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The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) G\"odel-L\"ob logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong L\"ob logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic.

We propose a simple empirical representation of expectations such that: For a number of samples above a certain threshold, drawn from any probability distribution with finite fourth-order statistic, the proposed estimator outperforms the empirical average when tested against the actual population, with respect to the quadratic loss. For datasets smaller than this threshold, the result still holds, but for a class of distributions determined by their first four statistics. Our approach leverages the duality between distributionally robust and risk-averse optimization.

We examine the moments of the number of lattice points in a fixed ball of volume $V$ for lattices in Euclidean space which are modules over the ring of integers of a number field $K$. In particular, denoting by $\omega_K$ the number of roots of unity in $K$, we show that for lattices of large enough dimension the moments of the number of $\omega_K$-tuples of lattice points converge to those of a Poisson distribution of mean $V/\omega_K$. This extends work of Rogers for $\mathbb{Z}$-lattices. What is more, we show that this convergence can also be achieved by increasing the degree of the number field $K$ as long as $K$ varies within a set of number fields with uniform lower bounds on the absolute Weil height of non-torsion elements.

Modeling the behavior of biological tissues and organs often necessitates the knowledge of their shape in the absence of external loads. However, when their geometry is acquired in-vivo through imaging techniques, bodies are typically subject to mechanical deformation due to the presence of external forces, and the load-free configuration needs to be reconstructed. This paper addresses this crucial and frequently overlooked topic, known as the inverse elasticity problem (IEP), by delving into both theoretical and numerical aspects, with a particular focus on cardiac mechanics. In this work, we extend Shield's seminal work to determine the structure of the IEP with arbitrary material inhomogeneities and in the presence of both body and active forces. These aspects are fundamental in computational cardiology, and we show that they may break the variational structure of the inverse problem. In addition, we show that the inverse problem might have no solution even in the presence of constant Neumann boundary conditions and a polyconvex strain energy functional. We then present the results of extensive numerical tests to validate our theoretical framework, and to characterize the computational challenges associated with a direct numerical approximation of the IEP. Specifically, we show that this framework outperforms existing approaches both in terms of robustness and optimality, such as Sellier's iterative procedure, even when the latter is improved with acceleration techniques. A notable discovery is that multigrid preconditioners are, in contrast to standard elasticity, not efficient, where a one-level additive Schwarz and generalized Dryja-Smith-Widlund provide a much more reliable alternative. Finally, we successfully address the IEP for a full-heart geometry, demonstrating that the IEP formulation can compute the stress-free configuration in real-life scenarios.

Non-Hermitian topological phases can produce some remarkable properties, compared with their Hermitian counterpart, such as the breakdown of conventional bulk-boundary correspondence and the non-Hermitian topological edge mode. Here, we introduce several algorithms with multi-layer perceptron (MLP), and convolutional neural network (CNN) in the field of deep learning, to predict the winding of eigenvalues non-Hermitian Hamiltonians. Subsequently, we use the smallest module of the periodic circuit as one unit to construct high-dimensional circuit data features. Further, we use the Dense Convolutional Network (DenseNet), a type of convolutional neural network that utilizes dense connections between layers to design a non-Hermitian topolectrical Chern circuit, as the DenseNet algorithm is more suitable for processing high-dimensional data. Our results demonstrate the effectiveness of the deep learning network in capturing the global topological characteristics of a non-Hermitian system based on training data.

Covariance and Hessian matrices have been analyzed separately in the literature for classification problems. However, integrating these matrices has the potential to enhance their combined power in improving classification performance. We present a novel approach that combines the eigenanalysis of a covariance matrix evaluated on a training set with a Hessian matrix evaluated on a deep learning model to achieve optimal class separability in binary classification tasks. Our approach is substantiated by formal proofs that establish its capability to maximize between-class mean distance and minimize within-class variances. By projecting data into the combined space of the most relevant eigendirections from both matrices, we achieve optimal class separability as per the linear discriminant analysis (LDA) criteria. Empirical validation across neural and health datasets consistently supports our theoretical framework and demonstrates that our method outperforms established methods. Our method stands out by addressing both LDA criteria, unlike PCA and the Hessian method, which predominantly emphasize one criterion each. This comprehensive approach captures intricate patterns and relationships, enhancing classification performance. Furthermore, through the utilization of both LDA criteria, our method outperforms LDA itself by leveraging higher-dimensional feature spaces, in accordance with Cover's theorem, which favors linear separability in higher dimensions. Our method also surpasses kernel-based methods and manifold learning techniques in performance. Additionally, our approach sheds light on complex DNN decision-making, rendering them comprehensible within a 2D space.

We show that a constant number of self-attention layers can efficiently simulate, and be simulated by, a constant number of communication rounds of Massively Parallel Computation. As a consequence, we show that logarithmic depth is sufficient for transformers to solve basic computational tasks that cannot be efficiently solved by several other neural sequence models and sub-quadratic transformer approximations. We thus establish parallelism as a key distinguishing property of transformers.

{We analyze a general Implicit-Explicit (IMEX) time discretization for the compressible Euler equations of gas dynamics, showing that they are asymptotic-preserving (AP) in the low Mach number limit. The analysis is carried out for a general equation of state (EOS). We consider both a single asymptotic length scale and two length scales. We then show that, when coupling these time discretizations with a Discontinuous Galerkin (DG) space discretization with appropriate fluxes, an all Mach number numerical method is obtained. A number of relevant benchmarks for ideal gases and their non-trivial extension to non-ideal EOS validate the performed analysis.

We prove that functions over the reals computable in polynomial time can be characterised using discrete ordinary differential equations (ODE), also known as finite differences. We also provide a characterisation of functions computable in polynomial space over the reals. In particular, this covers space complexity, while existing characterisations were only able to cover time complexity, and were restricted to functions over the integers. We prove furthermore that no artificial sign or test function is needed even for time complexity. At a technical level, this is obtained by proving that Turing machines can be simulated with analytic discrete ordinary differential equations. We believe this result opens the way to many applications, as it opens the possibility of programming with ODEs, with an underlying well-understood time and space complexity.

Often the question arises whether $Y$ can be predicted based on $X$ using a certain model. Especially for highly flexible models such as neural networks one may ask whether a seemingly good prediction is actually better than fitting pure noise or whether it has to be attributed to the flexibility of the model. This paper proposes a rigorous permutation test to assess whether the prediction is better than the prediction of pure noise. The test avoids any sample splitting and is based instead on generating new pairings of $(X_i, Y_j)$. It introduces a new formulation of the null hypothesis and rigorous justification for the test, which distinguishes it from previous literature. The theoretical findings are applied both to simulated data and to sensor data of tennis serves in an experimental context. The simulation study underscores how the available information affects the test. It shows that the less informative the predictors, the lower the probability of rejecting the null hypothesis of fitting pure noise and emphasizes that detecting weaker dependence between variables requires a sufficient sample size.