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We characterize bifurcation values of polynomial functions by using the theory of perverse sheaves and their vanishing cycles. In particular, by introducing a method to compute the jumps of the Euler characteristics with compact support of…
In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic partial differential equations. Our approach is probabilistic. The theory of…
We prove that suitable properties of the twists by Dirichlet characters of an L-function of degree 2 imply that its Euler product is of polynomial type.
Given a subset $D$ of the Euclidean space, we study nonlocal quadratic forms that take into account tuples $(x,y) \in D \times D$ if and only if the line segment between $x$ and $y$ is contained in $D$. We discuss regularity of the…
We investigate the internal observability of the wave equation with Dirichlet boundary conditions in a triangular domain. More precisely, the domain taken into exam is the half of the equilateral triangle. Our approach is based on Fourier…
Epsilon-near-zero and epsilon near-pole materials enable reflective systems supporting a class of symmetry-protected and accidental embedded eigenstates (EE) characterized by a diverging phase-resonance. Here we show that pairs of…
In this paper, we revisit approximation properties of piecewise polynomial spaces, which contain more than ${\cal P}_{r-1}$ but not ${\cal P}_r$. We develop more accurate upper and lower error bounds that are sharper than those used in…
We consider multidimensional random walks in pyramidal cones (or multidimensional orthants), which are intersections of a finite number of half-spaces. We explore the connection between the existence of (positive) discrete harmonic…
This paper develops a threshold model with a time-varying threshold, represented using a wavelet series expansion. The model adequately captures irregular and abrupt variations, as well as smooth changes in the threshold parameter, allowing…
This paper presents division polynomials for twisted Edwards curves. Their chief property is that they characterise the $n$-torsion points of a given twisted Edwards curve. We also present results concerning the coefficients of these…
We study a family of Laguerre--Sobolev orthogonal polynomials associated with a Sobolev inner product arising from second--order boundary value problems on the semi--infinite interval $(0,+\infty)$. These polynomials generate an orthogonal…
In this paper, we propose a new two-dimensional directional discrete wavelet transform that can decompose an image into 12 multiscale directional edge components. The proposed transform is designed in a fully discrete setting and thus is…
We study singularity confinement phenomena in examples of delay-differential Painlev\'e equations, which involve shifts and derivatives with respect to a single independent variable. We propose a geometric interpretation of our results in…
This work is concerned with an inverse electromagnetic scattering problem in two dimensions. We prove that in the TE polarization case, the knowledge of the electric far-field pattern incited by a single incoming wave is sufficient to…
We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming--Viot process is a particular example. The defining property of finite dimensional polynomial processes considered by Cuchiero…
We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in R^n such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every…
In this paper we give a review on the computational methods used to characterize the complex singularities developed by some relevant PDEs. We begin by reviewing the singularity tracking method based on the analysis of the Fourier spectrum.…
We explore geometric and topological properties of 3d surfaces swept by Poncelet triangles, as well as tangles formed by associated points.
We study Poncelet's Theorem in finite projective coordinate planes over the field $GF(p)$ and concentrate on a particular pencil of conics. For pairs of such conics we investigate whether we can find polygons with $n$ sides, which are…
Based upon elements of the modern Pseudoanalytic Function Theory, we analyse a new method for numerically approaching the solution of the Dirichlet boundary value problem, corresponding to the two-dimensional Electrical Impedance Equation.…