This paper will review recent advances in the formulation of a discrete version of geometric mechanics, based on the discretization of Hamilton’s variational principle, and progress that has been made in reduction theory for discrete variational mechanics in the form of Discrete Routh Reduction. To place discrete geometric mechanics on a firm mathematical foundation, we propose to develop a dis...

In 1933, anticipating formal cohomology theory, van Kampen [5] gave a slightly rough description of an obstruction o(K) ∈ H Z/2(K ,Z) which vanishes if and only if an n-dimensional simplicial complex K admits a piecewise-linear embedding into R, n ≥ 3. Here K is the deleted product of a complex K. The cohomology in question is the Z/2-equivariant cohomology where Z/2 acts on the space by exchan...

For having a Poincaré duality via a cap product between the intersection homology of a paracompact oriented pseudomanifold and the cohomology given by the dual complex, G. Friedman and J. E. McClure need a coefficient field or an additional hypothesis on the torsion. In this work, by using the classical geometric process of blowing-up, adapted to a simplicial setting, we build a cochain complex...

We define analytic torsion τ (X,E ,H) ∈ detH(X, E ,H) for the twisted de Rham complex, consisting of differential forms on a compact Riemannian manifold X valued in a flat vector bundle E , with a differential given by ∇ +H ∧ · , where ∇ is a flat connection on E , H is an odd-degree closed differential form on X, and H(X, E ,H) denotes the cohomology of this Z2-graded complex. We show that whe...

For any natural d ≥ k ≥ 2 we calculate the cohomology groups of the space of homogeneous polynomials R → R of degree d, which do not vanish with multiplicity ≥ k on real lines. For k = 2 this problem provides the simplest example of the situation, when the “finite-order” invariants of nonsingular objects are not a complete system of invariants. The “affine” version of this problem (the calculat...

HOMOTOPY THEORY AND GENERALIZED SHEAF COHOMOLOGY BY KENNETHS. BROWN0) ABSTRACT. Cohomology groups Ha(X, E) are defined, where X is a topological space and £ is a sheaf on X with values in Kan's category of spectra. These groups generalize the ordinary cohomology groups of X with coefficients in an abelian sheaf, as well as the generalized cohomology of X in the usual sense. The groups are defin...

We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of indecomposable maximal Cohen-Macaulay modules of higher rank. A result of Klyachko states that the category of reflexive toric sheaves is equivalent to the category of vector spaces together with a certain family of filtrations. Within th...

In this work we study the framework of mathematical morphology on simplicial complex spaces. Simplicial complexes are widely used to represent multidimensional data, such as meshes, that are two dimensional complexes, or graphs, that can be interpreted as one dimensional complexes. Mathematical morphology is one of the most powerful frameworks for image processing, including the processing of d...

For any flag simplicial complex $K$, we describe the multigraded Poincare series, minimal number of relations and degrees these in Pontryagin algebra corresponding moment-angle $Z_K$. We compute LS-category $Z_K$ for complexes give a lower bound general case. The key observation is that Milnor-Moore spectral sequence collapses at second sheet $K$. also show results Panov Ray about algebras Davi...