1. Abelian functions by Poincaré. 1-1. If the variable x and a general solution y of a linear differential equation with polynomial coefficients are algebraically dependent, the periods of abelian integrals of the first kind associated with the curve f(x, y) = 0 satisfy certain relations. In his earliest works on abelian functions Poincaré examined such relations in some special cases. He also ...

<p style='text-indent:20px;'>Dumortier and Roussarie proposed a conjecture in their paper (2009, Discrete Con. Dyn. Sys., 2,723-781): For any <inline-formula><tex-math id="M1">\begin{document}$ q\in {\mathbb{N}} $\end{document}</tex-math></inline-formula>, the Abelian integrals id="M2">\begin{document}$ J_{2j+1}(h) = \int_{\gamma_h}x^{2j-1}\,\mathrm dy id="M3"&g...

In this paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the counting function $\pi(x)$ and Chebyshev $\vartheta$-function. Some of these depend correctness Riemann hypothesis nontrivial zeros zeta $\zeta(s)$.

A dynamic system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for general non-linear dynamical systems. In this paper, we investigated a class of non-linear systems under perturbations. We proved that the upper bound of the number of zeros of the related elliptic integrals of the given system is 7n + 5 including multiple ...

Let P (x) be a real polynomial of degree 2g + 1, H = y2 + P (x) and δ(h) be an oval contained in the level set {H = h}. We study complete Abelian integrals of the form

in this paper an algorithm is presented for the regularization of singular integrals with any degrees of singularity, which may be employed in all three-dimensional problems analyzed by boundary elements. the integrals in boundary integrals equations are inherently singular. for example, one can mention the integrals confronted in potential problems to evaluate the flow or the gradient of the f...