Invertibility of linear differential operators with closed range and poisson coefficients

The invertibility and injectivity properties of linear differential operators with closed range and Poisson coefficients are studied in the context of their equivalence in several spaces of vector functions defined on the axis. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 143–147, January, 1999.     

Kaup-Newell方程族等谱和一阶非等谱之间的规范变换

通过引入等谱和一阶非等谱Ablowitz-Kaup-Newell-Segur系统,得到了等谱KaupNewell(KN)谱问题和一阶非等谱KN谱问题的规范变换,并建立了等谱和一阶非等谱KN方程族之间的等价关系。     

Approximate Reasoning Based on Similarity

The connection between similarity logic and the theory of closure operators is examined. Indeed one proves that the consequence relation defined in [14] can be obtained by composing two closure operators and that the resulting operator is still a closure operator. Also, we extend any similarity into a similarity which is compatible with the logical equivalence, and we prove that this gives the same consequence relation.     

Complexity of finite sequences of zeros and ones and geometry of finite spaces of functions

The sequence 001 001 001 001 is simpler than 01 001 0111 001. We present a formal mathematical theory that gives an exact meaning to the preceding (intuitively clear) statement. Supported in part by the RFBR (Grant No. 05-01-00104). Read at the Moscow Mathematical Society meeting 22 November 2005, homepage: .     

The Cauchy problem for a spin-liquid model in three space dimensions

In this paper, we study the Cauchy problem for magnetic fluid of spin-liquid type with Mermin-Ho relation in the three-dimensional space and prove global existence and uniqueness of solutions. The proof is based on the equivalence relation between smooth solutions of the spin-liquid model and the systems of Schrödinger equations with Abelian gauge field. The Sobolev spaces with fractional derivatives are also used in our estimates.     

Maslov Complex Germ Method for Systems with First-Class Constraints

We develop the semiclassical mechanics of systems with first-class constraints. A convenient quantization method is the method based on modifying the inner product used in the theory. We consider semiclassical states of the wave-packet type (with small indeterminacies in both coordinates and momenta) that appear in the theory of the Maslov complex germ at a point. We show that these states have a nonzero norm only if the classical coordinates and momenta lie on the constraint surface. The set of semiclassical states of the wave-packet type forms a (semiclassical) bundle whose base is the set of admissible classical states and whose fibers are function spaces determining the form of the wave packet. In some cases, the difference between two semiclassical states has a zero norm; it is therefore possible to introduce the gauge equivalence relation. The semiclassical gauge transformations that are automorphisms of the semiclassical bundle form a Batalin quasigroup. We also study the action of semiclassical observables and of semiclassical evolution transformations. We show that they preserve the norm and the gauge equivalence relation and that the observables coinciding on the constraint surface act on semiclassical states similarly up to the gauge invariance.     

A hierarchy of generalized invariants for linear partial differential operators

We study invariants of linear partial differential operators in two variables under gauge transformations. Using the Beals-Kartashova factorization, we construct a hierarchy of generalized invariants for operators of an arbitrary order. We study the properties of these invariants and give some examples. We also show that the classic Laplace invariants correspond to some particular cases of generalized invariants. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 470–478, June, 2006.     

Indecomposability of treed equivalence relations

We define rigorously a “treed” equivalence relation, which, intuitively, is an equivalence relation together with a measurably varying tree structure on each equivalence class. We show, in the nonamenable, ergodic, measure-preserving case, that a treed equivalence relation cannot be stably isomorphic to a direct product of two ergodic equivalence relations.     

Singular Perturbation Elliptic Boundary-Value Problems in the Case of a Nonisolated Root of the Degenerate Equation

For a singularly perturbed elliptic equation (the Neumann boundary-value problem), we prove a theorem on the passage to the limit for the case in which the degenerate equation has a nonisolated root.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 26–36.Original Russian Text Copyright © 2005 by V. F. Butuzov, M. A. Terent’ev.     

Canonical form of Tarski sets in Zermelo-Fraenkel set theory

We establish the equivalence of the notions of an inaccessible cumulative set and uncountable Tarski set. In addition, the equivalence of these notions and that of a galactic set is proved.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 323–333.Original Russian Text Copyright © 2005 by E. I. Bunina, V. K. Zakharov.This revised version was published online in April 2005 with a corrected issue number.     

Unitary equivalence of one-parameter groups of Toeplitz and composition operators

The composition operators on H² whose symbols are hyperbolic automorphisms of the unit disk fixing ±1 comprise a one-parameter group and the analytic Toeplitz operators coming from covering maps of annuli centered at the origin whose radii are reciprocals also form a one-parameter group. Using the eigenvectors of the composition operators and of the adjoints of the Toeplitz operators, a direct unitary equivalence is found between the restrictions to zH² of the group of Toeplitz operators and the group of adjoints of these composition operators. On the other hand, it is shown that there is not a unitary equivalence of the groups of Toeplitz operators and the adjoints of the composition operators on the whole of H², but there is a similarity between them.     

On the spectrum of a differential operator of high-order

In this paper, sufficient conditions for the spectrum of the operator of high order to be discrete and unbounded below are obtained.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 188–193.Original Russian Text Copyright © 2005 by M. G. Gimadislamov.This revised version was published online in April 2005 with a corrected issue number.     

Generalized isoclinism and characters of finite groups

In this paper, we deal with equivalence relations on groups, inspired by the relation of isoclinism due to P. Hall. Isoclinism is — essentially — defined by the commutator map of groups. We take another map and define related equivalence relations. We shall show that modular character degrees and modular monomiality of finite groups are preserved under such an equivalence relation.     

Some Continuous Analogs of the Expansion in Jacobi Polynomials and Vector-Valued Orthogonal Bases

We obtain the spectral decomposition of the hypergeometric differential operator on the contour Re z = 1/2. (The multiplicity of the spectrum of this operator is 2.) As a result, we obtain a new integral transform different from the Jacobi (or Olevskii) transform. We also construct an ₃ F ₂-orthogonal basis in a space of functions ranging in ℂ². The basis lies in the analytic continuation of continuous dual Hahn polynomials with respect to the index n of a polynomial.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 31–46, 2005Original Russian Text Copyright © by Yu. A. Neretin     

A quantum model with one fermionic degree of freedom

A quantum model with one fermionic degree of freedom is discussed in detail. The operator action of the model has local operator gauge symmetry. A group of constrains on operator gauge potentialB ₀ and gauge transformation operatorU from some physical requirement are obtained. The Euler-Lagrange equation of motion of fermionic operator φ is just the usual equation of motion of fermion type. And the Euler-Lagrange equation of motion of operator gauge potentialB ₀ is just a constraint, which is just. the canonical quantization condition of fermion.     

Fixed point theorems in metric spaces

In this paper, we establish two general theorems for equivalence between the Meir–Keeler type contractive conditions and the contractive definitions involving gauge functions. One of these theorems is an extension of a recent result of Lim (On characterization of Meir–Keeler contractive maps, Nonlinear Anal. 46 (2001) 113–120).     

Complete integrability,gauge equivalence and Lax representation of inhomogeneous nonlinear evolution equations

The gauge equivalence between the inhomogeneous versions of the nonlinear Schrödinger and the Heisenberg ferromagnet equations is studied. An unexplicit criterion for integrability is proposed. Examples of gauge equivalent inhomogeneous nonlinear evolution equations are presented. It is shown that in the nonintegrable cases the M-operators in their Lax representations possess unremovable pole singularities lying on the spectrum of the L-operators.To the memory of Michail Constantinovich PolivanovInstitute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 3, pp. 374–386, September, 1992.     

Equivalence classes of functions between finite groups

Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser (“graph”) equivalence are inequivalent under the finer (“bundle”) equivalence. Here we relate graphs to transversals and splitting relative difference sets (RDSs) and introduce an intermediate relation, canonical equivalence, to aid in distinguishing the classes. We identify very precisely the conditions under which a graph equivalence determines a bundle equivalence, using transversals and extensions. We derive a new and easily computed algebraic measure of nonlinearity for a function f, calculated from the image of its coboundary ∂f. This measure is preserved by bundle equivalence but not by the coarser equivalences. It takes its minimum value if f is a homomorphism, and takes its maximum value if the graph of f contains a splitting RDS.     

D Functor for tensor cross products

A natural equivalence relation between crossing cochains and the concept of a strong homotopy of DGA-mappings of co-algebras is introduced. An obstruction theory is constructed for this equivalence relation and a homotopical classification theorem for tensor cross products is proved.Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 465–472, October, 1976.