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Neural abstractions have been recently introduced as formal approximations of complex, nonlinear dynamical models. They comprise a neural ODE and a certified upper bound on the error between the abstract neural network and the concrete dynamical model. So far neural abstractions have exclusively been obtained as neural networks consisting entirely of $ReLU$ activation functions, resulting in neural ODE models that have piecewise affine dynamics, and which can be equivalently interpreted as linear hybrid automata. In this work, we observe that the utility of an abstraction depends on its use: some scenarios might require coarse abstractions that are easier to analyse, whereas others might require more complex, refined abstractions. We therefore consider neural abstractions of alternative shapes, namely either piecewise constant or nonlinear non-polynomial (specifically, obtained via sigmoidal activations). We employ formal inductive synthesis procedures to generate neural abstractions that result in dynamical models with these semantics. Empirically, we demonstrate the trade-off that these different neural abstraction templates have vis-a-vis their precision and synthesis time, as well as the time required for their safety verification (done via reachability computation). We improve existing synthesis techniques to enable abstraction of higher-dimensional models, and additionally discuss the abstraction of complex neural ODEs to improve the efficiency of reachability analysis for these models.

We are interested in creating statistical methods to provide informative summaries of random fields through the geometry of their excursion sets. To this end, we introduce an estimator for the length of the perimeter of excursion sets of random fields on $\mathbb{R}^2$ observed over regular square tilings. The proposed estimator acts on the empirically accessible binary digital images of the excursion regions and computes the length of a piecewise linear approximation of the excursion boundary. The estimator is shown to be consistent as the pixel size decreases, without the need of any normalization constant, and with neither assumption of Gaussianity nor isotropy imposed on the underlying random field. In this general framework, even when the domain grows to cover $\mathbb{R}^2$, the estimation error is shown to be of smaller order than the side length of the domain. For affine, strongly mixing random fields, this translates to a multivariate Central Limit Theorem for our estimator when multiple levels are considered simultaneously. Finally, we conduct several numerical studies to investigate statistical properties of the proposed estimator in the finite-sample data setting.

In this work, we study the convergence and performance of nonlinear solvers for the Bidomain equations after decoupling the ordinary and partial differential equations of the cardiac system. Firstly, we provide a rigorous proof of the global convergence of Quasi-Newton methods, such as BFGS, and nonlinear Conjugate-Gradient methods, such as Fletcher--Reeves, for the Bidomain system, by analyzing an auxiliary variational problem under physically reasonable hypotheses. Secondly, we compare several nonlinear Bidomain solvers in terms of execution time, robustness with respect to the data and parallel scalability. Our findings indicate that Quasi-Newton methods are the best choice for nonlinear Bidomain systems, since they exhibit faster convergence rates compared to standard Newton-Krylov methods, while maintaining robustness and scalability. Furthermore, first-order methods also demonstrate competitiveness and serve as a viable alternative, particularly for matrix-free implementations that are well-suited for GPU computing.

Future wireless networks and sensing systems will benefit from access to large chunks of spectrum above 100 GHz, to achieve terabit-per-second data rates in 6th Generation (6G) cellular systems and improve accuracy and reach of Earth exploration and sensing and radio astronomy applications. These are extremely sensitive to interference from artificial signals, thus the spectrum above 100 GHz features several bands which are protected from active transmissions under current spectrum regulations. To provide more agile access to the spectrum for both services, active and passive users will have to coexist without harming passive sensing operations. In this paper, we provide the first, fundamental analysis of Radio Frequency Interference (RFI) that large-scale terrestrial deployments introduce in different satellite sensing systems now orbiting the Earth. We develop a geometry-based analysis and extend it into a data-driven model which accounts for realistic propagation, building obstruction, ground reflection, for network topology with up to $10^5$ nodes in more than $85$ km$^2$. We show that the presence of harmful RFI depends on several factors, including network load, density and topology, satellite orientation, and building density. The results and methodology provide the foundation for the development of coexistence solutions and spectrum policy towards 6G.

We provide a unified operational framework for the study of causality, non-locality and contextuality, in a fully device-independent and theory-independent setting. We define causaltopes, our chosen portmanteau of "causal polytopes", for arbitrary spaces of input histories and arbitrary choices of input contexts. We show that causaltopes are obtained by slicing simpler polytopes of conditional probability distributions with a set of causality equations, which we fully characterise. We provide efficient linear programs to compute the maximal component of an empirical model supported by any given sub-causaltope, as well as the associated causal fraction. We introduce a notion of causal separability relative to arbitrary causal constraints. We provide efficient linear programs to compute the maximal causally separable component of an empirical model, and hence its causally separable fraction, as the component jointly supported by certain sub-causaltopes. We study causal fractions and causal separability for several novel examples, including a selection of quantum switches with entangled or contextual control. In the process, we demonstrate the existence of "causal contextuality", a phenomenon where causal inseparability is clearly correlated to, or even directly implied by, non-locality and contextuality.

An Electroencephalogram (EEG) is a non-invasive exam that records the electrical activity of the brain. This exam is used to help diagnose conditions such as different brain problems. EEG signals are taken for the purpose of epilepsy detection and with Discrete Wavelet Transform (DWT) and machine learning classifier, they perform epilepsy detection. In Epilepsy seizure detection, mainly machine learning classifiers and statistical features are used. The hidden information in the EEG signal is useful for detecting diseases affecting the brain. Sometimes it is very difficult to identify the minimum changes in the EEG in the time and frequency domains purpose. The DWT can give a good decomposition of the signals in different frequency bands and feature extraction. We use the tri-dimensionality reduction algorithm.; Principal Component Analysis (PCA), Independent Component Analysis (ICA), and Linear Discriminant Analysis (LDA). Finally, features are selected by using a fusion rule and at the last step three different classifiers Support Vector Machine (SVM), Naive Bayes (NB) and K-Nearest-Neighbor(KNN) have been used individually for the classification. The proposed framework is tested on the Bonn dataset and the simulation results provide the accuracy for the combination of LDA and SVM 89.17%, LDA and KNN 80.42%, PCA and NB 89.92%, PCA and SVM 85.58%, PCA and KNN 80.42%, ICA and NB 82.33%, ICA and SVM 90.42%, and ICA and KNN 90%, LDA and NB 100%, accuracy. It shows the sensitivity, specificity, accuracy, Precision, and Recall of 100%, 100%, 100%, 100%, and 100%. This combination of LDA with NB method provides the accuracy of 100% outperforming all existing methods. The results prove the effectiveness of this model.

Penrose tilings are the most famous aperiodic tilings, and they have been studied extensively. In particular, patterns composed with hexagons ($H$), boats ($B$) and stars ($S$) were soon exhibited and many physicists published on what they later called $HBS$ tilings, but no article or book combines all we know about them. This work is done here, before introducing new decorations and properties including explicit substitutions. For the latter, the star comes in three versions so we have 5 prototiles in what we call the Star tileset. Yet this set yields exactly the strict $HBS$ tilings formed using 3 tiles decorated with either the usual decorations (arrows) or Ammann bar markings for instance. Another new tileset called Gemstones is also presented, derived from the Star tileset.

Perturbation analysis has emerged as a significant concern across multiple disciplines, with notable advancements being achieved, particularly in the realm of matrices. This study centers on specific aspects pertaining to tensor T-eigenvalues within the context of the tensor-tensor multiplication. Initially, an analytical perturbation analysis is introduced to explore the sensitivity of T-eigenvalues. In the case of third-order tensors featuring square frontal slices, we extend the classical Gershgorin disc theorem and show that all T-eigenvalues are located inside a union of Gershgorin discs. Additionally, we extend the Bauer-Fike theorem to encompass F-diagonalizable tensors and present two modified versions applicable to more general scenarios. The tensor case of the Kahan theorem, which accounts for general perturbations on Hermite tensors, is also investigated. Furthermore, we propose the concept of pseudospectra for third-order tensors based on tensor-tensor multiplication. We develop four definitions that are equivalent under the spectral norm to characterize tensor $\varepsilon$-pseudospectra. Additionally, we present several pseudospectral properties. To provide visualizations, several numerical examples are also provided to illustrate the $\varepsilon$-pseudospectra of specific tensors at different levels.

The details of second-order partial derivatives of rigid-body Inverse/Forward dynamics are provided. Several properties and identities using Spatial Vector Algebra are listed, along with their detailed derivations. The expressions build upon previous work by the author on first-order partial derivatives of inverse dynamics. The first/second-order derivatives are also extended for systems with external forces. Finally, the KKT Forward dynamics and Impact dynamics derivatives are derived.

The Information Bottleneck (IB) is a method of lossy compression of relevant information. Its rate-distortion (RD) curve describes the fundamental tradeoff between input compression and the preservation of relevant information embedded in the input. However, it conceals the underlying dynamics of optimal input encodings. We argue that these typically follow a piecewise smooth trajectory when input information is being compressed, as recently shown in RD. These smooth dynamics are interrupted when an optimal encoding changes qualitatively, at a bifurcation. By leveraging the IB's intimate relations with RD, we provide substantial insights into its solution structure, highlighting caveats in its finite-dimensional treatments. Sub-optimal solutions are seen to collide or exchange optimality at its bifurcations. Despite the acceptance of the IB and its applications, there are surprisingly few techniques to solve it numerically, even for finite problems whose distribution is known. We derive anew the IB's first-order Ordinary Differential Equation, which describes the dynamics underlying its optimal tradeoff curve. To exploit these dynamics, we not only detect IB bifurcations but also identify their type in order to handle them accordingly. Rather than approaching the IB's optimal curve from sub-optimal directions, the latter allows us to follow a solution's trajectory along the optimal curve under mild assumptions. We thereby translate an understanding of IB bifurcations into a surprisingly accurate numerical algorithm.