1,374 research outputs found
Asymptotics for rank and crank moments
Moments of the partition rank and crank statistics have been studied for
their connections to combinatorial objects such as Durfee symbols, as well as
for their connections to harmonic Maass forms. This paper proves a conjecture
due to Bringmann and Mahlburg that refined a conjecture of Garvan. Garvan's
conjecture states that the moments of the crank function are always larger than
the moments of the rank function, even though the moments have the same main
asymptotic term. The proof uses the Hardy-Ramanujan method to provide precise
asymptotic estimates for rank and crank moments and their differences.Comment: 11 page
Almost holomorphic Poincare series corresponding to products of harmonic Siegel-Maass forms
We investigate Poincar\'e series, where we average products of terms of
Fourier series of real-analytic Siegel modular forms. There are some (trivial)
special cases for which the products of terms of Fourier series of elliptic
modular forms and harmonic Maass forms are almost holomorphic, in which case
the corresponding Poincar\'e series are almost holomorphic as well. In general
this is not the case. The main point of this paper is the study of
Siegel-Poincar\'e series of degree attached to products of terms of Fourier
series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms. We
establish conditions on the convergence and nonvanishing of such
Siegel-Poincar\'e series. We surprisingly discover that these Poincar\'e series
are almost holomorphic Siegel modular forms, although the product of terms of
Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular
forms (in contrast to the elliptic case) is not almost holomorphic. Our proof
employs tools from representation theory. In particular, we determine some
constituents of the tensor product of Harish-Chandra modules with walls
Fine-Grained Complexity Theory: Conditional Lower Bounds for Computational Geometry
Fine-grained complexity theory is the area of theoretical computer sciencethat proves conditional lower bounds based on the Strong Exponential TimeHypothesis and similar conjectures. This area has been thriving in the lastdecade, leading to conditionally best-possible algorithms for a wide variety ofproblems on graphs, strings, numbers etc. This article is an introduction tofine-grained lower bounds in computational geometry, with a focus on lowerbounds for polynomial-time problems based on the Orthogonal Vectors Hypothesis.Specifically, we discuss conditional lower bounds for nearest neighbor searchunder the Euclidean distance and Fr\'echet distance.<br
Inequalities for differences of Dyson's rank for all odd moduli
Kathrin Bringmann (Mathematisches Institut, UniversitÀt Köln, Weyertal 86-90, D-50931 Köln, Germany)
Ben Kane (Wiskunde Afdeling, Radboud Universiteit, Postbus 9010, 6500 GL, Nijmegen, Netherlands)postprin
Modular local polynomials
In this paper, we consider modular local polynomials. These functions satisfy modularity while they are locally defined as polynomials outside of an exceptional set. We prove an inequality for the dimension of the space of such forms when the exceptional set is given by certain natural geodesics related to binary quadratic forms of (positive) discriminant D. We furthermore show that the dimension is the largest possible if and only if D is an even square. Following this, we describe how to use the methods developped in this paper to establish an algorithm which explicitly determines the space of modular local polynomials for each D.postprin
Approximability of the Discrete {Fr\'echet} Distance
<p>The Fréchet distance is a popular and widespread distance measure for point sequences and for curves. About two years ago, Agarwal et al. [SIAM J. Comput. 2014] presented a new (mildly) subquadratic algorithm for the discrete version of the problem. This spawned a flurry of activity that has led to several new algorithms and lower bounds.</p><p>In this paper, we study the approximability of the discrete Fréchet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound showing that strongly subquadratic algorithms for the discrete Fréchet distance are unlikely to exist, even in the one-dimensional case and even if the solution may be approximated up to a factor of 1.399.</p><p>This raises the question of how well we can approximate the Fréchet distance (of two given -dimensional point sequences of length ) in strongly subquadratic time. Previously, no general results were known. We present the first such algorithm by analysing the approximation ratio of a simple, linear-time greedy algorithm to be . Moreover, we design an -approximation algorithm that runs in time , for any . Hence, an -approximation of the Fréchet distance can be computed in strongly subquadratic time, for any \varepsilon > 0.</p
New identities involving sums of the tails related to real quadratic fields
In previous work, the authors discovered new examples of q-hypergeometric series related to the arithmetic of and . Building on this work, we construct in this paper sum of the tails identities for which some which some of these functions occur as error terms. As an application, we obtain formulas for the generating function of a certain zeta functions for real quadratic fields at negative integers.postprin
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