The paper is devoted to classify all irreducible tempered representations of $\mathit{SL}(n)$ $p$-adic which appear in the spectral side of J. Arthur's local trace formula. We give description of such representations which are elliptic or irreducibly induced from elliptic representations of proper Levi subgroups with certain condition on the Weyl group.

Harris proved that there is an indecomposable bimodule with a trivial source which induces a Morita equivalence between Glauberman-Watanabe corresponding block algebras of finite groups with normal defect groups and the Glauberman correspondence of characters in corresponding blocks. We note an implication of the Puig correspondence in the conext of the Glauberman-Watanabe correspondence and then, using this, show Harris's theorem in two ways.

We confirm R.H. Fox's trapezoidal conjecture for alternating knots of genus two by a method different from P. Ozsváth and Z. Szabó's one. As an application, we determine the alternating knots of genus two whose Alexander polynomials have minimal coefficients equal to one or two.

A class of interacting superprocesses on $\mathbb{R}$, called superprocesses with dependent spatial motion (SDSMs), were introduced and studied in Wang [32] and Dawson et al. [9]. In the present paper, we extend this model to allow particles moving in a bounded domain in $\mathbb{R}^{d}$ with killing boundary. We show that under a proper re-scaling, a class of discrete SPDEs for the empirical measure-valued processes generated by branching particle systems subject to the same white noise converge in $L^{2}(\Omega, \mathcal{F}, \mathbb{P})$ to the SPDE for an SDSM on a bounded domain and the corresponding martingale problem for the SDSMs on a bounded domain is well-posed.

In this paper we present several discreteness criterion for a non-elementary group $G$ in $\mathit{SL}(2,C)$ by using a test map which need not to be in $G$.

We study subelliptic harmonic morphisms i.e. smooth maps $\phi\colon \Omega \to \tilde{\Omega}$ among domains $\Omega \subset \mathbb{R}^{N}$ and $\tilde{\Omega} \subset \mathbb{R}^{M}$, endowed with Hörmander systems of vector fields $X$ and $Y$, that pull back local solutions to $H_{Y} v = 0$ into local solutions to $H_{X} u = 0$, where $H_{X}$ and $H_{Y}$ are Hörmander operators. We show that any subelliptic harmonic morphism is an open mapping. Using a subelliptic version of the Fuglede-Ishihara theorem (due to E. Barletta, [5]) we show that given a strictly pseudoconvex CR manifold $M$ and a Riemannian manifold $N$ for any heat equation morphism $\Psi\colon M \times (0, \infty) \to N \times (0, \infty)$ of the form $\Psi (x,t) = (\phi (x), h(t))$ the map $\phi\colon M \to N$ is a subelliptic harmonic morphism.

We consider a $2m$-th-order elliptic operator of divergence form in a domain $\Omega$ of $\mathbb{R}^{n}$, assuming that the coefficients are Hölder continuous of exponent $r \in (0,1]$. For the self-adjoint operator associated with the Dirichlet boundary condition we improve the asymptotic formula of the spectral function $e(\tau^{2m},x,y)$ for $x=y$ to obtain the remainder estimate $O(\tau^{n-\theta}+\dist(x,\partial\Omega)^{-1}\tau^{n-1})$ with any $\theta \in (0,r)$, using the $L^{p}$ theory of elliptic operators of divergence form. We also show that the spectral function is in $C^{m-1,1-\varepsilon}$ with respect to $(x,y)$ for any small $\varepsilon > 0$. These results extend those for the whole space $\mathbb{R}^{n}$ obtained by Miyazaki [19] to the case of a domain.

Kawashita, Nakazawa, and Soga [3] give a necessary condition for the uniform energy decay of the dissipative wave equation whose principal term has constant coefficients. In their proof, they construct asymptotic solutions for a suitable family of the Cauchy data. In this paper, instead of the asymptotic solutions, we consider the semiclassical measure associated with the family and extend this result to the variable coefficient case. Moreover we give some lower bound estimate for the energy decay.

The $n$-th modular equation for the elliptic modular function $j(z)$ has large coefficients even for small $n$, and those coefficients grow rapidly as $n \to \infty$. The growth of these coefficients was first obtained by Cohen ([5]). And, recently Cais and Conrad ([1], \S7) considered this problem for the Hauptmodul $j_{5}(z)$ of the principal congruence group $\Gamma(5)$. They found that the ratio of logarithmic heights of $n$-th modular equations for $j(z)$ and $j_{5}(z)$ converges to 60 as $n \to \infty$, and observed that 60 is the group index $[\overline{\Gamma(1)} : \overline{\Gamma(5)}]$. In this paper we prove that their observation is true for Hauptmoduln of somewhat general Fuchsian groups of the first kind with genus zero.

We reveal a relationship between the non-local operator $\mathcal{L}$ with variable order having $n$ as a Lévy-type kernel and the symmetric quadratic form defined by the kernel $n$. The relationship is obtained through the carré du champ operator relative to $\mathcal{L}$.

A covariant functor on the elliptic curves with complex multiplication is constructed. The functor takes values in the noncommutative tori with real multi-plication. A conjecture on the rank of an elliptic curve is formulated.

We consider the Noetherian properties of the ring of differential operators of an affine semigroup algebra. First we show that it is always right Noetherian. Next we give a condition, based on the data of the difference between the semigroup and its scored closure, for the ring of differential operators being anti-isomorphic to another ring of differential operators. Using this, we prove that the ring of differential operators is left Noetherian if the condition is satisfied. Moreover we give some other conditions for the ring of differential operators being left Noetherian. Finally we conjecture necessary and sufficient conditions for the ring of differential operators being left Noetherian.

The first step in the fundamental Clifford theoretic approach to general block theory of finite groups reduces to: $H$ is a subgroup of the finite group $G$ and $e$ is a central idempotent of $H$ such that $e({}^{g}e)=0$ for all $g \in G-H$. Then $\Tr_{H}^{G}(e)$ is a central idempotent of $G$ and induction from $H$ to $G$, $\Ind_{H}^{G}$, is part of a Morita equivalence between the categories of $e$-modules and of $\Tr_{H}^{G}(e)$-modules. Let $W$ be an indecomposable $e$-module, so that $V=\Ind_{H}^{G}(W)$ is an indecomposable $\Tr_{H}^{G}(e)$-module. We present results that relate the Green correspondents of $W$ and $V$ via induction and restriction.

We obtain the affirmative answer for a special case of the linearization problem for algebraic embeddings of $\mathbb{C}^{2}$ into $\mathbb{C}^{3}$. Indeed, we determine all the compactifications $(X,Y)$ of $\mathbb{C}^{2}$ such that $X$ are normal quartic hypersurfaces in $\mathbb{P}^{3}$ without triple points and $Y$ are hyperplane sections of $X$. Moreover, for each $(X,Y)$, we construct a tame automorphism of $\mathbb{C}^{3}$ which transforms the hypersurface $X\setminus Y$ onto a coordinate hyperplane.