We formulate and prove reguralized double shuffle and derivation relations for multiple $L$-values. A description of principal part of a multiple $L$-function is also given.

This paper is concerned with positive solutions of semilinear diffusion equations $u_t={\epsilon }^2∆u+u^p$ in $\Omega$ with small diffusion under the Neumann boundary condition, where $p>1$ is a constant and $\Omega$ is a bounded domain in $R^N$ with $C^2$ boundary. For the ordinary differential equation $u_t=u^p$, the solution $u^0$ with positive initial data $u_0∊C\left(\overline{\Omega }\right)$ has a blow-up set $S^0=\{x∊\overline{\Omega }|u_0\left(x\right)={\max }_{y∊\overline{\Omega }}{u}_0\left(y\right)\}$ and a blowup profile $$u_{*}^0\left(x\right)=\left(u_0\right(x{)}^{-(p-1)}-\left(\underset{y∊\overline{\Omega }}{\max }{u}_0\right(y){)}^{-(p-1)}{)}^{-1/(p-1)}$$ outside the blow-up set $S^0$. For the diffusion equation $u_t={\epsilon }^2∆u+u^p$ in $\Omega$ under the boundary condition $\partial u/\partial v=0$ on $\partial \Omega$, it is shown that if a positive function $u_0∊C^2\left(\overline{\Omega }\right)$ satisfies $\partial u_0/\partial v=0$ on $\partial \Omega$, then the blow-up profile $u_{*}^{\epsilon }\left(x\right)$ of the solution $u^{\epsilon }$ with initial data $u_0$ approaches $u_{*}^0\left(x\right)$ uniformly on compact sets of $\overline{\Omega }\setminus S^0$ as $\mathrm{\epsilon }\to +0$.

This paper is concerned with positive solutions of the semilinear diffusion equation $u_t=∆u+u^p$ in $\Omega$ under the Neumann boundary condition, where $p>1$ is a constant and $\Omega$ is a bounded domain in $R^N$ with $C^2$ boundary. This equation has the constant solution $(p-1{)}^{-1/(p-1)}(T_0-t{)}^{-1/(p-1)}$ with the blow-up time $T_0>0$. It is shown that for any $\epsilon >0$ and open cone $\Gamma$ in $\{f∊C(\overline{\Omega }\left)\right|f\left(x\right)>0\}$, there exists a positive function $u_0\left(x\right)$ in $\overline{\Omega }$ with $\partial u_0/\partial v=0$ on $\partial \Omega$ and $\left|\right|u_0\left(x\right)-$ such that the blow-up time of the solution $u(x,t)$ with initial data $u(x,0)=u_0\left(x\right)$ is larger than $T_0$ and the function $u(x,T_0)$ belongs to the cone $\Gamma$. A theorem on the blow-up profile is also given.

Without any assumption on the characteristic exponents, we give fundamental solutions of linear Fuchsian partial differential equations.

We study foliations on CR manifolds and show the following. (1) For a strictly pseudoconvex CR manifold $M$, the relationship between a foliation $\mathcal{F}$ on $M$ and its pullback ${\pi }^{*}\mathcal{F}$ on the total space $C\left(M\right)$ of the canonical circle bundle of $M$ is given, with emphasis on their interrelation with the Webster metric on $M$ and the Fefferman metric on $C\left(M\right)$, respectively. (2) With a tangentially CR foliation $\mathcal{F}$ on a nondegenerate CR manifold $M$, we associate the basic Kohn-Rossi cohomology of $(M,\mathcal{F})$ and prove that it gives the basis of the $E_2$-term of the spectral sequence naturally associated to $\mathcal{F}$. (3) For a strictly pseudoconvex domain $\Omega$ in a complex Euclidean space and a foliation $\mathcal{F}$ defined by the level sets of the defining function of $\Omega$ on a neighborhood $U$ of $\partial \Omega$, we give a new axiomatic description of the Graham-Lee connection, a linear connection on $U$ which induces the Tanaka-Webster connection on each leaf of $\mathcal{F}$. (4) For a foliation $\mathcal{F}$ on a nondegenerate CR manifold $M$, we build a pseudohermitian analogue to the theory of the second fundamental form of a foliation on a Riemannian manifold, and apply it to the flows obtained by integrating infinitesimal pseudohermitian transformations on $M$.

We consider Riemann surfaces of infinite type and their reduced Teichmüller spaces. The reduced Teichmüller space admits the action of the reduced mapping class group. Generally, the action is not discrete while it is faithful. We give sufficient conditions for the discreteness of the action in terms of the geometry of Riemann surfaces.

We consider holomorphic mappings of complex manifolds with ball model into complex manifolds which are quotients of bounded domains and estimate the dimension of the moduli space of holomorphic mappings in terms of the essential boundary dimension of target manifolds. For this purpose, we generalize a classical uniqueness theorem of Fatou-Riesz for bounded holomorphic functions on the unit disk to one for bounded holomorphic mappings on a bounded $C^2$ domain. This generalization enables us to establish rigidity and finiteness theorems for holomorphic mappings. We also discuss the rigidity for holomorphic mappings into quotients of some symmetric bounded domains. In the final section, we construct examples related to our results.

In the framework of microlocal analysis, a boundary value morphism is defined for solutions to the regular-specializable system of analytic linear partial differential equations. This morphism can be regarded as a microlocal counterpart of the boundary value morphism for hyperfunction solutions due to Monteiro Fernandes, and the injectivity of this morphism (that is, the Holmgren type theorem) is proved. Moreover, under a kind of hyperbolicity condition, it is proved that this morphism is surjective (that is, the solvability).

If $X$ is a smooth complex projective 3-fold with ample canonical divisor $K$, then the inequality $K^3\ge (2/3)(2p_g-7)$ holds, where $p_g$ denotes the geometric genus. This inequality is nearly sharp. We also give similar, but more complicated, inequalities for general minimal 3-folds of general type.

Let $A$ be a Buchsbaum local ring with the maximal ideal $\mathfrak{m}$ and let $e\left(A\right)$ denote the multiplicity of $A$. Let $Q$ be a parameter ideal in $A$ and put $I=Q$ : $\mathfrak{m}$. Then the equality $I^2=QI$ holds true, if $e\left(A\right)=2$ and $\mathrm{depth}A>0$. The assertion is no longer true, unless $e\left(A\right)=2$. Counterexamples are given.

We show the existence of solutions for Dirichlet problem of evolutionary surfaces of prescribed mean curvature. Usually the lateral boundary needs to satisfy a kind of convexity, more precisely $H$-convexity condition. But in this article we do not assume it on a portion $S$ of the lateral boundary. Under some assumptions on the exterior forces and the shape of $S$ we prove that the solution satisfies Dirichlet boundary condition on $S$ in a weak sense.

Transversely piecewise linear foliations of codimension one on closed 3-manifolds are deformed by surgery. We will show the surgery formula for the discrete Godbillon-Vey invariant of these foliations.

In this note, we study the Laplacian comparison theorem and the sub-mean-value theorem for a special type of Hermitian manifolds called multiplier Hermitian manifolds. By conformal change of the metrics, this covers much wider objects than in the case of ordinary Kähler manifolds.

For each Painlevé system $P_J$ except the first one, we have a Bäcklund transformation group which is a lift of an alfine Weyl group. In this PaPer, we show that the Bäcklund transformation groups for $J=V,$$IV,$$III,$$II$ are successively obtained from that for $J=VI$ by the well known degeneration or confluence processes.

This paper concerns projectively Anosov flows ${\phi }_t$ with smooth stable and unstable foliations ${\mathcal{F}}^S$ and ${\mathcal{F}}^{\mathsf{U}}$ on a Seifert fibered 3-manifold $M$ over a hyperbolic orbifold. We show that if the foliations ${\mathcal{F}}^S$ and ${\mathcal{F}}^{\mathsf{U}}$ do not have compact leaves, then after changing the parameter, ${\phi }_t$ is differentiably isotopic to a quasi-Fuchsian flow lifted to a finite cover.