In this paper, we introduce the definition of a convex real valued function $f$ defined on the set of integers, $\mathbb{Z}$ . We prove that $f$ is convex on $\mathbb{Z}$ if and only if ${{\Delta }^{2}}f\,\ge \,0$ on $\mathbb{Z}$ . As a first application of this new concept, we state and prove discrete Hermite–Hadamard inequality using the basics of discrete calculus (i.e., the calculus on $\mathbb{Z}$ ). Second, we state and prove the discrete fractional Hermite–Hadamard inequality using the basics of discrete fractional calculus. We close the paper by defining the convexity of a real valued function on any time scale.

The classification of Euclidean frieze groups into seven conjugacy classes is well known, and many articles on recreational mathematics contain frieze patterns that illustrate these classes. However, it is only possible to draw these patterns because the subgroup of translations that leave the pattern invariant is (by definition) cyclic, and hence discrete. In this paper we classify the conjugacy classes of frieze groups that contain a non-discrete subgroup of translations, and clearly these groups cannot be represented pictorially in any practicalway. In addition, this discussion sheds light onwhy there are only seven conjugacy classes in the classical case.

In this paper, the quaternionic hyperbolic ideal triangle groups are parametrized by a real one-parameter family $\{{{\phi }_{s}}\,:\,s\,\in \,\mathbb{R}\}$ . The indexing parameter s is the tangent of the quaternionic angular invariant of a triple of points in $\partial \mathbf{H}_{\mathbb{H}}^{2}$ forming this ideal triangle. We show that if $s>\sqrt{125/3},$ then ${{\phi }_{S}}$ is not a discrete embedding, and if $s\,\le \,\sqrt{3\text{5}}$ , then ${{\phi }_{S}}$ is a discrete embedding.

Let $R$ be a prime ring of characteristic diòerent from $2$, let ${{Q}_{r}}$ be its right Martindale quotient ring, and let $C$ be its extended centroid. Suppose that $F$ is a generalized skew derivation of $R,\,L$ a non-central Lie ideal of $R,\,0\,\ne \,a\,\in \,R,\,m\,\ge \,0$ and $n,\,s\,\ge \,1$ fixed integers. If

$$a{{\left( {{u}^{m}}F\left( u \right){{u}^{n}} \right)}^{s}}\,=\,0$$

for all $u\,\in \,L$, then either $R\,\subseteq \,{{M}_{2}}\left( C \right)$, the ring of $2\,\times \,2$ matrices over $C$, or $m\,=\,0$ and there exists $b\,\in \,{{Q}_{r}}$ such that $F\left( x \right)\,=\,bx$, for any $x\,\in \,R$, with $ab\,=\,0$.

Let $R\,=\,{{\oplus }_{n\ge 0}}{{R}_{n}}$ be a graded Noetherian ring with local base ring $\left( {{R}_{0}},{{\text{m}}_{0}} \right)$ and let ${{R}_{+}}\,=\,{{\oplus }_{n>0}}{{R}_{n}}$ . Let $M$ and $N$ be finitely generated graded $R$ -modules and let $\mathfrak{a}\,=\,{{\mathfrak{a}}_{0}}\,+\,{{R}_{+}}$ an ideal of $R$ . We show that $H_{\mathfrak{b}0}^{j}\,\left( H_{\mathfrak{a}}^{i}\left( M,\,N \right) \right)$ and ${H_{\mathfrak{a}}^{i}\left( M,\,N \right)}/{{{\mathfrak{b}}_{0}}H_{\mathfrak{a}}^{i}\left( M,\,N \right)}\;$ are Artinian for some $i\text{ s}$ and $j\,\text{s}$ with a specified property, where ${{\mathfrak{b}}_{o}}$ is an ideal of ${{R}_{0}}$ such that ${{\mathfrak{a}}_{0}}\,+\,{{\mathfrak{b}}_{0}}$ is an ${{\mathfrak{m}}_{0}}$ -primary ideal.

Using a recent result by S. Papadima and A. Suciu, we show that the equivariant Poincaré– Deligne polynomial of the Milnor fiber of a projective line arrangement having only double and triple points is combinatorially determined.

An incomplete pairwise balanced design is equivalent to a pairwise balanced design with a distinguished block, viewed as a ‘hole’. If there are v points, a hole of size $w$ , and all (other) block sizes equal $k$ , this is denoted $\text{IPBD}\left( \left( v;w \right),\,k \right)$ . In addition to congruence restrictions on $v$ and $w$ , there is also a necessary inequality: $v\,>\,\left( k\,-\,1 \right)w$ . This article establishes two main existence results for $\text{IPBD}\left( \left( v;w \right),\,k \right)$ : one in which $w$ is fixed and $v$ is large, and the other in the case $v>\,\left( k-1+\varepsilon \right)w$ when $w$ is large (depending on $\varepsilon$ ). Several possible generalizations of the problemare also discussed.

In this paper we study the zero sets of harmonic functions on open sets in ${{\mathbb{R}}^{N}}$ and holomorphic functions on open sets in ${{\mathbb{C}}^{N}}$. We show that the non-extendability of such zero sets is a generic phenomenon.

We show that the product rank of the $3\,\times \,3$ determinant ${{\det }_{3}}$ is $5$ , and the product rank of the $3\,\times \,3$ permanent $\text{per}{{\text{m}}_{3}}$ is $4$ . As a corollary, we obtain that the tensor rank of ${{\det }_{3}}$ is $5$ and the tensor rank of $\text{per}{{\text{m}}_{3}}$ is $4$ . We show moreover that the border product rank of $\text{per}{{\text{m}}_{3}}$ is larger than $n$ for any $n\,\ge \,3$ .

In this paper, we study uniform perturbations of von Neumann subalgebras of a von Neumann algebra. Let $M$ and $N$ be von Neumann subalgebras of a von Neumann algebra with finite probabilistic index in the sense of Pimsner and Popa. If $M$ and $N$ are sufficiently close, then $M$ and $N$ are unitarily equivalent. The implementing unitary can be chosen as being close to the identity.

In this paper, we develop a generalized Jordan canonical form theorem for a certain class of operators in $L\left( H \right)$ . A complete criterion for similarity for this class of operators in terms of $K$ -theory for Banach algebras is given.

We study an associative algebra $A$ over an arbitrary field that is a sum of two subalgebras $B$ and $C$ (i.e., $A\,=\,B+C$ ). We show that if $B$ is a right or left Artinian $PI$ algebra and $C$ is a $PI$ algebra, then $A$ is a $PI$ algebra. Additionally, we generalize this result for semiprime algebras $A$ . Consider the class of all semisimple finite dimensional algebras $A\,=\,B+C$ for some subalgebras $B$ and $C$ that satisfy given polynomial identities $f\,=\,0$ and $g\,=\,0$ , respectively. We prove that all algebras in this class satisfy a common polynomial identity.

We study and generalize a classical theoremof L. Bers that classifies domains up to biholomorphic equivalence in terms of the algebras of holomorphic functions on those domains. Then we develop applications of these results to the study of domains with noncompact automorphism group.

Let $T$ be a quadratic operator on a complex Hilbert space $H$ . We show that $T$ can be written as a product of two positive contractions if and only if $T$ is of the form

$$aI\,\oplus \,bI\,\oplus \left( \begin{matrix} aI & P \\ 0 & bI \\ \end{matrix} \right)\,\text{on}\,{{H}_{1}}\,\oplus \,{{H}_{2}}\,\oplus \,\left( {{H}_{3\,}}\,\oplus \,{{H}_{3}} \right)$$

for some $a,\,b\,\in \,\left[ 0,\,1 \right]$ and strictly positive operator $P$ with $\left\| P \right\|\,\le \,\left| \sqrt{a}-\sqrt{b} \right|\sqrt{\left( 1-a \right)\left( 1-b \right)}$ . Also, we give a necessary condition for a bounded linear operator $T$ with operator matrix $\left( \begin{matrix} {{T}_{1}} & {{T}_{3}} \\ 0 & {{T}_{2}} \\ \end{matrix} \right)$ on $H\,\oplus \,K$ that can be written as a product of two positive contractions.

In this paper, a nonlinear stage-structured model for Lyme disease is considered. The model is a system of diòerential equations with two time delays. We derive the basic reproductive rate ${{R}_{0}}\left( {{\tau }_{1}},\,{{\tau }_{2}} \right)$. If ${{R}_{0}}\left( {{\tau }_{1}},\,{{\tau }_{2}} \right)\,<\,1$, then the boundary equilibrium is globally asymptotically stable. If ${{R}_{0}}\left( {{\tau }_{1}},\,{{\tau }_{2}} \right)\,>\,1$, then there exists a unique positive equilibrium whose local asymptotic stability and the existence of Hopf bifurcations are established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived by using the normal form and the center manifold theory. Some numerical simulations are performed to confirm the correctness of theoretical analysis. Finally, some conclusions are given.

A doubly stochastic measure on the unit square is a Borel probability measure whose horizontal and vertical marginals both coincide with the Lebesgue measure. The set of doubly stochasticmeasures is convex and compact so its extremal points are of particular interest. The problem number 111 of Birkhoò is to provide a necessary and suõcient condition on the support of a doubly stochastic measure to guarantee extremality. It was proved by Beneš and Štepán that an extremal doubly stochastic measure is concentrated on a set which admits an aperiodic decomposition. Hestir and Williams later found a necessary condition which is nearly sufficient by further refining the aperiodic structure of the support of extremal doubly stochastic measures. Our objective in this work is to provide a more practical necessary and nearly sufficient condition for a set to support an extremal doubly stochastic measure.

We investigate whether the total character of a finite group $G$ is a polynomial in a suitable irreducible character of $G$ . When $\left( G,\,Z\left( G \right) \right)$ is a generalized Camina pair, we show that the total character is a polynomial in a faithful irreducible character of $G$ if and only if $Z\left( G \right)$ is cyclic.

Let $\mathfrak{a}$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\text{f}{{\text{d}}_{R}}X,\,\text{Gf}{{\text{d}}_{R}}X$ and ${{\text{G}}_{\text{C}}}-\text{f}{{\text{d}}_{R}}\,X\,\text{by}\,\text{T}\left( X \right)$. Let $M$ be an $R$-module such that $\text{H}_{\mathfrak{a}}^{i}\left( M \right)\,=\,0$ for all $i\,\ne \,n$. It is proved that if $T\left( M \right)\,<\,\infty$, then $\text{T}\left( \text{H}_{\mathfrak{a}}^{n}\left( M \right) \right)\,\le \,\text{T}\left( M \right)\,+\,n$, and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen–Macaulay modules, dualizing modules, and Gorenstein rings.

In this paper, we consider the quasi-linear elliptic problem

$$-M\left( {{\int }_{{{\mathbb{R}}^{N}}}}{{\left| x \right|}^{-ap}}{{\left| {{\nabla }_{u}} \right|}^{p}}dx \right)\,\text{div}\left( {{\left| x \right|}^{-ap}}{{\left| \nabla u \right|}^{p-2}}\nabla u \right)=\frac{\alpha }{\alpha +\beta }H\left( x \right){{\left| u \right|}^{\alpha -2}}u{{\left| v \right|}^{\beta }}+\text{ }\lambda \text{ }{{\text{h}}_{1}}\left( x \right){{\left| u \right|}^{q-2}}u,$$

$$-M\left( {{\int }_{{{\mathbb{R}}^{N}}}}{{\left| x \right|}^{-ap}}{{\left| \nabla v \right|}^{p}}dx \right)\,\text{div}\left( {{\left| x \right|}^{-ap}}{{\left| \nabla v \right|}^{p-2}}\nabla v \right)=\frac{\beta }{\alpha +\beta }H\left( x \right){{\left| v \right|}^{\beta -2}}v{{\left| u \right|}^{\alpha }}+\mu {{h}_{2}}\left( x \right){{\left| v \right|}^{q-2}}v,$$

$$u\left( x \right)>0,v\left( x \right)>0,x\in {{\mathbb{R}}^{N}},$$

where $\text{ }\lambda \text{ ,}\mu >\text{0,}\text{1}<\text{p}<\text{N,}\text{1}<\text{q}<\text{p}<\text{p}\left( \tau +1 \right)<\alpha +\beta <{{p}^{*}}=\frac{{{N}_{p}}}{N-p},0\le a<\frac{N-p}{p},a\le b<a+1,d=a+1-b>0,M\left( s \right)=k+l{{s}^{\tau }},k>0,l,\tau \ge 0$ and the weight $H\left( x \right),\,{{h}_{1}}\left( x \right),\,{{h}_{2}}\left( x \right)$ are continuous functions that change sign in ${{\mathbb{R}}^{N}}$ . We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler functional for this problem.

For any ${{C}^{*}}$ -algebra $A$ with an approximate unit of projections, there is a smallest ideal $I$ of $A$ such that the quotient $A$ / $I$ is stably finite. In this paper a sufficient and necessary condition for an ideal of a ${{C}^{*}}$ -algebra with real rank zero to be this smallest ideal is obtained by using $K$ -theory