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1
Hyers-Ulam stability of quadratic forms in 2-normed spaces
EN
In this paper, we obtain Hyers-Ulam stability of the functional equations f(x+y, z+w) + f(x-y, z-w) = 2f(x, z) + 2f(y, w), f(x+y, z-w) + f(x-y, z+w) = 2f(x, z) + 2f(y, w) and f(x+y, z-w) + f(x-y, z+w) = 2f(x, z) - 2f(y, w) in 2-Banach spaces. The quadratic forms ax2+bxy+cy2, ax2+by2 and axy are solutions of the above functional equations, respectively.
2
Bounded solutions of a generalized Gołąb-Schinzel equation
EN
Let X be a linear space over the field K of real or complex numbers. We characterize solutions f : X - > K and M : K - > K of the equation f(x+M)(f)y)=f(x)f(y) in the case where the set {x is an element of X : f (x) = 0} has an algebraically interior point. As a consequence we give solutions of the equation such that f is bounded on this set.
3
One-to-one solutions of generalized Gołąb-Schinzel equation
EN
Let K be the field of real or complex numbers and let X be a nontrivial linear space over K. Assume that [...]. We give a necessary and sufficient condition for functions f and M to satisfy the equation The functional equation f(x+M(f(x))y)=f(x)f(y) is a generalization of the well-known Gołąb-Schinzel functional equation f(x+f(x)y)=f(x)f(y).
4
Tresses of polygons
EN
A class of configurations which can be considered as series of suitably inscribed closed polygons is introduced and some fundamental properties of them are established.
5
Multiple perspectives and generalizations of the Desargues configuration
EN
We introduce a class of finite confgurations, which we call combinatorial Grassmannians, and which generalize the Desargues configuration. Fundamental geome- tric properties of them are established, in particular we determine their automorphisms, correlations, mutual embedability, and prove that no one of them contains a Pascal or Pappus figure.
6
EN
The theory of right invertible operators was started with works of D. Przeworska-Rolewicz and then it has been developed by M. Tasche, H. von Trotha, Z. Binderman and many other mathematicians (see [10]). Nguyen Dinh Quyet (in [5, 7]), has considered the controllability of linear system described by right invertible operators where the resolving operator is invertible. These results were generalized by A. Pogorzelec in the case of one-sized invertible resolving operator (see [9]) and by Nguyen Van Mau for the system described by generalized invertible operator (see [3]). However, for the degenerate systems, the problem has not been investigated. In this paper, we deal with the initial value problem for degenerate system of the form (2.7)-(2.8) and the controllability of this system.
7
On set-theoretic and cyclic representation of the structure of barycentres
EN
In the paper certain abstract combinatorial structures are studied as representations of the structure of barycentres of all the subsimplices of a given simplex in an arbitrary Desarguesian affine space. General properties of the configuration of barycentres are characterized in terms of those combinatorial structures. Most essential parameters of these structures are established and relevant automorphism groups are characterized.
8
Strict 2-convexity and strict convexity
EN
In this paper, we give some new characterizations of strict convexity and strict 2-convexity in linear 2-normed spaces.
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