diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzncav" "b/data_all_eng_slimpj/shuffled/split2/finalzzncav" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzncav" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nSensors are employed by unmanned autonomous vehicles to navigate through their surroundings, with a substantial dependence on vision-based sensors like RGB cameras. As these sensors are impacted by bad weather conditions, perception pipelines require considerable training on diverse data to increase the robustness on downstream tasks. One specific scenario which causes distortion of images is adverse weather conditions, like heavy snowfall, haze and dust tornados. In these critical situations, weather corruptions can hinder the object detectability and pose a serious threat to navigation and reliability. Thus, there is a need for efficient denoising, deraining and restoration techniques.\n\nHowever, denoising techniques are often evaluated using Image similarity metrics like PSNR, SSIM \\cite{a37} and not by their effectiveness in achieving results for the targeted application. It is possible that the output image of these methods has high image quality but is contextually irrelevant for the object detection task. In this work we evaluate the effectiveness of restoration techniques for denoising images with the intention of better object detection. By introducing a contrastive approach towards restoration evaluation, a method for guiding the training of GANs and restoration progress is proposed. Additionally, attention maps are leveraged for understanding why these techniques assist object detection and why certain classes are easily recognized or ignored.\n\nPrimary contributions of this work are :\n\\begin{enumerate}\n \\item Exploring two new data-based generative adversarial network techniques for denoising weather corrupted images. \n \\item Proposing a novel contrastive approach using a weighted loss for evaluating the training progress and post-training performance of the highlighted restoration techniques. \n \\item Explaining the training progress of the proposed generative denoising methods using attention maps and validating the results using object detection evaluation metrics.\n \\item Evaluating the optimal noise level of the trained Restormer denoising methods (color and grayscale) using attention maps and validating the results using object detection evaluation metrics.\n\\end{enumerate}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth,height=5in]{images\/image1.png}\n\\end{center}\n \\caption{RestoreX-AI: Proposed contrastive approach for evaluating restoration models.}\n\\label{fig1}\n\\end{figure*}\n\n\n\n\n\n\\section{Related Work}\n\nThe rapid development and deployment of autonomous vehicles have exposed several critical challenges in computer vision, one of which being object detection robust to weather corruptions. At the foundation of this challenge is the classic vision task of object detection, the progress of which we summarize in the following sub-section. Ultimately, when these modules are integrated in autonomous driving systems, their suggestively \"black-box\" nature, triggers the concerns of industry professionals, car-makers and users alike. Without human-tailored explanations for the vehicle's behavior, even the most advanced systems fail to benefit from widespread adoption due to concerns about safety and reliability. Especially in adverse weather, when even human drivers exercise extra caution, these robust models need to provide irrefutable explanations for the detections made at every state. We discuss the progress of previous works in these 3 directions which set the background of our work in restoration, detection and explainability. \n\n\n\n\n\\subsection{Multi-weather corruption and restoration}\n\nThe thermal variations accompanying weather change can adversely impact the optical, electronic, and mechanical components used in capturing visual data, thus harming the performance of visual recognition systems \\cite{a4}. Frigid temperatures, snowfall or dense fog, for example, can cause condensation on the lens, further blurring the view and obscuring the object boundaries; rain streaks on car windows can generate glares or act as a double lens \\cite{a36}. For an autonomous car, it is critical and essential to overcome the effects of weather conditions to ensure reliability. Number of approaches for this have been reported. For example, one study \\cite{a5} looked at the performance of gated cameras, while another \\cite{a6} expanded the research to include stereo, gated, and thermal cameras, as well as Radar and LiDAR scanners, and found considerable increases in car recognition in varying levels of fog, and other adverse weather conditions. The use of unique methodologies like domain adaptation to transform the weather conditions while keeping objects of interest intact. For example, \\cite{a7} investigates the consequences of synthetic weather images on road segmentation and traffic object detection, whereas \\cite{a8} shows that using synthetic time-of-day (night imagery) improves localization, and \\cite{a9} proposes a de-raining model to improve semantic segmentation. For their efficiency in image restoration and effectiveness in removing weather corruptions, image restoration employing restomers \\cite{a10} and image denoising algorithms \\cite{a11} are becoming increasingly popular.\n\n\n\\subsection{Object Detection}\n\nThe introduction of fast detectors like SSD, Faster RCNN, and YOLO transformed the face of object detection \\cite{a22,a23,a24}. Taking 2D detectors forward, 3D detection expanded with Stereo RCNN \\cite{a25}, AVOD \\cite{a26}, MVLidarNet \\cite{a27} and MVF algorithm \\cite{a28} bringing new perspectives to the task of object detection. Specific tasks like multiscale object detection \\cite{a29}, pedestrian detection in crowds \\cite{a30} and detection under adverse weather \\cite{a31} have also been solved using ensemble methods and data augmentation.\n\n\n\n\\subsection{Explainability in Object Detection}\n\nThe early rise of deployable machine learning technologies was accompanied by criticism of the \"black-box\"-like nature of ML models. Particularly in sensitive applications like medical diagnosis, self-driving cars and algorithmic test checking, there is a need for thorough explainability in models to ensure the trust and safety of users. To meet these requirements, many techniques were proposed for explaining models \\cite{a12,a13,a14,a15,a16}. In \\cite{a17}, explainability is demonstrated by studying what each of its neurons has learned to detect. \\cite{a18} focused on individual predictions, using the technique of heatmaps to highlight important pixels. Some works also interpret classifiers by identifying representative training examples \\cite{a19,a20}. \\cite{a21} introduced a new perspective to this challenge by making CNN-based models more transparent by producing visual explanations. As newer machine learning systems are rapidly adopted, the demand for explainable models which incorporate diverse approaches is gaining attention of the community. \n\n\n\\section{Methodology}\n\nThis work proposes a contrastive approach (loss) for monitoring the training progress of restoration models in the context of object detection. To put forward a diverse range of training samples, new restoration techniques using GANs have also been proposed. First, the restoration models were trained on different tasks and tested on the DAWN dataset \\cite{a35}. As the models trained, their progress was monitored using the contrastive approach and simultaneously attention maps were generations to support the detection task. Finally, the OD performance of all the methods is compared using standard evaluation metrics (class AP and mAP). The 4 restoration techniques experimented with in this study are:\n\n\\begin{enumerate}\n \\item Weather-NightGAN:Conditional GANs trained on night-to-day task for multi-weather corruption restoration.\n \\item Weather-RainGAN:Conditional GANs trained on rain-to-clear task for multi-weather corruption restoration.\n \\item Restormer (Gaussian color denoising) for multi-weather corruption restoration.\n \\item Restormer (Gaussian grayscale denoising) for multi-weather corruption restoration.\n\\end{enumerate}\n\n\n\n\\subsection{Conditional Generative Adversarial Networks}\n\nGANs or Generative Adversarial networks are generative models that learn mapping between noisy z and output image y, G : z \u2192 y. Conditional GANs learn a mapping from observed image x and random noise vector z, to y, G : {x, z} \u2192 y \\cite{a1}. The generator G is trained to produce images similar to the \"real\" images, as compared by an adversarially trained discriminator, D, which is used for detecting the \"fakes\". The final objective of the conditional GAN can be expressed as: \n\n\\begin{equation*}\n G^{*} = arg min_{G} max_{D} \\mathcal{L}_{cGAN} (G, D) + \\lambda \\mathcal{L}_{L1} (G) \n\\end{equation*}\n\nwhere G tries to minimize this objective against an adversarial D that tries to maximize it, i.e. $G^{*} = arg min_{G} max_{D} L_{cGAN} (G, D).$\n\n\nIn this study, we propose 2 new use-cases of the conditional GAN for restoration purposes. In the first case, the GAN is trained on night-to-day images and tested for multi-weather corruption tasks. The intuition behind this is the similarity in corruptions of night and bad weather images, like poor lighting and condensation on lens. In the second use-case, the GAN is trained on only rain-to-clear images (synthetically generated) and tested for multi-weather corruption tasks. The intuition behind this is the similarity in corruptions of rain and bad weather images, like snow and rain streaks. There is an additional challenge which notes if the single-weather trained conditional GAN can adapt to multi-weather corruptions. \n\n\n\n\\subsection{Restormer}\n\nThe restormer is a highly efficient transformer that was proposed for denoising tasks in image restoration \\cite{a2}. It consists of a multi-Dconv head transposed attention (MDTA) and a gated-Dconv feed-forward network (GDFN). These proposed architectural changes gave it the ability to capture long-range pixel interactions, while still remaining applicable to large images. It is both computationally efficient, and has the capacity to handle high-resolution images, a feature critical for a task like adverse weather object detection. In this work, we study the effects of trained noise levels (15, 25 and 50) on the denoising performance of color and grayscale Restormers on the DAWN dataset. The goal is to study how the noise level affects the model performance in OD task and which level is optimal for generating explainable detections.\n\n\\subsection{Grad-CAM}\n\nGradient-weighted Class Activation Mapping (Grad-CAM) is a technique that produces visual explanations for the purpose of making CNN-based models more transparent \\cite{a3}. For getting the class discriminative localization map Grad-CAM ${L_{Grad-CAM}^c}$ $\\epsilon R ^ {u * v}$ of width u and height v for any class c , first comes the computation of gradient of the score for class c, $y^c$ (before the softmax), according to feature maps $A^k$ of a convolutional layer, i.e. $\\frac{\\partial{\\mathbf{y^{c}}}}{\\partial A_{}^k}$ . The neuron importance weights $a_c^k$ are attained by global-average-pooling these gradients flowing back. The 'importance' of feature map k for a target class c is captured by this weight $a_c^k$. A weighted combination of forward activation maps is performed, and followed by a ReLU to obtain,\nThis results in a coarse heat-map of the same size as the convolutional feature maps. Grad-CAM is used for the purpose of explaining object detection in the restored images of different techniques compared in this study. We additionally use the Grad-CAM model's detection probability in calculating the contrastive metric for monitoring training progress.\n\n\n\\subsection{Proposed Approach: RestoreX-AI}\n\nDue to the instability of training of GANs, over-or-under training does not always lead to the perfect solution images for object detection. Parallelly, tuning on high noise levels does not always provide the best images from the Restormer model. Even after producing good images by standard metrics (PSNR, SSIM), their applicability for the OD task remains uncertain, which brings the need for a new evaluation standard. We propose using a weighted sum of the explainability results (detection probability of class provided by the Grad-CAM model) and the similarity of the predicted and actual label to define this new standard. This weighted sum is calculated for every stage of training (one stage can be a user-defined set of epochs), and then used to monitor the progress of the model. We introduce this new parameter for assessing the quality of restoration which is calculated using equation 1. \n\n\\begin{equation}\n \\Delta \\phi = \\Delta ( \\Sigma (S(p,a)*d)\/N )\n \\end{equation}\n \n Here S is the similarity of labels that be measured either by grouping the objects (cars, race cars and taxis have similarity 1, person, groom have 1 and so on), or by strict parameters (cars and race cars have similarity 0). p and a are the predicted and actual labels of the object under detection. N is the number of training samples generated in that stage, which are used to evaluate the current progress of that model. Refer to Appendix \\ref{appendix:a} Table \\ref{tab2} to view the measure of similarity of objects grouped together for this study. The explanation probability is d or the value returned by the Grad-CAM model, which is its prediction of what is present in the image. The quality of restoration between stages can be denoted as $ \\Delta \\phi $, or the difference between qualities at consecutive stages.\n\n\n\\subsection{Datasets}\n\nFor training the Weather-RainGAN and Weather-NightGAN, corresponding images of the same scene in rain-clear and night-day conditions were required. For the Weather-RainGAN, we used Rain 100L \\cite{a32}, which is a synthesized data of rain streaks with corresponding rain-free images. For the Weather-NightGAN we used Transient Attributes dataset \\cite{a34}, which used a high-level image editing method which allows a user to adjust the attributes of a scene, e.g. change a scene to be \"night\" or \"day\". The final testing of all restoration methods required a multi-weather dataset with high-resolution images, for which the DAWN dataset was selected. The DAWN dataset is a large vehicle detection dataset which has captured images of driving scenes in adverse weather conditions \\cite{a35}. It consists of 1000 images from real-traffic scenes as seen in multiple adverse weather conditions including fog, snow, rain, and sandstorms. The images have been annotated with 2D annotations(boxes) with 6 object classes namely car, bus, truck, motorcycle, person and bicycle.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth,height=8.5in]{images\/image2.png}\n\\end{center}\n \\caption{Foggy weather condition: Grad-CAM Attention Maps for (a) Restormer Grayscale Denoising (b) Restormer Colour Denoising (c) Weather-RainGAN and (d) Weather-NightGAN.}\n\\label{fig2}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth,height=8.5in]{images\/image3.png}\n\\end{center}\n \\caption{Snowfall weather condition: Grad-CAM Attention Maps for (a) Restormer Grayscale Denoising (b) Restormer Colour Denoising (c) Weather-RainGAN and (d) Weather-NightGAN.}\n\\label{fig3}\n\\end{figure*}\n\n\n\\begin{table*}[h]\n \\centering\n\\begin{tabular}{llllll} \\hline \n\\textbf{Image Restoration Technique} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Class 1 AP\\\\ {[}car{]}\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Class 2 AP\\\\ {[}bus{]}\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Class 3 AP\\\\ {[}person{]}\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Class 4 AP\\\\ {[}motorcycle{]}\\end{tabular}} & \\textbf{mAP} \\\\ \\hline \n\\multicolumn{6}{l}{\\textbf{Gaussian Gray Denoising Restormer}} \\\\\\hline \nNoise 15 & 1 & 5 & 22 & 0 & 6 \\\\\nNoise 25 & 1 & 9 & 23 & 0 & 6 \\\\\nNoise 50 & 1 & 0 & 29 & 0 & 6 \\\\ \\hline \n\\multicolumn{6}{l}{\\textbf{Gaussian Color Denoising Restormer}} \\\\ \\hline \nNoise 15 & 1 & 3 & 21 & 0 & 5 \\\\\nNoise 25 & 1 & 5 & 22 & 0 & 6 \\\\\nNoise 50 & 1 & 11 & 24 & 0 & 7 \\\\ \\hline \n\\multicolumn{6}{l}{\\textbf{Weather-RainGAN}} \\\\\\hline \nStage 1 & 1 & 0 & 0 & 0 & 0 \\\\\nStage 2 & 1 & 17 & 0 & 0 & 4 \\\\\nStage 3 & 1 & 0 & 0 & 0 & 0 \\\\\nStage 4 & 1 & 0 & 0 & 76 & 15 \\\\\nStage 5 & 1 & 0 & 0 & 0 & 0 \\\\ \\hline \n\\multicolumn{6}{l}{\\textbf{Weather-NightGAN}} \\\\\\hline\nStage 1 & 11 & 0 & 0 & 0 & 2 \\\\\nStage 2 & 1 & 0 & 0 & 0 & 0 \\\\\nStage 3 & 1 & 0 & 0 & 0 & 0 \\\\\nStage 4 & 17 & 0 & 0 & 0 & 3 \\\\\nStage 5 & 48 & 0 & 0 & 0 & 10 \\\\ \\hline \n\\end{tabular}\n\\caption{Object detection performance of all 4 restoration methods as measured at the designed stages.}\n\\label{tab1}\n\\end{table*}\n\n\n\n\n\\section{Experiments and Results}\n\n\\subsection{Restormer-based Grayscale Denoising}\n\nSection 1 of Table \\ref{tab1} shows the object detection scores of Gaussian grayscale image denoising using the Restormer model. To compare the clarity and detection scores, models trained on different noise levels 15, 25 and 50 are included in testing. The purpose of this experimentation is to identify which noise level is optimal for the object detection task, additionally verified by the explainability setup shown in Figures \\ref{fig2} and \\ref{fig3}. The overall mAP of object detection remains constant on all noise levels, however individual class scores are observed to fluctuate. The AP for bus and person class increases from noise 15 to 25 by 4 AP and 1AP respectively. However the bus AP drops to 0 and person AP boosts to 29 when noise is set to 50. The increase in noise level would correspond to smoother images produced by restormer, however, it is observed that the bus AP drops when noise level is set to 50. This is very interesting to note and also observable in Figure \\ref{fig2}, where the attention maps shift from the bus to the car as the noise level increases.\n\n\\subsection{Restormer-based Colour Denoising}\n\n\nSection 2 of Table \\ref{tab1} shows the object detection scores of Gaussian color image denoising using the Restormer model. To compare the clarity and detection scores, models trained on different noise levels 15, 25 and 50 are included in testing. The purpose of this experimentation is to identify which noise level is optimal for the object detection task, additionally verified by the explainability setup. The overall mAP of object detection increases steadily by 1 with all increasing noise levels, and individual class scores are observed to increase as well. The AP for bus and person class increases from noise 15 to 25 by 2 AP and 1AP respectively. The bus AP further increases to 11 and person AP boosts to 24 when noise is set to 50. The increase in noise level is improving the image quality and object detection. As visible in Figure \\ref{fig3}, the attention maps on the people are focusing in the region of interest as the noise level increases.\n\n\n\\subsection{Weather-RainGAN}\n\nThe utilization of Pix2Pix GAN \\cite{a1} for deraining purposes in mapped rain-clear images is proposed, and validated using our proposed technique. The dataset used for this purpose was Rain 100L \\cite{a32}, which is a synthesized data of rain streaks with corresponding rain-free images. The images in Rain 100 L are originally from BSD 200 dataset \\cite{a33}. The GAN model is trained with the rainy images as source and clear images as target, expecting this technique to produce denoised images on test images of the DAWN dataset. The intuition behind this idea was the inherent similarity between weather corruptions and synthetic rain streaks, with the goal of the GAN learning to work similarly on the 2 tasks. The training epochs are divided into stages(1,2,3,4,5) to monitor the training progress of the GANs and produce results of OD and explainability at each stage. As expected, the GANs are producing highly unstable behavior with increasing and 0 AP at most of the stages. However, the sudden boost in AP at specific epochs inspired the formulation of our proposed weighted explainability measure. It can be seen in Section 3 of Table \\ref{tab1}, that the bus AP is 17 at Stage 4, but 0 at all other stages and the motorcycle AP is 76 at Stage 8 and 0 at other Stages. The car AP however, remains constant at 1 and overall mAP increases over time due to the individual class performance boosts. As these results are vague, we take a closer look at Figures \\ref{fig2} and \\ref{fig3}, to determine possible causes for this object detection performance. It can be observed that the GAN is actually performing quite well in denoising weather conditions like fog and snow, and the attention maps are converging towards the relevant objects as the training progresses. But while the object detection results are very disjoint, the attention maps progress continuously and show improvement. The effectiveness of this GAN can be observed as a deraining solution as the image not only gets clearer, but the object detection and attention maps get more precise over training as well. \n\n\n\n\n\n\n\n\\subsection{Weather-NightGAN}\n\nA new use-case of the PixtoPix GAN \\cite{a1} is proposed, trained on a different use-case for deraining purposes in mapped rain-clear images, and validated our proposed technique using the experimental procedure. The dataset used for this purpose was Transient Attributes dataset \\cite{a34}, which used a high-level image editing method which allows a user to adjust the attributes of a scene, e.g. change a scene to be \"night\" or \"day\". The GAN model is trained with the night images as source and day images as target, expecting this technique to produce denoised images on test images of the DAWN dataset. The intuition behind this idea was the inherent similarity between weather corruptions and dark night images. As expected, the GANs are producing highly unstable behavior with increasing and 0 AP at most of the stages as shown in Table \\ref{tab1}. This technique actually worked out only for 1 class (car) and boosted its AP from 11 to 48 over the training period. The remaining classes had a constant 0 AP. By taking a closer look at Figures \\ref{fig2} and \\ref{fig3}, possible causes for this object detection performance can be determined. The GAN is actually performing quite randomly in denoising weather conditions like fog and snow using night training images, but the attention maps are still converging towards the relevant objects as the training progresses. This is exactly aligned with the initial analysis and intuition for the discovery i.e. cases in which object detection is clear to a CV detector, although imperceptible to the human eye. While the object detection results are very disjoint, the attention maps progress continuously and show improvement. The effectiveness of this GAN can be observed in denoising weather conditions for cars, but not for other classes. The purpose of our methodology is not just to observe the restoration progress in a positive light, but to also stop training in case the GAN goes too far. As can be seen in this particular case, the GAN distorts the images after Stage 2, which may confuse detectors when tested against it. In Stage 4, the GAN produces images which are heavily distorted but oddly easier for car detection than even the original image. \n\nThe attention maps and results for all four restoration techniques and four weather conditions: fog, rain, snow and dust tornado have been displayed in Appendix \\ref{appendix:a}.\n\n\\section{Conclusion}\n\n\nThe goal of our work was to study the relationship between object detection performance, image clarity, explainability and training time in the context of novel versus established restoration techniques. The scope of this study covered 4 different restoration techniques, aiming to denoise images corrupted by over 6 different weather conditions presented in the DAWN dataset, namely fog, snow storm, haze, dust tornadoes, rainfall and mist. All 4 techniques worked differently, with the Restormer-based methods providing stable all-rounded results, while the GANs provided class-specific boosts in performance. The overall rise in mAP before and after applying the techniques was 0\\%, 40\\%, 275\\%, 400\\% respectively. Contrary to popular beliefs, greater denoising does not always guarantee better object detection results as observed in this paper. And conversely, poor denoising does not always guarantee worse object detection results. Particularly for specific tasks like bus and car detection, it can be seen that newer approaches like our proposed method (Weather-RainGAN and Weather-NightGAN) can boost the detector's performance with its resultant images. We present conditional GANs cases that perform superbly on diverse weather conditions ranging from dust tornadoes to snowfall, in spite of having trained on limited single weather conditions.\n\nWe present 2 very interesting observations obtained through this study:\n\\begin{enumerate}\n \\item GANs can generate images complex to the human eye, but comparatively interpretable for vision models post processing. This opens the possibility of exploring the capabilities of vision models beyond the scope of human vision and also warding off potential attacks which can cripple modern detectors. \n \\item Restoration and denoising methods which produce clearer images (as measured using standard image quality metrics like PSNR), may in fact present a greater challenge for machine perception in object detection.\n \n\\end{enumerate}\nUnderstanding how differently humans and detectors perceive information in the same image demands greater exploration of explainability. We would like to open the discussion for countless possibilities arising from these disparate perspectives, E.g. if a model sees a pedestrian on a rainy road which a human cannot see, or conversely a pedestrian visible to a passenger's eye which a model cannot capture. While using denoising techniques are a popular choice for tackling corruptions, it must also be acknowledged that not all techniques are suitable for all use-cases as seen for car detection. Certain classes may respond better to denoising depending on their physical characteristics as perceived by the detectors. Going ahead with building robust, explainable models, this problem must be studied from multiple perspectives in the future. We hope to inspire a new line of research in this direction which deals with the complexity of the task of weather corruptions and can solve it using diverse generalizable solutions. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{APPENDIX}\n\n\n\n\n\n\\bibliographystyle{ieeetr}\n\n\\section{Introduction}\n\nA very popular research discipline in control theory is the extension\nof control methodologies originally developed for systems that are\ndescribed by ordinary differential equations (ODEs) to systems governed\nby partial differential equations (PDEs); however, with regard to\nstability investigations this extension is accompanied by a significant\nrise of complexity, see e.g. \\cite{Luo1998} for a comprehensive framework\nfor the stability analysis of infinite-dimensional systems. Therefore,\na lot of research effort has been invested in this topic, where for\nexample the stability analysis of mechanical systems with certain\nboundary conditions has been addressed. For instance, in \\cite{Miletic2015}\nthe stability of an Euler-Bernoulli beam subjected to nonlinear damping\nand a nonlinear spring at the tip is analysed, whereas \\cite{Stuerzer2016}\nis concerned with the stability behaviour of a gantry crane with heavy\nchain and payload. Furthermore, the proof of stability of a Lyapunov-based\ncontrol law as well as a Lyapunov-based observer design for an in-domain\nactuated Euler-Bernoulli beam has been presented in \\cite{Henikl2012}.\n\nA well-known methodology, that has also been extended to the infinite-dimensional\nscenario, is the combination of the port-Hamiltonian (pH) system representation\nwith energy-based control. In this regard, in particular a pH-system\nrepresentation based on an underlying jet-bundle structure, see e.g.\n\\cite{Ennsbrunner2005,Schoeberl2012,Schoeberl2014a}, as well as a\nformulation exploiting Stokes-Dirac structures, see e.g. \\cite{Schaft2002,Gorrec2005},\nhave turned out to be especially suitable. For a detailed comparison\nof these approaches, where the main difference is the choice of the\nvariables, the interested reader is referred to \\cite{Schoeberl2013b}\nor \\cite{Malzer2020}. In fact, with respect to boundary-control systems,\na lot of literature is available, see e.g. \\cite{Schoeberl2011,Rams2017a}\nand \\cite{Macchelli2004,Macchelli2017}, where boundary controllers\nbased on the well-known energy-Casimir method are designed within\nthe jet-bundle and the Stokes-Dirac approach, respectively. Moreover,\nrecently the pH-system description has also been exploited with regard\nto the observer design, see e.g. \\cite{Toledo2019}, where a pH-based\nobserver-design procedure for boundary-control systems has been developed\nwithin the Stokes-Dirac scenario. In light of the observer design,\nof course stability investigations play an important role, since it\nmust be ensured that the observer error tends to zero.\n\nIn \\cite{Malzer2020}, a control-design procedure based on the energy-Casimir\nmethod together with an observer design exploiting the pH-system representation\nhas been presented within the jet-bundle framework as well as within\nthe Stokes-Dirac scenario for infinite-dimensional systems with in-domain\nactuation. Furthermore, the design procedures have been demonstrated\nand compared by means of an in-domain actuated vibrating string; however,\nthe investigation regarding the asymptotic stability of the observer\nerror -- which is of course essential -- has only been sketched.\nTherefore, the aim of this paper is to carry out the stability investigation\nof the observer error of this system in detail. To this end, first\nof all, in Section \\ref{sec:Observer_Design} we summarise the observer\ndesign that exploits the pH-system representation based on a jet-bundle\nstructure, while in Section \\ref{sec:Vibrating_String} the observer\ndesign is explicitly demonstrated for an in-domain actuated vibrating\nstring. Thus, the main contribution of this paper is to verify the\nasymptotic stability of the observer error, where i) it is necessary\nto investigate the well-posedness, see Subsection \\ref{subsec:Wellposedness},\nand ii) to apply LaSalle's invariance principle for infinite-dimensional\nsystems, see Subsection \\ref{subsec:LaSalle}.\n\n\\section{Observer Design based on a Port-Hamiltonian Framework\\label{sec:Observer_Design}}\n\nWith respect to the observer design, see \\cite[Sec. V]{Malzer2020},\nwe intend to exploit a pH-system description for infinite-dimensional\nsystems with $1$-dimensional spatial domain, which is equipped with\nthe spatial coordinate $z\\in[0,L]$. The system representation is\nbased on an underlying jet-bundle structure, and therefore, first\nof all we introduce the bundle $\\pi:\\mathcal{E}\\rightarrow\\mathcal{B}$,\nwhere the total manifold $\\mathcal{E}$ is equipped with the coordinates\n$(z,x^{\\alpha})$, with $x^{\\alpha}$, $\\alpha=1,\\ldots,n$, denoting\nthe dependent variables, while the base manifold $\\mathcal{B}$ possesses\nthe independent (spatial) coordinate $(z)$ solely. Next, we consider\nthe so-called vertical tangent bundle $\\nu_{\\mathcal{E}}:\\mathcal{V}(\\mathcal{E})\\rightarrow\\mathcal{E}$,\nequipped with the coordinates $(z,x^{\\alpha},\\dot{x}^{\\alpha})$,\nwhich is a subbundle of the tangent bundle $\\tau_{\\mathcal{E}}:\\mathcal{T}(\\mathcal{E})\\rightarrow\\mathcal{E}$,\npossessing the coordinates $(z,x^{\\alpha},\\dot{z},\\dot{x}^{\\alpha})$\ntogether with the fibre bases $\\partial_{z}=\\partial\/\\partial z$\nand $\\partial_{\\alpha}=\\partial\/\\partial x^{\\alpha}$. Thus, a vertical\nvector field $v=\\mathcal{E}\\rightarrow\\mathcal{V}(\\mathcal{E})$,\nin local coordinates given as $v=v^{\\alpha}\\partial_{\\alpha}$ with\n$v^{\\alpha}\\in C^{\\infty}(\\mathcal{E})$, i.e. $v^{\\alpha}$ is a\nsmooth function on $\\mathcal{E}$, is defined as a section. A further\nimportant differential-geometric object is the so-called co-tangent\nbundle $\\tau_{\\mathcal{E}}^{*}:\\mathcal{T}^{*}(\\mathcal{E})\\rightarrow\\mathcal{E}$,\npossessing the coordinates $(z,x^{\\alpha},\\dot{z},\\dot{x}_{\\alpha})$\ntogether with the fibre bases $\\mathrm{d}z$ and $\\mathrm{d}x^{\\alpha}$,\nwhich allows to introduce a one-form $w:\\mathcal{E}\\rightarrow\\mathcal{T}^{*}(\\mathcal{E})$\nas a section that can locally be given as $w=\\breve{w}\\mathrm{d}z+w_{\\alpha}\\mathrm{d}x^{\\alpha}$\nwith $\\breve{w},w_{\\alpha}\\in C^{\\infty}(\\mathcal{E})$. With respect\nto the pH-system representation, we are interested in densities $\\mathfrak{H}=\\mathcal{H}\\mathrm{d}z$\nwith $\\mathcal{H}\\in C^{\\infty}(\\mathcal{J}^{1}(\\mathcal{E}))$, where\nthese densities can be formed by sections of certain pullback bundles,\nwhose use is omitted here for ease of presentation. That is, $\\mathcal{H}$\nis a smooth function on the first jet manifold $\\mathcal{J}^{1}(\\mathcal{E})$,\nwhich is equipped with the coordinates $(z,x^{\\alpha},x_{z}^{\\alpha})$,\nwhere the $1$st-order jet variable $x_{z}^{\\alpha}$ corresponds\nto the derivative of $x^{\\alpha}$ with respect to $z$. Moreover,\nthe first prolongation of a vertical vector field reads as $j^{1}(v)=v^{\\alpha}\\partial_{\\alpha}+d_{z}(v^{\\alpha})\\partial_{\\alpha}^{z}$,\nwith $\\partial_{\\alpha}^{z}=\\partial\/\\partial x_{z}^{\\alpha}$, where\nwe exploit the total derivative $d_{z}=\\partial_{z}+x_{z}^{\\alpha}\\partial_{\\alpha}+x_{zz}^{\\alpha}\\partial_{\\alpha}^{z}+\\ldots$.\n\nHaving discussed this essential preliminaries, we are able to introduce\nthe pH-system representation including inputs and outputs on the spatial\ndomain as\\begin{subequations}\\label{eq:pH_sys_jetbundle}\n\\begin{align}\n\\dot{x} & =(\\mathcal{J}-\\mathcal{R})(\\delta\\mathfrak{H})+u\\rfloor\\mathcal{G}\\,,\\label{eq:pH_sys_dynamics}\\\\\ny & =\\mathcal{G}^{*}\\rfloor\\delta\\mathfrak{H}\\,,\n\\end{align}\n\\end{subequations}see e.g. \\cite{Ennsbrunner2005,Schoeberl2008a,Schoeberl2014},\nwhere $\\rfloor$ denotes the so-called Hook operator allowing for\nthe natural contraction between tensor fields. In (\\ref{eq:pH_sys_jetbundle}),\nthe variational derivative $\\delta\\mathfrak{H}=\\delta_{\\alpha}\\mathcal{H}\\mathrm{d}x^{\\alpha}\\wedge\\mathrm{d}z$,\nwith $\\wedge$ denoting the exterior (wedge) product, locally reads\nas $\\delta_{\\alpha}\\mathcal{H}=\\partial_{\\alpha}\\mathcal{H}-d_{z}(\\partial_{\\alpha}^{z}\\mathcal{H})$.\nFurthermore, the linear operators $\\mathcal{J},\\mathcal{R}:\\mathcal{T}^{*}(\\mathcal{E})\\wedge\\mathcal{T}^{*}(\\mathcal{B})\\rightarrow\\mathcal{V}(\\mathcal{E})$\ndescribe the internal power flow and the dissipation effects of the\nsystem, respectively. The coefficients $\\mathcal{J}^{\\alpha\\beta}$\nof the interconnection tensor $\\mathcal{J}$ meet $\\mathcal{J}^{\\alpha\\beta}=-\\mathcal{J}^{\\beta\\alpha}\\in C^{\\infty}(\\mathcal{J}^{2}(\\mathcal{E}))$,\nwhile we have $\\mathcal{R}^{\\alpha\\beta}=\\mathcal{R}^{\\beta\\alpha}\\in C^{\\infty}(\\mathcal{J}^{2}(\\mathcal{E}))$\nand $[\\mathcal{R}^{\\alpha\\beta}]\\geq0$ for the coefficient matrix\nof the symmetric and positive semi-definite dissipation mapping $\\mathcal{R}$.\nWith respect to the dual input and output bundles $\\rho:\\mathcal{U}\\rightarrow\\mathcal{J}^{2}(\\mathcal{E})$\nand $\\varrho:\\mathcal{Y}\\rightarrow\\mathcal{J}^{2}(\\mathcal{E})$,\nwe have the input map and its adjoint output map $\\mathcal{G}:\\mathcal{U}\\rightarrow\\mathcal{V}(\\mathcal{E})$\nand $\\mathcal{G}^{*}:\\mathcal{T}^{*}(\\mathcal{E})\\wedge\\mathcal{T}^{*}(\\mathcal{B})\\rightarrow\\mathcal{Y}$,\nrespectively, and thus, the relation $(u\\rfloor\\mathcal{G})\\rfloor\\delta\\mathfrak{H}=u\\rfloor(\\mathcal{G}^{*}\\rfloor\\delta\\mathfrak{H})=u\\rfloor y$\nholds, see \\cite[Sec. 4]{Ennsbrunner2005} or \\cite[Sec. 3]{Schoeberl2008a}.\nTo be able to determine the formal change of the Hamiltonian functional\n$\\mathscr{H}=\\int_{0}^{L}\\mathcal{H}\\mathrm{d}z$ along solutions\nof (\\ref{eq:pH_sys_dynamics}), we make use of the Lie-derivative\n$\\mathrm{L}_{j^{1}(v)}$, where we set $v=\\dot{x}$ with (\\ref{eq:pH_sys_dynamics}),\nsee \\cite[Sec. IV-A]{Schoeberl2011}, and thus, we obtain\n\\begin{equation}\n\\dot{\\mathscr{H}}=-\\int_{0}^{L}\\mathcal{R}(\\delta\\mathfrak{H})\\rfloor\\delta\\mathfrak{H}+\\int_{0}^{L}u\\rfloor y+\\left.(\\dot{x}\\rfloor\\delta^{\\partial}\\mathfrak{H})\\right|_{0}^{L}\\,\\label{eq:H_p}\n\\end{equation}\nby means of integration by parts and Stoke's theorem. If $\\mathscr{H}$\ncorresponds to the total energy of the system, then (\\ref{eq:H_p})\nstates a power-balance relation, where the first expression describes\nthe energy that is dissipated for example due to damping effects.\nMoreover, the expression $\\int_{0}^{L}u\\rfloor y$ denotes a collocation\nterm distributed over (a part of) the spatial domain. The last term\ncorresponds to collocation restricted to the boundary, which is indicated\nby $(\\cdot)|_{0}^{L}$, where the boundary operator locally reads\nas $\\delta_{\\alpha}^{\\partial}\\mathcal{H}=\\partial_{\\alpha}^{z}\\mathcal{H}$.\nNote that here we consider systems with trivial boundary conditions,\nimplying that the boundary ports $(\\dot{x}^{\\alpha}\\delta_{\\alpha}^{\\partial}\\mathcal{H})|_{0}^{L}$\nvanish.\n\nNext, the intention is to exploit the pH-formulation with respect\nto the observer design. In particular, the copy of the plant (\\ref{eq:pH_sys_dynamics})\nis extended by an error-injection term, and thus, by means of the\nobserver-energy density $\\hat{\\mathcal{H}}$, the observer system\nis locally given by\\begin{subequations}\\label{eq:observer_JB}\n\\begin{align}\n\\dot{\\hat{x}}^{\\hat{\\alpha}} & =(\\mathcal{J}^{\\hat{\\alpha}\\hat{\\beta}}-\\mathcal{R}^{\\hat{\\alpha}\\hat{\\beta}})\\delta_{\\hat{\\beta}}\\hat{\\mathcal{H}}+\\mathcal{G}_{\\xi}^{\\hat{\\alpha}}u^{\\xi}+\\mathcal{K}_{\\eta}^{\\hat{\\alpha}}u_{o}^{\\eta}\\,,\\label{eq:observer_JB_dynamics}\\\\\n\\hat{y}_{\\xi} & =\\mathcal{G}_{\\xi}^{\\hat{\\alpha}}\\delta_{\\hat{\\alpha}}\\hat{\\mathcal{H}}\\,,\\label{eq:observer_JB_output_densities}\n\\end{align}\n\\end{subequations}with $\\hat{\\alpha},\\hat{\\beta}=1,\\ldots,n$ and\n$\\xi,\\eta=1,\\ldots,m$, where we use Einstein's convention on sums.\nIn (\\ref{eq:observer_JB_dynamics}), we have the additional input\n$u_{o}^{\\eta}=\\delta^{\\eta\\xi}(\\bar{y}_{\\xi}-\\hat{\\bar{y}}_{\\xi})$\n-- with the Kronecker-Delta symbol meeting $\\delta^{\\xi\\eta}=1$\nfor $\\xi=\\eta$ and $\\delta^{\\xi\\eta}=0$ for $\\xi\\neq\\eta$ --,\nwhere $\\bar{y}_{\\xi}$ corresponds to the integrated output density\nof the plant according to $\\bar{y}_{\\xi}=\\int_{0}^{L}y_{\\xi}\\mathrm{d}z$,\nwhich is assumed to be available as measurement quantity, while $\\hat{\\bar{y}}_{\\xi}$\nrepresents the copy of the integrated plant-output according to $\\hat{\\bar{y}}_{\\xi}=\\int_{0}^{L}\\hat{y}_{\\xi}\\mathrm{d}z$\nwith (\\ref{eq:observer_JB_output_densities}). The aim is to design\nthe observer gain $\\mathcal{K}_{\\eta}^{\\hat{\\alpha}}$ such that the\nobserver error $\\tilde{x}=x-\\hat{x}$ tends to $0$, where it is beneficial\nto reformulate the observer-error dynamics $\\dot{\\tilde{x}}=\\dot{x}-\\dot{\\hat{x}}$\nas pH-system according to\\begin{subequations}\\label{eq:observer_error_system_JB}\n\\begin{align}\n\\dot{\\tilde{x}}^{\\tilde{\\alpha}} & =(\\mathcal{J}^{\\tilde{\\alpha}\\tilde{\\beta}}-\\mathcal{R}^{\\tilde{\\alpha}\\tilde{\\beta}})\\delta_{\\tilde{\\beta}}\\tilde{\\mathcal{H}}-\\mathcal{K}_{\\xi}^{\\tilde{\\alpha}}u_{o}^{\\xi}\\,,\\\\\n\\tilde{y}_{\\xi} & =-\\mathcal{K}_{\\xi}^{\\tilde{\\alpha}}\\delta_{\\tilde{\\alpha}}\\tilde{\\mathcal{H}}\\,.\\label{eq:observer_error_output_JB}\n\\end{align}\n\\end{subequations}with (\\ref{eq:observer_error_output_JB}) denoting\nthe collocated output density. If we investigate the formal change\nof the error-Hamiltonian $\\tilde{\\mathscr{H}}=\\int_{0}^{L}\\tilde{\\mathcal{H}}\\mathrm{d}z$,\nwhich follows to\n\\begin{multline*}\n\\dot{\\tilde{\\mathscr{H}}}=-\\int_{0}^{L}\\delta_{\\tilde{\\alpha}}(\\tilde{\\mathcal{H}})\\mathcal{R}^{\\tilde{\\alpha}\\tilde{\\beta}}\\delta_{\\tilde{\\beta}}(\\tilde{\\mathcal{H}})\\mathrm{d}z+\\ldots\\\\\n-\\int_{0}^{L}\\delta_{\\tilde{\\alpha}}(\\tilde{\\mathcal{H}})\\mathcal{K}_{\\xi}^{\\tilde{\\alpha}}\\delta^{\\xi\\eta}(\\bar{y}_{\\eta}-\\hat{\\bar{y}}_{\\eta})\\mathrm{d}z\\,,\n\\end{multline*}\nwe find that by means of a proper choice for the components $\\mathcal{K}_{\\xi}^{\\hat{\\alpha}}$\nwe are able to render $\\dot{\\tilde{\\mathscr{H}}}\\leq0$ . Hence, the\ntotal energy of the observer error $\\tilde{\\mathscr{H}}$ is an appropriate\ncandidate for a Lyapunov functional and therefore serves as basis\nwith respect to the stability analysis. Next, the observer-design\nprocedure is demonstrated by an example.\n\n\\section{Observer Design for an In-Domain Actuated Vibrating String\\label{sec:Vibrating_String}}\n\nIn this chapter, we design an infinite-dimensional observer for an\nin-domain actuated vibrating string by exploiting energy considerations.\nThe governing equation of motion of the system under consideration\nreads as\\begin{subequations}\n\\begin{equation}\n\\rho\\frac{\\partial^{2}w}{\\partial t^{2}}=T\\frac{\\partial^{2}w}{\\partial z^{2}}+f(z,t)\\,,\\label{eq:vib_String_eom}\n\\end{equation}\nwhere $w$ describes the vertical deflection of the string, $\\rho$\nthe mass density and $T$ Young's modulus. Regarding the boundary\nconditions, we have that the string is clamped at $z=0$ and free\nat $z=L$, i.e.\n\\begin{align}\nw(0,t) & =0\\,,\\quad T\\frac{\\partial w}{\\partial z}(L,t)=0\\,.\\label{eq:BC_VS}\n\\end{align}\n\\end{subequations}In (\\ref{eq:vib_String_eom}), the distributed\nforce $f(z,t)=g(z)u(t)$ is generated by an actuator behaving like\na piezoelectric patch, where the applied voltage $u(t)$ serves as\nmanipulated variable. The spatially dependent function $g(z)=h(z-L_{p_{1}})-h(z-L_{p_{2}})$,\nwhere $h(\\cdot)$ denotes the Heaviside function, describes the placement\nof the actuator between $z=L_{p_{1}}$ and $z=L_{p_{2}}$. In fact,\nthe force-distribution on the domain $L_{p_{1}}\\leq z\\leq L_{p_{2}}$\nis supposed to be constant and is scaled by $u(t)$.\n\nFirst, the intention is to find a pH-system representation that can\nbe exploited for the observer design. To this end, we introduce the\nunderlying bundle structure based on $\\pi:(z,w,p)\\rightarrow(z)$\ntogether with the generalised momenta $p=\\rho\\dot{w}$, and thus,\n(\\ref{eq:vib_String_eom}) can be rewritten as\n\\begin{equation}\n\\dot{p}=Tw_{zz}+g(z)u\\,.\\label{eq:VS_JB}\n\\end{equation}\nIf we use the Hamiltonian density $\\mathcal{H}=\\frac{1}{2\\rho}p^{2}+\\frac{1}{2}T(w_{z})^{2}\\in\\mathcal{J}^{1}(\\mathcal{E})$,\nwe obtain the appropriate pH-system formulation\\begin{subequations}\\label{eq:pH_formulation_VS}\n\\begin{align}\n\\left[\\begin{array}{c}\n\\dot{w}\\\\\n\\dot{p}\n\\end{array}\\right] & =\\left[\\begin{array}{cc}\n0 & 1\\\\\n-1 & 0\n\\end{array}\\right]\\left[\\begin{array}{c}\n\\delta_{w}\\mathcal{H}\\\\\n\\delta_{p}\\mathcal{H}\n\\end{array}\\right]+\\left[\\begin{array}{c}\n0\\\\\ng(z)\n\\end{array}\\right]u\\,,\\\\\ny & =\\left[\\begin{array}{cc}\n0 & g(z)\\end{array}\\right]\\left[\\begin{array}{c}\n\\delta_{w}\\mathcal{H}\\\\\n\\delta_{p}\\mathcal{H}\n\\end{array}\\right]=g(z)\\frac{p}{\\rho}\\,.\\label{eq:pH_form_VS_output_density}\n\\end{align}\n\\end{subequations}By taking the boundary conditions (\\ref{eq:BC_VS})\ninto account, one finds that the formal change of the Hamiltonian\nfunctional $\\mathscr{H}$ follows to $\\dot{\\mathscr{H}}=\\int_{0}^{L}g(z)\\frac{p}{\\rho}u\\mathrm{d}z$,\ni.e. we have a distributed port that can be used for control purposes.\nIn fact, for the system under consideration, in \\cite{Malzer2020}\na dynamic controller based on the energy-Casimir method has been designed.\nHowever, with regard to this control methodology, it should be mentioned\nthat it yields unsatisfactory results for uncertain initial conditions,\nsee e.g. \\cite{Rams2017b} where this problem is briefly discussed\nfor a boundary-control system. Therefore, in the following we intend\nto design an infinite-dimensional observer in order to overcome this\nobstacle.\n\nConcerning the observer design, it is assumed that the spatial integration\nof the distributed output density (\\ref{eq:pH_form_VS_output_density})\naccording to $\\bar{y}=\\int_{0}^{L}g(z)\\frac{p}{\\rho}\\mathrm{d}z$,\nwhich can be interpreted as the current through the actuator, is available\nas measurement quantity. Thus, if we use the observer density $\\hat{\\mathcal{H}}=\\frac{1}{2\\rho}\\hat{p}^{2}+\\frac{1}{2}T(\\hat{w}_{z})^{2}$\nand the copy of the plant output according to $\\hat{\\bar{y}}=\\int_{0}^{L}g(z)\\frac{\\hat{p}}{\\rho}\\mathrm{d}z$,\nwe are able to introduce an observer for the in-domain actuated vibrating\nstring in the form\n\\begin{align*}\n\\left[\\begin{array}{c}\n\\dot{\\hat{w}}\\\\\n\\dot{\\hat{p}}\n\\end{array}\\right]\\! & =\\!\\left[\\!\\begin{array}{cc}\n0\\! & \\!1\\\\\n-1\\! & \\!0\n\\end{array}\\!\\right]\\!\\left[\\begin{array}{c}\n\\delta_{\\hat{w}}\\hat{\\mathcal{H}}\\\\\n\\delta_{\\hat{p}}\\hat{\\mathcal{H}}\n\\end{array}\\right]\\!+\\!\\left[\\begin{array}{c}\n0\\\\\ng(z)\n\\end{array}\\right]\\!u\\!+\\!\\left[\\begin{array}{c}\nk_{1}\\\\\nk_{2}\n\\end{array}\\right]\\!(\\bar{y}-\\hat{\\bar{y}}),\n\\end{align*}\nwhere the governing equations are restricted to the boundary conditions\n$\\hat{w}(0)=0$ and $T\\hat{w}_{z}(L)=0$. Next, by means of the error\ncoordinates $\\tilde{w}=w-\\hat{w}$, $\\tilde{p}=p-\\hat{p}$, the observer-error\ndynamics can be deduced to\\begin{subequations}\\label{eq:observer_error_dynamics_VS}\n\\begin{align}\n\\dot{\\tilde{w}} & =\\dot{w}-\\dot{\\hat{w}}=\\frac{1}{\\rho}\\tilde{p}-k_{1}(\\bar{y}-\\hat{\\bar{y}})\\,,\\label{eq:observer_error_dyn_w}\\\\\n\\dot{\\tilde{p}} & =\\dot{p}-\\dot{\\hat{p}}=T\\tilde{w}_{zz}-k_{2}(\\bar{y}-\\hat{\\bar{y}})\\,,\\label{eq:observer_error_dyn_p}\n\\end{align}\nwhere the boundary conditions\n\\begin{align}\n\\tilde{w}(0) & =0\\,,\\quad T\\tilde{w}_{z}(L)=0\\label{eq:BC_observer_error_left}\n\\end{align}\n\\end{subequations}hold. With respect to the determination of $k_{1}$\nand $k_{2}$, it is beneficial to reformulate (\\ref{eq:observer_error_dyn_w})\nand (\\ref{eq:observer_error_dyn_p}) as the pH-system\\begin{subequations}\\label{eq:observer_error_VS_JB-1}\n\\begin{align}\n\\left[\\begin{array}{c}\n\\dot{\\tilde{w}}\\\\\n\\dot{\\tilde{p}}\n\\end{array}\\right]\\! & =\\!\\left[\\begin{array}{cc}\n0 & 1\\\\\n-1 & 0\n\\end{array}\\right]\\!\\left[\\begin{array}{c}\n\\delta_{\\tilde{w}}\\tilde{\\mathcal{H}}\\\\\n\\delta_{\\tilde{p}}\\tilde{\\mathcal{H}}\n\\end{array}\\right]\\!-\\!\\left[\\begin{array}{c}\nk_{1}\\\\\nk_{2}\n\\end{array}\\right]\\!(\\bar{y}-\\hat{\\bar{y}}),\\\\\n\\tilde{y} & \\!=\\!-\\!\\left[\\begin{array}{cc}\nk_{1} & k_{2}\\end{array}\\right]\\!\\left[\\begin{array}{c}\n\\delta_{\\tilde{w}}\\tilde{\\mathcal{H}}\\\\\n\\delta_{\\tilde{p}}\\tilde{\\mathcal{H}}\n\\end{array}\\right]\\!=\\!k_{1}T\\tilde{w}_{zz}\\!-\\!k_{2}\\frac{\\tilde{p}}{\\rho},\\label{eq:pH_observer_error_output}\n\\end{align}\n\\end{subequations}where the energy density of the observer error\nreads as $\\tilde{\\mathcal{H}}=\\frac{1}{2\\rho}\\tilde{p}^{2}+\\frac{1}{2}T(\\tilde{w}_{z})^{2}$\nand (\\ref{eq:pH_observer_error_output}) states the corresponding\noutput density. If we investigate the formal change of the error-Hamiltonian\nfunctional $\\tilde{\\mathscr{H}}$, which can be deduced to\n\\begin{align}\n\\dot{\\tilde{\\mathscr{H}}} & =\\int_{0}^{L}(T\\tilde{w}_{zz}k_{1}(\\bar{y}-\\hat{\\bar{y}})-\\frac{\\tilde{p}}{\\rho}k_{2}(\\bar{y}-\\hat{\\bar{y}}))\\mathrm{d}z\\,,\\label{eq:H_tilde_p}\n\\end{align}\nand take into account that $(\\bar{y}-\\hat{\\bar{y}})=\\int_{0}^{L}g(z)\\frac{1}{\\rho}\\tilde{p}\\mathrm{d}z$,\nwe find that the choice $k_{1}=0$ and $k_{2}=kg(z)$ with $k>0$,\nyields\n\\begin{equation}\n\\dot{\\tilde{\\mathscr{H}}}(\\tilde{w},\\tilde{p})=-k(\\bar{y}-\\hat{\\bar{y}})^{2}\\leq0\\,.\\label{eq:H_tilde_p_fin}\n\\end{equation}\nHowever, the fact that $\\tilde{\\mathscr{H}}>0$ and $\\dot{\\tilde{\\mathscr{H}}}\\leq0$\nhold is not sufficient for the convergence of the observer, and therefore,\nin the following, detailed stability investigations are carried out\nto verify that the observer error is asymptotically stable.\n\n\\section{Observer Convergence\\label{sec:Stability_Analysis}}\n\nIn this section, based on functional analysis the convergence of the\nobserver error is proven in two steps. First, we address the well-posedness\nof the observer-error system making heavy use of the well-known Lumer-Phillips\ntheorem, see e.g. \\cite{Liu1999}. Afterwards, LaSalle's invariance\nprinciple for infinite-dimensional systems is applied to show the\nasymptotic stability of the observer error, where beforehand it is\nnecessary to verify the precompactness of the solution trajectories.\n\n\\subsection{Well-posedness of the Observer-Error System\\label{subsec:Wellposedness}}\n\nNow, a careful investigation of the well-posedness of the observer-error\nsystem is carried out. To this end, we reformulate (\\ref{eq:observer_error_dynamics_VS})\nas an abstract Cauchy problem and show that the operator under consideration\ngenerates a $C_{0}$-semigroup of contractions.\n\nFirst, we define the state vector $\\chi=\\left[\\chi^{1},\\chi^{2}\\right]^{T}=\\left[\\tilde{w},\\tilde{p}\\right]^{T}$\ntogether with the state space $\\mathcal{X}=H_{C}^{1}(0,L)\\times L^{2}(0,L)$,\nwhere $H_{C}^{1}(0,L)=\\{\\chi^{1}\\in H^{1}(0,L)|\\chi^{1}(0)=0\\}$,\nwith $H^{l}(0,L)$ denoting a Sobolev space of functions whose derivatives\nup to order $l$ are square integrable, see \\cite{Adams2003} for\na detailed introduction of Sobolev spaces. Thus, the state space $\\mathcal{X}$\nis equipped with the standard norm\n\\begin{equation}\n\\left\\Vert \\chi\\right\\Vert _{n}^{2}=\\left\\langle \\tilde{w},\\tilde{w}\\right\\rangle _{L^{2}}+\\left\\langle \\tilde{w}_{z},\\tilde{w}_{z}\\right\\rangle _{L^{2}}+\\left\\langle \\tilde{p},\\tilde{p}\\right\\rangle _{L^{2}}\\,.\\label{eq:standard_norm}\n\\end{equation}\nNext, to be able to rewrite the observer-error dynamics as an abstract\nCauchy problem of the form $\\dot{\\chi}(t)=\\mathcal{A}\\chi(t)$ with\n$\\chi(0)=\\chi_{0}$, we introduce the linear operator $\\mathcal{A}:\\mathcal{D}(\\mathcal{A})\\subset\\mathcal{X}\\rightarrow\\mathcal{X}$\naccording to\n\\[\n\\mathcal{A}:\\left[\\begin{array}{c}\n\\tilde{w}\\\\\n\\tilde{p}\n\\end{array}\\right]\\rightarrow\\left[\\begin{array}{c}\n\\frac{1}{\\rho}\\tilde{p}\\\\\nT\\tilde{w}_{zz}-kg(z)\\int_{0}^{L}g(z)\\frac{1}{\\rho}\\tilde{p}\\mathrm{d}z\n\\end{array}\\right]\\,,\n\\]\nwhere the (dense) domain of $\\mathcal{A}$ is defined as\n\\begin{multline}\n\\mathcal{D}(\\mathcal{A}):=\\{\\chi\\in\\mathcal{X}|\\tilde{w}\\in(H^{2}(0,L)\\cap H_{C}^{1}(0,L)),\\\\\n\\tilde{p}\\in H_{C}^{1}(0,L),T\\tilde{w}_{z}(L)=0\\}\\,.\\label{eq:domain_A}\n\\end{multline}\nThus, the intention is to investigate the operator $\\mathcal{A}$\nregarding some properties such that a variant of the well-known Lumer-Phillips\ntheorem \\cite[Thm. 1.2.4]{Liu1999} can be applied. With respect to\nthis forthcoming investigations, it is beneficial to introduce\n\\begin{equation}\n\\left\\Vert \\chi\\right\\Vert _{\\mathcal{\\mathcal{X}}}^{2}=T\\left\\langle \\tilde{w}_{z},\\tilde{w}_{z}\\right\\rangle _{L^{2}}+\\frac{1}{\\rho}\\left\\langle \\tilde{p},\\tilde{p}\\right\\rangle _{L^{2}}\\,,\\label{eq:energy_norm}\n\\end{equation}\nwhich is called energy norm due to the equivalence $\\tilde{\\mathscr{H}}=\\frac{1}{2}\\left\\Vert \\chi\\right\\Vert _{\\mathcal{\\mathcal{X}}}^{2}$.\nBecause $\\tilde{w}(0)=0$ and further $\\tilde{w}(z)=\\int_{0}^{z}\\tilde{w}_{z}\\mathrm{d}y_{1}$\nholds, we find constants $c_{1},c_{2}$, which have to meet $00$, such\nthat $c_{1}\\left\\Vert \\chi\\right\\Vert _{n}^{2}\\leq\\left\\Vert \\chi\\right\\Vert _{\\mathcal{X}}^{2}\\leq c_{2}\\left\\Vert \\chi\\right\\Vert _{n}^{2}$\nis fulfilled, and hence, the energy norm (\\ref{eq:energy_norm}) is\nequivalent to the standard norm (\\ref{eq:standard_norm}). Similar\nto the proof of Lemma 2.2 in \\cite{Stuerzer2016}, where they exploit\nthe dense inclusion $H^{2}(0,L)\\subset H^{1}(0,L)$ and modify the\nboundary values of $\\tilde{w}$ and its derivatives in a proper manner,\nit can be shown that the domain $\\mathcal{D}(\\mathcal{A})$ given\nin (\\ref{eq:domain_A}) is dense in $\\mathcal{X}$. Thus, according\nto \\cite[Def. 1.1.1]{Liu1999}, -- since we have the equivalence\n$\\tilde{\\mathscr{H}}=\\frac{1}{2}\\left\\Vert \\chi\\right\\Vert _{\\mathcal{\\mathcal{X}}}^{2}$\n-- the relation (\\ref{eq:H_tilde_p_fin}) implies that $\\mathcal{A}$\nis dissipative.\n\nIn the following, we show that the inverse operator $\\mathcal{A}^{-1}$\nexists and is bounded, i.e. for every $\\bar{\\chi}=\\left[f,h\\right]^{T}\\in\\mathcal{X}$\nand $\\chi=\\left[\\tilde{w},\\tilde{p}\\right]^{T}\\in\\mathcal{D}(\\mathcal{A})$,\nwe can uniquely solve\n\\begin{equation}\n\\mathcal{A}\\!\\left[\\!\\begin{array}{c}\n\\tilde{w}\\\\\n\\tilde{p}\n\\end{array}\\!\\right]\\!=\\!\\left[\\begin{array}{c}\n\\frac{1}{\\rho}\\tilde{p}\\\\\nT\\tilde{w}_{zz}-kg(z)\\int_{0}^{L}g(z)\\frac{1}{\\rho}\\tilde{p}\\mathrm{d}z\n\\end{array}\\right]\\!=\\!\\left[\\begin{array}{c}\nf\\\\\nh\n\\end{array}\\right],\\label{eq:Calc_A-1}\n\\end{equation}\nand prove that $\\mathcal{A}^{-1}$ maps bounded sets in $\\mathcal{X}$\ninto bounded sets in $\\mathcal{K}:=(H^{2}(0,L)\\cap H_{C}^{1}(0,L))\\times H_{C}^{1}(0,L)$.\nFrom the $1$st line of (\\ref{eq:Calc_A-1}) it follows that $\\tilde{p}=\\rho f\\in H_{C}^{1}(0,L)$.\nMoreover, an integration of the $2$nd line of (\\ref{eq:Calc_A-1})\nyields\n\\begin{multline}\n\\tilde{w}_{z}(z)=-\\frac{1}{T}(\\int_{z}^{L}h(y_{2})\\mathrm{d}y_{2}+\\ldots\\\\\n+\\int_{z}^{L}kg(y_{2})\\int_{0}^{L}g(y_{1})f(y_{1})\\mathrm{d}y_{1}\\mathrm{d}y_{2})\\label{eq:w_z_calc}\n\\end{multline}\nas $\\tilde{w}_{z}(L)=0$ holds. If we further integrate (\\ref{eq:w_z_calc}),\nwe obtain\n\\begin{multline}\n\\tilde{w}(z)=-\\frac{1}{T}(\\int_{0}^{z}\\int_{y_{3}}^{L}h(y_{2})\\mathrm{d}y_{2}\\mathrm{d}y_{3}+\\ldots\\\\\n+\\int_{0}^{z}\\int_{y_{3}}^{L}kg(y_{2})\\int_{0}^{L}g(y_{1})f(y_{1})\\mathrm{d}y_{1}\\mathrm{d}y_{2}\\mathrm{d}y_{3})\\label{eq:w_calc}\n\\end{multline}\nas $\\tilde{w}(0)=0$, and thus, $\\tilde{w}(z)$ is uniquely defined\nby $\\bar{\\chi}$. Since we have shown that the inverse operator $\\mathcal{A}^{-1}$\nexists, it remains to investigate the boundedness. To this end, it\nis verified that the norm of $\\chi=\\mathcal{A}^{-1}\\bar{\\chi}$ in\n$\\mathcal{K}$ is bounded by $\\left\\Vert \\bar{\\chi}\\right\\Vert _{\\mathcal{X}}$.\nFirst, we state an inequality that is often used in the sequel; in\nfact, for a -- basically arbitrary -- function $f$, by means of\nthe Cauchy-Schwarz inequality we find the important relation\n\\begin{equation}\n(\\int_{0}^{L}f\\mathrm{d}z)^{2}\\leq C\\int_{0}^{L}\\left|f\\right|^{2}\\mathrm{d}z\\,,\\label{eq:inequalitiy_square}\n\\end{equation}\nwhere it should be mentioned that here and in the following $C$\ndenotes positive, not necessarily equal constants. Next, we investigate\nthe norm $\\left\\Vert \\tilde{w}_{z}\\right\\Vert _{L^{2}}$. Therefore,\nwe substitute (\\ref{eq:w_z_calc}) in $\\left\\Vert \\tilde{w}_{z}\\right\\Vert _{L^{2}}=(\\int_{0}^{L}\\left|\\tilde{w}_{z}\\right|^{2}\\mathrm{d}z)^{1\/2}$\nand apply the Triangle inequality, which yields\n\\begin{multline*}\n\\left\\Vert \\tilde{w}_{z}\\right\\Vert _{L^{2}}\\leq(\\int_{0}^{L}\\frac{1}{T^{2}}(\\int_{z}^{L}h(y_{2})\\mathrm{d}y_{2})^{2}\\mathrm{d}z)^{\\frac{1}{2}}+\\\\\n(\\int_{0}^{L}\\frac{1}{T^{2}}(\\int_{z}^{L}kg(y_{2})\\int_{0}^{L}g(y_{1})f(y_{1})\\mathrm{d}y_{1}\\mathrm{d}y_{2})^{2}\\mathrm{d}z)^{\\frac{1}{2}}.\n\\end{multline*}\n Thus, by means of (\\ref{eq:inequalitiy_square}) and due to the fact\nthat $\\int_{z}^{L}h^{2}\\mathrm{d}y_{2}\\leq\\int_{0}^{L}h^{2}\\mathrm{d}z=\\left\\Vert h\\right\\Vert _{L_{2}}^{2}$\nholds, we obtain\n\\begin{multline}\n\\left\\Vert \\tilde{w}_{z}\\right\\Vert _{L^{2}}\\leq C\\left\\Vert h\\right\\Vert _{L_{2}}\\frac{1}{T}L^{\\frac{1}{2}}+\\\\\nC(\\int_{0}^{L}\\frac{1}{T^{2}}\\int_{z}^{L}k^{2}g^{2}(y_{2})(\\int_{0}^{L}g(y_{1})f(y_{1})\\mathrm{d}y_{1})^{2}\\mathrm{d}y_{2}\\mathrm{d}z)^{\\frac{1}{2}}.\\label{eq:ineq_w_z_L_2}\n\\end{multline}\nNext, we apply the Cauchy-Schwarz inequality to the second term of\nthe right-hand side in (\\ref{eq:ineq_w_z_L_2}), which enables us\nto find the estimate $\\left\\Vert \\tilde{w}_{z}\\right\\Vert _{L^{2}}\\leq C(\\left\\Vert f\\right\\Vert _{H^{1}}+\\left\\Vert h\\right\\Vert _{L^{2}})$.\nSimilarly, by means of the $2$nd line of (\\ref{eq:Calc_A-1}) we\nare able to deduce $\\left\\Vert \\tilde{w}_{zz}\\right\\Vert _{L^{2}}\\leq C(\\left\\Vert f\\right\\Vert _{H^{1}}+\\left\\Vert h\\right\\Vert _{L^{2}})$.\nMoreover, if we substitute (\\ref{eq:w_calc}) in $\\left\\Vert \\tilde{w}\\right\\Vert _{L^{2}}=(\\int_{0}^{L}\\left|\\tilde{w}\\right|^{2}\\mathrm{d}z)^{1\/2}$,\nwe find $\\left\\Vert \\tilde{w}\\right\\Vert _{L^{2}}\\leq C(\\left\\Vert f\\right\\Vert _{H^{1}}+\\left\\Vert h\\right\\Vert _{L^{2}})$,\nand hence, we have $\\left\\Vert \\tilde{w}\\right\\Vert _{H^{2}}\\leq C(\\left\\Vert f\\right\\Vert _{H^{1}}+\\left\\Vert h\\right\\Vert _{L^{2}})$.\nSince from the first line in (\\ref{eq:Calc_A-1}) we immediately get\n$\\left\\Vert \\tilde{p}\\right\\Vert _{H^{1}}=\\rho\\left\\Vert f\\right\\Vert _{H^{1}}$,\nwe can state the important estimate\n\\[\n\\left\\Vert \\tilde{w}\\right\\Vert _{H^{2}}+\\left\\Vert \\tilde{p}\\right\\Vert _{H^{1}}\\leq C(\\left\\Vert f\\right\\Vert _{H^{1}}+\\left\\Vert h\\right\\Vert _{L^{2}})\\,,\n\\]\nwhich shows that $\\mathcal{A}^{-1}$ maps bounded sets in $\\mathcal{X}$\ninto bounded sets in $\\mathcal{K}$. \n\nThe boundedness of $\\mathcal{A}^{-1}$ implies that $\\lambda=0$ cannot\nbe an eigenvalue of $\\mathcal{A}$, and hence, it follows that $0\\in\\rho(\\mathcal{A})$,\nthe resolvent set of $\\mathcal{A}$. Furthermore, since $\\mathcal{D}(\\mathcal{A})$\nis dense in $\\mathcal{X}$ and $\\mathcal{A}$ is dissipative, all\nrequirements for the variant of the Lumer-Phillips theorem according\nto \\cite[Thm. 1.2.4]{Liu1999} are met, and therefore, we are able\nto show that $\\mathcal{A}$ is the infinitesimal generator of a $C_{0}$-semigroup\nof contractions on $\\mathcal{X}$. That is, the norm $\\left\\Vert \\chi(t)\\right\\Vert _{\\mathcal{X}}$\nremains bounded for $t\\rightarrow\\infty$; however, with respect to\nthe observer error it is necessary that it tends to $0$, which is\nshown in the following subsection.\n\n\\subsection{Asymptotic Stability of the Observer-Error System\\label{subsec:LaSalle}}\n\nNow, the objective is to apply LaSalle's invariance principle in order\nto prove the asymptotic stability of the observer error, where the\nproof follows the intention of \\cite[Sec. 3]{Guo2011}. However, the\napplicability of LaSalle's invariance principle according to \\cite[Thm. 3.64]{Luo1998}\nrequires the precompactness of the solution trajectories, which is\nnot ensured in the infinite-dimensional scenario. Since in the previous\nsection we have shown that $\\mathcal{A}^{-1}$ is bounded, by means\nof the Sobolev embedding theorem, it follows that $\\mathcal{A}^{-1}$\nis compact (see proof of Lemma 2.4 in \\cite{Stuerzer2016} or \\cite[p. 201]{Luo1998}),\nwhich further implies the precompactness of the trajectories, see\n\\cite[Rem. 4.2]{Miletic2015}. \n\nIn light of LaSalle's invariance principle, we investigate the set\n$\\mathcal{S}=\\{\\chi\\in\\mathcal{X}|\\dot{\\tilde{\\mathscr{H}}}=0\\}$,\nwhere $\\dot{\\tilde{\\mathscr{H}}}(\\tilde{w},\\tilde{p})=-k(\\int_{0}^{L}g(z)\\frac{1}{\\rho}\\tilde{p}\\mathrm{d}z)^{2}=0$\nimplies $\\int_{0}^{L}g(z)\\frac{1}{\\rho}\\tilde{p}\\mathrm{d}z=0$. In\nthe set $\\mathcal{S}$ we have\\begin{subequations}\\label{eq:eom_S}\n\\begin{align}\n\\rho\\tilde{w}_{tt} & =T\\tilde{w}_{zz}\\,,\\label{eq:eom_S_pde}\\\\\n\\tilde{w}(0,t) & =0\\,,\\label{eq:eom_S_BC_left}\\\\\nT\\tilde{w}_{z}(L,t) & =0\\,,\\label{eq:eom_S_BC_right}\n\\end{align}\n\\end{subequations}which is similar to the problem considered in \\cite[Sec. 3]{Guo2011};\nhowever, the restriction describing the set $\\mathcal{S}$, which\nis constrained to the boundary there, is completely different. To\nbe able to show that the only possible solution in $\\mathcal{S}$\nis the trivial one, we need to investigate the general solution of\n(\\ref{eq:eom_S}). To this end, like in \\cite[Sec. 3]{Guo2011}, we\nfirst focus on determining the eigenvalues and eigenfunctions of (\\ref{eq:eom_S}),\ni.e. we consider\n\\begin{equation}\n\\bar{\\mathcal{A}}\\left[\\begin{array}{cc}\n\\phi & \\kappa\\end{array}\\right]^{T}=\\left[\\begin{array}{cc}\n\\frac{\\kappa}{\\rho} & T\\phi_{zz}\\end{array}\\right]=\\lambda\\left[\\begin{array}{cc}\n\\phi & \\kappa\\end{array}\\right]^{T}\\,.\\label{eq:eigen_equation}\n\\end{equation}\nFrom (\\ref{eq:eigen_equation}) we obtain $\\kappa=\\rho\\lambda\\phi$\nand furthermore\\begin{subequations}\\label{eq:eigen_problem}\n\\begin{align}\n\\phi_{zz} & =\\frac{\\lambda^{2}}{\\vartheta^{2}}\\phi\\,,\\\\\n\\phi(0) & =0\\,,\\label{eq:eigen_problem_BC_left}\\\\\n\\phi_{z}(L) & =0\\,,\\label{eq:eigen_problem_BC_right}\n\\end{align}\n\\end{subequations}where $\\vartheta^{2}=\\frac{T}{\\rho}$. To find\nthe solution of (\\ref{eq:eigen_problem}), we have to investigate\nthe three cases $\\lambda^{2}>0$, $\\lambda^{2}=0$ and $\\lambda^{2}<0$\nin the following. For $\\lambda^{2}>0$ and $\\lambda^{2}=0$, we have\nthe ansatz $\\phi(z)=Ae^{\\frac{\\lambda}{\\vartheta}z}+Be^{-\\frac{\\lambda}{\\vartheta}z}$\nand $\\phi(z)=Az+B$, respectively, where by means of the boundary\nconditions (\\ref{eq:eigen_problem_BC_left}) and (\\ref{eq:eigen_problem_BC_right}),\none can easily deduce that for both cases only the trivial solution\n$\\phi(z)=0$ exists. Thus, we focus on the case $\\lambda^{2}<0$,\nand consequently, due to the fact that $\\lambda$ has an imaginary\ncharacter then, as ansatz for the eigenfunctions we have $\\phi(z)=A\\sin(\\frac{\\left|\\lambda\\right|}{\\vartheta}z)+B\\cos(\\frac{\\left|\\lambda\\right|}{\\vartheta}z)$.\nTo fulfil the boundary condition (\\ref{eq:eigen_problem_BC_left}),\n$B=0$ must be valid, and hence, the ansatz simplifies to $\\phi(z)=A\\sin(\\frac{\\left|\\lambda\\right|}{\\vartheta}z)$.\nFurthermore, by means of the boundary condition (\\ref{eq:eigen_problem_BC_right}),\nwe find $\\partial_{z}\\phi(L)=\\frac{\\left|\\lambda\\right|}{\\vartheta}A\\cos(\\frac{\\left|\\lambda\\right|}{\\vartheta}L)=0$,\nwhich exhibits infinitely many non-trivial solutions for\n\\begin{equation}\n\\left|\\lambda_{k}\\right|=(k-\\frac{1}{2})\\frac{\\pi}{L}\\vartheta\\,,\\label{eq:eigenvalues}\n\\end{equation}\nwith $k=1,2,\\ldots$. With regard to the investigation of the set\n$\\mathcal{S}$, the velocity of the vibrating string is of particular\ninterest. Consequently, since we deduced the (imaginary) eigenvalues\n$\\lambda_{k}=\\pm i\\omega_{k}\\vartheta$ with $\\omega_{k}=(k-\\frac{1}{2})\\frac{\\pi}{L}$,\nthe ansatz for the general solution of the velocity can be given according\nto\\begin{subequations}\\label{eq:solution_a_b}\n\\begin{align}\n\\tilde{w}_{t}(z,t) & \\!=\\!\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}\\!(a_{k}\\cos(\\omega_{k}\\vartheta t)\\!+\\!b_{k}\\sin(\\omega_{k}\\vartheta t))\\varphi_{k}(z),\\label{eq:solution_w_t}\n\\end{align}\nwhere the coefficients $A_{k}$ are hidden in $a_{k}$ and $b_{k}$,\nand therefore, for the eigenfunctions we use $\\varphi_{k}(z)=\\sin(\\omega_{k}z)$\nhere and in the sequel. Hence, an integration of (\\ref{eq:solution_w_t})\nyields\n\\begin{equation}\n\\tilde{w}(z,t)\\!=\\!\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}\\!(a_{k}\\sin(\\omega_{k}\\vartheta t)\\!-\\!b_{k}\\cos(\\omega_{k}\\vartheta t))\\frac{\\varphi_{k}(z)}{\\omega_{k}\\vartheta}.\\label{eq:solution_w}\n\\end{equation}\n\\end{subequations}By means of $\\sin(x)=\\frac{1}{2i}(e^{ix}-e^{-ix})$\nand $\\cos(x)=\\frac{1}{2}(e^{ix}+e^{-ix})$, after a straightforward\ncomputation we can beneficially rewrite (\\ref{eq:solution_a_b}) according\nto\n\\begin{multline*}\n\\left[\\begin{array}{c}\n\\tilde{w}(z,t)\\\\\n\\tilde{w}_{t}(z,t)\n\\end{array}\\right]=\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}c_{k}e^{i\\omega_{k}\\vartheta t}\\left[\\begin{array}{c}\n-i\\frac{\\varphi_{k}}{\\omega_{k}\\vartheta}\\\\\n\\phi_{k}\n\\end{array}\\right]+\\ldots\\\\\n\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}c_{-k}e^{-i\\omega_{k}\\vartheta t}\\left[\\begin{array}{c}\ni\\frac{\\varphi_{k}}{\\omega_{k}\\vartheta}\\\\\n\\phi_{k}\n\\end{array}\\right]\\,,\n\\end{multline*}\nwhere the coefficients $c_{k}=\\frac{1}{2}\\left(a_{k}-ib_{k}\\right)$\nand $c_{-k}=\\frac{1}{2}\\left(a_{k}+ib_{k}\\right)$ fulfil (see \\cite[Eq. (3.19)]{Guo2011})\n\\begin{equation}\n\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}\\left|c_{\\pm k}\\right|^{2}=\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}\\left(a_{k}^{2}+b_{k}^{2}\\right)<\\infty\\,,\\label{eq:coefficients_bounded}\n\\end{equation}\nwhich will play an important role later. Thus, we are able to write\n$\\int_{0}^{L}g(z)\\frac{1}{\\rho}\\tilde{p}\\mathrm{d}z=\\int_{L_{p_{1}}}^{L_{p_{2}}}\\tilde{w}_{t}\\mathrm{d}z=0$\nas\n\\begin{equation}\n\\int_{L_{p_{1}}}^{L_{p_{2}}}\\overset{\\infty}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}(c_{k}e^{i\\omega_{k}\\vartheta t}+c_{-k}e^{-i\\omega_{k}\\vartheta t})\\sin(\\omega_{k}z)\\mathrm{d}z=0\\,.\\label{eq:condition_S}\n\\end{equation}\n\nNow, we show that the only solution in $\\mathcal{S}$ is the trivial\none, i.e. $c_{\\pm k}=0\\forall k\\geq1$ is valid. Otherwise, if there\nexists a $k_{0}$ with $\\left|c_{k_{0}}\\right|\\neq0$, due to (\\ref{eq:coefficients_bounded})\nwe can find a $K>k_{0}$ such that\\begin{subequations}\\label{eq:bound_inf_coeffs}\n\\begin{align}\n\\left|\\int_{L_{p_{1}}}^{L_{p_{2}}}\\overset{\\infty}{\\underset{\\mathop{k=K}}{\\mathop{\\sum}}}c_{k}\\varphi_{k}\\mathrm{d}z\\right| & <\\left|\\frac{c_{k_{0}}}{4}\\int_{L_{p_{1}}}^{L_{p_{2}}}\\varphi_{k_{0}}\\mathrm{d}z\\right|\\\\\n\\left|\\int_{L_{p_{1}}}^{L_{p_{2}}}\\overset{\\infty}{\\underset{\\mathop{k=K}}{\\mathop{\\sum}}}c_{-k}\\varphi_{k}\\mathrm{d}z\\right| & <\\left|\\frac{c_{k_{0}}}{4}\\int_{L_{p_{1}}}^{L_{p_{2}}}\\varphi_{k_{0}}\\mathrm{d}z\\right|\n\\end{align}\n\\end{subequations}holds, i.e. the sum of the coefficients from $K$\nto $\\infty$ multiplied with their corresponding eigenfunctions can\nbe bounded by $c_{k_{0}}$ and $\\varphi_{k_{0}}$. Here, it is assumed\nthat $\\int_{L_{p_{1}}}^{L_{p_{2}}}\\sin(\\omega_{k_{0}})\\mathrm{d}z\\neq0$,\ni.e. $\\omega_{k_{0}}\\neq\\frac{2\\pi}{L_{p_{2}}-L_{p_{1}}}j$ with $j\\in\\mathbb{N}_{+}$.\nHowever, if we consider the absolute value of the eigenvalues (\\ref{eq:eigenvalues}),\nwe find that this is ensured for a proper choice of the length of\nthe in-domain actuator according to $L_{p_{2}}-L_{p_{1}}\\neq\\frac{4L}{2k-1}$.\nConsequently, because $\\omega_{k}\\neq\\omega_{l}\\forall k\\neq l$,\nfor $t>0$ we can reformulate (\\ref{eq:condition_S}) as\n\\begin{multline}\n-c_{k_{0}}\\int_{L_{p_{1}}}^{L_{p_{2}}}\\varphi_{k_{0}}\\mathrm{d}z=\\int_{L_{p_{1}}}^{L_{p_{2}}}\\{\\overset{K}{\\underset{\\mathop{k=1,k\\neq k_{0}}}{\\mathop{\\sum}}}c_{k}e^{i(\\omega_{k}-\\omega_{k_{0}})\\vartheta t}\\varphi_{k}+\\\\\n+\\overset{\\infty}{\\underset{\\mathop{k=K+1}}{\\mathop{\\sum}}}c_{k}e^{i(\\omega_{k}-\\omega_{k_{0}})\\vartheta t}\\varphi_{k}+\\overset{K}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}c_{-k}e^{-i(\\omega_{k}+\\omega_{k_{0}})\\vartheta t}\\varphi_{k}+\\\\\n+\\overset{\\infty}{\\underset{\\mathop{k=K+1}}{\\mathop{\\sum}}}c_{-k}e^{-i(\\omega_{k}+\\omega_{k_{0}})\\vartheta t}\\varphi_{k}\\}\\mathrm{d}z\\,.\\label{eq:condition_S_ref}\n\\end{multline}\nNext, the idea is to integrate (\\ref{eq:condition_S_ref}) with respect\nto the time $t$ and to investigate the absolute value. Hence, we\nfind that the right-hand side of\n\\begin{multline}\n\\left|c_{k_{0}}\\int_{L_{p_{1}}}^{L_{p_{2}}}\\varphi_{k_{0}}\\mathrm{d}z\\right|t\\leq\\\\\n2\\left|\\int_{0}^{t}\\int_{L_{p_{1}}}^{L_{p_{2}}}\\{\\overset{K}{\\underset{\\mathop{k=1,k\\neq k_{0}}}{\\mathop{\\sum}}}c_{k}e^{i(\\omega_{k}-\\omega_{k_{0}})\\vartheta\\tau}\\varphi_{k}\\}\\mathrm{d}z\\mathrm{d}\\tau\\right|+\\\\\n+2\\left|\\int_{0}^{t}\\int_{L_{p_{1}}}^{L_{p_{2}}}\\{\\overset{K}{\\underset{\\mathop{k=1}}{\\mathop{\\sum}}}c_{-k}e^{-i(\\omega_{k}+\\omega_{k_{0}})\\vartheta\\tau}\\varphi_{k}\\}\\mathrm{d}z\\mathrm{d}\\tau\\right|\\,,\\label{eq:inequaltiy_coefficients}\n\\end{multline}\nwhere we used (\\ref{eq:bound_inf_coeffs}) to obtain an estimation\nfor the sums from $k=K+1$ to $k=\\infty$, is bounded for all $t\\geq0$.\nSince for an appropriate choice of the actuator-length it is ensured\nthat the integral on the left-hand side cannot vanish, the only possibility\nthat inequality (\\ref{eq:inequaltiy_coefficients}) holds for $t\\rightarrow\\infty$\nis that $c_{k_{0}}=0$ is valid. Thus, it is shown that the only possible\nsolution in $\\mathcal{S}$ is the trivial one, which finally proves\nthe asymptotic stability of the observer error and therefore justifies\nthe application of the observer developed in \\cite{Malzer2020}. Furthermore,\nin Figure (\\ref{fig:Comp_w_state_obs}), the comparison of the string\ndeflection $w(L,t)$ and the observer state $\\hat{w}(L,t)$ is depicted,\nwhere the tip of the string is moved from $w(L,0)=0$ to $w(L,t_{end})=0.1$\nand the observer state is initialised as $\\hat{w}(L,0)=0.1$.\n\\begin{figure}\n\\input{w_L_Obs.tex}\\caption{\\label{fig:Comp_w_state_obs}Comparison of the string deflection $w(L,t)$\nand the observer state $\\hat{w}(L,t)$.}\n\\end{figure}\n\n\n\\section{Conclusion and Outlook}\n\nIn this paper, the asymptotic stability of an observer error of an\nin-domain actuated vibrating string, where the observer has been developed\nin \\cite{Malzer2020}, was investigated. First, we showed that the\nlinear operator, which describes the observer error as an abstract\nCauchy problem, is the infinitesimal generator of a contraction semigroup.\nSecond, by means of LaSalle's invariance principle the asymptotic\nstability of the observer error was proven. In fact, by choosing the\nlength of the actuator properly, it was shown that the only possible\nsolution for $\\dot{\\tilde{\\mathscr{H}}}=0$ is the trivial one, which\nimplies that the observer error tends to zero. Future-research tasks\nmight deal with the stability analysis of the closed loop obtained\nby the controller design presented in \\cite{Malzer2020}, or even\nwith the stability investigation of the combination of controller\nand observer.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe observation of the sky with space-born instruments, equipped with detectors working at those electro-magnetic frequencies that cannot be accessed from the ground, revealed the existence of several classes of high energy radiation sources. With their location in distant galaxies, Active Galactic Nuclei (AGNs, in brief) turned out to be the most powerful non transient sources of such radiation. Although AGNs appear with many different properties, they share an extreme intrinsic luminosity, ranging between $10^{41}\\, {\\rm erg\\, s^{-1}}$ and $10^{46}\\, {\\rm erg\\, s^{-1}}$, comparable to or greater than the energy output of large galaxies, but released from a region that is smaller than 1~pc in radius. To explain this property we assume that large amounts of matter, in the order of some solar masses per year, are conveyed to the nuclear regions of active galaxies, where they are accreted by a Super Massive Black Hole (SMBH). It is now well established that SMBHs with masses between $10^6\\, {\\rm M_\\odot}$ and some $10^9\\, {\\rm M_\\odot}$ reside in the nuclei of every massive galaxy (Ferrarese \\& Merrit, 2000; Shankar, 2009). In addition, we know that accretion of fuel into their gravitational field leads to the conversion of gravitational binding energy into radiation with very high efficiency (Blandford \\& Znajek, 1977; Shields, 1978).\n\nDue to the presence of relativistic plasmas and strong magnetic fields in the vicinity of the black hole's accretion flow, the spectrum of the emitted radiation results from a combination of thermal and non-thermal components that cover several orders of magnitude in frequency, sometimes extending from radio wavelengths all the way up to $\\gamma$-ray energies. The Fermi Large Area Telescope (Fermi-LAT; see Atwood et al., 2009) gave new life to the study of the $\\gamma$-ray sky, producing, after 4 years of scientific observations, a map of $\\gamma$-ray detections with unprecedented resolution and sensitivity in the energy range between 100~MeV and 300~GeV (Acero et al., 2015). Thanks to this result, a large number of $\\gamma$-ray sources can now be associated with lower energy counterparts. In the extra-Galactic environment, it turned out that the most commonly detected objects are AGNs belonging to the blazar class. Blazars are extremely variable, highly polarized, radio-loud sources, dominated by power-law continuum spectra of type $F_\\nu \\propto \\nu^{-\\alpha}$, with a typical radio spectral slope $\\alpha \\leq 0.5$. Their properties are the consequence of the relativistic beaming of the synchrotron radiation produced by a jet that is collimated and accelerated close to our line of sight (Blandford \\& K\\\"onigl, 1979). They are classically separated into BL Lac objects (BLL), whose optical spectra show a nearly featureless power-law continuum, and Flat Spectrum Radio Quasars (FSRQs), which, instead, are characterized by strong emission lines.\n\nIn addition to the blazars that dominate the extra-Galactic $\\gamma$-ray population, other types of AGNs, together with a large number of sources without a firm classification, are detected as well. In this contribution we describe our investigation on the nature of $\\gamma$-ray AGNs of undetermined type, through the observation of their optical spectra. We focus our attention on targets whose spectral energy distributions (SED) are consistent with those of blazars, although they still lack firm spectroscopic classification, and they are therefore called Blazar Candidates of Undetermined type (BCU in 3FGL terminology, Acero et al., 2015). In this report, we discuss the spectral classification of some BCUs, with respect to the associated SEDs, rather than giving a full list of observations. In the following sections, we describe the techniques used in the attempt to identify the low energy counterparts of the $\\gamma$-ray sources, the details of how we collected the optical spectra and their classification criteria. Finally, we draw a sketch of the $\\gamma$-ray emission in the different classes of objects that we observe, compared with the multiple wavelength properties of their SEDs.\n\n\\section{Association of $\\gamma$-ray sources}\nAt the energies of $\\gamma$-ray photons it is not possible to focus radiation through reflecting or refracting optical devices. The Fermi-LAT, instead, measures the production of $e^\\pm$ pairs, through the conversion of photons in the detector (Atwood et al., 2009). The pair properties are used to reconstruct the energy and the direction of the incoming photon. The precision that can be achieved in the measurement depends mainly on the incoming photon incidence angle and on its energy. It can be roughly estimated that, for a photon of energy between 10 GeV and 100 GeV, hitting the detector at normal incidence, the 95\\% containment angle is approximately 0.5$^\\circ$ (Ackermann et al., 2012). Although the detection of multiple photons from a source can improve the performances, the contribution of nearby sources and background noise, combined with less than optimal detection conditions, results in a still significant uncertainty in the localization of the sources. Thus, in general, the task to associate a low energy counterpart to $\\gamma$-ray sources is not trivial.\n\nIn the case of AGNs, the expected broad band emission provides a reliable way to better constrain the source position. Since $\\gamma$-rays from AGNs arise from Inverse Compton (IC) scattering of low energy seed photons by relativistic plasma particles confined in strong magnetic fields, the $\\gamma$-ray production occurs together with synchrotron radiation. If the energy distribution of the plasma particles is extended enough to support significant $\\gamma$-ray emission, we expect that powerful synchrotron radiation is produced at radio and x-ray energies, as well. Taking advantage from the angular resolution of instruments working at these frequencies, which can measure the position of radiation sources down to a few arcseconds, we are able to identify candidate counterparts to $\\gamma$-ray emission by looking for coincident x-ray and radio sources within the $\\gamma$-ray detection uncertainty radius (Ackermann et al., 2011; Gasparrini et al., 2012). Furthermore, the existence of a connection between $\\gamma$-ray emission and lower frequency radiation implies that large flux variations, that characterize AGNs particularly in the high energies, can be observed in different frequencies. Thanks to the monitoring strategy of Fermi-LAT observations, covering the entire sky approximately every 3~hr, the detection of important flaring activity can be matched with follow-up observations that are able to identify the possible correlated variations of the source in other frequencies. Once the source position is determined by either of the techniques down to a few arcseconds, it is possible to search for the optical counterpart and to obtain its spectrum.\n\n\\section{Optical observations}\nThe latest catalog of Fermi-LAT detected AGNs (3LAC, see Ackermann et al., 2015) includes blazars, radio galaxies, steep spectrum radio quasars (SSRQs), Seyfert galaxies and Narrow Line Seyfert 1 galaxies. In addition to the classified sources, however, a number of undetermined type objects, generally called AGNs or BCUs, still exists. Focusing our attention on these unclassified objects, we carried out a spectroscopic study that combines publicly available spectra, like those extracted from the latest data release of the Sloan Digital Sky Survey (SDSS-DR 12; Alam et al., 2015) and of the 6dF Galaxy Redshift Survey (6dFGRS-DR 3; Jones et al., 2004, 2009), together with new observations performed at the 1.22m and the 1.82m telescopes of the Asiago Astrophysical Observatory (Ciroi et al., 2014).\n\nFig. 1 illustrates examples of spectra obtained in this study. The main purpose of this optical spectroscopic analysis is to identify the AGN class for the source associated to the $\\gamma$-ray emission and to determine its redshift, through the identification of known emission or absorption lines. In general, the detection of emission lines ensures the most reliable measurement of the source redshift, because they are easier to detect and they are originated very close to central SMBH. Absorption lines, on the contrary, are produced either in the host galaxy or along the light path towards us. They are much more difficult to detect and can only be seen if the AGN is not overwhelmingly dominant. It follows that the presence of emission lines characterizes AGNs with radiatively efficient accretion activity, therefore surrounded by nuclear and circum-nuclear regions of ionized gas, like quasars and Seyfert galaxies. BLLs, on the contrary, just give raise to a featureless power-law continuum, where some absorption features can show up, if the jet power is relatively low with respect to the host galaxy luminosity, or when some intervening medium happens to lie along the line of sight to the source.\n\n\\begin{table}[t]\n\\caption{List of sources included in this report. The table provides the $\\gamma$-ray source identifier, the associated counterpart name, the counterpart position (J2000.0), the object redshift and its spectroscopic classification.}\n\\begin{footnotesize}\n\\begin{center}\n\\begin{tabular}{llcccr}\n\\hline\n\\hline\nId. & Counterpart name & R. A. & Dec. & {\\it z} & Class \\\\\n\\hline\n3FGL J$0134.5+2638$ & 1RXS J$013427.2+263846$ & $01:34:28.3$ & $+26:38:45.0$ & $\\geq$0.108 & BL Lac \\\\\n3FGL J$0339.2-1738$ & PKS $0336-177$ & $03:39:13.7$ & $-17:36:00.6$ & 0.065 & Elliptical \\\\\n3FGL J$0904.3+4240$ & S4 $0900+42$ & $09:04:17.1$ & $+42:37:59.0$ & 1.342 & FSRQ \\\\\n3FGL J$1031.0+7440$ & S5 $1027+74$ & $10:31:22.0$ & $+74:41:58.3$ & 0.122 & FSRQ \\\\\n3FGL J$1315.4+1130$ & 1RXS J$131531.9+113327$ & $13:15:32.0$ & $+11:33:27.0$ & $\\geq$0.730 & BL Lac \\\\\n3FGL J$1412.0+5249$ & SBS $1410+530$ & $14:11:49.4$ & $+52:49:00.2$ & 0.076 & Elliptical \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{footnotesize}\n\\end{table}\n\\section{Results and models}\nA summary of our spectroscopic observations is presented in Table 1. According to the spectra collected in Fig. 1, our targets belong to different families of optical sources. Together with examples of classic high luminosity blazar objects, characterized by prominent emission lines (FSRQs) or dominant power-law featureless spectra (BLLs), we notice associations with some weaker sources, where the spectrum of the host galaxy, with its characteristic stellar continuum and absorption features, plays an increasing role. Looking at the spectra presented here, we note that in the high redshift regime (approximately from $z \\geq 0.15$) we are able to detect $\\gamma$-ray emission only from objects with powerful jets. At smaller distances, however, $\\gamma$-ray sources likely arising from jets of weaker power or misaligned orientation, which do not significantly contribute to the optical spectra, can be identified as well.\n\nIn order to estimate the power of the jets, which are likely producing the observed $\\gamma$-ray signal, and to compare it with the characteristics of the corresponding optical spectra, we reconstructed the SEDs associated to our targets. We used the ASI Space Data Center SED Builder tool to retrieve multiple frequency data points and to interpolate them with jet radiation models.\\footnote{The SED Builder tool is available at \\texttt{http:\/\/tools.asdc.asi.it\/SED\/}} We took into account archival data available from the literature, including in particular observations performed within extensive sky survey programs, to cover as an extended frequency range as possible. The results of this selection are plotted in Fig. 2. The broad band electro-magnetic emission is generally consistent with the classic two-hump blazar SED, very likely connected with jet activity. There are, however, some objects where an IR to optical radiation excess clearly occurs. The presence of such an excess is made particularly evident by comparing the observations with simple Synchrotron Self-Compton (SSC) models, which we used to interpolate the data. Due to the possibility that external radiation fields may give a relevant contribution to the IC effect (a scenario that is referred to as External Compton, or EC, and is very likely more appropriate, especially in the case of FSRQs), the SSC approach might be an oversimplified interpretation of the SEDs. However, the non simultaneous nature of the different data points collected in this work did not allow to consider more advanced models.\n\nTable 2 reports details of the SED models applied to our targets. The residuals were calculated without taking into account the thermal components. The SSC models are able to reproduce most of the sources with acceptable residuals ($\\chi^2_{red} \\leq 1.5$), while the most notable exceptions are probably due to the difficulties that such models have in explaining the high energy IC tail. Intrinsic source variability at the different observation epochs is also expected to affect the scatter. In general, we observe that jets of large scale and power are required to interpolate classic BLL and FSRQ blazars, while weaker targets are also explained by less powerful jets.\n\n\\subsection{Notes on single sources}\n{\\bf 3FGL J0134.5+2638}, observed with the Asiago 1.22m telescope, reveals a power law continuum spectrum with faint absorption lines. Absence of emission lines having equivalent width $EW \\geq 5\\,$\\AA leads to a BLL classification. Its SED shows the characteristic two-hump behavior of blazars and it is appreciably well interpolated by a SSC model. \\\\\n{\\bf 3FGL J0339.2--1738} is associated with an elliptical galaxy, with spectrum available from 6dFGRS. Flux calibration of the spectrum was obtained deriving an average sensitivity curve for the 6dF instrument, based on the flux calibrated spectra of IC 5135 and UGC 842. The resulting spectrum shows the characteristic continuum and absorption lines of an old stellar population, typical of ellipticals. The associated SED is a two-hump distribution with a prominent radiation excess in the optical window. \\\\\n{\\bf 3FGL J0904.3+4240}, detected by the SDSS, shows the highest redshift in this sample, which brings the strong UV emission lines of C~IV~$\\lambda$1549, C~III]~$\\lambda$1909 and Mg~II~$\\lambda$2798 of quasar spectra into the optical domain. The SED is characteristic of blazars, but with a dominant IC component over the Synchrotron part. \\\\\n{\\bf 3FGL J1031.0+7440} was observed in Asiago, with the 1.82m telescope. It shows the prominent emission lines of a Seyfert 1 galaxy, with a full width at half the maximum ${\\rm FWHM}({\\rm H}\\beta) = 2286 \\pm 350\\, {\\rm km\\, s^{-1}}$. In a standard $\\Lambda$CDM cosmology with $H_0 = 70\\, {\\rm km\\, s^{-1}\\, Mpc^{-1}}$, $\\Omega_\\Lambda = 0.7$ and $\\Omega_M = 0.3$, its redshift corresponds to a distance of 569.8~Mpc. From an apparent magnitude $V = 17.2$, we infer an absolute magnitude $M_V = -21.6$, placing this object on the border between faint quasar and bright Seyfert 1 activity. The blazar SED is accompanied by a small radiation excess in the optical domain, suggesting that the jet power and the thermal contribution from the central engine are comparable in this object. \\\\\n{\\bf 3FGL J1315.4+1130}, also detected by the SDSS, is characterized by the BLL power law continuum and by faint absorption lines. Identification of the strongest features as a Ca~II doublet is consistent with the presence of an absorption feature at the predicted wavelength of Mg~I. Fewer data points are available to reconstruct the SED of this source and we do not appreciate any deviation from a two component blazar SED. \\\\\n{\\bf 3FGL J1412.0+5249} is detected by the SDSS and it shows the characteristics of an elliptical galaxy. Its counterpart is actually a giant elliptical located in a galaxy cluster. The associated SED is the most complex of this sample, featuring a strong optical excess, emitted by the bright host galaxy, and a high energy IC component that is hardly reproduced by SSC models. \\\\\n\n\\begin{table}[t]\n\\caption{SSC model parameters. The table columns report, respectively, the 3FGL source name, the electron energy distribution power-law index before break $\\alpha_{el}^{(1)}$, the electron distribution index after break $\\alpha_{el}^{(2)}$, the logarithm of the break energy (in units of $m_e c^2$), the magnetic field $B$ (expressed in Gauss), the Doppler factor $\\delta$, the jet radius expressed in parsec, and the reduced residuals.}\n\\begin{footnotesize}\n\\begin{center}\n\\begin{tabular}{cccccccc}\n\\hline\n\\hline\nId. & $\\alpha_{el}^{(1)}$ & $\\alpha_{el}^{(2)}$ & $\\log E_{break}$ & $B$ & $\\delta$ & $R_{jet}$ & ${\\chi_{red}^2}^{\\rm a}$ \\\\\n\\hline\n3FGL J$0134.5+2638$ & 1.5 & 4.7 & 4.0 & 1.00 & 10 & 0.001 & 1.180 \\\\\n3FGL J$0339.2-1738$ & 1.5 & 5.0 & 4.0 & 0.75 & 15 & 0.001 & 1.097 \\\\\n3FGL J$0904.3+4240$ & 2.3 & 3.6 & 3.5 & 0.05 & 30 & 0.003 & 1.455 \\\\\n3FGL J$1031.0+7440$ & 1.8 & 5.0 & 4.1 & 1.00 & 15 & 0.001 & 1.339 \\\\\n3FGL J$1315.4+1130$ & 1.6 & 4.8 & 4.8 & 0.60 & 20 & 0.002 & 1.732 \\\\\n3FGL J$1412.0+5249$ & 1.5 & 4.7 & 4.0 & 1.00 & 10 & 0.001 & 1.889 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n$^{\\rm a}$ Residuals of SSC models are computed without taking into account the thermal excess data points.\n\\end{footnotesize}\n\\end{table}\n\\section{Conclusions}\nIn this study we presented the optical spectra of six targets that have been associated to $\\gamma$-ray sources of still undetermined type in 3LAC. In some cases, the spectra of this type of objects can still be obtained from large spectroscopic surveys, such as the SDSS, above all, but, in others, we need specifically planned observations. The increase of sensitivity and resolution achieved by the Fermi-LAT at $\\gamma$-ray energies is now providing better opportunities to identify candidate counterparts to extra-Galactic $\\gamma$-ray emission and, therefore, to improve the selection of targets.\n\nWith an improved ability to detect $\\gamma$-rays from faint sources we can now investigate the occurrence of nuclear activity on different power scales. The detection of faint blazar-like activity in low luminosity AGNs or even apparently normal galaxies opens a new window on the demographics of $\\gamma$-ray sources, as well as on the mechanisms that contribute to black hole growth and jet formation. The possibility that such objects may represent an important contribution to the $\\gamma$-ray radiation of undetermined origin deserves further investigation. Searching for radiation from hidden AGNs is a fundamental science case for instruments designed to observe high energy photons and other hints of jet activity, such as light polarization. Therefore, we plan to further investigate the spectroscopic properties of candidate counterparts to $\\gamma$-ray emission, taking possibly into account light polarization studies as well, through an extensive observational campaign designed for middle class telescopes.\n\n\\section*{Acknowledgements}\nThe \\textit{Fermi}-LAT Collaboration acknowledges support for LAT development, operation and data analysis from NASA and DOE (United States), CEA\/Irfu and IN2P3\/CNRS (France), ASI and INFN (Italy), MEXT, KEK, and JAXA (Japan), and the K.A.~Wallenberg Foundation, the Swedish Research Council and the National Space Board (Sweden). Science analysis support in the operations phase from INAF (Italy) and CNES (France) is also gratefully acknowledged.\n\nThis work is based on observations collected at Copernico (or\/and Schmidt) telescope(s) (Asiago, Italy) of the INAF - Osservatorio Astronomico di Padova. Part of this work is based on archival data, software or online services provided by the ASI SCIENCE DATA CENTER (ASDC).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nPositive feedback is ubiquitous in biochemical networks and can lead to a bifurcation from a monostable to a bistable cellular state \\cite{mitrophanov2008positive, tkavcik2012optimizing, das2009digital, vogel2016dichotomy}. Near the bifurcation point, the bistable state often reflects a choice between two accessible but opposing cell fates. For example, in T cells, the distribution of doubly phosphorylated ERK (ppERK) can be bimodal \\cite{vogel2016dichotomy}. ppERK is a protein that initiates cell proliferation and is implicated in the self\/non-self decision between mounting an immune response or not \\cite{vogel2016dichotomy, altan2005modeling}.\n\nThe bifurcation point is similar to an Ising-type critical point in physical systems such as fluids, magnets, and superconductors, where a disordered state transitions to one of two ordered states at a critical temperature \\cite{goldenfeld1992lectures}. In fact, universality tells us that the two should not just be similar, they should be the same: because they are both bifurcating systems, both types of systems should exhibit the same critical scaling exponents and therefore belong to the same universality class \\cite{goldenfeld1992lectures}. Although this powerful idea has allowed diverse physical phenomena to be united into specific behavioral classes, the application of universality to biological systems is still developing \\cite{mora2011biological, munoz2018colloquium, salman2012universal, brenner2015universal, pal2014non, ridden2015entropy, qian2016framework, hidalgo2014information}.\n\nBiological tools such as flow cytometry, fluorescence microscopy, and RNA sequencing allow reliable experimental estimates of abundance distributions, inspiring researchers to seek to apply insights from statistical physics to biological data. In particular, recent studies have demonstrated that biological systems on many scales, from molecules \\cite{mora2010maximum}, to cells \\cite{kastner2015critical, krotov2014morphogenesis, de2017critical, chen2012scale, aguilar2018critical, wan2018time}, to populations \\cite{bialek2014social, attanasi2014finite, cavagna2017dynamic}, exhibit signatures consistent with physical systems near a critical point. However, some of these studies have come under scrutiny because some of the signatures, particularly scaling laws, can arise far from or independent of a critical point \\cite{schwab2014zipf, touboul2017power, newman2005power}. Part of the problem is that the identification of appropriate scaling variables from data can be ambiguous, and one is often left looking for scaling relationships in an unguided way. \n\nTypical approaches to the interpretation of abundance distributions include fitting to either detailed mechanistic models of the underlying reaction scheme, or to an effective description of the data such as a Gaussian or lognormal mixture model. The former approach is usually difficult to parameterize and difficult to generalize to other systems. The latter approach often suffers from numerical issues (the likelihood is unbounded and the expectation-maximization algorithm can lead to spurious solutions \\cite{biernacki2003choosing}). Moreover, the vicinity of a bifurcation point is precisely where a mixture analysis is most likely to fail. In contrast, mapping to a statistical physics framework is expected to be universal, in the sense that the precise microscopic details of a broad range of biochemical models are unimportant near the bifurcation point, as they are coarse-grained rather than particular reaction parameters.\n\nHere we provide a framework for mapping well-mixed stochastic models of biochemical feedback to the mean-field Ising model and apply it to published data on T cells. This allows us to extract effective thermodynamic quantities from experimental data without needing to fit to a parametric model of the system. This makes the theory applicable to a broad class of biological datasets without worrying about model selection or goodness-of-fit criteria. The theory provides insights on how T cells respond to drugs and reveals distinctions between one type of drug response and another. Furthermore, we find that one of the thermodynamic quantities (the heat capacity) provides a novel way to estimate absolute molecule number from fluorescence level in bifurcating systems. We demonstrate that our results can be extended to cases where feedback is indirect and discuss further extensions, including to spatiotemporal dynamics.\n\n\n\\section{Results}\n\nWe consider a reaction network in a cell where $X$ is the molecular species of interest, and the other species $A$, $B$, $C$, etc.\\ form a chemical bath for $X$ [Fig.\\ \\ref{fig:setup}(a)]. The reactions of interest produce or degrade an $X$ molecule, can involve the bath species, and in principle are reversible. We allow for nonlinear feedback on $X$, meaning that the production of an $X$ molecule in a particular reaction might require a certain number of $X$ molecules as reactants. This leads to an arbitrary number of reactions of the form\n\\begin{equation}\n\\label{eq:rxns}\nj_rX + Y_r^+ \\xrightleftharpoons[k_r^-]{k_r^+} (j_r+1)X + Y_r^-,\n\\end{equation}\nwhere in the $r$th reaction, $j_r$ are stoichiometric integers describing the nonlinearity, $k_r^\\pm$ are the forward ($+$) and backward ($-$) reaction rates, and $Y_r^\\pm$ represent bath species involved as reactants ($+$) or products ($-$). A simple and well-studied special case of Eq.\\ \\ref{eq:rxns} is Schl\\\"ogl's second model \\cite{schlogl1972chemical, dewel1977renormalization, nicolis1980systematic, brachet1981critical, grassberger1982phase, prakash1997dynamics, liu2007quadratic, vellela2009stochastic}, in which $X$ is either produced spontaneously from bath species $A$, or in a trimolecular reaction from two existing $X$ molecules and bath species $B$ (i.e., $R = 2$, $j_1 = 0$, $j_2 = 2$, $Y_1^+ = A$, $Y_2^+ = B$, and $Y_1^- = Y_2^- = \\emptyset$).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fig1}\n\\caption{Setup and behavior of the model. (a) We consider well-mixed stochastic biochemical networks described by an effective feedback function $f_n$. (b) Feedback produces either one or two stable steady states. (c) The molecule number distribution is peaked around these states or flat at the bifurcation point. (d) Mapping to the Ising model reveals that the effective reduced temperature drives the distribution to the unimodal ($\\theta > 0$) or bimodal ($\\theta < 0$) state (see c), while the effective field $h$ biases the distribution toward high ($h > 0$) or low ($h < 0$) molecule number. Parameters: $H=3$ and $n_c = 100$ in b, c, and d; $h = 0$ in b and c; and $\\theta = 0$ in d (see also Appendix \\ref{app:param}).}\n\\label{fig:setup}\n\\end{figure}\n\nWe assume that molecules are well-mixed and that the numbers of bath molecules are constant. The latter assumption is equivalent to integrating out all species but $X$, such that the feedback on $X$ arises directly from $X$ itself (Eq.\\ \\ref{eq:rxns}). However, in general the feedback will be indirect, with $X$ regulating dynamic species in the bath that in turn regulate $X$ (this is almost certainly the case in the T cells we study here). Therefore, we consider this more general case later in Section \\ref{sec:indirect} and show that the results discussed below remain unchanged.\n\nThe master equation for the probability of observing $n$ molecules of species $X$ according to Eq.\\ \\ref{eq:rxns} is\n\\begin{equation}\n\\label{eq:me}\n\\dot{p}_n = b_{n-1}p_{n-1} + d_{n+1}p_{n+1} - (b_n + d_n)p_n,\n\\end{equation}\nwhere $b_n = \\sum_{r=1}^R J_{rn}^+$ and $d_n = \\sum_{r=1}^R J_{rn}^-$ are the total birth and death propensities, and $J_{rn}^+ = k_r^+ n_r^+ n!\/(n-j_r)!$ and $J_{rn}^- = k_r^- n_r^- n!\/(n-j_r-1)!$ are the forward and backward propensities of each reaction pair. Here $n_r^\\pm$ are the numbers of molecules of the bath species involved in reaction $r$, and the factorials account for the number of ways that $X$ molecules can meet in a reaction. The steady state of Eq.\\ \\ref{eq:me} is \\cite{van1992stochastic, gardiner1985handbook}\n\\begin{equation}\n\\label{eq:pn}\np_n = p_0 \\prod_{j = 1}^n \\frac{b_{j-1}}{d_j} = \\frac{p_0}{n!} \\prod_{j=1}^n f_j,\n\\end{equation}\nwhere $p_0^{-1} = \\sum_{n=0}^\\infty(1\/n!)\\prod_{j=1}^n f_j$ is set by normalization. In the second step of Eq.\\ \\ref{eq:pn} we define an effective birth propensity $f_n \\equiv nb_{n-1}\/d_n$ corresponding to spontaneous death with propensity $n$ [Fig.\\ \\ref{fig:setup}(a)]. In general, $f_n$ is an arbitrary, nonlinear feedback function governed by the reaction network. For the Schl\\\"ogl model, it is $f_n = [aK^2 + s(n-1)(n-2)]\/[(n-1)(n-2)+K^2]$, where we have introduced the dimensionless quantities $a \\equiv k_1^+n_A\/k_1^-$, $s \\equiv k_2^+ n_B\/k_2^-$, and $K^2 \\equiv k_1^-\/k_2^-$. As a ubiquitous example we also consider the Hill function $f_n = a + sn^H\/(n^H+K^H)$ with coefficient $H$. Importantly, the inverse of Eq.\\ \\ref{eq:pn},\n\\begin{equation}\n\\label{eq:fn}\nf_n = \\frac{np_n}{p_{n-1}},\n\\end{equation}\nallows calculation of the feedback function from the distribution \\cite{walczak2009stochastic}, as utilized when analyzing the experimental data later in Section \\ref{sec:immune}.\n\nThe quantity $f_n-n$ determines the dynamic stability: there can be either one or two stable states $n_*$ [Fig.\\ \\ref{fig:setup}(b)], and the transition from a monostable to a bistable regime occurs at a bifurcation point [Fig.\\ \\ref{fig:setup}(c) inset]. These deterministic regimes correspond stochastically to unimodal and bimodal distributions $p_n$, respectively, with maxima at $n_*$, while the bifurcation point corresponds to a distribution that is flat on top [Fig.\\ \\ref{fig:setup}(c)].\n\n\n\\subsection{Ising mapping and scaling exponents}\n\nTo understand the scaling behavior near the bifurcation point, we expand the stability condition $f_{n_*}-n_*=0$ to third order around a point $n_c$ satisfying $f''_{n_c} = 0$. This choice of $n_c$ eliminates the quadratic term in the dynamic forcing $f_n-n$, equivalent to eliminating the cubic term in an effective potential as in Ginzburg--Landau theory \\cite{kopietz2010introduction}. Defining the parameters\n\\begin{equation}\n\\label{eq:cparam}\nm \\equiv \\frac{n_*-n_c}{n_c}, \\quad\nh \\equiv \\frac{2(f_{n_c} - n_c)}{-f'''_{n_c}n_c^3}, \\quad\n\\theta \\equiv \\frac{2(1-f'_{n_c})}{-f'''_{n_c}n_c^2},\n\\end{equation}\nthe expansion $f_{n_c} + f'_{n_c}(n_*-n_c) + f'''_{n_c}(n_*-n_c)^3\/3! - n_* = 0$ becomes $h - \\theta m - m^3\/3 = 0$. This expression is equivalent to the expansion of the Ising mean field equation $m = \\tanh[(m+h)\/(1+\\theta)]$ for small magnetization $m$, where $\\theta = (T-T_c)\/T_c$ is the reduced temperature, and $h$ is the dimensionless magnetic field \\cite{kopietz2010introduction}. Therefore, in our system we interpret $m$ as the order parameter, $\\theta$ as an effective reduced temperature, and $h$ as an effective field. Explicit expressions for $n_c$, $\\theta$, and $h$ in terms of the biochemical parameters and vice versa are given for the Schl\\\"ogl and Hill models in Appendix \\ref{app:param}.\n\nWe see in Fig.\\ \\ref{fig:setup}(c) and (d) that $n_c$ determines where the distribution is centered, that $\\theta$ drives the system to the unimodal ($\\theta > 0$) or bimodal ($\\theta < 0$) state, and that $h$ biases the system to high ($h > 0$) or low ($h < 0$) molecule numbers. Note that unlike in the Ising model, even when $h=0$ an asymmetry persists between the high and low states [see the purple distribution in Fig.\\ \\ref{fig:setup}(c)]. The reason is that in the master equation (Eq.\\ \\ref{eq:me}), unlike in Ginzburg--Landau theory, fluctuations scale with molecule number, such that the high state is wider than the low state.\n\nThe equivalence between our system and the Ising mean-field equation near the critical point (Eq.\\ \\ref{eq:cparam}) implies that our system has the same scaling exponents $\\beta=1\/2$, $\\gamma=1$, and $\\delta=3$ as the Ising universality class in its mean-field limit \\cite{kopietz2010introduction}. For completeness, we verify in Appendix \\ref{app:scalings} that these scalings are indeed obeyed by the Schl\\\"ogl and Hill models.\n\nHowever, Eq.\\ \\ref{eq:cparam} does not explicitly determine the value of the exponent $\\alpha$. The reason is that, unlike $\\beta$, $\\gamma$, and $\\delta$, the exponent $\\alpha$ depends on the entire distribution $p_n$, not just the maxima. Specifically, $\\alpha$ concerns the heat capacity, $C|_{h=0}\\sim |\\theta|^{-\\alpha}$, which depends on the entropy $S$ and thus $p_n$. The equilibrium definition $C = T\\partial_T S$ generalizes to a nonequilibrium system like ours when one uses the Shannon entropy $S = -k_{\\rm B}\\sum_n p_n \\log p_n$ \\cite{mandal2013nonequilibrium}. Since $T = (1+\\theta)T_c$, we have $C = (1+\\theta)\\partial_\\theta S$, or\n\\begin{equation}\n\\label{eq:C}\n\\frac{C}{k_{\\rm B}} = -(1+\\theta)\\sum_{n=0}^\\infty p_n (1+\\log p_n) \\Bigg( \\psi_n - \\sum_{j=0}^\\infty p_j \\psi_j \\Bigg),\n\\end{equation}\nwhere $\\psi_n \\equiv (1\/2)f'''_{n_c}n_c^2\\sum_{j=1}^n(j-n_c)\/f_j$. Eq.\\ \\ref{eq:C} follows from performing the $\\theta$ derivative using the expression in Eq.\\ \\ref{eq:pn}, the expansion below Eq.\\ \\ref{eq:cparam}, and the definition of $\\theta$ (Eq.\\ \\ref{eq:cparam}). We see in Fig.\\ \\ref{fig:heat1}(a) that when $h=0$, $C$ exhibits a minimum at $\\theta^*$. We see in Fig.\\ \\ref{fig:heat1}(b) that $\\theta^*$ vanishes as the system size increases, $n_c\\to\\infty$. This implies that $C|_{h=0}\\sim |\\theta|^0$ to sub-quadratic order in $\\theta$, or $\\alpha = 0$, again consistent with the Ising universality class in its mean-field limit. Interestingly, whereas $C$ is discontinuous in the mean-field Ising model \\cite{kopietz2010introduction} and constant in the van der Waals model of a fluid \\cite{goldenfeld1992lectures}, it is minimized here; nevertheless, in all cases $\\alpha = 0$. Note from Fig.\\ \\ref{fig:heat1}(a) that $C$ is negative near $\\theta = 0$; negative heat capacity is a well-known feature of nonequilibrium steady states \\cite{zia2002getting, boksenbojm2011heat, bisquert2005master}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fig2}\n\\caption{(a) Heat capacity (Eq.\\ \\ref{eq:C}) is minimized at the bifurcation point, corresponding to exponent $\\alpha = 0$. (b) The location of the minimum approaches $\\theta^*\\to0$ as $n_c\\to\\infty$, as expected. Parameters: $H=3$, $n_c = 500$, and $h=0$.}\n\\label{fig:heat1}\n\\end{figure}\n\n\n\n\\subsection{Application to immune cell data}\n\\label{sec:immune}\n\nTo demonstrate the utility of our theory, we apply it to published data from T cells \\cite{vogel2016dichotomy}. In these experiments, chemotherapy drugs inhibit the enzymes MEK and SRC in the biochemical networks of the cells. The inhibition results in bimodal (low dose) or unimodal (high dose) distributions of ppERK abundance, which is measured as fluorescence intensity $I$ by flow cytometry. The distributions are shown for a range of drug doses in Fig.\\ \\ref{fig:experiment}(a) and (b) (the insets show distributions of log intensity for clarity). Experimental details are given in the original publication \\cite{vogel2016dichotomy} and are summarized in Appendix \\ref{app:expt}, along with the drugs and dose amounts.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fig3}\n\\caption{Application of the theory to immune cell data. Upon administration of either (a) MEK or (b) SRC inhibitor, experimental distributions of T cell ppERK fluorescence intensity are unimodal (bimodal) for high (low) doses. Insets show distributions of log intensity for clarity. (c, d) Feedback functions calculated from the experimental distributions correspondingly exhibit either one or two stable states. (e-g) Effective thermodynamic quantities calculated from the data vary with drug dose in distinct ways for each drug. The results in (c-g) corroborate those in \\cite{vogel2016dichotomy}, but with a much simpler framework that has three parameters instead of five and requires no fitting or prior biological knowledge of the system. Error bars: standard error from filter windows $25 \\le W \\le 35$ (see Appendix \\ref{app:analysis}).}\n\\label{fig:experiment}\n\\end{figure}\n\nFirst, we compute the feedback function $f$ from each distribution using Eq.\\ \\ref{eq:fn} (see Appendix \\ref{app:analysis}). Fig.\\ \\ref{fig:experiment}(c) and (d) show the corresponding forcing functions [compare to Fig.\\ \\ref{fig:setup}(b)]. As expected, in each case we see that the forcing function transitions from two stable states to one stable state as the drug is applied.\n\nThen, we compute $I_c$ (the analog of $n_c$ in units of fluorescence intensity), $\\theta$, and $h$ from the feedback function using Eq.\\ \\ref{eq:cparam} (see Appendix \\ref{app:analysis}). These quantities are shown as a function of drug dose in Fig.\\ \\ref{fig:experiment}(e)-(g). We see that the behavior is different depending on whether MEK inhibitor (MEKi) or SRC inhibitor (SRCi) is applied. Specifically, MEKi decreases $I_c$, increases $\\theta$, and decreases $h$; whereas SRCi only decreases $h$, leaving the other quantities unchanged. Thus, the effective thermodynamic quantities can differentiate cellular responses to different perturbations, such as the application of different drugs.\n\nFurthermore, the mapping provides an intuitive interpretation of the drug responses. MEKi causes a transition from a bimodal to a unimodal state in the expected way: by increasing the reduced temperature $\\theta$ from a negative to a positive value [Fig.\\ \\ref{fig:experiment}(f)]. In the process, $I_c$ decreases [Fig.\\ \\ref{fig:experiment}(e)], meaning that the unimodal state is shifted to lower molecule number, near the lower mode of the bimodal state [Fig.\\ \\ref{fig:experiment}(a) inset]. In contrast, SRCi causes a transition from a bimodal to a unimodal state in a different way: by decreasing the field while leaving $\\theta$ and $I_c$ unchanged [Fig.\\ \\ref{fig:experiment}(e)-(g)]. In essence, the distribution remains bimodal and unshifted, except that the field causes the high mode to diminish in weight [Fig.\\ \\ref{fig:experiment}(b) inset]. Interestingly, the mean dose-response curves are similar for the two drugs \\cite{vogel2016dichotomy}, but our mapping elucidates precisely how the transitions are different at the distribution level. Related conclusions were drawn in \\cite{vogel2016dichotomy}, but those conclusions relied on fitting the distributions to a five-parameter Gaussian mixture model, which is expected to fail near the bifurcation point. Here we use only three parameters and no fitting, and we emerge with an intuitive interpretation in terms of thermodynamic quantities.\n\nFinally, we note that for both drugs the effective field is negative at all doses [Fig.\\ \\ref{fig:experiment}(g)]. The reason is that the fluorescence distributions have long tails (which is why they are often easier to visualize in log space); see Fig.\\ \\ref{fig:experiment}(a) and (b). In the theory, a long tail is indistinguishable from a low-molecule-number bias in the peak, which corresponds to $h < 0$. We address the possible origins and implications of the long tails in the Discussion (Section \\ref{sec:discussion}).\n\n\n\\subsection{Estimation of molecule number}\n\nWe now apply the theory to compute the heat capacity from the T cell data.\nSpecifically, we compute $C$ using Eq.\\ \\ref{eq:C} (see Appendix \\ref{app:analysis}) for all drugs and doses used in the experiments \\cite{vogel2016dichotomy} (Appendix \\ref{app:expt}). Unlike the other thermodynamic quantities, $C$ requires a conversion from fluorescence intensity to molecule number because it depends explicitly on the distribution $p_n$ (Eq.\\ \\ref{eq:C}). Therefore we compute $C$ for various values of the conversion factor $I_1$, where $n = I\/I_1$. The results are shown in Fig.\\ \\ref{fig:heat2}. We see that irrespective of $I_1$ over four orders of magnitude, the data closest to $h=0$ (yellow) exhibit a global minimum in $C$ at $\\theta = 0$, as expected from Fig.\\ \\ref{fig:heat1}(a). However, we also see that the depth of the minimum agrees with that of the theory only for the particular choice $I_1 \\approx 0.1$ [Fig.\\ \\ref{fig:heat2}(c)].\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fig4}\n\\caption{Estimation of molecule number by comparing heat capacity between theory and experiments. (a-d) Rough estimate of fluorescence-to-molecule-number conversion factor $I_1$ (see titles) obtained by comparing depths of theory and experimental minima. ``Hill'' refers to the theoretical curve produced by Hill-function feedback as in Eq.~\\ref{eq:Hill}. Different symbols correspond to different drugs. See Appendix \\ref{app:expt} for drugs (shape) and doses (size). (e) More precise estimate obtained from plotting sum of squared errors (SSE) for data within $-\\Delta\\theta \\le \\theta \\le \\Delta\\theta$ and fitting to parabola (see Appendix D for details). Here $\\Delta\\theta = 0.05$. (f) Estimate is insensitive to value of $\\Delta\\theta$. Theory parameters: $H = 4$, $h = 0$, and $n_c = \\bar{I}_c\/I_1$, where $\\bar{I}_c = 730$ is the average value across all experiments.}\n\\label{fig:heat2}\n\\end{figure}\n\nTo obtain a more precise estimate of $I_1$, we plot the sum of squared errors between the data and the theory as a function of $I_1$ in Fig.\\ \\ref{fig:heat2}(e). We focus on the bifurcation region by considering only values of $\\theta$ within $-\\Delta\\theta \\le \\theta \\le \\Delta\\theta$, and we find that our results are not sensitive to the choice of $\\Delta\\theta$ [Fig.\\ \\ref{fig:heat2}(f)]. This procedure (see the details in Appendix \\ref{app:analysis}) results in an estimate of $I_1 = 0.5$ $\\pm$ $0.2$, as seen in Fig.\\ \\ref{fig:heat2}(f). This value of $I_1$ corresponds to $\\bar{n}_*=$ 170,000 $\\pm$ 70,000 ppERK molecules in the high mode averaged across all cases with no inhibitor. It is possible to compare this value with previous measurements on these cells. In two separate experiments, it was estimated that there are approximately 100,000 \\cite{altan2005modeling} and 214,000 \\cite{hukelmann2016cytotoxic} ERK molecules per cell, and that only about 50\\% of these molecules are doubly phosphorylated during T cell receptor activation \\cite{altan2005modeling} (see Appendix \\ref{app:analysis}). These considerations give a range of roughly 50,000$-$107,000 ppERK molecules, which is consistent with our estimate of 170,000 $\\pm$ 70,000. The agreement is especially notable given that T cell protein abundances generally span six orders of magnitude, from tens to tens of millions of molecules per cell \\cite{hukelmann2016cytotoxic}.\n\nWhy does the heat capacity extract the conversion between fluorescence intensity and molecule number? As mentioned above, $\\alpha$ is the only exponent that is a function of $p_n$ instead of just its maxima. This means that the plot of $C$ vs.\\ $\\theta$ contains information not only about means or modes, but also about fluctuations. The notion that fluctuation information is essential for converting from intensity to molecule number can be seen with a simpler example: a Poisson distribution. Here we would have $\\sigma_I^2\/\\bar{I}^2 = \\sigma_n^2\/\\bar{n}^2 = 1\/\\bar{n} = I_1\/\\bar{I}$. From this relation it is clear that information about not only the mean ($\\bar{I}$) but also the fluctuations ($\\sigma_I^2$) in intensity is necessary and sufficient to infer the conversion factor $I_1$. In our case, the heat capacity is extracting similar information, but for a bifurcating system.\n\n\n\\subsection{Generalization to indirect feedback}\n\\label{sec:indirect}\n\nIn the T cells, it is well known that ppERK does not apply feedback to its own activation directly, but rather indirectly via upstream components \\cite{vogel2016dichotomy, shin2009positive, altan2005modeling}. Therefore, we seek to determine the extent to which the above results are sensitive to our assumption in the theory that the feedback is direct. To this end, we construct a minimal extension of the model in Eq.\\ \\ref{eq:rxns} in which the feedback is indirect:\n\\begin{align}\n& \\emptyset \\xrightleftharpoons[k_2]{k_1} X, \\qquad\n2X \\xrightleftharpoons[k_4]{k_3} D, \\nonumber \\\\\n& D \\xrightarrow{k_5} D + A, \\qquad\nA \\xrightarrow{k_6} A + X, \\qquad\nA \\xrightarrow{k_7} \\emptyset, \\nonumber \\\\\n\\label{eq:indirect}\n& D \\xrightarrow{k_8} D + B, \\qquad\nB + X \\xrightarrow{k_9} B, \\qquad\nB \\xrightarrow{k_{10}} \\emptyset.\n\\end{align}\nHere $X$ is produced, is degraded, and reversibly dimerizes (first line); the dimer $D$ produces a species $A$ that produces $X$ and is degraded (second line); and the dimer also produces a species $B$ that degrades $X$ and is degraded (third line). Eq.\\ \\ref{eq:indirect} is an extension of Eq.\\ \\ref{eq:rxns} because there are multiple stochastic variables ($X$, $D$, $A$, and $B$), there are irreversible reactions, and $X$ feeds back on itself indirectly through $D$, $A$, and $B$ instead of directly.\n\nThe deterministic steady state of Eq.\\ \\ref{eq:indirect} is\n\\begin{equation}\n\\label{eq:multidet}\n0 = \\dot{n}\/k_2 = c_0 - n_* + c_2 n_*^2 - c_3 n_*^3,\n\\end{equation}\nwhere $c_0 \\equiv k_1\/k_2$, $c_2 \\equiv k_3k_5k_6\/(k_2k_4k_7)$, $c_3 \\equiv k_3k_8k_9\/(k_2k_4k_{10})$, and the molecule numbers of $D$, $A$, and $B$ have been eliminated in favor of $n_*$ by setting their own time derivatives to zero. Because Eq.\\ \\ref{eq:multidet} is cubic in $n_*$, we see immediately that it has the same form as the expanded Ising mean field equation $h - \\theta m - m^3\/3 = 0$ (see Eq.\\ \\ref{eq:cparam}). Specifically, defining $m = (n_*-n_c)\/n_c$ as in Eq.\\ \\ref{eq:cparam}, the choice $n_c = c_2\/(3c_3)$ eliminates the term quadratic in $m$ and implies $\\theta = 3c_3\/c_2^2 - 1$ and $h = 9c_0c_3^2\/c_2^3 - 3c_3\/c_2^2 +2\/3$. It immediately follows that this model has the same exponents $\\beta=1\/2$, $\\gamma=1$, and $\\delta=3$ as the mean-field Ising universality class.\n\nTo test whether the heat capacity for this model exhibits the same features as that for the direct feedback model in Fig.\\ \\ref{fig:heat1}(a), we compute the steady state marginal distribution $p_n$ using stochastic simulations \\cite{gillespie1977exact} of Eq.\\ \\ref{eq:indirect}. Specifically, we set $k_3\/k_4 = 1\/n_c$ and $k_5\/k_7 = k_8\/k_{10} = 1$ to ensure that the numbers of $D$, $A$, and $B$ molecules, respectively, are on the order of $n_c$. We then set $k_4\/k_2 = k_7\/k_2 = k_{10}\/k_2 = \\rho$, where $\\rho$ is a free parameter that determines whether the degradation timescales of $D$, $A$, and $B$, respectively, are faster ($\\rho > 1$) or slower ($\\rho < 1$) than that of $X$. These conditions, along with the definitions of $n_c$, $\\theta$, and $h$ above, constitute nine equations for nine reaction rates, plus $k_2$ which sets the units of time. Solving these equations yields expressions for the rates in terms of $n_c$, $\\theta$, $h$, and $\\rho$ that we use in the simulations.\n\nFig.\\ \\ref{fig:indirect}(a) shows the heat capacity $C$ as a function of $\\theta$ for $h = 0$, $n_c = 100$, and $\\rho = \\{0.1, 1, 10\\}$, where $C = (1+\\theta)\\partial_\\theta S$ is computed from the entropy $S = -k_{\\rm B}\\sum_n p_n \\log p_n$ by numerical derivative. We see that for all $\\rho$ values, the curves exhibit a minimum at $\\theta = 0$, implying $\\alpha = 0$, and they rise more steeply for negative than for positive $\\theta$ as in Fig.\\ \\ref{fig:heat1}(a).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fig5}\n\\caption{Verification that indirect feedback does not qualitatively change modeling assumptions or results. (a) $C$ and $\\theta$ calculated from extended model with indirect feedback. (b) $C$ and $\\theta$ inferred assuming the feedback is direct (Eq.\\ \\ref{eq:fn}). Compare with Fig.\\ \\ref{fig:heat1}(a). Parameters: $n_c=100$ and $h=0$.}\n\\label{fig:indirect}\n\\end{figure}\n\nWe then investigate whether Eq.\\ \\ref{eq:rxns} remains valid as a coarse-grained description of the extended model in Eq.\\ \\ref{eq:indirect}. To answer this question, we infer values of $n_c$, $\\theta$, $h$, and $C$ directly from the simulation data $p_n$ using the same protocol as for the experimental data. That is, we compute $f_n$ via Eq.\\ \\ref{eq:fn}, and then compute $\\theta$, $h$, and $C$ from its derivatives at $n_c$ according to Eqs.\\ \\ref{eq:cparam} and \\ref{eq:C}, where $n_c$ satisfies $f''_{n_c} = 0$. As with the experimental data (see Appendix \\ref{app:analysis}), derivatives are calculated using a Savitsky-Golay filter \\cite{savitzky1964smoothing}, although here we apply the filter directly to $f_n$ and perform the analysis directly in $n$ space, not log space.\n\nFig.\\ \\ref{fig:indirect}(b) shows the result of this procedure for the inferred heat capacity $C$ as a function of the inferred $\\theta$. We see that, as with the exact $C$ and $\\theta$ [Fig.\\ \\ref{fig:indirect}(a)], the data exhibit a minimum at $\\theta = 0$ and rise more steeply for negative than for positive $\\theta$. Note that the values of $C$ and $\\theta$ are different in (a) and (b), which is expected because the shape of $p_n$ is not quantitatively the same in the two models of Eqs.\\ \\ref{eq:rxns} and \\ref{eq:indirect}; nonetheless, the shape of the $C$ vs.\\ $\\theta$ curves remains the same. We have checked that the inferred values of $n_c$ and $h$ are distributed around their known values of $100$ and $0$, respectively, and that the shape persists across a range of filter window sizes.\n\nThese results suggest that the main findings above are not sensitive to our assumption that feedback is direct, and therefore that we are justified in using Eq.\\ \\ref{eq:rxns} as a coarse-grained model to analyze the T cell data.\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nWe have employed the fact that a feedback-induced bifurcation exhibits the scaling properties of the mean-field Ising universality class to provide a simple prescription for modeling and analyzing biological data. Contrary to existing mixture-model approaches, our method is most valuable near the bifurcation point, which is where biologically significant cell-fate decisions are expected to take place. Our approach provides the effective order parameter, reduced temperature, magnetic field, and heat capacity from experimental distributions without fitting or needing to know the molecular details. By applying the approach to T cell flow cytometry data, we discovered that these quantities discriminate between cellular responses in an intuitive, interpretable way, and that the heat capacity allows estimation of the molecule number from fluorescence intensity for a bifurcating system. By generalizing the theory to include indirect feedback, we demonstrated the capacity to model realistic signaling cascades where indirect feedback is common. Our approach should be applicable to other systems observed to undergo a pitchfork-like bifurcation and the associated unimodal-to-bimodal transition in abundance distributions, but not to systems which have an absorbing or extinction state, as they are expected to fall under a different universality class \\cite{ohtsuki1987nonequilibrium, grassberger1978reggeon}.\n\nThe theory assumes only birth-death reactions and neglects more complex mechanisms such as bursting \\cite{friedman2006linking, mugler2009spectral} or parameter fluctuations \\cite{shahrezaei2008colored, horsthemke1984noise}. These mechanisms are known to produce long tails and may be responsible for the long tails observed in the experimental data [Fig.\\ \\ref{fig:experiment}(a) and (b)]. Cell-to-cell variability (CCV) may also contribute to the long tails, as it is known to be present in T cell populations \\cite{cotari2013cell}. Our theory neglects CCV and instead assumes that the distribution of molecule numbers across the population is the same as that traced out by a single cell over time. Although CCV may play an important role, one generically expects the role of intrinsic fluctuations to be amplified near a critical point, and models that ignore CCV have been shown to be sufficient to explain both the bimodality \\cite{das2009digital} and variance properties \\cite{prill2015noise} of ppERK in T cells. Moreover, the fact that our theory provides an estimate of the molecule number that is consistent with other estimates suggests that intrinsic fluctuations play a large role. Distinguishing between intrinsic fluctuations and long-lived CCV is an important topic for future work.\n\nOur work provides key tools that can be used for a broader exploration of biological systems. The approach is applicable to any experimental dataset that exhibits unimodal and bimodal abundance distributions, and could lead to a unified picture of diverse cell types and environmental perturbations in terms of effective thermodynamic quantities. At the same time, several extensions of our work are natural. For example, the dynamics of the theory could be probed to investigate the consequences of critical slowing down for driven or dynamically perturbed systems with feedback. Alternatively, the theory could be generalized to systems that are not well-mixed, such as intracellular compartments or communicating populations, to investigate space-dependent universal behavior and its biological implications.\n\n\n\n\\section{Data availability}\nData and code for all figures and the MIFlowCyt record are available at\\\\\n\\url{https:\/\/github.com\/AmirErez\/UniversalImmune}.\n\n\n\n\\section*{Acknowledgments}\nThis work was supported by Human Frontier Science Program grant LT000123\/2014 (Amir Erez), National Institutes of Health (NIH) grant R01 GM082938 (A.E.), Simons Foundation grant 376198 (T.A.B.\\ and A.M.), and the Intramural Research Program of the NIH, Center for Cancer Research, National Cancer Institute.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:Introduction}\nThe use of Low Temperature Detectors (LTDs) for sensing \n X-ray and $\\gamma$-ray signals is quite widespread and well \nestablished~\\cite{Ullom-2015}.\nLTDs are also widely used in the field of fundamental physics, especially for \nDouble Beta Decay (DBD), and Dark Matter (DM) searches~\\cite{Pirro-2017}.\nIn these surveys the need for a hybrid detector, in which an energy \nrelease can be measured through different mechanisms, is of primary importance in order \nto distinguish the nature of interacting particles. For instance hybrid detectors can help identify \nand reject events caused by the natural background. With thermal detectors this can be \nachieved using scintillating or luminescent crystals. The simultaneous and \n\\it independent \\rm readout of the heat and the (escaping) light produced by the \ninteraction reveals the nature of the interacting particles thanks to \nthe different scintillation yields of $n$, $\\alpha$ and $\\gamma\/\\beta$ events.\nThis discrimination technique is presently used for DM \nsearches~\\cite{CRESST-2016,Angloher:2017sft,Angloher:2016hbv}, \nDBD searches~\\cite{CUPID-0-2018,Cupid-Mo-2017,Amore-2017}, and it can be \nalso implemented for rare nuclear decays~\\cite{Alfa-1,Alfa-2,Pattavina:2018nhk}.\n\nAt milli-Kelvin temperatures, the light detectors are usually bolometers \nthemselves: a \\it dark \\rm thin crystal absorbs the photons, producing heat (phonons) that is\nmeasured by a suitable thermometer. \nThe main difference among the various Bolometric Light Detector (BLD) instruments currently in use \nis the choice of the thermometer element, e.g. Transition Edge Sensors \n(TES)~\\cite{TES_LD_CRESST}, Neutron Transmutation Doped (NTD) \nthermistors~\\cite{NTD_LD_Lucifer-2013} or Micro Magnetic Calorimeters (MMC)~\\cite{MMC_LD-2015}. \n\nThe work presented here was performed within the CUPID \nframework~\\cite{CUORE-IHE-2014,CUPID-2015}, the future follow up of \nCUORE~\\cite{CUORE-2018} that represents the largest world-wide bolometric experiment to date. \nThe aim was to develop NTD-based BLDs with improved performance in terms of \nsensitivity, time response and simplified packaging for large arrays. \nUsing the tiny Cherenkov light emission of TeO$_2$~\\cite{Tabarelli-2010,Enriched-TeO2-Cherenkov-2017} to \ndecrease by two order of magnitude the $\\alpha$-induced background, requires a BLD with a S\/N ratio \nof the order of $\\sim$5~\\cite{CUPID-2015}: this corresponds to a RMS baseline resolution of the BLD of the \norder of $\\sim$20~eV being the Cherenkov light signal of the order of 100 eV.\nActually one can work towards the optimization of the light collection~\\cite{Casali-2017} and\/or \ntowards the energy resolution of the BLD or -as we made in this work- both.\nAdditionally, in case of $^{100}$Mo-based compounds, beside the same need to suppress the surface \n$\\alpha$-induced background, a fast time response of the BLD ($\\leq$~1~ms) is mandatory to suppress\nthe background induced the pile-up of the 2$\\nu$ DBD~\\cite{2_nu-Pile_up-2012}: also in \nthis case the S\/N ratio will play an important role~\\cite{2_nu-Pile_up-2016}.\n\n\n\nOur work has therefore focused on two aspects of BLD performance: (1) improving the response of \nthe NTD thermometer and (2) increasing the light collection.\nWhile the first aspect is strictly related to a specific technique, the second aspect \nis worthy of additional remarks. The working principle of a BLD is \nirrespective of the sensor: a thin crystal wafer (usually Si or Ge) absorbs \nthe emitted photons and converts them into heat. \nUnlike a conventional bolometric approach, we have to avoid the optical coupling \nbetween crystal and BLD made with optical grease or similar substance since the unavoidable \nheat flow through the optical coupling and the increase of the heat capacity of the system would reduce the \nindependence of the two detectors, eliminating the possibility of particle \ndiscrimination afforded by the different scintillation yields. Therefore the thermal contact between \nthe luminescent crystal and BLD has to be avoided,\nespecially in the case of extremely low scintillation yields. This is true for most \nof the Mo-based compounds~\\cite{Cupid-Mo-2017} and, even more importantly in case of \nCherenkov signals. A 2615~keV $\\gamma$-ray energy release in a CUORE-like TeO$_2$ absorber \nproduces a light signal in the BLD on the order of $\\sim$100~eV~\\cite{Casali-2017}. \nFor this reason the BLD is always facing the scintillating crystal without directly contacting \nit via a coupling medium.\n \nIn the following section it is shown that if the BLD is simply resting on the crystal \nsurface, held in position only by gravity, the thermal coupling between the BLD and \nthe crystal is almost negligible and the leakage of the BLD thermal signal through the scintillating crystal vanishes. \nThis fact can be explained considering the acoustic mismatch described in \nthe diffused mismatch model whereby the heat carriers (phonons) in insulating materials are \nscattered at the interfaces~\\cite{Matsumoto-1977,Swartz-1989}. This approach shows that the \nthermal resistance between two dielectric crystals is strongly dependent on the surface \nstate, on the different phonon characteristics in the two materials (density and Debye \ntemperature), and on the applied force. This latter parameter has a significant effect. \nWhen two solids are placed in contact with each other, the actual contact area can be much smaller \nthan the cross sections involved due to surface irregularities. By rising the applied force \nbetween the materials, a plastic or permanent deformation occurs and the \"real\" contact surface \narea increases. The result of this action is that the thermal conductance of the contact is directly \nproportional to the applied force~\\cite{Barucci-2001,Ventura-2008}.\n\nAlthough such simple stand will clearly not produce a so-called \"optical matching,\" the \nlight collection will be definitively larger due to geometrical factors~\\footnote{For instance if \nthe BLD is held in its own structure, depending on the mounting scheme, there are generally a \nfew mm of distance from the BLD to the scintillating crystal. \nThis increases the chance for photon escape or absorption by the holding structure rather than the BLD.}.\nIn addition, removing the BLD mounting structure decreases the presence of materials and surfaces close\nto the detector which reduces possible radioactive contamination, a fundamental aspect of dealing with rare event searches.\n\n\n\\section{Bolometric Light Detectors}\n\nOur BLDs are usually constituted by electronic grade undoped Ge wafers, coupled with Ge NTD thermistors. \nWe started to develop these detectors coupled with several scintillating DBD \ncrystals~\\cite{PIRRO-2005} and we deeply characterized their operation and \nperformances~\\cite{light-detectors-2013} to finally realize the LUCIFER~\\cite{LUCIFER-2013} experiment, \nwhich has been renamed CUPID-0~\\cite{CUPID-0-detector_2018}.\n\nEach BLD of CUPID-0 (totalling 26 detectors) was made by a double side polished electronic \ngrade undoped Ge wafer (44.5~mm diameter, 0.17~mm thick). The NTD thermistor, with dimension of \n(2.85~$\\times$~2~$\\times$~0.5)~mm$^3$, \nis glued through six small glue dots ($\\sim$~0.5~mm diameter, 0.05~mm height) made with \nAraldit\\textsuperscript{\\textregistered} Rapid glue.\nThe performance of six of these detectors was evaluated in a dedicated test run~\\cite{LUCIFER-2016} and \nthe results are summarized in Tab.~\\ref{tab-cupid-0-LD}.\n\nTo further optimize our BLDs, we produced a set of devices based on the pioneering work of \nCoron et.al.~\\cite{Coron-2004}. For this study we (1) decreased the heat capacity (size) of the \nthermistor, (2) increased the thermal conductance between the thermistor and the Ge wafer, and (3) \ndecreased the thermal conductance to the thermal bath. \nWith respect to the thermistor size, we used thermistors with a dimension of\n(2.85~$\\times$~1~$\\times$ 0.4)~mm$^3$, roughly 2.5 times smaller than the CUPID-0 devices. \nWe also decided to replace the six glue dots with an uniform glue layer, thus increasing the thermal \nconductance between the thermistors and light-absorbing Ge wafer. \n\nIt should be noted that in our experience the use of glue dots \ninstead of a \\it more effective \\rm thin gluing layer is preferred when coupling inherently different \nmaterials (e.g. TeO$_2$ crystals and Ge thermistors). The dot approach reduces the mechanical stresses \ninduced by differential thermal contraction of the materials when cooled. \nIn such cases, and especially when working with larger-sized thermistors, we sometimes observed cracks on \nthe crystal surface after a cooling cycle. This phenomenon is greatly reduced in our case since we glue \nGermanium thermistors to Germanium light absorbers and use smaller thermistors. Even in this \ncase, however, there are some small unavoidable stresses due to misorientation between the thermistor\nand absorber crystallographic planes, but we have found that these effects never led to visible cracks.\n\nWith respect to the mounting (i.e. the conductance to the thermal bath), there are many ways \nto hold the BLD in place. In earlier work we adopted two~\\cite{light-detectors-2013} or \nthree~\\cite{CUPID-0-detector_2018} small PTFE clamps that squeeze the edge of the Ge, keeping \nit fixed in a Cu standalone holder. \nPTFE is a common material also used by other groups working with NTD \nsensors~\\cite{Lumineu-2017} and with MMC detectors~\\cite{MMC_LD-2015}. \nOther clamping schemes and material choices have been demonstrated by the CRESST group. \nThese include bronze clamps and Silicon or CaWO$_4$-based sticks~\\cite{Strauss-2018}. \nThe design used in~\\cite{Coron-2004}, however, is probably the most complex from a construction point \nof view, using several ultra thin superconductive wires to suspend the Ge wafer from a copper frame to \nproduce a negligible thermal link that maximizes the heat flow from the wafer to the NTD. \n\\begin{table}[t]\n\\centering\n\\caption{Mean performance of six CUPID-0-like light detectors~\\cite{LUCIFER-2016}.\n\\textbf{R$_{work}$} refers to the resistance of the NTD Ge thermistor in working conditions, \n\\textbf{Response} refers to the absolute voltage \ndrop (in $\\mu$V) produced by an energy release of 1\\,keV, \\textbf{Baseline RMS} is the resolution\nafter signal filtering~\\cite{Gatti-1986:1,Alduino-2016:045503}. {\\bf$\\tau_{r}$} and {\\bf$\\tau_{d}$} are the \nrise and decay times, computed as the time difference between the 90$\\%$ and 10$\\%$ of the leading\nedge and as the time difference between the 30$\\%$ and 90$\\%$ of the trailing edge,\nrespectively. The Bessel cut-off frequency is 200 Hz (see last remarks of \nSec.~\\ref{sec:results}).}\n\\label{tab-cupid-0-LD}\n\\begin{tabular}{ccccc}\n\\hline\\noalign{\\smallskip}\nR$_{work}$ &Response &Baseline RMS &$\\tau_{r}$ &$\\tau_{d}$ \\\\\n\n[M$\\Omega$] &[$\\mu$V\/keV] &[eV] &[ms] &[ms]\\\\\n\\hline\n 0.87 & 1.36 & 43 & 1.77 & 5.06 \\\\\n\\hline\n\\end{tabular}\n\\end{table} \n\nWe decided to avoid any kind of holding structure whatsoever so we laid the BLD \ndirectly on the crystal, kept in position only by its weight ($\\sim$1.1 g). \nIn this configuration the main thermal link between the BLD and the cryostat is represented by \nthe thin gold NTD thermistor wires ( 2 $\\times$ 15 mm length, 25 $\\mu$m diameter). As mentioned \nabove, the expected thermal conductance to the scintillating crystal is negligible. The crystal \nchosen for this test was a (50.5~$\\times$~50.5~$\\times$~50.5)~mm$^3$ \nTeO$_2$ crystal. The aim was to test the new setup with a light signal on the order \nof few tens of eV. The Ge light-absorbing wafer \nbelongs to the batch used for CUPID-0, which include a 70~nm SiO anti-reflecting coating~\\cite{Mancuso-2014} \nthat was deposited on the side that rests on the TeO$_2$ crystal.\n\n\\section{Experimental details}\n\\label{sec:experimental_details}\nThe TeO$_2$ crystal was mounted in a similar way as described \nin~\\cite{Enriched-TeO2-Cherenkov-2017,Casali-2014} with the only exception that the TeO$_2$ crystal was standing on \nthe reflecting foil and both TeO$_2$ and BLD were not equipped with Si heaters.\nThese heaters were normally glued \non the bolometer to inject pulsed thermal signals for gain stabilization.\nThe TeO$_2$ face supporting the BLD and the opposite one were polished at (nearly) optical level. \nThe remaining four lateral faces were matted in order to increase light collection~\\cite{Casali-2017}. \n\nThe TeO$_2$ crystal is held by four S-shaped PTFE supports that are fixed to Cu columns. \nThe PTFE contracts upon cooling, creating a tensioned support that maintains the crystal \nposition. \n\nIn order to maximize light collection, the crystal is completely surrounded by a plastic \nreflecting sheet (3M Vikuiti\\textsuperscript{TM}), in the same way as \nin~\\cite{Enriched-TeO2-Cherenkov-2017,Casali-2014}. A photograph of the detectors is presented \nin Fig.~\\ref{fig_0-setup}.\n\\begin{figure}[hbt] \n\\centering \n\\includegraphics[width=0.48\\textwidth]{fig_0-setup.pdf}\n\\caption{Photograph of the detectors. The BLD is simply resting on the TeO$_2$ and the four \nPTFE supports (as well as the thermistor glued on the TeO$_2$) do not hold the BLD in any way: they \nsimply avoid the BLD to lean out from the top surface, as a mere translation constraints. \nThe gold wires of both NTDs are then crimped within micro Cu tubes to ensure the electrical contact \nas well as the thermal conductance to the heat sink. The $^{55}$Fe X-ray source is attached \nto the top reflecting cover sheet that encloses the detectors (with a clearance of $\\sim$4~mm from the BLD) \nand can be observed -reflected by the Ge wafer surface- between the two NTDs.}\n\\label{fig_0-setup}\n\\end{figure}\nThe entire setup was enclosed in a Cu box and thermally coupled to the mixing chamber \nof the CUPID R\\&D cryostat, a $^3$He\/$^4$He dilution refrigerator installed \ndeep underground within Hall C of the Laboratori Nazionali del Gran Sasso, Italy. \nTo avoid vibrations reaching the detectors, the box is mechanically decoupled from the \ncryostat by utilizing a two-stage pendulum system~\\cite{Pirro-2006}. \n\n\nThe thermistors of the detectors are biased with a quasi-constant current produced by applying a \nfixed voltage through large (27+27 or 2+2 G$\\Omega$) load resistors~\\cite{Arnaboldi-2002:1808}. \nWhen light is absorbed in the Ge wafer, a thermal pulse is produced which is subsequently \ntransferred to the NTD sensor, changing the resistance of the thermistor. This, in turn, creates a\nvoltage change across the current-biased NTD which is amplified using \nfront end electronics located just outside the cryostat~\\cite{Arnaboldi-2004}. The signals are \nthen filtered by an anti-aliasing 6-pole Bessel filter (with a cutoff frequency of 16~Hz \nfor the TeO$_2$ crystal and 550~Hz for the BLD) and finally fed into a NI PXI-6284 18-bit ADC.\n\nThe sampling rate of the ADC was 1~kHz for the TeO$_2$ crystal and 8 kHz for the BLD. \nThe two independent triggers are software generated such that when a trigger fires, the \ncorresponding waveform is recorded. Moreover, when the trigger of the \nTeO$_2$ crystal fires, the corresponding waveform of the BLD is always \nrecorded, irrespective of its trigger. A detailed description of the DAQ system can \nbe found in~\\cite{DiDomizio:2018ldc}.\nThe amplitude and the shape of the voltage pulses are then determined via off-line analysis. \nThe pulse amplitude of the thermal signals is estimated by the Optimum Filtering (OF) \ntechnique~\\cite{Gatti-1986:1,Alduino-2016:045503}, that maximizes the signal-to-noise ratio \nin a way that improves the energy resolution and lowers the threshold of the detector. \nThe amplitude of the light signal, however, is evaluated from the filtered waveform \nat a fixed time delay with respect to the TeO$_2$ bolometer, as described in detail \nin~\\cite{Piperno-2001:10005}.\\newline\nThe amplitude of the acquired TeO$_2$ heat signals is energy-calibrated using several \n$\\gamma$-ray peaks from a $^{228}$Th source. \nThe BLD, on the contrary, is calibrated thanks to the 5.9~keV and 6.5~keV X-ray \nquanta produced by a $^{55}$Fe X-ray source permanently faced to the detector.\n\n\\section{Data analysis and results}\n\\subsection{BLD performance}\n\\label{sec:results}\nThe crystals were tested at a cryostat base temperature of $\\sim$11~mK. \nIn order to obtain a fast response, we operated the BLD in the so-called \"over-biased\" \nconfiguration whereby \nthe biasing current of the circuit is set much larger than the current that would ensure the highest \nabsolute thermal response~\\cite{NTD_LD_Lucifer-2013}. This choice ensures a small working \nresistance, thus minimizing the effect of the low pass filtering induced by the overall \ncapacity ($\\sim$200 pF) of the front end readout wires. \n\nIn Fig.~\\ref{fig_1-55Fe} we show the $^{55}$Fe calibration spectrum obtained with the BLD.\nThe baseline energy resolution (ie, the absolute sensitivity) of the BLD is given by \nthe width of randomly acquired baselines (noise) after the application of OF. \nAs is typical for this style of detectors, the energy resolution of monochromatic energy \nabsorption events is much worse than the baseline resolution, irrespective \nof the type of sensor~\\cite{NTD_LD_Lucifer-2013,TES_LD_CRESST}. \n\n\\begin{figure}[hbt] \n\\centering \n\\includegraphics[width=0.48\\textwidth]{fig_1-55Fe.pdf}\n\\caption{Energy distribution of the random sampled noise. The width of the distribution \n($\\sigma\\approx$20 eV) represents the baseline energy resolution of our BLD. The right inset shows\nthe $^{55}$Fe calibration spectrum of the BLD. The x-axis units represent the absolute voltage drop \nacross the thermistor. \nThe RMS resolution on the 5.9 keV and 6.5 keV X-ray peaks is 59 eV (see text).}\n\\label{fig_1-55Fe}\n\\end{figure}\nThe noise and signal power spectra of the BLD are presented in Fig.~\\ref{fig_2-NPS}.\n\n\\begin{figure}[hbt] \n\\centering \n\\includegraphics[width=0.48\\textwidth]{fig_2-NPS.pdf}\n\\caption{Noise power spectrum (black line) and signal power spectrum (blue line) of the BLD. \nThe y-axis scale is in absolute values for the noise. The signal spectrum is scaled in \narbitrary units, being the roll-off induced by the Bessel filter the same between noise and signal.\nThe working resistance of the thermistor is 1.47 M$\\Omega$, biased with a current of 3.7 nA \nthorough (2+2) G$\\Omega$ metallic load resistors. The peaks are due to the microphonic \nnoise induced by the vibration of the readout wires.}\n\\label{fig_2-NPS}\n\\end{figure}\nThe bump that can be observed in Fig.~\\ref{fig_2-NPS} at $\\sim$400 Hz arises from a \nresonance that enhances the thermal noise generated within the thermistor. This occurs \nwhen the impedance of the parasitic capacitance of the link becomes smaller than that of the \nthermistor, which is a fed-backed device~\\cite{Arnaboldi-2005}. \nThe bump is found at the border of the bandwidth of the signal and is rejected from the \noptimum filter algorithm.\n\n\nFig.~\\ref{fig_3-rise-decay} shows the corresponding rise and decay times of $^{55}$Fe X-rays absorption events.\n\\begin{figure}[hbt] \n\\centering \n\\includegraphics[width=0.48\\textwidth]{fig_3-rise-decay.pdf}\n\\caption{Rise and decay times distributions corresponding to the $^{55}$Fe X-rays. The \nBessel cut-off frequency of the Front-End is 550~Hz.}\n\\label{fig_3-rise-decay}\n\\end{figure}\nThe measured rise time shown in Fig.~\\ref{fig_3-rise-decay} is most likely slower than the intrinsic \nrise time of the detector since it contains contributions from the Bessel filter (independent from the \nthermistor impedance) and from the capacitance of the readout wires. This last \ncontribution is difficult to measure since it involves the dynamic resistance of the \nthermistor. The contribution of the 550~Hz Bessel filter to the rise time was evaluated \nin~\\cite{NTD_LD_Lucifer-2013} and reported as 0.65~ms. Thus, after applying a quadratic deconvolution, the \n\\it intrinsic \\rm rise time of our BLD should be of the order of 0.5~ms, compatible with the \nexpectation of ~\\cite{Coron-2004}.\nThe overall performance of the BLD is summarized in Tab.~\\ref{tab-new-BLD}.\n\n\\begin{table}\n\\centering\n\\caption{Performances of the BLD of this work, to be compared with the ones of \nTab.~\\ref{tab-cupid-0-LD}.}\n\\label{tab-new-BLD}\n\\begin{tabular}{ccccc}\n\\hline\nR$_{work}$ &Response &Baseline RMS &$\\tau_{r}$ &$\\tau_{d}$ \\\\\n\n[M$\\Omega$] &[$\\mu$V\/keV] &[eV] &[ms] &[ms]\\\\\n\\hline\n1.47 &3.86 &20 & 0.83 &1.63 \\\\\n\\hline\n\\end{tabular}\n\\end{table} \n\\subsection{Heat and Light measurement}\n\\label{scatter-section}\nIn order to evaluate the long-term discriminatory performance of our BLD, we performed a 70 h run that \nincluded two event-generating calibration sources embedded into the setup. A $^{228}$Th source was placed\na few cm away from the TeO$_2$ crystal and \na \\it smeared \\rm $^{238}$U $\\alpha$ source was applied to the inside of the light reflector facing the TeO$_2$. \nThe aim of the $\\alpha$ source was to directly measure the discrimination capability between \n$\\alpha$ and $\\beta\/\\gamma$ in the DBD region of interest of $^{130}$Te. \nThe source was made using 2 $\\mu l$ of a standard calibrated solution (0.1 \\%) of $^{238}$U, and the dried \nsource deposition was covered with a 6 $\\mu m$ aluminized Mylar foil to smear the \n$\\alpha$ energy.\n\nThe light vs heat scatter plot is presented in Fig.~\\ref{fig_4-scatter-plot} and shows an \nunexpected feature.\n\\begin{figure}[hbt] \n\\centering \n\\includegraphics[width=0.48\\textwidth]{fig_4-scatter-plot.pdf}\n\\caption{Light vs heat scatter plot obtained in a 70 h measurement with the TeO$_2$ exposed \nto a $^{228}$Th source and a smeared $^{238}$U $\\alpha$ source. Unfortunately $\\alpha$ \nenergy loss in the Mylar -constituting the smearing medium- results in a tiny, but \nmeasurable, light emission that increases towards lower energies, i.e. at larger energy \nloss in the Mylar. The events above 4~MeV, on the contrary, are due to internal and\/or \nsurface contaminations and their light emission is compatible with zero \n(see text).}\n\\label{fig_4-scatter-plot}\n\\end{figure}\nThe $^{238}$U $\\alpha$-events arising from the smeared source clearly show a tiny light \nemission that increases towards lower energies. This feature can only be ascribed to an \nenergy loss in the Mylar which emits few scintillation photons. To avoid this effect \nwe usually face the aluminized surface of the Mylar towards the crystal so as to reflect the \n(very few) photons that could be produced in this plastic. This time however, \nwe mistakenly mounted the Mylar with the uncoated side towards the detector. This was confirmed after\nsubsequently opening the cryostat and checking.\n\nThe result is shown in Fig.~\\ref{fig_4-scatter-plot}: the amount of Cherenkov \nlight, produced by a 2615~keV $\\gamma$, that is collected with this new set-up is \n(151~$\\pm$~4)~eV, 50 \\% larger with respect to all our previous measurements with \nmassive crystals~\\cite{Casali-2017}, as well as roughly 50 \\% larger with respect to a \nmeasurement recently performed with a NTD-based light detector~\\cite{Lumineu-2017} of the \nsame type (considering the 40 \\% reduced transmission area between BLD and crystal, as \ndeclared in the article).\nThe light distribution of the 74 events belonging to the internal\n$^{210}$Po $\\alpha$ at 5407~keV (5304~keV $\\alpha$ + 103~keV nucleus recoil) shows a \nmean value of (5.8~$\\pm$~3.3)~eV, still compatible with zero (see Sec.~\\ref{sec:thermal_interference}) \nas it should be if the light only arises from the Cherenkov effect. More importantly, the width of \nthe light distribution of $\\alpha$'s is $\\sigma_{\\alpha}$=(22.7~$\\pm$~ 2.7)~eV, fully compatible \nwith the RMS noise of the BLD of Tab.~\\ref{tab-new-BLD}.\nThe light signal induced by the 2615~keV $\\gamma$ -on the contrary- shows a width of \n$\\sigma_{\\gamma\/\\beta}$=(31.5~$\\pm$~4.3)~eV which is \na result of the photostatistics and the light collection. \n\nIn order to evaluate the Discrimination Power (DP) that can be obtained between the \n$\\alpha$ and $\\beta\/\\gamma$ distributions at 2528~keV (the Q$_{\\beta\\beta}$-value of \nthe DBD of $^{130}$Te) we use the same formula and arguments used \nin~\\cite{Enriched-TeO2-Cherenkov-2017,Lumineu-2017}: the DP can be quantified as the \ndifference between the average values of the two distributions normalized to the square \nroot of the quadratic sum of their widths: \n\\begin{equation} \nDP = \\frac{|\\mu_{\\gamma\/\\beta}-\\mu_{\\alpha}|}{\\sqrt{\\sigma^{2}_{\\gamma\/\\beta}+\n\\sigma^{2}_{\\alpha}}}.\n\\label{eq:DP}\n\\end{equation} \nRe-scaling the light signal from 2615 to 2528~keV, we obtain DP=3.6, using one highly likely \nassumption that an $\\alpha$ particle at 2528~keV will show a light signal equal than the same \nparticle at~5304 keV ($^{210}$Po).\nThis DP is the best ever achieved with large mass TeO$_2$ crystals (M $>$ 7 g) and without \nthe need for additional Neganov-Luke \namplification~\\cite{Lumineu-2017,Casali:2015gya,Gironi:2016nae}, or\nmore sophisticated TES sensors~\\cite{Karo-2014} or both~\\cite{Willers-2014}.\n\n\\section{Thermal conductance}\\label{sec:thermal_interference}\nAs stated in Sec.~\\ref{sec:Introduction}, the actual goal of this work was to experimentally \ndemonstrate that the BLD can rest on the scintillating or luminescent crystal without heat sinking to it. \nUsing the results in the previous section we can now calculate a limit on the heat flow through \nthe Ge wafer and the TeO$_2$. If one assumes that a 5407 keV energy release in the TeO$_2$ produces\na mean value BLD signal that only depends on the heat flow (assuming no light emission), then we have \nan upper limit for the ratio of the heat flow through TeO$_2$ and Ge: 5.8~eV\/5407~keV$\\sim$10$^{-6}$.\n\nIn our case, an extremely low heat conductance was determined experimentally using static conditions. We measured \nthe base resistance of the BLD as 223.5 M$\\Omega$ (corresponding to 11.8~mK), keeping the \nTeO$_2$ thermistor unbiased (i.e. no power dissipation in it). We then gave the maximum \n(allowed by our biasing set-up) bias to the TeO$_2$ thermistor, corresponding to \n4.8 nA, and the TeO$_2$ thermistor changed its resistance from 626 M$\\Omega$ \n(bias~$\\rightarrow$~0) to 1.71 M$\\Omega$. The power dissipated on the TeO$_2$ was therefore 40 pW. \nThe base resistance of the BLD decreased to 222.8 M$\\Omega$, which corresponds to a temperature increase of \nonly $\\approx$~4.3 $\\pm$ 0.2 ~$\\mu$K. \nThe same operation was performed with the BLD in working condition, i.e. bias current \nof 3.7 nA and a resistance of 1.47~M$\\Omega$ (corresponding to $\\sim$23~mK), and no variation\nof the baseline of the BLD was registered.\nA further investigation of the thermal conductance between a Ge-BLD and a TeO$_2$ crystal was performed\nby exploiting a small TeO$_2$ crystal ($20~\\times~20~\\times~14$~mm$^{3}$, 34~g mass).\nWe used a standard BLD, i.e., the same thickness and height as in the previous discussion, but with the Ge wafer\nheld with PTFE clamps in a stand-alone Cu mounting~\\cite{NTD_LD_Lucifer-2013}.\nFor this experiment we rested the $20\\times20$~mm$^{2}$ surface of the 34~g crystal \n on the Ge wafer. The NTD thermistor-equipped TeO$_2$ crystal was surrounded with the same reflecting \nfoil and we performed the same measurement described in Sec.~\\ref{scatter-section} with the same \noverall setup.\nThis time a 5304~keV $^{210}$Po decay occurring in the TeO$_2$ created a mean signal in the BLD \nof (317~$\\pm$~29)~eV, definitively not compatible with the result of Sec.~\\ref{scatter-section}.\nThe mean (light) signal registered in coincidence with the 2615~keV $\\gamma$-line of $^{208}$Tl \nwas (336~$\\pm$~5)~eV. \nThe $\\alpha$-induced signal in the BLD, therefore, has to be ascribed to an effective thermal \ntransfer from the TeO$_2$ to the BLD.\nWe can make a very rough estimation of the size of this transfer using the \nresults of the measurement of Sec.~\\ref{scatter-section}. If we assume the heat conductance \nto be linearly proportional to the pressure force between the two mediums, then we may\nsimply compare the weight differences: 1.1 g in the case of the wafer resting onto the TeO$_2$ \ncrystal versus 34~g in this last configuration.\nTheir ratio, i.e. 31, should be, in first approximation, the ratio between the thermal \nconductance in the two setups. Ascribing the $\\alpha$ signal of Sec.~\\ref{scatter-section} \nexclusively to thermal transfer we would expect a thermal transfer signal of \n(180~$\\pm$~90)~eV, which is compatible with the 317~eV observed during this measurement. On the other\nhand, under the same assumption, we can evaluate the 2615-keV induced Cherenkov light signal\nof this crystal as the difference between the observed signal and the re-scaled thermal transfer \nevaluated from the $\\alpha$. In this way we observe that the energy of the Cherenkov light \nemission in this 34~g crystal is (185~$\\pm$~15)~eV.\n\n\\section{Conclusions}\nWe have demonstrated the possibility of mounting BLDs by simply resting them on the surface of the \ncorresponding scintillating crystal. With this new mounting method the light collection can increase up\nto 50\\% with respect to standard setups. We do not observe appreciable heat flow between the\nscintillating crystal and BLD. \nWe also improved the time response of our thermistor-based light detectors, reaching a rise\ntime of 0.8 ms and demonstrating that 0.5 ms is achievable. This time response is necessary \nto remove the background induced by the pile-up of the 2$\\nu$-DBD mode in the case \nof $^{100}$Mo-based crystals. We reached a baseline resolution\nof 20~eV RMS, more than 2 times better than the average value our previous CUPID-0-like detectors. \nThanks to these developments, we definitively demonstrated that standard thermistor-based\nBLDs can be used for CUPID, both to read out the tiny Cherenkov light of TeO$_2$ as well as to\nread out the Mo-based scintillating crystals. \n\nWe do believe that this simplified technique could be applied to any kind of BLD, irrespective\nof the sensor type. The first approximation thermal conductance between crystal and BLD \ndoes not depend upon the energy of the phonons, so we would expect that thermal transfer \nwould be as negligible in TES or MMC devices as it is in our NTDs. \nMore generally this new technique could be also applied in the case of stacked, standard small\nbolometers, provided that the weight does not exceed a\nfew grams. However, since the measured thermal transfer is rather small, the weight of the\nbolometer will not be a significant limiting factor in low energy threshold applications. \n\n\\section{Acknowledgments}\nThis work was performed within the CUPID experiment founded by INFN and supported by\nthe National Science Foundation under Grant NSF-PHY-1614611.\n\nWe thank the CUPID-0 and the CUORE collaborations for the overall support and for sharing their \nDAQ and software.\nWe express our gratitude to LNGS for the generous hospitality and, in particular, to\nthe mechanical workshop personnel including E. Tatananni, A. Rotilio, A. Corsi, and B.\nRomualdi for their continuous and constructive help. We are also grateful to M. Guetti \nfor his invaluable support and expertise in the cryostat facility maintenance. \nWe acknowledge Dr. C. Arnaboldi for his precious \nsupport, even though he has left this field of research many years ago. We are especially \ngrateful to E. Ferri for her kind support in the thermistor wire-bonding.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}