This paper discusses the notion of cointegrating space for linear processes integrated any order. It first shows that notions (polynomial) vectors and root functions coincide. Second, it how can be defined (i) as a vector polynomial over complex scalars, (ii) free module scalar polynomials, or finally (iii) rational scalars. Third, canonical set used basis various space. Fourth, reviews results...

A simple expression for the Smagorinsky length scale is derived using arguments in physical space. This expression is sensitive to the choice of the basis functions employed in a numerical method and can be used with the finite volume or the finite element method on fully unstructured grids and meshes. For a polynomial basis, the dependence of the Smagorinsky length scale on the order of comple...

The quest for more accurate numerical simulations of the Navier Stokes equations has motivated the study of high-order piecewise-polynomial basis functions, also known as k-version nite element methods, as a means to attain this accuracy in the most cost eeective manner. Stabilized nite element methods (e.g. SUPG) have been proven to attain optimal rates of convergence for any polynomial order ...

in this paper we present a new method for solving fractional optimal control problems with delays in state and control. this method is based upon bernstein polynomial basis and using feedback control. the main advantage of using feedback or closed-loop controls is that they can monitor their effect on the system and modify the output accordingly. in this work, we use bernstein polynomials to tr...

We show the equivalence of the Pieri formula for ag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a nite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a uniied deenition o...

We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified defini...

In Gaussian process regression, both the basis functions and their prior distribution are simultaneously specified by the choice of the covariance function. In certain problems one would like to choose the covariance independently of the basis functions (e. g., in polynomial signal processing or Wiener and Volterra analysis). We propose a solution to this problem that approximates the desired c...

We present the SP package which provides some functions on Schubert polynomials. These polynomials contain as a subfamily the Schur functions that are the fundamental symmetric functions. They are indexed by permutations and provide a convenient basis of the polynomial ring.