Abstract:Let denote the error incurred by approximating the number of -free integers less than by . It is well known that , and widely conjectured that . By establishing weak linear independence of some subsets of zeros of the Riemann zeta function, we establish an effective proof of the lower bound, with significantly larger bounds on the constant compared to those obtained in prior work. For example

Abstract:We show that a positive proportion of the primes have the property that if any one of its digits in base , including its infinitely many leading 0 digits, is replaced by a different digit, then the resulting number is composite.

Abstract:We find the minimal value of the Fried average entropy by proving new lower bounds for regulators of totally real number fields.

Abstract:We give algorithms to compute isomorphism classes of ordinary abelian varieties defined over a finite field whose characteristic polynomial (of Frobenius) is square-free and of abelian varieties defined over the prime field whose characteristic polynomial is square-free and does not have real roots. In the ordinary case we are also able to compute the polarizations and the group of automorphisms

Abstract:We prove that a -dimensional selfdual Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to -dimensional -adic selfdual representations with the ground field not equal to . The proof makes use of the Faltings-Serre method

Abstract:We study the strong convergence of certain multidimensional continued fraction algorithms. In particular, in the two- and three-dimensional case, we prove that the second Lyapunov exponent of Selmer's algorithm is negative and bound it away from zero. Moreover, we give heuristic results on several other continued fraction algorithms. Our results indicate that all classical multidimensional

Abstract:We make explicit a theorem of Pintz, which gives a version of the prime number theorem with error term roughly square-root of that which was previously known. We apply this to a long-standing problem concerning an inequality studied by Ramanujan.

Abstract:We study and polynomials , called Newman and Littlewood polynomials, that have a prescribed number of zeros in the open unit disk . For every pair , where and , we prove that it is possible to find a -polynomial of degree with non-zero constant term , such that and on the unit circle . On the way to this goal, we answer a question of D. W. Boyd from 1986 on the smallest degree Newman polynomial

Abstract:In this article, we introduce the notion ``-probabilistic convergence" ( ) of stochastic Bernstein polynomials built upon order statistics of identically, independently, and uniformly distributed random variables on . We establish power and exponential convergence rates in terms of the modulus of continuity of a target function . For in the range we obtain Gaussian tail bounds for the corresponding

Abstract:In a recent paper by the same authors, we provided a theoretical foundation for the component-by-component (CBC) construction of lattice algorithms for multivariate approximation in the worst case setting, for functions in a periodic space with general weight parameters. The construction led to an error bound that achieves the best possible rate of convergence for lattice algorithms. Previously

Abstract:Many hyperbolic and kinetic equations contain a non-stiff convection/transport part and a stiff relaxation/collision part (characterized by the relaxation or mean free time ). To solve this type of problems, implicit-explicit (IMEX) multistep methods have been widely used and their performance is understood well in the non-stiff regime ( ) and limiting regime ( ). However, in the intermediate

Abstract:The first statement of Lemma 11 in our recent paper [KG18] (Math. Comp. 87 (2018), no. 314, 2611-2640) is incorrect: For the sequence of nested admissible partitions produced by the adaptive discontinuous Galerkin method (ADGM) we have , and . In the first line of the proof of [KG18, Lemma 11 on p. 2620], we used that where denotes the Lebesgue measure. This, however, is not true in general

Abstract:The focus of this work is a posteriori error estimation for stochastic Galerkin approximations of parameter-dependent linear elasticity equations. The starting point is a three-field partial differential equation model with the Young modulus represented as an affine function of a countable set of parameters. We introduce a weak formulation, establish its stability with respect to a weighted

Abstract:The so-called FEM-FCT (finite element method flux-corrected transport) scheme for evolutionary scalar convection-dominated equations leads in each time instant to a nonlinear problem. For sufficiently small time steps, the existence and uniqueness of a solution of these problems is shown. Moreover, the convergence of a semi-smooth Newton's method is studied.

Abstract:For a generalized Hodge Laplace equation, we prove the quasi-optimal convergence rate of an adaptive mixed finite element method. This adaptive method can control the error in the natural mixed variational norm when the space of harmonic forms is trivial. In particular, we obtain new quasi-optimal adaptive mixed methods for the Hodge Laplace, Poisson, and Stokes equations. Comparing to existing

Abstract:We propose a finite element scheme for numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker-Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general

Abstract:The Canham-Helfrich-Evans models of biomembranes consist of a family of geometric constrained variational problems. In this article, we compare two classes of numerical methods for these variational problems based on piecewise linear (PL) and subdivision surfaces (SS). Since SS methods are based on spline approximation and can be viewed as higher order versions of PL methods, one may expect

Abstract:In their previous work, the authors studied the abelian sandpile model on graphs constructed from a growing piece of a plane or space tiling, given periodic or open boundary conditions, and identified spectral parameters which govern the asymptotic spectral gap and asymptotic mixing time. This paper gives a general method of determining the spectral parameters either computationally or asymptotically

Abstract:A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical description is given by a witness collection, whose structure is more involved. We build on recent work to develop a toolkit for the numerical manipulation

Abstract:Previously, we established lower bounds for the usual measures (trace, length, Mahler measure) of totally positive algebraic integers, i.e., all of whose conjugates are positive real numbers. We used the method of explicit auxiliary functions and we noticed that the house of most of the totally positive polynomials involved in our functions are bounded by 5.8. Thanks to this observation, we

Abstract:We say a power series is multiplicative if the sequence is so. In this paper, we consider multiplicative power series such that is also multiplicative. We find a number of examples for which is a rational function or a theta series and prove that the complete set of solutions is the locus of a (probably reducible) affine variety over . The precise determination of this variety turns out to

Abstract:We consider a reconstruction problem for ``spike-train'' signals of an a priori known form from their moments We assume that the moments , , are known with an absolute error not exceeding . This problem is essentially equivalent to solving the Prony system We study the ``geometry of error amplification'' in reconstruction of from in situations where the nodes near-collide, i.e., form a cluster

Abstract:We present recent results on the existence of a continuous time limit for Ensemble Kalman Filter algorithms. In the setting of continuous signal and observation processes, we apply the original Ensemble Kalman Filter algorithm proposed by Burgers, van Leeuwen, and Evensen [Monthly Weather Review 126 (1998), pp. 1719-1724] as well as a recent variant of de Wiljes, Reich, and Stannat [SIAM J

Abstract:In this paper, we provide a detailed convergence analysis for a first order stabilized linear semi-implicit numerical scheme for the nonlocal Cahn-Hilliard equation, which follows from consistency and stability estimates for the numerical error function. Due to the complicated form of the nonlinear term, we adopt the discrete norm for the error function to establish the convergence result

Abstract:This paper addresses the three concepts of consistency, stability and convergence in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of ``balance laws''. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and

Abstract:Inspired by so-called TVD limiter-based second-order schemes for hyperbolic conservation laws, we develop a formally second-order accurate numerical method for multi-dimensional aggregation equations. The method allows for simulations to be continued after the first blow-up time of the solution. In the case of symmetric, -convex potentials with a possible Lipschitz singularity at the origin

Abstract:Analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media has been investigated extensively in the last several decades. Of particular interest in practical applications is the lowest-order Galerkin-mixed method, in which a linear Lagrange FE approximation is used for the concentration and the lowest-order Raviart-Thomas FE approximation is used for the velocity/pressure

Abstract:In this paper, we propose a simple way of constructing HDG+ projections on polyhedral elements. The projections enable us to analyze the Lehrenfeld-Schöberl HDG (HDG+) methods in a very concise manner, and make many existing analysis techniques of standard HDG methods reusable for HDG+. The novelty here is an alternative way of constructing the projections without using -decompositions as

Abstract:The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or computational fluid dynamics formulation, amounts to writing the optimal transport problem as the optimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. Several discretizations of this problem have been proposed, leading to

Abstract:Optimal transport (OT) problems arise in a wide range of applications, from physics to economics. Getting a numerical approximate solution of these problems is a challenging issue of practical importance. In this work, we investigate the relaxation of the OT problem when the marginal constraints are replaced by some moment constraints. Using Tchakaloff's theorem, we show that the moment constrained

Abstract:For a matrix , we provide an analytic formula that keeps track of an orthonormal basis for the range of under rank-one modifications. More precisely, we consider rank-one adaptations of a given with known matrix factorization , where is column-orthogonal and is invertible. Arguably, the most important methods that produce such factorizations are the singular value decomposition (SVD), where

Abstract:The aim of this paper is to derive pointwise global and local best approximation type error estimates for biharmonic problems using the interior penalty method. The analysis uses the technique of dyadic decompositions of the domain, which is assumed to be a convex polygon. The proofs require local energy estimates and new pointwise Green's function estimates for the continuous problem which

Abstract:We derive the implicit equations for certain parametric surfaces in three-dimensional projective space termed tensor product surfaces. Our method computes the implicit equation for such a surface based on the knowledge of the syzygies of the base point locus of the parametrization by means of constructing an explicit virtual projective resolution.

Abstract:In this paper, we develop a new discontinuous Galerkin method for solving several types of partial differential equations (PDEs) with high order spatial derivatives. We combine the advantages of a local discontinuous Galerkin (LDG) method and the ultraweak discontinuous Galerkin (UWDG) method. First, we rewrite the PDEs with high order spatial derivatives into a lower order system, then apply

Abstract:Most adaptive finite element strategies employ the Dörfler marking strategy to single out certain elements of a triangulation for refinement. In the literature, different algorithms have been proposed to construct , where usually two goals compete. On the one hand, should contain a minimal number of elements. On the other hand, one aims for linear costs with respect to the cardinality of

Abstract:A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract

Abstract:In this paper we consider numerical approximation to the generalised solutions to the Monge-Ampère type equations with general source terms. We first give some important propositions for the border of generalised solutions. Then, for both the classical and weak Dirichlet boundary conditions, we present well-posed numerical methods for the generalised solutions with general source terms. Finally

Abstract:We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the

Abstract:This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors represented through the density matrix, i.e., the orthogonal projector on the

Abstract:Bruin and Najman [LMS J. Comput. Math. 18 (2015), no. 1, 578-602] and Ozman and Siksek [Math. Comp. 88 (2019), no. 319, 2461-2484] have recently determined the quadratic points on each modular curve of genus 2, 3, 4, or 5 whose Mordell-Weil group has rank 0. In this paper we do the same for the of genus 2, 3, 4, and 5 and positive Mordell-Weil rank. The values of are 37, 43, 53, 61, 57, 65

Abstract:In this work, we consider the numerical integration of the nonlinear Dirac equation and the Dirac-Poisson system (NDEs) under rough initial data. We propose an ultra low-regularity integrator (ULI) for solving the NDEs which enables optimal first-order time convergence in for solutions in , i.e., without requiring any additional regularity on the solution. In contrast to classical methods

Abstract:We focus on establishing an algorithm to solve the tensor eigenvalue complementarity problem (TEiCP), and we have two contributions in this paper. First, a smoothing Newton-type algorithm is proposed for the TEiCP based on the CHKS smoothing function. Its global convergence is established under some mild conditions. Numerical experiments are reported to show that the proposed algorithm is

Abstract:Cartan matrices and quasi-Cartan matrices play an important role in such areas as Lie theory, representation theory, and algebraic graph theory. It is known that each (connected) positive definite quasi-Cartan matrix is -equivalent with the Cartan matrix of a Dynkin diagram, called the Dynkin type of . We present a symbolic, graph-theoretic algorithm to compute the Dynkin type of , of the

Abstract:We prove that the Galerkin finite element solution of the Laplace equation in a convex polyhedron , with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree , satisfies the following weak maximum principle: with a constant independent of the mesh size . By using this result, we show that the Ritz projection operator is stable in norm uniformly

Abstract:Quantiles and expected shortfalls are usually used to measure risks of stochastic systems, which are often estimated by Monte Carlo methods. This paper focuses on the use of the quasi-Monte Carlo (QMC) method, whose convergence rate is asymptotically better than Monte Carlo in the numerical integration. We first prove the convergence of QMC-based quantile estimates under very mild conditions

Abstract:An initial-boundary value problem with a Caputo time derivative of fractional order is considered, solutions of which typically exhibit a singular behaviour at an initial time. An L2-type discrete fractional-derivative operator of order is considered on nonuniform temporal meshes. Sufficient conditions for the inverse-monotonicity of this operator are established, which yields sharp pointwise-in-time

Abstract:We propose an algorithm to compute the GIT-fan for torus actions on affine varieties with symmetries. The algorithm combines computational techniques from commutative algebra, convex geometry, and group theory. We have implemented our algorithm in the SINGULAR library GITFAN.LIB. Using our implementation, we compute the Mori chamber decomposition of  .

Abstract:As a generalization of orthonormal wavelets in , tightframelets (also called tight wavelet frames) are of importance in wavelet analysis and applied sciences due to their many desirable properties in applications such as image processing and numerical algorithms. Tight framelets are often derived from particular refinable functions satisfying certain stringent conditions. Consequently, a large

Abstract:We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution, generalizing spherical harmonics to the surface of a cone, hyperboloid, and paraboloid. We use this construction to develop cubature and fast approximation

Abstract:We give a formula for the eventual multiplicities of irreducible representations appearing in a finitely presented -module over the rational numbers. The result relies on structure theory due to Sam-Snowden [Trans. Amer. Math. Soc. 146 (2018), no. 10, pp. 4117-4126].

Abstract:Let be a number field, let be a finite-dimensional semisimple -algebra, and let be an -order in . It was shown in previous work that, under certain hypotheses on , there exists an algorithm that for a given (left) -lattice either computes a free basis of over or shows that is not free over . In the present article, we generalize this by showing that, under weaker hypotheses on , there exists

Abstract:Numerical methods for stochastic partial differential equations typically estimate moments of the solution from sampled paths. Instead, we shall directly target the deterministic equations satisfied by the mean and the spatio-temporal covariance structure of the solution process. In the first part, we focus on stochastic ordinary differential equations. For the canonical examples with additive

Abstract:We relate the classical and post-Lie Magnus expansions. Intertwining algebraic and geometric arguments allows us to place the classical Magnus expansion in the context of Lie group integrators.

Abstract:The Coleman integral is a -adic line integral that plays a key role in computing several important invariants in arithmetic geometry. We give an algorithm for explicit Coleman integration on curves, using the algorithms of the second author [Math. Comp. 85 (2016), pp. 961-981] and [Finite Fields Appl. 45 (2019), pp. 301-322] to compute the action of Frobenius on -adic cohomology. We present

Abstract:Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron to obtain periodic representations for algebraic irrationals, analogous to the case of simple continued fractions and quadratic irrationals. Continued fractions have been studied in the field of -adic numbers . MCFs have also been recently introduced in , including, in particular, a -adic Jacobi-Perron algorithm

Abstract:We describe a new nonconstructive technique to show that squares are avoidable by an infinite word even if we force some letters from the alphabet to appear at certain occurrences. We show that as long as forced positions are at a distance at least 19 (resp., 3, resp., 2) from each other, then we can avoid squares over 3 letters (resp., 4 letters, resp., 6 or more letters). We can also deduce

Abstract:This article considers the computational (acoustic) wave propagation in strongly heterogeneous structures beyond the assumption of periodicity. A high contrast between the constituents of microstructured multiphase materials can lead to unusual wave scattering and absorption, which are interesting and relevant from a physical viewpoint, for instance, in the case of crystals with defects. We