The character table of $G=Sp_4(F_{q}),$ $q$ odd, was calculated by B. Srinivasan in 1968 [22]. The rational and the real Schur indices of each complex irreducible character of $G$ were calculated by A. Przygocki in 1982 [21]. We calculate the Hasse invariants of each simple component of the group algebra $Q[G]$ of $G$ over $Q$.

Here we prove the following result and a few related statements. Let $V$ be a Banach space with countable unconditional basis and the localizing property, $Q\subset P(V)$ a quadric hypersurface with finite-dimensional singular locus and $E$ a holomorphic vector bundle of finite rank on $Q$. Then $E\cong\oplus_{1\leq i\leq r}0_{Q}(a_{i})$ for some integers $a_{i}$ and $h^{1}(Q, E(t))=0$ for every integer $t$.

The aim of this paper is to characterize $n$-Gorenstein rings in terms of Gorenstein injective modules and the Ext functor. We will show that if $R$ is a left and right noetherian ring and $n$ is a positive integer, then $R$ is $n$-Gorenstein if and only if $M$ being Gorenstein injective means that Ext$^{1}$ $(L, M )=0$ for all countably generated $R$-modules $L$ of projective dimension at most $n$. In particular, if $R$ is $n$-Gorenstein, then an $R$-module $M$ is Gorenstein injective if and only if it is $U$-Gorenstein injective whenever $U$ is a free $R$-module with a countable base.

In this work, we develop a global calculus for a class of Fourier integral operators with symbols $a(x, \xi)$ having exponential growth in $R_{x,\xi}^{2n}$. The functional frame is given by the spaces of type S of Gelfand and Shilov. As an application, we construct a parametrix and prove the existence of a solution for the Cauchy problem associated to SG-hyperbolic operators with one characteristic of constant multiplicity.

In this paper, we will show that classical resultants-which are generalizations of Sylvester determinants-are closely related to Chow-forms of projective varieties. As a result, we will see that the maximal degree of the coefficients of a Chow-form in the coefficients of defining equations may be assumed to be bounded from above.

Let $H$ be a Hopf $\pi$-coalgebra and let $A$ be a right group $H$-comodule algebra with a total integral $\phi$. In this article we will find some sufficient conditions under which an epimorphism of $(H, A)$-Hopf $\pi$-comodule splits if it splits $A$-linearly. As an application, we obtain a characterization for an $(H, A)$-Hopf $\pi$-comodule to be projective as an $A$-module, generalizing the one of the Maschke type theorem found in [D2].

This paper studies the simultaneous Diophantine phenomena of commuting systems of vector fields and diffeomorphisms. We consider how these conditions are related with the Diophantine phenomena of every element of the system.

Geometrical and structural properties are proved for manifolds which are sturctured by the presence of a local conformal section.

Building on work by L. Capogna & D. Danielli & N. Garofalo (cf. [7]-[8] ), G. Citti & N. Garofalo & E. Lanconelli (cf. [10]) and P. Hájlasz & P. Strzelecki (cf. [16]) we study local properties of weak subelliptic $F$-harmonic maps (cf. [4]) of a nondegenerate CR manifold into a sphere $S^{m}$, where $\rho(t)=F^{\prime}(t/2)$, $F\in C^{2},$ $F(t)\geq 0,$ $F^{\prime}(t)>0,$ $t\geq 0$. If $\Omega\subset R^{n}$ is a bounded domain and $X$ is a Hörmander system on $R^{n}$, we show that any weak solution $\phi\in W_{X}^{1,D}(\Omega, S^{m})$ to the nonlinear subelliptic system $-X^{*}\cdot(\rho(|X\phi|^{2})X\phi)=\rho(|X\phi|^{2})\phi|X\phi|^{2}$ is locally Hölder continuous, where $D$ is a homogeneous dimension of $\Omega$ with respect to $X$, provided that $t^{p}/K\leq p(t)\leq Kt^{P}$ for some $0 \lt p \lt (D-2)/2$.

We introduce and study the $n$-Dimensional Perfect Homotopy Approximation Property (briefly $n$-PHAP) equivalent to the discrete $n$-cells property in the realm of $LC^{n}$-spaces. It is shown that the product $X\times Y$ of a space $X$ with $n$-PHAP and a space $Y$ with $m$-PHAP has $(n+m+1)$-PHAP. We derive from this that for a (nowhere locally compact) locally connected Polish space $X$ without free arcs and for each $n\geq 0$ the power $X^{n+1}$ contains a closed topological copy of each at most $n$-dimensional compact (resp. Polish) space.