Description of the Structure of Irregularity Sets of Linear Differential Systems with a Linear Parameter

Differential Equations - We consider the class of linear parametric differential systems $$\dot {x}=\mu A(t)x $$ defined on the half-line $$t\geq 0 $$ , where $$\mu \in \mathbb {R} $$ is a...     

Maps preserving a certain product of operators

Let $$\mathcal {A}$$ be a standard operator algebra on a Banach space $$\mathcal {X}$$ with $$ \dim \mathcal {X}\ge 3$$. In this paper, we determine the form of the bijective maps $$\phi :\mathcal {A}\longrightarrow \mathcal {A}$$ satisfying $$\begin{aligned} \phi \left( \frac{1}{2}(AB^2+B^2A)\right) = \frac{1}{2}[\phi (A)\phi (B)^{2}+\phi (B)^{2}\phi (A)], \end{aligned}$$for every $$A,B \in \mathcal {A}$$.     

A formula for the conductor of a semimodule of a numerical semigroup with two generators

Semigroup Forum - We provide an expression for the conductor $$c(\varDelta )$$ of a semimodule $$\varDelta $$ of a numerical semigroup $$\varGamma $$ with two generators in terms of the syzygy...     

Inverse Boundary Value Problem of Determining Up to a Second Order Tensor Appear in the Lower Order Perturbation of a Polyharmonic Operator

Journal of Fourier Analysis and Applications - We consider the following perturbed polyharmonic operator $$\mathcal {L}(x,D)$$ of order 2m defined in a bounded domain $$\Omega \subset \mathbb...     

Numerical semigroups bounded by the translation of a plane monoid

Aequationes mathematicae - Let $$\mathbb {N}$$ be the set of nonnegative integer numbers. A plane monoid is a submonoid of $$(\mathbb {N}^2,+)$$ . Let M be a plane monoid and $$p,q\in \mathbb {N}$$...     

On the structure of partition which the difference of their representation function is a constant

Periodica Mathematica Hungarica - Let $$\mathbb {N}$$ be the set of nonnegative integers. For any set $$A \subset \mathbb {N}$$ , let $$R_1(A, n)$$ , $$R_2(A, n)$$ and $$R_3(A, n)$$ be the number...     

A result on the sum of element orders of a finite group

Let G be a finite group and $$\psi (G)=\sum _{g\in {G}}{o(g)}$$. There are some results about the relation between $$\psi (G)$$ and the structure of G. For instance, it is proved that if G is a group of order n and $$\psi (G)>\dfrac{211}{1617}\psi (C_n)$$, then G is solvable. Herzog et al. in (J Algebra 511:215–226, 2018) put forward the following conjecture: Conjecture. If G is a non-solvable group of order n, then $$\begin{aligned} {\psi (G)}\,{\le }\,{{\dfrac{211}{1617}}{\psi (C_n)}}, \end{aligned}$$with equality if and only if $$G \cong A_5$$. In particular, this inequality holds for all non-Abelian simple groups. In this paper, we prove a modified version of Herzog’s Conjecture.     

The depth of a reflexive polytope

Given arbitrary integers d and r with $$d \ge 4$$ and $$1 \le r \le d + 1$$ , a reflexive polytope $${\mathscr {P}}\subset {\mathbb R}^d$$ of dimension d with $$\mathrm{depth}\,K[{\mathscr {P}}] = r$$ for which its dual polytope $${\mathscr {P}}^\vee $$ is normal will be constructed, where $$K[{\mathscr {P}}]$$ is the toric ring of $${\mathscr {P}}$$ .     

An isoperimetric inequality in the plane with a log-convex density

Ricerche di Matematica - Given a positive lower semi-continuous density f on $$\mathbb {R}^2$$ the weighted volume $$V_f:=f\mathscr {L}^2$$ is defined on the $$\mathscr {L}^2$$ -measurable sets in...     

Asymptotic behavior and representation of solutions to a Volterra kind of equation with a singular kernel

Semigroup Forum - Let A be a densely defined closed, linear $$\omega$$ -sectorial operator of angle $$\theta \in [0,\frac{\pi }{2})$$ on a Banach space X, for some $$\omega \in \mathbb...     

On the irreducibility of the Severi variety of nodal curves in a smooth surface

Archiv der Mathematik - Let X be a smooth projective surface and $$L\in \mathrm {Pic}(X)$$ . We prove that if L is $$(2k-1)$$ -spanned, then the set $${\tilde{V}}_k(L)$$ of all nodal and...     

The primitive spectrum of a semigroup of Markov operators

Positivity - For a semigroup $$\mathcal {S}$$ of Markov operators on a space of continuous functions, we use $$\mathcal {S}$$-invariant ideals to describe qualitative properties of $$\mathcal {S}$$...     

On the Spectrum of a Non-Self-Adjoint Quasiperiodic Operator

Doklady Mathematics - We study the operator $$\mathcal{A}$$ acting in $${{l}^{2}}(\mathbb{Z})$$ by the formula $${{(\mathcal{A}u)}_{l}} = {{u}_{{l + 1}}} + {{u}_{{l - 1}}} + \lambda {{e}^{{ - 2\pi...     

Nonlocal Dezin Problem for a Mixed Type Equation of the Second Kind

Differential Equations - For the equation $$u_{{xx}}+yu_{{yy}}+\alpha u_y=0 $$ with parameter $$\alpha \leq -1/2 $$ given in the mixed domain that is the rectangle $$[0,1]\times [-\beta ,\gamma ]$$...     

Exponent of a finite group of odd order with an involutory automorphism

Let $$w = w(x_1, \ldots , x_n)$$ be a non-trivial word of n-variables. The word map on a group G that corresponds to w is the map $$\widetilde{w}: G^n\rightarrow G$$ where $$\widetilde{w}((g_1, \ldots , g_n)) := w(g_1, \ldots , g_n)$$ for every sequence $$(g_1, \ldots , g_n)$$ . Let $$\mathcal G$$ be a simple and simply connected group which is defined and split over an infinite field K and let $$G =\mathcal G(K)$$ . For the case when $$w = w_1w_2 w_3 w_4$$ and $$w_1, w_2, w_3, w_4$$ are non-trivial words with independent variables, it has been proved by Hui et al. (Israel J Math 210:81–100, 2015) that $$G{\setminus } Z(G) \subset {{\text { Im}}}\,\widetilde{w}$$ where Z(G) is the center of the group G and $${{\text { Im}}}\, {\widetilde{w}}$$ is the image of the word map $$\widetilde{w}$$ . For the case when $$G = {{\text {SL}}}_n(K)$$ and $$n \ge 3$$ , in the same paper of Hui et al. (2015) it was shown that the inclusion $$G{\setminus } Z(G)\subset {{\text { Im}}}\,\widetilde{w}$$ holds for a product $$w = w_1w_2 w_3$$ of any three non-trivial words $$ w_1, w_2, w_3$$ with independent variables. Here we extent the latter result for every simple and simply connected group which is defined and split over an infinite field K except the groups of types $$B_2, G_2$$ .     

On the Optimal Value Function of a Linearly Perturbed Quadratic Program

The optimal value function of the quadratic program , where is a given symmetric matrix, a given matrix, and are the linear perturbations, is considered. It is proved that is directionally differentiable at any point in its effective domain . Formulae for computing the directional derivative of at in a direction are obtained. We also present an example showing that, in general, is not piecewise linear-quadratic on W. The preceding (unpublished) example of Klatte is also discussed.     

On a divisor of the central binomial coefficient

The Ramanujan Journal - It is well known that for all $$n\ge 1$$ the number $$n+1$$ is a divisor of the central binomial coefficient $${2n\atopwithdelims ()n}$$ . Since the nth central binomial...     

On the Second Smallest and the Largest Normalized Laplacian Eigenvalues of a Graph

Acta Mathematicae Applicatae Sinica, English Series - Let G be a simple connected graph with order n. Let $$\mathcal{L}(G)$$ and $$\mathcal{Q}(G)$$ be the normalized Laplacian and normalized...     

A Bilogarithmic Criterion for the Existence of a Regular Minorant that Does Not Satisfy the Bang Condition

Mathematical Notes - Problems of constructing regular majorants for sequences $$\mu=\{\mu_n\}_{n=0}^{\infty}$$ of numbers $$\mu_n\ge0$$ that are the Taylor coefficients of integer transcendental...