In this paper it is proven that analytic Jacobi Eisenstein series always admit a meromorphic continuation on the whole complex plane and statements about the location of possible poles are given. Moreover a new interpretation of Shimura’s approach to the standard L-function of a Siegel modular form is presented.

The Schrödinger operator $H = -d^{2}/dx^{2}+V (x)$ on an interval $[0, a]$ with Dirichlet or Neumann boundary conditions has discrete spectrum $E_{1}[V] < E_{2}[V] < E_{3}[V] < \cdots$, for bounded $V$. In this paper, we apply the perturbation theory of discrete eigenvalues to obtain upper bounds for $\sum_{j=1}^{k} E_{j}[V]$, where $k$ is any positive integer. Our results include the following:

(i) $\sum_{j=1}^{k} E_{j}[V]\leq \sum_{j=1}^{k} E_{j}[V_{s}]$, where $V_{s}(x) = [V (x)+V (a-x)]/2$, with equality if and only if $V$ is symmetric about $x = a/2$.

(ii) If $V$ is convex, then the Dirichlet eigenvalues satisfy \[ \sum_{j=1}^{k} E_{j}[V]\leq \sum_{j=1}^{k} E_{j}[0]+\frac{k}{a}\int _{0}^{a}V (x)dx \] with equality if and only if V is constant.

(iii) If $V$ is concave, then the Neumann eigenvalues satisfy \[ \sum_{j=1}^{k} E_{j}[V]\leq \sum_{j=1}^{k} E_{j}[0]+\frac{k}{a}\int _{0}^{a}V (x)dx \] with equality if and only if $V$ is constant.

We consider the isotropic elasticity system: \[ \begin{array}{cclcc} \rho \partial _{t}^{2}\mathbf{u}&-&\mu (\Delta\mathbf{u}+\nabla (\nabla ^{T}\mathbf{u})-\nabla (\lambda\nabla ^{T}\mathbf{u})&&\\ &-&\sum _{j=1}^{3}\nabla\mu\cdot (\nabla u_{j}+\partial _{j}\mathbf{u})\mathbf{e}_{j}=0 &\text{in}& \Omega\times (0,T) \end{array} \] for the displacement vector $\mathbf{u} = (u_{1}, u_{2}, u_{3})$ depending on $x \in \Omega$ and $t \in (0, T)$ where $\Omega$ is a bounded domain in $\mathbb{R}^{3}$ with the $C^{2}$-boundary, and we assume the density $\rho \in C^{2}(\Bar{\Omega}\times[0, T])$ and the Lamé parameters $\mu , \lambda \in C^{3}(\Bar{\Omega}\times[0, T])$. We will give Lipschitz stability estimates for solutions $\mathbf{u}$ to the above elasticity system with the lateral boundary data \[ \begin{array}{cc} \mathbf{u} = \mathbf{g} \textrm{ on } \partial\Omega\times (0, T),& \partial _{\nu}\mathbf{u} = \mathbf{h} \textrm{ on } \Gamma \times (0, T) \end{array} \] where $\Gamma$ is some part of $\partial\Omega$. Our proof is based on (1) a Carleman estimate with boundary data, (2) cut-off technique, and (3) principal diagonalization of the Lamé system.

We will investigate geodesics in 2-step Carnot groups. And, by applying this, we will prove that every isometry of 2-step Carnot group $N$ fixing the identity is an automorphism of Lie group $N$. Our argument will be elementary.

Finally, we will see that the same conclusion holds for general Carnot groups.

Let $Z$ be an $\mathbb{R}^{d}$-valued Lévy process with strong finite $p$-variation for some $p < 2$. We prove that the “decompensated” process $\Tilde{Z}$ obtained from $Z$ by annihilating its generalized drift has a small deviations property in $p$-variation. This property means that the null function belongs to the support of the law of $\Tilde{Z}$ with respect to the $p$-variation distance. Thanks to the continuity results of T. J. Lyons/D. R. E. Williams [19], [35], this allows us to prove a support theorem with respect to the $p$-Skorohod distance for canonical SDE’s driven by $Z$ without any assumption on $Z$, improving the results of H. Kunita [15]. We also give a criterion ensuring the small deviation property for $Z$ itself, noticing that the characterization under the uniform distance, which we had obtained in [26], no more holds under the $p$-variation distance.

Let $\mathbf{X} = \{ X(t) :t \leq 0\}$ be a one-dimensional generalized diffusion process with initial state $X(0) > 0$, hitting time $\tau _{\mathbf{X}}(0)$ at the origin and speed measure m which is regularly varying at infinity with exponent $1/\alpha -1 > 0$. It is proved that, for a suitable function $u(c)$, the probability law of $\{ u(c)^{-1}X(ct) : 0 < t \leq 1\}$ conditioned by $\{\tau _{\mathbf{X}}(0) > c \}$ converges as $c \to \infty$ to the conditioned $2(1-\alpha )$-dimensional Bessel excursion on natural scale and that the latter is equivalent to the $2(1-\alpha )$-dimensional Bessel meander up to a scale transformation. In particular, the distribution of $u(c)^{-1}X(c)$ converges to the Weibull distribution. From the conditional limit theorem we also derive a limit theorem for some of regenerative process associated with $\mathbf{X}$.

If $G$ is a toral group, i.e. an extension of a torus by a finite group, and $X$ is a $G$-CW complex we prove that there exists a $G$-homotopy equivalent CW complex $Y$ with the property that the action map $\rho : G\times Y \to Y$ is a cellular map.

Homotopy normality of Lie groups is studied by using the adjoint action on the space of based loops.

For a smooth quartic plane curve $C$ we show an existence of a family of Galois closure curves $\phi : S \longrightarrow C$, where $S$ is a nonsingular projective surface and $\phi ^{-1}(P)$ is isomorphic to the Galois closure curve $C_{P}$ for a general point $P \in C$. Moreover we determine the types of singular fibers. As a corollary we can say that $C_{P}$ is not isomorphic to $C_{Q}$ if $P$ is close to $Q$.

We show that there is an essential phantom map $f : K(Z, n) \to \Sigma Y$ for a suitable $n$ if $H_{i}(Y;\mathbb{Q})\neq 0$ for some $i > 0$. The localized version of this problem is also considered. The ingredient of the proof is the computation of the Morava K-theories of the Eilenberg-MacLane spaces by Ravenel and Wilson.