Mathematicshttps://qmro.qmul.ac.uk/xmlui/handle/123456789/336972019-02-16T07:27:32Z2019-02-16T07:27:32ZExplicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree levelDickson, MPitale, ASaha, ASchmidt, Rhttps://qmro.qmul.ac.uk/xmlui/handle/123456789/553082019-02-14T11:23:30Z2019-01-01T00:00:00ZExplicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level
Dickson, M; Pitale, A; Saha, A; Schmidt, R
We formulate an explicit refinement of B\"ocherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of L-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan-Gross-Prasad conjecture for Bessel periods as proposed by Yifeng Liu. We note several consequences of our conjecture to arithmetic and analytic properties of L-functions and Fourier coefficients of Siegel modular forms.
2019-01-01T00:00:00ZBeyond the clustering coefficient: A topological analysis of node neighbourhoods in complex networksKartun-Giles, APBianconi, Ghttps://qmro.qmul.ac.uk/xmlui/handle/123456789/552882019-02-13T11:49:31Z2019-01-01T00:00:00ZBeyond the clustering coefficient: A topological analysis of node neighbourhoods in complex networks
Kartun-Giles, AP; Bianconi, G
In Network Science node neighbourhoods, also called ego-centered networks have attracted large attention. In particular the clustering coefficient has been extensively used to measure their local cohesiveness. In this paper, we show how, given two nodes with the same clustering coefficient, the topology of their neighbourhoods can be significantly different, which demonstrates the need to go beyond this simple characterization. We perform a large scale statistical analysis of the topology of node neighbourhoods of real networks by first constructing their clique complexes, and then computing their Betti numbers. We are able to show significant differences between the topology of node neighbourhoods of real networks and the stochastic topology of null models of random simplicial complexes revealing local organisation principles of the node neighbourhoods. Moreover we observe that a large scale statistical analysis of the topological properties of node neighbourhoods is able to clearly discriminate between power-law networks, and planar road networks.
2019-01-01T00:00:00ZThe density of states of 1D random band matrices via a supersymmetric transfer operatorDisertori, MLohmann, MSODIN, Ahttps://qmro.qmul.ac.uk/xmlui/handle/123456789/552322019-02-08T11:42:12Z2019-01-01T00:00:00ZThe density of states of 1D random band matrices via a supersymmetric transfer operator
Disertori, M; Lohmann, M; SODIN, A
Recently, T. and M. Shcherbina proved a pointwise semicircle law for the density of states of one-dimensional
Gaussian band matrices of large bandwidth. The main step of their proof is a new method to study the spectral
properties of non-self-adjoint operators in the semiclassical regime. The method is applied to a transfer operator
constructed from the supersymmetric integral representation for the density of states.
We present a simpler proof of a slightly upgraded version of the semicircle law, which requires only standard
semiclassical arguments and some peculiar elementary computations. The simplification is due to the use of supersymmetry, which manifests itself in the commutation between the transfer operator and a family of transformations
of superspace, and was applied earlier in the context of band matrices by Constantinescu. Other versions of this
supersymmetry have been a crucial ingredient in the study of the localization–delocalization transition by theoretical
physicists.
2019-01-01T00:00:00ZLarge deviation theory of percolation on multiplex networksBianconi, Ghttps://qmro.qmul.ac.uk/xmlui/handle/123456789/552292019-02-08T11:42:20Z2019-01-01T00:00:00ZLarge deviation theory of percolation on multiplex networks
Bianconi, G
Recently increasing attention has been addressed to the fluctuations observed in percolation defined in single and multiplex networks. These fluctuations are extremely important to characterize the robustness of real finite networks but cannot be captured by the traditionally adopted mean-field theory of percolation. Here we propose a theoretical framework and a message passing algorithm that is able to fully capture the large deviation of percolation in interdependent multiplex networks with a locally tree-like structure. This framework is here applied to study the robustness of single instance multiplex networks and compared to the results obtained using extensive simulations of the initial damage. For simplicity the method is here developed for interdependent multiplex networks without link overlap, however it can be generalized to treat multiplex networks with link overlap.
2019-01-01T00:00:00Z