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We consider the $h$-version of the finite-element method, where accuracy is increased by decreasing the meshwidth $h$ while keeping the polynomial degree $p$ constant, applied to the Helmholtz equation. Although the question "how quickly must $h$ decrease as the wavenumber $k$ increases to maintain accuracy?" has been studied intensively since the 1990s, none of the existing rigorous wavenumber-explicit analyses take into account the approximation of the geometry. In this paper we prove that for nontrapping problems solved using straight elements the geometric error is order $kh$, which is then less than the pollution error $k(kh)^{2p}$ when $k$ is large; this fact is then illustrated in numerical experiments. More generally, we prove that, even for problems with strong trapping, using degree four (in 2-d) or degree five (in 3-d) polynomials and isoparametric elements ensures that the geometric error is smaller than the pollution error for most large wavenumbers.

This paper presents new upper bounds on the rate of linear $k$-hash codes in $\mathbb{F}_q^n$, $q\geq k$, that is, codes with the property that any $k$ distinct codewords are all simultaneously distinct in at least one coordinate.

In this paper, we study optimal quadrature errors, approximation numbers, and sampling numbers in $L_2(\Bbb S^d)$ for Sobolev spaces ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with logarithmic perturbation on the unit sphere $\Bbb S^d$ in $\Bbb R^{d+1}$. First we obtain strong equivalences of the approximation numbers for ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with $\alpha>0$, which gives a clue to Open problem 3 as posed by Krieg and Vyb\'iral in \cite{KV}. Second, for the optimal quadrature errors for ${\rm H}^{\alpha,\beta}(\Bbb S^d)$, we use the "fooling" function technique to get lower bounds in the case $\alpha>d/2$, and apply Hilbert space structure and Vyb\'iral's theorem about Schur product theory to obtain lower bounds in the case $\alpha=d/2,\,\beta>1/2$ of small smoothness, which confirms the conjecture as posed by Grabner and Stepanyukin in \cite{GS} and solves Open problem 2 in \cite{KV}. Finally, we employ the weighted least squares operators and the least squares quadrature rules to obtain approximation theorems and quadrature errors for ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with $\alpha>d/2$ or $\alpha=d/2,\,\beta>1/2$, which are order optimal.

In this paper we consider an initial-boundary value problem with a Caputo time derivative of order $\alpha\in(0,1)$. The solution typically exhibits a weak singularity near the initial time and this causes a reduction in the orders of convergence of standard schemes. To deal with this singularity, the solution is computed with a fitted difference scheme on a graded mesh. The convergence of this scheme is analysed using a discrete maximum principle and carefully chosen barrier functions. Sharp error estimates are proved, which show an enhancement in the convergence rate compared with the standard L1 approximation on uniform meshes, and also indicate an optimal choice for the mesh grading. This optimal mesh grading is less severe than the optimal grading for the standard L1 scheme. Furthermore, the dependence of the error on the final time forms part of our error estimate. Numerical experiments are presented which corroborate our theoretical results.

We establish several properties of (weighted) generalized $\psi$-estimators introduced by Barczy and P\'ales in 2022: mean-type, monotonicity and sensitivity properties, bisymmetry-type inequality and some asymptotic and continuity properties as well. We also illustrate these properties by providing several examples including statistical ones as well.

We consider the success probability of the $L_0$-regularized box-constrained Babai point, which is a suboptimal solution to the $L_0$-regularized box-constrained integer least squares problem and can be used for MIMO detection. First, we derive formulas for the success probability of both $L_0$-regularized and unregularized box-constrained Babai points. Then we investigate the properties of the $L_0$-regularized box-constrained Babai point, including the optimality of the regularization parameter, the monotonicity of its success probability, and the monotonicity of the ratio of the two success probabilities. A bound on the success probability of the $L_0$-regularized Babai point is derived. After that, we analyze the effect of the LLL-P permutation strategy on the success probability of the $L_0$-regularized Babai point. Then we propose some success probability based column permutation strategies to increase the success probability of the $L_0$-regularized box-constrained Babai point. Finally, we present numerical tests to confirm our theoretical results and to show the advantage of the $L_0$ regularization and the effectiveness of the proposed column permutation algorithms compared to existing strategies.

In the context of two-player games over graphs, a language $L$ is called half-positional if, in all games using $L$ as winning objective, the protagonist can play optimally using positional strategies, that is, strategies that do not depend on the history of the play. In this work, we describe the class of parity automata recognising half-positional languages, providing a complete characterisation of half-positionality for $\omega$-regular languages. As corollaries, we establish decidability of half-positionality in polynomial time, finite-to-infinite and 1-to-2-players lifts, and show the closure under union of prefix-independent half-positional objectives, answering a conjecture by Kopczy\'nski.

We consider the community detection problem in a sparse $q$-uniform hypergraph $G$, assuming that $G$ is generated according to the Hypergraph Stochastic Block Model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al. (2015). We characterize the spectrum of the non-backtracking operator for the sparse HSBM and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, community detection for the sparse HSBM on $n$ vertices can be reduced to an eigenvector problem of a $2n\times 2n$ non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. To the best of our knowledge, this is the first provable and efficient spectral algorithm that achieves the conjectured threshold for HSBMs with $r$ blocks generated according to a general symmetric probability tensor.

In this paper, we combine the stabilizer free weak Galerkin (SFWG) method and the implicit $\theta$-schemes in time for $\theta\in [\frac{1}{2},1]$ to solve the fourth-order parabolic problem. In particular, when $\theta =1$, the full-discrete scheme is first-order backward Euler and the scheme is second-order Crank Nicolson scheme if $\theta =\frac{1}{2}$. Next, we analyze the well-posedness of the schemes and deduce the optimal convergence orders of the error in the $H^2$ and $L^2$ norms. Finally, numerical examples confirm the theoretical results.

In this paper we show that every graph $G$ of bounded maximum average degree ${\rm mad}(G)$ and with maximum degree $\Delta$ can be edge-colored using the optimal number of $\Delta$ colors in quasilinear expected time, whenever $\Delta\ge 2{\rm mad}(G)$. The maximum average degree is within a multiplicative constant of other popular graph sparsity parameters like arboricity, degeneracy or maximum density. Our algorithm extends previous results of Chrobak and Nishizeki [J. Algorithms, 1990] and Bhattacharya, Costa, Panski and Solomon [arXiv, 2023].