Isolation of the singular part of the electromagnetic field scatteredby objects with sharp edges and conic points

In this article we propose a method for overcoming a number of problems that may arise when solving scattering problems on objects of complex structure. This approach can be used for mathematical and numerical modeling of waveguide, emitting and scattering systems of various complexity.

bel and Euler Summation Formulas for SBV(R)

The purpose of this paper is to show that the natural setting for various Abel and Euler-Maclaurin summation formulas is the class of special function of bounded variation. A function of one real variable is of bounded variation if its distributional derivative is a Radom measure. Such a function decomposes uniquely as sum of three components: the first one is a convergent series of piece-wise constant function, the second one is an absolutely continuous function and the last one is the so-called singular part, that is a continuous function whose derivative vanishes almost everywhere. A function of bounded variation is special if its singular part vanishes identically. We generalize such space of special function of bounded variation to include higher order derivatives and prove that the functions of such spaces admit a Euler-Maclaurin summation formula. Such a result is obtained by deriving in this setting various integration by part formulas which generalizes various classical Abel summation formulas.

Dynamics and Stability of Non-Smooth Dynamical Systems with Two Switches

One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. At this work, we leverage a methodology that, through blow-ups and singular perturbation, allows the extension of Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, non-linear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto-like theorems characterizing semi-local structural stability.

A Nonhomogeneous Boundary Value Problem for Steady State Navier-Stokes Equations in a Multiply-Connected Cusp Domain

The boundary value problem for the steady Navier–Stokes system is considered in a 2D multiply-connected bounded domain with the boundary having a power cusp singularity at the point O. The case of a boundary value with nonzero flow rates over connected components of the boundary is studied. It is also supposed that there is a source/sink in O. In this case the solution necessarily has an infinite Dirichlet integral. The existence of a solution to this problem is proved assuming that the flow rates are “sufficiently small” . This condition does not require the norm of the boundary data to be small. The solution is constructed as the sum of a function with the finite Dirichlet integral and a singular part coinciding with the asymptotic decomposition near the cusp point.

Numerical simulation of rigid particles in Stokes flow: lubrication correction for general shape of particles

We address the problem of numerical simulation of suspensions of rigid particles in a Stokes flow. We focus on the inclusion of the singular short range interaction effects (lubrication effects) in the simulations when the particles come close one to another. As in LefebvreMerletNguyen2015, the key idea is to decompose the velocity and pressure flows in a sum of a singular and a regular part. In this article, the singular part is computed using an explicit asymptotic expansion of the solution when the distance goes to zero. This expansion is similar to the asymptotic expansion proposed in HillairetKelai2015 but is more appropriate for numerical simulations of suspensions. It can be computed for any locally convex (particles convex close to the contact point) and regular shape of particles. Using HillairetKelai2015 as an intermediate result, we prove that the remaining part is regular in the sense that it is bounded independently of the distance. As a consequence, only a small number of degrees of freedom are necessary to obtain accurate results. The method is tested in dimension 2 for clusters of two or three aligned particles with general rigid velocities. We show that, as expected, the convergence is independent of the distance.

Pansu–Wulff shapes in ℍ1

We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

Asymptotic analysis of the Korteweg-de Vries equation by the nonlinear WKB technique

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative. The non-linear WKB technique has been used to construct the asymptotic step-like solution to the equation. Such a solution contains regular and singular parts of the asymptotics. The regular part of the solution describes the background of the wave process, while its singular part reflects specific features associated with soliton properties. The singular part of the searched asymp\-totic solution has the main term that, like the soliton solution, is the quickly decreasing function of the phase variable $\tau$. In contrast, other terms do not possess this property. An algorithm of constructing asymptotic step-like solutions to the singularly perturbed Korteweg--de Vries equation with variable coefficients is presented. In some sense, the constructed asymptotic solution is similar to the soliton solution to the Korteweg-de Vries equation $u_t+uu_x+u_{xxx}=0$. Statement on the accuracy of the main term of the asymptotic solution is proven.

A Generator of Bivariate Distributions: Properties, Estimation, and Applications

In 2020, El-Morshedy et al. introduced a bivariate extension of the Burr type X generator (BBX-G) of distributions, and Muhammed presented a bivariate generalized inverted Kumaraswamy (BGIK) distribution. In this paper, we propose a more flexible generator of bivariate distributions based on the maximization process from an arbitrary three-dimensional baseline distribution vector, which is of interest for maintenance and stress models, and expands the BBX-G and BGIK distributions, among others. This proposed generator allows one to generate new bivariate distributions by combining non-identically distributed baseline components. The bivariate distributions belonging to the proposed family have a singular part due to the latent component which makes them suitable for modeling two-dimensional data sets with ties. Several distributional and stochastic properties are studied for such bivariate models, as well as for its marginals, conditional distributions, and order statistics. Furthermore, we analyze its copula representation and some related association measures. The EM algorithm is proposed to compute the maximum likelihood estimations of the unknown parameters, which is illustrated by using two particular distributions of this bivariate family for modeling two real data sets.

Lorenz-generated bivariate Archimedean copulas

A novel generating mechanism for non-strict bivariate Archimedean copulas via the Lorenz curve of a non-negative random variable is proposed. Lorenz curves have been extensively studied in economics and statistics to characterize wealth inequality and tail risk. In this paper, these curves are seen as integral transforms generating increasing convex functions in the unit square. Many of the properties of these “Lorenz copulas”, from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating Lorenz curve. For instance, by selecting random variables with a lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An “alchemy” of Lorenz curves that can be used as general framework to build multiparametric families of copulas is also discussed.