We construct the quantum $s$-tuple subfactors for an AFD II$_{1}$
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 <| a^x - b^y | < \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
a more 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 $(A)$ 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 $\Lp$.
Let $f\in\Lp$ and $f\poly$ 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
$\lambda\pi/2$, where $\lambda\in \mathbb R$ is a parameter
involved in the equation. It is known that there exists
$\lambda^{*}<0$ such that the equation with suitable boundary
conditions has at least one positive solution for each $\lambda\ge
\lambda^{*}$ and has no positive solutions for
$\lambda<\lambda^{*}$. The known numerical result shows
$\lambda^{*}=-0.1988$. In this paper, $\lambda^{*}\in
[-0.4,-0.12]$ 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
$\gcd(b_1, b_2)=1$ and $c_1|b_1|<|b_2|\leq c_2|b_1|$, where
$c_1, c_2$ are any given positive constants. Let $n$ be any
integer satisfying $\{gcd(n, b_i)=1$, $i=1,2$. 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 $b_1p+b_2m=n$
is solvable in prime $p$ and almost prime $m=P_k$, $k\geq 3$
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 $m=P_k, k\geq 3$.

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
$IJ=Ib$ 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 $\textbf{Q} _S$ with distances in $S$.

The consideration of tensor products of $0$-Hecke algebra modules
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 $\Max(M)$ are
characterized algebraically. The relationships among the maximal
spectra of $M$, $\Soc(M)$ and $\Ass(M)$ are studied. It is shown
that $\Soc(M)$ is exactly the set of all elements of $M$ which
belongs to every maximal submodule of $M$ except for a finite
number. If $\Max(M)$ is infinite, $\Max(M)$ is a one-point
compactification of a discrete space if and only if $M$ is Gelfand and for
some maximal submodule $K$, $\Soc(M)$ 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
$\Max(M)$ is an almost discrete space. The set of uniform
submodules of $M$ and the set of minimal submodules of $M$
coincide. $\Ann(\Soc(M))M$ is a summand submodule of $M$ if and only if
$\Max(M)$ is the union of two disjoint open subspaces $A$ and
$N$, where $A$ is almost discrete and $N$ is dense in itself. In
particular, $\Ann(\Soc(M))=\Ann(M)$ if and only if $\Max(M)$ 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 $R=\bigoplus_{i=1}^{\infty}R_{i}$ be a graded nil ring. It is shown
that primitive ideals in $R$ are homogeneous. Let
$A=\bigoplus_{i=1}^{\infty}A_{i}$ 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 K\"ahler 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.