We show that an asymptotic property of the determinants of certain matrices whose entries are finite sums of cotangents with rational arguments is equivalent to the GRH for odd Dirichlet characters. This is th...
Authors: John Lewis and Don Zagier
Citation:Research in the Mathematical Sciences
2018
6:4
The present paper introduces a novel notion of ‘(effective) computability,’ called viability, of strategies in game semantics in an intrinsic (i.e., without recourse to the standard Church–Turing computability),
Authors: Norihiro Yamada
Citation:Research in the Mathematical Sciences
2018
6:3
We study the general properties of certain rank 4 rigid local systems considered by Goursat. We analyze when they are irreducible, give an explicit integral description as well as the invariant Hermitian form H w...
Authors: Danylo Radchenko and Fernando Rodriguez Villegas
Citation:Research in the Mathematical Sciences
2018
5:38
We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp...
Authors: Francis Brown
Citation:Research in the Mathematical Sciences
2018
5:34
In this paper, we address the following two closely related questions. First, we complete the classification of finite symmetry groups of type IIA string theory on
...
Authors: Miranda C. N. Cheng, Sarah M. Harrison, Roberto Volpato and Max Zimet
Citation:Research in the Mathematical Sciences
2018
5:32
Given a tensor f in a Euclidean tensor space, we are interested in the critical points of the distance function from f to the set of tensors of rank at most k, which we call the critical rank-at-most-k tensors fo...
Authors: Jan Draisma, Giorgio Ottaviani and Alicia Tocino
Citation:Research in the Mathematical Sciences
2018
5:27
Period polynomials have long been fruitful tools for the study of values of L-functions in the context of major outstanding conjectures. In this paper, we survey some facets of this study from the perspective of ...
Authors: Nikolaos Diamantis and Larry Rolen
Citation:Research in the Mathematical Sciences
2018
5:9
This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite disti...
Authors: Francis Brown
Citation:Research in the Mathematical Sciences
2018
5:7
This is a slightly expanded version of my lecture at the conference Modular forms are everywhere at Bonn, May 2017, taking into account remarks by Don Zagier and Shaul Zemel, and suggestions of the referees.
Authors: Roelof Bruggeman
Citation:Research in the Mathematical Sciences
2018
5:5
We prove that certain degenerate abelian varieties that include compactified Jacobians, namely stable semiabelic varieties, satisfy autoduality. We establish this result by proving a comparison theorem that re...
Authors: Jesse Leo Kass
Citation:Research in the Mathematical Sciences
2017
4:27
In this paper, we discuss partial differential equations with multiple scales for which scale resolution is needed in some subregions, while a separation of scale and numerical homogenization is possible in th...
Authors: Assyr Abdulle and Orane Jecker
Citation:Research in the Mathematical Sciences
2017
4:28
We introduce the sparse operator compression to compress a self-adjoint higher-order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using loc...
Authors: Thomas Y. Hou and Pengchuan Zhang
Citation:Research in the Mathematical Sciences
2017
4:24
How heterogeneous multiscale methods (HMM) handle fluctuations acting on the slow variables in fast–slow systems is investigated. In particular, it is shown via analysis of central limit theorem (CLT) and larg...
Authors: David Kelly and Eric Vanden-Eijnden
Citation:Research in the Mathematical Sciences
2017
4:23
The Laplace–Beltrami operator, a fundamental object associated with Riemannian manifolds, encodes all intrinsic geometry of manifolds and has many desirable properties. Recently, we proposed the point integral...
Authors: Zuoqiang Shi and Jian Sun
Citation:Research in the Mathematical Sciences
2017
4:22
We investigate the impact of adding inner nodes for a Filon-type method for highly oscillatory quadrature. The error of Filon-type method is composed of asymptotic and interpolation errors, and the interplay b...
Authors: Jing Gao and Arieh Iserles
Citation:Research in the Mathematical Sciences
2017
4:21
We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to
...
Authors: Olivia Beckwith
Citation:Research in the Mathematical Sciences
2017
4:20
We propose a new algorithm for solving Helmholtz equations in exterior domains with implicitly represented boundaries. The algorithm not only combines the advantages of implicit surface representation and the ...
Authors: Chieh Chen and Richard Tsai
Citation:Research in the Mathematical Sciences
2017
4:19
A fourientation of a graph is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. Fixing a total order on the edges and a reference orienta...
Authors: Spencer Backman and Sam Hopkins
Citation:Research in the Mathematical Sciences
2017
4:18
In this paper we study the stochastic Galerkin approximation for the linear transport equation with random inputs and diffusive scaling. We first establish uniform (in the Knudsen number) stability results in ...
Authors: Shi Jin, Jian-Guo Liu and Zheng Ma
Citation:Research in the Mathematical Sciences
2017
4:15
This paper describes versions of OPAL, the Occam-Plausibility Algorithm (Farrell et al. in J Comput Phys 295:189–208, 2015) in which the use of Bayesian model plausibilities is replaced with information-theoretic...
Authors: Kathryn Farrell-Maupin and J. T. Oden
Citation:Research in the Mathematical Sciences
2017
4:14
A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: For any prime p, reduce its coefficients mod p and consider its action on the field
...
Authors: Andrew Bridy and Derek Garton
Citation:Research in the Mathematical Sciences
2017
4:13
What is the probability that the smallest part of a random integer partition of N is odd? What is the expected value of the smallest part of a random integer partition of N? It is straightforward to see that the ...
Authors: Hieu T. Ngo and Robert C. Rhoades
Citation:Research in the Mathematical Sciences
2017
4:17
The hierarchical interpolative factorization (HIF) offers an efficient way for solving or preconditioning elliptic partial differential equations. By exploiting locality and low-rank properties of the operator...
Authors: Yingzhou Li and Lexing Ying
Citation:Research in the Mathematical Sciences
2017
4:12
We implement new techniques involving Artin fans to study the realizability of tropical stable maps in superabundant combinatorial types. Our approach is to understand the skeleton of a fundamental object in l...
Authors: Dhruv Ranganathan
Citation:Research in the Mathematical Sciences
2017
4:11
For a prime number p, we study the zeros modulo p of divisor polynomials of rational elliptic curves E of conductor p. Ono (CBMS regional conference series in mathematics, 2003, vol 102, p. 118) made the observat...
Authors: Matija Kazalicki and Daniel Kohen
Citation:Research in the Mathematical Sciences
2017
4:10
We present a ray-based finite element method for the high-frequency Helmholtz equation in smooth media, whose basis is learned adaptively from the medium and source. The method requires a fixed number of grid ...
Authors: Jun Fang, Jianliang Qian, Leonardo Zepeda-Núñez and Hongkai Zhao
Citation:Research in the Mathematical Sciences
2017
4:9
Given a family X of complex varieties degenerating over a punctured disk, one is interested in computing related invariants called the motivic nearby fiber and the refined limit mixed Hodge numbers, both of which...
Authors: Alan Stapledon
Citation:Research in the Mathematical Sciences
2017
4:8
We investigate the effect of anisotropic elastic energy on defect patterns of liquid crystals confined in a three-dimensional spherical domain within the framework of Landau–de Gennes model. Two typical strong...
Authors: Dong An, Wei Wang and Pingwen Zhang
Citation:Research in the Mathematical Sciences
2017
4:7
Bringmann et al. (Trans Am Math Soc 364(5):2393–2410, 2012) showed how to ‘regularize’ mock modular forms by a certain linear combination of the Eichler integral of their shadows in order to obtain p-adic modular...
Authors: Luca Candelori and Francesc Castella
Citation:Research in the Mathematical Sciences
2017
4:5
We present a spectrally accurate method for the rapid evaluation of free-space Stokes potentials, i.e., sums involving a large number of free space Green’s functions. We consider sums involving stokeslets, str...
Authors: Ludvig af Klinteberg, Davoud Saffar Shamshirgar and Anna-Karin Tornberg
Citation:Research in the Mathematical Sciences
2017
4:1
We study the qualitative homogenization of second-order Hamilton–Jacobi equations in space-time stationary ergodic random environments. Assuming that the Hamiltonian is convex and superquadratic in the momentu...
Authors: Wenjia Jing, Panagiotis E. Souganidis and Hung V. Tran
Citation:Research in the Mathematical Sciences
2017
4:6
We consider high-frequency waves satisfying the scalar wave equation with highly oscillatory initial data. The wave speed, and the phase and amplitude of the initial data are assumed to be uncertain, described...
Authors: G. Malenová, M. Motamed and O. Runborg
Citation:Research in the Mathematical Sciences
2017
4:3
In this paper, we prove the existence of an infinite-dimensional graded supermodule for the finite sporadic Thompson group Th whose McKay–Thompson series are weakly holomorphic modular forms of weight
Authors: Michael J. Griffin and Michael H. Mertens
Citation:Research in the Mathematical Sciences
2016
3:36
In this work, we study the convergence of an efficient iterative method, the fast sweeping method (FSM), for numerically solving static convex Hamilton–Jacobi equations. First, we show the convergence of the F...
Authors: Songting Luo and Hongkai Zhao
Citation:Research in the Mathematical Sciences
2016
3:35
We investigate Poincaré 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 ser...
Authors: Kathrin Bringmann, Olav K. Richter and Martin Westerholt-Raum
Citation:Research in the Mathematical Sciences
2016
3:30
In this paper, we provide an explicit construction of weight 0 meromorphic modular forms. Following work of Petersson, we build these via Poincaré series. There are two main aspects of our investigation which ...
Authors: Kathrin Bringmann and Ben Kane
Citation:Research in the Mathematical Sciences
2016
3:24