This paper is devoted to the study of Wirtinger-type inequalities for the Lebesgue $\Delta$ -integral on an arbitrary time scale $\mathbb{T}$ . We prove a general inequality for a class of absolutely continuous functions on closed subintervals of an adequate subset of $\mathbb{T}$ . By using this expression and by assuming that $\mathbb{T}$ is bounded, we deduce that a general inequality is valid for every absolutely continuous function on $\mathbb{T}$ such that its $\Delta$ -derivative belongs to $L_{\Delta }^{2}\,([a,\,b)\,\cap \,\mathbb{T})$ and at most it vanishes on the boundary of $\mathbb{T}$ .

In this paper we obtain quantitative results on the occurrence of consecutive large gaps between $B$ -free numbers, and consecutive large gaps between nonzero Fourier coefficients of a class of newforms without complex multiplication.

In this paper we describe the group gradings by a finite abelian group $G$ of the matrix algebra ${{M}_{n}}(F)$ over an algebraically closed field $F$ of characteristic different from 2, which respect an involution (involution gradings). We also describe, under somewhat heavier restrictions on the base field, all $G$ -gradings on all finite-dimensional involution simple algebras.

We give a necessary and sufficient condition for a composition operator on an $\alpha$ -Bloch space with $\alpha \,\ge \,1$ to be bounded below. This extends a known result for the Bloch space due to P. Ghatage, J. Yan, D. Zheng, and H. Chen.

We prove that for every function $f\,:\,X\,\to \,Y$ , where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\,\in \,\overset{\sim }{\mathop{\mathcal{A}}}\,$ such that $f$ is Gâteaux differentiable at all $x\,\in \,S\left( f \right)\backslash A$ , where $S\left( f \right)$ is the set of points where $f$ is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every $K$ -monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to $\tilde{C}\,;$ this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f\,:\,X\,\to \,\mathbb{R}$ cone monotone, $g\,:\,X\,\to \,\mathbb{R}$ continuous convex, then there exists a point in $X$ , where $f$ is Hadamard differentiable and $g$ is Fréchet differentiable.

We study a second order nonlinear system driven by the vector $p$ -Laplacian, with a multivalued nonlinearity and defined on the positive time semi-axis ${{\mathbb{R}}_{+}}.$ Using degree theoretic techniques we solve an auxiliary mixed boundary value problem defined on the finite interval $\left[ 0,\,n \right]$ and then via a diagonalization method we produce a solution for the original infinite time horizon system.

Let $\Omega$ be a domain in ${{\mathbb{R}}^{n}}\,(n\,\ge \,2).$ We find a necessary and sufficient topological condition on $\Omega$ such that, for any measure $ $ on ${{\mathbb{R}}^{n}}$ , there is a function $u$ with specified boundary conditions that satisfies the Poisson equation $\Delta u\,=\,\mu$ on $\Omega$ in the sense of distributions.

Let ${{B}_{p}}$ be the unit ball in ${{\mathbb{L}}_{p}}$ , $0\,<\,p\,<\,1$ , and let $\Delta _{+}^{s}$ , $s\,\in \,\mathbb{N}$ , be the set of all $s$ -monotone functions on a finite interval $I$ , i.e., $\Delta _{+}^{s}$ consists of all functions $x\,:\,I\,\mapsto \,\mathbb{R}$ such that the divided differences $[x;\,{{t}_{0}},\,...\,,\,{{t}_{s}}]$ of order $s$ are nonnegative for all choices of $\left( s\,+\,1 \right)$ distinct points ${{t}_{0}},\,.\,.\,.\,,{{t}_{s}}\,\in \,I.$ For the classes $\Delta _{+}^{s}{{B}_{P}}\,:=\,\Delta _{+}^{s}\,\cap \,{{B}_{P}},$ we obtain exact orders of Kolmogorov, linear and pseudo-dimensional widths in the spaces ${{\mathbb{L}}_{q}},$ $0\,<\,q\,<\,p\,<\,1$ :

$${{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{psd}}\asymp {{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{kol}}\asymp {{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{lin}}\asymp {{n}^{-s}}.$$

Let $M$ be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue $\text{ }\!\!\lambda\!\!\text{ }.$ We give upper and lower bounds on the inner radius of the type $C/{{\lambda }^{\alpha }}{{(\log \lambda )}^{\beta }}.$ Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman and a Poincaré type inequality proved by Maz’ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.

An $n$ -dimensional quantum torus is a twisted group algebra of the group ${{\mathbb{Z}}^{n}}.$ It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational $n$ -dimensional quantum tori over any field. Moreover, we show that for $n\,=\,2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field.

We prove that for a generic hypersurface in ${{\mathbb{P}}^{2n+1}}$ of degree at least $2\,+\,2/n$ , the $n$ -th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing.

Let $KG$ be a non-commutative strongly Lie solvable group algebra of a group $G$ over a field $K$ of positive characteristic $p$ . In this note we state necessary and sufficient conditions so that the strong Lie derived length of $KG$ assumes its minimal value, namely $\left\lceil {{\log }_{2}}(p\,+\,1) \right\rceil$ .

In this paper we prove that the Kostrikin radical of an extended affine Lie algebra of reduced type coincides with the center of its core, and use this characterization to get a type-free description of the core of such algebras. As a consequence we get that the core of an extended affine Lie algebra of reduced type is invariant under the automorphisms of the algebra.

The homotopy groups of a finite partially ordered set (poset) can be described entirely in the context of posets, as shown in a paper by $\text{B}$ . Larose and $\text{C}$ . Tardif. In this paper we describe the relative version of such a homotopy theory, for pairs $\left( X,\,A \right)$ where $X$ is a poset and $A$ is a subposet of $X$ . We also prove some theorems on the relevant version of the notion of weak homotopy equivalences for maps of pairs of such objects. We work in the category of reflexive binary relational structures which contains the posets as in the work of Larose and Tardif.