In this paper, we investigate a certain fusion scheme $\tilde{\mathcal{X}}\left(A_n\right)$ of the group association scheme $\mathcal{X}\left(A_n\right)$ of the alternating group of arbitrary degree $n$. In particular, under some extra assumption on 'geometry of maximal cliques', we characterize $\tilde{\mathcal{X}}\left(A_n\right)$ by parameters.

We classify all selfinjective finite dimensional algebras over an algebraically closed field which are socle equivalent to the tame selfinjective algebras which admit simply connected Galois coverings and whose Auslander-Reiten quiver consists only of stable tubes.

The simplest quartic fields are the real cyclic quartic number fields defined by the irreducible quartic polynomials $x^4-m{x}^3-6x^2+mx+1$, where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. We give an explicit lower bound for their class numbers which is much better than the previous known ones obtained by A. Lazarus. Then, using it, we determine the simplest quartic fields with ideal class groups of exponents $\le 2$.

We give a Fourier-Ehrenpreis integral representation formula that expresses a harmonic function in a ball with a prescribed boundary value by superposition of harmonic exponentials.

Let $K$ be a division ring with a $\sigma$-derivation $\delta$, where $\sigma$ is an endomorphism of $K$ and $K(X;\sigma ,\delta )$ be the quotient division ring of the Ore extension $K[X;\sigma ,\delta ]$ over $K$ in an indeterminate $X$. First, we describe non-commutative valuation rings of $K(X;\sigma ,\delta )$ which contain $K[X;\sigma ,\delta ]$ . Suppose that $(\sigma ,\delta )$ is compatible with $V$, where $V$ is a total valuation ring of $K$, then $R^{\left(1\right)}=V[X;\sigma ,\delta {]}_{J\left(V\right)[X;\sigma ,\delta ]}$, the localization of $V[X;\sigma ,\delta ]$ at $J\left(V\right)[X;\sigma ,\delta ]$, is a total valuation ring of $K(X;\sigma ,\delta )$. Applying the description above, then, second, we describe non-commutative valuation rings $B$ of $K(X;\sigma ,\delta )$ such that $B\cap K=V,$$X\in B$ and $B\subseteq R^{\left(1\right)}$, which is the aim of this paper. In the end of each section we give several examples to display some of the various phenomena.

We consider a simple model which is a caricature of a crystal interacting with a radiation field. The model has two bands of continuous spectrum and the particle can pass from the upper one to the lower by radiating a photon, the coupling between the excited and deexcited states being of a Friedrichs type. Under suitable regularity and analyticity assumptions we find the continued resolvent and show that for weak enough coupling it has a curve-type singularity in the lower halfplane which is a deformation of the upper-band spectral cut. We then find a formula for the decay amplitude and show that for a fixed energy it is approximately exponential at intermediate times, while the tail has a power-like behaviour.

We construct $Z_p$-extensions whose Iwasawa modules have prescribed structure. Specifically, we give a $Z_p$-extension with prescribed finite Iwasawa module. Also we show that there exists a $Z_p$-extension with arbitrarily given Iwasawa $\mu$-invariant. We apply the construction of such $Z_p$-extensions to a certain capitulation problem.

Let $T$ be the billiard map for a two-dimensional dispersing billiard without eclipse. We show that the nonwandering set ${\Omega }^{+}$ for $T$ has a hyperbolic structure quite similar to that of the horseshoe. We construct a sort of stable foliation for $({\Omega }^{+},T)$ each leaf of which is a $K$-decreasing curve. We call the foliation a $K$-stable foliation for $({\Omega }^{+},T)$. Moreover, we prove that the foliation is Lipschitz continuous with respect to the Euclidean distance in the so called $(r,\varphi )$-coordinates. It is well-known that we can not always expect the existence of such a Lipschitz continuous invariant foliation for a dynamical system even if the dynamical system itself is smooth. Therefore, we keep our construction as elementary and self-contained as possible so that one can see the concrete structure of the set ${\Omega }^{+}$ and why the $K$-stable foliation turns out to be Lipschitz continuous.

We prove an analog to the classical Mikhlin-Hörmander multiplier theorem for Markov chains.

We define a class of Besov type spaces which is a generalization of that defined by Kenig-Ponce-Vega ([4], [5]) in their study on $KdV$ equation and nonlinear Schrödinger equation. Using these spaces, we prove the following results: the 1-dimendional semilinear Schrödinger equation with the nonlinear term $c_1{u}^2+c_2{\overline{u}}^2$ has a unique local-in-time solution for the initial data $∊B_{2,1}^{-3/4}$, and that with $cu\overline{u}$ has a unique local-in-time solution for the initial data $\in B_{2,1}^{-1/4, \sharp}$. Note that $B_{2,1}^{-1/4, \sharp}(\mathbf{R}) \supset B_{2,1}^{-1/4}(\mathbf{R}) \supset H^5(\mathbf{R})$ for any $s > -1/4$.

We realize simple complex Lie algebras as quotient Lie algebras defined via Hall algebras of tame hereditary algebras.

In this paper, we give a necessary condition for the Cauchy problem for parabolic equation in order to be uniquely solvable in the analytic class.

The simultaneous universality of twisted automorphic $L$-functions, associated with a new form with respect to a congruence subgroup of $SL(2,Z)$ and twisted by Dirichlet characters, is proved. Applications to the functional independence and the zero density of linear combinations of those $L$-functions are given.

If a normal quartic surface admits a singular point that is not a rational double point, then the surface is determined by the triplet $(M,D,E)$ consisting of the minimal desingularization $M$, the pullback $D$ of a general hyperplane section, and a nonzero effective anti-canonical divisor $E$ of $M$. Geometric constructions of all the possible triplets $(M,D,E)$ are given.