On some generalizations of the factorization method

The classical factorization method reduces the study of a system of ordinary differential equations U_t=[U₊, U] to solving algebraic equations. Here U(t) belongs to a Lie algebra which is the direct sum of its subalgebras and , where “+” signifies the projection on . We generalize this method to the case . The corresponding quadratic systems are reducible to a linear system with variable coefficients. It is shown that the generalized version of the factorization method can also be applied to Liouville equation-type systems of partial differential equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 339–350, March, 1997.     

Quantum duality in quantum deformations

In accordance with the quantum duality principle, the twisted algebra is equivalent to the quantum group and has two preferred bases: one inherited from the universal enveloping algebra and the other generated by coordinate functions of the dual Lie group . We show howthe transformation can be explicitly obtained for any simple Lie algebra and a factorable chain of extended Jordanian twists. In the algebra , we introduce a natural vector grading , compatible with the adjoint representation of the algebra. Passing to the dual-group coordinates allows essentially simplifying the costructure of the deformed Hopf algebra , considered as a quantum group . The transformation can be used to construct new solutions of the twist equations. We construct a parameterized family of extended Jordanian deformations and study it in terms of ; we find new realizations of the parabolic twist. Dedicated to the birthday of my teacher, Yurii Novozhilov __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 112–125, July, 2006.     

Dispersion of Lagrangian trajectories in a random large-scale velocity field

We study the distribution of the distance R(t) between two Lagrangian trajectories in a spatially smooth turbulent velocity field with an arbitrary correlation time and a non-Gaussian distribution. There are two dimensionless parameters, the degree of deviation from the Gaussian distribution α and β=τD, where τ is the velocity correlation time and D is a characteristic velocity gradient. Asymptotically, R(t) has a lognormal distribution characterized by the mean runaway velocity and the dispersion Δ. We use the method of higher space dimensions d to estimate and Δ for different values of α and β. It was shown previously that for β≪ 1 and for β≫ 1. The estimate of Δ is then nonuniversal and depends on details of the two-point velocity correlator. Higher-order velocity correlators give an additional contribution to Δ estimated as αD²τ for β≪1 and αβ/τ for β≫1. For α above some critical value σ_cr, the values of and Δ are determined by higher irreducible correlators of the velocity gradient, and our approach loses its applicability. This critical value can be estimated as for β≪1 and for β≫1. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 456–467, March, 2000.     

Miura-type transformations of nonlinear partial differential equations integrable by the\bar \partial-problem methodmethod

A new method is suggested for constructing Miura-type transformations for systems of nonlinear equations of mathematical physics integrable by the -problem method using a generalized normalization function in the -problem. Examples of these transformations for the Kadomtsev-Petviashvili equation are considered. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 47–54, April, 1999.     

Orthogonal polynomials of a discrete variable and Lie algebras of complex-size matrices

We give a uniform interpretation of the classical continuous Chebyshev and Hahn orthogonal polynomials of a discrete variable in terms of the Feigin Lie algebra for λ∈ℂ. The Chebyshev and Hahn q-polynomials admit a similar interpretation, and orthogonal polynomials corresponding to Lie superalgebras can be introduced. We also describe quasi-finite modules over , real forms of this algebra, and the unitarity conditions for quasi-finite modules. Analogues of tensors over are also introduced. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 205–236, May, 2000.     

Zamolodchikov-Faddeev algebras for Yangian doubles at level 1

The representation theory of centrally extended Yangian doubles is investigated. The intertwining operators are constructed for infinite dimensional representations of , which are deformed analogs of the highest weight representations of the affine algebra at level 1. We give bosonized expressions for the intertwining operators and verify that they generate an algebra isomorphic to the Zamolodchikov-Faddeev algebra for the SU(2)-invariant Thirring model. From them, we compose L-operators by Miki’s method and verify that they coincide with L-operators constructed from the universal R-matrix. The matrix elements of the product of these operators are calculated explicitly and are shown to satisfy the quantum (deformed) Knizhnik-Zamolodchikov equation associated with the universal R-matrix for . This paper was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 25–45. January, 1997.     

Galois groups and complete domains

Consider a domain that is complete with respect to a non-zero prime ideal. This paper proves two Galois-theoretic results about such rings. Using Grothendieck’s Existence Theorem we prove that every finite group occurs as the Galois group of a Galois extension of . This generalizes results of David Harbater who proved the result in the case where the ideal is maximal and the domain is normal. As a consequence, we deduce that if is a Noetherian domain that is complete with respect to a non-zero prime ideal, then every finite group occurs as a Galois group over . This proves the Noetherian case of a conjecture posed by Moshe Jarden.     

Martin boundary of unlimited covering surfaces

LetW be an open Riemann surface and ap-sheeted (1<p<∞) unlimited covering surface ofW. Denote by Δ¹ (resp., ) the minimal Martin boundary ofW (resp., ). For ζ ∈ Δ, let ζ be the (cardinal) number of the set of pionts which lie over ζ and the class of open connected subsetsM ofW such thatM∪{ζ} is a minimal fine neighborhood of ζ. Our main result is the following: , where is the number of components of π^-1 M and π is the projection of ontoW. Moreover, some applications of the above results are discussed whenW is the unit disc.     

Polarization Tomography of Quantum Radiation: Theoretical Aspects and Operator Approach

We present theoretical foundations for the quantum tomography of polarization states of light fields as a method for measuring their polarization density operator , which characterizes only the polarization degrees of freedom of the radiation. We mainly attend to the method in which the tomographic observables (the “measurement instruments”) are polarizable in nature. We show that the quantum nature of this method can be adequately expressed using the quasispectral tomographic decompositions of in special operator bases, which are finite sums of partially orthogonal projection operators determining the probability distributions of tomographic observables as the decomposition coefficients. We obtain the matrix versions of such “tomographic” representations of , in particular, by projecting them on semiclassical operator bases determining the polarization quasiprobability functions. We briefly discuss the information aspects of the schemes considered in the paper. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 3, pp. 344–357, December, 2005.     

Some properties of the absolute galois group of a hilbertian field

LetK be a hilbertian field,G(K) its absolute Galois group. IfK is countable, then for a.a. inG(K) ^e , and there is no intermediate field with . Let ∈G(K) ^e . Then for a.a. in .     

3-Folds in ℙ5 of degree 12of degree 12

Let be a 3-dimensional submanifold of ℙ⁵ of degree 12. This article gives, up to one case, a complete classification of the deformation classes of those 3-folds. The main tools used are methods already applied in the classification of degrees 9 to 11 and adjunction theoretic results. We show here how the 2^nd reduction of can be applied to analyse the birational structure of or even exclude the existence of .     

The closure diagram for nilpotent orbits of the split real form of E8

Let and be adjoint nilpotent orbits in a real semisimple Lie algebra. Write ≥ if is contained in the closure of . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E ₈. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible components of the boundaries and of the intersections .     

Necessary and sufficient conditions for inclusion relations for absolute summability

We obtain a set of necessary and sufficient conditions for to imply for 1 <k ≤ s < ∞. Using this result we establish several inclusion theorems as well as conditions for the equivalence of and .     

Nichtlineares invariantes Differential

Zusammenfassung Das invariante Differential eines Vektors ist selbst ein Vektor, der vom Orte xⁱ, von dxⁱ, und von selbst abh?ngig ist. Es wird nun infolge dieser Definition in einigen F?llen die m?gliche Form des invarianten Differentials bestimmt. Enrico Bompiani zu seinem wissenschaftlichen Jubil?um.     

Asymptotic properties of the maximum likelihood estimate in the first order autoregressive process

Summary In this paper we obtain an asymptotic expansion of the distribution of the maximum likelihood estimate (MLE) based onT observations from the first order Gaussian process up to the term of orderT ⁻¹. The expansion is used to compare with a generalized estimate including the least square estimate (LSE) , based on the asymptotic probabilities around the true value of the estimates up to the terms of orderT ⁻¹. It is shown that (or the modified MLE ) is better than (or the modified estimate ). Further, we note that does not attain the bound for third order asymptotic median unbiased estimates.     

Quantum integrability and quantum chaos in the micromaser

The time-dependent quantum Hamiltonians

Application des inequations quasi-variationnelles a quelques problemes non lineaires de la physique des plasmas

This paper deals with some problems arising in plasma physics. The typical example is the following: where is the (neither local, nor monotone, nor continuous) operator: . Using a quasi-variational approach, we prove the existence of minimal and maximal solutions for a weak form of this problem, involving a multi-valued operator β. Various generalizations are treated.      

Eine Verallgemeinerung derCW-Komplexe

In the definition ofCW-complexes, the one-point spaceP, respectively the spaceP∪* with basepoint *, play the roll of the only “building-stone”. Let be a family of compact spaces. Then the definition of a generalizedCW-complex over is obtained from the definition of aCW-complex by replacingP by the spaces of and formation of the mapping cone by a slightly modified construction. LetCW * denote the category of all pointed spaces which have the homotopy type of a generalizedCW-complex over . If , thenCW * is the category of all pointedCW-spaces.CW * is closed under the formation of direct sums and of mapping cones, cylinders and tori, and is formally characterized as the smallest such subcategory of Top * containing the spaces W∪*, . Following the methods of E. H. Brown, it is proved, that any half exact homotopy functor onCW * is representable, and any cohomology theory onCW is naturally equivalent to the cohomology theory of an Ω-spectrum; for example, the singular cohomo logy is representable onCW for any family of compact spaces.      

Primitive systems of elements in the variety\mathfrak{A}\mathfrak{N}_2 and some locally finite varieties of groupsand some locally finite varieties of groups

Part of any basis of a relatively free group in the variety is called a primitive system of elements. We provide a criterion of being primitive for , where is a variety of Abelian groups satisfying x^m=1, and a variety generated by a finite group. Let be a variety of nilpotent groups of class ≤c. It is proved that, for the group , the property of being primitive for an element g is stronger than the condition of being unimodular on a vector composed of values of Fox derivatives in the ring . The group is not residually finite whenever a system of elements is primitive. Supported by RFFR grant No. 96-01-01948. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 687–699, November–December, 1998.     

Representations of infinite-dimensional algebras and conformal field theory: FromN = 2 to\widehat{sl}(2|1)(2|1)

We review several constructions that are realized in bosonic and N = 2 strings and which relate the affine Lie algebra (2), affine superalgebra (2|1), and the superconformal N = 2 algebra. This paper was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 2, pp. 195–240, August, 1997.