Institute of Mathematics AS CR, v. v. i.
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Regularity of powers of binomial edge ideals of complete multipartite graphs
summary:Let be a complete multipartite graph on with and being its binomial edge ideal. It is proved that the Castelnuovo-Mumford regularity is for any positive integer
On the least almost-prime in arithmetic progression
summary:Let denote an almost-prime with at most prime factors, counted according to multiplicity. Suppose that and are positive integers satisfying . Denote by the least almost-prime which satisfies . It is proved that for sufficiently large , there holds This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range in place of
A determinant formula from random walks
summary:One usually studies the random walk model of a cat moving from one room to another in an apartment. Imagine now that the cat also has the possibility to go from one apartment to another by crossing some corridors, or even from one building to another. That yields a new probabilistic model for which each corridor connects the entrance rooms of several apartments. This article computes the determinant of the stochastic matrix associated to such random walks. That new model naturally allows to compute the determinant of a large class of matrices. Two examples involving digraphs and hyperplane arrangements are provided
On the regularity of bilinear maximal operator
summary:We study the regularity properties of bilinear maximal operator. Some new bounds and continuity for the above operators are established on the Sobolev spaces, Triebel-Lizorkin spaces and Besov spaces. In addition, the quasicontinuity and approximate differentiability of the bilinear maximal function are also obtained
A class of quantum doubles of pointed Hopf algebras of rank one
summary:We construct a class of quantum doubles of pointed Hopf algebras of rank one . We describe the algebra structures of by generators with relations. Moreover, we give the comultiplication , counit and the antipode , respectively
Majority choosability of 1-planar digraph
summary:A majority coloring of a digraph with colors is an assignment such that for every we have for at most half of all out-neighbors . A digraph is majority -choosable if for any assignment of lists of colors of size to the vertices, there is a majority coloring of from these lists. We prove that if is a 1-planar graph without a 4-cycle, then is majority 3-choosable. And we also prove that every NIC-planar digraph is majority 3-choosable
Nonoscillatory solutions of discrete fractional order equations with positive and negative terms
summary:This paper aims at discussing asymptotic behaviour of nonoscillatory solutions of the forced fractional difference equations of the form \begin{align} \Delta ^{\gamma }u(\kappa )&+\Theta [\kappa +\gamma ,w(\kappa +\gamma )]\\=&\Phi (\kappa +\gamma )+\Upsilon (\kappa +\gamma )w^{\nu }(\kappa +\gamma ) +\Psi [\kappa +\gamma ,w(\kappa +\gamma )],\quad \kappa \in \mathbb {N}_{1-\gamma },\\ u_{0} =&c_{0}, \end{align} where , , is a Caputo-like fractional difference operator. Three cases are investigated by using some salient features of discrete fractional calculus and mathematical inequalities. Examples are presented to illustrate the validity of the theoretical results
Practical -stability behavior of time-varying nonlinear systems
summary:We deal with the problem of practical uniform -stability for nonlinear time-varying perturbed differential equations. The main aim is to give sufficient conditions on the linear and perturbed terms to guarantee the global existence and the practical uniform -stability of the solutions based on Gronwall's type integral inequalities. Several numerical examples and an application to control systems with simulations are presented to illustrate the applicability of the obtained results