We construct the quantum $s$ -tuple subfactors for an AFD $\text{I}{{\text{I}}_{1}}$ subfactor with finite index and depth, for an arbitrary natural number $s$ . This is a generalization of the quantum multiple subfactors by Erlijman and Wenzl, which in turn generalized the quantum double construction of a subfactor for the case that the original subfactor gives rise to a braided tensor category. In this paper we give a multiple construction for a subfactor with a weaker condition than braidedness of the bimodule system.

In this note, we give a new short proof of the fact, recently discovered by Ye, that all (finite) values are equidistributed by the Riemann zeta function.

We apply the hypergeometric method of Thue and Siegel to prove that if $a$ and $b$ are positive integers, then the inequality $0\,<\,\left| {{a}^{x}}\,-\,{{b}^{y}} \right|\,<\,\frac{1}{4}\,\max \{{{a}^{x/2}},\,{{b}^{y/2}}\}$ has at most a single solution in positive integers $x$ and $y$ . This essentially sharpens a classic result of LeVeque.

Perturbation theory is a fundamental tool in Banach space theory. However, the applications of the classical results are limited by the fact that they force the perturbed sequence to be equivalent to the given sequence. We will develop amore general perturbation theory that does not force equivalence of the sequences.

Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type $\left( A \right)$ in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator.

A new semilocal convergence result for the Picard method is presented, where the main required condition in the contraction mapping principle is relaxed.

In this paper, we generalize a result recently obtained by the author. We characterize the cyclic vectors in $L_{a}^{p}\,\left( \mathbb{C},\,\phi\right)$ . Let $f\,\in \,L_{a}^{p}\,\left( \mathbb{C},\,\phi\right)$ and $fC$ be contained in the space. We show that $f$ is non-vanishing if and only if $f$ is cyclic.

The well-known Falkner–Skan equation is one of the most important equations in laminar boundary layer theory and is used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past wedge shaped bodies of angles related to $\text{ }\!\!\lambda\!\!\text{}\!\!\pi\!\!\text{ /2}$ , where $\text{ }\!\!\lambda\!\!\text{ }\,\in \,\mathbb{R}$ is a parameter involved in the equation. It is known that there exists ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,<\,0$ such that the equation with suitable boundary conditions has at least one positive solution for each $\text{ }\!\!\lambda\!\!\text{ }\,\ge \,{{\text{ }\!\!\lambda\!\!\text{ }}^{*}}$ and has no positive solutions for $\text{ }\!\!\lambda\!\!\text{ }\,<\,{{\text{ }\!\!\lambda\!\!\text{ }}^{*}}$ . The known numerical result shows ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,=\,-0.1988$ . In this paper, ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,\in \,[-0.4,\,-0.12]$ is proved analytically by establishing a singular integral equation which is equivalent to the Falkner–Skan equation. The equivalence result provides new techniques to study properties and existence of solutions of the Falkner–Skan equation.

Let ${{b}_{1}}$ , ${{b}_{2}}$ be any integers such that $\gcd \left( {{b}_{1}},{{b}_{2}} \right)=1$ and ${{c}_{1}}\left| {{b}_{1}} \right|\,<\,\left| {{b}_{2}} \right|\,\le \,{{c}_{2}}\left| {{b}_{1}} \right|$ , where ${{c}_{1}}$ , ${{c}_{2}}$ are any given positive constants. Let $n$ be any integer satisfying $\gcd \left( n,\,{{b}_{i}} \right)\,=\,1$ , $i\,=\,1,\,2$ . Let ${{P}_{k}}$ denote any integer with no more than $k$ prime factors, counted according to multiplicity. In this paper, for almost all ${{b}_{2}}$ , we prove (i) a sharp lower bound for $n$ such that the equation ${{b}_{1}}p\,+\,{{b}_{2}}m\,=\,n$ is solvable in prime $p$ and almost prime $m\,=\,{{P}_{k}}$ , $k\,\ge \,3$ whenever both ${{b}_{i}}$ are positive, and (ii) a sharp upper bound for the least solutions $p$ , $m$ of the above equation whenever ${{b}_{i}}$ are not of the same sign, where $p$ is a prime and $m\,=\,{{P}_{k}}$ , $k\,\ge \,3$ .

This paper deals with the concepts of condensed and strongly condensed domains. By definition, an integral domain $R$ is condensed (resp. strongly condensed) if each pair of ideals $I$ and $J$ of $R$ , $IJ\,=\,\{ab/a\,\in \,I,\,b\,\in \,J\}$ (resp. $IJ\,=\,aJ$ for some $a\,\in \,I\,or\,I\,J\,=\,Ib$ for some $b\,\in \,J$ ). More precisely, we investigate the ideal theory of condensed and strongly condensed domains in Noetherian-like settings, especially Mori and strong Mori domains and the transfer of these concepts to pullbacks.

Given a countable set $S$ of positive reals, we study finite-dimensional Ramsey-theoretic properties of the countable ultrametric Urysohn space ${{\text{Q}}_{S}}$ with distances in $S$ .

The consideration of tensor products of 0-Hecke algebramodules leads to natural analogs of the Bessel $J$ -functions in the algebra of noncommutative symmetric functions. This provides a simple explanation of various combinatorial properties of Bessel functions.

An $R$ -module $M$ is called a multiplication module if for each submodule $N$ of $M,\,N\,=\,IM$ for some ideal $I$ of $R$ . As defined for a commutative ring $R$ , an $R$ -module $M$ is said to be semiprimitive if the intersection of maximal submodules of $M$ is zero. The maximal spectra of a semiprimitive multiplication module $M$ are studied. The isolated points of $\text{Max}\left( M \right)$ are characterized algebraically. The relationships among the maximal spectra of $M$ , $\text{Soc}\left( M \right)$ and $\text{Ass}\left( M \right)$ are studied. It is shown that $\text{Soc}\left( M \right)$ is exactly the set of all elements of $M$ which belongs to every maximal submodule of $M$ except for a finite number. If $\text{Max}\left( M \right)$ is infinite, $\text{Max}\left( M \right)$ is a one-point compactification of a discrete space if and only if $M$ is Gelfand and for some maximal submodule $K$ , $\text{Soc}\left( M \right)$ is the intersection of all prime submodules of $M$ contained in $K$ . When $M$ is a semiprimitive Gelfand module, we prove that every intersection of essential submodules of $M$ is an essential submodule if and only if $\text{Max}\left( M \right)$ is an almost discrete space. The set of uniform submodules of $M$ and the set of minimal submodules of $M$ coincide. $\text{Ann}\left( \text{Soc}\left( M \right) \right)M$ is a summand submodule of $M$ if and only if $\text{Max}\left( M \right)$ is the union of two disjoint open subspaces $A$ and $N$ , where $A$ is almost discrete and $N$ is dense in itself. In particular, $\text{Ann}\left( \text{Soc}\left( M \right) \right)\,=\,\text{Ann}\left( M \right)$ if and only if $\text{Max}\left( M \right)$ is almost discrete.

Biharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed.

Let $R\,=\,\oplus _{i=1}^{\infty }\,{{R}_{i}}$ be a graded nil ring. It is shown that primitive ideals in $R$ are homogeneous. Let $A\,=\,\oplus _{i=1}^{\infty }\,{{A}_{i}}$ be a graded non-PI just-infinite dimensional algebra and let $I$ be a prime ideal in $A$ . It is shown that either $I\,=\,\{0\}$ or $I\,=\,A$ . Moreover, $A$ is either primitive or Jacobson radical.

In this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact Kähler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.