Generating function

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except that La

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Le carré de la fonction des diviseurs

Nicolas Bersier

Let d(n) denote Dirichlet's divisor function for positive integer numbers. This work is primarily concerned with the study of We are interested, in the error term where Ρ3 is a polynomial of degree 3 ; more precisely xΡ3(log x) is the residue of in s = 1. A. Ivić showed that E(x) = Ο(x1/2+ε) for all ε > 0 (cf. [9], p.394). We will prove that for all x > 0, we have With this intention, we apply Perron's formula to the generating function ζ4(s)/ ζ (2s) and Landau's finite difference method. It was conjectured that E(x) = Ο(x1/4+ε) for ε > 0. The existence of non-trivial zeros of the Riemann ζ function implies that we cannot do better, that is The study of the Riesz means for ρ sufficiently large shows that their error term, is an infinite series , on the zeros of the Riemann Zeta function added with a development , into a series of Hardy-Voronoï's type, both being convergent. To find the "meaning" of , one could consider the difference But the series (probably) doesn't converge.We will thus substract only a finite part of , weighted by a smooth function ω, the number of terms of the finite part depending on x. If we consider this new error term , we obtain, using a classical method due to Hardy, that for x ≥ 1.

EPFL2007

Homology of the three flag Hilbert Scheme

Daniele Boccalini

We prove the existence of an affine paving for the three-step flag Hilbert scheme that parametrizes flag of three 0-dimensional subschemes of length, respectively, n, n+1 and n+2 that are supported at the origin of the affine plane. This is done by showing that the space stratifies in smooth subvarieties, the Hilbert-Samuel's strata, each of which has an affine paving with cells of known dimension, indexed by marked Young diagrams. The affine pavings of the Hilbert-Samuel's strata allow us to prove that the Poincaré polynomials for our spaces satisfy a generating function. In the process of proving the formula for the generating function we relate combinatorially the homology of our spaces with that of known smooth subspaces of another Hilbert scheme of flags, this time of length n and n+2. As a corollary we find an affine paving and a combinatorial formula for the Poincaré of these last ambient spaces.

EPFL2016

On the fundamental domain of affine Springer fibers

Zongbin Chen

Let G be a connected reductive algebraic group over an algebraically closed field k,gamma is an element of g( k(( epsilon ))) a semisimple regular element, we introduce a fundamental domain F gamma for the affine Springer fibers X gamma. We show that the purity conjecture of X gamma is equivalent to that of F gamma via the Arthur-Kottwitz reduction. We then concentrate on the unramified affine Springer fibers for the group GL(d). It turns out that their fundamental domains behave nicely with respect to the root valuation of gamma. We formulate a rationality conjecture about a generating series of their Poincare polynomials, and study them in detail for the group GL(3). In particular, we pave them in affine spaces and we prove the rationality conjecture.

Springer Heidelberg2017

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In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 a

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In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers le

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In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma functio

CS-101: Advanced information, computation, communication I

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics as diverse as mathematical reasoning, combinatorics, discrete structures & algorithmic thinking.

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L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et de démontrer rigoureusement les résultats principaux de ce sujet.

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L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.