The shifted Schur measure introduced in [TW2] is a measure on the set of all strict partitions, which is defined by Schur $Q$-functions. The main aim of this paper is to calculate the correlation function of this measure, which is given by a pfaffian. As an application, we prove that a limit distribution of parts of partitions with respect to a shifted version of the Plancherel measure for symmetric groups is identical with the corresponding distribution of the original Plancherel measure. In particular, we obtain a limit distribution of the length of the longest ascent pair for a random permutation. Further we give expressions of the mean value and the variance of the size of partitions with respect to the measure defined by Hall-Littlewood functions.

The 3-transposition groups that act on vertex operator algebras in the way described by Miyamoto in [Mi] are classified under the assumption that the group is centerfree and the VOA carries a positive-definite invariant Hermitian form.

We study a divisorial contraction $\pi :Y\to X$ such that $\pi$ contracts an irreducible divisor $E$ to a point $P$ and that the discrepancy of $E$ is $1$ when $P\in X$ is a $3$-dimensional terminal singularity of type (cD/2) and (cE/2).

We study essential norms of differences of composition operators on the Banach algebra $H^{\mathrm{\infty }}$ of bounded analytic functions on the unit disk.

In this paper we treat the regularity problem for minimizers $u\left(x\right):\mathrm{\Omega }\subset {\mathbf{R}}^m\to {\mathbf{R}}^n$ of quadratic growth functionals ${\int }_{\mathrm{\Omega }}A(x,u,Du)dx$. About the dependence on the variable $x$ we assume only that $A(\cdot ,u,p)$ is in the class $VMO$ as a function of $x$. Namely, we do not assume the continuity of $A(x,u,p)$ with respect to $x$. We will prove a partial regularity result for the case $m\le 4$.

In this paper we describe a compact set $K$ in ${\mathbf{R}}^d$ which is typical from the point of view of Baire categories, as it appears when seen from some point $x$ of ${\mathbf{R}}^d$. It matters whether $x$ belongs to $K$ or not! If $x\notin K$ , then $K$ looks porous (this is easily seen). If $x\in K$, then $K$ looks only $\sigma$-porous, but dense at least in a half-sphere (of directions from $x$). If $x$ is a typical point of $K$, then $K$ looks even dense (in the whole sphere).

A statement of Weierstrass is known for meromorphic functions which admit an algebraic addition theorem. We give its precise formulation and prove it complex analytically. In fact, we show that if $K$ is a non-degenerate algebraic function field in $n$ variables over $\mathbf{C}$ which admits an algebraic addition theorem, then any $f\in K$ is a rational function of some coordinate functions and abelian functions of other variables.

Existence of complex codimension-one transverse structure is studied using the complex dilatation. As an application, a version of quasiconformal surgeries of foliations is considered.

For sequences of stochastic integrals ${\int }_0^{\cdot }{K}_{s-}^{n}d{X}_s^n$, functional limit theorems are presented. And stability of strong solutions of stochastic differential equations of type $$X^n=H^n+{\int }_0^{\cdot }f\left(X_{s-}^n\right)d{Y}_s^n,\forall n\ge 1$$ is discussed under jointly weak convergence of driving processes $\left\{\right(H^n,Y^n){\}}_{n\ge 1}$. Where $Y^n$ is an $\mathbf{H}$-valued semimartingale, $H^n$ is a $\mathbf{G}$-valued càdlàg adapted process, $K^n$ is an $ℒ(\mathbf{H},\mathbf{G})$-valued càdlàg adapted process and $f:\mathbf{G}\mapsto ℒ(\mathbf{H},\mathbf{G})$ satisfies a Lipschitz condition.

Let $G$ be a simply connected Lie group and consider a Lie $G$ foliation $ℱ$ on a closed manifold $M$ whose leaves are all dense in $M$. Then the space of ends $ℰ\left(F\right)$ of a leaf $F$ of $ℱ$ is shown to be either a singleton, a two points set, or a Cantor set. Further if $G$ is solvable, or if $G$ has no cocompact discrete normal subgroup and $ℱ$ admits a transverse Riemannian foliation of the complementary dimension, then $ℰ\left(F\right)$ consists of one or two points. On the contrary there exists a Lie $\widetilde{SL}(2,\mathbf{R})$ foliation on a closed 5-manifold whose leaf is diffeomorphic to a 2-sphere minus a Cantor set.

In this paper we investigate $a$-Bloch, Hardy, Bergman, $BM{O}_p$ and Dirichlet spaces of harmonic functions on the open unit ball in ${\mathbf{R}}^n$, and the boundedness of the Hardy-Littlewood operator on these spaces.

We define notions of isotropic, coisotropic and lagrangian submanifolds of Dirac manifolds. Notion of Dirac manifolds, Dirac maps and Dirac relations are defined. Extending the isotropic calculus on presymplectic manifolds and the coisotropic calculus on Poisson manifolds to Dirac manifolds,we construct the lagrangian calculus on Dirac manifolds as an extension of the one on symplectic manifolds. We see that there are three natural categories of Dirac manifolds.

Let $S$ be a non-empty finite set of prime numbers and, for each $p$ in $S$, let ${\mathbf{Z}}_p$ denote the ring of $p$-adic integers. Let $F$ be an abelian extension over the rational field such that the Galois group of $F$ over some subfield of $F$ with finite degree is topologically isomorphic to the additive group of the direct product of ${\mathbf{Z}}_p$ for all $p$ in $S$. We shall prove that each of certain arithmetic progressions contains only finitely many prime numbers $l$ for which the $l$-class group of $F$ is nontrivial. This result implies our conjecture in [3] that the set of prime numbers $l$ for which the $l$-class group of $F$ is trivial has natural density $1$ in the set of all prime numbers.

We develop reductions for classifications of singularities of orbit closures in module varieties. Then we show that the orbit closures for representations of Dynkin quivers are regular in codimension two.

In this paper, we study various topological properties of generic smooth maps between manifolds whose regular fibers are disjoint unions of homotopy spheres. In particular, we show that if a closed $4$-manifold admits such a generic map into a surface,then it bounds a $5$-manifold with nice properties. As a corollary, we show that each regular fiber of such a generic map of the $4$-sphere into the plane is a homotopy ribbon $2$-link and that any spun $2$-knot of a classical knot can be realized as a component of a regular fiber of such a map.

It is a crucial step how to describe the relationship between the strain, the stress and the temperature field, when we consider the mathematical modelling for shape memory alloy materials. From the experimental results we know that the relationship can be described by the hysteresis operators. In this paper we propose a new system consisting of differential equations as a mathematical model for shape memory alloy materials occupying the three dimensional domain. The key of the modelling is the characterization for the generalized stop operators by using the ordinary differential equations including the subdifferential of the indicator function for the closed interval. Also, we give a proof of the well-posedness of the system.