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The trace-dev-div inequality in $H^s$ controls the trace in the norm of $H^s$ by that of the deviatoric part plus the $H^{s-1}$ norm of the divergence of a quadratic tensor field different from the constant unit matrix. This is well known for $s=0$ and established for orders $0\le s\le 1$ and arbitrary space dimension in this note. For mixed and least-squares finite element error analysis in linear elasticity, this inequality allows to establish robustness with respect to the Lam\'e parameter $\lambda$.

This paper deals with the equation $-\Delta u+\mu u=f$ on high-dimensional spaces $\mathbb{R}^m$, where the right-hand side $f(x)=F(Tx)$ is composed of a separable function $F$ with an integrable Fourier transform on a space of a dimension $n>m$ and a linear mapping given by a matrix $T$ of full rank and $\mu\geq 0$ is a constant. For example, the right-hand side can explicitly depend on differences $x_i-x_j$ of components of $x$. Following our publication [Numer. Math. (2020) 146:219--238], we show that the solution of this equation can be expanded into sums of functions of the same structure and develop in this framework an equally simple and fast iterative method for its computation. The method is based on the observation that in almost all cases and for large problem classes the expression $\|T^ty\|^2$ deviates on the unit sphere $\|y\|=1$ the less from its mean value the higher the dimension $m$ is, a concentration of measure effect. The higher the dimension $m$, the faster the iteration converges.

We survey recent developments in the field of complexity of pathwise approximation in $p$-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion. First, we briefly review the case of equations with globally Lipschitz continuous coefficients, for which an error rate of at least $1/2$ in terms of the number of evaluations of the driving Brownian motion is always guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate that giving up the global Lipschitz continuity of the coefficients may lead to a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an arbitrary slow decay of the smallest possible error that can be achieved on the basis of finitely many evaluations of the driving Brownian motion. Finally, we turn to recent positive results for equations with a drift coefficient that is not globally Lipschitz continuous. Here we focus on scalar equations with a Lipschitz continuous diffusion coefficient and a drift coefficient that satisfies piecewise smoothness assumptions or has fractional Sobolev regularity and we present corresponding complexity results.

We study the problem of adaptive variable selection in a Gaussian white noise model of intensity $\varepsilon$ under certain sparsity and regularity conditions on an unknown regression function $f$. The $d$-variate regression function $f$ is assumed to be a sum of functions each depending on a smaller number $k$ of variables ($1 \leq k \leq d$). These functions are unknown to us and only few of them are nonzero. We assume that $d=d_\varepsilon \to \infty$ as $\varepsilon \to 0$ and consider the cases when $k$ is fixed and when $k=k_\varepsilon \to \infty$, $k=o(d)$ as $\varepsilon \to 0$. In this work, we introduce an adaptive selection procedure that, under some model assumptions, identifies exactly all nonzero $k$-variate components of $f$. In addition, we establish conditions under which exact identification of the nonzero components is impossible. These conditions ensure that the proposed selection procedure is the best possible in the asymptotically minimax sense with respect to the Hamming risk.

We present some basic elements of the theory of generalised Br\`{e}gman relative entropies over nonreflexive Banach spaces. Using nonlinear embeddings of Banach spaces together with the Euler--Legendre functions, this approach unifies two former approaches to Br\`{e}gman relative entropy: one based on reflexive Banach spaces, another based on differential geometry. This construction allows to extend Br\`{e}gman relative entropies, and related geometric and operator structures, to arbitrary-dimensional state spaces of probability, quantum, and postquantum theory. We give several examples, not considered previously in the literature.

The logics $\mathsf{CS4}$ and $\mathsf{IS4}$ are the two leading intuitionistic variants of the modal logic $\mathsf{S4}$. Whether the finite model property holds for each of these logics have been long-standing open problems. It was recently shown that $\mathsf{IS4}$ has the finite frame property and thus the finite model property. In this paper, we prove that $\mathsf{CS4}$ also enjoys the finite frame property. Additionally, we investigate the following three logics closely related to $\mathsf{IS4}$. The logic $\mathsf{GS4}$ is obtained by adding the G\"odel--Dummett axiom to $\mathsf{IS4}$; it is both a superintuitionistic and a fuzzy logic and has previously been given a real-valued semantics. We provide an alternative birelational semantics and prove strong completeness with respect to this semantics. The extension $\mathsf{GS4^c}$ of $\mathsf{GS4}$ corresponds to requiring a crisp accessibility relation on the real-valued semantics. We give a birelational semantics corresponding to an extra confluence condition on the $\mathsf{GS4}$ birelational semantics and prove strong completeness. Neither of these two logics have the finite model property with respect to their real-valued semantics, but we prove that they have the finite frame property for their birelational semantics. Establishing the finite birelational frame property immediately establishes decidability, which was previously open for these two logics. Our proofs yield NEXPTIME upper bounds. The logic $\mathsf{S4I}$ is obtained from $\mathsf{IS4}$ by reversing the roles of the modal and intuitionistic relations in the birelational semantics. We also prove the finite frame property, and thereby decidability, for $\mathsf{S4I}$

We present an asymptotic-preserving (AP) numerical method for solving the three-temperature radiative transfer model, which holds significant importance in inertial confinement fusion. A carefully designedsplitting method is developed that can provide a general framework of extending AP schemes for the gray radiative transport equation to the more complex three-temperature radiative transfer model. The proposed scheme captures two important limiting models: the three-temperature radiation diffusion equation (3TRDE) when opacity approaches infinity and the two-temperature limit when the ion-electron coupling coefficient goes to infinity. We have rigorously demonstrated the AP property and energy conservation characteristics of the proposed scheme and its efficiency has been validated through a series of benchmark tests in the numerical part.

A finite-energy signal is represented by a square-integrable, complex-valued function $t\mapsto s(t)$ of a real variable $t$, interpreted as time. Similarly, a noisy signal is represented by a random process. Time-frequency analysis, a subfield of signal processing, amounts to describing the temporal evolution of the frequency content of a signal. Loosely speaking, if $s$ is the audio recording of a musical piece, time-frequency analysis somehow consists in writing the musical score of the piece. Mathematically, the operation is performed through a transform $\mathcal{V}$, mapping $s \in L^2(\mathbb{R})$ onto a complex-valued function $\mathcal{V}s \in L^2(\mathbb{R}^2)$ of time $t$ and angular frequency $\omega$. The squared modulus $(t, \omega) \mapsto \vert\mathcal{V}s(t,\omega)\vert^2$ of the time-frequency representation is known as the spectrogram of $s$; in the musical score analogy, a peaked spectrogram at $(t_0,\omega_0)$ corresponds to a musical note at angular frequency $\omega_0$ localized at time $t_0$. More generally, the intuition is that upper level sets of the spectrogram contain relevant information about in the original signal. Hence, many signal processing algorithms revolve around identifying maxima of the spectrogram. In contrast, zeros of the spectrogram indicate perfect silence, that is, a time at which a particular frequency is absent. Assimilating $\mathbb{R}^2$ to $\mathbb{C}$ through $z = \omega + \mathrm{i}t$, this chapter focuses on time-frequency transforms $\mathcal{V}$ that map signals to analytic functions. The zeros of the spectrogram of a noisy signal are then the zeros of a random analytic function, hence forming a Point Process in $\mathbb{C}$. This chapter is devoted to the study of these Point Processes, to their links with zeros of Gaussian Analytic Functions, and to designing signal detection and denoising algorithms using spatial statistics.

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a compact space, Gr\"ochenig in \cite{G} discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all $1\le p\le\infty$, we develop weighted least $\ell_p$ approximation induced by a sequence of Marcinkiewicz-Zygmund inequalities in $L_p$ on a compact smooth Riemannian manifold $\Bbb M$ with normalized Riemannian measure (typical examples are the torus and the sphere). In this paper we derive corresponding approximation theorems with the error measured in $L_q,\,1\le q\le\infty$, and least quadrature errors for both Sobolev spaces $H_p^r(\Bbb M), \, r>d/p$ generated by eigenfunctions associated with the Laplace-Beltrami operator and Besov spaces $B_{p,\tau}^r(\Bbb M),\, 0<\tau\le \infty, r>d/p $ defined by best polynomial approximation. Finally, we discuss the optimality of the obtained results by giving sharp estimates of sampling numbers and optimal quadrature errors for the aforementioned spaces.

We present a new method to compute the solution to a nonlinear tensor differential equation with dynamical low-rank approximation. The idea of dynamical low-rank approximation is to project the differential equation onto the tangent space of a low-rank tensor manifold at each time. Traditionally, an orthogonal projection onto the tangent space is employed, which is challenging to compute for nonlinear differential equations. We introduce a novel interpolatory projection onto the tangent space that is easily computed for many nonlinear differential equations and satisfies the differential equation at a set of carefully selected indices. To select these indices, we devise a new algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes any tensor train and its tangent space with tensor cross interpolants. We demonstrate the proposed method with applications to tensor differential equations arising from the discretization of partial differential equations.