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We analyze a Discontinuous Galerkin method for a problem with linear advection-reaction and $p$-type diffusion, with Sobolev indices $p\in (1, \infty)$. The discretization of the diffusion term is based on the full gradient including jump liftings and interior-penalty stabilization while, for the advective contribution, we consider a strengthened version of the classical upwind scheme. The developed error estimates track the dependence of the local contributions to the error on local P\'eclet numbers. A set of numerical tests supports the theoretical derivations.

The classical Heawood inequality states that if the complete graph $K_n$ on $n$ vertices is embeddable in the sphere with $g$ handles, then $g \ge\dfrac{(n-3)(n-4)}{12}$. A higher-dimensional analogue of the Heawood inequality is the K\"uhnel conjecture. In a simplified form it states that for every integer $k>0$ there is $c_k>0$ such that if the union of $k$-faces of $n$-simplex embeds into the connected sum of $g$ copies of the Cartesian product $S^k\times S^k$ of two $k$-dimensional spheres, then $g\ge c_k n^{k+1}$. For $k>1$ only linear estimates were known. We present a quadratic estimate $g\ge c_k n^2$. The proof is based on beautiful and fruitful interplay between geometric topology, combinatorics and linear algebra.

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.

We investigate a linearised Calder\'on problem in a two-dimensional bounded simply connected $C^{1,\alpha}$ domain $\Omega$. After extending the linearised problem for $L^2(\Omega)$ perturbations, we orthogonally decompose $L^2(\Omega) = \oplus_{k=0}^\infty \mathcal{H}_k$ and prove Lipschitz stability on each of the infinite-dimensional $\mathcal{H}_k$ subspaces. In particular, $\mathcal{H}_0$ is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearised) Calder\'on problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmic-type in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert-Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fr\'echet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general $L^2(\Omega)$ perturbation onto the $\mathcal{H}_k$ spaces, hence reconstructing any $L^2(\Omega)$ perturbation.

Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this paper, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository.

We present a randomized algorithm for estimating the permanent of an $M \times M$ real matrix $A$ up to an additive error. We do this by viewing the permanent $\mathrm{perm}(A)$ of $A$ as the expectation of a product of centered joint Gaussian random variables with a particular covariance matrix $C$. The algorithm outputs the empirical mean $S_{N}$ of this product after sampling $N$ times. Our algorithm runs in total time $O(M^{3} + M^{2}N + MN)$ with failure probability \begin{equation*} P(|S_{N}-\text{perm}(A)| > t) \leq \frac{3^{M}}{t^{2}N} \prod^{2M}_{i=1} C_{ii}. \end{equation*} In particular, we can estimate $\mathrm{perm}(A)$ to an additive error of $\epsilon\bigg(\sqrt{3^{2M}\prod^{2M}_{i=1} C_{ii}}\bigg)$ in polynomial time. We compare to a previous procedure due to Gurvits. We discuss how to find a particular $C$ using a semidefinite program and a relation to the Max-Cut problem and cut-norms.

A new area of application of methods of algebra of logic and to valued logic, which has emerged recently, is the problem of recognizing a variety of objects and phenomena, medical or technical diagnostics, constructing modern machines, checking test problems, etc., which can be reduced to constructing an optimal extension of the logical function to the entire feature space. For example, in logical recognition systems, logical methods based on discrete analysis and propositional calculus based on it are used to build their own recognition algorithms. In the general case, the use of a logical recognition method provides for the presence of logical connections expressed by the optimal continuation of a k-valued function over the entire feature space, in which the variables are the logical features of the objects or phenomena being recognized. The goal of this work is to develop a logical method for object recognition consisting of a reference table with logical features and classes of non-intersecting objects, which are specified as vectors from a given feature space. The method consists of considering the reference table as a logical function that is not defined everywhere and constructing an optimal continuation of the logical function to the entire feature space, which determines the extension of classes to the entire space.

We derive entropy bounds for the absolute convex hull of vectors $X= (x_1 , \ldots , x_p)\in \mathbb{R}^{n \times p} $ in $\mathbb{R}^n$ and apply this to the case where $X$ is the $d$-fold tensor matrix $$X = \underbrace{\Psi \otimes \cdots \otimes \Psi}_{d \ {\rm times} }\in \mathbb{R}^{m^d \times r^d },$$ with a given $\Psi = ( \psi_1 , \ldots , \psi_r ) \in \mathbb{R}^{m \times r} $, normalized to that $ \| \psi_j \|_2 \le 1$ for all $j \in \{1 , \ldots , r\}$. For $\epsilon >0$ we let ${\cal V} \subset \mathbb{R}^m$ be the linear space with smallest dimension $M ( \epsilon , \Psi)$ such that $ \max_{1 \le j \le r } \min_{v \in {\cal V} } \| \psi_j - v \|_2 \le \epsilon$. We call $M( \epsilon , \psi)$ the $\epsilon$-approximation of $\Psi$ and assume it is -- up to log terms -- polynomial in $\epsilon$. We show that the entropy of the absolute convex hull of the $d$-fold tensor matrix $X$ is up to log-terms of the same order as the entropy for the case $d=1$. The results are generalized to absolute convex hulls of tensors of functions in $L_2 (\mu)$ where $\mu$ is Lebesgue measure on $[0,1]$. As an application we consider the space of functions on $[0,1]^d$ with bounded $q$-th order Vitali total variation for a given $q \in \mathbb{N}$. As a by-product, we construct an orthonormal, piecewise polynomial, wavelet dictionary for functions that are well-approximated by piecewise polynomials.

We construct an estimator $\widehat{\Sigma}$ for covariance matrices of unknown, centred random vectors X, with the given data consisting of N independent measurements $X_1,...,X_N$ of X and the wanted confidence level. We show under minimal assumptions on X, the estimator performs with the optimal accuracy with respect to the operator norm. In addition, the estimator is also optimal with respect to direction dependence accuracy: $\langle \widehat{\Sigma}u,u\rangle$ is an optimal estimator for $\sigma^2(u)=\mathbb{E}\langle X,u\rangle^2$ when $\sigma^2(u)$ is ``large".

A set $C$ of vertices in a graph $G=(V,E)$ is an identifying code if it is dominating and any two vertices of $V$ are dominated by distinct sets of codewords. This paper presents a survey of Iiro Honkala's contributions to the study of identifying codes with respect to several aspects: complexity of computing an identifying code, combinatorics in binary Hamming spaces, infinite grids, relationships between identifying codes and usual parameters in graphs, structural properties of graphs admitting identifying codes, and number of optimal identifying codes.