On a pythagorean-type functional equation with exponent 3 involving finite fields

We solve the functional equationf(x ³ +y ³) =f(x)³ +f(y)³ for maps of a finite field into itself.     

On the stability of the quadratic functional equation on bounded domains

A result of Skof and Terracini will be generalized; More precisely, we will prove that if a functionf : [-t, t]ⁿ →E satisfies the inequality (1) for some δ > 0 and for allx, y ∈ [-t, t]ⁿ withx + y, x - y ∈ [-t, t]ⁿ, then there exists a quadratic functionq: ℝⁿ →E such that ∥f(x) -q(x)∥ < (2912n² + 1872n + 334)δ for anyx ∈ [-t, t] ⁿ .     

Regularization of a three-element functional equation

In this paper we study the three-element functional equation

$(V\Phi )(z) \equiv \Phi (iz) + \Phi ( - iz) + G(z)\Phi \left( {\frac{1}{z}} \right) = g(z), z \in R,$

, subject to

$R: = \{ z:\left| z \right| < 1, \left| {\arg z} \right| < \frac{\pi }{4}\} .$

We assume that the coefficients G(z) and g(z) are holomorphic in R and their boundary values G ⁺(t) and g ⁺(t) belong to H(Γ), G(t)G(t ^?1) = 1. We seek for solutions Φ(z) in the class of functions holomorphic outside of \(\bar R\) such that they vanish at infinity and their boundary values Φ^?(t) also belong to H(Γ). Using the method of equivalent regularization, we reduce the problem to the 2nd kind integral Fredholm equation.

     

The probabilistic stability for a functional equation in a single variable

We discuss the probabilistic stability of the equation μ ∘ f ∘ η = f, by using the fixed point method.      

The Gellerstedt problem for a parabolic-hyperbolic equation of the second kind

In this paper we consider the Gellerstedt problem for a parabolic-hyperbolic equation of the second kind. We prove the unique solvability of this problem by means of a new representation for a solution to the modified Cauchy problem in a generalized class R.     

Stability of the generalized alternative cauchy equation

Let G be a commutative semigroup and letL be a complete Archimedean Riesz Space. Suppose thatF: G → L satisfies for somee ∈ L ₊ the inequality

On a class of slowly changing functions

We study the slowly changing functions and use them in the study of entire functions. An erratum to this article is available at .     

On the universal A.S. central limit theorem

Let (X _k ) be a sequence of independent r.v.’s such that for some measurable functions gk : R ^k → R a weak limit theorem of the form

On the general solution of a nonsymmetric partial difference functional equation analogous to the wave equation

We give the general solution of the nonsymmetric partial difference functional equationf(x + t,y) + f(x – t,y) – 2f(x,y)/t ² =f(x,y + s) + f(x,y – s) – 2f(x,y)/s ² (N) analogous to the well-known wave equation ( ²/x ² – ²/y ²)f(x,y) = 0 with the aid of generalized polynomials when no regularity assumptions are imposed onf. The result is as follows. Theorem.Let R be the set of all real numbers. A function f: R × R R satisfies the functional equation (N)for all x, y R, s, t R\{0}, and s t if and only if there exist

On continuation of a topology from a system of generators of a group to a topology on the group

In this paper we establish conditions under which a topology is continued from a subset of a group to a topology on the group coordinated with the group operation.     

Solution of fifth-degree equations

In this paper we establish a relationship between two approaches to the solution of algebraic fifth-degree equations, namely, the Hermite-Kronecker method (based on the modular elliptic equation) and the Mellin method (based on hypergeometric series).     

On the dual Stiefel-Whitney classes of some Grassmann manifolds

We present some non-vanishing dual Stiefel-Whitney classes of the Grassmann manifolds O(n)/O(4) × O(n − 4) for n = 2 ^s + 2 and n = 2 ^s + 3 (s ≧ 3), providing a supplement to results of Hiller, Stong, and Oproiu. Some applications are also mentioned. Part of this research was carried out while the first author was a member of three research teams supported in part by the grant agencies VEGA and APVV, and the second author was a member of a research team supported in part by VEGA.     

Stability of the Quadratic Equation of Pexider Type

We will investigate the stability problem of the quadratic equation (1) and extend the results of Borelli and Forti, Czerwik, and Rassias. By applying this result and an improved theorem of the author, we will also prove the stability of the quadratic functional equation of Pexider type,f ₁ (x +y) + f₂(x -y) =f ₃(x) +f ₄(y), for a large class of functions.     

On the relationships of seven numerical sequences

We analyze the relationships of four classical and three recently defined classes of numerical sequences. This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant # T042462 and TS44782.     

On a convex body with odd hadwiger number

We construct a convex body K ⊃ ℝ³ such that the maximum number of mutually nonoverlapping translates of K which touch K is 15. The work is in part supported by grant no. T043556 of the Hungarian National Science Foundation (OTKA).     

Generalized bisymmetry on a restricted domain

Let X ⊂ ℝ be an interval of positive length and define the set Δ = {(x, y) ∈ X × X | x ≧ y}. We give the solution of the equation

On the complexity of a family related to the Legendre symbol

Ahlswede, Khachatrian, Mauduit and A. Sárközy introduced the notion of family-complexity of families of binary sequences. They estimated the family-complexity of a large family related to Legendre symbol introduced by Goubin, Mauduit and Sárközy. Here their result is improved, and apart from the constant factor the best lower bound is given for the family-complexity.