In this paper, we derive an explicit formula for the bivariate Lagrange basis polynomials of a general set checkerboard nodes. This generalizes existing results at Padua nodes, Chebyshev Morrow-Patterson and Geronimus We also construct subspace spanned by linearly independent vanishing that vanish nodes prove uniqueness in quotient space defined as with certain degree over polynomials.

The Grothendieck compactness principle states that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In this article, an analogue of the Grothendieck compactness principle is considered when the norm topology of a Banach space is replaced by its weak topology. It is shown that every weakly compact subset of a Banach space is contained in...

The electron spin resonance (ESR) of nanoscale molecular magnet V15 is studied. Since the Hamiltonian of V15 has a large Hilbert space and numerical calculations of the ESR signal evaluating the Kubo formula with exact diagonalization method is difficult, we implement the formula with the help of the random vector technique and the Chebyshev polynominal expansion, which we name the double Cheby...

Abstract. This paper discusses a method based on Laguerre polynomials combined with a Filtered Conjugate Residual (FCR) framework to compute the product of the exponential of a matrix by a vector. The method implicitly uses an expansion of the exponential function in a series of orthogonal Laguerre polynomials, much like existing methods based on Chebyshev polynomials do. Owing to the fact that...

Claim of Johnson-Lindenstrauss Theorem: The Euclidean metric on any finite set X (a bunch of high dimensional points) can be embedded with distortion D = 1 + ε in R for k = O(ε−2 log n). If we lose ε (ε = 0), it becomes almost impossible to do better than that in R. Nevertheless, it is not hard to construct a counter example to this: a simplex of n+ 1 points. The Johnson-Lindenstrauss theorem g...

We consider Markov decision processes (MDP) as generators of sequences of probability distributions over states. A probability distribution is psynchronizing if the probability mass is at least p in a single state, or in a given set of states. We consider four temporal synchronizing modes: a sequence of probability distributions is always p-synchronizing, eventually p-synchronizing, weakly psyn...

The following estimate for the Rayleigh–Ritz method is proved: |λ̃−λ||(ũ,u)| ≤ ‖Aũ− λ̃ũ‖sin∠{u;Ũ}, ‖u‖= 1. Here A is a bounded self-adjoint operator in a real Hilbert/euclidian space, {λ,u} one of its eigenpairs, Ũ a trial subspace for the Rayleigh–Ritz method, and {λ̃, ũ} a Ritz pair. This inequality makes it possible to analyze the fine structure of the error of the Rayleigh–Ritz method, in part...

We give a nonstandard variant of Jordan’s proof of the Jordan curve theorem which is free of the defects his contemporaries criticized and avoids the epsilontic burden of the classical proof. The proof is selfcontained, except for the Jordan theorem for polygons taken for granted.

In this paper a new technique is proposed for the clustering and classification of spatio-temporal object trajectories extracted from video motion clips. The trajectories are represented as motion time series and modelled using Chebyshev polynomial approximations. Trajectory clustering is then performed to discover patterns of similar object motion. The coefficients of the basis functions are u...

We prove the hardness of weakly learning halfspaces in the presence of adversarial noise using polynomial threshold functions (PTFs). In particular, we prove that for any constants d ∈ Z and ε > 0, it is NP-hard to decide: given a set of {−1, 1}-labeled points in Rn whether (YES Case) there exists a halfspace that classifies (1−ε)-fraction of the points correctly, or (NO Case) any degree-d PTF ...