Convex solutions of a functional equation arising in information theory

Given a convex function f defined for positive real variables, the so-called Csiszár f-divergence is a function If defined for two n-dimensional probability vectors p=(p₁,…,pn) and q=(q₁,…,qn) as . For this generalized measure of entropy to have distance-like properties, especially symmetry, it is necessary for f to satisfy the following functional equation: for all x>0. In the present paper we determine all the convex solutions of this functional equation by proposing a way of generating all of them. In doing so, existing usual f-divergences are recovered and new ones are proposed.     

A functional partial differential equation arising in a cell growth model with dispersion

In this paper we solve an initial‐boundary value problem that involves a pde with a nonlocal term. The problem comes from a cell division model where the growth is assumed to be stochastic. The deterministic version of this problem yields a first‐order pde; the stochastic version yields a second‐order parabolic pde. There are no general methods for solving such problems even for the simplest cases owing to the nonlocal term. Although a solution method was devised for the simplest version of the first‐order case, the analysis does not readily extend to the second‐order case. We develop a method for solving the second‐order case and obtain the exact solution in a form that allows us to study the long time asymptotic behaviour of solutions and the impact of the dispersion term. We establish the existence of a large time attracting solution towards which solutions converge exponentially in time. The dispersion term does not appear in the exponential rate of convergence.     

A note on a singular integral equation arising in water wave scattering

A quick method of solution of a singular integral equationof the first kind involving both logarithmic singularity aswell as Cauchy-type singularity is explained.     

A note on rational solutions to a Hirota-Satsuma-like equation

With the generalized bilinear operators based on a prime number $p=3$, a Hirota-Satsuma-like equation is proposed. Rational solutions are generated and graphically described by using symbolic computation software Maple.     

A functional equation arising in dynamic programming

In this paper, we utilize some fixed point theorems of contractive type to present a few existence and uniqueness theorems for a functional equation arising in dynamic programming of continuous multistage decision processes.     

On the solutions of a functional equation arising from multiplication of quantum integers

This paper is the first of several papers in which we prove, for the case where the fields of coefficients are of characteristic zero, four open problems posed in the work of Melvyn Nathanson (2003) [1] concerning the solutions of a functional equation arising from multiplication of quantum integers q[n]=qn−1+qn−2+?+q+1. In this paper, we prove one of the problems. The next papers, namely [002], [003] and [004] by Lan Nguyen, contain the solutions to the other 3 problems.     

A note on the D'Alembert functional equation

It is proved that the analog of DAlemberts equation for a group has symmetric solutions which are central functions only in the commutative case. Necessary and sufficient conditions are given for the representability of the solutions in the form of a semisum of well-defined homomorphisms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 182–183, 1974.     

A note on a nonlinear functional equation and its application

This paper treats the following type of nonlinear functional equations

     

On a functional equation arising in information theory

We derive the representation of generalized additive measures of information, depending on two probability distributions and having the sum property with a regular generating function. This family of measures includes the Shannon entropy, the entropy of degree (,), the generalized information energy, the sine entropy and the generalized directed divergence.     

非线性Schodinger方程多重正解的一个注记

Some results about the multiplicity of positive solutions for a nonlinear SchrOdinger equation are given by means of variational methods.     

A note on an integrated cauchy functional equation

In characterizing the semistable law, [Shimizu reduced the problem to solving the equation ,x0 where andv are given positive measures on [0, ). In this note, we obtain a simple proof and show that some of his conditions can be weakened.     

A probabilistic note on the Cauchy functional equation

Aequationes mathematicae - In this short note we show that a weak version of Bernstein’s characterization of the normal distribution implies the local integrability of a measurable solution...     

A note on the multiplicity of solutions for a second-order difference equation with a parameter

We undertake the question of the multiplicity of solutions to the discrete Emden-Fowler BVP by correcting some results contained in [2].     

A functional equation arising in multi-agent statistical decision theory

Using recent results of Járai we show that the measurable solutions of the functional equationf(x ₁ y ₁,...,x _n y _n )f((1?x ₁)(1?y ₁),..., (1?x _n )(1?y _n ))=f(x ₁(1?y ₁),...,x _n (1 ? (y _n ))f(y ₁(1?x ₁),...,y _n(1 ?x _n )), wheref: (0, 1) ⁿ → (0, ∞) and 0<x _i ,y _i <1,i=1,...,n, are of the form $$f(x_1 ,...,x_n ) = c \exp \left( {\sum\limits_{i = 1}^n {a_i (x_1 - x_1^2 ))} \prod\limits_{i = 1}^n {x_i^{b_1 } ,} } \right.$$ wherec>0,a ₁,...,a _n andb ₁,..., _b are arbitrary real constants. This result enables one to characterize certain independence-preserving methods of aggregating probability distributions over four alternatives.     

A note on periodic solutions of a differential equation with two parameters

In [1] (p. 215), the authors Andronov, Leontovich-Andronova, Gordon, and Maier, consider the following equation: $$\left\{ \begin{gathered} \tfrac{{dx}}{{dt}} = y, \hfill \\ \tfrac{{dy}}{{dt}} = x + x^2 - \left( {\varepsilon _1 + \varepsilon _2 x} \right)y, \hfill \\ \end{gathered} \right.$$ whereε ₁ andε ₂ are real constants andε ₁ andε ₂ are not both zero. They proved that there are no non-trivial periodic solutions except possibly for the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ . They left that case as an open problem. In this note we prove that there are indeed no non-trivial periodic solutions in the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ either. Our method of proof consists essentially of constructing a Dulac function (see [6] and [9]) and using the conception of Duff's rotated vector field (see [4], [7], [8], [10], and [11]).     

A note on a functional equation on groups with involutions in two variables

Aequationes mathematicae - In this paper, we determine the central solutions $$f:Gtimes Grightarrow {{mathbb {C}}}$$ of the functional equation $$begin{aligned} f(x_1sigma y_1,x_2tau...     

A note on solutions to pomeron functional equations

We construct exact solutions to the pomeron functional equation and ageneralization of it. These results clarify and extend previous workon this problem.