26 research outputs found
Coefficients of Sylvester's Denumerant
For a given sequence of positive integers, we consider
the combinatorial function that counts the nonnegative
integer solutions of the equation , where the right-hand side is a varying
nonnegative integer. It is well-known that is a
quasi-polynomial function in the variable of degree . In combinatorial
number theory this function is known as Sylvester's denumerant.
Our main result is a new algorithm that, for every fixed number , computes
in polynomial time the highest coefficients of the quasi-polynomial
as step polynomials of (a simpler and more explicit
representation). Our algorithm is a consequence of a nice poset structure on
the poles of the associated rational generating function for
and the geometric reinterpretation of some rational
generating functions in terms of lattice points in polyhedral cones. Our
algorithm also uses Barvinok's fundamental fast decomposition of a polyhedral
cone into unimodular cones. This paper also presents a simple algorithm to
predict the first non-constant coefficient and concludes with a report of
several computational experiments using an implementation of our algorithm in
LattE integrale. We compare it with various Maple programs for partial or full
computation of the denumerant.Comment: minor revision, 28 page
Top Coefficients of the Denumerant
International audienceFor a given sequence of positive integers, we consider the combinatorial function that counts the nonnegative integer solutions of the equation , where the right-hand side is a varying nonnegative integer. It is well-known that is a quasipolynomial function of of degree . In combinatorial number theory this function is known as the . Our main result is a new algorithm that, for every fixed number , computes in polynomial time the highest coefficients of the quasi-polynomial as step polynomials of . Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Experiments using a implementation will be posted separately.Considérons une liste de entiers positifs. Le dénumérant est lafonction qui compte le nombre de solutions en entiers positifs ou nuls de l’équation , où varie dans les entiers positifs ou nuls. Il est bien connu que cette fonction est une fonction quasi-polynomiale de , de degré . Nous donnons un nouvel algorithme qui calcule, pour chaque entier fixé (mais n’est pas fixé, les plus hauts coefficients du quasi-polynôme en termes de fonctions en dents de scie. Notre algorithme utilise la structure d’ensemble partiellement ordonné des pôles de la fonction génératrice de . Les plus hauts coefficients se calculent à l’aide de fonctions génératrices de points entiers dans des cônes polyèdraux de dimension inférieure ou égale à
Bits and Pieces in Logarithmic Conformal Field Theory
These are notes of my lectures held at the first School & Workshop on
Logarithmic Conformal Field Theory and its Applications, September 2001 in
Tehran, Iran.
These notes cover only selected parts of the by now quite extensive knowledge
on logarithmic conformal field theories. In particular, I discuss the proper
generalization of null vectors towards the logarithmic case, and how these can
be used to compute correlation functions. My other main topic is modular
invariance, where I discuss the problem of the generalization of characters in
the case of indecomposable representations, a proposal for a Verlinde formula
for fusion rules and identities relating the partition functions of logarithmic
conformal field theories to such of well known ordinary conformal field
theories.
These two main topics are complemented by some remarks on ghost systems, the
Haldane-Rezayi fractional quantum Hall state, and the relation of these two to
the logarithmic c=-2 theory.Comment: 91 pages, notes of lectures delivered at the first School and
Workshop on Logarithmic Conformal Field Theory and its Applications, Tehran,
September 2001. Amendments in Introductio
Relations between elliptic multiple zeta values and a special derivation algebra
We investigate relations between elliptic multiple zeta values and describe a
method to derive the number of indecomposable elements of given weight and
length. Our method is based on representing elliptic multiple zeta values as
iterated integrals over Eisenstein series and exploiting the connection with a
special derivation algebra. Its commutator relations give rise to constraints
on the iterated integrals over Eisenstein series relevant for elliptic multiple
zeta values and thereby allow to count the indecomposable representatives.
Conversely, the above connection suggests apparently new relations in the
derivation algebra. Under https://tools.aei.mpg.de/emzv we provide relations
for elliptic multiple zeta values over a wide range of weights and lengths.Comment: 43 pages, v2:replaced with published versio
Identities for Partitions of N with Parts from A Finite Set
We show for a prime power number of parts m that the first differences of partitions into at most m parts can be expressed as a non-negative linear combination of partitions into at most m – 1 parts. To show this relationship, we combine a quasipolynomial construction of p(n,m) with a new partition identity for a finite number of parts. We prove these results by providing combinatorial interpretations of the quasipolynomial of p(n,m) and the new partition identity. We extend these results by establishing conditions for when partitions of n with parts coming from a finite set A can be expressed as a non-negative linear combination of partitions with parts coming from a finite set B. We extend this work into Gaussian Polynomials and show that the techniques used can prove asymptotic formulas of partitions with parts from a finite set A
Tau functions: theory and applications to matrix models and enumerative geometry
In this thesis we study partition functions given by matrix integrals from the point of view of isomon-
odromic deformations, or more generally of Riemann\u2013Hilbert problems depending on parameters
Time evolution of entanglement for holographic steady state formation
Within gauge/gravity duality, we consider the local quench-like time
evolution obtained by joining two 1+1-dimensional heat baths at different
temperatures at time t=0. A steady state forms and expands in space. For the
2+1-dimensional gravity dual, we find that the shockwaves expanding the
steady-state region are of spacelike nature in the bulk despite being null at
the boundary. However, they do not transport information. Moreover, by adapting
the time-dependent Hubeny-Rangamani-Takayanagi prescription, we holographically
calculate the entanglement entropy and also the mutual information for
different entangling regions. For general temperatures, we find that the
entanglement entropy increase rate satisfies the same bound as in the
"entanglement tsunami" setups. For small temperatures of the two baths, we
derive an analytical formula for the time dependence of the entanglement
entropy. This replaces the entanglement tsunami-like behaviour seen for high
temperatures. Finally, we check that strong subadditivity holds in this
time-dependent system, as well as further more general entanglement
inequalities for five or more regions recently derived for the static case.Comment: 57 pages, 25 figures. v2: Minor revisions and references added. v3:
Referee's comments included. The numerical codes described in this paper are
available in the ancillary files directory (anc/) of this submissio