We define invariants of two dimensional $C^2$ projectively Anosov diffeomorphisms. The invariants are defined by the topology of the space of circles tangent to an invariant subbundle and are preserved under homotopy of projectively Anosov diffeomorphisms. As an application, we show that the invariant subbundle is not uniquely integrable and two distinct periodic orbits exist if certain invariants do not vanish.

The definitions of the Feynman path integral for the Pauli equation and more general equations in configuration space and in phase space are proposed, probably for the first time. Then it is proved rigorously that the Feynman path integrals are well-defined and are the solutions to the corresponding equations. These Feynman path integrals are defined by the time-slicing method through broken line paths, which is familiar in physics. Our definitions of these Feynman path integrals and our results give the extension of ones for the Schrödinger equation.

Motivated by recent developments on visible action on complex manifolds, we raise a question whether or not the multiplication of three subgroups $L, G',$ and $H$ surjects a Lie group $G$ in the setting that $G / H$ carries a complex structure and contains $G' / G' \cap H$ as a totally real submanifold.

Paticularly important cases are when $G / L$ and $G / H$ are generalized flag varieties, and we classify pairs of Levi subgroups $(L,H)$ such that $LG'H / G$, or equivalently, the real generalized flag variety $G^{\prime} / \cap G^{\prime}$ meets every $L$-orbit on the complex generalized flag variety $G / H$ in the setting that $(G,G') = (U(n),O(n))$. For such pairs $(L,H)$, we introduce a herringbone stitch method to find a generalized Cartan decomposition for the double coset space $L \backslash G / H$, for which there has been no general theory in the non-symmetric case. Our geometric results provides a unified proof of various multiplicity-free theorems in representation theory of general linear groups.

We locate all of the zeros of the Eisenstein series associated with the Fricke groups ${\mathrm{\Gamma }}_0^{*}\left(2\right)$ and ${\mathrm{\Gamma }}_0^{*}\left(3\right)$ in their fundamental domains by applying and expanding the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer ("On the zeros of Eisenstein series", 1970).

In this article, we provide a new method solving the Fekete-Szegö problem for classes of close-to-convex functions defined in terms of subordination. As an example, we apply it to the class of strongly close-to-convex functions.

We describe several classes of Montesinos links up to mutation and 5-move equivalence, and obtain from this a Jones and Kauffman polynomial test for a Montesinos link.

We study open books (or open book decompositions) of a closed oriented 3-manifold which support overtwisted contact structures. We focus on a simple closed curve along which one can perform Stallings twist, called "twisting loop". We show that the existence of a twisting loop on the fiber surface of an open book is equivalent up to positive stabilization to the existence of an overtwisted disk in the contact manifold supported by the open book. We also show a criterion for overtwistedness using a certain arc properly embedded in the fiber surface, which is an extension of Goodman's one.

Given a positive integer $n\ge 2$, an arbitrary field $K$ and an $n$-block $q=\left[q^{\left(1\right)}\right|\cdots \left|q^{\left(n\right)}\right]$ of $n\times n$ square matrices $q^{\left(1\right)},\dots ,q^{\left(n\right)}$ with coefficients in $K$ satisfying certain conditions, we define a multiplication $._q : \mathbf{M}_n(K) \otimes_K \mathbf{M}_n(K) \rightarrow \mathbf{M}_n(K)$ on the $K$-module ${\mathbf{M}}_n\left(K\right)$ of all square $n\times n$ matrices with coefficients in $K$ in such a way that ${\cdot }_q$ defines a $K$-algebra structure on ${\mathbf{M}}_n\left(K\right)$. We denote it by ${\mathbf{M}}_n^q\left(K\right)$, and we call it a minor $q$-degeneration of the full matrix $K$-algebra ${\mathbf{M}}_n\left(K\right)$. The class of minor degenerations of the algebra ${\mathbf{M}}_n\left(K\right)$ and their modules are investigated in the paper by means of the properties of $q$ and by applying quivers with relations. The Gabriel quiver of ${\mathbf{M}}_n^q\left(K\right)$ is described and conditions for $q$ to be ${\mathbf{M}}_n^q\left(K\right)$ a Frobenius algebra are given. In case $K$ is an infinite field, for each $n\ge 4$ a one-parameter $K$-algebraic family $\{C_{\mu }{\}}_{\mu \in K^{*}}$ of basic pairwise non-isomorphic Frobenius $K$-algebras of the form $C_{\mu }={\mathbf{M}}_n^{q_{\mu }}\left(K\right)$ is constructed. We also show that if $A_q={\mathbf{M}}_n^q\left(K\right)$ is a Frobenius algebra such that $J(A_q{)}^3=0$, then $A_q$ is representation-finite if and only if $n=3$, and $A_q$ is tame representation-infinite if and only if $n=4$.

Submanifolds satisfying $\mathrm{\Delta }\overrightarrow{H}=\lambda \overrightarrow{H}$ are named by B. Y. Chen submanifolds with proper mean curvature vector. We prove that a hypersurface of the pseudo-Euclidean space $E_s^4$ with $\mathrm{\Delta }\overrightarrow{H}=\lambda \overrightarrow{H}$ and diagonalizable shape operator, has constant mean curvature.

Let $S$ be a non-empty finite set of prime numbers, and 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 the $p$-adic integer rings for all $p$ in $S$. Let $m$ be a positive integer that is neither congruent to $2$ modulo $4$ nor divisible by any prime number outside $S$ but divisible by all prime numbers in $S$. Let $\mathrm{\Omega }$ denote the composite of $p^n$-th cyclotomic fields for all $p$ in $S$ and all positive integers $n$. In our earlier paper [3], it is shown that there exist only finitely many prime numbers $l$ for which the $l$-class group of $F$ is nontrivial and the $m$-th cyclotomic field contains the decomposition field of $l$ in $\mathrm{\Omega }$. We shall prove more precise results providing us with an effective upper bound for a prime number $l$ such that the $l$-class group of $F$ is nontrivial and that the $m$-th cyclotomic field contains the decomposition field of $l$ in $\mathrm{\Omega }$.

Let $M$ be a complete glued surface whose sectional curvature is greater than or equal to $k$ and $\mathrm{▵}pqr$ a geodesic triangle domain with vertices $p,q,r$ in $M$. We prove a compression theorem that there exists a distance nonincreasing map from $\mathrm{▵}pqr$ onto the comparison triangle domain $\widetilde{\mathrm{▵}}pqr$ in the two-dimensional space form with sectional curvature $k$. Using the theorem, we also have some compression theorems and an application to a circular billiard ball problem on a surface.

We consider the nonstationary Stokes and Navier-Stokes equations in an aperture domain $\mathrm{\Omega }\subset {\mathbf{R}}^n$, $n\ge 2$. For this purpose, we prove $L^p$-$L^q$ type estimate of the Stokes semigroup in the aperture domain. Our proof is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the resolvent of the Stokes operator near the origin. We apply them to the Navier-Stokes initial value problem in the aperture domain. As a result, we can prove the global existence of a unique solution to the Navier-Stokes problem with the vanishing flux condition and some decay properties as $t\to \mathrm{\infty }$, when the initial velocity is sufficiently small in the $L^n$ space. Moreover we can prove the time-local existence of a unique solution to the Navier-Stokes problem with the non-trivial flux condition.

We consider the initial-boundary value problem \begin{equation}\qquad\left\{ \begin{array}{lll} \frac{\partial}{\partial t}u = \Delta u-V(|x|)u & \mbox{in}& \Omega_L\times(0,\infty), \\ \mu u+(1-\mu)\frac{\partial}{\partial\nu}u = 0 & \mbox{on}& \partial\Omega_L\times(0,\infty), \\ u(\cdot,0) = \phi(\cdot)\in L^p(\Omega_L), & p \ge 1,& \end{array}\right. \tag{P}\end{equation} where $\Omega_L = \{ |x| \in \mathbf{R}^{N} : |x\ > L\}, N \geq 2, L > 0, 0 \leq \mu \leq 1, v$ is the outer unit normal vector to $\partial \Omega_L$, and $V$ is a nonnegative smooth function such that $V(r) = O(r^{-2})$ as $r \to \infty$. In this paper, we study the decay rates of the derivatives $\bigtriangledown^j_x u$ of the solution $u$ to $(P)$ as $t \to \infty$.

For a holomorphic map $F: X \rightarrow Y$ and a non-singular foliation on $Y$, we give a residue formula which measures singularities of the lifted foliation on $X$ by applying the Baum-Bott residue formula.