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
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
We analyse multivalued stochastic differential equations driven by semimartingales. Such equations are understood as the corresponding multivalued stochastic integral equations. Under suitable conditions, it is shown that the considered multivalued stochastic differential equation admits at least one solution. Then we prove that the set of all solutions is closed and bounded., Marek T. Malinowski, Ravi P. Agarwal., and Obsahuje bibliografii
A subgroup H of a finite group G is said to be ss-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is s-permutable in K. In this paper, we first give an example to show that the conjecture in A.A. Heliel’s paper (2014) has negative solutions. Next, we prove that a finite group G is solvable if every subgroup of odd prime order of G is ss-supplemented in G, and that G is solvable if and only if every Sylow subgroup of odd order of G is ss-supplemented in G. These results improve and extend recent and classical results in the literature., Jiakuan Lu, Yanyan Qiu., and Obsahuje seznam literatury
We present new sharp embedding theorems for mixed-norm analytic spaces in pseudoconvex domains with smooth boundary. New related sharp results in minimal bounded homogeneous domains in higher dimension are also provided. Last domains we consider are domains which are direct generalizations of the well-studied so-called bounded symmetric domains in Cn. Our results were known before only in the very particular case of domains of such type in the unit ball. As in the unit ball case, all our proofs are heavily based on nice properties of the r-lattice. Some results of this paper can be also obtained in some unbounded domains, namely tubular domains over symmetric cones., Romi F. Shamoyan, Olivera R. Mihić., and Obsahuje seznam literatury