Related papers: Newell-Whitehead-Segel equation,A Simpler Proof
Noether's problem is classical and very important problem in algebra. It is an intrinsically interesting problem in invariant theory, but with far reaching applications in the sutdy of moduli spaces, PI-algebras, and the Inverse problem of…
This paper offers two elementary yet precise derivations of an exact formula \[ W(n) = \sum_{i=1} ^{n} \lceil \lg i \rceil = n \lceil \lg n \rceil - 2^{\lceil \lg n \rceil} + 1 \] for the maximum number $ W(n) $ of comparisons of keys…
We establish the arithmetic Siegel-Weil formula on the modular curve $\mathcal{X}_{0}(N)$ for arbitrary level $N$, i.e., we relate the arithmetic degrees of special cycles on $\mathcal{X}_{0}(N)$ to the derivatives of Fourier coefficients…
Two main results are presented: 1) a new class of applied problems that lead to equations with $(p,q)$-Laplace is presented; 2) a method for solving nonlinear boundary value problems involving $(p,q)$-Laplace with measurable unbounded…
The purpose of this article is to verify the conjectures of the previous paper in the particular case of $GL(4)$. We accomplish this in general, but observe two failures of the conjectures: First, that the Strong Interchange of Integrals…
We consider the reflection-transmission problem in a waveguide with obstacle. At certain frequencies, for some incident waves, intensity is perfectly transmitted and the reflected field decays exponentially at infinity. In this work, we…
We notice a remarkable connection between Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four…
We show that finite-dimensional Lie algebras over a field of characteristic zero such that their high-degree cohomology in any finite-dimensional non-trivial irreducible module vanishes, are, essentially, direct sums of semisimple and…
In this work, a class of non-linear weakly singular fractional integro-differential equations is considered, and we first prove existence, uniqueness, and smoothness properties of the solution under certain assumptions on the given data. We…
We present numerical methods based on the fast Fourier transform (FFT) to solve convolution integral equations on a semi-infinite interval (Wiener-Hopf equation) or on a finite interval (Fredholm equation). We extend and improve a FFT-based…
The Fourier transform of the indicator function of arbitrary polygons and polyhedra is computed for complex wavevectors. Using the divergence theorem and Stokes' theorem, closed expressions are obtained. Apparent singularities, all…
We study the representations of the W-algebra W(g) associated to an arbitrary finite-dimensional simple Lie algebra g via the quantized Drinfeld-Sokolov reductions. The characters of irreducible representations with non-degenerate highest…
This paper discusses a generalization of spectral representations related to convex one-homogeneous regularization functionals, e.g. total variation or $\ell^1$-norms. Those functionals serve as a substitute for a Hilbert space structure…
In this note is presented a method, given nodal values on multidimensional nonconforming spectral elements, for calculating global Fourier-series coefficients. This method is ``exact'' in that given the approximation inherent in the…
The nonlinear integral equation P(x)=\int_alpha^beta dy w(y) P(y) P(x+y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions P(x). For polynomial solutions, this nonlinear…
In this paper, we study one typical Einstein-Weyl equation. It arises from Ferapontov and Kruglikov's investigation on the integrability of several dispersionless partial differential equations and the geometry of their formal…
We prove that every subset of $\{1,\dots, N\}$ which does not contain any solutions to the equation $x+y+z=3w$ has at most $\exp(-c(\log N)^{1/5+o(1)})N$ elements, for some $c>0$. This theorem improves upon previous estimates. Additionally,…
We show that there are many (compact) convex semi-algebraic sets in euclidean space that do not have a semidefinite representation. This gives a negative answer to a question by Nemirovski, resp. it shows that the Helton-Nie conjecture is…
Given a finite-dimensional FTvN system $(\mathbb{V},\mathbb{W},\lambda)$, we study the convexification of the spectral set $\lambda^{-1}(\mathcal{C})$ induced by a set $\mathcal{C} \subseteq \mathbb{W}$. While the case of invariant…
The transmission eigenvalue problem (TEP) plays a central role in inverse scattering theory. Despite substantial theoretical progress, the numerical solution of direct and inverse TEP in spherically symmetric domains with variable…