Multi-degree splines are piecewise polynomial functions having sections of different degrees. They offer significant advantages over the classical uniform-degree framework, as they allow for modeling complex geometries with fewer degrees freedom and, at same time, a more efficient engineering analysis. Moreover possess set basis similar properties to standard B-splines. In this paper we develop...

Here presented is the interrelationship between Eulerian polynomials, Eulerian fractions and Euler-Frobenius polynomials, Euler-Frobenius fractions, Bsplines, respectively. The properties of Eulerian polynomials and Eulerian fractions and their applications in B-spline interpolation and evaluation of Riemann zeta function values at odd integers are given. The relation between Eulerian numbers a...

Abstract In this paper, we explore the interrelationship between Eulerian numbers and B splines. Specifically, using B splines, we give the explicit formulas of the refined Eulerian numbers, and descents polynomials. Moreover, we prove that the coefficients of descent polynomials D d (t) are logconcave. This paper also provides a new approach to study Eulerian numbers and descent polynomials.

A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well known B-splines, play a key role in computer aided geometric designs. Refinable splines have been studied in several papers, most noticably in [7] for integer dilations and [3] for real dilations. There are general characterizations in these papers, but these characte...

The main result of this paper is a generalization of the property that, for smooth u, uxy = 0 implies (*) u(x, y) = a(x) + b(y). Any function having generalized unsymmetric mixed partial derivative identically zero is of the form (*). There is a function with generalized symmetric mixed partial derivative identically zero not of the form (*), but (*) does follow here with the additional assumpt...

Multi-degree Tchebycheffian splines are with pieces drawn from extended (complete) Tchebycheff spaces, which may differ interval to interval, and possibly of different dimensions. These a natural extension multi-degree polynomial splines. Under quite mild assumptions, they can be represented in terms so-called B-spline (MDTB-spline) basis; such basis possesses all the characterizing properties ...

We explore the subrings in trigonometric polynomial rings and their factorization properties. Consider the ring S′ of complex trigonometric polynomials over the field Q(i) (see [11]). We construct the subrings S′ 1, S′ 0 of S′ such that S′ 1 ⊆ S′ 0 ⊆ S′. Then S′ 1 is a Euclidean domain, whereas S′ 0 is a Noetherian HFD. We also characterize the irreducible elements of S′ 1, S′ 0 and discuss amo...

In this study we explore the subrings in trigonometric polynomial rings. Consider the rings T and T ′ of real and complex trigonometric polynomials over the fields R and its algebraic extension C respectively ( see [6]). We construct the subrings T0 of T and T ′ 0, T ′ 1 of T ′. Then T0 is a BFD whereas T ′ 0 and T ′ 1 are Euclidean domains. We also discuss among these rings the Condition : Let...

We introduce isogeometric collocation methods based on generalized B-splines and we analyze their performance through numerical examples for univariate and multivariate scalarand vector-valued problems. In particular, advection–diffusion and linear elasticity model problems are addressed. The resultingmethod combines the favorable properties of isogeometric collocation and the geometrical and a...

Since the 1970’s B-splines have evolved to become the de facto standard for use in curve and surface representation. This resulted in a relatively recent proliferation of nonrigid image registration techniques based on B-splines. These techniques fall under the general Free-Form Deformation (FFD) approach in which the object to be registered is embedded within a B-spline object. The deformation...