Institute of Mathematics AS CR, v. v. i.
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    Regularity of powers of binomial edge ideals of complete multipartite graphs

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    summary:Let G=Kn1,n2,,nrG=K_{n_1,n_2,\ldots ,n_r} be a complete multipartite graph on [n][n] with n>r>1n>r>1 and JGJ_G being its binomial edge ideal. It is proved that the Castelnuovo-Mumford regularity reg(JGt){\rm reg}(J^t_G) is 2t+12t+1 for any positive integer tt

    Foreword to proceedings of Equadiff 15

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    On the least almost-prime in arithmetic progression

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    summary:Let Pr\mathcal {P}_r denote an almost-prime with at most rr prime factors, counted according to multiplicity. Suppose that aa and qq are positive integers satisfying (a,q)=1(a,q)=1. Denote by P2(a,q)\mathcal {P}_2(a,q) the least almost-prime P2\mathcal {P}_2 which satisfies P2a(modq)\mathcal {P}_2\equiv a\pmod q. It is proved that for sufficiently large qq, there holds P2(a,q)q1.8345. \mathcal {P}_2(a,q)\ll q^{1.8345}. This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range 1.8451.845 in place of 1.83451.8345

    A determinant formula from random walks

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    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

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    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

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    summary:We construct a class of quantum doubles D(HDn)D(H_{D_n}) of pointed Hopf algebras of rank one HDH_{\mathcal {D}}. We describe the algebra structures of D(HDn)D(H_{D_n}) by generators with relations. Moreover, we give the comultiplication ΔD\Delta _{D}, counit εD\varepsilon _D and the antipode SDS_{D}, respectively

    Majority choosability of 1-planar digraph

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    summary:A majority coloring of a digraph DD with kk colors is an assignment π ⁣:V(D){1,2,,k}\pi \colon V(D) \rightarrow \{1,2,\cdots ,k\} such that for every vV(D)v\in V(D) we have π(w)=π(v)\pi (w)=\pi (v) for at most half of all out-neighbors wN+(v)w\in N^+(v). A digraph DD is majority kk-choosable if for any assignment of lists of colors of size kk to the vertices, there is a majority coloring of DD from these lists. We prove that if U(D)U(D) is a 1-planar graph without a 4-cycle, then DD 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

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    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 N1γ={1γ,2γ,3γ,}\mathbb {N}_{1-\gamma }=\{1-\gamma ,2-\gamma ,3-\gamma ,\cdots \}, 0<γ10<\gamma \leq 1, Δγ\Delta ^{\gamma } 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 hh-stability behavior of time-varying nonlinear systems

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    summary:We deal with the problem of practical uniform hh-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 hh-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

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    Institute of Mathematics AS CR, v. v. i. is based in Czechia
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