Abstract We study the average condition number for polynomial eigenvalues of collections of matrices drawn from some random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with random Gaussian entries are very well conditioned on the average.

Abstract This paper presents two algorithms. The first decides the existence of a pointed homotopy between given simplicial maps \(f {},\,g:X\rightarrow Y\), and the second computes the group \([\varSigma X,Y]^*\) of pointed homotopy classes of maps from a suspension; in both cases, the target Y is assumed simply connected. More generally, these algorithms work relative to \(A\subseteq X\).

Abstract We outline an algorithm to recover the canonical (or, coarsest) stratification of a given finite-dimensional regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively localizing the poset of cells about a family of subposets; these subposets are in turn determined by a collection of cosheaves which capture variations in cohomology

Abstract In this paper, we present a deterministic polynomial time algorithm for testing whether a symbolic matrix in non-commuting variables over \({\mathbb {Q}}\) is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the

Abstract Differential (Ore) type polynomials with “approximate” polynomial coefficients are introduced. These provide an effective notion of approximate differential operators, with a strong algebraic structure. We introduce the approximate greatest common right divisor problem (GCRD) of differential polynomials, as a non-commutative generalization of the well-studied approximate GCD problem. Given

Abstract In this paper, we introduce a randomized version of the backward Euler method that is applicable to stiff ordinary differential equations and nonlinear evolution equations with time-irregular coefficients. In the finite-dimensional case, we consider Carathéodory-type functions satisfying a one-sided Lipschitz condition. After investigating the well-posedness and the stability properties of

Abstract We present a probabilistic Las Vegas algorithm for solving sufficiently generic square polynomial systems over finite fields. We achieve a nearly quadratic running time in the number of solutions, for densely represented input polynomials. We also prove a nearly linear bit complexity bound for polynomial systems with rational coefficients. Our results are obtained using the combination of

Abstract Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. This matrix is readily computed from an input dataset, and its eigen decomposition can then be used to identify algebraic properties of the support or density/support estimates with the Christoffel function. It is

Abstract In this paper we study a class of fast geometric image inpainting methods based on the idea of filling the inpainting domain in successive shells from its boundary inwards. Image pixels are filled by assigning them a color equal to a weighted average of their already filled neighbors. However, there is flexibility in terms of the order in which pixels are filled, the weights used for averaging

Abstract A family of effective equations for wave propagation in periodic media for arbitrary timescales \(\mathcal {O}(\varepsilon ^{-\alpha })\), where \(\varepsilon \ll 1\) is the period of the tensor describing the medium, is proposed. The well-posedness of the effective equations of the family is ensured without requiring a regularization process as in previous models (Benoit and Gloria in Long-time

Abstract We propose a new approach to the theory of conditioning for numerical analysis problems for which both classical and stochastic perturbation theories fail to predict the observed accuracy of computed solutions. To motivate our ideas, we present examples of problems that are discontinuous at a given input and even have infinite stochastic condition number, but where the solution is still computed

Abstract We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that non-symplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate

Abstract Monotone operators and (firmly) nonexpansive mappings are fundamental objects in modern analysis and computational optimization. It was shown in 2012 that if finitely many firmly nonexpansive mappings have or “almost have” fixed points, then the same is true for compositions and convex combinations. More recently, sharp information about the minimal displacement vector of compositions and

Abstract A modified version of Schiffer’s conjecture on a regular pentagon states that Neumann eigenfunctions of the Laplacian do not change sign on the boundary. In a companion paper by Bartłomiej Siudeja it was shown that eigenfunctions which are strictly positive on the boundary exist on regular polygons with at least 6 sides, while on equilateral triangles and cubes it is not even possible to find

Abstract We introduce a new methodology to design uniformly accurate methods for oscillatory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing for standard schemes to retain their usual orders of convergence. Overall, high-order numerical

Abstract This work considers the question: what convergence guarantees does the stochastic subgradient method have in the absence of smoothness and convexity? We prove that the stochastic subgradient method, on any semialgebraic locally Lipschitz function, produces limit points that are all first-order stationary. More generally, our result applies to any function with a Whitney stratifiable graph

Abstract This paper presents convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature

Abstract In this paper, we prove necessary and sufficient conditions for a hybridizable discontinuous Galerkin method to satisfy a multisymplectic conservation law, when applied to a canonical Hamiltonian system of partial differential equations. We show that these conditions are satisfied by the “hybridized” versions of several of the most commonly used finite element methods, including mixed, nonconforming

Abstract We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of closed semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size

Abstract We analyze the propagation properties of the numerical versions of one- and two-dimensional wave equations, semi-discretized in space by finite difference schemes. We focus on high-frequency solutions whose propagation can be described, both at the continuous and at the semi-discrete levels, by micro-local tools. We consider uniform and non-uniform numerical grids as well as constant and variable

Abstract Folding grid value vectors of size \(2^L\) into Lth-order tensors of mode size \(2\times \cdots \times 2\), combined with low-rank representation in the tensor train format, has been shown to result in highly efficient approximations for various classes of functions. These include solutions of elliptic PDEs on nonsmooth domains or with oscillatory data. This tensor-structured approach is attractive

Abstract We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions and Conley–Morse graphs of the combinatorial

Abstract Can an ideal I in a polynomial ring \(\Bbbk [\mathbf {x}]\) over a field be moved by a change of coordinates into a position where it is generated by binomials \(\mathbf {x}^\mathbb A- \lambda \mathbf {x}^\mathbf {b}\) with \(\lambda \in \Bbbk \), or by unital binomials (i.e., with \(\lambda = 0\) or 1)? Can a variety be moved into a position where it is toric? By fibering the G-translates

Abstract In this paper, we explore orthogonal systems in \(\mathrm {L}_2({\mathbb R})\) which give rise to a real skew-symmetric, tridiagonal, irreducible differentiation matrix. Such systems are important since they are stable by design and, if necessary, preserve Euclidean energy for a variety of time-dependent partial differential equations. We prove that there is a one-to-one correspondence between

Abstract Consider a finite system of non-strict polynomial inequalities with solution set \(S\subseteq \mathbb R^n\). Its Lasserre relaxation of degree d is a certain natural linear matrix inequality in the original variables and one additional variable for each nonlinear monomial of degree at most d. It defines a spectrahedron that projects down to a convex semialgebraic set containing S. In the best

Abstract In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group \(\mathrm {O}_{3}\) on the space \(\mathbb {R}[x,y,z]_{2d}\) of ternary forms of even degree 2d. The construction relies on two key ingredients: on the one hand, the Slice Lemma allows us to reduce the problem to determining the invariants for the action on a subspace

A shape sensitive, variational approach for the matching of surfaces considered as thin elastic shells is investigated. The elasticity functional to be minimized takes into account two different types of nonlinear energies: a membrane energy measuring the rate of tangential distortion when deforming the reference shell into the template shell, and a bending energy measuring the bending under the deformation

The convergence of full discretisations by implicit Runge-Kutta and nonconforming Galerkin methods applied to nonlinear evolutionary inequalities is studied. The scope of applications includes differential inclusions governed by a nonlinear operator that is monotone and fulfills a certain growth condition. A basic assumption on the considered class of stiffly accurate Runge-Kutta time discretisations

In this paper we study the formal algebraic structure underlying the intrinsic classification algorithm, recently introduced in Singer et al. (SIAM J. Imaging Sci. 2011, accepted), for classifying noisy projection images of similar viewing directions in three-dimensional cryo-electron microscopy (cryo-EM). This preliminary classification is of fundamental importance in determining the three-dimensional