We announce a new result on determining the Conley index of the Poincaré map for a time-periodic non-autonomous ordinary differential equation. The index is computed using some singular cycles related to an index pair of a small-step discretization of the equation. We indicate how the result can be applied to computer-assisted proofs of the existence of bounded and periodic solutions. We provide also some comments on computer-assisted proving in dynamics.

We show that the solution to the orthogonal additivity problem in real inner product spaces depends continuously on the given function and provide an application of this fact.

A set S⊂V is a dominating set of a graph G = (V,E) if every vertex v∈V which does not belong to S has a neighbour in S. The domination number γ(G) of the graph G is the minimum cardinality of a dominating set in G. A dominating set S is a γ-set in G if jSj = γ(G).
Some graphs have exponentially many γ-sets, hence it is worth to ask a question if a γ-set can be obtained by some transformations from another γ-set. The study of gamma graphs is an answer to this reconfiguration problem. We give a partial answer to the question which graphs are gamma graphs of trees. In the second section gamma graphs γ.T of trees with diameter not greater than five will be presented. It will be shown that hypercubes Q_k are among γ.T graphs. In the third section γ.T graphs of certain trees with three pendant vertices will be analysed. Additionally, some observations on the diameter of gamma graphs will be presented, in response to an open question, published by Fricke et al., if diam(T(γ)) = O(n)?

In this paper, we investigate the general solution and Hyers–Ulam–Rassias stability of a new mixed type of additive and quintic functional equation of the form
f(3x + y) - 5f(2x + y) + f(2x - y) + 10f(x + y) - 5f(x - y) = 10f(y) + 4f(2x) - 8f(x)
in the set of real numbers.

The abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem u_t - Δu = u|u|^s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

Let f(z) = Σ_n=0^∞α_nzⁿ be a function defined by power series with complex coefficients and convergent on the open disk D(0,R)⊂ℂ, R>0. For any x,y∈????, a Banach algebra, with ‖x‖,‖y‖<R we show among others that
‖f(y)-f(x)‖ ≤ ‖y-x‖∫₀¹f'_a(‖(1-t)x+ty‖)dt
where f_a(z) = Σ_n=0^∞|α_n|zⁿ. Inequalities for the commutator such as
‖f(x)f(y) - f(y)f(x)‖ ≤ 2f_a(M)f'_a(M)‖y-x‖,
if ‖x‖,‖y‖≤M<R, as well as some inequalities of Hermite–Hadamard type are also provided.

We examine the class of spaces in which the second player has a winning strategy in the open-open game. We show that this spaces are not universally Kuratowski–Ulam. We also show that the games G and G₇ introduced by P. Daniels, K. Kunen, H. Zhou [Fund. Math. 145 (1994), no. 3, 205–220] are not equivalent.

The paper is devoted to the communication complexity of lattice operations in linearly ordered finite sets. All well known techniques ([4, Chapter 1]) to determine the communication complexity of the infimum function in linear lattices disappoint, because a gap between the lower and upper bound is equal to ????(log₂n), where n is the cardinality of the lattice. Therefore our aim will be to investigate the communication complexity of the function more carefully. We consider a family of so called interval protocols and we construct the interval protocols for the infimum. We prove that the constructed protocols are optimal in the family of interval protocols. It is still open problem to compute the communication complexity of constructed protocols but the numerical experiments show that their complexity is less than the complexity of known protocols for the infimum function.

In this paper, a general fixed point theorem for cyclic multi-valued mappings satisfying an implicit relation from [19] different from implicit relations used in [13] and [23], generalizing some results from [22], [15], [13], [14], [16], [10] and from other papers, is proved.

The main goal of the paper is to examine the dimension of the vector space spanned by powers of linear forms. We also find a lower bound for the number of summands in the presentation of zero form as a sum of d-th powers of linear forms.

Markov operators arising from graph directed constructions of iterated function systems are considered. Exponential convergence to an invariant measure is proved.

Report of Meeting. The Fifteenth Katowice-Debrecen Winter Seminar, Będlewo (Poland), January 28–31, 2015.