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Invariant theory provides more efficient tools, such as Molien generating functions and integrity bases, than basic group theory, that relies on projector techniques for the construction of symmetry--adapted polynomials in the symmetry…
The focus of this article is the approximation of functions which are analytic on a compact interval except at the endpoints. Typical numerical methods for approximating such functions depend upon the use of particular conformal maps from…
We consider families of finite elements on polygonal meshes, that are defined implicitly on each mesh cell as solutions of local Poisson problems with polynomial data. Functions in the local space on each mesh cell are evaluated via…
The accuracy of finite element solutions is closely tied to the mesh quality. In particular, geometrically nonlinear problems involving large and strongly localized deformations often result in prohibitively large element distortions. In…
Polymers, macromolecules formed from covalently bonded monomers, underpin countless technologies and are indispensable to modern life. While deep learning is advancing polymer science, existing methods typically represent the whole polymer…
We investigate asymptotic polynomial approximation for a class of weighted Bloch functions in the unit disc. Our main result is a structural theorem on asymptotic polynomial approximation in the unit disc, in the flavor of the classical…
Localized features such as singularities, sharp gradients, discontinuities, and moving sources require adaptive finite element discretizations. Conventional refinement strategies introduce significant computational overhead through…
We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…
Cyclic and non-wellfounded proofs are now increasingly employed to establish metalogical results in a variety of settings, in particular for type systems with forms of (co)induction. Under the Curry-Howard correspondence, a cyclic proof can…
Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required…
The interpolation-regression approximation is a powerful tool in numerical analysis for reconstructing functions defined on square or triangular domains from their evaluations at a regular set of nodes. The importance of this technique lies…
In this paper, we propose a family of nonconforming finite elements for $2m$-th order partial differential equations in $\mathbb{R}^n$ on simplicial grids when $m=n+1$. This family of nonconforming elements naturally extends the elements…
We consider nonparametric regression with functional covariates, that is, they are elements of an infinite-dimensional Hilbert space. A locally polynomial estimator is constructed, where an orthonormal basis and various tuning parameters…
Some properties of generalized convexity for sets and for functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is…
This paper addresses a multi-scale finite element method for second order linear elliptic equations with arbitrarily rough coefficient. We propose a local oversampling method to construct basis functions that have optimal local…
In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set…
Discrete de Rham (DDR) methods provide non-conforming but compatible approximations of the continuous de Rham complex on general polytopal meshes. Owing to the non-conformity, several challenges arise in the analysis of these methods. In…
In this paper, we introduce a superconvergent approximation method that employs radial basis functions (RBFs) in the numerical solution of conservation laws. The use of RBFs for interpolation and approximation is a well developed area of…
Complex polynomial optimization has recently gained more and more attention in both theory and practice. In this paper, we study the optimization of a real-valued general conjugate complex form over various popular constraint sets including…
We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.