Stability of a simple Levi–Civitá functional equation on non-unital commutative semigroups

In this paper, we study the Hyers–Ulam stability of a simple Levi–Civitá functional equation f(x+y)=f(x)h(y)+f(y) and its pexiderization f(x+y)= g(x) h(y)+k(y) on non-unital commutative semigroups by investigating the functional inequalities |f(x+y)?f(x)h(y)?f(y)|≤?? and |f(x+y)?g(x)h(y)?k(y)|≤??, respectively. We also study the bounded solutions of the simple Levi–Civitá functional inequality.     

The Kac–Bernstein functional equation on Abelian groups

Aequationes mathematicae - We consider the Kac–Bernstein functional equation $$\begin{aligned} f(x+y)g(x-y)=f(x)f(y)g(x)g(-y), \quad x, y\in X, \end{aligned}$$ on an arbitrary Abelian group...     

A motion with a permanent segment-areal speed and the Levi-Cività functional equation

Summary. It is proved that a motion with constant segmential speed is possible only along conics. The proof is based on the theory of Levi-Cività functional equations.     

General solution of the Bagley–Torvik equation with fractional-order derivative

This paper investigates the general solution of the Bagley–Torvik equation with 1/2-order derivative or 3/2-order derivative. This fractional-order differential equation is changed into a sequential fractional-order differential equation (SFDE) with constant coefficients. Then the general solution of the SFDE is expressed as the linear combination of fundamental solutions that are in terms of α-exponential functions, a kind of functions that play the same role of the classical exponential function. Because the number of fundamental solutions of the SFDE is greater than 2, the general solution of the SFDE depends on more than two free (independent) constants. This paper shows that the general solution of the Bagley–Torvik equation involves actually two free constants only, and it can be determined fully by the initial displacement and initial velocity.     

A Levi-Civitá Equation on Compact Groups and Nonabelian Fourier Analysis

In this paper, we study the following Levi-Civitá equation $ w(xy) + w(yx) = \sum_{i=1}^{m} f_{i}(x)g_{i}(y) \quad \quad \quad ({\rm LC})$ on a compact group G, where w, f _i ’s, and g _i ’s are continuous complex-valued functions to determine. Our main ingredient is (nonabelian) Fourier analysis on compact groups. We apply the Fourier transform to Eq. (LC) on the product group G × G so that we obtain its several equivalent operator equations. Using those equivalent equations, we derive some crucial properties of solutions to Eq. (LC). Consequently, Eq. (LC) with m ≤ 2 is completely solved. In particular, a Wilson type equation arising from and playing a central role in [4] is solved on compact groups.     

Estimates on the solution of a schrödinger equation and some applications

Dedicated to Elliott Lieb on the occasion of his 60th birthday.     

The non-abelian simple groups g, |g| < 105— presentations

Of the 56 isomorphism classes of simple groups of order up to 10⁶ there are 10 of order up to 10⁵ distinct from PSL(2,q). Presentations for these groups are given here in the form G = < a, b; a² = b^m = 1, {r_i (a, b)}_iεI = 1 > where m is minimal (with respect to a). They are complete in the sense that any pair (a, b) of generators of G satisfying the relations a² = b^m = 1, with m minimal, will satisfy the defining relations of just one presentation listed here.     

Bright soliton solution of a Gross–Pitaevskii equation

We theoretically and numerically study the bright soliton solutions of a Gross–Pitaevskii equation governing one-dimensional (1D)(cigar-shaped) Bose–Einstein condensates (BEC) trapped in an optical lattice of 1D structure. The analytical expression of bright soliton is derived by using the variational approximation, which completely matches the numerical results with a range of potential’s parameters. Moreover, we determined the parameter domains for the persistence and non-persistence of bright soliton solutions.     

The non-abelian simple groups g, |g| < 106— character tables

Character tables are presented for all the 56 non-abelian simple groups of order up to 10⁶, including three tables for the family PSL(2,q) and twenty tables for individual groups (with some overlap). Information presented includes the power maps, the orders of the centralizers of elements, and tables of structure constants for each class of involutions.     

On the solution of a generalized Schrödinger-type equation

Lu(t)+(u,F)g(t)=f(t), tS. , ( F, g). .

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The authors wish to thank Professor Yu. A. Rozanov for his help and discussions.     

Approximate grid solution of a nonlocal boundary value problem for Laplace’s equation on a rectangle

A nonlocal boundary value problem for Laplace’s equation on a rectangle is considered. Dirichlet boundary conditions are set on three sides of the rectangle, while the boundary values on the fourth side are sought using the condition that they are equal to the trace of the solution on the parallel midline of the rectangle. A simple proof of the existence and uniqueness of a solution to this problem is given. Assuming that the boundary values given on three sides have a second derivative satisfying a Hölder condition, a finite difference method is proposed that produces a uniform approximation (on a square mesh) of the solution to the problem with second order accuracy in space. The method can be used to find an approximate solution of a similar nonlocal boundary value problem for Poisson’s equation.     

A note on “Quasi-analytical solution of two-dimensional Helmholtz equation”

The recent paper of Van Hirtum in this journal repeats a number of misconceptions about the use of conformal mappings in solving the two-dimensional Helmholtz equation. These are discussed, as is the fact that the numerical approach presented does not lead to accurate results. In general conformal mapping is not useful in solving Helmholtz’s equation. Other, accurate, techniques are briefly reviewed.     

On a complex fundamental solution of the Schrödinger equation

A second-order Schrödinger differential operator of parabolic type is considered, for which an explicit form of a fundamental solution is derived. Such operators arise in many problems of physics, and the fundamental solution plays the role of the Feynman propagation function.