Fell topology on the space of functions with closed graph

LetX andY beT ₁ topological spaces andG(X, Y) the space of all functions with closed graph. Conditions under which the Fell topology and the weak Fell topology coincide onG(X,Y) are given. Relations between the convergence in the Fell topologyτF, Kuratowski and continuous convergence are studied too. Characterizations of a topological spaceX by separation axioms of (G(X, R), τF) and topological properties of (G(X, R), τF) are investigated.     

Ruled Weingarten surfaces in the Galilean space

In this paper we study ruled Weingarten surfaces in the Galilean space. Weingarten surfaces are surfaces having a nontrivial funcional relation between their Gaussian and mean curvature. We describe some further examples of Weingarten surfaces.      

A certain boundary-value problem with shifts in a four-dimensional space

We consider a problem with shifts in boundary conditions for the Bianchi equation in a four-dimensional space. We establish sufficient conditions which allow one to evaluate its unique solution in terms of the Riemann function.     

Continuous functions taking every value a given number of times

We give necessary and sufficient conditions for a function f: [0, 1] → {1,2,...,w, c} under which there exists a continuous function F: [0, 1] → [0, 1] such that for every y ɛ [0, 1], |F ⁻¹ (y)| = f(y).      

Copositive approximation of periodic functions

Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y _i ∈ Y:= {y _i } _i∈ℤ such that for x ∈ [y _i , y _i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P _n of order ≦ n, changing its sign at the same points y _i ∈ Y as f, and

Nowhere monotone functions and microscopic sets

We investigate how large a set can be on which a continuous nowhere monotone function is one-to-one. We consider the σ-ideal of microscopic sets, which is situated between the countable sets and the sets of Hausdorff dimension zero and prove that the typical function in C[0, 1] (in the sense of Baire) is nowhere monotone and one-to-one except on some microscopic set. We also give an example of a continuous nowhere monotone function of bounded variation on [0, 1], which is one-to-one except on some microscopic set, so it is not a typical function.      

Continuous operators on asymmetric normed spaces

If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖⁺‖, q(y) = ‖y ⁺‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases. The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain and FEDER, under grant MTM2006-14925-C02-01 and Generalitat Valenciana under grant GV/2007/198.     

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 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 .     

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.     

On the nonextendability of germs of holomorphic functions

We construct explicit supporting manifolds and local holomorphic peak functions as obstructions to the extendability of holomorphic functions on a class of domains not necessarily pseudoconvex in C^N, N >2.     

Dual affine connections on a quadratic hyperband distribution in a projective-metric space and their applications

In this paper, we develop fundamentals of the dual theory of quadratic hyperband distributions H of m-dimensional line elements in a projective-metric space K _n (m < n ? 1). In particular, we show that, on a dual normalized distribution H, there are induced two dual affine connections and indicate some applications of these connections to the geometry of m-webs on H.     

On hyperbolically convex functions

Let be the unit disk of the complex plane. A conformai map of into itself is called hyperbolically convex if the non-Euclidean segment between any two points of also belongs to . In this paper we prove several inequalities that are analogous to inequalities about (Euclidean) convex univalent functions. We show that if ƒ (0) = 0, then Re zf′/f > 1/2. This inequality is the key for the results of this paper. In particular we deduce a three-variable inequality corresponding to that of Ruscheweyh and Sheil-Small. The sharp bound for the Schwarzian derivative remains open.     

On the derivability of Lipschitz functions

We show that given any Borel measure onR, every Lipschitz function is μ-a.e. differentiable with respect to μ.     

Subspaces and Parallelity in semiaffine partial linear spaces

In the paper we characterize subspaces and strong subspaces of a semiaffine partial linear space and consider definability of projective and semiaffine planes, affine lines and parallelity in terms of projective lines. We also give some construction of a wide class of semiaffine partial linear spaces.     

Collapsing monoids consisting of permutations and constants

In this paper we determine all collapsing transformation monoids that contain at least one unary constant operation and whose nonconstant operations are permutations. Furthermore, we find an infinite family of transformation monoids that consist of at least three unary constant operations and some permutations for which the corresponding monoidal intervals are 2-element chains. This research is supported by Hungarian National Foundation for Scientific Research grant nos. T 37877 and K 60148.     

Some remarks on trigonometric sums

Let

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.     

An application of index forms in cryptography

We investigate the possibility of using index forms as basic ingredients of cryptographically important functions. We suggest the use of a hash function based on index forms and we prove some important properties of the suggested function. The research was supported in part by the Hungarian Academy of Sciences, by grants T048791 and K67580 of the Hungarian NFSR, by the National Office for Research and Technology and by the grant JP-26/2006.