and Applied Analysis 3 2. Symmetry of Power Sum and the Generalized Bernoulli Polynomials Let χ be the Dirichlet character with conductor d ∈ N. From 1.3 , we note that ∫ X χ x edx t ∑d−1 i 0 χ i e it edt − 1 ∞ ∑ n 0 Bn,χ t n! , 2.1 where Bn,χ x are the nth generalized Bernoulli numbers attached to χ. Now, we also see that the generalized Bernoulli polynomials attached to χ are given by ∫ X χ (...

In efforts to discover disease mechanisms and improve clinical diagnosis of tumors, it is useful to mine profiles for informative genes with definite biological meanings and to build robust classifiers with high precision. In this study, we developed a new method for tumor-gene selection, the Chi-square test-based integrated rank gene and direct classifier (χ(2)-IRG-DC). First, we obtained the ...

This note relates to bounds on the chromatic number χ(R) of the Euclidean space, which is the minimum number of colors needed to color all the points in R so that any two points at the distance 1 receive different colors. In [6] a sequence of graphs Gn in R was introduced showing that χ(R) ≥ χ(Gn) ≥ (1 + o(1)) 2 6 . For many years, this bound has been remaining the best known bound for the chro...

We improve upper bounds on the chromatic number proven independently in [1] and [5]. Our main lemma gives a sufficient condition for two paths in graph to be completely joined. Using this, we prove that if a graph has an optimal coloring with more than ω2 singleton color classes, then it satisfies χ ≤ ω+∆+1 2 . It follows that a graph satisfying n − ∆ < α + ω−1 2 must also satisfy χ ≤ ω+∆+1 2 ,...

Let Q(n, χ) denote the minimum clique size an n-vertex graph can have if its chromatic number is χ . Using Ramsey graphs we give an exact, albeit implicit, formula for the case χ ≥ (n + 3)/2.

The determination of the second-order susceptibility (χ(2)) of thin film samples can be a delicate matter since well-established χ(2) measurement methodologies such as the Maker fringe technique are best suited for nonlinear materials with large thicknesses typically ranging from tens of microns to several millimeters. Here we compare two different second-harmonic generation setups and the corr...

Let p be a puncture of a punctured sphere, and let Q be the set of all other punctures. We prove that the maximal cardinality of a set A of arcs pairwise intersecting at most once, which start at p and end in Q, is |χ|(|χ| + 1). We deduce that the maximal cardinality of a set of arcs with arbitrary endpoints pairwise intersecting at most twice is |χ|(|χ|+ 1)(|χ|+ 2).

abstract background: the negative side effects of infertility affect not only the relations between the couple but also other aspects of their lives and destroy their personal, social, and professional functions. this research aims to study the social consequences of infertility on families in the city of isfahan. sociological and psychological theories provide the theoretical framework of the ...

We will show that any complex minimal surface of general type with c1 = 2χ− 1 having non-trivial 2-torsion divisors, where c 2 1 and χ are the first Chern number and the Euler characteristic of the structure sheaf respectively, has the Euler characteristic χ not exceeding 4. We will also give a complete description for the surfaces of the case χ = 4.