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 vector We give explicit systems of generators of the algebras of invariant polynomials in arbitrary many vector variables for the classical reflection groups (including the dihedral groups). The least upper bound for the degrees of elements in a system of generators turns out to be independent of the number of vector variables. (HereM3 denotes the vector space of 3×3 matrices over k andp>amp;gt;3.) The method of proof involves an induction, and is potentially of wide applicability. A basis is calledmonomial if each of its elements is the result of applying to a (fixed) highest weight vector a monomial in the Chevalley basis elementsYα, α a simple root, in the opposite Borel subalgebra. The proof is based on a variant of Moser's method using time-dependent vector fields. Structure of some ?-graded lie superalgebras of vector fields Recently one of the authors obtained a classification of simple infinite dimensional Lie superalgebras of vector fields which extends the well known classification of E. From Lie algebras of vector fields to algebraic group actions The set ${\mathcal A}$ of all non-associative algebra structures on a fixed 2-dimensional real vector space $A$ is naturally a ${\mbox{\rm GL}}(2,{\mbox{\bf R}})$-module. Then one can associate to every vector bundle $E$ of rank $r$ over $X$ a vector bundle $E_\rho$ with fibre $V$. We would like to study triples $(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ over $X$, $L$ is a line bundle over $X$, and $\phi\colon E_\rho\rightarrow L$ is a nontrivial homomorphism. This setup comprises well known objects such as framed vector bundles, Higgs bundles, and conic bundles. Standard monomial bases, Moduli spaces of vector bundles, and Invariant theory A notion of a mixed representation of a quiver can be derived from ordinary quiver representation by considering the dual action of groups on "vertex" vector spaces together with their usual action. We are interested in the determination of the vector invariants of H. Let p be a prime and let V be a finite-dimensional vector space over the field $\mathbb{F}_p$. The frame operator of this sequence is expressed as a matrix-valued function multiplying a vector-valued function. The proof of the results uses an extension of the Littlewood-Paley-Stein theory of square functions to the vector-valued case and inequalities previously proved by one of the authors in the context of the Riesz transforms of order one. Vector Potential Theory on Nonsmooth Domains in R3 and Applications to Electromagnetic Scattering Vector potential theory on nonsmooth domains in R3 and applications to electromagnetic scattering

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