Nonlinear Maxwell Equations in Inhomogeneous Media

We study the basic properties of the Maxwell equations for nonlinear inhomogeneous media. Assuming the classical nonlinear optics representation for the nonlinear polarization as a power series, we show that the solution exists and is unique in an appropriate space if the excitation current is not too large. The solution to the nonlinear Maxwell equations is represented as a power series in terms of the solution of the corresponding linear Maxwell equations. This representation holds at least for the time period inversely proportional to the appropriate norm of the solution to the linear Maxwell equation. We derive recursive formulas for the terms of the power series for the solution including an explicit formula for the first significant term attributed to the nonlinearity.     

On p-Extensions of Fields of Characteristic p

It is proved that a Galois extension of a field of characteristic p is completely determined by its Galois group and the endomorphism of the additive group of the group algebra that corresponds to the raising to the power p.     

Automorphisms of Green Orders and Their Derived Categories

In an earlier paper, Raphaël Rouquier and the author introduced the group of self-equivalences of a derived category. In the case of a Brauer tree algebra, we determined a nontrivial homomorphism of the Artin braid group to this group of self-equivalences. The class of Brauer tree algebras include blocks of finite group rings over a large enough field with cyclic defect groups. In the present paper we give an integral version of this homomorphism. Moreover, we identify some interesting arithmetic subgroups with natural groups of self-equivalences of the derived category.     

Polarized Fluorescence of Jet-Cooled indole and Carbazole and Their Complexes with Water

The spectra of the fluorescence excitation within the rotational contours of the bands of the pure electronic long-wavelength S ₀-S ₁ transitions of jet-cooled indole and carbazole molecules and their complexes with water are measured. For the carbazole-water complex, a contour with three maxima is registered, which is possibly related to the occurrence of two isomers, differing in a slight displacement of hydrogen between the nitrogen atom of the imine group of carbazole and the oxygen atom of the water molecule. The degrees of polarization of integral fluorescence upon excitation within the rotational contours of the S ₀-S ₁ electronic transition bands of the above molecules and their complexes with water are determined for the first time. The coincidence of the calculated (7.7%) and measured (7.3%) values of the degree of polarization upon excitation in the rotational Q branch of the ^b L ₁-A electronic transition of indole confirms the accepted intramolecular orientation of the transition dipole moment at an angle of 38.3° with respect to the principal axis of inertia A. Upon excitation of indole, its complex with water, and carbazole into the P and R branches, the measured and calculated degrees of polarization are also close to each other and amount to 2–3%. This confirms the occurrence of contributions to the fluorescence polarization due to the rotations of the indole molecules around the principal axes of inertia A and C.     

Continuous modified Newton’s-type method for nonlinear operator equations

A nonlinear operator equation F(x)=0, F:H→H, in a Hilbert space is considered. Continuous Newton’s-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point x ₀ and becoming asymptotically close to a solution of F(x)=0 as t→+∞ are discussed. Well-posed and ill-posed problems are investigated. Received: June 29, 2001; in final form: February 26, 2002?Published online: February 20, 2003 This paper was finished when AGR was visiting Institute for Theoretical Physics, University of Giessen. The author thanks DAAD for support     

Generic polynomials for transitive subgroups of the group S 4 over fields of characteristic 2

Generic polynomials for the symmetric group S₄, the Klein group V₄, the cyclic group of order 4, the dihedral group D₄, and the alternating group A₄ over fields of characteristic 2 are described explicitly. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 247–258.