In our present paper we study the duality theory and linear identities for real polynomials and functions on Banach spaces, which allows for a unified treatment and generalization of some classical results in the area. The basic idea is to exploit point evaluations of polynomials, as e.g. in [Rez93]. As a by-product we also obtain a curious generalization of the well-known Hilbert lemma on the ...

Abstract This paper studies convex generalized Nash equilibrium problems that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization computing equilibria (GNEs). The Moment-SOS hierarchy of semidefinite relaxations used solve the optimization. Under some general assumptions, we prove method can find a GNE if t...

This paper proposes a novel fully nonlinear refined beam element for pre-twisted structures undergoing large deformation and finite untwisting. The present model is constructed in the twisted basis to account effects of geometrical nonlinearity initial twist. Cross-sectional allowed by introducing Lagrange polynomials framework Carrera unified formulation. principle virtual work applied obtain ...

In this paper, an algorithm based on the collocation and homotopy analysis methods, for solving initial-boundary value problems, is introduced. The application of this algorithm is based on the approximation and interpolation of the dependent variables by using suitable functions or polynomials according to their values in the collocation points corresponding to a suitable discretization of the...

We describe the computations of some intrinsic constants associated to an n-dimensional normed space V, namely the N -th “allometry” constants κ∞(V) := inf { ‖T‖ · ‖T ′‖, T : `∞ → V, T ′ : V → `∞, TT ′ = IdV } . These are related to Banach–Mazur distances and to several types of projection constants. We also present the results of our computations for some low-dimensional spaces such as sequenc...

Let R = K[x;σ] be a skew polynomial ring over a division ring K. We introduce the notion of derivatives of skew polynomial at scalars. An analogous definition of derivatives of commutative polynomials from K[x] as a function of K[x] → K[x] is not possible in a non-commutative case. This is the reason why we have to define the derivative of a skew polynomial at a scalar. Our definition is based ...

Spectra and pseudospectra of matrix polynomials are of interest in geometric intersection problems, vibration problems, and analysis of dynamical systems. In this note we consider the effect of the choice of polynomial basis on the pseudospectrum and on the conditioning of the spectrum of regular matrix polynomials. In particular, we consider the direct use of the Lagrange basis on distinct int...

We study Legendrian singularities arising in complex contact geometry. We define a one-parameter family of bases in the ring of Legendrian characteristic classes such that any Legendrian Thom polynomial has nonnegative coefficients when expanded in these bases. The method uses a suitable Lagrange Grassmann bundle on the product of projective spaces. This is an extension of a nonnegativity resul...

Abstract. A k-generalized Dyck path of length n is a lattice path from (0, 0) to (n, 0) in the plane integer lattice Z × Z consisting of horizontal-steps (k, 0) for a given integer k ≥ 0, up-steps (1, 1), and down-steps (1,−1), which never passes below the x-axis. The present paper studies three kinds of statistics on k-generalized Dyck paths: ”number of u-segments”, ”number of internal u-segme...