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Müntz Legendre polynomials: Approximation properties and applications Math. Comp. (IF 2.2) Pub Date : 2024-05-11 Tengteng Cui, Chuanju Xu
The Müntz Legendre polynomials are a family of generalized orthogonal polynomials, defined by contour integral associated with a complex sequence Λ = { λ 0 , λ 1 , λ 2 , ⋯ } \Lambda =\{\lambda _{0},\lambda _{1},\lambda _{2},\cdots \} . In this paper, we are interested in two subclasses of the Müntz Legendre polynomials. Precisely, we theoretically and numerically investigate the basic approximation
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A random active set method for strictly convex quadratic problem with simple bounds Math. Comp. (IF 2.2) Pub Date : 2024-05-11 Ran Gu, Bing Gao
The active set method aims at finding the correct active set of the optimal solution and it is a powerful method for solving strictly convex quadratic problems with bound constraints. To guarantee the finite step convergence, existing active set methods all need strict conditions or some additional strategies, which can significantly impact the efficiency of the algorithm. In this paper, we propose
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Fourier optimization and Montgomery’s pair correlation conjecture Math. Comp. (IF 2.2) Pub Date : 2024-05-11 Emanuel Carneiro, Micah Milinovich, Antonio Pedro Ramos
Assuming the Riemann hypothesis, we improve the current upper and lower bounds for the average value of Montgomery’s function F ( α , T ) F(\alpha , T) over long intervals by means of a Fourier optimization framework. The function F ( α , T ) F(\alpha , T) is often used to study the pair correlation of the non-trivial zeros of the Riemann zeta-function. Two ideas play a central role in our approach:
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Virtual element methods for Biot–Kirchhoff poroelasticity Math. Comp. (IF 2.2) Pub Date : 2024-05-11 Rekha Khot, David Mora, Ricardo Ruiz-Baier
This paper analyses conforming and nonconforming virtual element formulations of arbitrary polynomial degrees on general polygonal meshes for the coupling of solid and fluid phases in deformable porous plates. The governing equations consist of one fourth-order equation for the transverse displacement of the middle surface coupled with a second-order equation for the pressure head relative to the solid
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Analysis of stochastic probing methods for estimating the trace of functions of sparse symmetric matrices Math. Comp. (IF 2.2) Pub Date : 2024-05-11 Andreas Frommer, Michele Rinelli, Marcel Schweitzer
We consider the problem of estimating the trace of a matrix function f ( A ) f(A) . In certain situations, in particular if f ( A ) f(A) cannot be well approximated by a low-rank matrix, combining probing methods based on graph colorings with stochastic trace estimation techniques can yield accurate approximations at moderate cost. So far, such methods have not been thoroughly analyzed, though, but
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Schinzel-type bounds for the Mahler measure on lemniscates Math. Comp. (IF 2.2) Pub Date : 2024-05-11 Ryan Looney, Igor Pritsker
We study the generalized Mahler measure on lemniscates, and prove a sharp lower bound for the measure of totally real integer polynomials that includes the classical result of Schinzel expressed in terms of the golden ratio. Moreover, we completely characterize many cases when this lower bound is attained. For example, we explicitly describe all lemniscates and the corresponding quadratic polynomials
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Extensible grid sampling for quantile estimation Math. Comp. (IF 2.2) Pub Date : 2024-05-11 Jingyu Tan, Zhijian He, Xiaoqun Wang
Quantiles are used as a measure of risk in many stochastic systems. We study the estimation of quantiles with the Hilbert space-filling curve (HSFC) sampling scheme that transforms specifically chosen one-dimensional points into high dimensional stratified samples while still remaining the extensibility. We study the convergence and asymptotic normality for the estimate based on HSFC. By a generalized
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Numerical solution of Poisson partial differential equation in high dimension using two-layer neural networks Math. Comp. (IF 2.2) Pub Date : 2024-05-11 Mathias Dus, Virginie Ehrlacher
The aim of this article is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of the high-dimensional Poisson partial differential equation with Neumann boundary condition. Using Barron’s representation of the solution [IEEE Trans. Inform. Theory 39 (1993), pp. 930–945] with a probability measure defined on the set of parameter values, the energy is
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Error bounds for Gauss–Jacobi quadrature of analytic functions on an ellipse Math. Comp. (IF 2.2) Pub Date : 2024-05-11 Hiroshi Sugiura, Takemitsu Hasegawa
For the Gauss–Jacobi quadrature on [ − 1 , 1 ] [-1,1] , the location is estimated where the kernel of the error functional for functions analytic on an ellipse and its interior in the complex plane attains its maximum modulus. For the Jacobi weight ( 1 − t ) α ( 1 + t ) β (1-t)^\alpha (1+t)^\beta ( α > − 1 \alpha >-1 , β > − 1 \beta >-1 ) except for the Gegenbauer weight ( α = β \alpha =\beta ), the
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Uniform error estimate of an asymptotic preserving scheme for the Lévy-Fokker-Planck equation Math. Comp. (IF 2.2) Pub Date : 2024-05-01 Weiran Sun, Li Wang
We establish a uniform-in-scaling error estimate for the asymptotic preserving (AP) scheme proposed by Xu and Wang [Commun. Math. Sci. 21 (2023), pp. 1–23] for the Lévy-Fokker-Planck (LFP) equation. The main difficulties stem not only from the interplay between the scaling and numerical parameters but also the slow decay of the tail of the equilibrium state. We tackle these problems by separating the
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Optimal analysis of finite element methods for the stochastic Stokes equations Math. Comp. (IF 2.2) Pub Date : 2024-04-29 Buyang Li, Shu Ma, Weiwei Sun
Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations in the L ∞ ( 0 , T ; L 2 ( Ω ; L 2 ) ) L^\infty (0, T; L^2(\Omega ; L^2)) norm all suffer from the order reduction with respect to the
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Towards a classification of isolated 𝑗-invariants Math. Comp. (IF 2.2) Pub Date : 2024-04-25 Abbey Bourdon, Sachi Hashimoto, Timo Keller, Zev Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, Himanshu Shukla
We develop an algorithm to test whether a non-complex multiplication elliptic curve E / Q E/\mathbf {Q} gives rise to an isolated point of any degree on any modular curve of the form X 1 ( N ) X_1(N) . This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to E E . Running this algorithm on all elliptic curves presently in the
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Cellular approximations to the diagonal map Math. Comp. (IF 2.2) Pub Date : 2024-04-24 Khaled Alzobydi, Graham Ellis
We describe an elementary algorithm for recursively constructing diagonal approximations on those finite regular CW-complexes for which the closure of each cell can be explicitly collapsed to a point. The algorithm is based on the standard proof of the acyclic carrier theorem, made constructive through the use of explicit contracting homotopies. It can be used as a theoretical tool for constructing
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Low-regularity exponential-type integrators for the Zakharov system with rough data in all dimensions Math. Comp. (IF 2.2) Pub Date : 2024-04-22 Hang Li, Chunmei Su
We propose and analyze a type of low-regularity exponential-type integrators (LREIs) for the Zakharov system (ZS) with rough solutions. Our LREIs include a first-order integrator and a second-order one, and they achieve optimal convergence under weaker regularity assumptions on the exact solution compared to the existing numerical methods in literature. Specifically, the first-order integrator exhibits
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Polynomial preserving recovery for the finite volume element methods under simplex meshes Math. Comp. (IF 2.2) Pub Date : 2024-04-19 Yonghai Li, Peng Yang, Zhimin Zhang
The recovered gradient, using the polynomial preserving recovery (PPR), is constructed for the finite volume element method (FVEM) under simplex meshes. Regarding the main results of this paper, there are two aspects. Firstly, we investigate the supercloseness property of the FVEM, specifically examining the quadratic FVEM under tetrahedral meshes. Secondly, we present several guidelines for selecting
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Adaptive fast multiplication of ℋ²-matrices Math. Comp. (IF 2.2) Pub Date : 2024-04-19 Steffen Börm
Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. H 2 \mathcal {H}^2 -matrices refine this representation following the ideas of fast multipole methods in order to achieve linear, i.e., optimal complexity for a variety of important algorithms. The matrix multiplication, a key component of many more advanced
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Convergence of the numerical approximations and well-posedness: Nonlocal conservation laws with rough flux Math. Comp. (IF 2.2) Pub Date : 2024-04-19 Aekta Aggarwal, Ganesh Vaidya
We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow. The discontinuous coefficient of the flux function is assumed to be of bounded variation (BV) and bounded away from zero, and hence the spatial discontinuities of the flux function can be infinitely many with possible accumulation points. Strong compactness of the Godunov and
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Super-polynomial accuracy of multidimensional randomized nets using the median-of-means Math. Comp. (IF 2.2) Pub Date : 2024-04-18 Zexin Pan, Art Owen
We study approximate integration of a function f f over [ 0 , 1 ] s [0,1]^s based on taking the median of 2 r − 1 2r-1 integral estimates derived from independently randomized ( t , m , s ) (t,m,s) -nets in base 2 2 . The nets are randomized by Matousek’s random linear scramble with a random digital shift. If f f is analytic over [ 0 , 1 ] s [0,1]^s , then the probability that any one randomized net’s
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Discrete tensor product BGG sequences: Splines and finite elements Math. Comp. (IF 2.2) Pub Date : 2024-04-17 Francesca Bonizzoni, Kaibo Hu, Guido Kanschat, Duygu Sap
In this paper, we provide a systematic discretization of the Bernstein-Gelfand-Gelfand diagrams and complexes over cubical meshes in arbitrary dimension via the use of tensor product structures of one-dimensional piecewise-polynomial spaces, such as spline and finite element spaces. We demonstrate the construction of the Hessian, the elasticity, and div div \operatorname {div}\operatorname {div}
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Wavenumber-explicit stability and convergence analysis of ℎ𝑝 finite element discretizations of Helmholtz problems in piecewise smooth media Math. Comp. (IF 2.2) Pub Date : 2024-03-29 M. Bernkopf, T. Chaumont-Frelet, J. Melenk
We present a wavenumber-explicit convergence analysis of the h p hp Finite Element Method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber k k . Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers.
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Convergence proof for the GenCol algorithm in the case of two-marginal optimal transport Math. Comp. (IF 2.2) Pub Date : 2024-03-26 Gero Friesecke, Maximilian Penka
The recently introduced Genetic Column Generation (GenCol) algorithm has been numerically observed to efficiently and accurately compute high-dimensional optimal transport (OT) plans for general multi-marginal problems, but theoretical results on the algorithm have hitherto been lacking. The algorithm solves the OT linear program on a dynamically updated low-dimensional submanifold consisting of sparse
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Six-dimensional sphere packing and linear programming Math. Comp. (IF 2.2) Pub Date : 2024-03-20 Matthew de Courcy-Ireland, Maria Dostert, Maryna Viazovska
We prove that the Cohn–Elkies linear programming bound for sphere packing is not sharp in dimension 6. The proof uses duality and optimization over a space of modular forms, generalizing a construction of Cohn–Triantafillou [Math. Comp. 91 (2021), pp. 491–508] to the case of odd weight and non-trivial character.
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Convergence of Langevin-simulated annealing algorithms with multiplicative noise Math. Comp. (IF 2.2) Pub Date : 2024-03-15 Pierre Bras, Gilles Pagès
We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for V : R d → R V : \mathbb {R}^d \to \mathbb {R} a potential function to minimize, we consider the stochastic differential equation d Y t = − σ σ ⊤ ∇ V ( Y t ) dY_t = - \sigma \sigma ^\top \nabla V(Y_t) d t + a ( t ) σ ( Y t ) d W t + a ( t ) 2 Υ ( Y t ) d t dt + a(t)\sigma (Y_t)dW_t + a(t)^2\Upsilon
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Stochastic nested primal-dual method for nonconvex constrained composition optimization Math. Comp. (IF 2.2) Pub Date : 2024-03-13 Lingzi Jin, Xiao Wang
In this paper we study the nonconvex constrained composition optimization, in which the objective contains a composition of two expected-value functions whose accurate information is normally expensive to calculate. We propose a STochastic nEsted Primal-dual (STEP) method for such problems. In each iteration, with an auxiliary variable introduced to track the inner layer function values we compute
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Uniform accuracy of implicit-explicit Runge-Kutta (IMEX-RK) schemes for hyperbolic systems with relaxation Math. Comp. (IF 2.2) Pub Date : 2024-03-13 Jingwei Hu, Ruiwen Shu
Implicit-explicit Runge-Kutta (IMEX-RK) schemes are popular methods to treat multiscale equations that contain a stiff part and a non-stiff part, where the stiff part is characterized by a small parameter ε \varepsilon . In this work, we prove rigorously the uniform stability and uniform accuracy of a class of IMEX-RK schemes for a linear hyperbolic system with stiff relaxation. The result we obtain
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Large-scale Monte Carlo simulations for zeros in character tables of symmetric groups Math. Comp. (IF 2.2) Pub Date : 2024-03-09 Alexander Miller, Danny Scheinerman
This is a brief report on some recent large-scale Monte Carlo simulations for approximating the density of zeros in character tables of large symmetric groups. Previous computations suggested that a large fraction of zeros cannot be explained by classical vanishing results. Our computations eclipse previous ones and suggest that the opposite is true. In fact, we find empirically that almost all of
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Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms Math. Comp. (IF 2.2) Pub Date : 2024-03-09 T. Hangelbroek, C. Rieger
Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal rates for the whole RKHS are often known. Schaback added the doubling trick [Math. Comp. 68 (1999), pp. 201–216], which shows that functions having double the smoothness
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On discrete ground states of rotating Bose–Einstein condensates Math. Comp. (IF 2.2) Pub Date : 2024-03-09 Patrick Henning, Mahima Yadav
The ground states of Bose–Einstein condensates in a rotating frame can be described as constrained minimizers of the Gross–Pitaevskii energy functional with an angular momentum term. In this paper we consider the corresponding discrete minimization problem in Lagrange finite element spaces of arbitrary polynomial order and we investigate the approximation properties of discrete ground states. In particular
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CM points, class numbers, and the Mahler measures of 𝑥³+𝑦³+1-𝑘𝑥𝑦 Math. Comp. (IF 2.2) Pub Date : 2024-03-09 Zhengyu Tao, Xuejun Guo
We study the Mahler measures of the polynomial family Q k ( x , y ) = x 3 + y 3 + 1 − k x y Q_k(x,y) = x^3+y^3+1-kxy using the method previously developed by the authors. An algorithm is implemented to search for complex multiplication points with class numbers ⩽ 3 \leqslant 3 , we employ these points to derive interesting formulas that link the Mahler measures of Q k ( x , y ) Q_k(x,y) to L L -values
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GMRES, pseudospectra, and Crouzeix’s conjecture for shifted and scaled Ginibre matrices Math. Comp. (IF 2.2) Pub Date : 2024-03-09 Tyler Chen, Anne Greenbaum, Thomas Trogdon
We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted N × N N\times N matrix whose entries are independent complex Gaussians. When the right-hand side of this linear system is independent of this random matrix, the N → ∞ N\to \infty behavior of the GMRES residual error can be determined exactly. To handle cases where the right hand side depends on the random
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Stochastic alternating structure-adapted proximal gradient descent method with variance reduction for nonconvex nonsmooth optimization Math. Comp. (IF 2.2) Pub Date : 2024-03-08 Zehui Jia, Wenxing Zhang, Xingju Cai, Deren Han
The blocky optimization has gained a significant amount of attention in far-reaching practical applications. Following the recent work (M. Nikolova and P. Tan [SIAM J. Optim. 29 (2019), pp. 2053–2078]) on solving a class of nonconvex nonsmooth optimization, we develop a stochastic alternating structure-adapted proximal (s-ASAP) gradient descent method for solving blocky optimization problems. By deploying
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Numerical analysis of a mixed-dimensional poromechanical model with frictionless contact at matrix–fracture interfaces Math. Comp. (IF 2.2) Pub Date : 2024-03-07 Francesco Bonaldi, Jérôme Droniou, Roland Masson
We present a complete numerical analysis for a general discretization of a coupled flow–mechanics model in fractured porous media, considering single-phase flows and including frictionless contact at matrix–fracture interfaces, as well as nonlinear poromechanical coupling. Fractures are described as planar surfaces, yielding the so-called mixed- or hybrid-dimensional models. Small displacements and
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Analysis of the boundary conditions for the ultraweak-local discontinuous Galerkin method of time-dependent linear fourth-order problems Math. Comp. (IF 2.2) Pub Date : 2024-02-22 Fengyu Fu, Chi-Wang Shu, Qi Tao, Boying Wu
In this paper, we study the ultraweak-local discontinuous Galerkin (UWLDG) method for time-dependent linear fourth-order problems with four types of boundary conditions. In one dimension and two dimensions, stability and optimal error estimates of order k + 1 k+1 are derived for the UWLDG scheme with polynomials of degree at most k k ( k ≥ 1 k\ge 1 ) for solving initial-boundary value problems. The
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Generalized Pohst inequality and small regulators Math. Comp. (IF 2.2) Pub Date : 2024-02-22 Francesco Battistoni, Giuseppe Molteni
Current methods for the classification of number fields with small regulator depend mainly on an upper bound for the discriminant, which can be improved by looking for the best possible upper bound of a specific polynomial function over a hypercube. In this paper, we provide new and effective upper bounds for the case of fields with one complex embedding and degree between five and nine: this is done
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A new div-div-conforming symmetric tensor finite element space with applications to the biharmonic equation Math. Comp. (IF 2.2) Pub Date : 2024-02-22 Long Chen, Xuehai Huang
A new H ( div div ) H(\operatorname {div}\operatorname {div}) -conforming finite element is presented, which avoids the need for supersmoothness by redistributing the degrees of freedom to edges and faces. This leads to a hybridizable mixed method with superconvergence for the biharmonic equation. Moreover, new finite element divdiv complexes are established. Finally, new weak Galerkin and C 0 C^0
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Double-variable trace maximization for extreme generalized singular quartets of a matrix pair: A geometric method Math. Comp. (IF 2.2) Pub Date : 2024-02-16 Wei-Wei Xu, Zheng-Jian Bai
In this paper, we consider the problem of computing an arbitrary generalized singular value of a Grassman or real matrix pair and a triplet of associated generalized singular vectors. Based on the QR factorization, the problem is reformulated as two novel trace maximization problems, each of which has double variables with unitary constraints or orthogonal constraints. Theoretically, we show that the
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Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping Math. Comp. (IF 2.2) Pub Date : 2024-02-14 Katherine Baker, Lehel Banjai, Mariya Ptashnyk
We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of nonlocal in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and
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A posteriori error estimates for the Richards equation Math. Comp. (IF 2.2) Pub Date : 2024-02-14 K. Mitra, M. Vohralík
The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection–reaction–diffusion equation that exhibits both parabolic–hyperbolic and parabolic–elliptic kind of degeneracies. In this study, we provide reliable, fully computable, and locally space–time efficient a posteriori
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High-order splitting finite element methods for the subdiffusion equation with limited smoothing property Math. Comp. (IF 2.2) Pub Date : 2024-02-07 Buyang Li, Zongze Yang, Zhi Zhou
In contrast with the diffusion equation which smoothens the initial data to C ∞ C^\infty for t > 0 t>0 (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is
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Construction of diagonal quintic threefolds with infinitely many rational points Math. Comp. (IF 2.2) Pub Date : 2024-02-07 Maciej Ulas
In this note we present a construction of an infinite family of diagonal quintic threefolds defined over Q \mathbb {Q} each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples B = ( B 0 , B 1 , B 2 , B 3 ) B=(B_{0}, B_{1}, B_{2}, B_{3}) of co-prime integers such that for a suitable chosen integer b b (depending on B B ), the equation B 0
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Quinary forms and paramodular forms Math. Comp. (IF 2.2) Pub Date : 2024-02-07 N. Dummigan, A. Pacetti, G. Rama, G. Tornaría
We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of Rösner and Weissauer
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Effective homology and periods of complex projective hypersurfaces Math. Comp. (IF 2.2) Pub Date : 2024-02-07 Pierre Lairez, Eric Pichon-Pharabod, Pierre Vanhove
We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwines with a new method for computing an explicit basis of the singular homology of the hypersurface. It is based on Picard–Lefschetz theory and relies on the computation of the monodromy action induced by a one-parameter family of hyperplane sections on the homology of a given section
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Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations Math. Comp. (IF 2.2) Pub Date : 2024-02-07 Constantin Carle, Marlis Hochbruck
This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe
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Energy diminishing implicit-explicit Runge–Kutta methods for gradient flows Math. Comp. (IF 2.2) Pub Date : 2024-02-07 Zhaohui Fu, Tao Tang, Jiang Yang
This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge–Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique
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Convergence, finiteness and periodicity of several new algorithms of 𝑝-adic continued fractions Math. Comp. (IF 2.2) Pub Date : 2024-01-25 Zhaonan Wang, Yingpu Deng
Classical continued fractions can be introduced in the field of p p -adic numbers, where p p -adic continued fractions offer novel perspectives on number representation and approximation. While numerous p p -adic continued fraction expansion algorithms have been proposed by the researchers, the establishment of several excellent properties, such as the Lagrange’s Theorem for classic continued fractions
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Explicit calculations for Sono’s multidimensional sieve of 𝐸₂-numbers Math. Comp. (IF 2.2) Pub Date : 2024-01-23 Daniel Goldston, Apoorva Panidapu, Jordan Schettler
We derive explicit formulas for integrals of certain symmetric polynomials used in Keiju Sono’s multidimensional sieve of E 2 E_2 -numbers, i.e., integers which are products of two distinct primes. We use these computations to produce the currently best-known bounds for gaps between multiple E 2 E_2 -numbers. For example, we show there are infinitely many occurrences of four E 2 E_2 -numbers within
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Few hamiltonian cycles in graphs with one or two vertex degrees Math. Comp. (IF 2.2) Pub Date : 2024-01-17 Jan Goedgebeur, Jorik Jooken, On-Hei Solomon Lo, Ben Seamone, Carol Zamfirescu
Inspired by Sheehan’s conjecture that no 4 4 -regular graph contains exactly one hamiltonian cycle, we prove results on hamiltonian cycles in regular graphs and nearly regular graphs. We fully disprove a conjecture of Haythorpe on the minimum number of hamiltonian cycles in regular hamiltonian graphs, thereby extending a result of Zamfirescu, as well as correct and complement Haythorpe’s computational
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Identifying the source term in the potential equation with weighted sparsity regularization Math. Comp. (IF 2.2) Pub Date : 2024-01-17 Ole Elvetun, Bjørn Nielsen
We explore the possibility for using boundary measurements to recover a sparse source term f ( x ) f(x) in the potential equation. Employing weighted sparsity regularization and standard results for subgradients, we derive simple-to-check criteria which assure that a number of sinks ( f ( x ) > 0 f(x)>0 ) and sources ( f ( x ) > 0 f(x)>0 ) can be identified. Furthermore, we present two cases for which
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On median filters for motion by mean curvature Math. Comp. (IF 2.2) Pub Date : 2024-01-17 Selim Esedoḡlu, Jiajia Guo, David Li
The median filter scheme is an elegant, monotone discretization of the level set formulation of motion by mean curvature. It turns out to evolve every level set of the initial condition precisely by another class of methods known as threshold dynamics. Median filters are, in other words, the natural level set versions of threshold dynamics algorithms. Exploiting this connection, we revisit median filters
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Convergence analysis of Laguerre approximations for analytic functions Math. Comp. (IF 2.2) Pub Date : 2024-01-17 Haiyong Wang
Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree n n
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Faster truncated integer multiplication Math. Comp. (IF 2.2) Pub Date : 2024-01-10 David Harvey
We present new algorithms for computing the low n n bits or the high n n bits of the product of two n n -bit integers. We show that these problems may be solved in asymptotically 75 75% of the time required to compute the full 2 n 2n -bit product, assuming that the underlying integer multiplication algorithm relies on computing cyclic convolutions of sequences of real numbers.
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Rational group algebras of generalized strongly monomial groups: Primitive idempotents and units Math. Comp. (IF 2.2) Pub Date : 2024-01-03 Gurmeet Bakshi, Jyoti Garg, Gabriela Olteanu
We present a method to explicitly compute a complete set of orthogonal primitive idempotents in a simple component with Schur index 1 of a rational group algebra Q G \mathbb {Q}G for G G a finite generalized strongly monomial group. For the same groups with no exceptional simple components in Q G \mathbb {Q}G , we describe a subgroup of finite index in the group of units U ( Z G ) \mathcal {U}(\mathbb
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Kac-Rice formulas and the number of solutions of parametrized systems of polynomial equations Math. Comp. (IF 2.2) Pub Date : 2022-08-11 Elisenda Feliu, AmirHosein Sadeghimanesh
Abstract:Kac-Rice formulas express the expected number of elements a fiber of a random field has in terms of a multivariate integral. We consider here parametrized systems of polynomial equations that are linear in enough parameters, and provide a Kac-Rice formula for the expected number of solutions of the system when the parameters follow continuous distributions. Combined with Monte Carlo integration
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Finite element/holomorphic operator function method for the transmission eigenvalue problem Math. Comp. (IF 2.2) Pub Date : 2022-08-11 Bo Gong, Jiguang Sun, Tiara Turner, Chunxiong Zheng
Abstract:The transmission eigenvalue problem arises from the inverse scattering theory for inhomogeneous media. It plays a key role in the unique determination of inhomogeneous media. Furthermore, transmission eigenvalues can be reconstructed from the scattering data and used to estimate the material properties of the unknown object. The problem is posted as a system of two second order partial differential
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Explicit Tamagawa numbers for certain algebraic tori over number fields Math. Comp. (IF 2.2) Pub Date : 2022-08-09 Thomas Rüd
Abstract:Given a number field extension $K/k$ with an intermediate field $K^+$ fixed by a central element of $\operatorname {Gal}(K/k)$ of prime order $p$, there exists an algebraic torus over $k$ whose rational points are elements of $K^\times$ sent to $k^\times$ by the norm map $N_{K/K^+}$. The goal is to compute the Tamagawa number such a torus explicitly via Ono’s formula that expresses it as a
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Recovery of Sobolev functions restricted to iid sampling Math. Comp. (IF 2.2) Pub Date : 2022-08-09 David Krieg, Erich Novak, Mathias Sonnleitner
Abstract:We study $L_q$-approximation and integration for functions from the Sobolev space $W^s_p(\Omega )$ and compare optimal randomized (Monte Carlo) algorithms with algorithms that can only use identically distributed (iid) sample points, uniformly distributed on the domain. The main result is that we obtain the same optimal rate of convergence if we restrict to iid sampling, a common assumption
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Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate Math. Comp. (IF 2.2) Pub Date : 2022-08-05 Josef Dick, Takashi Goda, Kosuke Suzuki
Abstract:We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich [J. Approx. Theory 240 (2019), pp. 96–113] shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted
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ℚ-Curves, Hecke characters and some Diophantine equations Math. Comp. (IF 2.2) Pub Date : 2022-08-04 Ariel Pacetti, Lucas Villagra Torcomian
Abstract:In this article we study the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$ for positive square-free values of $d$. A Frey curve over $\mathbb {Q}(\sqrt {-d})$ is attached to each primitive solution, which happens to be a $\mathbb {Q}$-curve. Our main result is the construction of a Hecke character $\chi$ satisfying that the Frey elliptic curve representation twisted by $\chi$ extends to $Gal_\mathbb
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Improved computation of fundamental domains for arithmetic Fuchsian groups Math. Comp. (IF 2.2) Pub Date : 2022-08-03 James Rickards
Abstract:A practical algorithm to compute the fundamental domain of an arithmetic Fuchsian group was given by Voight, and implemented in Magma. It was later expanded by Page to the case of arithmetic Kleinian groups. We combine and improve on parts of both algorithms to produce a more efficient algorithm for arithmetic Fuchsian groups. This algorithm is implemented in PARI/GP, and we demonstrate the
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Anti-Gaussian quadrature formulae of Chebyshev type Math. Comp. (IF 2.2) Pub Date : 2022-08-03 Sotirios Notaris
Abstract:We prove that there is no positive measure $d\sigma$ on the interval $[a,b]$ such that the corresponding anti-Gaussian quadrature formula is also a Chebyshev quadrature formula. We also show that the only positive and even measure $d\sigma (t)=d\sigma (-t)$ on the symmetric interval $[-a,a]$, for which the anti-Gaussian formula has the form $\int _{-a}^{a}f(t)d\sigma (t)=\frac {\mu _{0}}{