Related papers: Poincar\'e series for modular graph forms at depth…
We construct Fourier transforms relating functions and distributions on finite height $p$-divisible rigid analytic groups and objects in a dual category of $\mathbb{Z}_p$-local systems with analyticity conditions. Our Fourier transforms are…
Two methods can be used to calculate explicitly the Killing form on the Lie algebras. The first one is a direct calculation of the traces of the generators in a matrix representation of the algebra, and the second one is the usage of the…
The Molien-Weyl integral formula and the Hilbert-Poincar\'e series have proven to be powerful mathematical tools in relation to gauge theories, allowing to count the number of gauge invariant operators. In this paper, we show that these…
The first half of this paper is largely expository, wherein we present a systematic combinatorial approach to the theory of polynomial (semi)invariants and multilinear invariants of several vectors and covectors, for the classical groups.…
We consider certain elliptic modular graph functions that arise in the asymptotic expansion around the non--separating node of genus two string invariants that appear in the integrand of the $D^8 R^4$ interaction in the low momentum…
There exist several equivalent equations for the Poincar\'e series of a collection of valuations on the ring of germs of functions on a complex analytic variety. We give definitions of the Poinca\'e series of a collection of valuations in…
An automated treatment of iterated integrals based on letters induced by real-valued quadratic forms and Kummer--Poincar\'e letters is presented. These quantities emerge in analytic single and multi--scale Feynman diagram calculations. To…
We give theoretical and practical information on the Pari/GP modular forms package available since the spring of 2018. Thanks to the use of products of two Eisenstein series, this package is the first which can compute Fourier expansions at…
In a previous paper, the authors introduced a filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety defined by arcs on the singularity and called the arc filtration. The Poincar\'e…
Consider the classical action of ${\rm GL}_n$ on a sum of $q$ copies of the defining representation and $p$ copies of its dual; by Howe duality, the polynomial functions on this space decompose under the joint action of ${\rm GL}_n$ and…
A formula for computation of the bivariate Poincar\'e series $\mathcal{P}_d(z,t)$ for the algebra of covariants of binary $d$-form is found.
We propose that the full Poincar\'{e} beam with any polarization geometries can be pictorially described by the hybrid-order Poincar\'{e} sphere whose eigenstates are defined as a fundamental-mode Gaussian beam and a Laguerre-Gauss beam. A…
The Poincare function is a compact form of counting moduli in local geometric problems. We discuss its property in relation to V.Arnold's conjecture, and derive this conjecture in the case when the pseudogroup acts algebraically and…
We consider the space of germs of Fedosov structures at a point, together with the group of origin-preserving diffeomorphisms acting on it. We calculate dimensions of moduli spaces of $k$-jets of generic structures and construct Poincar\'e…
We prove non-trivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the P\'olya-Vinogradov range. We then derive applications to the second moment of holomorphic cusp forms twisted…
In string theories, one-loop scattering amplitudes are characterized by integrals over genus-one surfaces using the Kronecker-Eisenstein series. A recent methodology proposed a genus-one basis formed from products of these series of chain…
The temperature inversion properties of the internal energy, E, on odd spheres, and its derivatives, together with their expression in elliptic terms, as expounded in previous papers, are extended to the integrals of E, thence making…
In this paper we prove discrete Poincar\'e inequalities that are uniform in the mesh size for the discrete de Rham complex of differential forms developed in [Bonaldi, Di Pietro, Droniou, and Hu, An exterior calculus framework for polytopal…
Moduli spaces of stable coherent sheaves on a surface are of much interest for both mathematics and physics. Yoshioka computed generating functions of Poincare polynomials of such moduli spaces if the surface is the projective plane P2 and…
We show, for levels of the form $N = p^a q^b N'$ with $N'$ squarefree, that in weights $k \geq 4$ every cusp form $f \in \mathcal{S}_k(N)$ is a linear combination of products of certain Eisenstein series of lower weight. In weight $k=2$ we…