Newton polyhedra, unstable faces and the poles of Igusa's local zeta function

In this paper we examine when the order of a pole of Igusa's local zeta function associated to a polynomial is smaller than ``expected'. We carry out this study in the case that is sufficiently non-degenerate with respect to its Newton polyhedron , and the main result of this paper is a proof of one of the conjectures of Denef and Sargos. Our technique consists in reducing our question about the polynomial to the same question about polynomials , where are faces of depending on the examined pole and is obtained from by throwing away all monomials of whose exponents do not belong to . Secondly, we obtain a formula for Igusa's local zeta function associated to a polynomial , with unstable, which shows that, in this case, the upperbound for the order of the examined pole is obviously smaller than ``expected'.

     

Selberg-Type Integrals Associated with SL3

We present several formulae for Selberg-type integrals associated with the Lie algebra .     

Calculation of Two-Center Nuclear Attraction Integrals over Integer and Noninteger n-Slater Type Orbitals in Nonlined-Up Coordinate Systems

Two-center nuclear attraction integrals over Slater type orbitals with integer and noninteger principal quantum numbers in nonlined up coordinate systems have been calculated by means of formulas in our previous work (T. Özdoan and M. Orbay, Int. J. Quant. Chem. 87 (2002) 15). The computer results for integer case are in best agreement with the prior literature. On the other hand, the results for noninteger case are not compared with the literature due to the scarcity of the literature, but also compared with the limit of integer case and good agreements are obtained. The proposed algorithm for the calculation of two-center nuclear attraction integrals over Slater type orbitals with noninteger principal quantum numbers in nonlined-up coordinate systems permits to avoid the interpolation procedure used to overcome the difficulty introduced by the presence of noninteger principal quantum numbers. Finally, numerical aspects of the presented formulae are analyzed under wide range of quantum numbers, orbital exponents and internuclear distances.     

On Approximations of First Integrals for Strongly Nonlinear Oscillators

In this paper strongly nonlinear oscillator equations will be studied.It will be shown that the recently developed perturbation method based onintegrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how in a rather efficient way the existence and stability oftime-periodic solutions can be obtained from these approximations. In particularthe generalized Rayleigh oscillator equation will be studied in detail, and it will beshown that at least five limit cycles can occur.     

Existence of positive solutions of nonlinear fractional differential equations

Existence of positive solutions for the nonlinear fractional differential equation D^su(x)=f(x,u(x)), 0<s<1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804-812), where D^s denotes Riemann-Liouville fractional derivative. In the present work we study existence of positive solutions in case of the nonlinear fractional differential equation:

L(D)u=f(x,u),u(0)=0,0<x<1,     

Weighted Young Inequalities for Convolutions

Sufficient conditions for some weighted Young inequalities type are obtained.AMS Subject Classification (1991): 42A85, 42B20, 47G10     

First integrals and reduction of a class of nonlinear higher order ordinary differential equations

We propose a method for constructing first integrals of higher order ordinary differential equations. In particular third, fourth and fifth order equations of the form are considered. The relation of the proposed method to local and nonlocal symmetries are discussed.     

Singular integrals with rough kernels along real-analytic submanifolds in

mapping properties will be established in this paper for singular Radon transforms with rough kernels defined by translates of a real-analytic submanifold in .

     

Modular Hypergeometric Residue Sums of Elliptic Selberg Integrals

It is shown that the residue expansion of an elliptic Selberg integral gives rise to an integral representation for a multiple modular hypergeometric series. A conjectural evaluation formula for the integral then implies a closed summation formula for the series, generalizing both the multiple basic hypergeometric ₈₇ sum of Milne-Gustafson type and the (one-dimensional) modular hypergeometric ₈₇ sum of Frenkel and Turaev. Independently, the modular invariance ensures the asymptotic correctness of our multiple modular hypergeometric summation formula for low orders in a modular parameter.     

On Nonlinear Integro-differential Operators in Generalized Orlicz–Sobolev Spaces

A nonlinear integral operator T of the form (Tf)(s)=∫_G K(t, f (σ(s, t))) dμ(t), for sG, is defined and investigated in the measure space (G, Σ, μ), where f and K are vector-valued functions with values in normed linear spaces E and F, respectively. The results are applied to the case of integro-differential operators in generalized Orlicz–Sobolev spaces. There are studied problems of existence, embeddings, and approximation by means of T.