Abstract:Fermat’s well-known factorization algorithm is based on finding a representation of natural numbers $N$ as the difference of squares. In 1895, Lawrence generalized this idea and applied it to multiples $kN$ of the original number. A systematic approach to choose suitable values for $k$ was introduced by Lehman in 1974, which resulted in the first deterministic factorization algorithm considerably

Abstract:We present an algorithmic embedded desingularization of arithmetic surfaces bearing in mind implementability. Our algorithm is based on work by Cossart-Jannsen-Saito, though our variant uses a refinement of the order instead of the Hilbert-Samuel function as a measure for the complexity of the singularity. We particularly focus on aspects arising when working in mixed characteristics. Furthermore

Abstract:In 1980, the first and third authors proposed a probabilistic primality test that has become known as the Baillie-PSW primality test. Its power to distinguish between primes and composites comes from combining a Fermat probable prime test with a Lucas probable prime test. No odd composite integers have been reported to pass this combination of primality tests if the parameters are chosen in

Abstract:We study the recovery of functions in real spline spaces from unsigned sampled values. We consider two types of recovery. The one is to recover functions locally from finitely many unsigned samples. And the other is to recover functions on the whole line from infinitely many unsigned samples. In both cases, we give characterizations for a set of points to be a phaseless sampling set, at which

Abstract:We consider rank-$1$ lattices for integration and reconstruction of functions with series expansion supported on a finite index set. We explore the connection between the periodic Fourier space and the non-periodic cosine space and Chebyshev space, via tent transform and then cosine transform, to transfer known results from the periodic setting into new insights for the non-periodic settings

Abstract:The existence and uniqueness of weak solutions to dynamical low-rank evolution problems for parabolic partial differential equations in two spatial dimensions is shown, covering also non-diagonal diffusion in the elliptic part. The proof is based on a variational time-stepping scheme on the low-rank manifold. Moreover, this scheme is shown to be closely related to practical methods for computing

Abstract:A block FETI–DP/BDDC preconditioner for mixed formulations of almost incompressible elasticity is constructed and analyzed; FETI–DP (Finite Element Tearing and Interconnecting Dual-Primal) and BDDC (Balancing Domain Decomposition by Constraints) are two very successful domain decomposition algorithms for a variety of elliptic problems. The saddle point problems of the mixed problems are discretized

Abstract:$L^2$ stable discontinuous Galerkin method with a family of numerical fluxes was studied for the one-dimensional wave equation by Cheng, Chou, Li, and Xing in [Math. Comp. 86 (2017), pp. 121–155]. Although optimal convergence rates were numerically observed with wide choices of parameters in the numerical fluxes, their error estimates were only proved for a sub-family with the construction

Abstract:We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poisson manifold structure which encodes the physical properties, symmetries, and conservation laws of the dynamics. The goal is to design reduced basis methods

Abstract:In this work we propose a new class of entropy consistent schemes for hyperbolic systems of conservation laws (HCL). The schemes developed so far in the classic works of Tadmor, Johnson and their collaborators start from an appropriate entropy conservative formulation of the system. Then entropy diminishing schemes are obtained by adding appropriate artificial diffusion terms. This program

Abstract:In this paper, the first family of conforming discrete three dimensional Gradgrad-complexes consisting of finite element spaces is constructed. These discrete complexes are exact in the sense that the range of each discrete map is the kernel space of the succeeding one. These spaces can be used in the mixed form of the linearized Einstein-Bianchi system.

Abstract:When solving 2D linear elastodynamic equations in homogeneous isotropic media, a Helmholtz decomposition of the displacement field decouples the equations into two scalar wave equations that only interact at the boundary. It is then natural to look for numerical schemes that independently solve the scalar equations and couple the solutions at the boundary. The case of rigid boundary condition

Abstract:For the discretization of the integral fractional Laplacian $(-\Delta )^s$, $0 < s < 1$, based on piecewise linear functions, we present and analyze a reliable weighted residual a posteriori error estimator. In order to compensate for the lack of $L^2$-regularity of the residual in the regime $3/4 < s < 1$, this weighted residual error estimator includes as an additional weight a power of

Abstract:We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of the bending energy and the tangent-point functional. Based on estimates for the second derivative of the latter and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution

Abstract:In this paper we obtain a complete list of imaginary -quadratic fields with class groups of exponent and under extended Riemann hypothesis for every positive integer where an -quadratic field is a number field of degree represented as the composite of -quadratic fields.

Abstract:We give explicit upper and lower bounds for , the number of zeros of a Dirichlet -function with character and height at most . Suppose that has conductor , and that . If , then We give slightly stronger results for small and . Along the way, we prove a new bound on for .

Abstract:Let denote . A polynomial is a Littlewood polynomial (LP) of length if the are for , and for . An LP has order if it is divisible by . The problem of finding the set of lengths of LPs of order is equivalent to finding the lengths of spectral-null codes of order , and to finding such that admits a partition into two subsets whose first moments are equal. Extending the techniques and results

Abstract:We develop the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. One key ingredient of this algorithm is our new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Our implementation can solve problem instances

Abstract:In the present paper we study quasi-Monte Carlo rules for approximating integrals over the -dimensional unit cube for functions from weighted Sobolev spaces of regularity one. While the properties of these rules are well understood for anchored Sobolev spaces, this is not the case for the ANOVA spaces, which are another very important type of reference spaces for quasi-Monte Carlo rules. Using

Abstract:Several convergent expansions are available for most of the special functions of the mathematical physics, as well as some asymptotic expansions [NIST Handbook of Mathematical Functions, 2010]. Usually, both type of expansions are given in terms of elementary functions; the convergent expansions provide a good approximation for small values of a certain variable, whereas the asymptotic expansions

Abstract:Based on the Delsarte-Yudin linear programming approach, we extend Levenshtein's framework to obtain lower bounds for the minimum -energy of spherical codes of prescribed dimension and cardinality, and upper bounds on the maximal cardinality of spherical codes of prescribed dimension and minimum separation. These bounds are universal in the sense that they hold for a large class of potentials

Abstract:We introduce and study a cone which consists of a class of generalized polynomial functions and which provides a common framework for recent non-negativity certificates of polynomials in sparse settings. Specifically, this -cone generalizes and unifies sums of arithmetic-geometric mean exponentials (SAGE) and sums of non-negative circuit polynomials (SONC). We provide a comprehensive characterization

Abstract:Approximation algorithms like sum-up rounding that allow to compute integer-valued approximations of the continuous controls in a weak sense have attracted interest recently. They allow to approximate (optimal) feasible solutions of continuous relaxations of mixed-integer control problems (MIOCPs) with integer controls arbitrarily close. To this end, they use compactness properties of the

Abstract:We propose two approximate versions of the first-order primal-dual algorithm (PDA) to solve a class of convex-concave saddle point problems. The introduced approximate criteria are easy to implement in the sense that they only involve the subgradient of a certain function at the current iterate. The first approximate PDA solves both subproblems inexactly and adopts the absolute error criteria

Abstract:We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as well as multiple-network models used in medical applications. The semi-explicit approach decouples the system such that each time step requires the solution

Abstract:In this paper, we study the behaviour of some staggered discretization based numerical schemes for the barotropic Navier-Stokes equations at low Mach number. Three time discretizations are considered: the implicit-in-time scheme and two non-iterative pressure correction schemes. The two latter schemes differ by the discretization of the convection term: linearly implicit for the first one

Abstract:For the Landau-Lifshitz-Gilbert (LLG) equation of micromagnetics we study linearly implicit backward difference formula (BDF) time discretizations up to order combined with higher-order non-conforming finite element space discretizations, which are based on the weak formulation due to Alouges but use approximate tangent spaces that are defined by -averaged instead of nodal orthogonality constraints

Abstract:The paper studies a geometrically unfitted finite element method (FEM), known as trace FEM or cut FEM, for the numerical solution of the Stokes system posed on a closed smooth surface. A trace FEM based on standard Taylor–Hood (continuous $\mathbf P_2$–$P_1$) bulk elements is proposed. A so-called volume normal derivative stabilization, known from the literature on trace FEM, is an essential

Abstract:We analyze rates of convergence for quasi-Monte Carlo (QMC) integration for Bayesian inversion of linear, elliptic partial differential equations with uncertain input from function spaces. Adopting a Riesz or Schauder basis representation of the uncertain inputs, function space priors are constructed as product measures on spaces of (sequences of) coefficients in the basis representations

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:We address here a class of staggered schemes for the compressible Euler equations; this scheme was introduced in recent papers and possesses the following features: upwinding is performed with respect to the material velocity only and the internal energy balance is solved, with a correction term designed on consistency arguments. These schemes have been shown in previous works to preserve

Abstract:The variable two-step backward differentiation formula (BDF2) is revisited via a new theoretical framework using the positive semi-definiteness of BDF2 convolution kernels and a class of orthogonal convolution kernels. We prove that, if the adjacent time-step ratios , the adaptive BDF2 time-stepping scheme for linear reaction-diffusion equations is unconditionally stable and (maybe, first-order)

Abstract:We prove the convergence of the hydrostatic reconstruction scheme with kinetic numerical flux for the Saint Venant system with continuous topography with locally integrable derivative. We use a recently derived fully discrete sharp entropy inequality with dissipation, that enables us to establish an estimate in the inverse of the square root of the space increment of the norm of the gradient

Abstract:New superconvergence structures are introduced by the finite volume element method (FVEM), which gives us the freedom to choose the superconvergent points of the derivative (for odd order schemes) and the superconvergent points of the function value (for even order schemes) for . The general orthogonal condition and the modified M-decomposition (MMD) technique are established to prove the

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