Let $R$ be an associative ring with identity. First we prove some results about zero-divisor graphs of reversible rings. Then we study the zero-divisors of the skew power series ring $R\left[\!\left[ x;\,\alpha \right]\!\right]$, whenever $R$ is reversible $\alpha$-compatible. Moreover, we compare the diameter and girth of the zero-divisor graphs of $\Gamma \left( R \right),\,\Gamma \left( R[x;\,\alpha ,\,\delta ] \right)$, and $\Gamma \left( R\left[\!\left[ x;\,\alpha \right]\!\right] \right)$, when $R$ is reversible and $\left( \alpha ,\,\delta \right)$-compatible.

A commuting family of subnormal operators need not have a commuting normal extension. We study when a representation of an abelian semigroup can be extended to a normal representation, and show that it suffices to extend the set of generators to commuting normals. We also extend a result due to Athavale to representations on abelian lattice ordered semigroups.

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 give a simple proof and strengthening of a uniqueness theorem for functions in the extended Selberg class.

As an application of the theory of graph-like Legendrian unfoldings, relations of the hidden structures of caustics, and wave front propagations are revealed.

Alfred Schild established conditions where Lorentz transformations map world-vectors $\left( ct,\,x,\,y,\,z \right)$ with integer coordinates onto vectors of the same kind. The problem was dealt with in the context of tensor and spinor calculus. Due to Schild’s number-theoretic arguments, the subject is also interesting when isolated from its physical background.

Schild’s paper is not easy to understand. Therefore, we first present a streamlined version of his proof which is based on the use of null vectors. Then we present a purely algebraic proof that is somewhat shorter. Both proofs rely on the properties of Gaussian integers

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.

An action of a Lie group $G$ on a smooth manifold $M$ is called cohomogeneity one if the orbit space ${M}/{G}\;$ is of dimension 1. A Finsler metric $F$ on $M$ is called invariant if $F$ is invariant under the action of $G$. In this paper, we study invariant Randers metrics on cohomogeneity one manifolds. We first give a sufficient and necessary condition for the existence of invariant Randers metrics on cohomogeneity one manifolds. Then we obtain some results on invariant Killing vector fields on the cohomogeneity one manifolds and use them to deduce some sufficient and necessary conditions for a cohomogeneity one Randers metric to be Einstein.

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 $\mathbf{A}$ be a density matrix in ${{\mathbb{M}}_{m}}\,\otimes \,{{\mathbb{M}}_{n}}$. Audenaert [J. Math. Phys. 48(2007) 083507] proved an inequality for Schatten $p$-norms:

$$1\,+\,||\mathbf{A}|{{|}_{p}}\,\ge \,{{\left\| \text{T}{{\text{r}}_{1}}\,\mathbf{A} \right\|}_{p}}\,+\,||\text{T}{{\text{r}}_{2}}\,\mathbf{A}|{{|}_{p}},$$

where $\text{T}{{\text{r}}_{1}}$ and $\text{T}{{\text{r}}_{2}}$ stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality

$$1\,+\,\det \,\mathbf{A}\,\ge \,\det {{\left( \text{T}{{\text{r}}_{1}}\mathbf{A} \right)}^{m}}\,+\,\det {{\left( \text{T}{{\text{r}}_{2}}\mathbf{A} \right)}^{2}}.$$

A master formula of transformation formulas for bilinear sums of basic hypergeometric series is proposed. It is obtained from the author’s previous results on a transformation formula for Milne’s multivariate generalization of basic hypergeometric series of type $A$ with different dimensions and it can be considered as a generalization of the Whipple–Sears transformation formula for terminating balanced $_{4}{{\phi }_{3}}$ series. As an application of the master formula, the one-variable cases of some transformation formulas for bilinear sums of basic hypergeometric series are given as examples. The bilinear transformation formulas seem to be new in the literature, even in the one-variable case.

On a real hypersurface $M$ in a non-flat complex space form there exist the Levi–Civita and the $k$-th generalized Tanaka–Webster connections. The aim of this paper is to study three dimensional real hypersurfaces in non-flat complex space forms, whose Lie derivative of the structure Jacobi operatorwith respect to the Levi–Civita connection coincides with the Lie derivative of it with respect to the $k$-th generalized Tanaka-Webster connection. The Lie derivatives are considered in direction of the structure vector field and in direction of any vector field orthogonal to the structure vector field.

We present an elementary method for studying the problem of getting an asymptotic formula that is better than Hooley's and Heath-Brown's results for certain cases.

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.

We formulate a conjectural hard Lefschetz property for Chow groups and prove it in some special cases, roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional Griõths group. An appendix includes related statements that follow from results of Vial.

We show that the conjectural criterion of $p$-incompressibility for products of projective homogeneous varieties in terms of the factors, previously known in a few special cases only, holds in general. Actually, the proof goes through for a wider class of varieties, including the norm varieties associated with symbols in Galois cohomology of arbitrary degree.

Let ${{a}_{1}},\,.\,.\,.\,,\,{{a}_{9}}$ be non-zero integers and $n$ any integer. Suppose that ${{a}_{1}}\,+\,.\,.\,.\,+\,{{a}_{9}}\,\equiv \,n$$\left( \bmod \,2 \right)$ and $\left( {{a}_{i}},\,{{a}_{i}} \right)\,=\,1$ for $1\,\le \,i\,<\,j\le \,9$. In this paper we prove that

• (i) if ${{a}_{j}}$ are not all of the same sign, then the cubic equation ${{a}_{1}}p_{1}^{3}\,+\,.\,.\,.\,+\,{{a}_{9}}p_{9}^{3}\,=\,n$ has prime solutions satisfying ${{p}_{j}}\,\ll \,{{\left| n \right|}^{{1}/{3}\;}}\,+\,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{8+\varepsilon }}$;

• (ii) if all ${{a}_{j}}$ are positive and $n\,\gg \,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{25+\varepsilon }}$ , then ${{a}_{1}}p_{1}^{3}\,+\,.\,.\,.\,+\,{{a}_{j}}p_{9}^{3}\,=\,n$ is soluble in primes $Pj$.

These results improve our previous results with the bounds $\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{14+\varepsilon }}$ and $\max \,{{\left\{ \left| {{a}_{j}} \right| \right\}}^{43+\varepsilon }}$ in place of $\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{8+\varepsilon }}$ and $\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{25+\varepsilon }}$ above, respectively.

Anstee, Przytycki, and Rolfsen introduced the idea of rotants, pairs of links related by a generalised form of link mutation. We exhibit infinitely many pairs of rotants that can be distinguished by Khovanov homology, but not by the Jones polynomial.

In this paper, using Calderón’s reproducing formula and almost orthogonality estimates, we prove the lifting property and the embedding theorem of the Triebel–Lizorkin and Besov spaces associated with Zygmund dilations.

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.