An incidence of a left superior vena cava with a left azygos vein was found in a cadaver during an Osaka Dental University student's dissection practice session in 2002. The left superior vena cava began from the confluence between the right and left brachiocephalic veins, and extended directly into the left superior edge of the coronary sinus. No right superior vena cava was observed. The left...

It is well known that a cardinal B-spline of order m,m ∈ N, is a piecewise polynomial function. In this note we propose an effective method for calculating the coefficients of polynomials which constitute a cardinal B-spline. © 2010 Elsevier Ltd. All rights reserved.

Odd degree Chebyshev polynomials over a ring of modulo 2 have two kinds of period. One is an “orbital period”. Odd degree Chebyshev polynomials are bijection over the ring. Therefore, when an odd degree Chebyshev polynomial iterate affecting a factor of the ring, we can observe an orbit over the ring. The “ orbital period ” is a period of the orbit. The other is a “degree period”. It is observe...

We show that, given a histogram with n bins—possibly non-contiguous or consisting of single points—there exists a unique polynomial histopolant (polynomial of order n with bin averages equal to the histogram’s). We also present histopolating cardinal functions for histograms with contiguous bins. The Hodge star operator from the theory of differential forms is the mapping which puts kand (d − k...

In this paper, we propose a method to approximate the solution of a linear Fredholm integro-differential equation by using the Chebyshev wavelet of the first kind as basis. For this purpose, we introduce the first Chebyshev operational matrix of integration. Chebyshev wavelet approximating method is then utilized to reduce the integro-differential equation to a system of algebraic equations. Il...

where f, g are two differentiable functions and f ′, g′ ∈ L∞(a, b) and p is a positive and integrable function on [a, b]. Other interesting corollaries are also presented in [12]. Many researchers have given considerable attention to (1) and several inequalities related to this functional have appeared in the literature, to mention a few, see [1, 3, 8, 15, 16, 19, 20, 22] and the references cit...

In this paper, a Chebyshev polynomial approximation for the solution of second-order partial differential equations with two variables and variable coefficients is given. Also, Chebyshev matrix is introduced. This method is based on taking the truncated Chebyshev expansions of the functions in the partial differential equations. Hence, the result matrix equation can be solved and approximate va...

We systematically investigate the farthest distance function, farthest points, Klee sets, and Chebyshev centers, with respect to Bregman distances induced by Legendre functions. These objects are of considerable interest in Information Geometry and Machine Learning; when the Legendre function is specialized to the energy, one obtains classical notions from Approximation Theory and Convex Analys...

A Chebyshev knot C(a, b, c, φ) is a knot which has a parametrization of the form x(t) = Ta(t); y(t) = Tb(t); z(t) = Tc(t + φ), where a, b, c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ ∈ R. We show that any two-bridge knot is a Chebyshev knot with a = 3 and also with a = 4. For every a, b, c integers (a = 3, 4 and a, b coprime), we describe an algorithm that gives all Cheb...

The articles [11], [7], [14], [13], [15], [5], [6], [2], [12], [1], [10], [8], [9], [3], and [4] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: A, B denote ordinal numbers, K, M, N denote cardinal numbers, x, y, z, X , Y , Z, Z1, Z2 denote sets, n denotes a natural number, and f , g denote functions. Let I1 be a function. We say that I1 is car...