In this paper, we investigate when the unit, unitary and total graphs are ring graphs, and also we study the case that they are outerplanar.

Let $$ be a commutative ring with identity. For $$, define $$ and $$ to be \textit{associates}, denoted $$, if $$ and $$, so $$ and $$ for some $$. We are interested in the case where $$ and $$ can be taken or must be taken to be non zero-divisors or units. We study rings, $$, called \textit{strongly regular associate}, that have the property that, whenever $$ for $$, then there exist non zero-divisors $$ with $$ and $$ and rings $$, called \textit{weakly pr\'{e}simplifiable}, that have the property that, for nonzero $$ with $$, whenever $$ and $$, then $$ and $$ must be non zero-divisors.

Suppose $$ is a unital ring having an idempotent element $$ which satisfies $$ implies $$ and $$ implies $$. In this paper, we aim to characterize the map $$, $$ is surjective and $$ for all $$. It is shown that $$ for all $$, where $$, $$, and $$ is a map from $$ into $$. As an application, a characterization of nonlinear surjective maps preserving strong commutativity on von Neumann algebras with no central summands of type $$ is obtained.

In this paper, we define weakly coherent rings and examine the transfer of this property to homomorphic images, trivial ring extensions, localizations and finite direct products. These results provide examples of weakly coherent rings that are not coherent rings. We show that the class of weakly coherent rings is not stable under localization. Also, we show that the class of weakly coherent rings and the class of strongly $$-coherent rings are not comparable.

In this paper, we obtain coefficient criteria for a normalized harmonic function defined in the unit disk to be close-to-convex and fully starlike, respectively. Using these coefficient conditions, we present different classes of harmonic close-to-convex (respectively, fully starlike) functions involving Gaussian hypergeometric functions. In addition, we present a convolution characterization for a class of univalent harmonic functions discussed recently by Mocanu, and later by Bshouty and Lyzzaik in 2010. Our approach provides examples of harmonic polynomials that are close-to-convex and starlike, respectively.

Let $$ be smoothly bounded doubly connected regions in the complex plane. We establish a transformation law for the Szeg\H{o} kernel under proper holomorphic mappings. This extends known results concerning biholomorphic mappings between multiply connected regions as well as proper holomorphic mappings from multiply connected regions to simply connected regions.

We derive expressions for the number of factorization of positive integers into squarefree factors with order not counting and for the asymptotic average of these factorizations.

We define and provide some basic analysis of various types of crossed products by semimultiplicative sets, and then prove a $$-theoretical descent homomorphisms for semimultiplicative sets in accord with the descent homomorphism for discrete groups.

Let $$ be an elliptic curve with CM by $$. We determine a formula for (a generalization of) the arithmetic local constant of [{\bf5}] at almost all primes of good reduction. We apply this formula to the CM curves defined over $$ and are able to describe extensions $$ over which the $$-rank of $$ grows.

We provide a concrete class of rings in which there exists a primary ideal with respect to the maximal ideal that has only one Rees valuation.

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The term $$-algebra means a generalized $$-algebra and corresponds to an unbounded generalization of a standard von Neumann algebra. It was introduced by the second named author in 1978 for developing the Tomita-Takesaki theory in algebras of unbounded operators. In this note we consider tensor products of unbounded operator algebras resulting in a $$-algebra. Existence and uniqueness of the $$-tensor product is encountered, while ``properly $$-infinite" $$-algebras are introduced and their structure is investigated.

This work is motivated by R\u{a}dulescu's result~\cite{R˘ad04} on the comparison of $$-tensor norms on $$.

For unital $$-algebras $$ and $$, there are natural inclusions of $$ and $$ into the unital free product $$, the maximal tensor product $$ and the minimal tensor product $$. These inclusions define three operator system structures on the internal sum $$. Partly using ideas from quantum entanglement theory, we prove various interrelations between these three operator systems. As an application, the present results yield a significant improvement over R\u{a}dulescu's bound. At the same time, this tight comparison is so general that it cannot be regarded as evidence for the QWEP conjecture.

We consider a three point fourth order boundary value problem of focal type. Some upper and lower estimates for positive solutions of the problem are obtained. Sufficient conditions for the existence and nonexistence of positive solutions for the problem are established. An example is included to illustrate the results.

In this paper, we deal with the following integral boundary problem of nonlinear fractional differential equations with $$-Laplacian operator $$ where $$, $$, $$ is the standard Riemann-Liouville fractional derivative, $$, $$, $$. By the properties of the Green function, the lower and upper solution method and fixed-point theorem in partially ordered sets, some new existence and uniqueness of positive solutions to the above boundary value problem are established. As applications, examples are presented to illustrate the main results.

In this paper, first, we make the estimates for a class of fourth order elliptic equations in different domains and boundary conditions. Consequently, we study the qualitative properties of solutions with prescribed $$-curvature. Finally, we also will obtain some radially symmetric results by using moving plane methods.

Margalit and Schleimer \cite{MS} discovered a nontrivial root of the Dehn twist about a nonseparating curve on a closed oriented connected surface. We give a complete set of conjugacy invariants for such a root by using a classification theorem of Matsumoto and Montesinos \cite{MM1, MM2} for pseudo-periodic maps of negative twists. As an application, we determine the range of degree for roots of a Dehn twist.

Let $$ be the class of all normalized analytic and univalent functions in the unit disk $$. In this paper, we determine condition so that each section $$ of $$ is starlike in the disk $$. In particular, $$ is starlike in $$ for $$.

Let $$ be a hyperelliptic curve of good reduction defined over a discrete valuation field $$ with algebraically closed residue field $$. Assume moreover that $$. Given $$, we introduce an explicit description of the minimal regular model of the quadratic twist of $$ by $$. As an application, we show that if $$ is a nonsingular hyperelliptic curve given by $$ with $$ an irreducible polynomial, there exists a positive density family of prime quadratic twists of $$ which are not everywhere locally soluble.

We prove that a function $$ may be simultaneously approximated by a H\"older continuous function both pointwise and in the Sobolev norm.

For a linear partial differential operator with constant coefficients on $$, we investigate topological properties like barrelledness or bornolo\-gicity (which allow applications of fundamental principles like the Banach-Steinhaus or the open mapping theorem) of its kernel. Using recent functional analytic results inspired by homological algebra we prove that almost all barrelledness type conditions are equivalent in this situation and provide two distinct sufficient conditions which, in particular, are satisfied if the operator is surjective or hypoelliptic. This last case generalizes a classical result of Malgrange and H\"ormander.