Attila GilányiUniversity of Debrecen
Attila Gilányi
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67
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896
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Citations since 2017
Publications
Publications (67)
In this paper, we consider the condition $$\sum _{i=0}^{n+1}\varphi _i(r_ix+q_iy)\in {\mathbb {Z}}$$ ∑ i = 0 n + 1 φ i ( r i x + q i y ) ∈ Z for real valued functions defined on a linear space V . We derive necessary and sufficient conditions for functions satisfying this condition to be decent in the following sense: there exist functions $$f_i:V\...
A felsőoktatási tevékenység innovatív megközelítése A coaching lehetőségei és szerepe a tánctanárok munkája során, a tanulók, a hallgatók tanulási folyamatának hatékony segítése.
In the present paper, a general class of linear functional equations is considered and a computer program is described, which determines the exact solutions of systems of equations belonging to this class.
In this paper, we investigate set-valued maps of strongly and approximately
Jensen convex and Jensen concave type. We present counterparts of the
Bernstein--Doetsch Theorem with Tabor type error terms.
Parkinson's disease is a common disorder of the central nervous system. It is incurable currently, but several effective treatments are related to it. In this paper, in connection with physiotherapeutic care of the patients, we consider possible ways of therapies based on virtual reality systems.
In this paper, we consider some educational aspects related to mathability. Our main goal is to approach an answer to the question what are computer assisted ways to assimilate mathematical knowledge and possess mathematical abilities.
In connection with investigations related to mathability and to applications of computer assisted methods for studying mathematical problems, an animation of the m-convex hull of finite sets of points on the Cartesian plane is presented.
Motivated by results on strongly convex and strongly Jensen-convex functions by R. Ger and K. Nikodem in [Strongly convex functions of higher order, Nonlinear Anal. 74 (2011), 661-665] we investigate strongly Wright-convex functions of higher order and we prove decomposition and characterization theorems for them. Our decomposition theorem states t...
In this paper, we outline the scope and goals of a new branch of CogInfoCom, which we refer to as ‘mathability’.
In this paper, we present a three-dimensional virtual system which visualizes some interesting rooms of the University of Debrecen and makes the most cherished and carefully guarded treasures of the Collection of Rare and Early Printed Books of the University and National Library of the University of Debrecen virtually available. Our system is base...
The aim of this note is to give a type of characterization of Banach spaces in terms of the stability of functional equations. More precisely, we prove that a normed space X is complete if there exists a functional equation of the type Sigma(n)(i = 1) a(i)f (phi(i) (x(1), . . . , x(k))) - 0 (x(1), . . . , x(k) is an element of D) with given real nu...
In the recent years, subquadratic functions have been investigated by several authors. However, two different concepts of subquadraticity have been considered. Based on a simple modification of the geometric notion of concave functions a function f:[0,∞[ →ℝ is called subquadratic if, for each x≥0, there exists a constant c
x
∈ℝ such that the inequa...
Related to the theory of convex and subadditive functions, we investigate weakly subquadratic mappings, that is, solutions of the inequality for real-valued functions defined on a topological group G=(G,+). Especially, we study the lower and upper hulls of such functions and we prove Bernstein–Doetsch type theorems for them.
In the present paper, we prove the stability of the functional equation max{f((x o y) o y), f(x)} = f(x o y) + f(y) for real valued functions defined on a square-symmetric groupoid with a left unit element. As a consequence, we obtain the known result about the stability of the equation max{f(x + y), f(x - y)} = f(x) + f(y) for real valued function...
Based on J. L. W. V. Jensen’s concept of convex functions as well on its generalization by E. M. Wright and related to T. Popoviciu’s convexity notions, higher-order convexity properties of real functions are introduced and surveyed.
The concept of generalized convex functions introduced by Beckenbach [E.F. Beckenbach, Generalized convex functions, Bull. Amer. Math. Soc. 43 (1937) 363–371] is extended to the two-dimensional case. Using three-parameter families, we define generalized convex (midconvex, M-convex) functions f:D⊆R2→R and show some continuity properties of them.
Given a function f mapping a groupoid (X,
à{\diamond}
) into a metric groupoid (Y, * ,d) and satisfying the inequality
d(f(x ày),f(x)*f(y)) £ e(x,y) (x,y Î X), d(f(x \diamond y),f(x)*f(y))\leq \varepsilon(x,y)\quad (x,y \in X),
the problem of stability in the sense of Hyers-Ulam is to construct a solution g of the functional equation
g(x ày) =...
We solve the functional equation F1(t)−F1(t+s)=F2[F3(t)+F4(s)] for real functions defined on intervals, assuming that F2 is positive valued and strictly monotonic and that F3 is continuous. The equation arose from the equivalence problem of utility representations under assumptions of separability, homogeneity and segregation (e-distributivity).
In this paper various quasiconvexity notions are considered and compared. The main goal is to show that, under the assumption of upper semicontinuity, Jensen-type quasiconvexity properties are equivalent to the corresponding ordinary quasiconvexity property. The results thus obtained are analogous to the classical theorem of Bernstein and Doetsch f...
Concerning a problem raised by K. Nikodem, we prove the following statement. If G is an Abelian group divisible by 2, H is a Hilbert space and ε is a nonnegative real number and a function f: G → H satisfies ||f(x-y) - 2f(x) -2f(y)|| ≤ ||f(x + y)|| + ε (x,y ∈ G), then there exists a function g: G → H fulfilling g(x + y) + g(x - y) -2g(x) -2g(y) = 0...
In this paper we prove that a function f mapping from the set of the reals into a Banach space satisfying
$$\lim \frac{{\parallel \Delta _y^nf(x) - n!f(y)\parallel }}{{|y{|^\alpha }}} = 0$$
for a positive integer n, a real number α < n and as (x, y) tends to (−∞, ∞), (∞, −∞), (∞, ∞) or (−∞, −∞) can be approximated by a monomial function of degree n...
In this paper it is proved that, for a function $ f : G\to E $ mapping from an abelian group G divisible by 2 into an inner product space E, the functional inequality¶¶$ \Vert2f(x)+2f(y)-f(x y^{-1})\Vert\leq\Vert f(x y)\Vert \ \ \ (x,y\in G) $¶implies the parallelogram equation¶$ f(x y)+f(x y^{-1})-2f(x)-2f(y)=0 \ \ \ (x,y\in G) $.
In this paper a regularity theorem for the functional equation¶¶
$ f(x+y)-f(x) + \sum_{i=1}^{n}\varphi_i[g_i(y+z_i)-g_i(y)]¶¶ = \sum_{i=1}^{n}\psi_i[g_i(y+x+z_i)-g_i(y+x)-g_i(y+z_i)+g_i(y)] $
is proved.
In this paper higher-order convexity properties of real functions are
characterized in terms of a Dinghas-type derivative. The main tool used is a
mean value inequality for Dinghas-type derivatives.
We characterize probabilistic choice models that have two (or more) fixed ratio scales associated with each choice option. A version of Luce’s choice model with fixed scales and exponents, which we also characterize, is included. The characterizing properties are: Simple scalability, i.e. the choice probabilities depend upon a priori arbitrary comb...
In the present paper a certain form of the Hyers-Ulam stability of monomial functional equations is studied. This kind of stability was investigated in the case of additive functions by TH. M. RASSIAS and Z. GAJDA.
In the present paper the Hyers-Ulam stability of monomial functional equations for functions defined on a power-associative, power-symmetric groupoid is proved.
The main result of this paper is the following: if α ≥ 0, α ≠ 2 and a real function f satisfies f(x + 2y) - 2f(x + y) + f(x) - 2f(y) = o(y α) ((x, y) → (0, 0), x ≤ 0 ≤ x + 2y), then there exists a real function q such that q(x + 2y) - 2q(x + y) + q(x) - 2q(y) = 0 (x, y ∈ ℝ) and f(x) - q(x) = o(|x| α) (x → 0).
We present a computer program developed in the computer algebra system Maple, which determines the solutions of systems of linear functional equations of two variables.
The aim of this paper is to give a characterization of monomial functions with the aid of an operator
[(D)\tilde]\tilde D
introduced in Section 1. The present results were inspired by A. Dinghas [2] and A. Simon and P. Volkmann [5], who treated
the characterization of polynomial functions by the so called Dinghas interval-derivatives.
A támogatott kutatás keretében a függvényegyenletek és a függvényalgebrák területén stabilitási, regularitás-elméleti, konvexitási, reflexivitási, lineáris megőrzési, a Wigner-féle unitér-antiunitér tétellel kapcsolatos, valamint effekt-algebrákra vonatkozó problémákkal foglalkoztunk. Stabilitási vizsgálataink során általános struktúrán értelmezett...
A kutatásban 19 fő vett részt. A kutatócsoport tagjai 2003 és 2006 között 3 könyvet, 103 referált nemzetközi folyóiratban, vagy konferenciakiadványban megjelent tudományos dolgozatot, további 15 közlésre elfogadott dolgozatot, 6 PhD, 1 habilitációs és 2 MTA doktori értekezést készítettek, továbbá 208 tudományos előadást tartottak. A nem-iteratív fü...
A Clarke-féle általánosított derivált fogalmat (amely a lokálisan Lipschitz véges dimenziós normált terek között ható függvényekhez társít egy mátrix-halmazértékű deriváltat), általánosítottuk végtelen dimenziós normált tereken értelmezett és duális térként előálló Banach-terekbe képező lokálisan Lipschitz függvényekre, továbbá kidolgoztuk erre az...
Projects
Projects (3)
In a project from autumn 2021 to spring 2023, with the support of the Visegrad Fund, we will test the ballet historical MaxWhere presentation room in university and college education, as a space for presentation and collaboration. We also provide ballet history materials, which we offer as ready-made MaxWhere presentations.
The project explores and examines the birth and development of the stage dance in Hungary. The first and second project (Aurora 1-2) introduces the ballet art in the Hungarian National Theater (the pre Opera House period), by work of two significant artists: Federico Campilli, choreographer and ballet master; Emília Aranyváry, the first real ballerina, and first female choreographer in Hungary. The Aurora 3 deals with the period around the turn of the century. This period was particularly productive for ballets in the history of Hungarian ballet. A special repertoire has evolved, similar to the international ballet trends of the era.
