius, $r_\\text{min}^3$, so that\n\tall six orbits are of comparable magnitude.  The dot-dashed (black) line represents the SSF for a\n\tresonant geodesic with an initial resonant phase of $\\b_\\th \\bar{q}_0=q_{\\th 0}=0$, while the solid (red)\n\tline represents the SSF for a resonance with the same orbital parameters but an initial resonant phase\n\tof $\\b_\\th \\bar{q}_0=q_{\\th 0}=-\\pi\/2$.  The shaded grey region represents all of the SSF values produced\n\tby varying the initial phase parameter $\\bar{q}_0$ from $0$ to $2\\pi$.}\n\\label{fig:ssfRVar}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.95\\textwidth]{Figures\/ssf_t_var_rmin_save}\n\t\\caption{Time component of the SSF as a function of the resonant\n\tangle variable $\\bar{q}$, i.e., $\\bar{F}^\\text{res}_t(\\bar{q};\\bar{q}_0)$,\n\tfor the six resonant geodesics listed in Table \\ref{tab:orbits}.  The dot-dashed\n\t(black) line represents the SSF for a resonant geodesic with an initial resonant\n\tphase of $\\b_\\th \\bar{q}_0=q_{\\th 0}=0$, the solid (red) line represents an initial\n\tresonant phase of $\\b_\\th \\bar{q}_0=q_{\\th 0}=-\\pi\/2$, and the shaded grey region\n\trepresents all of the SSF values produced by varying $\\bar{q}_0$ from $0$ to $2\\pi$.}\n\t\\label{fig:ssfTVar}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.95\\textwidth]{Figures\/ssf_th_var_rmin_save}\n\t\\caption{Polar component of the SSF as a function of the resonant angle\n\tvariable $\\bar{q}$, i.e., $\\bar{F}^\\text{res}_\\th(\\bar{q};\\bar{q}_0)$, for\n\tthe six resonant geodesics listed in Table \\ref{tab:orbits}.  The dot-dashed\n\t(black) line represents the SSF for a resonant geodesic with an initial resonant\n\tphase of $\\b_\\th \\bar{q}_0=q_{\\th 0}=0$, the solid (red) line represents an initial\n\tresonant phase of $\\b_\\th \\bar{q}_0=q_{\\th 0}=-\\pi\/2$, and the shaded grey region\n\trepresents all of the SSF values produced by varying $\\bar{q}_0$ from $0$ to $2\\pi$.}\n\t\\label{fig:ssfThVar}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.95\\textwidth]{Figures\/ssf_ph_var_rmin_save}\n\t\\caption{Azimuthal component of the SSF as a function of the resonant angle variable\n\t$\\bar{q}$, i.e., $\\bar{F}^\\text{res}_\\vp(\\bar{q};\\bar{q}_0)$, for the six resonant geodesics\n\tlisted in Table \\ref{tab:orbits}.  The dot-dashed (black) line represents the SSF for a\n\tresonant geodesic with an initial resonant phase of $\\b_\\th \\bar{q}_0=q_{\\th 0}=0$, the\n\tsolid (red) line represents an initial resonant phase of $\\b_\\th \\bar{q}_0=q_{\\th 0}=-\\pi\/2$,\n\tand the shaded grey region represents all of the SSF values produced by varying $\\bar{q}_0$\n\tfrom $0$ to $2\\pi$.}\n\t\\label{fig:ssfPhVar}\n\\end{figure*}\n\n\\subsection{Scalar self-force as a function of $\\bar{q}$}\n\\label{sec:FresQQ0}\n\nFor a resonant orbit we can present the SSF in a simple line plot as a function of the net angle variable\n$\\bar{q}$, as depicted in Figs.~\\ref{fig:ssfRVar}, \\ref{fig:ssfTVar}, \\ref{fig:ssfThVar}, and \\ref{fig:ssfPhVar}.\nIn these plots the SSF has been weighted by the cube of the pericentric radius of the orbit (i.e.,\n$r_\\text{min}^3\\,\\bar{F}^\\text{res}_\\a$), which more tightly bounds the variations in the SSF and facilitates\ncomparisons across different models.  Each plot shows the SSF variation with $\\bar{q}$ for two different initial\nconditions (i.e., values of $\\bar{q}_0$).  The dot-dashed (black) curves show the SSF when the initial polar phase\nis $\\b_\\th \\bar{q}_0=q_{\\th 0}=0$ (i.e., initial conditions $x^\\mu_p(\\la=0)=(0,r_\\text{min},\\th_\\text{min},0)$ and\n$u^r(0)=u^\\th(0)=0$), while the solid (red) curves show the SSF when $\\b_\\th \\bar{q}_0=q_{\\th 0}=-\\pi\/2$ (i.e., initial\nconditions $x^\\mu_p(\\la=0)=(0,r_\\text{min},\\pi\/2,0)$, $u^r(0)=0$, and $u^\\th(0)<0$).\\footnote{Note that $q_{\\th 0}$\nis held constant rather than $\\bar{q}_0$, because the same value of $\\bar{q}_0$ will generate different initial\nconditions for resonances with different values of $\\b_\\th$.}  The shaded grey regions depict the range of SSF\nvalues that result from varying the initial phases---either $q_{\\th 0}$ or $\\bar{q}_0$---through their entire range.\n\nThe SSF is, of course, periodic with respect to $\\bar{q}$, but interestingly for the 2:3 resonances\n$\\bar{F}^\\text{res}_t$, $\\bar{F}^\\text{res}_r$, and $\\bar{F}^\\text{res}_\\vp$ are additionally periodic on the half\ninterval $[0,\\pi]$.  This behavior arises in the Kerr background because the time, radial, and azimuthal components\nof the SSF are invariant under parity transformations (i.e., reflections $\\th_p \\rightarrow \\pi-\\th_p$), while the\npolar component flips sign \\cite{Warb15} (equally true of the gravitational self-force \\cite{Vand18}).  For a\n2:3 resonance, the radial motion of the orbit is identical on the intervals $[0,\\pi]$ and $[\\pi,2\\pi]$, while the\npolar motion is related by the parity transformation.  From this fact follows the repetition in $\\bar{F}^\\text{res}_t$,\n$\\bar{F}^\\text{res}_r$, and $\\bar{F}^\\text{res}_\\vp$, while also giving the reflection behavior\n$\\bar{F}^\\text{res}_\\th(\\bar{q};\\bar{q}_0)=-\\bar{F}^\\text{res}_\\th(\\bar{q}+\\pi;\\bar{q}_0)$.\n\nThese symmetries in the geodesic motion also manifest themselves in the number of low-frequency oscillations\nthat appear in the SSF components, particularly in the low-eccentricity orbits.  Focusing on $\\bar{F}^\\text{res}_r$\nin Fig.~\\ref{fig:ssfRVar}, the SSF locally peaks 6 times for the $e02.13$ and $e02.23$ models and 4 times in\nthe $e02.12$ case.  The peaks closely align with the epochs at which each orbit passes through its polar extrema.\nA similar behavior is also seen for $\\bar{F}^\\text{res}_t$, $\\bar{F}^\\text{res}_\\vp$, and the higher eccentricity\nmodels, though for $e=0.5$ it is more difficult to identify local peaks, particularly as the orbit approaches\napocenter.  For $\\bar{F}^\\text{res}_\\th$ in Fig.~\\ref{fig:ssfThVar}, the peaks align with the passage of the source\nthrough $\\th_\\text{min}$, while the troughs align with its passages through $\\pi-\\th_\\text{min}$.\n\nThe degree to which the SSF varies with respect to changes in initial phase depends primarily on which component\nof the SSF vector we consider.  The time component, $\\bar{F}^\\text{res}_t$ (Fig.~\\ref{fig:ssfTVar}), displays the\nleast effect of varying the initial conditions.  The azimuthal component, $\\bar{F}^\\text{res}_\\vp$\n(Fig.~\\ref{fig:ssfPhVar}), shows slightly greater variations with respect to initial conditions.  The radial\ncomponent, $\\bar{F}^\\text{res}_r$ (Fig.~\\ref{fig:ssfRVar}), is still more affected.  Finally, the polar angular\ncomponent, $\\bar{F}^\\text{res}_\\th$ (Fig.~\\ref{fig:ssfThVar}), displays the most significant variations.  To\nunderstand these variations, recall that the radial and polar position of the resonant source, $\\bar{r}_p$\nand $\\bar{\\th}_p$, depend on the angle variables according to\n\\begin{align}\n\t\\bar{r}_p&=\\hat{r}_p(q_r)=\\hat{r}(\\b_r \\bar{q}), \\\\\n\t\\bar{\\th}_p&=\\hat{\\th}_p(q_\\th+q_{\\th 0})=\\hat{\\th}(\\b_\\th \\bar{q}+\\b_\\th \\bar{q}_0).\n\\end{align}\nConsequently, a broader grey band indicates a stronger dependence on the polar motion.  Thus, $\\bar{F}^\\text{res}_t$\nprimarily depends on the radial motion of the source, while $\\bar{F}^\\text{res}_r$ is sensitive to both polar and\nradial motions.  In behavior opposite of $\\bar{F}^\\text{res}_t$, $\\bar{F}^\\text{res}_\\th$ is primarily dependent on\nthe polar motion of the orbit.  Finally, $\\bar{F}^\\text{res}_\\vp$ depends mostly on radial motion of the source,\nthough the polar position becomes important near pericenter.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.89\\textwidth]{Figures\/ssf_t_contour_save}\n\t\\caption{Time component of the SSF $\\hat{F}_t$ projected on the poloidal motion two-torus for the six\nsources listed in Table~\\ref{tab:orbits}.  The SSF is normalized by the cube of each source's pericenter distance.\nColors correspond to values of the self-force (see colorbar).  The self-force is constant along each (solid) contour\nline with tic labels in the colorbar giving the values on those contours.  The dot-dashed lines depict the resonant\nmotion for fiducial initial conditions ($\\bar{q}_0=0$).}\n\\label{fig:ssfTTorus}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.89\\textwidth]{Figures\/ssf_r_contour_save}\n\t\\caption{Radial component of the scalar self-force $\\hat{F}_r$ for the six orbits listed in\nTable~\\ref{tab:orbits}.}\n\t\\label{fig:ssfRTorus}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.9\\textwidth]{Figures\/ssf_th_contour_save}\n\t\\caption{Polar component of the scalar self-force $\\hat{F}_\\th$ for the six orbits listed in\n\tTable~\\ref{tab:orbits}.}\n\t\\label{fig:ssfThTorus}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.9\\textwidth]{Figures\/ssf_ph_contour_save}\n\t\\caption{Azimuthal component of the scalar self-force $\\hat{F}_\\vp$ for the six orbits listed in\n\tTable~\\ref{tab:orbits}.}\n\t\\label{fig:ssfPhTorus}\n\\end{figure*}\n\n\\subsection{Scalar self-force as a function of $q_r$ and $q_\\th$}\n\\label{sec:FresQrQth}\n\nAn alternative way to visualize the dependence of the SSF on the radial and polar motion of resonant orbits is\nto project the SSF components onto the two-torus spanned by $q_r+q_{r0}$ and $q_\\th+q_{\\th 0}$.  This projection\nis depicted in Figs.~\\ref{fig:ssfTTorus}, \\ref{fig:ssfRTorus}, \\ref{fig:ssfThTorus}, and \\ref{fig:ssfPhTorus}.\nWe again weight the SSF components by the cube of the pericentric radius of the orbit.  The dot-dashed (black) lines\ntrace the motion of an orbit with initial conditions $\\b_\\th \\bar{q}_0=q_{\\th 0}=0$.  Sampling the SSF as the source\nmoves along these tracks reproduces the black dot-dashed curves in Figs.~\\ref{fig:ssfRVar}, \\ref{fig:ssfTVar},\n\\ref{fig:ssfThVar}, and \\ref{fig:ssfPhVar}.  Maintaining previous notation, we refer to the SSF parametrized by\n$q_r+q_{r0}$ and $q_\\th+q_{\\th 0}$ as $\\hat{F}_\\a$.\n\nAs observed in Sec.~\\ref{sec:FresQQ0}, $\\hat{F}_t$ (shown in Fig.~\\ref{fig:ssfTTorus}) primarily depends on the\nradial motion, with little variation as the orbit advances along the $q_\\th$ axis.  As the contours show in\nFigs.~\\ref{fig:ssfRTorus} and \\ref{fig:ssfPhTorus}, the radial and azimuthal components, $\\hat{F}_r$ and\n$\\hat{F}_\\vp$ are sensitive to both the radial and polar motion, especially near pericenter.  Finally, the contours\nof the SSF seen in Fig.~\\ref{fig:ssfThTorus} clearly demonstrate the antisymmetry across the equatorial plane\nof $\\hat{F}_\\th$, as discussed in the previous section.\n\nIn agreement with previous investigations \\cite{WarbBara10,WarbBara11,Warb15,NasiOsbuEvan19} of the SSF, we see\nthat $\\hat{F}_t$ is strictly positive.  This contrasts with the gravitational self-force case where the time component\ncan become negative in both radiation and Lorenz gauge \\cite{Vand16,Vand18}.\\footnote{We do not try to draw any\nphysical interpretation from this behavior since the SSF and gravitational self-force are coordinate and (in the\nGSF case) gauge-dependent results.}  On the other hand, $\\hat{F}_r$ is predominantly negative across the entire\ntorus, though it becomes slightly positive near apocenter.  This behavior is consistent with the observation\n\\cite{WarbBara10} that higher black hole spin leads to an attractive radial SSF.  Large inclinations, on the other\nhand, lead predominantly to positive values of the SSF, as seen in SSF results for spherical orbits \\cite{Warb15}.\nHowever, those prior observations involved inclinations $x\\gtrsim 0.5$, which we did not consider here.\n\nInterestingly, while all of the SSF components peak in magnitude following pericenter passage, the magnitude of\nthese peaks grows for $\\hat{F}_t$ and $\\hat{F}_r$ as $r_\\text{min}$ decreases, while the peaks grow for\n$\\hat{F}_\\th$ and $\\hat{F}_\\vp$ as $r_\\text{min}$ increases.  The latter behavior is actually due to the factor of\n$r_\\text{min}^3$.  If one removes this weighting, then closer pericenter passages excite larger peaks in the SSF\nfor all components.  This suggests that the leading order behavior of $\\hat{F}_\\th$ and $\\hat{F}_\\vp$ is closer\nto $1\/r^2$, which one might expect based on dimensional analysis ($[F_{\\th,\\vp}\/F_{t,r}]_\\text{dim}\n\\sim [M]_\\text{dim}$).\n\n\\section{Evolution of the orbital constants}\n\\label{sec:flux}\n\n\\subsection{Overview}\n\nIn the presence of radiative losses and the self-force, the ordinarily constant quantities $\\mathcal{E}$,\n$\\mathcal{L}_z$, and $\\mathcal{Q}$ are perturbed and gradually evolve according to\n\\begin{gather}\n\\label{eqn:edot}\n\t\\dot{\\mathcal{E}} = - \\frac{q^2}{u^t} a_t, \\qquad \\qquad\n\t\\dot{\\mathcal{L}}_z = \\frac{q^2}{u^t} a_\\vp,\n\t\\\\ \\label{eqn:qdot}\n\t\\dot{\\mathcal{Q}}=\n\t \\frac{2}{u^t} \\left[q^2 K_{\\mu\\nu}u^\\mu a^\\nu - (\\mathcal{L}_z\n\t-a\\mathcal{E})(\\dot{\\mathcal{L}}_z-a\\dot{\\mathcal{E}})\n\t\\right],\n\\end{gather}\nwhere an overdot represents a derivative with respect to Boyer-Lindquist time, and the self-acceleration $a^\\mu$ is given\nby $\\mu a^\\nu = (g^{\\mu\\nu}+u^\\mu u^\\nu)F_\\mu = F^\\nu - q^{-2}u^\\nu d\\mu\/d\\tau$.  Note that the lack of orthogonality between\n$F_\\nu$ and $u^\\nu$ drives changes in the mass $\\mu$ (see\n\\ref{app:RestMass}).\n\nThe changes $\\dot{\\mathcal{E}}$, $\\dot{\\mathcal{L}}_z$, and $\\dot{\\mathcal{Q}}$ consist of both secularly growing\nand oscillating parts, with the secular piece found by orbit-averaging \\eqref{eqn:edot} and \\eqref{eqn:qdot} with\nrespect to $t$.  For a nonresonant orbit, the averaging is over a long timescale,\n\\begin{equation} \\label{eqn:orbitAvg}\n\t\\langle \\dot{\\mathcal{X}} \\rangle \\equiv \\lim_{T\\rightarrow \\infty}\n\t \\frac{1}{T} \\int_0^T \\dot{\\mathcal{X}} dt,\n\t\\qquad \\mathcal{X} =\n\t\\mathcal{E},\\, \\mathcal{L}_z,\\, \\mathcal{Q} .\n\\end{equation}\nThese averages produce the leading order, adiabatic evolution of the system \\cite{HindFlan08}.  The time integrals\ncan be reexpressed in terms of the angle variables that are used to parametrize the self-force.  Then the averaging\nis done over the motion on the torus \\cite{DrasHugh04,GrosLeviPere13}.  For nonresonant orbits, $\\dot{\\mathcal{E}}$,\n$\\dot{\\mathcal{L}}_z$, and $\\dot{\\mathcal{Q}}$ are averaged over the entire two-torus by integrating with equal weight\nover all $q_r$ and $q_\\th$.  For resonances, these orbit-averages are carried out over a single, one-dimensional\nclosed track on the torus, reducing \\eqref{eqn:orbitAvg} to a single integral over the resonant phase variable\n$\\bar{q}$,\n\\begin{align} \\label{eqn:work}\n\t\\mu \\langle \\dot{\\mathcal{E}} \\rangle & =\n\t- \\frac{q^2}{\\G} \\int_0^{2\\pi}  \\frac{d\\bar{q}}{2\\pi}\\,\n\t\\bar{\\Sig}_p\\,\n\t\\bar{F}^\\text{res}_t = q^2\\mathcal{W},\n\t\\\\ \\label{eqn:torque}\n\t\\mu \\langle \\dot{\\mathcal{L}}_z \\rangle &=\n\t\\frac{q^2}{\\G} \\int_0^{2\\pi}  \\frac{d\\bar{q}}{2\\pi}\\,\n\t\\bar{\\Sig}_p\\,\n\t\\bar{F}^\\text{res}_\\vp = q^2\\mathcal{T},\n\t\\\\  \\label{eqn:qdotAvg}\n\t\\mu \\langle \\dot{\\mathcal{Q}} \\rangle &=\n\t2q^2\\Big[- (\\mathcal{L}_z\n\t-a\\mathcal{E})(\\dot{\\mathcal{T}}-a\\dot{\\mathcal{W}})\n\t\\\\\n\t& \\qquad \\qquad \\; +\n\t\\frac{1}{\\G} \\int_0^{2\\pi}  \\frac{d\\bar{q}}{2\\pi}\\,\n\t\\bar{\\Sig}_p\\,\\bar{K}_p^{\\mu\\nu}\n\t\\bar{u}_\\mu \\bar{F}^\\text{res}_\\nu \\Big] .\n\t\\notag\n\\end{align}\nIn the expressions above, all quantities with an overbar are understood to be functions of $\\bar{q}$ and\nparametrized by $\\bar{q}_0$ [e.g.,\n$\\bar{\\Sig}_p=\\bar{\\Sig}(\\bar{q}; \\bar{q}_0)=\\bar{r}_p^2(\\bar{q})+a^2\\cos^2\\bar{\\th}_p(\\bar{q}+\\bar{q}_0)$].\nThe changes $\\langle \\dot{\\mathcal{E}} \\rangle$ and $\\langle \\dot{\\mathcal{L}}_z \\rangle$ are directly related to\nthe average rate of work $\\mathcal{W}$ and torque $\\mathcal{T}$ done on the small body by the SSF (per charge squared)\nand incorporate the fact that the average change in $\\mu$ vanishes (see \\ref{app:RestMass}).\n\nFor nonresonant orbits the conservative components of the self-force vanish when averaged over the entire torus.\nThis fact can be seen from the symmetries of \\eqref{eqn:consSSF} and \\eqref{eqn:ssfRet2Adv}, combined with the\nexpressions for $\\dot{\\mathcal{E}}$, $\\dot{\\mathcal{L}}_z$, and $\\dot{\\mathcal{Q}}$.  Only the dissipative\nself-force contributes to the leading order adiabatic evolution of the system when it is off resonance.  When on\nresonance, we cannot make use of these same symmetries to discard the conservative component of the self-force in\n\\eqref{eqn:work}, \\eqref{eqn:torque}, and \\eqref{eqn:qdotAvg}.  However, flux-balance conditions do confirm that\nconservative contributions to $\\langle \\dot{\\mathcal{E}}\\rangle$ and $\\langle \\dot{\\mathcal{L}}_z\\rangle$ continue\nto vanish, as we further discuss in Sec.~\\ref{sec:EnAndLz}.  Additionally, the averages over an $r\\th$ resonance\nretain their dependence on $\\bar{q}_0$, meaning that they vary according to the initial phase at which the system\nenters a resonance, as demonstrated previously \\cite{FlanHughRuan14,BerrETC16}.  Thus different initial conditions\ncan either diminish or enhance the averaged evolution of $\\mathcal{E}$, $\\mathcal{L}_z$, and $\\mathcal{Q}$ during\na resonance.  The following subsections detail this behavior in the scalar case and provide numerical data on how\nthe conservative and dissipative components of the SSF contribute to $\\langle \\dot{\\mathcal{E}} \\rangle$,\n$\\langle \\dot{\\mathcal{L}}_z \\rangle$, and $\\langle \\dot{\\mathcal{Q}} \\rangle$.\n\n\\subsection{Energy and angular momentum changes for a resonant orbit}\n\\label{sec:EnAndLz}\n\nFlux-balance equates the average changes in the orbital energy and angular momentum,\n$\\langle \\dot{\\mathcal{E}} \\rangle$ and $\\langle \\dot{\\mathcal{L}}_z \\rangle$, to the average radiative fluxes\n\\cite{Galt82,QuinWald99,Mino03}.  For energy, the average work $\\mathcal{W}$ done by the SSF balances the total flux\n$\\langle \\dot{E}\\rangle^\\mathrm{tot}$ radiated by the scalar field to infinity and down the horizon, with the\non-resonance fluxes having slightly modified expressions\n\\begin{align}\n\\label{eqn:fluxEBalance}\n\t-\\mathcal{W} &= \\langle \\dot{E} \\rangle^\\mathrm{tot} \\equiv\n\t\\langle \\dot{E} \\rangle^\\mathcal{H}\n\t+ \\langle \\dot{E} \\rangle^\\infty,\n\t\\\\\n\t\\langle \\dot{E} \\rangle^\\mathcal{H} & = \\frac{1}{4\\pi}\n\t\\sum_{\\hat{l} = 0}^\\infty \\sum_{m=-\\hat{l}}^{\\hat{l}} \\sum_{N=-\\infty}^\\infty\n\t\\o_{mN} \\g_{mN} |\\bar{C}^-_{\\hat{l} mN}|^2,\n\t\\\\\n\t\\langle \\dot{E} \\rangle^\\infty & = \\frac{1}{4\\pi}\n\t\\sum_{\\hat{l} = 0}^\\infty \\sum_{m=-\\hat{l}}^{\\hat{l}} \\sum_{N=-\\infty}^\\infty\n\t\\o_{mN}^2 |\\bar{C}^+_{\\hat{l} mN}|^2 .\n\\end{align}\nHere $\\langle \\dot{E} \\rangle^\\mathcal{H}$ is the energy flux (per charge squared) through the horizon, and\n$\\langle \\dot{E} \\rangle^\\infty$ is the energy flux (per charge squared) at infinity, with\n$\\g_{mN}\\equiv \\o_{mN} - ma\/(2Mr_+)$ being the spatial frequency of the radial modes at the horizon and\n$r_+ \\equiv M+\\sqrt{M^2-a^2}$ denoting the radius of the outer horizon.  In the resonant case, the fluxes include\na single sum over the net harmonic number $N$ of the net amplitudes $\\bar{C}^\\pm_{\\hat{l} mN}$.  The net amplitudes\nare themselves sums \\eqref{eqn:ClmN2Clmkn} over amplitudes $\\hat{C}^\\pm_{\\hat{l} mkn}$ with individual radial and\npolar harmonic numbers $n$ and $k$.  These underlying sums reflect the coherence between all harmonics of the\nradial and polar librations that contribute to a given $N$.  In this way, interference terms appear in the flux\nthat would otherwise average to zero in the off-resonance case.\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[width=0.42\\textwidth]{Figures\/enLzFlux_save.pdf}\n\t\\caption{Residual variations in the energy (top panel) and angular\n\tmomentum (bottom panel) fluxes as a function of the initial phase\n\t$q_{\\th 0}=\\b_\\th \\bar{q}_0$ for the $e02.23$ (solid black curves),\n\t$e05.23$ (dot-dashed blue curves), and\n\t$e05.12$ (dotted red curves) orbits in in Table \\ref{tab:orbits}.\n\t}\n\t\\label{fig:fluxRes}\n\\end{figure}\n\n{\\renewcommand{\\arraystretch}{1.75}\n\\begin{table*}[!bthp]\n\\caption{Energy and angular momentum fluxes for the resonant-orbit models listed in Table \\ref{tab:orbits}.\nFluxes through the horizon, $\\langle\\dot{E}\\rangle^\\mathcal{H}$ and $\\langle\\dot{L}_z\\rangle^\\mathcal{H}$,\nand infinity, $\\langle\\dot{E}\\rangle^\\infty$ and $\\langle\\dot{L}_z\\rangle^\\infty$, are included.  Each model contains\na row of fluxes for an orbit with initial phase $q_{\\th 0} = \\b_\\th\\bar{q}_0=0$ and a row of fluxes for an orbit with\n$q_{\\th 0} = \\b_\\th\\bar{q}_0= -\\pi\/2$.  A third row in each case shows the $\\bar{q}_0$ averages of the fluxes as\ndefined by \\eqref{eqn:fluxAvg}, which ignores constructive and destructive interference terms.  The reported\nprecision in each flux indicates the accuracy of each calculation (though we truncate more accurate results at nine\ndecimal places).  The total fluxes are also compared to the local work and torque due to the SSF, $\\mathcal{W}$ and\n$\\mathcal{T}$, to illustrate the (orbit-averaged) fractional errors in the flux balance relations.  These errors\nrange from $\\sim 10^{-11}$ to $\\sim 10^{-5}$, reflecting the numerical accuracy of our SSF results.}\n\t\\label{tab:fluxes}\n\t\\centering\n\t\\begin{tabular*}{\\textwidth}{c @{\\extracolsep{\\fill}} c c c c c c c}\n\t\t\\hline\n\t\t\\hline\n\t\t\\vspace{-10pt}\n\t\t\\\\\n\t\tModel\n\t\t& $\\b_\\th\\bar{q}_0$\n\t\t& $\\langle\\dot{E}\\rangle^\\mathcal{H}\\times {10^5}$\n\t\t& $\\langle\\dot{L}_z\\rangle^\\mathcal{H}\\times {10^4}$\n\t\t& $\\langle\\dot{E}\\rangle^\\infty\\times {10^3}$\n\t\t& $\\langle\\dot{L}_z\\rangle^\\infty\\times {10^3}$\n\t\t& $\\left|1+\\frac{\\langle\\dot{E}\\rangle^\\text{tot}}{\\mathcal{W}}\\right|$\n\t\t& $\\left|1+\\frac{\\langle\\dot{L}_z\\rangle^\\text{tot}}\n\t\t{\\mathcal{T}}\\right|$\n\t\t\\\\\n\t\t\\vspace{-10pt}\n\t\t\\\\\n\t\t\\hline\n\t\n\t\t$e02.13$ & $0$ & $-4.411457095$ & $-7.017966266$\n\t\t& $1.301535\\phantom{000}$ & $7.677846\\phantom{000}$ & $9\\times 10^{-7}$ &\n\t\t$8\\times 10^{-7}$\n\t\t\\\\\n\t\t& $-\\pi\/2$ & $-4.411497781$ & $-7.017992874$\n\t\t& $1.301534\\phantom{000}$ & $7.677831\\phantom{000}$ & $7\\times 10^{-7}$ &\n\t\t$6\\times 10^{-7}$\n\t\t\\\\\n\t\t& avg & $-4.411477437$ & $-7.017979570$\n\t\t& $1.301535\\phantom{000}$ & $7.677838\\phantom{000}$ & - & -\n\t\t\\\\ \\hline\n\t\t$e02.12$ & 0 & $-2.021123696$ & $-3.395925026$\n\t\t& $0.5737075\\phantom{00}$ & $4.843929\\phantom{000}$ & $2\\times 10^{-8}$ &\n\t\t$2\\times 10^{-8}$\n\t\t\\\\\n\t\t& $-\\pi\/2$ & $-2.021127357$ & $-3.396083610$\n\t\t& $0.5736988\\phantom{00}$ & $4.843824\\phantom{000}$ & $2\\times 10^{-8}$ &\n\t\t$2\\times 10^{-8}$\n\t\t\\\\\n\t\t& avg & $-2.021125529$ & $-3.396004318$\n\t\t& $0.5737031\\phantom{00}$ & $4.483877\\phantom{000}$ & - & -\n\t\t\\\\ \\hline\n\t\t$e02.23$ & $0$ & $-0.324830139$ & $-0.877896699$\n\t\t& $0.134964247$ & $1.762343845$ & $6\\times 10^{-9}$ &\n\t\t${}^{\\phantom{1}}3\\times 10^{-11}$\n\t\t\\\\\n\t\t& $-\\pi\/2$ & $-0.325170299$ & $-0.877675562$\n\t\t& $0.134984611$ & $1.762586419$ & $6\\times 10^{-9}$ &\n\t\t${}^{\\phantom{1}}3\\times 10^{-11}$\n\t\t\\\\\n\t\t& avg & $-0.325000220$ & $-0.877786129$\n\t\t& $0.134974429$ & $1.762465132$ & - & -\n\t\t\\\\ \\hline\n\t\t$e05.13$ & $0$ & $-0.482340196$ & $-6.445882073$\n\t\t& $1.45739\\phantom{0000}$ & $7.28846\\phantom{0000}$ & $9\\times 10^{-5}$ &\n\t\t$8\\times 10^{-5}$\n\t\t\\\\\n\t\t& $-\\pi\/2$ & $-0.480744834$ & $-6.448440589$\n\t\t& $1.45724\\phantom{0000}$ & $7.28681\\phantom{0000}$ & $9\\times 10^{-5}$ &\n\t\t$8\\times 10^{-5}$\n\t\t\\\\\n\t\t& avg & $-0.481543289$ & $-6.447161427$\n\t\t& $1.45731\\phantom{0000}$ & $7.28763\\phantom{0000}$ & - & -\n\t\t\\\\ \\hline\n\t\t$e05.12$ & $0$ & $-0.364726314$ & $-2.915401042$\n\t\t& $0.590229\\phantom{000}$ & $3.85672\\phantom{0000}$ & $3\\times 10^{-5}$ &\n\t\t$2\\times 10^{-5}$\n\t\t\\\\\n\t\t& $-\\pi\/2$ & $-0.361475840$ & $-2.919310921$\n\t\t& $0.589990\\phantom{000}$ & $3.85408\\phantom{0000}$ & $3\\times 10^{-5}$ &\n\t\t$2\\times 10^{-5}$\n\t\t\\\\\n\t\t& avg & $-0.363102237$ & $-2.917355986$\n\t\t& $0.590110\\phantom{000}$ & $3.85540\\phantom{0000}$ & - & -\n\t\t\\\\ \\hline\n\t\t$e05.23$ & $0$ & $\\phantom{-}0.092800200$ &\n\t\t$-0.713304108$\n\t\t& $0.12832694\\phantom{0}$ & $1.3990094\\phantom{00}$ & $3\\times 10^{-7}$ &\n\t\t$2\\times 10^{-7}$\n\t\t\\\\\n\t\t& $-\\pi\/2$ & $\\phantom{-}0.090053088$ & $-0.710893569$\n\t\t& $0.12855813\\phantom{0}$ & $1.4017751\\phantom{00}$ & $3\\times 10^{-7}$ &\n\t\t$2\\times 10^{-7}$\n\t\t\\\\\n\t\t& avg & $\\phantom{-}0.091426409$ & $-0.712098512$\n\t\t& $0.12844253\\phantom{0}$ & $1.4003923\\phantom{00}$ & - & -\n\t\t\\\\\n\t\t\\hline\n\t\t\\hline\n\t\\end{tabular*}\n\\end{table*}\n}\n\nIn a similar way the average torque $\\mathcal{T}$ applied by the SSF balances the sum of the angular momentum flux\nat the horizon $\\langle \\dot{L}_z \\rangle^\\mathcal{H}$ and at infinity $\\langle \\dot{L}_z \\rangle^\\infty$,\n\\begin{align}\n\\label{eqn:fluxLzBalance}\n\t-\\mathcal{T} &= \\langle \\dot{L}_x \\rangle^\\mathrm{tot} \\equiv\n\t\\langle \\dot{L}_z \\rangle^\\mathcal{H}\n\t+ \\langle \\dot{L}_z \\rangle^\\infty,\n\t\\\\\n\t\\langle \\dot{L}_z \\rangle^\\mathcal{H} & = \\frac{1}{4\\pi}\n\t\\sum_{\\hat{l} = 0}^\\infty \\sum_{m=-\\hat{l}}^{\\hat{l}} \\sum_{N=-\\infty}^\\infty\n\tm \\g_{mN} |\\bar{C}^-_{\\hat{l} mN}|^2,\n\t\\\\ \\label{eqn:fluxLzBalanceLast}\n\t\\langle \\dot{L}_z \\rangle^\\infty & = \\frac{1}{4\\pi}\n\t\\sum_{\\hat{l} = 0}^\\infty \\sum_{m=-\\hat{l}}^{\\hat{l}} \\sum_{N=-\\infty}^\\infty\n\tm\\o_{mN} |\\bar{C}^+_{\\hat{l} mN}|^2 .\n\\end{align}\nRecall from \\eqref{eqn:ClmN2Clmkn} that the net amplitude $\\bar{C}^\\pm_{\\hat{l} mN}= \\bar{C}^\\pm_{\\hat{l} mN}(\\bar{q}_0)$\nis a function of $\\bar{q}_0$, which captures the effect on the fluxes of the phase of the resonant orbit.\n\nIn a numerical calculation the fluxes tend to converge exponentially with increasing numbers of modes and only\nrequire calculation of the matching coefficients $\\bar{C}^\\pm_{\\hat{l} mN}$, not the full SSF.  The effect is that the\nfluxes can usually be computed to high accuracy.  In Table \\ref{tab:fluxes} we report numerical values for the\ntotal fluxes (at infinity and the horizon) for two different phase parameters, $q_{\\th 0} = \\b_\\th \\bar{q}_0 = 0$\nand $-\\pi\/2$, and for each of the models outlined in Table \\ref{tab:orbits}.  Consistent with calculations of\ngravitational fluxes \\cite{Misn72}, most of the horizon fluxes are negative due to superradiant scattering (each\nmodel has primary spin of $a\/M = 0.9$).  As expected, orbits with smaller pericentric distances $r_\\text{min}$ tend\nto produce larger fluxes, while eccentricity has a smaller effect.\n\nIn Table \\ref{tab:fluxes} we also list the computed average over $\\bar{q}_0$ of the resonant-orbit fluxes\n\\begin{equation} \\label{eqn:fluxAvg}\n\t\\langle\\langle \\dot{X}\\rangle\\rangle_{\\bar{q}_0}\n\t\\equiv \\frac{1}{2\\pi}\\int_0^{2\\pi} \\langle \\dot{X} \\rangle\\;\n\td\\bar{q}_0,\n\\end{equation}\nwhere $X = E, L_z$.  This average over $\\bar{q}_0$ (i.e., a double averaging) gives the flux that would be seen in\na system with nearly the same orbital parameters but infinitesimally off resonance so that its motion ergodically\nfills the torus.  It is equivalent to computing the fluxes with the normal incoherent sum of terms with\n$\\vert \\hat{C}^\\pm_{\\hat{l} mkn} \\vert^2$ over all $n$ and $k$.  With \\eqref{eqn:fluxAvg} giving a background average,\nwe can define the residual variation (enhancement or diminishment) in the fluxes that arise on resonance,\n\\begin{equation} \\label{eqn:resVar}\n\t\\langle \\d \\dot{X} \\rangle \\equiv\n\t\\langle \\dot{X} \\rangle - \\langle\\langle\n\t\\dot{X}\\rangle\\rangle_{\\bar{q}_0}.\n\\end{equation}\nWe plot $\\langle \\d\\dot{E}\\rangle^\\mathrm{tot}$ and\n$\\langle \\d\\dot{L}_z\\rangle^\\mathrm{tot}$ for the $e02.12$ (solid black lines), $e02.23$ (dashed red lines), and\n$e05.23$ (dot dashed lines) orbits in Fig.~\\ref{fig:fluxRes}.  By comparing these figures to the values in Table\n\\ref{tab:fluxes}, we see that the residual variations are relatively small compared to the magnitudes of the total\nfluxes, but they are still greater than the $\\sim 10^{-8}$ fractional error in our numerical calculations.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.9\\textwidth]{Figures\/workTorque_allOrbits_save.pdf}\n\t\\caption{Separate contributions from the dissipative and conservative components of the self-force to the\n\tresidual variations $\\langle \\d \\dot{\\mathcal{E}} \\rangle$ in energy (top row) and\n\t$\\langle \\d \\dot{\\mathcal{L}}_z \\rangle$ in angular momentum (bottom row) for the $e02.12$ (left),\n\t$e02.23$ (middle), and $e05.23$ (right) orbits plotted as functions of the resonant-orbit phase.\n\tContributions from the dissipative self-force are given by the solid (black) curves, while contributions from\n\tthe conservative self-force are depicted by the dashed (red) curves.  The (grey) shaded region represents the\n\tformal uncertainty in our calculation of the conservative SSF, $\\langle \\d \\dot{\\mathcal{Q}} \\rangle^\\mathrm{cons} \\pm \\s_\\mathrm{cons}$, due to fitting for\n\thigher-order regularization parameters.  The calculation of $\\s_\\mathrm{cons}$ is discussed below\n\t(in Sec.~\\ref{sec:EnAndLz}).  As expected, the residual variations in energy and angular momentum arising from\n\tthe conservative SSF (dashed\/red curves) are consistent with zero.\n\t}\n\t\\label{fig:workTorque}\n\\end{figure*}\n\nIn the case of the 2:3 resonances plotted as dashed (red) and dot-dashed (blue) curves in Fig.~\\ref{fig:fluxRes},\nthe energy and angular momentum fluxes are minimized when the motion possesses simultaneous turning points in $r$\nand $\\th$ (i.e., $\\b_{\\th}\\bar{q}_{0}=0,\\pi$).  While not plotted here, the other 2:3 resonance ($e05.23$) shares\nthis behavior.  On the other hand, for the 1:2 resonance in Fig.~\\ref{fig:fluxRes}, the energy and angular momentum\nfluxes are maximized when the motion possesses simultaneous turning points, a feature which is shared by the other\n1:2- and 1:3-resonant orbits.  Evidence of this behavior can also be found in Table \\ref{tab:fluxes}.  Following\nthe work of \\cite{FlanHughRuan14}, we also report in Appendix \\ref{app:fracVarFlux} the total fractional variations\n$\\D \\dot{E}$ and $\\D \\dot{L}_z$ as defined by (4.5) in \\cite{FlanHughRuan14}.\n\nAdditionally, we can make use of the flux-balance laws to test the accuracy of our SSF data.  The fractional errors\nbetween the fluxes and the work $\\mathcal{W}$ and torque $\\mathcal{T}$, computed via \\eqref{eqn:fluxEBalance}\nand \\eqref{eqn:fluxLzBalance}, are given in the last two columns of Table \\ref{tab:fluxes}.  We find good agreement,\nwith fractional errors of $\\sim 10^{-11}$-$10^{-5}$.  Recalling the numerical convergence of the SSF displayed in\nFig.~\\ref{fig:convergenceFph}, these fractional errors are in line with the predicted numerical accuracy of our\nSSF data.\n\nThe good agreement between our flux and SSF results is a further way of seeing that the conservative component of\nthe SSF does not contribute to $\\langle \\dot{\\mathcal{E}} \\rangle$ and $\\langle \\dot{\\mathcal{L}}_z \\rangle$.  The\nfluxes are \\emph{purely dissipative} quantities, and for the flux-balance laws to hold, only the dissipative component\nof the SSF can contribute to the averages $\\langle \\dot{\\mathcal{E}} \\rangle$ and\n$\\langle \\dot{\\mathcal{L}}_z \\rangle$,\neven during resonances \\cite{IsoyETC19}.\n\nTo verify this, we calculate separately the contributions of the\ndissipative and conservative SSF to the residual variations $\\langle \\d \\dot{\\mathcal{E}} \\rangle$ and\n$\\langle \\d \\dot{\\mathcal{L}}_z \\rangle$ by replacing $\\bar{F}_\\a^\\mathrm{res}$ with $\\bar{F}_\\a^\\mathrm{diss}$\nand $\\bar{F}_\\a^\\mathrm{cons}$ in \\eqref{eqn:work} and \\eqref{eqn:torque}.\nIn Fig.~\\ref{fig:workTorque} we plot the residual variations for the $e02.12$ (left), $e02.23$ (middle), and\n$e05.23$ (right) orbits.  The solid (black) curves correspond to the dissipative contributions, which share the same\nvarying behavior as seen in Fig.~\\ref{fig:fluxRes} [note the opposite sign in \\eqref{eqn:fluxEBalance} and\n\\eqref{eqn:fluxLzBalance}].  The dashed (red) curves correspond to the conservative contributions, and the filled\n(grey) regions plot the estimated uncertainty of the conservative contributions due to truncation of the\nregularization procedure, which affects the conservative SSF.\n\nThe uncertainty in the conservative contributions originates in our mode-sum regularization of the SSF.  As summarized\nin Sec.~\\ref{sec:ssfReg} [and discussed with additional detail in \\cite{NasiOsbuEvan19}, Sec.~IV A, the paragraph\nfollowing Eq.~(4.11)], we regularize our SSF data via mode-sum regularization, but must numerically fit for\nhigher-order regularization parameters in order to improve the numerical convergence of our $l$-mode sum.  Without\nextrapolating these higher-order terms, our regularized results would be dominated by truncation errors of\n$O(l_\\mathrm{max}^{-1})$.  Our extrapolation procedure typically enhances truncation error scaling to\n$O(l_\\mathrm{max}^{-7})$ but it also introduces systematic uncertainties, since our extrapolated results depend on\nhow many terms we include in our numerical fits and which multipole SSF modes we use to produce those fits.  These\nsystematic uncertainties tend to be much larger than what the improved truncation error scaling would naively suggest, and\nthus become the dominant form of error in our numerical conservative SSF results.\\footnote{Recall that only the\nconservative component of the SSF needs to be regularized.}  Therefore, any quantity that depends on the conservative\nself-force will also have an estimated uncertainty associated with the fitting.  We estimate the uncertainty of the\nnewly calculated quantity by propagating uncertainties in additive terms,\ni.e.,\n\\begin{equation}\n\t\\sigma_f^2 = \\left\\vert \\frac{\\partial f}{\\partial x_{0}} \\right\\vert^2\n\t\\sigma_{x_0}^2 + \\left\\vert \\frac{\\partial f}{\\partial x_1} \\right\\vert^2\n\t\\sigma_{x_1}^2 + \\cdots\n\t+ \\left\\vert \\frac{\\partial f}{\\partial x_n} \\right\\vert^2\n\t\\sigma_{x_n}^2 ,\n\\end{equation}\nwhere $\\sigma_f$ is the propagated uncertainty of a quantity $f$ due to its dependence on $n$ (assumed independent)\nparameters $(x_0, x_1, \\dots, x_n)$ with (assumed uncorrelated) errors\n$(\\sigma_{x_0}, \\sigma_{x_1}, \\dots, \\sigma_{x_n})$.  For example,\n$\\langle \\d \\dot{\\mathcal{E}} \\rangle^\\mathrm{cons}$ and its uncertainty are explicitly computed via the sums\n\\begin{align}\n\t\\langle \\d \\dot{\\mathcal{E}} \\rangle^\\mathrm{cons}(\\bar{q}_0) &= \\frac{q^2}{N}\n\t\\sum_{n = 0}^{N-1} \\bar{\\Sigma}_p\\left(\\frac{2\\pi i n}{N};\\bar{q}_0\\right)\n\t\\bar{F}^\\mathrm{cons}_t\\left(\\frac{2\\pi i n}{N}; \\bar{q}_0\\right), \\notag\n\t\\\\\n\t\\sigma_\\mathrm{cons}^2(\\bar{q}_0) &= \\frac{q^4}{N^2} \\sum_{n=0}^{N-1}\n\t\\bar{\\Sigma}_p^2\\left(\\frac{2\\pi i n}{N};\\bar{q}_0\\right)\n\t\\sigma_{t}^2\\left(\\frac{2\\pi i n}{N};\\bar{q}_0\\right), \\label{eqn:propErrorEn}\n\\end{align}\nwhere $\\sigma_{t}(\\bar{q};\\bar{q}_0)$ is the estimated uncertainty of $\\bar{F}_t^\\mathrm{cons}(\\bar{q};\\bar{q}_0)$\nfrom our fitting procedure.  The first line of \\eqref{eqn:propErrorEn} is obtained by replacing the integrand of\n\\eqref{eqn:work} with its discrete Fourier transform.\\footnote{Recall that $\\langle\\langle \\dot{\\mathcal{E}} \\rangle\\rangle_{\\bar{q}_0}$\nvanishes exactly due to the symmetries of Kerr geodesics, and, thus,\n$\\langle\\d \\dot{\\mathcal{E}}\\rangle^\\mathrm{cons} = \\langle \\dot{\\mathcal{E}}\\rangle^\\mathrm{cons}$.}\n\nNo uncertainty estimates for the dissipative\ncontributions are included, as these are orders of magnitude smaller.  While the conservative\npart leaves behind a nonzero numerical result, these variations fall well below our estimated uncertainty and are\nthus consistent with zero, as expected.  The estimated uncertainty is much larger for the $e02.12$ model due to the\nslower convergence of the SSF for orbits that lie deeper in the strong field (see Fig.~\\ref{fig:convergenceFph}).\nIn that model, our formal uncertainty far exceeds not only the conservative contributions but even the dissipative\nvariations, which may point to the formal uncertainties being too conservative.\n\n\\subsection{Carter constant and the integrability conjecture}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{Figures\/Qdot_allOrbits_errors.pdf}\n\t\\caption{Contributions from the dissipative and conservative components of the self-force to\n\t$\\langle \\d \\dot{\\mathcal{Q}}\\rangle$ for the $e02.12$ (left), $e02.23$ (middle), and $e05.23$ (right)\n\torbits.  Contributions from the dissipative self-force are given by the solid (black) curves, while\n\tcontributions from the conservative self-force are depicted by the dashed (red) curves.  The (grey) shaded\n\tregion represents the formal uncertainty in our calculation of the conservative SSF, $\\langle \\d \\dot{\\mathcal{Q}} \\rangle^\\mathrm{cons} \\pm \\s_\\mathrm{cons}$, due\n\tto fitting for higher-order regularization parameters.  The calculation of $\\s_\\mathrm{cons}$ is discussed in detail\n\tin Sec.~\\ref{sec:EnAndLz}.\n\t}\n\t\\label{fig:dQdot}\n\\end{figure*}\n\nUnlike $\\langle \\dot{\\mathcal{E}}\\rangle$ and $\\langle \\dot{\\mathcal{L}}_z\\rangle$, $\\langle \\dot{\\mathcal{Q}}\\rangle$\nis not associated with a radiation flux.  Instead, we must directly calculate the orbit-averaged rate of change of\nthe Carter constant from the self-force,\n\\begin{align} \\label{eqn:QdotAvgMino}\n\t\\langle \\dot{\\mathcal{Q}} \\rangle\n\t= \\frac{1}{\\G} \\left\\langle \\Sig \\frac{d\\mathcal{Q}}{d\\tau}\n\t \\right\\rangle_\\la .\n\\end{align}\nHere $\\langle \\mathcal{X}\\rangle_\\la$ refers to an average over $\\mathcal{X}(\\la)$ with respect to Mino time $\\la$,\nand the proper time derivative of $\\mathcal{Q}$ is given by\n\\begin{align}\n\t\\frac{\\mu}{2}\\frac{d\\mathcal{Q}}{d\\tau} &= \\label{eqn:QTauDotTh}\n\tq^2 (\\csc^2\\th \\mathcal{L}_z-a \\mathcal{E})(F_\\vp+a\\sin^2\\th F_t)\n\t\\\\ \\notag\n\t& \\qquad + q^2 u_\\th F_\\th - q^2(\\mathcal{L}_z-a\\mathcal{E})\n\t\\left(F_\\vp+aF_t\\right)\n\t\\\\ \\notag\n\t&\\qquad \\qquad \\qquad \\qquad\n\t- q^2(\\mathcal{Q} -  a^2\\cos^2\\th) u^\\a F_\\a,\n\t\\\\\n\t&= \\label{eqn:QTauDotR}\n\tq^2 \\D^{-1}(a \\mathcal{L}_z-\\varpi^2 \\mathcal{E})\n\t\\left(a F_\\vp+\\varpi^2 F_t\\right)\n\t\\\\ \\notag\n\t&\\qquad -  q^2 \\D u_r F_r - q^2(\\mathcal{L}_z-a\\mathcal{E})\n\t\\left(F_\\vp+aF_t \\right)\n\t\\\\ \\notag\n\t&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad\n\t- q^2(\\mathcal{Q}+  r^2 ) u^\\a F_\\a.\n\\end{align}\n(See Appendix \\ref{app:Qdot}.)\n\nFor nonresonant orbits, the conservative contributions to $\\langle \\dot{\\mathcal{Q}} \\rangle$ vanish due to\nsymmetries of the motion.  This can be seen by reexpressing \\eqref{eqn:QdotAvgMino} as a two-dimensional integral\nover $q_r$ and $q_\\theta$ \\cite{DrasHugh04,GrosLeviPere13},\n\\begin{equation} \\label{eqn:QdotAverageNonRes}\n\t\\langle \\dot{\\mathcal{Q}} \\rangle = \\frac{1}{\\G} \\int_0^{2\\pi} \\frac{dq_r}{2\\pi}\n\t\\int_0^{2\\pi} \\frac{dq_\\th}{2\\pi} \\left(\\Sig \\frac{d\\mathcal{Q}}{d\\tau}\n\t\\right).\n\\end{equation}\nRecall from \\eqref{eqn:consSSF} and \\eqref{eqn:ssfRet2Adv} that the two components $F^\\mathrm{cons}_{t,\\vp}$ are\nantisymmetric on the $q_r$-$q_{\\th}$ torus, while the other two components $F^\\mathrm{cons}_{r,\\theta}$ are\nsymmetric on the torus, so that\n\\begin{align}\n\tF^\\mathrm{cons}_{t,\\vp}(2\\pi-q_r, 2\\pi-q_\\theta)\n&= - F^\\mathrm{cons}_{t,\\vp}(q_r, q_\\theta),\n\\\\\n\tF^\\mathrm{cons}_{r,\\theta}(2\\pi-q_r, 2\\pi-q_\\theta)\n&= +F^\\mathrm{cons}_{r,\\theta}(q_r, q_\\theta).\n\\end{align}\nHence, if we only make use of the conservative components of the self-force in \\eqref{eqn:QTauDotTh} or\n\\eqref{eqn:QTauDotR}, then $dQ^\\mathrm{cons}\/d\\tau$ is also antisymmetric.  Because $\\Sig$ is symmetric with respect\nto the angle variables, the conservative contributions must vanish when integrated over the entire torus in\n\\eqref{eqn:QdotAverageNonRes}.  By disregarding these conservative perturbations and relating\n$\\langle \\dot{\\mathcal{Q}}\\rangle$ to the purely radiative (dissipative) piece of the perturbing field\n\\cite{Mino03,SagoETC06}, \\eqref{eqn:QdotAvgMino} reduces to a weighted mode sum over the field's asymptotic\namplitudes \\cite{DrasFlanHugh05,SagoETC06,FlanHughRuan14}, akin to\n\\eqref{eqn:fluxEBalance}-\\eqref{eqn:fluxLzBalanceLast}.\n\nFor $r\\theta$ resonances, the Mino time average reduces to the single integral over $\\bar{q}$ in \\eqref{eqn:qdotAvg},\n\\begin{equation}\n\\label{eqn:QdotAverageRes}\n\t\\langle \\dot{\\mathcal{Q}} \\rangle = \\frac{1}{\\G} \\int_0^{2\\pi}\n\t\\frac{d\\bar{q}}{2\\pi} \\left(\\bar{\\Sig} \\frac{d\\mathcal{Q}}{d\\tau}\n\t\\right) .\n\\end{equation}\nWhile the integrand in the above expression is still antisymmetric with respect to both $q_r$ and $q_\\theta$, it is\nnot, for a general choice of the initial resonant phase $\\bar{q}_0$, antisymmetric with respect to just $\\bar{q}$\n(though there may be special values of $\\bar{q}_0$ where it is).  When integrated over a single closed track on the\ntorus, \\eqref{eqn:QdotAverageRes} is not guaranteed to vanish, in contrast with the nonresonant case.\n\nHowever, Flanagan and Hinderer \\cite{FlanHind12} conjecture that dynamics driven by the conservative piece of the\nself-force are always integrable in Kerr spacetime,\\footnote{Conservative perturbations are integrable in\nSchwarzschild spacetime, but their integrability has not been fully demonstrated for Kerr\n\\cite{VineFlan15,FujiETC17}.} and cannot therefore drive secular evolution of the perturbed system through a\nresonance.  If this integrability conjecture for conservative perturbations holds true, then $r\\th$-resonant dynamics\nwill be driven purely by the dissipative self-force (at adiabatic order), and $\\langle \\dot{\\mathcal{E}}\\rangle$,\n$\\langle \\dot{\\mathcal{L}}_z\\rangle$, and $\\langle \\dot{\\mathcal{Q}}\\rangle$ can be computed via efficient mode-sum\nexpressions [e.g., \\eqref{eqn:fluxEBalance}-\\eqref{eqn:fluxLzBalanceLast}] during resonant (and nonresonant) motion,\nas demonstrated in \\cite{FlanHughRuan14}.\\footnote{Note that the authors of \\cite{FlanHughRuan14} made no claim about\nthe validity of the integrability conjecture, but instead adopted it as a matter of practicality, since calculations of\nthe conservative GSF were unavailable at that time (Hughes, private communications).}\n\nOn the other hand, Isoyama \\textit{et al.}~\\cite{IsoyETC13,IsoyETC19} also derived mode-sum expressions for\n$\\langle \\dot{\\mathcal{E}}\\rangle$, $\\langle \\dot{\\mathcal{L}}_z\\rangle$, and $\\langle \\dot{\\mathcal{Q}}\\rangle$ during\nresonances using a Hamiltonian formulation, but they found that their expression for $\\langle \\dot{\\mathcal{Q}} \\rangle$\ndepends on the conservative (or what they call the symmetric) component of the perturbed Hamiltonian (for the scalar\ncase, see Eqs.~(49)-(51) in \\cite{IsoyETC13} and for the gravitational case, see Eqs.~(63) and (75) in\n\\cite{IsoyETC19}).  Unless this term vanishes upon averaging due to further symmetries, the integrability conjecture\nmust break down during resonances, and conservative perturbations will also drive the adiabatic evolution of $Q$.\n\nThis issue can in principle be tested using numerical modeling.  To date numerical calculations of\n$\\langle \\dot{\\mathcal{Q}} \\rangle$ for $r\\th$-resonant orbits \\cite{FlanHughRuan14,RuanHugh14,BerrETC16} have not\nincorporated the full first-order self-force.\\footnote{Note that the results of Isoyama \\textit{et al.}~\\cite{IsoyETC13,IsoyETC19}\nwere purely analytical.  No one has made use of the formalisms outlined in \\cite{IsoyETC13,IsoyETC19} to numerically\nevaluate $\\langle\\dot{\\mathcal{Q}}\\rangle$, and our methods differ enough from the Hamiltonian formulation and Green's\nfunction methods of \\cite{IsoyETC13,IsoyETC19} that we cannot easily compare our numerical results to these works.}\nThus there is no numerical evidence to support or negate the integrability conjecture.  The scalar self-force model can\npotentially provide some insight.  We test\nthe conjecture by measuring the relative contributions of the conservative and dissipative components of the SSF to\n$\\langle \\dot{\\mathcal{Q}} \\rangle$.  As we did in Sec.~\\ref{sec:EnAndLz}, we replace $\\bar{F}^\\mathrm{res}_\\a$ in\n\\eqref{eqn:qdotAvg} with a breakdown in terms of $\\bar{F}^\\mathrm{cons}_\\a$ and $\\bar{F}^\\mathrm{diss}_\\a$ to\ncalculate separately the conservative and dissipative contributions to $\\langle \\dot{\\mathcal{Q}}\\rangle$.  To compare\nthese quantities, we then calculate their residual variations $\\langle \\d \\dot{\\mathcal{Q}}\\rangle^\\mathrm{cons}$\nand $\\langle \\d \\dot{\\mathcal{Q}}\\rangle^\\mathrm{diss}$ as functions of the resonant-orbit phase using \\eqref{eqn:resVar}.\n\nIn Fig.~\\eqref{fig:dQdot}, we compare the variations $\\langle \\d \\dot{\\mathcal{Q}}\\rangle^\\mathrm{diss}$ (solid\nblack curves) and $\\langle \\d \\dot{\\mathcal{Q}}\\rangle^\\mathrm{cons}$ (dashed red curves) for the $e02.12$ (left),\n$e02.23$ (middle), and $e05.23$ (right) orbits, along with our formal uncertainty estimate $\\s_\\mathrm{cons}$ due to\nfitting for high-order regularization parameters in the conservative component of the self-force.  Across all three\norbits, the numerical calculation gives a smoothly varying conservative contribution to\n$\\langle \\d \\dot{\\mathcal{Q}}\\rangle$.  The amplitudes of the conservative contributions are slightly smaller but on\nthe same order as the dissipative variations.  \\emph{The smooth variations of the conservative contributions and their\ncomparable magnitudes to the dissipative variations strongly suggest that $\\langle \\dot{\\mathcal{Q}}\\rangle$ does depend\non conservative scalar perturbations during a resonance.}  However, in each case the conservative contributions from our\nSSF data fall below or nearly within the formal uncertainty estimates.  Furthermore, for all three orbits, the estimated\nuncertainty in our calculations of $\\langle \\d \\dot{\\mathcal{Q}}\\rangle^\\mathrm{cons}$ are significantly larger than\nthe estimated uncertainties in $\\langle \\d \\dot{\\mathcal{E}}\\rangle^\\mathrm{cons}$ and\n$\\langle \\d \\dot{\\mathcal{L}}_z\\rangle^\\mathrm{cons}$, as seen from comparing Figs.~\\ref{fig:workTorque} and\n\\ref{fig:dQdot}.  In fact, for the $e02.12$ and $e05.23$ orbits, our estimated uncertainty is typically large enough\nthat even if the conservative contribution dominated the variations in the dissipative contributions, they could\nstill be consistent with zero.\n\nIn the case of the $e02.12$ model, the truncation in the regularization of the conservative part of the SSF suggests\na formal uncertainty that is an order of magnitude greater than the numerically determined values of both the\nconservative and dissipative contributions to $\\langle \\d \\dot{\\mathcal{Q}}\\rangle$, and the grey region engulfs\nthe entire left panel of Fig.~\\ref{fig:dQdot}.  The size of the uncertainty in this model is similar to what was seen\nin $\\langle \\d \\dot{\\mathcal{E}}\\rangle$ and $\\langle \\d \\dot{\\mathcal{L}}_z \\rangle$ in Fig.~\\ref{fig:workTorque}.\nIn the $e05.23$ orbit, the calculated conservative contribution not only falls consistently below the formal\nuncertainty, but contains high-frequency oscillations that are indicative of numerical noise.  This noise appears\nalso in the formal uncertainty estimate itself.  In the $e02.23$ model,\n$\\langle \\d \\dot{\\mathcal{Q}}\\rangle^\\mathrm{cons}$ still primarily falls within the formal uncertainty, but is\nclosest, in this case, to being a significant nonzero result.  The numerical calculations, in this model at least,\nare close to providing evidence of the integrability conjecture's failure, but a reduction in the\nregularization errors by an order of magnitude or two would be required to be sure.\n\nWhen compared to calculating $\\langle \\d \\dot{\\mathcal{E}}\\rangle^\\mathrm{cons}$ and\n$\\langle \\d \\dot{\\mathcal{L}}_z\\rangle^\\mathrm{cons}$, the estimated uncertainty proves to be much higher when\nanalyzing the evolution of the Carter constant.  In part, this is because\n$\\langle \\d \\dot{\\mathcal{Q}}\\rangle^\\mathrm{cons}$ depends on $F_\\th$, which tends to be the most difficult\ncomponent of the SSF to accurately extrapolate with our numerical regularization procedure.  The $\\th$ component\ncouples to even higher spheroidal modes than the other SSF components as a result of the window function described\nin Sec.~\\ref{sec:ssf}.  If we truncate all mode calculations at a particular $l_\\mathrm{max}$, then we can only\ncalculate the multipoles of $F_\\th$ up to $l_\\mathrm{max}-3$, as seen from \\eqref{eqn:ssfRetMino} and\n\\eqref{eqn:delTheta}.  Since the higher $l$ modes are beneficial for extracting the higher-order regularization\nparameters, missing this higher-mode information for $F_\\th$ hampers our ability to fit for its (regularized)\nconservative component.  We still are able to calculate $F_\\th$ to 3 or 4 significant digits, which in the\nabsence of resonances is accurate enough to provide its contribution to EMRI evolutions that are phase accurate to\nless than a radian.  As this analysis indicates, additional accuracy will be needed to quantify with certainty\npossible contributions from the conservative sector to the secular evolution of the Carter constant.  In principle,\nwe could calculate additional modes to improve the accuracy of $F_\\th$.  In practice, this is difficult because\nhigher $l$ modes are harder to calculate accurately due to rapid oscillatory behavior of the integrands in the\nsource integrations.  This behavior is mirrored in the gravitational case \\cite{Vand16}.\n\nThe plotted uncertainty regions in Fig.~\\eqref{fig:dQdot} provide an estimate of how well we have fit for the\nhigher-order regularization parameters and regularized our self-force data.  Any nonvanishing, and potentially\nsmooth, variations within these uncertainty bounds could be a result of residual contributions of the singular field\nthat were not removed during regularization.  This issue is unique to averages over $r\\th$-resonant orbit.  In the\nnonresonant case, the singular field---much like the conservative contribution---vanishes when averaged over the\nentire two-torus.  Even if we do not regularize the conservative component of the SSF,\n$\\langle \\d \\dot{\\mathcal{Q}}\\rangle^\\mathrm{cons}$ will still exactly vanish for nonresonant orbits.  On the other\nhand, averages over the singular contributions, or any antisymmetric function on the two-torus, are not guaranteed\nto vanish for an arbitrary $r\\th$ resonance.\n\nWe conclude that our conservative SSF results are not yet accurate enough to show a definitive conflict with the\nintegrability conjecture.  That being said, our uncertainty regions may overestimate residual contributions of the\nsingular field that were not properly regularized.  If this is the case, then the nondissipative variations that we\nsee in Fig.~\\ref{fig:dQdot} may very well be physical.  Further tests of the integrability conjecture are necessary\nbut will require improved regularization of the self-force.  This might come from either more extensive numerical\ncalculations and regularization parameter fitting or from the added input of analytically determined higher-order\nregularization parameters.\n\n\\section{Summary}\n\\label{sec:conclusion}\n\nWe considered point scalar charges following $r\\th$-resonant geodesics in Kerr spacetime and calculated, for the\nfirst time, the resulting strong-field SSF experienced by these charges. This work serves as a\nfirst step in understanding the still unquantified behavior of the gravitational self-force during transient orbital\n$r\\th$ resonances.  To calculate the SSF we used a \\textsc{Mathematica} code previously developed in\n\\cite{NasiOsbuEvan19}.  In constructing our SSF data, we derived a simple shifting relation, \\eqref{eqn:ssfqq0}, that\nallows us to calculate the self-force during $r\\th$ resonances using self-force data that assumes the geodesic\nsources are nonresonant.  This mapping provides an efficient method for analyzing the self-force as a function of\nthe initial phase at which a system enters resonance.\n\nWhen calculating the SSF, we focused on six different $r\\th$-resonant orbits: each scalar charge followed a 1:3, 1:2,\nor 2:3 $r\\th$-resonant geodesic and each orbit either had an eccentricity of $0.2$ or $0.5$.  The full set of source\nparameters can be found in Table \\ref{tab:orbits}.  In Figs.~\\ref{fig:ssfRVar}-\\ref{fig:ssfPhVar}, we demonstrated\nhow varying the initial phase of $r\\th$-resonant orbits impacts the evolution of the self-force.  We then projected\nour SSF data onto invariant two-tori in Figs.~\\ref{fig:ssfTTorus}-\\ref{fig:ssfPhTorus} to display the dependence of\nthe self-force on the radial and polar motion of the source, regardless of initial conditions and phases.  We\nvalidated our SSF data by analyzing the convergence properties of our regularized self-force multipoles, as shown\nin Fig.~\\ref{fig:convergenceFph}, and by comparing the radiation fluxes to the rate of work and torque done on each\nscalar-charge source by the SSF via flux-balance laws, which are reported in Table \\ref{tab:fluxes}.\n\nWith these novel self-force calculations we also analyzed the impact of the conservative scalar self-force on the\norbit-averaged evolution of the orbital energy $\\langle \\dot{\\mathcal{E}} \\rangle$, $z$ component of the angular\nmomentum $\\langle \\dot{\\mathcal{L}}_z \\rangle$, and Carter constant $\\langle \\dot{\\mathcal{Q}} \\rangle$.  As expected\nfrom flux-balance arguments, the contributions to $\\langle \\dot{\\mathcal{E}} \\rangle$ and\n$\\langle \\dot{\\mathcal{L}}_z \\rangle$ from the conservative SSF are negligible and consistent with zero, as shown\nin Fig.~\\ref{fig:workTorque}.  On the other hand, our conservative SSF data substantially contribute to\n$\\langle \\dot{\\mathcal{Q}} \\rangle$, as displayed in Fig.~\\ref{fig:dQdot}, though these contributions are on the\norder of, or much less than, our estimated uncertainty.  This uncertainty is a result of the numerical regularization\nof the conservative SSF.  Because these contributions fall within these uncertainty bounds, we cannot distinguish\nwhether these nondissipative contributions are due to the residual singular field or the regularized conservative\nself-force.  If these contributions are in fact from the conservative SSF, then this would indicate that the\nintegrability conjecture proposed by Flanagan and Hinderer \\cite{FlanHind12} breaks down during $r\\th$ resonances,\nand conservative contributions will need to be considered as predicted by Isoyama \\textit{et al.} \\cite{IsoyETC19}.  The\npresence of these conservative contributions would then introduce a new numerical challenge for adiabatic\ncalculations: to account for the evolution through resonances, adiabatic codes would need to incorporate\nregularization procedures to accurately quantify $\\langle \\dot{\\mathcal{Q}} \\rangle$.  Furthermore, these regularized\ncontributions would need to be known to high levels of precision, at least to $O(\\eps^{-1\/2})$, which we have not\nbeen able to achieve with our current numerical implementations.\n\nIn future work, we will determine whether or not these nondissipative contributions to\n$\\langle \\dot{\\mathcal{Q}} \\rangle$ in the scalar case are physical, or merely systematic errors, by implementing\nalternative, improved regularization schemes and by deriving analytic expressions for the higher-order regularization\nparameters.  If the conservative SSF contributions that we have observed are indeed physical, then we hypothesize that\nduring $r\\theta$ resonances the conservative GSF will also contribute to $\\langle \\dot{\\mathcal{Q}} \\rangle$, as first suggested in \\cite{IsoyETC19}. Therefore, we are also constructing a code to calculate the GSF experienced by EMRI systems as they encounter $r\\th$ resonances to test this hypothesis.\n\n\\acknowledgements\n\nThis work was supported by NSF Grant Nos.~PHY-1806447 and PHY-2110335 to the University of North Carolina--Chapel Hill,\nand by the North Carolina Space Grant Graduate Research Fellowship.  Z.N.~acknowledges additional support from NSF Grant\nNo.~DMS-1439786 while in residence as a postdoctoral fellow for the Institute for Computational and Experimental\nResearch in Mathematics in Providence, Rhode Island, during the Advances in Computational Relativity semester program.\nZ.N.~also acknowledges support by appointment to the NASA Postdoctoral Program at the Goddard Space Flight Center,\nadministered by Universities Space Research Association under contract with NASA.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Traveling Salesman Problem (TSP) is the problem of finding a least-cost Hamiltonian cycle in an edge-weighted graph.\nIt is one of the most widely studied hard combinatorial optimization problems.\nFor details on the TSP, we refer to the well-known books~\\cite{ApplegateBCC11, GutinP06, LawlerLRS85, Reinelt94}.\nFor clarity of discussion, we will refer to this problem as the \\emph{linear TSP}.\n\nLet $G = (V,E)$ be a graph (directed or undirected) on the vertex set $ V = \\{1, \\ldots ,n\\}$.\nFor each pair $(e,f)$ of edges in $E$, a cost $q_{ef}$ is prescribed.\nLet $\\mathcal{T}_n$ be the set of all Hamiltonian cycles (tours) in $G$.\nThe quadratic cost $Q[\\tau]$ of a tour $\\tau \\in \\mathcal{T}_n$ is given by $ Q[\\tau] := \\sum \\limits_{(e,f) \\in \\tau \\times \\tau} q_{ef}$.\nThen the \\emph{quadratic traveling salesman problem} (QTSP) is to find a tour $\\tau \\in \\mathcal{T}_n$ such that $Q[\\tau]$ is minimum.\n\nThe problem QTSP received only limited attention in the literature.\nA special case of the \\emph{adjacent quadratic TSP} in which $q_{ef}$ is assumed to be zero if edges $e$ and $f$ are not adjacent, is frequently also called quadratic TSP.\nThis problem has been studied by various authors recently~\\cite{Fischer14, Fischer16, FischerFJKMG14, FischerH13, JagerM08, RostamiMBG16, AichholzerFFMPPS17}.\nWhile QTSP is known to be strongly $\\cplxNP$-hard for Halin graphs, the adjacent quadratic TSP can be solved in $\\orderO{n}$ time on this class of graphs~\\cite{WoodsP17Halin}.\nQTSP defined over the set of pyramidal tours is also known to be strongly $\\cplxNP$-hard~\\cite{WoodsP17PQ}.\nOther special cases of QTSP are studied in~\\cite{WoodsP17} along with various complexity results.\nThe $k$-neighbour TSP~\\cite{WoodsPS17}, Angular metric TSP~\\cite{AggarwalKMS97}, Dubins vehicle TSP~\\cite{SavlaFB08} are also related to QTSP.\n\nFor the rest of this paper, unless otherwise stated, we assume that $G$ is a complete directed graph.\nLet $Q = (q_{ef})$ be the matrix of size $n(n-1)\\times n(n-1)$ consisting of the cost of the edge pairs.\nNote that $(e,e)$ is a permitted pair.\nAn instance of QTSP is completely specified by $Q$.\nThe matrix $Q$ is said to be \\emph{tour-linearizable} (or simply linearizable) if there exists a matrix $C = (c_{ij})$ of size $n \\times n$ such that $Q[\\tau] = C(\\tau)$ for all $\\tau \\in \\mathcal{T}_n$, where $C(\\tau) := \\sum_{(i,j)\\in \\tau} c_{ij}$.\nSuch a cost matrix $C$ is called a \\emph{linearization} of $Q$ for the QTSP.\n\nThe \\emph{QTSP linearization problem} can be stated as follows:\n``Given an instance of QTSP with cost matrix $Q$, verify if it is linearizable and if yes, compute a linearization $C$ of $Q$.''\n\nIt may be noted that since (the decision versions of) the QTSP and the TSP are both $\\cplxNP$-complete problems, there exists a polynomial-time reduction from one problem to the other.\nWe are not attempting to find a reduction from QTSP to TSP for some special cases, although such a result is a byproduct.\nThe question of linearization is more involved and requires the development of structural properties of linearizable matrices.\nUnlike QTSP or TSP, there are no immediate certificates for linearizability or non-linearizability of matrices.\nHence, even containment of the linearization problem in $\\cplxNP$ or $\\cplxCoNP$ is challenging.\n\nLinearization problems have been studied in the literature for various quadratic combinatorial optimization problems~\\cite{AdamsW14, cCelaDW16, CusticSPB17, CusticP18, KabadiP11, LendlP17, PunnenK13, PunnenP18, HuS18, HuS18a} and polynomial-time algorithms are available in all these cases.\nUnfortunately, the simple characterization established for quadratic spanning trees~\\cite{CusticP18} and for various bilinear type problems~\\cite{cCelaDW16, CusticSPB17, LendlP17} do not seem to extend to the case of QTSP.\nThe characterization of linearizable instances of the quadratic assignment problem studied in~\\cite{KabadiP11} have some similarities to the case of QTSP that we study here.\nHowever, to solve the linearization problem for QTSP one needs to overcome more challenges, which makes it an interesting and relevant problem to investigate.\nA family of linearizable cost matrices can be used to obtain strong continuous relaxations of quadratic 0-1 integer programming problems, as demonstrated in~\\cite{HuS18a, PunnenP18}.\nStrong reformulations of quadratic 0-1 integer programming problems are reported in~\\cite{BillionnetEP09, PornNSW17} exploiting linear equality constraints.\nMoreover, strong reformulations of quadratic 0-1 integer programming using quadratic equality constraints are reported in~\\cite{GalliL14}.\nAs observed in~\\cite{PunnenP18}, linearizable cost matrices immediately give redundant quadratic equality constraints and, exploiting the results of~\\cite{GalliL14}, strong reformulations of quadratic combinatorial optimization problems can be obtained.\nFurther, since linearizable cost matrices for any quadratic combinatorial optimization problem form a linear subspace of the space of all $n(n-1) \\times n(n-1)$ matrices, any basis of this subspace can be used to obtain strong reformulations~\\cite{PunnenP18}.\n\nWith this motivation, we study the QTSP linearization problem in this paper.\nNecessary and sufficient conditions are presented for a quadratic cost matrix $Q$ associated with QTSP being linearizable.\nWe show that these conditions can be verified by an $\\orderO{n^5}$-time algorithm that also constructs a linearization of $Q$ whenever one exists.\nHowever, it may be noted that the input size of a general QTSP is $\\orderO{n^4}$ and hence the complexity of our algorithms is almost linear in the input size in the general case.\nOur characterization extends directly to the case of complete undirected graphs.\nFinally, some easily verifiable sufficient conditions for linearizability are also given along with some open problems.\n\nThe terminology ``linearization'' has also been used in a different context where a quadratic integer program is written as a linear integer program by adding new variables and constraints~\\cite{AdamsFG04,Glover75}.\nWe want to emphasize that the concept of linearization used in this paper is different.\n\nWe conclude this section by presenting the most-relevant notation used in the paper.\nLet $N := \\{ 1,2,\\ldots,n\\}$ and $N_i := N \\setminus \\{ i\\}$ for $i \\in N$.\nAll matrices are represented using capital letters (sometimes with superscripts, hats, overbars, etc.) and all elements of the matrix are represented by the corresponding subscripted lower-case letters (along with the corresponding superscripts, hats, overbars, etc.), where the subscripts denoting row and column indices.\nWhen the rows and columns are indexed by elements of $N \\times N$, the $((i,j),(k,l))$'th element of the matrix $Q$ is represented by $q_{ijkl}$, of $Q^R$ by $q^R_{ijkl}$, etc.\nVectors in $\\RR^n$ are represented by bold lower-case letters (sometimes with superscripts, hats, overbars, etc.).\nThe $i$'th component of vector $\\mathbf{a}$ is $a_i$, of vector $\\bar{\\mathbf{b}}$, is $\\bar{b}_i$, etc.\nRows and columns of all matrices of size $n \\times n$ are indexed by $N$, whereas rows and columns of all $n(n-1) \\times n(n-1)$ matrices are indexed by edges of $G$ (i.e., by ordered pairs $(i,j) \\in N \\times N$, $i \\neq j$).\nThe transpose of a matrix $Q$ is denoted by $\\transpose{Q}$.\nThe vector space of all real-valued $n \\times n$ matrices with standard matrix addition and scalar multiplication is denoted by $\\mathbb{M}^n$.\nAccordingly, $\\mathbb{M}^{n(n-1)}$ is the vector space of all $n(n-1) \\times n(n-1)$ matrices.\n\n\\section{The QTSP linearization problem}\n\nLet us first introduce some general properties and definitions used in the paper.\nA matrix $C \\in\\mathbb{M}^n$ satisfies the \\emph{constant value property} (CVP) if there exists a constant $K$ such that $C(\\tau) = K$ for all $\\tau \\in \\mathcal{T}_n$.\nA matrix $C$ is said to be a \\emph{weak sum matrix} if there exist vectors $\\mathbf{a},\\mathbf{b}\\in \\mathbb{R}^n$ such that $c_{ij} = a_i + b_j$ for $i,j \\in N$ with $i \\neq j$.\n\n\\begin{lemma}[\\cite{Berenguer79, Gabovich76, KabadiP03, KabadiP06}]\n  A matrix $C \\in \\mathbb{M}^n$ satisfies the CVP if and only if it is a weak sum matrix (i.e., $c_{ij} = a_i + b_j$ for all $i,j \\in N$ with $i \\neq j$).\n  In this case, the constant value of the tours is given by $\\sum_{i\\in N}(a_i + b_i)$.\n\\end{lemma}\n\nTwo matrices $Q^1$ and $Q^2 \\in \\mathbb{M}^{n(n-1)}$ are \\emph{equivalent modulo a linear matrix} if there exists a matrix $H \\in \\mathbb{M}^n$ such that $Q^1[\\tau] = Q^2[\\tau] + H(\\tau)$ for all $\\tau \\in \\mathcal{T}_n$.\nUsing the fact that the value of a tour depends on the sum $q_{ijk\\ell} + q_{k\\ell ij}$ and not the individual values $q_{ijk\\ell}$ and $q_{k\\ell ij}$, we obtain the following lemma.\n\\begin{lemma}\n  \\label{LemmaInvariantTransformations}\n  Let $Q \\in \\mathbb{M}^{n(n-1)}$.\n  Then the following statements are equivalent.\n  \\begin{enumerate}\n  \\item\n    $Q$ is linearizable.\n  \\item\n    $\\transpose{Q}$ is linearizable.\n  \\item\n    $\\frac{1}{2}\\left(Q+\\transpose{Q} \\right)$ is linearizable.\n  \\item\n    $Q+D+S$ is linearizable, where $D$ is any diagonal matrix and $S$ is any skew-symmetric matrix of the same dimension as $Q$.\n  \\item\n    Any matrix equivalent to $Q$ modulo a linear matrix is linearizable.\n  \\end{enumerate}\n\\end{lemma}\n\nRecall that a matrix $C \\in \\mathbb{M}^{n}$ is a \\emph{linearization} of $Q \\in \\mathbb{M}^{n(n-1)}$ if $Q[\\tau] = C(\\tau)$ for all $\\tau \\in \\mathcal{T}_n$.\nIn this case we say that $Q$ is a \\emph{quadratic form} of $C$ and that the matrix $Q$ is \\emph{linearizable}.\nNote that the sum of two linearizable matrices in $\\mathbb{M}^{n(n-1)}$ is linearizable and the scalar multiple of a linearizable matrix is linearizable.\nHence, the collection of all linearizable matrices in $\\mathbb{M}^{n(n-1)}$ forms a linear subspace of $\\mathbb{M}^{n(n-1)}$.\n\nThe coefficients $q_{iji\\ell}$ with $j \\neq \\ell$, the coefficients $q_{ijkj}$ with $i \\neq k$ and the coefficients $q_{ijji}$ can never contribute a non-zero value to the objective function $Q[\\tau]$ of a tour $\\tau$.\nThese elements of the matrix $Q$ they play no role in the optimization problem, therefore, we assume subsequently that $q_{ijk\\ell} = 0$ for all indices $i,j,k$ and $\\ell$ that satisfy $i = k$ and $j \\neq \\ell$ or $i \\neq k$ and $j = \\ell$ or $i = \\ell$ and $j = k$.\n\nTo characterize the linearizability of $Q$, working with the original matrix $Q$ seems tedious because of several reasons.\nFor example, as Lemma~\\ref{LemmaInvariantTransformations} indicates, we can add any skew-symmetric matrix to $Q$, maintaining linearizability.\nIt is important to restrict $Q$ in an appropriate way to bring some kind of uniqueness while the generality is maintained.\nFor this purpose, we say that a matrix $Q \\in \\mathbb{M}^{n(n-1)}$ is in \\emph{quadratic reduced form} if all of the following conditions are satisfied:\n\n\\begin{enumerate}\n\\item\n  $Q$ is symmetric.\n\\item\n  The diagonal elements of $Q$ are zeros.\n\\item\n  All elements of $Q$ with rows and columns indexed by $\\{ (n,p),(p,n) : p\\in N_n\\}$ are zeros.\n\\end{enumerate}\n\nLet us now prove a useful decomposition theorem.\n\n\\begin{theorem}\n  \\label{TheoremReducedFormDecomposition}\n  Any matrix $Q \\in \\mathbb{M}^{n(n-1)}$ can be decomposed in $\\orderO{n^4}$ time into a pair $(Q^R, L)$ of matrices such that $Q^R \\in \\mathbb{M}^{n(n-1)}$ is in quadratic reduced form, $L \\in \\mathbb{M}^n$ and $Q[\\tau] = Q^R[\\tau] + L(\\tau)$ for all $\\tau \\in \\mathcal{T}_n$.\n\\end{theorem}\n\n\\begin{proof}\n  Let $\\tilde{Q}$ and $\\tilde{L}$ be defined as\n  \\[\n    \\tilde{q}_{ijkl} =\n    \\begin{cases}\n      q_{ijk\\ell} - q_{ijkn} - q_{ijn\\ell} - q_{ink\\ell} + q_{inkn} + q_{inn\\ell} - q_{njk\\ell} + q_{njkn} +q_{njn\\ell} & \\text{ if } n \\notin \\{i,j,k,\\ell\\} \\\\\n      0 & \\text{ otherwise}\n    \\end{cases}\n  \\]\n  and\n  \\[\n    \\tilde{l}_{ij} =\n    \\begin{cases}\n      \\sum \\limits_{k=1}^{n-1}(q_{ijkn} + q_{ijnk} + q_{knij} - q_{knin} - q_{knnj} + q_{nkij} - q_{nkin} - q_{nknj}) & \\text{ if } i \\neq j \\\\\n      0 & \\text{ otherwise.}\n    \\end{cases}\n  \\]\n  It can be verified that all entries of rows and columns of $\\tilde{Q}$ indexed by $(n,p)$ or $(p,n)$ for $p \\in N_n$ are zeros.\n  Further, with some algebra, it can be verified that $\\tilde{Q}$ is equivalent to $Q$ modulo the linear matrix $\\tilde{L}$.\n  Let $D \\in \\mathbb{M}^{n(n-1)}$ be the diagonal matrix with $d_{ijij} = \\tilde{q}_{ijij}$.\n  Define $Q^R= \\frac{1}{2}(\\tilde{Q}+\\transpose{\\tilde{Q}}) - D$ and choose $L \\in \\mathbb{M}^n$ such that $l_{ij} = \\tilde{l}_{ij} + d_{ijij}$.\n  From the discussion above and Lemma~\\ref{LemmaInvariantTransformations} we have $Q[\\tau] = Q^R[\\tau] + L(\\tau)$ for all $\\tau \\in \\mathcal{T}_n $ and that $Q^R$ is in quadratic reduced form.\n  Finally, it is easy to see that only $\\orderO{n^4}$ arithmetic operations are required to compute $Q^R$ and $L$.\n\\end{proof}\n\nThe matrix $Q^R$ constructed in the proof of Theorem~\\ref{TheoremReducedFormDecomposition} is referred to as the \\emph{quadratic reduced form of $Q$}.\nThe decomposition $(Q^R,L)$ of $Q$ is referred to as \\emph{quadratic reduced form decomposition} or simply \\emph{QRF decomposition}.\nThe proof of Theorem~\\ref{TheoremReducedFormDecomposition} provides one way of constructing a QRF decomposition of $Q$.\nIt may be noted that equivalent transformations discussed in~\\cite{Burkard07,KabadiP11} in the context of the quadratic assignment problem can be modified appropriately to get another method for constructing a QRF decomposition.\n\n\\begin{theorem}\n  Let $Q \\in \\mathbb{M}^{n(n-1)}$ and $(Q^R,L)$ be a QRF decomposition of $Q$.\n  Then $Q$ is linearizable if and only if $Q^R$ is linearizable.\n  Further, if $P \\in \\mathbb{M}^n$ is a linearization of $Q^R$, then $P + L$ is a linearization of $Q$.\n\\end{theorem}\n\n\\begin{proof}\n  Since $(Q^R,L)$ is a QRF decomposition of $Q$, $Q[\\tau] = Q^R[\\tau] + L(\\tau)$ holds for all $\\tau \\in \\mathcal{T}_n$.\n  By Lemma~\\ref{LemmaInvariantTransformations}, $Q$ is linearizable if and only if $Q^R$ linearizable.\n  If $P$ is a linearization of $Q^R$, it follows that $Q[\\tau] = (P + L)(\\tau)$ for all $\\tau \\in \\mathcal{T}_n$.\n\\end{proof}\n\nThus, to study the linearization problem of $Q$, it is sufficient to study the linearization problem of the quadratic reduced form $Q^R$ of $Q$.\nThe lemma below is a variation of the corresponding result proved in~\\cite{KabadiP11} for the quadratic assignment problem.\nSince it is used in our main theorem, a complete proof within the context of QTSP is given below.\n\\begin{lemma}\n  \\label{LemmaLinearizationWithZeros}\n  Suppose $Q^R \\in \\mathbb{M}^{n(n-1)}$ is linearizable and is in quadratic reduced form.\n  Then there exists a linearization $C$ of $Q^R$ such that $c_{in} = 0$ and $c_{ni} = 0$ for all $i \\in N_n$.\n\\end{lemma}\n\n\\begin{proof}\n  Suppose $C'$ is some linearization of $Q^R$.\n  Let $\\alpha := \\sum_{i\\in N_n}c'_{in} \/ (n-2)$ and $\\beta := \\sum_{i\\in N_n}c'_{ni}\/(n-2)$.\n  Define $a_i := -c'_{in} + \\beta$, $b_i := -c'_{ni} + \\alpha$ for all $i \\in N_n$ as well as $a_n := -\\alpha$ and $b_n := -\\beta$.\n  Moreover, let $c_{ij} := c'_{ij} + a_i + b_j$ for all $i,j \\in N$.\n  Then $\\sum_{i\\in N}(a_i+b_i) = 0$ and hence $C(\\tau) = C^{\\prime}(\\tau)$ for all $\\tau \\in \\mathcal{T}_n$.\n  Now it can be verified that $c_{in} = c_{ni} = 0$ for all $i \\in N_n$, which concludes the proof.\n\\end{proof}\n\nLet $\\bar{Q}$ be the principal submatrix obtained from $Q^R$ by deleting rows and columns indexed by the elements of the set $\\{ (n,p),(p,n) : p\\in N_n\\}$.\nNote that $\\bar{Q} \\in \\mathbb{M}^{(n-1)(n-2)}$.\nFor any $i,j \\in N_n$ with $i \\neq j$, define $Z^{ij} \\in \\mathbb{M}^{n-1}$ such that\n\\begin{equation*}\n  z^{ij}_{uv} =\n  \\begin{cases}\n     0 & \\text{ if } (i,j) = (u,v), \\\\\n     \\bar{q}_{ijuv}  & \\text{if } (i,j) \\neq (u,v).\n  \\end{cases}\n\\end{equation*}\nOur next theorem characterizes the linearizability of a matrix in quadratic reduced form.\nNote that, for $n \\leq 3$, any quadratic cost matrix is linearizable.\nSo, we restrict our attention to the case when $n \\geq 4$.\n\n\\begin{theorem}\n  \\label{TheoremCharacterization}\n  Let $Q^R \\in \\mathbb{M}^{n(n-1)}$ be in quadratic reduced form and $n \\geq 4$.\n  Then $Q^R$ is linearizable if and only if there exists a matrix $F \\in \\mathbb{M}^{n-1}$ (with elements $f_{ij}$) such that\n  \\begin{enumerate}\n  \\item\n    \\label{TheoremCharacterizationFirst}\n    $Z^{ij}(\\tau) - Z^{k\\ell}(\\tau) = \\frac{1}{2} \\left(f_{ij} - f_{kl}\\right )$ holds for every $\\tau \\in \\mathcal{T}_{n-1}$ and for all $(i,j), (k,\\ell) \\in \\tau $, and\n  \\item\n    \\label{TheoremCharacterizationSecond}\n    $\\frac{n-2}{n-3}F$ is a linearization of $\\bar{Q}\\in \\mathbb{M}^{(n-1)(n-2)}$.\n  \\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nSuppose $Q^R$ is linearizable.\nLet $C$ be a linearization of $Q^R$ satisfying the condition of Lemma~\\ref{LemmaLinearizationWithZeros} and let $C'$ be the submatrix obtained from $C$ by deleting its $n$'th row and column.\nDefine $F$ via $f_{ij} := c_{ij}$ for all $i,j \\in N_n$ with $i\\neq j$.\nWe will show that $F$ satisfies the conditions of the theorem.\nConsider any $\\tau \\in \\mathcal{T}_{n-1}$.\nFor any $(i,j) \\in \\tau$, we define the tour $\\tau^{ij} \\in \\mathcal{T}_n$ obtained by deleting arc $(i,j)$ from $\\tau$ and introducing arcs $(i,n)$ and $(n,j)$ (See Figure~\\ref{FigureTours}).\n\n\\begin{figure}[H]\n  \\centering\n  \\begin{tikzpicture}[>=stealth, decoration={markings, mark=at position 0.5 with {\\arrow{>}}}]\n\t\\node [vrtx] (v1) at (-3, 1.5){};\n\t\\node [vrtx] (v2) at (-1.8273, 0.9352) {};\n    \\node [vrtx] (v3) at (-1.5376, -0.3338) {};\n    \\node [vrtx, label=below right:{\\footnotesize$j$}] (v4) at (-2.3492, -1.3515) {};\n    \\node [vrtx, label=below left:{\\footnotesize$i$}] (v5) at (-3.6508, -1.3515) {};\n    \\node [vrtx] (v6) at (-4.4624, -0.3338) {};\n    \\node [vrtx] (v7) at (-4.1727, 0.9352) {};\n\t\\draw[postaction={decorate}] (v5) -- (v4);\n\t\\draw[postaction={decorate}] (v6) -- (v5);\n\t\\draw[postaction={decorate}] (v7) -- (v6);\n\t\\draw[postaction={decorate}] (v1) -- (v7);\n\t\\draw[postaction={decorate}] (v2) -- (v1);\n\t\\draw[postaction={decorate}] (v3) -- (v2);\n\t\\draw[postaction={decorate}] (v4) -- (v3);\n\n\t\\node [vrtx] (u1) at (3, 1.5){};\n\t\\node [vrtx] (u2) at (4.1727, 0.9352) {};\n    \\node [vrtx] (u3) at (4.4624, -0.3338) {};\n    \\node [vrtx, label=below right:{\\footnotesize$j$}] (u4) at (3.6508, -1.3515) {};\n    \\node [vrtx, label=below left:{\\footnotesize$i$}] (u5) at (2.3492, -1.3515) {};\n    \\node [vrtx] (u6) at (1.5376, -0.3338) {};\n    \\node [vrtx] (u7) at (1.8273, 0.9352) {};\n    \\node [vrtx, label=above:{\\footnotesize$n$}] (n) at (3, -0.5) {};\n\t\\draw[postaction={decorate}] (u5) -- (n);\n\t\\draw[postaction={decorate}] (u6) -- (u5);\n\t\\draw[postaction={decorate}] (u7) -- (u6);\n\t\\draw[postaction={decorate}] (u1) -- (u7);\n\t\\draw[postaction={decorate}] (u2) -- (u1);\n\t\\draw[postaction={decorate}] (u3) -- (u2);\n\t\\draw[postaction={decorate}] (u4) -- (u3);\n\t\\draw[postaction={decorate}] (n)-- (u4);\n\t\\draw[dashed, postaction={decorate}] (u5) -- (u4);\n  \\end{tikzpicture}\n  \\caption{Tours $\\tau \\in \\mathcal{T}_{n-1}$ (shown on the left) and $\\tau^{ij}\\in \\mathcal{T}_n$ (shown on the right) with $n = 8$.}\n  \\label{FigureTours}\n\\end{figure}\n\nNote that $Q^R[\\tau^{ij}] = \\bar{Q}[\\tau] - 2Z^{ij}(\\tau)$ and $C(\\tau^{ij}) = C^{\\prime}(\\tau) - c_{ij}$.\nBut $Q^R[\\tau_{ij}] = C(\\tau_{ij})$, and we obtain\n\\begin{equation}\n  \\label{TheoremCharacterizationZ1}\n  Z^{ij}(\\tau) = \\frac{1}{2} \\left( \\bar{Q}[\\tau] - C^{\\prime}(\\tau) +c_{ij}\\right).\n\\end{equation}\nSimilarly, for $(k,\\ell) \\in \\tau$,\n\\begin{equation}\n  \\label{TheoremCharacterizationZ2}\n  Z^{k\\ell}(\\tau) = \\frac{1}{2} \\left( \\bar{Q}[\\tau] - C^{\\prime}(\\tau) +c_{k\\ell}\\right).\n\\end{equation}\nSubtracting~\\eqref{TheoremCharacterizationZ2} from~\\eqref{TheoremCharacterizationZ1} yields\n\\[\n  Z^{ij}(\\tau) - Z^{k\\ell}(\\tau) = \\frac{1}{2} \\left(f_{ij} - f_{kl}\\right),\n\\]\ni.e., condition~\\eqref{TheoremCharacterizationFirst} of the theorem is satisfied.\nNow, adding~\\eqref{TheoremCharacterizationZ1} for all $(i,j)\\in \\tau$, we obtain\n\\begin{align*}\n  \\bar{Q}[\\tau] & = \\sum_{(i,j)\\in \\tau}Z^{ij}(\\tau) =\n  \\sum_{(i,j)\\in \\tau} \\frac{1}{2} \\left( \\bar{Q}[\\tau] - C^{\\prime}(\\tau) +c_{ij}\\right)\n  = \\frac{1}{2} \\left[(n-1)\\bar{Q}[\\tau] - (n-2)C^{\\prime}(\\tau)\\right],\n\\end{align*}\nwhich implies $\\bar{Q}[\\tau] = \\frac{n-2}{n-3}C^{\\prime}(\\tau)$.\nThus, $\\bar{Q}$ is tour-linearizable and $\\frac{n-2}{n-3}C^{\\prime}(\\tau)$ is a linearization.\n\n\\medskip\n\nConversely, suppose $Q^R$ satisfies conditions~\\eqref{TheoremCharacterizationFirst} and~\\eqref{TheoremCharacterizationSecond} of the theorem.\nWe will show that $Q^R$ is tour-linearizable.\nTo this end, define the matrix $C \\in \\mathbb{M}^n$ via\n\\begin{equation}\n  \\label{TheoremCharacterizationC}\n  c_{ij} :=\n  \\begin{cases}\n  f_{ij} & \\text{ if } i,j \\in N_n, \\\\\n  0 & \\text{ otherwise}.\n  \\end{cases}\n\\end{equation}\nLet $C^{\\prime}$ be the principal submatrix obtained from $C$ by deleting its $n$'th row and column. From condition~\\eqref{TheoremCharacterizationFirst} of the theorem, for every tour $\\tau \\in \\mathcal{T}_{n-1}$ and every pair $(i,j),(k,\\ell) \\in \\tau$ of edges in the tour,\n\\begin{equation}\n  \\label{TheoremCharacterizationPhi}\n  2Z^{ij}(\\tau) = c_{ij} + \\phi(\\tau)\n\\end{equation}\nholds, where $\\phi(\\tau) = 2Z^{k\\ell}(\\tau) - c_{k\\ell}$.\nFrom condition~\\eqref{TheoremCharacterizationSecond}, for every $\\tau \\in \\mathcal{T}_{n-1}$, $\\bar{Q}[\\tau] = \\frac{n-2}{n-3}C^{\\prime}(\\tau)$, which implies\n\\begin{align}\n  0 &= (n-1)\\bar{Q}[\\tau] - 2\\bar{Q}[\\tau] - (n-2)C^{\\prime}(\\tau) \\nonumber \\\\\n    &= (n-1)\\bar{Q}[\\tau] - \\sum_{(i,j)\\in \\tau}2Z^{ij}(\\tau) - (n-2)C^{\\prime}(\\tau). \\label{TheoremCharacterizationInsertZ}\n\\end{align}\nSubstituting equation~\\eqref{TheoremCharacterizationPhi} in~\\eqref{TheoremCharacterizationInsertZ} yields\n\\begin{align}\n  0 &= (n-1)\\bar{Q}[\\tau] - \\sum_{(i,j)\\in \\tau}\\left[c_{ij}+\\phi(\\tau) \\right]- (n-2)C^{\\prime}(\\tau) \\nonumber \\\\\n    &= (n-1)\\bar{Q}[\\tau] - C^{\\prime}(\\tau) - (n-1)\\phi(\\tau)- (n-2)C^{\\prime}(\\tau), \\nonumber\n\\end{align}\nwhich implies\n\\begin{equation}\n  \\phi(\\tau) = \\bar{Q}[\\tau] - C^{\\prime}(\\tau) \\label{TheoremCharacterizationPhiNew}\n\\end{equation}\nSubstituting equation~\\eqref{TheoremCharacterizationPhiNew} in~\\eqref{TheoremCharacterizationPhi}, we have, for every $(i,j) \\in \\tau$,\n\\begin{equation}\n  2Z^{ij}(\\tau) = c_{ij} + \\bar{Q}[\\tau] - C^{\\prime}(\\tau). \\label{TheoremCharacterizationZNew}\n\\end{equation}\nConsider any tour $\\tau^{ij} \\in \\mathcal{T}_n$, where $i$ and $j$ are the predecessor and successor of $n$ in $\\tau^{ij}$, respectively.\nLet $\\tau^0$ be the tour in $\\mathcal{T}_{n-1}$ obtained from $\\tau^{ij}$ by shortcutting the path $i \\rightarrow n \\rightarrow j$ using arc $(i,j)$.\nThen,\n\\begin{align*}\n  Q^R[\\tau^{ij}]  &= \\bar{Q}[\\tau^0] - 2Z^{ij}(\\tau^0) \\\\\n                  &= \\bar{Q}[\\tau^0] -\\left[c_{ij} + \\bar{Q}[\\tau^0] -C^{\\prime}(\\tau^0)\\right] \\\\\n                  &= C^{\\prime}(\\tau^0) - c_{ij} = C(\\tau^{ij}),\n\\end{align*}\nestablishing that $Q^R$ is linearizable with $C$ as a linearization.\n\\end{proof}\n\n\\begin{theorem}\n  The QTSP linearization problem can be solved in $\\orderO{n^5}$ time.\n  Moreover, a linearization of $Q$ can be constructed in $\\orderO{n^5}$ time whenever it exists.\n\\end{theorem}\n\n\\begin{proof}\n  Given $Q$, its QRF decomposition $(Q^R,L)$ can be obtained in $\\orderO{n^4}$ time using the construction from Theorem~\\ref{TheoremReducedFormDecomposition}.\n  We now test the linearizability characterization of $Q^R$ given in Theorem~\\ref{TheoremCharacterization}.\n  Let $\\Delta := \\{[(i,j),(k,\\ell)] : i,j,k,\\ell \\in N_n, k\\neq i, j\\neq \\ell, (i,j)\\neq (j,i), i\\neq j, k \\neq \\ell\\}$.\n  To verify condition~\\eqref{TheoremCharacterizationFirst} of Theorem~\\ref{TheoremCharacterization} and construct a linearization, if exists, we have to find $f_{ij}, (i,j) \\in N_n, i \\neq j$, such that for any $\\tau \\in \\mathcal{T}_n,$  $Z^{ij}(\\tau) - Z^{k\\ell}(\\tau) = \\frac{1}{2} \\left(f_{ij} - f_{kl}\\right )$ for all $(i,j), (k,\\ell) \\in \\tau$.\n  This can in principle be achieved by first testing if the matrix $P^{ijk\\ell} := Z^{ij}-Z^{kl}$ satisfies the CVP.\n  If it does not satisfy the CVP for some $[(i,j),(k,\\ell)] \\in \\Delta$, then condition~\\eqref{TheoremCharacterizationFirst} of Theorem~\\ref{TheoremCharacterization} is not satisfied.\n  If the CVP is satisfied for all $[(i,j),(k,\\ell)] \\in \\Delta$, let $\\eta_{ijk\\ell}$ be the constant value of tours obtained for the matrix $P^{ijk\\ell}$.\n  Now we need to solve the system\n  \\begin{equation}\n    \\label{EquationAlgorithmSystem}\n    f_{ij} - f_{k\\ell} = 2\\eta_{ijk\\ell} \\text{ for all } [(i,j),(k,\\ell)] \\in \\Delta.\n  \\end{equation}\n  If the system is inconsistent, condition~\\eqref{TheoremCharacterizationFirst} of Theorem~\\ref{TheoremCharacterization} is not satisfied.\n  Using the characterization of TSP with CVP~\\cite{Berenguer79,Gabovich76,KabadiP03,KabadiP06}, each of the $\\orderO{n^4}$-many $\\eta_{ijk\\ell}$-values can be obtained in $\\orderO{n^2}$ time.\n\n  Since this would lead to a running time of $\\orderO{n^6}$ just to compute the right-hand side of the system, we will now improve this part of the algorithm.\n  To this end, consider the graph $\\tilde{H} = (E,\\tilde{\\Delta})$, where\n  $\\tilde{\\Delta} := \\{ \\{(i,j),(k,\\ell) \\in E \\times E : [(i,j),(k,\\ell)] \\in \\Delta \\}$,\n  i.e., two edges $(i,j)$ and $(k,\\ell)$ are adjacent in $\\tilde{H}$ if and only if both arcs appear in at least one tour.\n  It is easy to check that $\\tilde{H}$ is connected and hence contains a spanning tree $\\tilde{T}$.\n  Moreover, by exploring the nodes from some arbitrary root node, one finds a sequence of $|E|-1$ rows of~\\eqref{EquationAlgorithmSystem},\n  each of which contains an $f$-variable that was not present in the previous rows.\n  Since the first row contains two $f$-variables, this submatrix has rank at least $|E|-1$.\n\n  We can find in $\\orderO{n^2}$ time a spanning tree of $\\tilde{H}$ and compute the corresponding $(|E|-1)$-many $\\eta$-values in $\\orderO{n^2 \\cdot |E|} = \\orderO{n^4}$ time.\n  Such $\\eta$-values may not exist, and in this case condition~\\ref{TheoremCharacterizationFirst} of Theorem~\\ref{TheoremCharacterization} is not satisfied.\n  If these $\\eta$-values exist, we proceed as follows.\n  Since every row of this system contains exactly two variables, it can be solved in time $\\orderO{n^2}$ using the algorithm of Aspvall and Shiloach~\\cite{AspvallS80}.\n  Note that due to the selection of the rows the system will always be consistent, that is, we will compute a candidate solution $\\tilde{F}$.\n  Moreover, the system is under-determined, since $\\tilde{F} + \\lambda \\onevec$ is also a solution for every $\\lambda \\in \\QQ$, where $\\onevec$ denotes the matrix having all entries equal to $1$.\n  This shows that system~\\eqref{EquationAlgorithmSystem} has in fact rank $|E|-1$.\n  Notice also that two rows of~\\eqref{EquationAlgorithmSystem} corresponding to $[(i,j),(k,\\ell)] \\in \\Delta$ and to $[(k,\\ell),(i,j)]$ are negatives of each other.\n  Hence, it does not matter which one we actually include in the subsystem.\n\n  We claim that our candidate solution $\\tilde{F}$ satisfies all (remaining) rows of~\\eqref{EquationAlgorithmSystem}.\n  To this end, consider a pair $[(i,j),(k,\\ell)] \\in \\Delta$ of arcs.\n  Let $(i_t,j_t)$ for $t = 0,1,\\dotsc,m$ be the (ordered) nodes in the unique $(k,\\ell)$-$(i,j)$-path in the spanning tree $\\tilde{T}$.\n  In particular, $(i_0,j_0) = (k,\\ell)$ and $(i_m,j_m) = (i,j)$.\n  By construction, $\\tilde{F}$ satisfies\n  \\begin{align*}\n    \\tilde{f}_{i_tj_t} - \\tilde{f}_{i_{t-1}j_{t-1}} &= 2\\eta_{i_tj_ti_{t-1}j_{t-1}}\n  \\end{align*}\n  for $t=1,2,\\dotsc,m$. The sum of the equations yields\n  \\begin{align*}\n    \\tilde{f}_{ij} - \\tilde{f}_{k\\ell} = \\tilde{f}_{i_mj_m} - \\tilde{f}_{i_0j_0} &= 2 \\sum_{t=1}^m \\eta_{i_tj_ti_{t-1}j_{t-1}}.\n  \\end{align*}\n  Similarly, for all tours $\\tau \\in \\mathcal{T}_{n-1}$ and for $t = 1,2,\\dotsc,m$, we have\n  \\begin{align*}\n    Z^{i_tj_t}(\\tau) - Z^{i_{t-1}j_{t-1}}(\\tau) &= \\eta_{i_tj_ti_{t-1}j_{t-1}}.\n  \\end{align*}\n  Again, the sum yields\n  \\begin{align*}\n    Z^{ij}(\\tau) - Z^{k\\ell}(\\tau) &= \\sum_{t=1}^m \\eta_{i_tj_ti_{t-1}j_{t-1}},\n  \\end{align*}\n  which establishes that the CVP is satisfied for $P^{ijk\\ell}$ and that $\\eta_{ijk\\ell} = \\sum_{t=1}^m \\eta_{i_tj_ti_{t-1}j_{t-1}}$.\n  Hence, $\\tilde{F}$ satisfies the row of~\\eqref{EquationAlgorithmSystem} corresponding to $[(i,j),(k,\\ell)]$, which shows that this row is redundant.\n\n  The parameter $\\lambda$ can be easily determined by enforcing $Q^R[\\tau] = (\\tilde{F} + \\lambda \\onevec)(\\tau)$ for an arbitrary tour $\\tau$.\n  Since $\\onevec(\\tau) = n$, we can set $F := \\tilde{F} + \\lambda \\onevec$ for $\\lambda := \\tfrac{1}{n}(Q^R[\\tau] - \\tilde{F}(\\tau))$.\n\n  To summarize, the overall complexity of verifying condition~\\eqref{TheoremCharacterizationFirst} of Theorem~\\ref{TheoremCharacterization} and computing a candidate solution $F$ is $\\orderO{n^4}$.\n\n  If condition~\\eqref{TheoremCharacterizationFirst} is satisfied, then we need to verify condition~\\eqref{TheoremCharacterizationSecond}.\n  This is achieved by testing if $\\bar{Q}$ is linearizable and if yes, by testing if $\\frac{n-2}{n-3}F$ is one of the linearizations.\n  If $\\bar{Q}$ is not linearizable, then condition~\\eqref{TheoremCharacterizationSecond} fails.\n  Suppose $H$ is a linearization of $\\bar{Q}$.\n  Then, to verify if $\\frac{n-2}{n-3}F$ is another linearization, it is enough to verify if the matrix $H - \\frac{n-2}{n-3}F$ satisfies the CVP with the constant tour value equal to zero.\n  Given $H$, this can be achieved in $\\orderO{n^2}$ time.\n\n  Thus, the problem of testing linearizability of $Q$ reduces to that of testing linearizability of $\\bar{Q}$, which has the size parameter $n$ reduced by 1 along with an additive $\\orderO{n^4}$ effort.\n  In the base case $n = 3$, $Q^R$ is trivially linearizable since there only exist two opposite tours that do not share edges.\n  Thus, if $g(n)$ is the complexity of testing linearizability of $Q$, we have $g(n) = g(n-1) + \\orderO{n^4}$, establishing $g(n) = \\orderO{n^5}$.\n\n  To obtain the linearization, construct the matrix $C$ from $F$ according to~\\eqref{TheoremCharacterizationC}.\n  Then $C + L$ is the required linearization of $Q$, where $L$ is provided by the QRF decomposition $(Q^R, L)$ of $Q$.\n\\end{proof}\n\n\\section{Simple sufficient conditions and extensions}\n\nThe complexity of testing linearizability of $Q$, although almost linear, imposes limits on applicability of the result outside the theoretical realm.\nIn this section we present some sufficient conditions for linearizability that can be verified more easily.\n\nThe cost matrix $C\\in \\mathbb{M}^n$ associated with a linear TSP on a complete directed graph on $n$ nodes satisfies the \\emph{$(k,\\ell)$-constant value property} ($(k,\\ell)$-CVP) if all tours containing the edge $(k,\\ell)$ have the same cost.\n\\begin{lemma}\n  A cost matrix $C \\in \\mathbb{M}^n$ satisfies the $(k,\\ell)$-CVP if and only if there exist $a_i \\in N_k$ and $b_i\\in N_{\\ell}$ such that $c_{ij} = a_i + b_j$ for all $i,j \\in N$ with $i\\neq j$, $i \\neq k$ and $j \\neq \\ell$.\n  Moreover, the constant value of the tours is given by $c_{k\\ell} + \\sum_{i\\in N_k} a_i + \\sum_{i \\in N_{\\ell}} b_i$.\n\\end{lemma}\nThe proof of this lemma can be obtained by making use of the characterization of matrices satisfying the CVP on complete digraphs.\nFor each row indexed by $(i,j)$ of $Q$, define $R^{ij}\\in \\mathbb{M}^n$ by\n\\begin{equation}\n  r^{ij}_{uv} :=\n  \\begin{cases}\n    q_{ijuv} & \\text{if } u \\neq v \\\\\n    0& \\text{ otherwise}.\n  \\end{cases}\n\\end{equation}\n\n\\begin{lemma}\n  If $R^{ij}$ satisfies the $(i,j)$-CVP for all $(i,j) \\in N \\times N$ with $i\\neq j$, then $Q$ is linearizable.\n  Moreover, if $L \\in \\mathbb{M}^n$ is a linearization, then $l_{ij}$ is equal to the $(i,j)$-CVP value of tours.\n\\end{lemma}\n\n\\begin{proof}\n  For any $\\tau \\in \\mathcal{T}_n$, $Q[\\tau] = \\sum_{(i,j)\\in \\tau}R^{ij}(\\tau) = \\sum_{(i,j)\\in\\tau}l_{ij} = L(\\tau)$.\n\\end{proof}\nA corresponding result can be derived by considering columns of $Q$.\nAnother simple sufficient condition for linearizability is that $Q$ is a weak-sum matrix.\nA more general version of this condition is provided below.\n\\begin{lemma}\n  If there exist $A,B,D,H \\in \\mathbb{R}^{n\\times n\\times n}$ such that $q_{ijkl} = a_{ijk} + b_{ijl} + d_{ikl} + h_{jkl}$ for all $i,j,k,l$ with $i\\neq j$ and $k\\neq l$ then $Q$ is linearizable.\n\\end{lemma}\n\\begin{proof}\n  Let $\\alpha_{ij} := \\sum \\limits_{k=1}^n a_{ijk}$, $\\beta_{ij} := \\sum \\limits_{k=1}^n b_{ijk}$, $\\gamma_{ij} := \\sum \\limits_{k=1}^n d_{kij}$, $\\delta_{ij} := \\sum \\limits_{k=1}^n h_{kij}$ and $c_{ij} := \\alpha_{ij} + \\beta_{ij} + \\gamma_{ij} + \\delta_{ij}$ for all $i,j$ with $i\\neq j$.\n  Also, let $x_{ij}$ be the binary decision variable which takes value $1$ if and only if the underlying tour contains arc $(i,j)$.\n  Thus, the values of $x_{ij}$ completely define a tour.\n  Now,\n\\begin{align*}\n  Q[\\tau]\n  &= \\sum \\limits_{i=1}^n \\sum \\limits_{j=1,j\\neq i}^n \\sum \\limits_{k=1}^n \\sum \\limits_{l=1,k\\neq l}^n q_{ijkl}x_{ij}x_{kl} \\\\\n  &= \\sum \\limits_{i=1}^n \\sum \\limits_{j=1,j\\neq i}^n \\sum \\limits_{k=1}^n \\sum \\limits_{l=1,k\\neq l}^n (a_{ijk}+b_{ijl}+d_{ikl}+h_{jkl})x_{ij}x_{kl} \\\\\n  &= \\sum \\limits_{i=1}^n \\sum \\limits_{j=1,j\\neq i}^n x_{ij} \\sum \\limits_{k=1}^n a_{ijk} \\sum \\limits_{l=1,k\\neq l}^n x_{kl} + \\sum \\limits_{k=1}^n \\sum \\limits_{l=1,k\\neq l}^n x_{kl} \\sum \\limits_{i=1}^n b_{ikl} \\sum \\limits_{j=1,i\\neq j}^n x_{ij} \\\\\n  &\\qquad+ \\sum \\limits_{i=1}^n \\sum \\limits_{j=1,j\\neq i}^n x_{ij} \\sum \\limits_{l=1}^n d_{ijl} \\sum \\limits_{k=1,k\\neq l}^n x_{kl} + \\sum \\limits_{k=1}^n \\sum \\limits_{l=1,k\\neq l}^n x_{kl} \\sum \\limits_{j=1}^n h_{jkl} \\sum \\limits_{i=1,i\\neq j}^n x_{ij} \\\\\n  &=\\sum \\limits_{i=1}^n \\sum \\limits_{j=1, i\\neq j}^n ( \\alpha_{ij} + \\beta_{ij} + \\gamma_{ij} + \\delta_{ij}) x_{ij}\n  = \\sum \\limits_{i=1}^n \\sum \\limits_{j=1, j\\neq i}^n c_{ij}x_{ij}\n  = C(\\tau),\n\\end{align*}\nwhich concludes the proof.\n\\end{proof}\n\nThe result in the previous lemma can be strengthened further by excluding the conditions of the lemma for some appropriate combinations of $i,j,k$ and $\\ell$.\n\nSo far we have been considering complete directed graphs.\nCorresponding results can be derived easily to solve the linearization problem for QTSP on a complete undirected graph $K_n$.\n\nAn interesting related question is to explore the linearization problem of QTSP on other meaningful classes of graphs.\nThe answer will very much depend on the structure of the graph.\nFor example, if the underlying graph is a wheel, any quadratic cost matrix $Q$ is linearizable.\nNote that a wheel on $n$ nodes contains $n-1$ tours and we can write a system of $n-1$ linear equations, where each equation corresponding to tour, and the variables are elements of a 'trial' linearization.\nIt can be shown that this system is always consistent.\nWe leave the question of characterizing linearizable quadratic cost matrices of specially structured graphs as a topic for further research.\n\n\\subsection*{Acknowledgement}\nThis project was initiated jointly with Santosh Kabadi, who passed away.\nWe acknowledge the significant contributions Santosh made in obtaining the results of this paper.\nThe work is supported by an NSERC discovery grant and an NSERC discovery accelerator supplement awarded to Abraham P.\\ Punnen.\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{I Introduction}\n\nThe quadratic Mandelbrot set is the set of non divergent points\nin the iteration $z_{n+1}:=z_n+c$,~$z_0:=(0,0)$,~$\\forall\nc$, over the binary numbers, with the same norm as in the\ncomplex case. Recently a convergence proof for the quadratic\nMandelbrot set was given \\cite{Met}. This set was discussed\nnumerically in \\cite{Sen}. The puzzling effect, that by changing\nonly one sign in the iteration formula results in a completely\ndifferent not even chaotic pictures, was expressed by the term\n''perplex numbers``, which are nothing but binary numbers\n\\cite{Maj,Fje}. Where the disappearance of the chaotic behavior\nwas expressed by ''The mystery of the quadratic Mandelbrot set``\nin \\cite{Met}.\n\nIn this note, we want to show how the two cases can be\nunderstood in a Clifford algebraic framework. This provide us\nwith several advantages over the usual picture.\n\nFirst we introduce the complex numbers as operator spinors\nacting on a unit reference vector in the Euclidean vector space\n{\\bf E(2)}. This is in analogy with Kustaanheimo, Stiefel\n\\cite{Kus}, Lounesto \\cite{Lou-mech} and the inspiring chapter 8\n''Spinor Mechanics`` of Hestenes' ''New Foundation of classical\nMechanics`` \\cite{Hes-mech}.\n\nIn a second step we utilize Clifford algebras with arbitrary\nnot necessarily symmetric or antisymmetric bilinear forms. Such\nalgebras where discussed by Chevalley \\cite{Che}, Riesz\n\\cite{Rie}, Oziewicz \\cite{Ozi} and  Lounesto \\cite{Lou-bili} in\na mathematical and geometrical manner.\n\nThis procedure is a general tool and was used in \\cite{Fau} in\nan entire other context. Here the geometrical point of view\nallows a generalization of the concept of operator spinors to\nthose vector spaces, which are equipped with general non\ndegenerate bilinear forms.\n\nThe third step stems from the observation, that the Clifford\nalgebra is strongly connected with a quadratic algebra. This\nstructure provides us with several existence and uniqueness\ntheorems. Separability provides us with a unique (pseudo)\nnormform, as with a unique trace \\cite{Hah}.\n\nSurprisingly we are left with a norm different from the one used\nin \\cite{Met}. But here the (pseudo) normform is compatible with\nthe algebraic structure and the geometrical meaning. These\n(pseudo) normforms and traces are therefore preferable.\n\nUsing these norms, we clearly see other images in numerical\nexperiments. The obtained pictures are in agreement with the\nphysical situation at hand. Several special cases will be\ndiscussed. The Mandelbrot dynamic is shown to be incompatible\nwith this structure and a better one should be derived from\ninhomogeneous {\\bf D$\\times$Ispin} or {\\bf Ispin} groups. The\nbreakdown of the dynamics in the hyperbolic case may be\ninterpreted as a decay process or an absorption of the particle\nconsidered.\n\nIn section II we introduce the operator spinor for {\\bf\nE(2)}\\footnote{See Hestenes \\cite{Hes-mech} for an enlargement\nto 3--dimensional space and the spinor gauge formulation of the\nKepler motion.}. In section III we introduce Clifford algebras\nwith nonsymmetric bilinear forms, described explicitly in\n\\cite{Ozi}. In section IV we recall several facts from the\ntheory of quadratic algebras as proposed in \\cite{Hah}. In\nsection V we discuss the Mandelbrot set over arbitrary\ntwo--component number systems using the (pseudo) normforms\ninduced by algebraic and geometric considerations. The\nconclusion summarizes our results and compares it to other work\ndone.\n\n\\section*{II Complex Numbers as Operator Spinors on {\\bf E(2)}}\n\nIn this section we introduce the concept of algebraic spinors in\nthe sense of Hestenes \\cite{Hes-mech,HesSob} and others\n\\cite{Che,Rie}. Therein a geometrical meaning is given to\nthe complex numbers. Indeed they are commonly identified with\ncoordinates, which is quite obscure. There is a long quest in\nmathematics to give a geometrical meaning to the complex entities.\nIn spite of their natural occurring in algebraic geometry they\nremained somehow mysterious \\cite{Kle}. An attempt in this direction\nseems very useful in the light of the ubiquitous appearance of\ncomplex numbers and {\\it complex coordinates} in physics.\n\nWe choose a standard orthonormal basis $e_i$,~$i\\in (1,2)$ of\n$CL(\\mbox{\\bf E(2)},\\delta)$ with the properties (algebraproduct\nby juxtaposition)\n\\begin{eqnarray}\ne_i e_i = e_i^2=1&&\\mbox{normalized} \\nonumber \\\\\ne_ie_j+e_je_i=0,&& i\\not= j;\\quad\\mbox{orthogonality.}\n\\end{eqnarray}\n\nThe standard involution (conjugation) may be defined by\n\\begin{eqnarray}\nJ &:& CL \\rightarrow CL \\nonumber \\\\\nJ &:& J(ab):=J(a)J(b) \\nonumber \\\\\nJ &:& J\\vert_{{\\bf K}\\oplus{\\bf E(2)}}:=id_{\\bf K}-id_{\\bf E(2)}\n\\nonumber \\\\\n  &&  J^2=id_{CL}.\n\\end{eqnarray}\nThe reversion is the main antiautomorphism defined by (see\n\\cite{BudTra})\n\\begin{eqnarray}\n\\tilde{~} &:& CL\\rightarrow CL \\nonumber \\\\\n\\tilde{~} &:& (ab)\\tilde{~} :=b\\tilde{~} a\\tilde{~} \\nonumber \\\\\n\\tilde{~} &:& \\tilde{~}\\vert_{{\\bf K}\\oplus{\\bf E(2)}}:=id_{\n{\\bf K}\\oplus{\\bf E(2)}}\\nonumber \\\\\n&& (\\tilde{~})\\tilde{~}=id_{CL},\n\\end{eqnarray}\nwhere {\\bf K} and {\\bf E(2)} are the images of the field (ring)\nand the vector space (module) in $CL$. Because there is a natural\ninjection we will not distinguish between this pictures.\n\nAn algebra basis is given by\n\\begin{eqnarray}\n\\{X_I \\} &:=& \\{1,e_2\\wedge e_1 ,e_1,e_2\\}\\label{X}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\ne_i\\wedge e_j&:=&\\frac{1}{2}(e_i e_j +J(e_j) e_i) \\nonumber \\\\\ne_i \\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$} e_j&:=&<e_i\\vert e_j>:=\\delta_{ij}=\\frac{1}{2}(e_1\ne_2-J(e_2) e_1).\\label{con}\n\\end{eqnarray}\nWe may use the symbol '$i$` to denote $e_2\\wedge e_1$, because of\nthe properties\n\\begin{eqnarray}\ni^2&=&(e_2\\wedge e_1)(e_2\\wedge e_1)=e_2 e_1 e_2\ne_1=-e_2^2e_1^2=-1\\nonumber \\\\{}%\n[i,X_I]_{deg\\vert X_I\\vert}&=&0.\n\\end{eqnarray}\nWhere $deg\\vert X_I\\vert$ means the {\\bf Z}$_2$--grade of the\nhomogeneous element $X_I$. For (odd) even elements we have the\n(anti)commutator. This graduation is induced by $J$ and has the\nstructure\n\\begin{eqnarray}\nCL&=CL_+\\oplus CL_-,\n\\end{eqnarray}\nwhere the even (+) part constitutes a subalgebra, whereas the\nodd (-) part has only a $CL_+$ module structure. In the {\\bf\nE(2)} case, $CL_+$ is generated by $\\{1,i\\}$ and is itself a\nClifford algebra isomorphic to the field ${\\bf C}$ of complex numbers..\n\nNow we want to emphasize the operator character of complex\nnumbers. Therefore we calculate the left action of the element\n'$i$` on the base vectors\n\\begin{eqnarray}\ni e_1 &=&  e_2\\nonumber \\\\\ni e_2 &=& -e_1,\n\\end{eqnarray}\nwhich is a counter clockwise rotation by $\\pi\/2$. If we choose\nan arbitrary unit reference vector $e$, $e^2=1$, we may\nwrite the elements in $CL_+$ as\n\\begin{eqnarray}\nCL_+\\ni z&:=&x+y i,\\quad x,y \\in {\\bf K}.\n\\end{eqnarray}\nSo we have\n\\begin{eqnarray}\nz=ze^2=(x e+ y ie)e=v e\n\\end{eqnarray}\nwith $v \\in {\\bf E(2)}$. From\n\\begin{eqnarray}\n\\{e, ie\\}_+=eie+iee=-ie^2+ie^2=0\n\\end{eqnarray}\nwe have $ e \\perp ie$. Thus $\\{e,ie\\}$ span {\\bf E(2)}, and\nwe are able to rotate the coordinate system by an orthogonal\ntransformation to achieve\n\\begin{eqnarray}\ne   =e_1\\nonumber \\\\\ni e =e_2.\n\\end{eqnarray}\nThe map $z \\rightarrow ze$ is a bijective map from $CL_+\n\\rightarrow {\\bf E(2)}$, because $e$ is invertible. On the other\nhand we may look on the map $z : {\\bf E(2)} \\rightarrow {\\bf\nE(2)}$ with $za \\rightarrow v$ for arbitrary $a \\in$ {\\bf E(2)}\nand fixed $z\\in CL_+$.\n\nOnly with the choice $e=e_1$ we are able to interpret the scalar\nand bivector part of an operator spinor to be coordinates.\n\nThe modulus may be defined as follows\n\\begin{eqnarray}\n\\vert z \\vert^2 &:=& zz\\tilde{~} =(x+iy)(x-iy)=x^2+y^2.\n\\end{eqnarray}\n\n$CL_+$ is isomorphic to the field ${\\bf C}$ and therefore\nalgebraically closed. Thus it is possible to find a root\n\\begin{eqnarray}\nz&=&w^2= zz\\tilde{~}\\frac{z}{zz\\tilde{~}}=\\vert z\\vert^2 u^2,\\quad\nuu\\tilde{~}=1,\\quad \\vert z\\vert \\in \\mbox{{\\bf K}}\\sim\\mbox{{\\bf D.}}\n\\end{eqnarray}\nWe can reformulate the map as\n\\begin{eqnarray}\nze&=&w^2e=\\vert z\\vert^2 u^2=\\vert z\\vert^2 ueu\\tilde{~}.\n\\end{eqnarray}\nThis decomposes the left action of $z$ into a dilation and a\nspinorial rotation. Because of $u\\tilde{~}=u^{-1}$, $u$ is a {\\bf\nspin(2)} transformation. The transformation obtained by the left\naction of $u$ is of half angle type.\n\nThis point of view is independent of the special vector $e$ and\nemphasizes as well the operator character of the iteration\nformula. Indeed the iteration is a sequence of maps from ${\\bf\nE(2)} \\rightarrow {\\bf E(2)}$, where $a \\in {\\bf E(2)}$ is given\nby $z_n(e)$.\n\nThis works well, because $CL_+ \\cong {\\bf C}$ is algebraically\nclosed and products as well as sums of $CL_+$ elements yields\nnew $CL_+$ elements, which can be interpreted as new operator\nspinors $z^\\prime$. In the general case, as in the hyperbolic\none, only the multiplicative structure forms a (Lipschitz) group\n{\\bf D}$\\times${\\bf spin(p,q)}, whereas the additive group is in\ngeneral incompatible with the geometric structure.\nFor example we find two timelike vectors in the forward\nlight cone, which become space like when added.\n\nA physically sound dynamic model should therefore have an\ninvariant, i.e. multiplicative structure. By studying\nmultiplicative structures in vector spaces admitting one higher\ndimension and performing a split \\cite{HesSob,HesZie} one can\nachive affine transformations with {\\bf D}$\\times${\\bf\nIspin(p,q)}. For clarity and brevity, as well as for comparison\nwith the results of the literature we omit this complication.\nHowever we take it into account, when we perform the actual\ncalculations.\n\n\\section*{III Clifford Algebras for Arbitrary Bilinear Forms}\n\nIn this section we give a brief account on Clifford algebras\nwith arbitrary bilinear forms. It's main purpose is to show the\nconnection to quadratic algebras. For a somehow polemic\ndiscussion of Grassmann or Clifford algebra as a basic tool in\nphysics we refer to \\cite{Ozi,Hes-clif}.\n\nAt first glance it is surprising to have non symmetric bilinear\nforms in Clifford algebras, because in the usual approach\n\\cite{Dir} they arise naturally with symmetric bilinear or\nsesquilinear forms. The situation looks even more puzzling when\nnoticing the universality of Clifford algebras, usually stated\nas: ''There is up to isomorphisms only one unique algebraic\nstructure compatible with a bilinear form of signature (p,q)\non the space {{\\cal V}}``.\n\nWhy is it worse to study isomorphic structures? In \\cite{Fau} it\nwas demonstrated that the physical content of a theory depends\nsensible on the embedding $\\bigwedge{{\\cal V}}\\rightarrow CL({{\\cal V}},B)$ of\nthe Grassmann or exterior algebra into the Clifford algebra.\nFor example normalordering is exactly such a change of this embedding.\n\nHere we will give the connection between such changes and\nthe properties of quadratic algebras. They provide us with\na deeper geometric understanding as described in section IV.\n\nChevalley \\cite{Che} was the first who decomposed the Clifford\nalgebra product into parts. These parts, then explicitly,\nexhibit the twofold structure of the Clifford elements.\n\nOne part acts like a derivation on the space $\\bigwedge{{\\cal V}}$ of\nexterior powers of ${{\\cal V}}$. Especially if $\\omega_x=x\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$}$ is a\nform of degree $1$ it constitutes a form on ${{\\cal V}}$ into the field\n{\\bf K} ($x,y \\in {{\\cal V}}$,~$\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$} : {{\\cal V}}\\rightarrow {{\\cal V}}^\\ast$),\n\\begin{eqnarray}\n\\omega_x(y)=x\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$} y&:=&B(x,y) \\in {\\bf K}.\n\\end{eqnarray}\nThus $\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$}$ is a dualisomorphism parameterized by $B$.\nThe second part of the Clifford product is simply the exterior\nmultiplication.\n\nMarcel Riesz \\cite{Rie} reexpressed the\ncontraction and the exterior multiplication with a grade\ninvolution $J$ and the Clifford product. He obtained for char\n{\\bf K} $\\not=$ 2, $x\\in{{\\cal V}},\\, u\\in CL$\n\\begin{eqnarray}\nx\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$}   u&:=&\\frac{1}{2}(xu-J(u)x)\\nonumber \\\\\nx\\wedge u&:=&\\frac{1}{2}(xu+J(u)x).\n\\end{eqnarray}\nOne is able to extend these operations to higher degrees of\nmultivectors by $(a,b,c \\in {{\\cal V}},\\, X\\in CL)$\n\\begin{eqnarray}\n(a\\wedge b)\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$} X&=&a\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$}(b\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$} X)\\nonumber \\\\\na\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$}(bc)&=&(a\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$} b)c+J(b)(a\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$} c).\n\\end{eqnarray}\nAssociativity of the wedge product and other properties are\nshown in \\cite{Rie}. A detailed account on such properties is\ngiven in \\cite{Ozi}.\n\nIn the following it is convenient to use explicit not\nnecessarily normed or orthogonal generating sets. Especially for\nlow dimensional examples this will be useful. Therefore we\ndefine the left--contraction and the exterior product\nas (${{\\cal V}}=span\\{e_1,\\ldots,e_n\\}$)\n\\begin{eqnarray}\ne_i\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$} e_j&:=&B(e_i,e_j)\\cong [B_{ij}]\\nonumber \\\\\ne_i\\wedge e_j&:=&\\frac{1}{2}(e_i e_j- e_j e_i)\n\\end{eqnarray}\nwith the standard grade involution $J(e_i)=-e_i$.\n\nNext we may decompose $B$ into symmetric and antisymmetric part\n\\begin{eqnarray}\nG+F&:=&B \\nonumber \\\\\nG^T& =&G \\nonumber \\\\\nF^T&=&-F\n\\end{eqnarray}\nwhere ${}^T$ is the matrix transposition. In the case of an\nalgebra over the complex numbers one has to use hermitean\nadjunction, but here we deal exclusively with real algebras.\n\nWe specialize now to 2 dimensions. Then we are left with 4\nparameters, 3 of the symmetric and 1 of the antisymmetric part.\n\\begin{eqnarray}\n[G]&=&\\kla{\\begin{array}{cc}\n            G_{11} & G_{12} \\\\\n            G_{12} & G_{22}\n           \\end{array}} \\nonumber \\\\\n{}[F]&=&\\kla{\\begin{array}{cc}\n            0      & F_{12} \\\\\n            -F_{12}& 0\n           \\end{array}}.\n\\end{eqnarray}\nThe even part of this algebra satisfies a quadratic equation for\nevery element $z=x+y e_2\\wedge e_1$ where\n\\begin{eqnarray}\n(e_2\\wedge e_1)^2&=&(e_2 e_1-B_{21})(e_2\\wedge e_1) \\nonumber \\\\\n                 &=&e_2(e_1\\mbox{$\\,$\\rule{1ex}{0.4pt}\\rule{0.4pt}{1ex}$\\,$} (e_2\\wedge e_1))-B_{21}e_2\\wedge\n                 e_1 \\nonumber \\\\\n                 &=&e_2(B_{12}e_1-B_{11}e_2)-B_{21}e_2\\wedge\n                 e_1 \\nonumber \\\\\n                 &=&(B_{12}-B_{21})e_2\\wedge\n                 e_1-(B_{11}B_{22}+B_{12}B_{21}) \\nonumber \\\\\n                 &=&-2F_{21}e_2\\wedge e_1-det(B). \\label{q1}\n\\end{eqnarray}\nNow we expand $det(B)$\n\\begin{eqnarray}\ndet(B)&=&B_{11}B_{22}-B_{12}B_{21} \\nonumber \\\\\n      &=&G_{11}G_{22}-(G_{12}+F_{12})(G_{12}-F_{12}) \\nonumber \\\\\n      &=&F_{12}^2+det(G) \\label{det}\n\\end{eqnarray}\nand we arrive at\n\\begin{eqnarray}\n(e_2\\wedge e_1)^2&=&2F_{12}e_2\\wedge e_1-det(G)-F_{12}^2.\n\\end{eqnarray}\nIf there is no antisymmetric part (e.g. $F_{12}\\equiv 0$), we\nspecialize to\n\\begin{eqnarray}\n(e_2\\wedge e_1)^2&=&-det(G)\n\\end{eqnarray}\nand the determinant of $G$ describes the geometry at hand.\nIf $det(G)$ is positive we arrive at a (anti) Euclidean geometry.\nIn the negative case the geometry is hyperbolic.\n\nWe are free to choose an other algebra basis via a new doted\nwedge product defined by\n\\begin{eqnarray}\ne_i\\dot{\\wedge} e_j&:=&F_{ij}+e_i\\wedge e_j.\n\\end{eqnarray}\nTherefore we have incorporated the whole antisymmetric part in\nthe wedge product and the Clifford product decomposes as\n\\begin{eqnarray}\ne_i e_j&=&B_{ij}+e_i\\wedge e_j \\nonumber \\\\\n       &=&G_{ij}+e_i\\dot{\\wedge} e_j.\n\\end{eqnarray}\nWe define a new set of elements spanning the algebra, which\nreads in two dimensions\n\\begin{eqnarray}\n\\{ Y_J\\} &:=&\\{1,e_2\\dot{\\wedge} e_1,e_1,e_2\\}\\label{Y}\n\\end{eqnarray}\nwhich leads to another quadratic relation\n\\begin{eqnarray}\n(e_2\\dot{\\wedge} e_1)^2&=&(e_2\\wedge e_1+F_{21})^2 \\nonumber \\\\\n       &=&(e_2\\wedge e_1)^2+2F_{21}e_2\\wedge e_1+F_{21}^2 \\nonumber \\\\\n       &=&2(F_{21}+F_{12})e_2\\wedge e_1\n       -det(G)-F_{12}^2+F_{21}^2 \\nonumber \\\\\n       &=&-det(G). \\label{q2}\n\\end{eqnarray}\nIn this construction the linear term has been absorbed in the\ndoted wedge. The quadratic relation has simplified to the\nhomogeneous case discussed above, but now not as a special case.\n\nNext we introduce a (pseudo) norm function on the\nquadratic subalgebra. In the Clifford algebra there are several\nsuch constructions, known as spinor norms \\cite{Lou-spin,Cru}.\n\nFor vector elements $a\\in {{\\cal V}}$ every of the three following maps\nhas the image in the field {\\bf K}. With the standard involution\n$J(a)=-a$ we have\n\\begin{eqnarray}\na^2\\rightarrow {\\bf K};\\quad\naJ(a\\tilde{~})=aJ(a)\\tilde{~}=-a^2\\rightarrow {\\bf K};\\quad\naa\\tilde{~}=a^2\\rightarrow {\\bf K}\n\\end{eqnarray}\nBut applying this to bivector elements, the first relation maps\nnot in the field, but is exactly the quadratic relation derived\nabove. If we allow arbitrary involutions, $a$ and $J(a)$ need\nnot be parallel vectors, as is the same for bivector elements.\nIn general we may not expect this map to be scalar valued. To\nanalyze the third map, we recognize\n\\begin{eqnarray}\n(e_i\\wedge e_j)\\tilde{~} &=& (e_i e_j-B_{ij})\\tilde{~}=e_j e_i-B_{ij} \\nonumber \\\\\n      &=&e_j\\wedge e_i+B_{ji}-B_{ij}  \\nonumber \\\\\n      &=&e_j\\wedge e_i-2F_{ij} \\nonumber \\\\\n      &=&- e_i\\wedge e_j-2F_{ij}.\n\\end{eqnarray}\nThus we have with (\\ref{q1})\n\\begin{eqnarray}\n(e_i\\wedge e_j)(e_i\\wedge e_j)\\tilde{~}&=&-(e_i\\wedge\n   e_j)^2-2F_{ij}e_i\\wedge e_j \\nonumber \\\\\n   &=&det(B)+2F_{ij}e_i\\wedge e_j-2F_{ij}e_i\\wedge ej \\nonumber \\\\\n   &=&det(B)\n\\end{eqnarray}\nas with (\\ref{q2}) in the same manner for the doted case\n\\begin{eqnarray}\n(e_i\\dot{\\wedge} e_j)(e_i\\dot{\\wedge} e_j)\\tilde{~} &=&-(e_i\\dot{\\wedge}\ne_j)(e_i\\dot{\\wedge} e_j)\\nonumber \\\\\n&=&det(G).\n\\end{eqnarray}\nThis reduces in the case of two dimensions to\n\\begin{eqnarray}\n(e_2\\dot{\\wedge} e_1)(e_2\\dot{\\wedge} e_1)\\tilde{~} &=&det(G).\n\\end{eqnarray}\nwhich is the discriminant of the quadratic equation (\\ref{q2}).\nIf the discriminant is positive, there are two roots in the\nalgebra (over {\\bf R}), whereas otherwise the field has to be\nalgebraically closed (i.e. {\\bf C}), or extracting the root is\nnot possible.\n\nWe introduce the abbreviation $X \\cong\\{e_2\\wedge e_1$ or\n$e_2\\dot{\\wedge} e_1\\}$, $a\\cong \\{2F_{12}$ or $0 \\}$, $b\\cong \\{\ndet(B)$ or $det(G) \\}$ and may\nwrite the quadratic algebra as\n\\begin{eqnarray}\n\\frac{K[X]}{X^2-aX-b},\n\\end{eqnarray}\nthe polynom algebra generated by $X$ over the field ${\\bf K}$\nmodulo the quadratic relation. This form is important for\ncomparison with the theory of quadratic algebras, but our aim\nis the exposition of the connection to the geometric relations\nencoded in this formula.\n\n\\section*{IV Quadratic Algebras, Conjugation and Special Elements}\n\nIn this section we give some results exposed in \\cite{Hah},\nwhich provide us with existence and uniqueness theorems. This\nsupplies more fondness to our somehow loosely construction above.\n\nThe constructions are valid in much more general settings, as\nClifford algebras over finite fields or over modules, which is\nyet not needed but quite interesting.\n\nLet ${{\\cal R}}$ be a commutative ring and ${{\\cal I}}$ an ideal of ${{\\cal R}}$, then\n${{\\cal R}}\\rightarrow {{\\cal R}}\/{{\\cal I}}={{\\cal A}}$ is in a natural way a ${{\\cal R}}$--algebra.\n\nA free quadratic algebra is obtained if one factorizes the\npolynom algebra in the indeterminate $X$ over ${{\\cal R}}$ by the ideal\n${{\\cal I}}=(X^2-aX-b)$\n\\begin{eqnarray}\nS&=&\\frac{{{\\cal R}}[X]}{X^2-aX-b}.\n\\end{eqnarray}\nThe identity is\n\\begin{eqnarray}\n1_S&=&1+{{\\cal I}}=1+(X^2-aX-b)\n\\end{eqnarray}\nand because $r\\in{{\\cal R}}$, $r\\rightarrow r1_S$ is injective we have\n${{\\cal R}}\\subseteq S$. A basis of $S$ as ${{\\cal R}}$--module is given by\n\\begin{eqnarray}\n\\{1&,& v=X+{{\\cal I}}=X+(X^2-aX-b) \\}.\n\\end{eqnarray}\nIt follows that\n\\begin{eqnarray}\nv^2&=&av+b.\n\\end{eqnarray}\nWe have several isomorphisms:\n\\begin{eqnarray}\nS_1&=&\\frac{{{\\cal R}}[X]}{X^2-X}\\cong {{\\cal R}}\\oplus {{\\cal R}}\n\\end{eqnarray}\nwith the diagonal product map. $\\phi : {{\\cal R}}\\oplus {{\\cal R}} \\rightarrow\nS_1$ is an isomorphism and we denote $S_1$ as trivial quadratic\n${{\\cal R}}$--algebra. This is the ''perplex`` case from above!\n\\begin{eqnarray}\nS_2&=&\\frac{{{\\cal R}}[X]}{X^2}\n\\end{eqnarray}\nconstitutes the algebra of dual numbers.\n\nDenoting the units of ${{\\cal R}}$ as ${{\\cal R}}^\\ast$, we can formulate the\nisomorphism criterion \\cite{Hah}(1.1).\n\n{\\it Let ${{\\cal R}}[X]\/(X^2-aX-b)$ and ${{\\cal R}}[X]\/(X^2-cX-d)$ be quadratic\nalgebras over ${{\\cal R}}$. Then\n\\begin{eqnarray}\n\\frac{{{\\cal R}}[X]}{X^2-aX-b}&\\cong&\\frac{{{\\cal R}}[X]}{X^2-cX-d} \\nonumber\n\\end{eqnarray}\niff there exist elements $r\\in{{\\cal R}}$ and $u\\in{{\\cal R}}^\\ast$ such that\n(i) $ c=ua+2r$ (ii) $d=u^2b-rua-r^2$.}\n\nNow, has $X^2-aX-b$ a root, say $\\gamma$ in ${{\\cal R}}$ then we have\n\\begin{eqnarray}\nX^2-aX-b&=&(X-\\gamma)(X-(a-\\gamma)) \\label{root}\n\\end{eqnarray}\nand $a-\\gamma$ is another root (in ${{\\cal R}}$) of the quadratic\nequation. If $a-\\gamma=\\gamma$ then $\\gamma$ is a double root.\nWe can state the following\n\\begin{eqnarray}\n\\begin{array}{ccccl}\n(1) & \\frac{{{\\cal R}}[X]}{X^2-aX-b} &\\cong& \\frac{{{\\cal R}}[X]}{X^2-taX-t^2b} &\n\\mbox{for $t\\in {{\\cal R}}^\\ast$} \\\\\n(2) & \\frac{{{\\cal R}}[X]}{X^2-aX-b} &\\cong& \\frac{{{\\cal R}}[X]}{X^2-cX} &\n\\Leftrightarrow X^2-aX-b\\quad \\mbox{has a root in ${{\\cal R}}$} \\\\\n(3) & \\frac{{{\\cal R}}[X]}{X^2-aX-b} &\\cong& \\frac{{{\\cal R}}[X]}{X^2} &\n\\Leftrightarrow X^2-aX-b\\quad \\mbox{has a double root in ${{\\cal R}}$}\n\\end{array} \\nonumber\n\\end{eqnarray}\n{\\bf Examples:}\n\\begin{enumerate}\n\\item ${{\\cal R}}\\cong {\\bf C}$: The only quadratic algebras are ${\\bf\nC}[X]\/(X^2-X)$ the complex trivial algebra and ${\\bf\nC}[X]\/(X^2)$ the algebra of complex dual numbers, because of the\nexistence of roots for every  $X\\in{\\bf C}$.\n\n\\item ${{\\cal R}}\\cong {\\bf R}$: We have three cases, because negative\nnumbers posses no roots in ${\\bf R}$.\n\\begin{eqnarray}\nS &\\cong&\n\\left\\{\\begin{array}{ccl}\n\\frac{{\\bf R}[X]}{X^2-X} & a^2+4b > 0 & \\mbox{trivial, ''perplex``} \\\\\n\\frac{{\\bf R}[X]}{X^2}   & a^2+4b = 0 & \\mbox{dual numbers} \\\\\n\\frac{{\\bf R}[X]}{X^2+1}\\cong {\\bf C} & a^2+4b < 0 &\n\\mbox{complex numbers}\n\\end{array}\\right.\n\\end{eqnarray}\nOf course, $a^2+4b$ is the discriminant of the quadratic relation.\n\n\\item ${{{\\cal R}}\\cong {\\bf Z}}$: Results in the infinitely many\nisomorphism classes\n\\begin{eqnarray}\n\\frac{{\\bf Z}[X]}{X^2-aX-b}&\\cong&\\left\\{\n\\begin{array}{cl}\n\\frac{{\\bf Z}[X]}{X^2-n} & \\mbox{if $a$ is even} \\\\\n\\frac{{\\bf Z}[X]}{X^2-X-n} & \\mbox{if $a$ is odd}\n\\end{array}\\right.\n\\end{eqnarray}\n\\end{enumerate}\nIt turns out, that simpler rings (as also finite Galois\nfields) bear much more structure.\n\nIf $\\alpha$ is an (anti) automorphism and $\\alpha^2=id_{{\\cal A}}$, then\n$\\alpha$ is an involution. Algebra homomorphisms which preserve\nsuch a structure are called graded homomorphisms. In physics the\nsuper symmetric transformations (mixing of Grassmann parity)\nare not grade perserving and thus minor symmetric.\n\nIn a quadratic algebra we may introduce the involution $\\sigma :\nS \\rightarrow S$ on the base\n\\begin{eqnarray}\n\\{ 1, v&=&X+(X^2-aX-b) \\} \\nonumber \\\\\n1^\\sigma &=& 1 \\nonumber \\\\\nv^\\sigma &=& (a-v)\n\\end{eqnarray}\nwhich results in\n\\begin{eqnarray}\n(x+yv)^\\sigma &=& (x+ya)+yv \\nonumber \\\\\n(x+yv)^{\\sigma\\sigma} &=& (x+yv).\n\\end{eqnarray}\nThis involution interchanges the roots of the quadratic relation\n(\\ref{root}). In the complex case this is the ordinary complex\nconjugation.\n\nNow this kind of involution is ''standard`` in\nquadratic algebras and induces the {\\bf Z}$_2$--grading of the\nClifford algebra.\nLet ${{\\cal A}}_i$, $i\\in (0,1)$ be ${{\\cal R}}$--submodules and ${{\\cal A}}=A_0\\oplus\n{{\\cal A}}_1$. As $\\R1 \\subseteq {{\\cal A}}_0$, ${{\\cal A}}$ is a {\\bf Z}$_2$--graded\nalgebra. Elements $a\\in {{\\cal A}}_i$ ($\\partial a=i$, grade of $a$) are\ncalled homogeneous.\n\nWe have two possibilities to introduce tensor products in graded\nalgebras via\n\\begin{eqnarray}\n{{\\cal A}} {\\otimes}_{{\\cal R}} {{\\cal B}} &:& (a\\otimes b)(a^\\prime \\otimes\nb^\\prime):=(aa^\\prime\\otimes bb^\\prime) \\nonumber \\\\\n{{\\cal A}} {\\hat\\otimes}_{{\\cal R}} {{\\cal B}} &:& (a\\otimes b)(a^\\prime \\otimes\nb^\\prime):=(-)^{\\partial a^\\prime\\partial b}(aa^\\prime\\otimes\nbb^\\prime).\n\\end{eqnarray}\nTo the algebra ${{\\cal A}}$ we find the opposite algebra ${{\\cal A}}^{op}$ by\nreversing the product, $(ab)^{op}=b^{op}a^{op}$, which is a map\nfrom ${{\\cal A}}$ into ${{\\cal A}}^{op}$. Hence we construct the enveloping\nalgebra ${{\\cal A}}^e:={{\\cal A}}\\otimes_{{\\cal R}} {{\\cal A}}^{op}$, as a (${{\\cal A}},{{\\cal A}}$)--bimodule.\nThere is a unique homomorphism $\\Phi : {{\\cal A}}^e\\rightarrow {{\\cal A}}$,\nwhich satisfies $\\Phi ( a\\otimes b^{op})=ab$. If there exists also\na homomorphism $\\Theta : {{\\cal A}}\\rightarrow {{\\cal A}}^e$ (coproduct) such that\n$\\Phi\\Theta=id_{{\\cal A}}$ then the algebra is called separable. It follows\nthen ${{\\cal A}}^e={{\\cal A}}\\otimes_{{\\cal R}}{{\\cal A}}^{op}=ker(\\Phi)\\oplus\\Theta({{\\cal A}})$. With\n\\cite{Hah}(2.1)(2.3) we state:\n\\begin{enumerate}\n\\item {\\it ${{\\cal A}}$ is a separable ${{\\cal R}}$--algebra iff ${{\\cal A}}$ has a\nseparability idempotent, $e=\\Theta(1)$.}\n\\item {\\it The separability idempotent of a free quadratic\nalgebra $S$ over ${{\\cal R}}$ is unique.}\n\\end{enumerate}\nNow it is possible to classify separable free quadratic\n${{\\cal R}}$--algebras by introducing the group $Qu_f({{\\cal R}})$. Therefore define\n\\begin{eqnarray}\nQ&:=&\\{(a,b) \\vert a^2+4b\\in {{\\cal R}}^\\ast\\}\n\\end{eqnarray}\nwith the product\n\\begin{eqnarray}\n(a,b)*(c,d)&:=&(ac,a^2d+c^2b+4bd)\n\\end{eqnarray}\nand the quotients $[a,b]=(a, b)\/{{\\cal R}}^{\\ast 2}$ form the group\n\\begin{eqnarray}\nQu_f({{\\cal R}})&:=&\\{[a,b]\\vert (a,b)\\in Q\\}.\n\\end{eqnarray}\nThe cardinality of $Qu_f({{\\cal R}})$ is the number of isomorphism\nclasses of (nontrivial) quadratic algebras, e.g.\n\\begin{eqnarray}\nQu_f({\\bf C})&\\cong&\\frac{{\\bf C}^\\ast}{{\\bf C}^{\\ast^2}}=1 \\nonumber \\\\\nQu_f({\\bf R})&\\cong&\\frac{{\\bf R}^\\ast}{{\\bf R}^{\\ast^2}}={\\bf Z_2}.\n\\end{eqnarray}\nThis can be extended to a graded version $QU_f({{\\cal R}})$.\n\nOne observes a connection between grading and standard\ninvolution via the special elements. To see this,\nlet $M=({{\\cal V}},B)$ be the pair of a vector space with a bilinear\nform (or a quadratic module), then we can build the Clifford\nalgebra $CL(M)=CL({{\\cal V}},B)$. We get now \\cite{Hah}(5.4)\n\n{\\it The decomposition $CL(M)=CL_0(M)\\oplus CL_1(M)$ is a ({\\bf\nZ}$_2$) grading of $CL(M)$. $CL_0(M)$ is a subalgebra and\n$CL_1(M)$ is a $CL_0(M)$ module.}\n\nWe call $\\sigma$ a standard involution, if $\\sigma$ is a\nantiinvolution and $aa^\\sigma\\in{{\\cal R}}$ $\\forall a\\in CL(M)$. In\nthis case we define a (pseudo) norm and a trace as\n\\begin{eqnarray}\nnr(a)&:=&aa^\\sigma (\\in {{\\cal R}}) \\nonumber \\\\\ntr(a)&:=&a+a^\\sigma (\\in {{\\cal R}}). \\label{norm}\n\\end{eqnarray}\n\nAn element $z\\in CL(M)$ is a special element if $\\{1,z\\}$ is a\nbasis of the centralizer $Cen_{CL(M)} CL_0(M)=\\{c\\in CL(M) \\vert\ncd=dc\\,\\, \\forall d\\in CL_0(M)\\}$. One can conclude\nthat\\footnote{See \\cite{Hah} (8.3)}\n\\begin{enumerate}\n\\item if rank $M$ is odd, then $z\\in CL_1(M)$, $z^\\sigma=-z$ and\n$z^2=b$ with $b\\in{{\\cal R}}$.\n\\item if rank $M$ is even, then $z\\in CL_0(M)$, $z^\\sigma=a-z$\nand $z^2=az-b$ with $a,b\\in{{\\cal R}}$, $a^2+4b\\in{{\\cal R}}^\\ast$.\n\\end{enumerate}\n\n{\\it If $\\gamma$ is a root of $X^2-aX-b$ in $S$, then $S$ has\nthe grading $S=S_0\\otimes S_1$, where\n\\begin{eqnarray}\nS_0&=&Cen_S(\\gamma)=\\{s\\in S \\vert \\gamma s= s\\gamma \\} \\nonumber \\\\\nS_1&=&\\{s\\in S \\vert \\gamma s+ s\\gamma=as \\}.\n\\end{eqnarray}}\n\nThe grading is trivial if $\\gamma$ is in the center of $S$.\n\nThese properties are strongly interwoven and can be used in\nconstructing representations of Clifford algebras \\cite{Dim}.\n\nThe existence of special elements in a general setting is proved\nin chapter 10 of \\cite{Hah}. All this constructions are possible\nin higher dimensions, but our above naive geometric\ninterpretation has then to be refined.\n\nAs a last topic we have a look at the representations of\nClifford algebras. A homomorphism $\\Phi$ of ${{\\cal R}}$--algebras\n\\begin{eqnarray}\n\\Phi &:& CL(M)\\rightarrow End_S(P)\n\\end{eqnarray}\nwhere $M$ is a quadratic module, $S$ a ${{\\cal R}}$--algebra and $P$ a\nright $S$--module is called an ($S$--)representation of $CL(M)$.\nIn \\cite{Hah}(8.7)(8.8) the connection between division\nalgebras, gradings and the quadratic algebra is shown and\nconnected with the existence of roots in $S$. The quadratic\nalgebra $S$ appears as tensor factor in such representations.\n\nDespite the universality of Clifford algebras, we need for\nphysical applications norms and traces, as the grading\n(vector space and {\\bf Z}$_2$). Therefore we have to distinguish\nbetween such homomorphisms preserving this additional structures\nand those doing this not. The free quadratic groups etc.\ncharacterize the isomorphism classes available, where the kernel\nof the factorization parameterizes distinct but isomorphic\nrepresentatives. Such a parameterization can be done equally well\nby parameterizing the isomorphic ideals in constructing Clifford\nalgebras. This was done in \\cite{Ozi-hecke}.\n\n\\section*{V The 2--dim. Case and Numeric Experiments}\n\nIn this section we discuss the 2--dim. case\nover {\\bf R}. Because of the even dimension we expect to have\ninhomogeneous terms and a rich structure. Using the natural\ngiven standard involution, we can construct (pseudo) norms and\ntraces as above and investigate the Mandelbrot set in each of the\nthree cases. We do not get a quadratic set, but a ''light cone``\nstructure in the hyperbolic case.\n\nLet us choose a generating set $\\{e_1, e_2\\}$ as in (\\ref{X}) or\nin (\\ref{Y}), and a bilinear form $B=G+F$ as above. We denote\nthe bivector element as $\\gamma$. Here one has to take care,\nbecause a change in the quadratic relation of $\\gamma$ is\nrelated with a change of the representation of the\nalgebra. In the same time this results in a redefinition of the\nwedge, as above done by using the two extreme cases $\\wedge$ and\n$\\dot{\\wedge}$.\n\nWe interpret the iteration formula in the operational sense\nexplained in section II and plot all figures in the\n$\\{1,\\gamma\\}$--plane. A translation into ${{\\cal V}}$ may be done as\nexplained also in section II. For comparison to other results\nthis is here {\\it not} done.\n\nWith Sylvester's theorem we could achieve a diagonal form for the\nsymmetric part of $G$ with $G\\equiv diag\\{\\pm 1, \\pm 1\\}$. But a\nrescaling of the base vectors by $\\sqrt{\\vert G_{11}\\vert}$ and\n$\\sqrt{\\vert G_{22}\\vert}$ would affect the magnitude of the\nnonsymmetric part also. This is contained in the isomorphism\ncriterion and will therefore be done only in the quadratic\nalgebra and not in the whole Clifford algebra.\n\nDefine $S={{\\cal R}}[\\gamma]\/(\\gamma^2-a\\gamma-b)$. We get from (\\ref{det})\n\\begin{eqnarray}\n\\gamma^2&=&2F_{12}\\gamma-det(B) \\nonumber \\\\\n        &=&2F_{12}\\gamma-det(G)-F_{12}^2.\n\\end{eqnarray}\nHence we set\n\\begin{eqnarray}\na&:=&2F_{12} \\nonumber \\\\\nb&:=&-det(B)=-det(G)-F_{12}^2.\n\\end{eqnarray}\nThe discriminant is connected to the metric via\n\\begin{eqnarray}\nd\\, =\\, a^2+4b&=&-4det(G).\n\\end{eqnarray}\nThe discriminant of the quadratic relation is thus $-4$ times\nthe determinant of the symmetric part of the bilinear form!\n\nIf $det(G)=0$ we have 2 roots in the algebra $S$. If the algebra\nis not dual (double root) we have $det(G)\\not= 0$.\n\nAs explained in section IV, there exists a standard involution\nif the algebra $S$ is separable. In our case this is the\nreversion in $CL$. It is constructed in the base $\\{1,\\gamma\\}$ by\n\\begin{eqnarray}\n1^\\sigma &=& 1 \\nonumber \\\\\n\\gamma^\\sigma &=& (a-\\gamma)=2F_{12}-\\gamma.\n\\end{eqnarray}\nThe inhomogeneous additive term is quite uncommon in usual\napproaches to Clifford algebras. Let us emphasize, that in the\nquantum mechanical case, where the field is complex, we are able\nto find {\\it always} an algebra isomorphism to achieve\n$\\gamma^{\\prime\\sigma}=\\gamma^\\prime$. Because there is only one\nnontrivial isomorphism class. This may be an argument for using\ncomplex numbers in quantum mechanics.\n\nThe iteration formula is now obtained as follows\n\\begin{eqnarray}\nz&:=&z_x+z_y\\gamma;\\quad c:=c_x+c_y\\gamma\n\\end{eqnarray}\nand with\n\\begin{eqnarray}\nz_{n+1}&:=&z_n^2+c\n\\end{eqnarray}\nwe obtain\n\\begin{eqnarray}\nz_{x(n+1)} &=& z_{x(n)}^2+bz_{y(n)}^2+c_x \\nonumber \\\\\nz_{y(n+1)} &=& 2z_{x(n)}z_{y(n)}+az_{y(n)}^2+c_y.\n\\end{eqnarray}\nThe spinor (pseudo) norm is then given by\n\\begin{eqnarray}\nnr^2(z)&=&zz\\tilde{~}=z_x^2-bz_y^2+az_xz_y.\n\\end{eqnarray}\nWe have:\n\\begin{itemize}\n\\item[i)] {\\it\n$nr^2(z)$ is positive definite in the complex isomorphism class}\n\\item[ii)] {\\it\n$nr^2(z)$ is positive semidefinite in the dual isomorphism class}\n\\item[iii)] {\\it\n$nr^2(z)$ is indefinite in the trivial isomorphism class\n(hyperbolic case)}\n\\end{itemize}\n{\\bf Proof:} We distinguish two cases\n\\begin{itemize}\n\\item[a)] Suppose that $y\\equiv 0$. We are left with\n$nr^2(z)=x^2$, which is positive definite in $x$ and $nr^2$ is\nsemidefinite in case iii), by isomorphy to the case $a=b=0$.\n\\item[b)] Suppose $y\\not= 0$. We introduce $\\rho=x\/y$ and have\n\\begin{eqnarray}\n\\frac{nr^2}{y^2}&=&\\rho^2-a\\rho-b\\nonumber \\\\\n                &=&(\\rho-\\frac{a}{2})^2-(\\frac{a^2}{4}+b)\\nonumber \\\\\n                &\\geq&-(\\frac{a^2}{4}+b)=-d=4\\, det(G).\n\\end{eqnarray}\nThus if the discriminant is negative the norm is positive\ndefinite. If $d=0$, the norm may be zero for non--null\nelements, that is positive semidefinite. Is $d > 0$ we are in the\nindefinite hyperbolic case. \\rule{1ex}{1ex}\n\\end{itemize}\n\nThe variable $\\rho$ is connected via an arctan or arctanh to the\nphase of $z$. But a polar decomposition is in the hyperbolic\ncase not obvious. See \\cite{Fje} and notice the appearance of\nthe Klein group.\n\nThe line ${{\\cal R}} 1_S$ is stabilized by $\\sigma$ via $z^\\sigma=z$\n$\\Rightarrow$ $y=0$. Whereas the trace maps $z$ onto ${{\\cal R}} 1_S$ as\n\\begin{eqnarray}\ntr(z)&=&\\frac{1}{2}(z+z^\\sigma)=x+\\frac{a}{2}y.\n\\end{eqnarray}\nWe define the operator spinor Mandelbrot set as\n\\begin{eqnarray}\nM&:=&\\{c\\vert z_0=(0,0),\\quad nr^2(z_n^c) > 0\\, \\forall n,\\quad\n\\lim_{n \\rightarrow\\infty}\\, nr^2(z^c_n)\\not\\rightarrow\\infty.\\}\n\\end{eqnarray}\n\nIn the complex case we set the parameters as\n\\begin{eqnarray}\na&=&0 \\nonumber \\\\\nb &=& -1,\\quad \\rightarrow\\quad \\gamma^2=-1.\n\\end{eqnarray}\nHence\n\\begin{eqnarray}\nd\\, =\\,a^2+4b&=&-4=-4\\, det(G)\\nonumber \\\\\ndet(G)&=&1,\n\\end{eqnarray}\nwhich results in a (anti) Euclidean geometry. Because of the\npositive discriminant we have always two roots in $S$. The norm\nyields\n\\begin{eqnarray}\nnr^2_{\\bf C}(z)&=&x^2+y^2.\n\\end{eqnarray}\nWe are left with the ordinary Mandelbrot set as shown in figure\n1. All pictures with $d= -4.0$, and arbitrary $a$ are isomorphic\nwithout a rescaling. But the coordinate interpretation is no\nlonger possible. If one looks at $d = -4.0$, $a=1.0$ (figure\n2.), one obtains up to an {\\bf SO(2)} transformation the\nbilinear form\n\\begin{eqnarray}\nB&=&G+F\\cong\n\\left(\n\\begin{array}{cc}\n1 & \\frac{1}{2} \\\\\n0 & 1\n\\end{array}\n\\right).\n\\end{eqnarray}\nThus we have now\n\\begin{eqnarray}\n\\gamma e_1 &=& (e_2 e_1-B_{21})e_1=e_1^2 e_2=e_2 \\nonumber \\\\\n\\gamma e_2 &=& \\frac{1}{2}e_2-1 e_1.\n\\end{eqnarray}\nThe new transformation obtained by $\\gamma$ : {\\bf\nE(2)}$\\rightarrow$ {\\bf E(2)} does not preserve angles, but areas.\nThe real axis ($\\sigma$ invariant points) is not affected by this.\nThe result is the deformed set of figure 2.\n\nIf the parameter $d$ is changed, this results in a scaling of\nthe $\\gamma$--axis. As in the above case the transition into the\nvector space is not unique. If one chooses the 1$_S$--axis to map\nonto $e_1$ then we arrive at a bilinear form like\n\\begin{eqnarray}\n[B]&=&\n\\left(\n\\begin{array}{cc}\nd & \\frac{a}{2} \\\\\n0 & 1\n\\end{array}\n\\right)\n=\\left(\n\\begin{array}{cc}\nG_{11} & F_{12} \\\\\n0 & 1\n\\end{array}\n\\right),\n\\end{eqnarray}\nwhich is a special case of the parameterization\n\\begin{eqnarray}\n[B] &=&d\n\\left(\n\\begin{array}{cc}\n\\lambda & \\frac{a}{2} \\\\\n0 & \\frac{1}{\\lambda}\n\\end{array}\n\\right).\n\\end{eqnarray}\nThis case is also isomorphic to the complex one, but this time\nwith an additional rescaling (figure 3). Areas are no longer\npreserved.\n\nThe second case is the dual one. Hence the algebra $S$ is no\nlonger separable and degenerated to a 1--dim. scheme. The\ncorresponding parameters are\n\\begin{eqnarray}\na&=&0 \\nonumber \\\\\nb&=&0,\\quad\\rightarrow\\quad\\gamma^2=0.\n\\end{eqnarray}\nThe discriminant vanishes. This case ($X^2=0$ as ideal) results\nin a degeneration of the dynamics. It is the limiting case of\nthe two other ones. The semidefinite norm is $nr^2(z)=(x-a\/2\\,\ny)^2$, which is sensitive only to one direction. In the case\nwith $a = 0$ this is the $x$--axis, see figure 4. To form a\ndefinite norm, and thus a physically meaning full situation, one\nhas to factor out the superfluous direction. So this case is\nessential one dimensional. This can take place even if $B$ is\nnondegenerate, but $G$ is still.\n\\begin{eqnarray}\n[B]&=&\n\\left(\n\\begin{array}{cc}\nG_{11} & F_{12} \\\\\n-F_{12}& 0\n\\end{array}\n\\right)=\n\\left(\n\\begin{array}{cc}\nG_{11} & 0 \\\\\n0      & 0\n\\end{array}\n\\right)\n+\\left(\n\\begin{array}{cc}\n0       & F_{12}\\\\\n-F_{12} & 0\n\\end{array}\n\\right)\n\\end{eqnarray}\nIn a physical context this case should be called trivial, but\nthis has not to be confused with the classification of the\nquadratic algebras, where this case is the dual one.\n\nThe hyperbolic (or ''perplex``) case is obtained with the\nparameter setting\n\\begin{eqnarray}\na&=&0 \\nonumber \\\\\nb&=&1,\\quad \\rightarrow\\quad \\gamma^2=1.\n\\end{eqnarray}\nThe discriminant becomes\n\\begin{eqnarray}\ndet(G)=-1,\n\\end{eqnarray}\nwhich corresponds to the hyperbolic geometry. Not every element\nin $S$ has a root in $S$ and especially $\\gamma$ has not.\n\nIn \\cite{Met} the convergence was proved with the norm $nr_{\\bf\nC}$ from above. With our considerations we get contrary\n\\begin{eqnarray}\nnr^2_{{\\bf R}\\oplus{\\bf R}}&=&x^2-y^2.\n\\end{eqnarray}\nWe recover the ''light cone``, which one is used to find in\nhyperbolic geometry. Hence $x$ is the timelike coordinate and\n$y$ the space like. Backward and forward light cone enclose the\ninvariant real line ${\\bf R} 1_S$.\n\nAs exposed in the introduction, the dynamic (iteration process)\ndoes not respect this structure. So timelike elements will\nbecome space like and vice versa. The pictures were done in such a\nway, that the iteration halted immediately whenever an element got\nspace like.\n\nThe most surprising effect is, that the light cones become\nseparated by the multi quadratic counter part of the Mandelbrot\nset. The two light cones are separated thus by a timelike\ndistance. On the real axis ${{\\cal R}} 1_S$ things doesn't change at all.\nSee figure 5.\n\nThe mono quadratic case would be reobtained if one would ignore the\nhyperbolic structure.\n\nThe hyperbolic case is the most interesting one, because of the\ndifference to the sets obtained in literature.\nThe asymptotic is as in the usual case. The separation results\nfrom a deformation of the backward cone light (negative abscissa)\nand a minor deformation of the forward cone. If the picture is\nscaled in such a way, that the separation distance is small, one\nobtains the ordinary cone structure. Points near the space like\nregion in the backward cone become space like during the\niteration. Points inside quadrangles which intersect the real\nline are non divergent points and thus the counter part to the\nmandelbrot set. The vertical cones without structure constitutes\nthe space like region.\n\nWhy is this dynamically interesting case called trivial?\n\nThis stems from the quadratic relation\n\\begin{eqnarray}\nX^2-cX&=&0.\n\\end{eqnarray}\nLet us assume that $c=1$\\footnote{The sign does not matter in\nthis case.}, then one arrives at\n\\begin{eqnarray}\nX^2&=&X,\n\\end{eqnarray}\nwhich is a projector equation. The algebra may then be\ndecomposed with help of $X$ into a direct sum.\n\\begin{eqnarray}\n1_S&=&X+(1-X)\\nonumber \\\\\nX(1-X)&=&0\\nonumber \\\\\nX^2=X && (1-X)^2=(1-X)\n\\end{eqnarray}\nTherefore $X$, $(1-X)$ are pair wise annihilating primitive\nidempotents.\n\nThe metric structure is then connected with\n\\begin{eqnarray}\n[B]=[G]+[F]&=&\n\\left(\n\\begin{array}{cc}\n0 & 0 \\\\\n0 & 0\n\\end{array}\n\\right)+\n\\left(\n\\begin{array}{cc}\n0   & \\frac{c}{2} \\\\\n\\frac{-c}{2} & 0\n\\end{array}\n\\right),\n\\end{eqnarray}\nwhich is a symplectic structure or equivalently\\footnote{With\nthe isomorphism criterion from section IV.} with\n\\begin{eqnarray}\n[B]=[G]&=&\n\\left(\n\\begin{array}{cc}\n\\frac{c^2}{4} & 0 \\\\\n0              & -1\n\\end{array}\n\\right),\n\\end{eqnarray}\nwithout any antisymmetric contribution! This corresponds to\n$X^2=c^2 > 0$. The decomposition is now obtained by the projectors\n\\begin{eqnarray}\ne_\\pm&:=&\\frac{1}{2c}(c\\pm X)\n\\end{eqnarray}\n\nThe parameter $a$ acts as in the complex case as is seen in\nfigure 6. The fact that the trivial case can always be splitted\ninto a direct sum with diagonal multiplicative structure was\nessential for the proof in \\cite{Met}.\n\nIn the pure hyperbolic case ($d = 4$, $a=0$) we have for example\n\\begin{eqnarray}\n[B]&=&\n\\left(\n\\begin{array}{cc}\n1 & 0 \\\\\n0 & -1\n\\end{array}\n\\right)\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\gamma e_1 &=& e_2\\nonumber \\\\\n\\gamma e_2 &=& e_1,\n\\end{eqnarray}\nwhich is not a rotation, but a space--time inversion. This\ntransformation flips also the orientation of ${{\\cal V}}$, thus a\nphysical interpretation should be charge or parity conjugation.\nBut there is a continuum of such transformations.\n\n\\section*{VI Conclusion}\n\nWe showed with numerical examples, that the multi quadratic\nMandelbrot set is superior to the quadratic one. The geometric\ninterpretation fits in all special cases, but then with a\ndistinguished (pseudo) norm. The operator spinor approach is the\nkey step in this consideration. In a first step we considered\nthe two dimensional case, which has to be enlarged to higher\ndimensions. Thereby the theorems from \\cite{Hah} provide us that\nthe same structure appears as tensor factor in the\nrepresentation theory of Clifford Algebras.\n\nThe strong connection between conjugation and (pseudo) norm, as\nwith the geometry of the underlying vector space (module \\ldots)\nwas shown. Thus a knowledge of the bilinear form in ${{\\cal V}}$ provides\nus with all information needed. One is able to choose even the\nspecial ideal out of the isomorphic ones. The dependence of the\nmultivector structure on this choices was shown.\n\nThe field {\\bf C} plays a special role, as the only one with a single\nnontrivial isomorphism class. This may be the origin of the\nusefulness of the complex numbers in quantum mechanics and\nnonlinear classical mechanics. This fact remains in higher\ndimensions.\n\nWe remarked the richness of this structure if the underlying\nspace is build up over rings as {\\bf Z} or finite fields as {\\bf\nF}$_q$. An investigation in this direction should result in much\nmore different cases. These cases are quite interesting in\nquantum theory, because they will be expected to be connected\nwith inequivalent representations. Normally this is achieved only\nwith infinitely many particles.\n\n\\section*{Appendix}\n\\begin{appendix}\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{A-\\arabic{equation}}\nThe figures are calculated with 800 $\\times$ 800 points\nresolution and 500 iterations in a window $[-5:5]$ for the $1_S$\n(hor.) and $\\gamma$ (vert.) axis. If the norms got negative, the\niteration had been stopped. The potential lines give the tendency\nof reaching infinity by surmounting a given threshold in\n$n$--steps. Twelve such lines are plotted. The interior of the\nMandelbrot set and the multi quadratic set consists of non\ndivergent points. $d=a^2+4b$ is the discriminant. $a$ is as in\nthe text the linear part of the quadratic relation.\n\n\\begin{figure}\n\\centerline{\\psfig{figure=fig1.ps,height=5.5cm,width=5.5cm}\n       \\hfil\\psfig{figure=fig2.ps,height=5.5cm,width=5.5cm}}\n\\centerline{\\hfil Fig.1 \\hfil\\hfil\\hfil\n                  Fig.2 \\hfil}\n\\centerline{\\psfig{figure=fig3.ps,height=5.5cm,width=5.5cm}\n       \\hfil\\psfig{figure=fig4.ps,height=5.5cm,width=5.5cm}}\n\\centerline{\\hfil Fig.3 \\hfil\\hfil\\hfil\n                  Fig.4 \\hfil}\n\\centerline{\\psfig{figure=fig5.ps,height=5.5cm,width=5.5cm}\n       \\hfil\\psfig{figure=fig6.ps,height=5.5cm,width=5.5cm}}\n\\centerline{\\hfil Fig.5 \\hfil\\hfil\\hfil\n                  Fig.6 \\hfil}\n\\end{figure}\n\\end{appendix}\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}