Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary form. In particular, we can show that for growing $d$ a positive proportion of the binary quadratic forms of discriminant $-d$ is represented by the given quaternary quadratic form.Comment: v5: Some typos correcte

On the critical pair theory in abelian groups : Beyond Chowla's Theorem

We obtain critical pair theorems for subsets S and T of an abelian group such that |S+T| < |S|+|T|+1. We generalize some results of Chowla, Vosper, Kemperman and a more recent result due to Rodseth and one of the authors.Comment: Submitted to Combinatorica, 23 pages, revised versio

Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function

We develop an analog of classical oscillation theory for Sturm-Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the finiteness of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established.Comment: 26 page

Local chromatic number of quadrangulations of surfaces

The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number

Small doubling in groups

Let A be a subset of a group G = (G,.). We will survey the theory of sets A with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous Freiman--Ruzsa theorem.Comment: 23 pages, survey article submitted to Proceedings of the Erdos Centenary conferenc

Handwritten Music Recognition for Mensural notation with convolutional recurrent neural networks

[EN] Optical Music Recognition is the technology that allows computers to read music notation, which is also referred to as Handwritten Music Recognition when it is applied over handwritten notation. This technology aims at efficiently transcribing written music into a representation that can be further processed by a computer. This is of special interest to transcribe the large amount of music written in early notations, such as the Mensural notation, since they represent largely unexplored heritage for the musicological community. Traditional approaches to this problem are based on complex strategies with many explicit rules that only work for one particular type of manuscript. Machine learning approaches offer the promise of generalizable solutions, based on learning from just labelled examples. However, previous research has not achieved sufficiently acceptable results for handwritten Mensural notation. In this work we propose the use of deep neural networks, namely convolutional recurrent neural networks, which have proved effective in other similar domains such as handwritten text recognition. Our experimental results achieve, for the first time, recognition results that can be considered effective for transcribing handwritten Mensural notation, decreasing the symbol-level error rate of previous approaches from 25.7% to 7.0%. (C) 2019 Elsevier B.V. All rights reserved.First author thanks the support from the Spanish Ministry "HISPAMUS" project (TIN2017-86576-R), partially funded by the EU. The other authors were supported by the European Union's H2020 grant "Recognition and Enrichment of Archival Documents" (Ref. 674943), by the BBVA Foundacion through the 2017-2018 and 2018-2019 Digital Humanities research grants "Carabela" and "HistWeather - Dos Siglos de Datos Cilmaticos", and by EU JPICH project "HOME - History Of Medieval Europe"(Spanish PEICTI Ref. PCI2018-093122).Calvo-Zaragoza, J.; Toselli, AH.; Vidal, E. (2019). Handwritten Music Recognition for Mensural notation with convolutional recurrent neural networks. Pattern Recognition Letters. 128:115-121. https://doi.org/10.1016/j.patrec.2019.08.021S11512112