Aequationes Mathematicae最新文献

The complex adjacency matrix ({A_{!mathbb {C}}}(G)) for a multidigraph G is introduced in Barik and Sahoo (AKCE Int J Graphs Comb 17(1):466–479, 2020). We study the rank of multidigraphs corresponding to the complex adjacency matrix and call it ({A_{!mathbb {C}}})-rank. It is known that a connected graph G has rank 2 if and only if G is a complete bipartite graph, and has rank 3 if and only if it is a complete tripartite graph (Cheng in Electron J Linear Algebra 16:60–67, 2007). We observe that these results hold as special cases for multidigraphs but are not sufficient. In this article, we characterize all multidigraphs with ({A_{!mathbb {C}}})-rank 2 and 3, respectively.

In the second part of his fifth problem Hilbert asks for functional equations “In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption.” In the case of the general functional equation

for the unknown function f under natural condition for the given functions it is proved on compact manifolds that (fin C^{-1}) implies (fin C^{infty }) and practically the general case can also be treated. The natural conditions imply that the dimension of x cannot be larger than the dimension of y. If we remove this condition, then we have to add another condition. In this survey paper a new problem for this second case is formulated and results are summarised for both cases.

For (nu in [0,1]) and a complex parameter (sigma ,) (Re, sigma >0,) we discuss a linear inhomogeneous functional difference equation with variable coefficients on a complex plane (zin {{mathbb {C}}}):

where (Omega (z)) and ({mathbb {F}}(z)) are given complex functions, while (a_{1}) and (a_{2}) are given real non-negative numbers. Under suitable conditions on the given functions and parameters, we construct explicit solutions of the equation and describe their asymptotic behavior as (|z|rightarrow +infty ). Some applications to the theory of functional difference equations and to the theory of boundary value problems governed by subdiffusion in nonsmooth domains are then discussed.

In the present paper we prove a generalized version of the famous decomposition theorem of Ng. We also focus on the problem posed by Zsolt Páles concerning the Wright-convex functions.

In this paper we give necessary conditions for quadratic functions ( f :mathbb {R}rightarrow mathbb {R}) that satisfy the additional equation ( y^2 f(x) = x^2 f(y) ) under the condition ( xy = 1 ,).

Let X and Y be Banach spaces. When A and B are linear relations in X and Y, respectively, we denote by (M_{C}) the linear relation in (Xtimes Y) of the form (left( begin{array}{cc} A &{} C 0 &{} B end{array} right) ), where 0 is the zero operator from X to Y and C is a bounded operator from Y to X. In this paper, by using properties of the SVEP, we study the defect set ((Sigma (A)cup Sigma (B))backslash Sigma (M_{C})), where (Sigma ) is the spectrum, the approximate point spectrum, the surjective spectrum, the Fredholm spectrum, the Weyl spectrum, the Browder spectrum, the generalized Drazin spectrum and the Drazin spectrum.

The paper is devoted to some extremal problems, related to convex polygons in the Euclidean plane and their perimeters. We present a number of results that have simple formulations, but rather intricate proofs. Related and still unsolved problems are also discussed.