In joint work with S.-T. Yau, we construct new cohomologies of differential forms and elliptic operators on symplectic manifolds. Their construction can be described simply following a symplectic decomposition of the exterior derivative operator into two first-order differential operators, which are analogous to the Dolbeault operators in complex geometry. These first-order operators lead to ne...

We discuss the algebraic renormalization of the Yang–Mills gauge field theory in the presence of translations. Due to the translations the algebra between Sorella’s δ–operator, the exterior derivative and the BRST–operator closes. Therefore, we are able to derive an integrated parameter formula collecting in an elegant and compact way all nontrivial solutions of the descent equations. Work supp...

The Whitney module of a matroid is a natural analogue of the tensor algebra of the exterior algebra of a vector space that takes into account the dependencies of the matroid. In this paper we indicate the role that tableaux can play in describing the Whitney module. We will use our results to describe a basis of the Whitney module of a certain class of matroids known as freedom (also known as S...

A very general measuretheoretic version of the Gauss-Green formula (for »-space) has recently been studied by De Giorgi in [DG 1 ] and [DG2]. Independently Fleming and Young have obtained related results (for 3-space) in [FY], employing the technique of "generalized surfaces." Here the work of De Giorgi will be supplemented through use of the geometric concept of exterior normal introduced in [...

The problem of decomposing the symmetric and exterior algebras of the adjoint representation of the Lie algebra gl N into sl N-irreducible components are considered. The exact formula for the principal specialization of the internal product of the Schur functions (similar to the formula for Kostka-Foulkes polynomials) is obtained by the purely combinatorial approach, based on the theory of rigg...

any language foremost is composed of words and words are one of the most important elements in conveying meaning and they have different diversities and levels in conveying this meaning. quranic words are not the exception; therefore, to discover and to understand the different levels of conveying meaning (exterior and inner) by words, the quran reader can be more aware of the meaning and the p...

The purpose of this article is to study the distribution of the trace on the unitary symplectic group. We recall its relevance to equidistribution results for the eigenvalues of the Frobenius in families of abelian varieties over finite fields, and to the limiting distribution of the number of points of curves. We give four expressions of the trace distribution if g = 2, in terms of special fun...

Estimating the extent of an animal's exterior surface, and its interface with the environment, is of prime importance in thermal biology. Regardless of the mode of transfer, heat fluxes across an animal's surface are often calculated as intensive quantities (e.g. W/m?) and converted to values for a particular animal by multiplying by surface area. Thus, the accuracy of heat-transfer estimates m...

We study the Schrödinger operator with a constant magnetic field in the exterior of a two-dimensional compact domain. Functions in the domain of the operator are subject to a boundary condition of the third type (Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulat...

Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to programming complicated geometrical operations. But there is a fundamental weakness in this approach — the Euclidean distance between points is not handled in a s...