Let $G$ be a finite group. A normal subgroup $N$ of $G$ is a union of several $G$-conjugacy classes, and it is called $n$-decomposable in $G$ if it is a union of $n$ distinct $G$-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5., Ruifang Chen, Xianhe Zhao., and Obsahuje bibliografické odkazy
We prove two Dyakonov type theorems which relate the modulus of continuity of a function on the unit disc with the modulus of continuity of its absolute value. The methods we use are quite elementary, they cover the case of functions which are quasiregular and harmonic, briefly hqr, in the unit disc., Miloš Arsenović, Miroslav Pavlović., and Seznam literatury
We propose a new method for studying stability of second order delay differential equations. Results we obtained are of the form: the exponential stability of ordinary differential equation implies the exponential stability of the corresponding delay differential equation if the delays are small enough. We estimate this smallness through the coefficients of this delay equation. Examples demonstrate that our tests of the exponential stability are essentially better than the known ones. This method works not only for autonomous equations but also for equations with variable coefficients and delays., Ravi P. Agarwal, Alexander Domoshnitsky, Abraham Maghakyan., and Obsahuje seznam literatury
An S-closed submodule of a module M is a submodule N for which M/N is nonsingular. A module M is called a generalized CS-module (or briefly, GCS-module) if any S-closed submodule N of M is a direct summand of M. Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right R-modules are projective if and only if all right R-modules are GCS-modules., Qingyi Zeng., and Obsahuje seznam literatury
Determination of heights with help of GPS in local geodetic networks is still more actual respecting the fact that the GPS technology becames more and more effective with hardware progress, with improvements in measuring and evaluating procedures, and with better modelling of the disturbing influences. In comparison with GPS the employment of classical terrestrial measuring technologies is often more difficult namely in broken mountain environment. In period 1998-2005 authors carried out repeated measurements of GPS baselines of various length and various height differences in local geodynamical network Sněžník and in other experimental areas. On ground of analyses of large GPS data sets the modified procedure for GPS observation was designed. The procedure is based on repetition of shorter static sessions separated by time intervals of optimal length. This technology represents an alternative to the ususal long static sessions, and is offering better effectivity of vertical GPS measurements with minimal loss of accuracy. The paper presents detailed description of the modified procedure together with some statistical analyses of results. The possibilities of elimination or mitigation of some disturbing influences are discussed. Two testing vertical profiles were marked in Sněžník network- longitudinal profile in N-S direction, and transversal profile in E-W direction - which were measured in course of several years by classical method of very precise levelling, and also by modified GPS heighting procedure in repeated sessions. Results obtained contributed to the local quasigeoid model creation., Otakar Švábenský, Josef Weigel and Radovan Machotka., and Obsahuje bibliografii
We study improper interval edge colourings, defined by the requirement that the edge colours around each vertex form an integer interval. For the corresponding chromatic invariant (being the maximum number of colours in such a colouring), we present upper and lower bounds and discuss their qualities; also, we determine its values and estimates for graphs of various families, like wheels, prisms or complete graphs. The study of this parameter was inspired by the interval colouring, introduced by Asratian, Kamalian (1987). The difference is that we relax the requirement on the original colouring to be proper., Peter Hudák, František Kardoš, Tomáš Madaras, Michaela Vrbjarová., and Obsahuje seznam literatury
We discuss the invariant subspace problem of polynomially bounded operators on a Banach space and obtain an invariant subspace theorem for polynomially bounded operators. At the same time, we state two open problems, which are relative propositions of this invariant subspace theorem. By means of the two relative propositions (if they are true), together with the result of this paper and the result of C. Ambrozie and V. Müller (2004) one can obtain an important conclusion that every polynomially bounded operator on a Banach space whose spectrum contains the unit circle has a nontrivial invariant closed subspace. This conclusion can generalize remarkably the famous result that every contraction on a Hilbert space whose spectrum contains the unit circle has a nontrivial invariant closed subspace (1988 and 1997)., Junfeng Liu., and Obsahuje bibliografii
Let G be an undirected connected graph with n, n\geqslant 3, vertices and m edges with Laplacian eigenvalues µ^{1}\geqslant µ_{2}\geq ...\geq µ_{n-1> µ_{n}}=0. Denote by {\mu _I} = {\mu _{{r_1}}} + {\mu _{{r_2}}} + \ldots + {\mu _{{r_k}}}, 1\leq k\leq n-2, 1\leq r_{1}< r_{2}< ...< r_{k} \leq n-1, the sum of k arbitrary Laplacian eigenvalues, with {\mu _{{I_1}}} = {\mu _1} + {\mu _2} + \ldots + {\mu _k} and {\mu _{{I_n}}} = {\mu _{n - k}} + \ldots + {\mu _{n - 1}}. Lower bounds of graph invariants {\mu _{{I_1}}} - {\mu _{{I_n}}} and {\mu _{{I_1}}}/{\mu _{{I_n}}} are obtained. Some known inequalities follow as a special case., Igor Ž. Milovanović, Emina I. Milovanović, Edin Glogić., and Obsahuje seznam literatury
A subgroup H of a finite group G is said to be conjugate-permutable if HHg = HgH for all g\in G. More generaly, if we limit the element g to a subgroup R of G, then we say that the subgroup H is R-conjugate-permutable. By means of the R-conjugatepermutable subgroups, we investigate the relationship between the nilpotence of G and the R-conjugate-permutability of the Sylow subgroups of A and B under the condition that G = AB, where A and B are subgroups of G. Some results known in the literature are improved and generalized in the paper., Xianhe Zhao, Ruifang Chen., and Obsahuje seznam literatury
Let $R$ be a commutative ring, $M$ an $R$-module and $G$ a group of $R$-automorphisms of $M$, usually with some sort of rank restriction on $G$. We study the transfer of hypotheses between $M/C_M(G)$ and $[M,G]$ such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose $[M,G]$ is $R$-Noetherian. If $G$ has finite rank, then $M/C_M(G)$ also is $R$-Noetherian. Further, if $[M,G]$ is $R$-Noetherian and if only certain abelian sections of $G$ have finite rank, then $G$ has finite rank and is soluble-by-finite. If $M/C_M(G)$ is $R$-Noetherian and $G$ has finite rank, then $[M,G]$ need not be $R$-Noetherian., Bertram A. F. Wehrfritz., and Obsahuje bibliografické odkazy