We shall show that any symmetric, forward and backward invariant lamination under $z\mapsto z^d$ is gap invariant in the sense of W. P. Thurston. As a corollary, we have the gap invariance of a quadratic invariant lamination.

It is shown that the set of finite regular Borel measures with natural spectra for a compact abelian group $\mathfrak{G}$ is closed under addition if and only if $\mathfrak{G}$ is discrete. If $G$ is a non-discrete locally compact abelian group, then there exists a finite regular Borel measure with natural spectrum such that the corresponding multiplication operator on $L^1(G)$ is not decomposable.

Regularity of infinite dimensional Lie groups was defined by Hideki Omori et al. and John Milnor. Up to now the only known sufficient conditions for regularity are analytic in nature and they are included in the definition of strong \textit{ILB}-Lie groups, since there are no existence theorems for ordinary differential equations on non-normable locally convex spaces. We prove that regularity can be characterized by the existence of a family of so called Lipschitz-metrics in all interesting cases of infinite dimensional Lie groups. On Lipschitz-metrizable groups all product integrals converge to the solutions of the respective equations if some weak conditions satisfied by all known Lie groups are given. Lipschitz-metrizable groups provide a framework to solve differential equations on infinite dimensional Lie groups. Furthermore Lipschitz-metrics are the non-commutative generalization of the concept of seminorms on a Fréchet-space viewed as abelian Lie group.

In this paper it will be shown that a potential $V(x)$ and a constant $\lambda$ are uniquely determined from the scattering operator $S$ associated with the nonlinear Schrödinger equation \[ i\frac{\partial u}{\partial t}+(-\Delta+V)u+\lambda(|x|^{-\sigma}*|u|^{2})u=0 , \] and the corresponding unperturbed equation \[ i\frac{\partial u}{\partial t}-\Delta u=0 . \]

We construct a new class of semi-selfsimilar processes as limiting processes of some random walks in random scenery, which extends the class of selfsimilar processes studied by Kesten and Spitzer (1979).

A (central) arrangement is a finite family of one-codimensional subspaces of a vector space $V$. We study the module of logarithmic forms with poles along the hyperplanes. We use a certain cochain complex and its cohomological groups to prove that cohomological properties of the module are closely related to the explicit structure of the module.

In this paper we prove that two compact spacelike surfaces in de Sitter space for which there exists an isometry preserving their mean curvature functions are necessarily congruent. As an application of this, we deduce that there exists no compact spacelike Bonnet surface in de Sitter space.

We consider direct sum decompositions $\beta=\beta_{-}+\beta_{+}$ and $L=L_{-}+L_{+}$ of two symplectic Hilbert spaces by Lagrangian subspaces with dense embeddings $\beta_{-}\hookrightarrow L-$ and $L_{+}\hookrightarrow\beta_{+}$. We show that such criss-cross embeddings induce a continuous mapping between the Fredholm Lagrangian Grassmannians $\mathcal{F}\mathcal{L}_{\beta_{-}}(\beta)$ and $\mathcal{F}\mathcal{L}_{L_{-}}(L)$ which preserves the Maslov index for curves. This gives a slight generalization and a new proof of the Yoshida-Nicolaescu Spectral Flow Formula for families of Dirac operators over partitioned manifolds.

A real hypersurface of a complex space form is called a Hopf hypersurface if the characteristic vector field $\xi=-JN$ on $M$ is a principal curvature vector. The main purpose of this paper is to obtain several simple geometric characterizations of all Hopf hypersurfaces with constant principal curvatures in nonflat complex space forms.

A graph $G$ is called edge-magic if there exists a bijection $f$ from $V(G)\cup E(G)$ to $\{1,2,\ldots,|V(G)|+|E(G)|\}$ such that $f(x)+f(y)+f(xy)=C$ is a constant for any $xy\in E(G)$. In this paper, we show that a wheel graph $W_n$ is edge-magic if $n\not\equiv 0 \pmod{4}$.

We show that any torsion class $e\in H^2(M;Z)$ of any closed manifold $M$ is realized as the Euler class of a smoothly foliated orientable circle bundle over $M$. In the case where $M$ is a 3-manifold, we construct the homomorphism $\pi_1(M)\rightarrow SO(2)\subset \text{Diff}_{+}^{\infty}(S^{1})$ explicitly whose Euler class is the given torsion class.

Let $\mathcal{M}_n$ be the moduli space of spatial polygons with $n$ edges. We consider the case of odd $n$. Let $K_{n}^{*}=\Lambda^{n-3}T\mathcal{M}_n$ be the dual bundle of the canonical bundle on $\mathcal{M}_n$. In this paper we determine the sheaf cohomology $H^*(\mathcal{M}_n,K_{n}^{*})$. We have $H^q(\mathcal{M}_n,K_{n}^{*})=0$ $(q\geq 1)$ and $\dim H^0(\mathcal{M}_n,K_{n}^{*})$ is equal to the number of lattice points in the convex polytope $\Delta_n$ in $\mathbf{R}^{n-3}$.

We study the Cauchy problem and the initial boundary value problem (IBVP) for nonlinear parabolic equations in $\mathcal{C}_b([0,T);L^p)$ and $L^q(0,T;L^p)$. We give a unified method to construct local mild solutions of the Cauchy problem or IBVP for a class of nonlinear parabolic equations in $\mathcal{C}_b([0,T);L^p)$ or $L^q(0,T;L^p)$ by introducing admissible triplet, generalized admissible triplet and establishing time space estimates for the solutions to the linear parabolic equations. Moreover, using our method, we also obtain the existence of global small solutions to the nonlinear parabolic equations.

Let $X$ be a Lévy process with negative drift starting from $x>0$, and let $\tau$ and $\tau_s$ be the first passage times to $(-\infty,0]$ and $(s,\infty)$, respectively. Under appropriate exponential moment conditions of $X$, we show that, for every $A\in\mathcal{F}_t$, the conditional laws $P_x(X\in A | \tau>s)$ and $P_x(X\in A | \tau>\tau_s)$ converge to different distributions as $s\rightarrow\infty$. Both of them can be regarded as the laws of $X$ conditioned to stay positive. We characterize these limit laws in terms of $h$-transforms, by the renewal functions, of some Lévy processes killed at the entrance time into $(-\infty,0]$.

We give a complete proof of Theorem 2.1 related to the Lie norm conjectured by Prof. Mitsuo Morimoto. His conjecture is a special case of our new theorem on convex functions on $\mathbf{R}^n$.

We prove that Vignéras' multiple gamma function does not satisfy any algebraic differential equation over $\mathbf{C}(z)$ by using a relation between logarithmic derivative of this function and the psi-function. Furthermore, we apply this relation to calculation of convergent factors in the Weierstrass product representation of Vignéras' multiple gamma function.