There has been some interest on how the average character degree affects structure of a finite group. We define, and denote by $ \mathrm{anz}(G) $, number zeros characters group G as in table divided irreducible $. show that if < 1 then is solvable also \frac{1}{2} supersolvable. characterise abelian groups showing \frac{1}{3} only abelian.

The aim of this paper is to introduce the theory of Abelian integrals for holomorphic foliations in a complex manifold of dimension two. We will show the importance of Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory. As an application we identify some irreducible components of the space of holomorphic foliations of a fixed degree and with a center singu...

Criterion for the Riemann hypothesis found by Báez-Duarte involves certain real coefficients ck defined as alternating binomial sums. These coefficients can be effectively investigated using Rice’s integrals. Their behavior exhibits typical trend (due to trivial zeros of zeta) and fading oscillations (due to complex zeros). This method enables to calculate ck for large values of k at least to k...

We consider the quadrature method developed by Kravanja and Van Barel (Computing 63(1):69–91, 1999) for computing all the zeros of an analytic function that lie inside the unit circle. The algorithm uses only the function values and no (first or higher order) derivatives. Information about the location of the zeros is obtained from certain integrals along the unit circle. In numerical computati...

We study triangulation schemes for the joint kernel of a diffusion process with uniformly continuous coefficients and an adapted, non-resonant Abelian process. The prototypical example of Abelian process to which our methods apply is given by stochastic integrals with uniformly continuous coefficients. The range of applicability includes also a broader class of processes of practical relevance,...

We reexamine and continue the work of J. Vosmanskyý [23] on the concept of continuous ranking of zeros of certain special functions from the point of view of the transformation theory of second order linear differential equations. This leads to results on higher monotonicity of such zeros with respect to the rank and to the evaluation of some definite integrals. The applications are to Airy, Be...

Riemann Zeta Function 602 25.2 Definition and Expansions . . . . . . . . 602 25.3 Graphics . . . . . . . . . . . . . . . . . . 603 25.4 Reflection Formulas . . . . . . . . . . . . 603 25.5 Integral Representations . . . . . . . . . 604 25.6 Integer Arguments . . . . . . . . . . . . 605 25.7 Integrals . . . . . . . . . . . . . . . . . . 606 25.8 Sums . . . . . . . . . . . . . . . . . . . 606 25....

Verification of new identity for the Green functions in N = 1 supersymmetric non-Abelian Yang–Mills theory with the matter fields. Abstract We investigate a new identity for Green functions using the higher covariant derivative regularization. It relates some coefficients in the vertex function of the matter superfield, in which one of external matter lines is not chiral. The calculation in the...