Communications in Contemporary Mathematics

Loss functions are an essential part in modern data-driven approaches, such as bi-level training scheme and machine learnings. In this paper, we propose a loss function consisting of a $r$-order (an)-isotropic total variation semi-norms ${\mathrm{\text{TV}}}^r$, $r\in {ℝ}^{+}$, defined via the Riemann–Liouville (RL) fractional derivative. We focus on studying key theoretical properties, such as the lower semi-continuity and compactness with respect to both the function and the order of derivative $r$, of such loss functions.

This paper proves new results on spectral and scattering theory for matrix-valued Schrödinger operators on the discrete line with non-compactly supported perturbations whose first moments are assumed to exist. In particular, a Levinson theorem is proved, in which a relation between scattering data and spectral properties (bound and half-bound states) of the corresponding Hamiltonians is derived. The proof is based on stationary scattering theory with prominent use of Jost solutions at complex energies that are controlled by Volterra-type integral equations.

We establish global-in-time averaging for the $L^2$-critical dispersion-managed nonlinear Schrödinger equation in the fast dispersion management regime. In particular, in the case of nonzero average dispersion, we establish averaging with any subcritical data, while in the case of a strictly positive dispersion map, we obtain averaging for data in $L^2$.

In the study of strong homology Mardešić and Prasolov isolated a certain inverse system of abelian groups $A$ indexed by elements of ${\omega }^{\omega }$. They showed that if strong homology is additive on a class of spaces containing closed subsets of Euclidean spaces then the higher derived limits ${lim}^{n}A$ must vanish, for $n>0$. They also proved that under the Continuum Hypothesis ${lim}^{1}A\ne 0$. The question whether ${lim}^{n}A$ vanishes, for all $n>0$, has attracted considerable interest from set theorists. Dow, Simon and Vaughan showed that under the Proper Forcing Axiom (PFA) ${lim}^{1}A=0$. Bergfalk showed that it is consistent that ${lim}^{2}A$ does not vanish. Later Bergfalk and Lambie-Hanson showed that, modulo a weakly compact cardinal, it is relatively consistent with ZFC that ${lim}^{n}A=0$, for all $n$. The large cardinal assumption was recently removed by Bergfalk, Hrušak and Lambie-Hanson. We complete the picture by showing that, for any $n>0$, it is relatively consistent with ZFC that ${lim}^{n}A\ne 0$.

We show that spatial patterns (“hotspots”) may form in the crime model

We do this by first constructing classical solutions to the nonlocal scalar problem

We prove a Steiner formula for regular surfaces with no characteristic points in 3D contact sub-Riemannian manifolds endowed with an arbitrary smooth volume. The formula we obtain, which is equivalent to a half-tube formula, is of local nature. It can thus be applied to any surface in a region not containing characteristic points. We provide a geometrical interpretation of the coefficients appearing in the expansion, and compute them on some relevant examples in three-dimensional sub-Riemannian model spaces. These results generalize those obtained in [Z. M. Balogh, F. Ferrari, B. Franchi, E. Vecchi and K. Wildrick, Steiner’s formula in the Heisenberg group, Nonlinear Anal. 126 (2015) 201–217; M. Ritoré, Tubular neighborhoods in the sub-Riemannian Heisenberg groups, Adv. Calc. Var. 14(1) (2021) 1–36] for the Heisenberg group.

We control the behavior of the Poincaré constant along the Polchinski renormalization flow using a dynamic version of $\mathrm{\Gamma }$-calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by Klartag and Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general ${\phi }^4$-measures and discuss the interpretation in terms of transport maps.

Motivated by the connection between the first eigenvalue of the Dirichlet–Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional rigidity, closely related to the Steklov eigenvalue. From a variational point of view, such a new object corresponds to the sharp constant for the trace embedding of $W^{1,2}(\mathrm{\Omega })$ into $L^1(\partial \mathrm{\Omega })$. We obtain various equivalent variational formulations, present some properties of the state function and obtain some sharp geometric estimates, both for planar simply connected sets and for convex sets in any dimension.

We show the existence and the regularity properties of (a class of) weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with density-dependent viscosity coefficients, by analyzing a fourth-order nonlinear elliptic equation for the stream function.

For some stationary symmetric flows, we reformulate the Navier–Stokes equations as ordinary differential equations and give explicit examples of weak solutions. We present some further (ir-)regularity results in the case of piecewise-constant viscosity coefficients.

We show that the natural bi-invariant cone structure on the linear symplectic group $\mathrm{\text{Sp}}(2n)$ is globally hyperbolic in the positively elliptic region ${\mathrm{\text{Sp}}}_{\mathrm{\text{ell}}}^{+}(2n)$. This answers a question by Abbondandolo, Benedetti and Polterovich and implies a formula for a bi-invariant Lorentzian distance function defined by these authors for elements in this region. Moreover we give a characterization of the positively elliptic region and of $-\mathrm{\text{id}}\in \mathrm{\text{Sp}}(2n)$ in terms of the causality of this cone structure.