Related papers: A Short Proof of Euler--Poincar\'e Formula
In this article, under mild constraints on the sectional curvature, we exploit a divergence formula for symmetric endomorphisms to deduce a general Poincar\'e type inequality. We apply such inequality to higher-order mean curvature of…
In this paper we introduce a combinatorial formula for the Ekeland-Hofer-Zehnder capacity of a convex polytope in $\mathbb{R}^{2n}$. One application of this formula is a certain subadditivity property of this capacity.
A domain $S\subset{\mathbb{R}}^d$ is said to fulfill the Poincar\'{e} cone property if any point in the boundary of $S$ is the vertex of a (finite) cone which does not otherwise intersects the closure $\bar{S}$. For more than a century,…
We revisit Allendoerfer-Weil's formula for the Euler characteristic of embedded hypersurfaces in constant sectional curvature manifolds, first taking some time to re-prove it while demonstrating techniques of [2] and then applying it to…
We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as the two toric polynomials introduced by Stanley, but allows different algebraic manipulations. The intertwined…
We provide a multidimensional weighted Euler--MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences…
Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial…
We study Euler-Poincare systems (i.e., the Lagrangian analogue of Lie-Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler-Poincare equations for a parameter dependent Lagrangian…
Braid Floer homology is an invariant of proper relative braid classes. Closed integral curves of 1-periodic Hamiltonian vector fields on the 2-disc may be regarded as braids. If the Braid Floer homology of associated proper relative braid…
A seminal result of E. Ehrhart states that the number of integer lattice points in the dilation of a rational polytope by a positive integer $k$ is a quasi-polynomial function of $k$ --- that is, a "polynomial" in which the coefficients are…
A theorem due to D. Bernstein states that Euler characteristic of a hypersurface defined by a polynomial f in (C\{0})^n is equal (upto a sign) to n! times volume of the Newton polyhedron of f. This result is related to algebaric torus…
In a 1967 paper, Banchoff stated that a certain type of polyhedral curvature, that applies to all finite polyhedra, was zero at all vertices of an odd-dimensional polyhedral manifold; one then obtains an elementary proof that…
We present a geometric proof of the Poincar\'e-Dulac Normalization Theorem for analytic vector fields with singularities of Poincar\'e type. Our approach allows us to relate the size of the convergence domain of the linearizing…
The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincar\'{e} duality complexes (PD complexes). The problem is that an arbitrary generalized manifold $X$ is always an ENR space, but it is…
In this paper, we use integration method to show that there is no existence of global $C^{2}$ solution with compact support, to the pressureless Euler-Poisson equations with attractive forces in $R^{N}$. And the similar result can be shown,…
There is a simple formula for the Ehrhart polynomial of a cyclic polytope. The purpose of this paper is to show that the same formula holds for a more general class of polytopes, lattice-face polytopes. We develop a way of decomposing any…
Partial generalizations of virtual polyhedra theory (sometimes under different names) appeared recently in the theory of torus manifolds. These generalizations look very different from the original virtual polyhedra theory. They are based…
Euler Maclaurin formulas for a polytope express the sum of the values of a function over the lattice points in the polytope in terms of integrals of the function and its derivatives over faces of the polytope or its expansions. Exact Euler…
We show how the Fourier transform of a shape in any number of dimensions can be simplified using Gauss's law and evaluated explicitly for polygons in two dimensions, polyhedra three dimensions, etc. We also show how this combination of…
It is well known that the Euler characteristic of an odd dimensional compact manifold is zero. An Euler complex is a combinatorial analogue of a compact manifold. We present here an elementary proof of the corresponding result for Euler…