Mathematics

The purpose of this paper is to introduce the concept of the automatic integration and present a new way of approximating definite integrals using the automatic integration based on an associative algebra with zero divisors.

Let x?? be a large number, let [x]=x?�{x} be the largest integer function, and let ?(n) be the Euler totient function. The result ??n?�x ?([x/n])=(6/ ? 2 )xlogx+O(x(logx ) 2/3 (loglogx ) 1/3 ) was proved very recently. This note presents a short elementary proof, and sharpen the error term to ??n?�x ?([x/n])=(6/ ? 2 )xlogx+O(x) . In addition, the first proofs of the asymptotics formulas for the finite sums ??n?�x ?([x/n])=(15/ ? 2 )xlogx+O(xloglogx) , and ??n?�x ?([x/n])=( ? 2 /6)xlogx+O(xloglogx) are also evaluated here.

The golden ratio and Fibonacci numbers are found to occur in various aspects of nature. We discuss the occurrence of this ratio in an interesting physical problem concerning center of masses in two dimensions. The result is shown to be independent of the particular shape of the object. The approach taken extends naturally to higher dimensions. This leads to ratios similar to the golden ratio and generalization of the Fibonacci sequence. The hierarchy of these ratios with dimension and the limit as the dimension tends to infinity is discussed using the physical problem.

It is shown that Hermite polynomials satisfy a Bessel type orthogonality relation, based on the zeros of a single index Hermite polynomial and with a finite integration interval. Because of the role of non-symmetric zeros in the final relation, its applicability covers Hermite polynomials P n (x) with n≥3 .

In this paper we introduce the notion of bi-slant submanifolds of a para Hermitian manifold. They naturally englobe CR, semi-slant and hemi-slant submanifolds. We study their first properties and present a whole gallery of examples.

In this paper we study biconservative immersions into the semi-Riemannian space form R 4 2 (c) of dimension 4, index 2 and constant curvature, where c∈{0,−1,1} . First, we obtain a characterization of quasi-minimal proper biconservative immersions into R 4 2 (c) . Then we obtain the complete classification of quasi-minimal biconservative surfaces in R 4 2 (0)= E 4 2 . We also obtain a new class of biharmonic quasi-minimal isometric immersion into E 4 2 .

In this paper, we extend the r-Dowling polynomials to their bivariate forms. Several properties that generalize those of the bivariate Bell and r-Bell polynomials are established. Finally, we obtain two forms of generalized Spivey's formula.

This paper provides bounds for the number of terms, denoted by f , of a harmonic sum with the condition that it starts from any arbitrary unit fraction 1 m , m>1 , until another unit fraction 1 m+f−1 such that the sum is the highest sum less than a particular positive integer q . Also, we consider the number of terms of Egyptian fractions whose terms are consecutive multiples of r , r≥1 , under the same above condition. We end the paper with a formula for the case: q=1 and r=1 .

In this paper, we derive new properties of the Mertens function and discuss a likely upper bound of the absolute value of the Mertens function logx! ????????????>|M(x)| when x>1 . Using this likely bound we show that we have a sufficient condition to prove the Riemann Hypothesis.

In the present paper, we define and study C -parallel and C -proper slant curves of S -manifolds. We prove that a curve γ in an S -manifold of order r≥3, under certain conditions, is C -parallel or C -parallel in the normal bundle if and only if it is a non-Legendre slant helix or Legendre helix, respectively. Moreover, under certain conditions, we show that γ is C -proper or C -proper in the normal bundle if and only if it is a non-Legendre slant curve or Legendre curve, respectively. We also give two examples of such curves in R 2m+s (−3s).