Related papers: Polar decomposition and Brion's theorem
We discuss and give elementary proofs of results of Brion and of Lawrence-Varchenko on the lattice-point enumerator generating functions for polytopes and cones. This largely expository note contains a new proof of Brion's Formula using…
We give an elementary geometric re-proof of a formula discovered by Michel Brion as well as two variants thereof. A subset of R^n gives rise to a formal Laurent series with monomials corresponding to lattice points in the set. Under…
Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an $H$-polar decomposition are found. In the process an equivalent to Witt's…
We give a new proof for the well-known Blaschke--Petkantschin formula which is based on the polar decomposition of rectangular matrices and may be of interest in random matrix theory.
A new application of polytope theory to Lie theory is presented. Exponential sums of convex lattice polytopes are applied to the characters of irreducible representations of simple Lie algebras. The Brion formula is used to write a polytope…
We find an elementary proof for Voiculescu's theorem on the polar decomposition of circular variables.
Brion's Formula realizes the Laurent polynomial of lattice points in a lattice polytope P as the sum of rational functions associated to the vertices of P. In this paper, we consider the special case where P is a generalized permutohedron.…
The description is presented for the dependence of the indirect exciton condensate density at the ring as a function of the polar angle at zero temperature with the involvement of the processes of formation and recombination of the…
We give in this note a weighted version of Brianchon-Gram's decomposition for a simple polytope. This weighted version is a direct consequence of the ordinary Brianchon-Gram formula.
The Ehrhart quasipolynomial of a rational polytope $P$ encodes the number of integer lattice points in dilates of $P$, and the $h^*$-polynomial of $P$ is the numerator of the accompanying generating function. We provide two decomposition…
A proof more elementary than the original one is given for Moretti's theorem that the usual polar decomposition of real matrices when applied to an orthochronous proper Lorentz matrix yields just its standard rotation-boost decomposition.…
In the paper, we give the proof of the polar decomposition of the Wiener measure according to the orbits of the group of diffeomorphisms.
In the work [Bull, Austr. Math. Soc. 85 (2012), 315-234], S.R. Moghadasi has shown how the decomposition of the $N$-fold product of Lebesgue measure on $\mathbb R^n$ implied by matrix polar decomposition can be used to derive the…
We make a full landscape analysis of the (generally non-convex) orthogonal Procrustes problem. This problem is equivalent to computing the polar factor of a square matrix. We reveal a convexity-like structure, which explains the already…
In the setting of adjointable operators on Hilbert $C^*$-modules, this paper deals with the polar decomposition of the product of three operators. The relationship between the polar decompositions associated with three operators is…
A subset K of R^n gives rise to a formal Laurent series with monomials corresponding to lattice points in K. Under suitable hypotheses, this series represents a rational function R(K). Michel Brion has discovered a surprising formula…
Provan and Billera introduced notions of (weak) decomposability of simplicial complexes as a means of attempting to prove polynomial upper bounds on the diameter of the facet-ridge graph of a simplicial polytope. Recently, De Loera and Klee…
We give an effective upper bound on the h^*-polynomial of a lattice polytope in terms of its degree and leading coefficient, confirming a conjecture of Batyrev. We deduce this bound as a consequence of a strong Cayley decomposition theorem…
This thesis examins a generalisation of polar decompositions to indefinite inner product spaces. The necessary general theory is studied and some general results are given. The main part of the thesis focuses on polar decompositions with…
The characters of simple Lie algebras are naturally decomposed into lattice polytope sums. The Brion formula for those polytope sums is remarkably similar to the Weyl character formula. Here we start to investigate if other character…