The notion of a blossom in extended Chebyshev spaces offers adequate generalizations and extra-utilities to the tools for free-form design schemes. Unfortunately, such advantages are often ...overshadowed by the complexity of the resulting algorithms. In this work, we show that for the case of Müntz spaces with integer exponents, the notion of a Chebyshev blossom leads to elegant algorithms whose complexities are embedded in the combinatorics of Schur functions. We express the blossom and the pseudo-affinity property in Müntz spaces in terms of Schur functions. We derive an explicit expression for the Chebyshev–Bernstein basis via an inductive argument on nested Müntz spaces. We also reveal a simple algorithm for dimension elevation. Free-form design schemes in Müntz spaces with Young diagrams as shape parameters are discussed.
Polynomials are incredibly useful mathematical tools that have a wide array of applications. This book provides a comprehensive overview of polynomials and recent developments in the field. It ...includes ten chapters that address such topics as polynomials-based cyclic coding, Hermite polynomials, Routh polynomials, fitting parametric polynomials with control point coefficients, the thermoelastic wave model, and much more.
This note presents explicit expressions for the inverses of the Gram matrix of the Bernstein basis and its principal submatrices, by taking the advantages of the transformations between the Bernstein ...basis and the constrained dual Bernstein basis. Using the symmetry property, fast calculation of these matrices and their inverses is achieved. Significant improvements are obtained for applications including polynomial approximation of functions and degree reduction of Bézier curves.
The normalized totally positive bases are widely used in many fields. Based on the generalized Vandermonde determinant, the normalized total positivity of a kind of generalized toric-Bernstein basis ...is proved, which is defined on a set of real points. By this result, then the progressive iterative approximation property of the generalized toric-Bézier curve is obtained and some examples are provided to illustrate this property.
An algorithm for the computation of the bidiagonal decomposition of strictly totally positive h-Bernstein-Vandermonde matrices is presented. These matrices are collocation matrices of h-Bernstein ...bases (a generalization of the Bernstein basis) of the space of polynomials of degree less than or equal to n. The algorithm is fast (its computational complexity is O(n2)) and has high relative accuracy. The computation of this bidiagonal decomposition is used as the initial stage for the accurate and efficient solution of several linear algebra problems with these matrices: linear system solving, eigenvalue computation and inverse computation. Numerical experiments are also included.
Consecutive-<inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>-out-of-<inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula>:F systems are one of the most ...well-studied types of networks when discussing reliability. They have been used from safety-critical environments, such as nuclear power plants or hospital's emergency backup power supplies, to classical transportation problems, such as public water systems and oil/gas pipelines. Exact formulae for the reliability polynomial of a consecutive system are known for quite a long time. In addition, several alternatives for computing exactly the reliability polynomial are also known. However, when dealing with large consecutive systems, exact calculations become prohibitive and approximations/bounds are the common route. We begin this article by providing an in-depth review of many known bounds. Next, we focus on the coefficients of the reliability polynomial of a consecutive system in its Bernstein form. By deriving shape properties of these coefficients, we are able to identify new bounds. Our approach is uncommon for this case, as none of the previously used bounding techniques has looked closely at each and every coefficient. This is probably the reason why we obtain tight bounds with low complexity costs. Finally, detailed simulations provide strong evidence of the fidelity of the proposed bounds.
PDF
