In this article, we derive a Brunn-Minkowski type theorem for sets bearing some relation to the causal structure on the Minkowski spacetime ${{\mathbb{L}}^{n+1}}$ . We also present an isoperimetric inequality in the Minkowski spacetime ${{\mathbb{L}}^{n+1}}$ as a consequence of this Brunn-Minkowski type theorem.

In this article we characterize the univalent harmonic mappings from the exterior of the unit disk, $\Delta $ , onto a simply connected domain $\Omega $ containing infinity and which are solutions of the system of elliptic partial differential equations $\overline{{{f}_{{\bar{z}}}}\left( Z \right)}=a\left( z \right){{f}_{z}}\left( z \right)$ where the second dilatation function $a\left( z \right)$ is a finite Blaschke product. At the end of this article, we apply our results to nonparametric minimal surfaces having the property that the image of its Gauss map is the upper half-sphere covered once or twice.

This article studies algebras $R$ over a simple artinian ring $A$ , presented by a quiver and relations and graded by a semigroup $\Sigma $ . Suitable semigroups often arise from a presentation of $R$ . Throughout, the algebras need not be finite dimensional. The graded ${{K}_{0}}$ , along with the $\Sigma $ -graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties.

A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert $\Sigma $ -series in the associated path incidence ring.

The rationality of the $\Sigma $ -Euler characteristic, the Hilbert $\Sigma $ -series and the Poincaré-Betti $\Sigma $ -series is studied when $\Sigma $ is torsion-free commutative and $A$ is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.

We study polynomial invariants of systems of vectors with respect to exceptional simple algebraic groups in their minimal linear representations. For each type we prove that the algebra of invariants is integral over the subalgebra of trace polynomials for a suitable algebraic system $\left( cf.\,[27],\,[28],\,[13] \right)$ .

Virasoro-toroidal algebras, ${{\tilde{J}}_{[n]}}$ , are semi-direct products of toroidal algebras ${{J}_{[n]}}$ and the Virasoro algebra. The toroidal algebras are, in turn, multi-loop versions of affine Kac-Moody algebras. Let $\Gamma $ be an extension of a simply laced lattice $\dot{Q}$ by a hyperbolic lattice of rank two. There is a Fock space $V\left( \Gamma\right)$ corresponding to $\Gamma $ with a decomposition as a complex vector space: $V\left( \Gamma\right)=\coprod{_{m\in z}K\left( m \right)}$ . Fabbri and Moody have shown that when $m\ne 0,\,K\left( m \right)$ is an irreducible representation of ${{\tilde{J}}_{[2]}}$ . In this paper we produce a filtration of ${{\tilde{J}}_{[2]}}$ -submodules of $K\left( 0 \right)$ . When $L$ is an arbitrary geometric lattice and $n$ is a positive integer, we construct a Virasoro-Heisenberg algebra $\tilde{H}\left( L,n \right)$ . Let $Q$ be an extension of $\dot{Q}$ by a degenerate rank one lattice. We determine the components of $V\left( \Gamma\right)$ that are irreducible $\tilde{H}\left( Q,1 \right)$ -modules and we show that the reducible components have a filtration of $\tilde{H}\left( Q,1 \right)$ -submodules with completely reducible quotients. Analogous results are obtained for $\tilde{H}\left( \dot{Q},2 \right)$ . These results complement and extend results of Fabbri and Moody.

Averages in weighted spaces $L_{\phi }^{p}[-1,1]$ defined by additions on $[-1,\,1]$ will be shown to satisfy strong converse inequalities of type $\text{A}$ and $\text{B}$ with appropriate $K$ -functionals. Results for higher levels of smoothness are achieved by combinations of averages. This yields, in particular, strong converse inequalities of type $\text{D}$ between $K$ -functionals and suitable difference operators.

A Banach space $X$ is said to be quotient hereditarily indecomposable if no infinite dimensional quotient of a subspace of $X$ is decomposable. We provide an example of a quotient hereditarily indecomposable space, namely the space ${{X}_{GM}}$ constructed by W. T. Gowers and B. Maurey in $[\text{GM}]$ . Then we provide an example of a reflexive hereditarily indecomposable space $\hat{X}$ whose dual is not hereditarily indecomposable; so $\hat{X}$ is not quotient hereditarily indecomposable. We also show that every operator on ${{\hat{X}}^{*}}$ is a strictly singular perturbation of an homothetic map.

We give necessary and sufficient conditions for a norm-compact subset of a Hilbert space to admit a ${{C}^{1}}$ embedding into a finite dimensional Euclidean space. Using quasibundles, we prove a structure theorem saying that the stratum of $n$ -dimensional points is contained in an $n$ -dimensional ${{C}^{1}}$ submanifold of the ambient Hilbert space. This work sharpens and extends earlier results of $\text{G}$ . Glaeser on paratingents. As byproducts we obtain smoothing theorems for compact subsets of Hilbert space and disjunction theorems for locally compact subsets of Euclidean space.

Let $L$ be a simple algebraic group and $P$ a parabolic subgroup with Abelian unipotent radical ${{P}^{u}}$ . Many familiar varieties (determinantal varieties, their symmetric and skew-symmetric analogues) arise as closures of $P$ -orbits in ${{P}^{u}}$ . We give a unified invariant-theoretic treatment of various properties of these orbit closures. We also describe the closures of the conormal bundles of these orbits as the irreducible components of some commuting variety and show that the polynomial algebra $k[{{P}^{u}}]$ is a free module over the algebra of covariants.

We prove a conjecture of Kudla and Rallis about the first occurrence in the theta correspondence, for dual pairs of the form $\left( U\left( p,q \right),\,U\left( r,s \right) \right)$ and most representations.

Let $L={{L}_{0}}+{{L}_{1}}$ be a ${{\mathbb{Z}}_{2}}$ -graded Lie algebra over a commutative ring with unity in which 2 is invertible. Suppose that ${{L}_{0}}$ is abelian and $L$ is generated by finitely many homogeneous elements ${{a}_{1}},.\,.\,.,{{a}_{k}}$ such that every commutator in ${{a}_{1}},.\,.\,.,{{a}_{k}}$ is ad-nilpotent. We prove that $L$ is nilpotent. This implies that any periodic residually finite ${2}'$ -group $G$ admitting an involutory automorphism $\phi $ with ${{C}_{G}}\left( \phi\right)$ abelian is locally finite.