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: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

We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex n-gon, our construction produces 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n + 1)/2 basis functions known to obtain quadratic convergence. The

We analyze a numerical algorithm for solving radiative transport equation with vacuum or reflection boundary condition that was proposed in [4] with angular discretization by finite element method and spatial discretization by discontinuous Galerkin or finite difference method.