It is well-known that the Euler product for the Riemann zeta function ζ(s) is still valid for !(s) = 1 and s "= 1. In this paper, we extend this result to zeta functions of number fields. In particular, the Dedekind zeta function ζk(s) for any algebraic number field k and the Hecke zeta function ζ(s,χ) for the rational number field are shown to have the Euler product on the line !(s) = 1 except...

Solutions to the equation ∂tE(x, t) − i 2m E(x, t) = λ|S(x, t)|2 E(x, t) are investigated, where S(x, t) is a complex Gaussian field with zero mean and specified covariance, and m = 0 is a complex mass with Im(m) ≥ 0. For real m this equation describes the backscattering of a smoothed laser beam by an optically active medium. Assuming that S(x, t) is the sum of a finite number of independent co...

In this letter we describe a new and simple approach for modulating the electromagnetic wave propagating in a dielectric medium in the form of solitons by considering the torque developed between the induced dipoles in the medium and external field without taking into account the nonlinear Kerr effect. A reductive perturbation method deduces the Maxwell equation coupled with the Larmor equation...

We discuss magnetoexcitons dispersion in single and coupled GaAs − (Ga,Al)As quantum wells using the Bethe-Salpeter (B-S) formalism. The B-S formalism in the case of quantum wells provides an equation for the exciton wave function which depends on two space variables plus the time variable, i.e. the B-S equation is 2 + 1-dimensional equation. We compare the results for magnetoexcitons dispersio...

to determine the coefficients of c1 and c in mitscherlich-bray equation, critical level and fertilizer recommendation for potassium (k) as by mitscherlich-bray (mb) equation, an experiment in a randomized complete block design of 5 treatments and 3 replications was carried out in 12 irrigated wheat (triticum aestivum l.) fields in sarpol-e-zohab region during the 2007-08 growing season. the tre...

Passive scalar equation is considered in a turbulent homogeneous incompressible Gaussian velocity field. The turbulent nature of the field results in non-smooth coefficients in the equation. A strong, in the stochastic sense, solution of the equation is constructed using the Wiener Chaos, and the properties of the solution are studied. The results apply to both viscous and conservative motions....

Passive scalar equation is considered in a turbulent homogeneous incompressible Gaussian velocity field. The turbulent nature of the field results in non-smooth coefficients in the equation. A strong, in the stochastic sense, solution of the equation is constructed using the Wiener Chaos, and the properties of the solution are studied. The results apply to both viscous and conservative motions....

In a space-time M with Killing vector field ξa which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of ξa gives a 3-dimension space S. An action of the fields on S is proposed which gives the same fields equations as those reduced from the vacuum Einstein equation on M . PACS number(s): 04.20.Fy, 04.20.Cv

We investigate a generalization of many functional equations. Namely, we consider the following equation ?S f(x + y t) d?(t) ?(y) = f(x) h(y), x, ? S, where (S,+) is an abelian semigroup, surjective endomorphism E linear space over field K {R, C} and ?,? are combinations Dirac measures. Under appropriate conditions on based Stetkar?s result [9], find characterize solutions previous equation.

Inverters provided non-sinusoidal voltage or current, and the tooth-slot and winding distribution of the motor could lead to a mass of space harmonic components, severely deteriorating the motor performance. In order to know the cause of electromagnetic thrust and mover velocity fluctuation of low-velocity PMLSM fed by SPWM-VS, the steady-state performances of low-velocity PMLSM are analyzed in...