For any two positive integers n and k\geqslant 2, let G(n, k) be a digraph whose set of vertices is {0, 1, ..., n − 1} and such that there is a directed edge from a vertex a to a vertex b if ak ≡ b (mod n). Let n = \prod\nolimits_{i = 1}^r {p_i^{{e_i}}} be the prime factorization of n. Let P be the set of all primes dividing n and let P_{1},P_{2} \subseteq P be such that P_{1\cup P_{2}}= P and P_{1\cup P_{2}}=\emptyset . A fundamental constituent of G(n, k), denoted by G_{{P_2}}^*(n,k), is a subdigraph of G(n, k) induced on the set of vertices which are multiples of \prod\nolimits_{{p_i} \in {P_2}} {{p_i}} and are relatively prime to all primes q\in P_{1}. L. Somer and M. Křižek proved that the trees attached to all cycle vertices in the same fundamental constituent of G(n, k) are isomorphic. In this paper, we characterize all digraphs G(n, k) such that the trees attached to all cycle vertices in different fundamental constituents of G(n, k) are isomorphic. We also provide a necessary and sufficient condition on G(n, k) such that the trees attached to all cycle vertices in G(n, k) are isomorphic., Amplify Sawkmie, Madan Mohan Singh., and Obsahuje seznam literatury
The packing constant is an important and interesting geometric parameter of Banach spaces. Inspired by the packing constant for Orlicz sequence spaces, the main purpose of this paper is calculating the Kottman constant and the packing constant of the Cesàro-Orlicz sequence spaces ({\text{ce}}{{\text{s}}_\varphi }) defined by an Orlicz function φ equipped with the Luxemburg norm. In order to compute the constants, the paper gives two formulas. On the base of these formulas one can easily obtain the packing constant for the Cesàro sequence space cesp and some other sequence spaces. Finally, a new constant \widetilde D (X), which seems to be relevant to the packing constant, is given., Zhen-Hua Ma, Li-Ning Jiang, Qiao-Ling Xin., and Obsahuje seznam literatury
In this paper we study parallel and totally geodesic hypersurfaces of two-step homogeneous nilmanifolds of dimension five. We give the complete classification and explicitly describe parallel and totally geodesic hypersurfaces of these spaces. Moreover, we prove that two-step homogeneous nilmanifolds of dimension five which have one-dimensional centre never admit parallel hypersurfaces. Also we prove that the only two-step homogeneous nilmanifolds of dimension five which admit totally geodesic hypersurfaces have three-dimensional centre., Mehri Nasehi., and Obsahuje seznam literatury
Každoroční týdenní soustředění fyzikálního korespondenčního semináře FYKOS se letos konalo na sklonku září na západě republiky v Zelené Lhotě. I tentokrát se tři desítky řešitelů rozhodly strávit celý týden svého času studiem fyziky a matematiky. and Vít Beran.
Let f: X → X be a continuous map with the specification property on a compact metric space X. We introduce the notion of the maximal Birkhoff average oscillation, which is the “worst” divergence point for Birkhoff average. By constructing a kind of dynamical Moran subset, we prove that the set of points having maximal Birkhoff average oscillation is residual if it is not empty. As applications, we present the corresponding results for the Birkhoff averages for continuous functions on a repeller and locally maximal hyperbolic set., Jinjun Li, Min Wu., and Obsahuje seznam literatury