On the discrete spectrum of a given SO(2) symmetry of multiparticle systems in potential and homogeneous magnetic fields

For a system of n identical particles in a homogeneous magnetic field, the discrete spectrum of the Hamiltonian H^{α, m} on the subspaces of functions with permutational symmetry α and rotational (SO(2)) symmetry m is studied as m→∞. It is proved that the discrete spectrum of the operator H^α,m contains only one eigenvalue if certain conditions are satisfied. The asymptotic behavior of this eigenvalue as m→∞ is found. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 28–41, 1992. Translated by A. V. Lyakhovskaya     

Mosaic approximations of discrete analogs of Calderón-Zygmund operators

Asymptotic estimates of the form mrA = O(InN · ln ^d ɛ ⁻¹), whered is the dimension of the initial space, for mosaic ranks of discrete analog of Calderón-Zygmund operators are obtained for various mosaic covers. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 81–94, January, 1998.     

Discretization of multidimensional submanifolds associated with spin-valued spectral problems

We present a large family of Spin(p, q)-valued discrete spectral problems. The associated discrete nets generated by the so-called Sym-Tafel formula are circular nets (i.e., all elementary quadrilaterals are inscribed into circles). These nets are discrete analogues of smooth multidimensional immersions in ℝ^m including isothermic surfaces, Guichard nets, and some other families of orthogonal nets. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 253–262, 2006.     

Geometric minimality conditions for systems of exponentials inL p [−π, π]

A proof is given of the stability theorem for minimal systems of exponentialse(Λ) = {e ^iλx }λ∈Λ inL _p [−π, π], where Λ ⊂ ℂ is a discrete subset. Geometric minimality conditions for such systems are obtained. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 773–777, November, 1995. I wish to express gratitude to A. A. Shkalikov, who posed the problem and paid constant attention to this work.     

Uniqueness of weak solutions of time-dependent 3-D Ginzburg-Landau model for superconductivity

We prove the uniqueness of weak solutions of the time-dependent 3-D Ginzburg-Landau model for superconductivity with (Ψ ₀, A ₀) ∈ L ²(Ω) initial data under the hypothesis that (Ψ, A) ∈ C([0, T]; L ³(Ω)) using the Lorentz gauge.      

Boundary values of holomorphic functions, singular unitary representations ofO(p,q), and their limits asq→∞

Let Ω be a bounded circular domain in ℂ ^N , let M be a submanifold in the boundary of Ω, and let H be a Hilbert space of holomorphic functions in Ω. We show that, under certain conditions stated in terms of the reproducing kernel of the space H, the restriction operator to the submanifold M is well defined for all functions from H. We apply this result to constructing a family of “singular” unitary representations of the groups SO(p,q). The singular representations arise as discrete components of the spectrum in the decomposition of irreducible unitary highest weight representations of the groups U(p,q) restricted to the subgroups SO(p,q). Another property of the singular representations is that they persist in the limit as q→∞. Bibliography: 70 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 9–91. Translated by B. Bekker.     

An A ∞ structure on simplicial complexes

We consider a discrete (finite-difference) analogue of differential forms defined on simplicial complexes, in particular, on triangulations of smooth manifolds. Various operations are explicitly defined on these forms including the exterior differential d and the exterior product ∧. The exterior product is nonassociative but satisfies a more general relation, the so-called A _∞ structure. This structure includes an infinite set of operations constrained by the nilpotency relation (d + ∧ + m + …)ⁿ = 0 of the second degree, n = 2. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 3–37, July, 2008.     

Spectral problems for pencils of polynomial matrices. Methods and algorithms. V

Methods and algorithms for the solution of spectral problems of singular and regular pencils D(λ, μ)=A(μ)-λB(μ) of polynomial matrices A(μ) and B(μ) are suggested (the separation of continuous and discrete spectra, the computation of points of a discrete spectrum with the corresponding, Jordan chains, the computation of minimal indices and a minimal basis of polynomial solutions, the computation of the determinant of a regular pencil). Bibliography: 13 titles. Translated by V. N. Kublanovskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 26–70     

Information complexity of projection algorithms for the solution of Fredholm equations of the first kind. I

We construct a new system of discretization of the Fredholm integral equations of the first kind with linear compact operators A and free terms from the set Range (A(A*A)^V), v > 1/2. The approach proposed enables one to obtain the optimal order of error on such classes of equations by using a considerably smaller amount of discrete information as compared with standard schemes. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 699–711, May, 1998.     

Bernstein widths of embeddings of Lebesgue spaces

Let (K, μ) be a measurable space with μ(K)=1. Let Ip,q: L^p (K, μ)→L^q (K, μ) be the embedding operator. The Bernstein widths of I_{p, q} are considered. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 166–169. Translated by S. V. Kislyakov.     

Groups generated by discrete sets of reflections

It is proved that a centroaffine group G generated by a discrete G-invariant set of reflections of Euclidean space E ³ is discrete. Affine nondiscrete groups generated by reflections in E ³ but containing only discrete sets of reflections are constructed. Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 59–66, 1991.     

Discrete analogs of Fisher information and least favorable distributions in robust estimation of location parameters

The analog of Fisher information for discrete lattice distributions is introduced. The variational problems of minimization of Fisher information are solved for the discrete analogs of the classes of finite distributions, nonsingular densities and ε-contaminated distributions. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 14–18, Perm, 1991.     

The Inverse Problem for a Discrete Periodic Schrodinger Operator

We study isospectral sets for a discrete 1D Schrodinger operator on ℤ with an (N + 1)-periodic potential. We show that for small odd potentials, the isospectral set consists of 2^{(N + 1)/2} elements, while for large potentials, the isospectral set consists of (N +1)! elements. Moreover, asymptotics for endpoints of the spectrum of the Schrodinger operator for small (and large) potentials are determined. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 315, 2004, pp. 96–101.     

Yang-baxterization of the quantum dilogarithm

A new solution of the Yang-Baxter equation with spectral parameter is found. The resulting R-matrix R(x) is an operator inH⊗H, whereH=L ₂(ℝ). This R-matrix is required to justify the solution of the sine-Gordon model on the discrete space-time. Bibliography: 18 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 146–154. Translated by B. Bekker.     

The initial boundary-value problem with a free surface condition for the penalized equations of aqueous solutions of polymers

In this paper, we study some nonlocal problems for the Kelvin-Voight equations (1) and the penalized Kelvin-Voight equations (2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove that these problems have global smooth solutions of the classW _∞ ¹ (ℝ⁺;W ₂ ^2+k (Ω)),k=1,2,...;Ω⊂ℝ³. Bibliography: 25 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 185–207. Translated by N. A. Karazeeva.     

Conservation and bifurcation of an invariant torus of a vector field

We consider small perturbations with respect to a small parameter ε≥0 of a smooth vector field in ℝ^n+m possessing an invariant torusT ^m. The flow on the torusT ^m is assumed to be quasiperiodic withm basic frequencies satisfying certain conditions of Diophantine type; the matrix Ω of the variational equation with respect to the invariant torus is assumed to be constant. We investigate the existence problem for invariant tori of different dimensions for the case in which Ω is a nonsingular matrix that can have purely imaginary eigenvalues. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 34–44, January, 1997. Translated by S. K. Lando     

On simultaneous diophantine approximations. Vectors of given diophantine type

For any monotone functionψ(y)=O(y ^1/s), we prove the existence of a continual family of vectors (α₁...,α_s) admitting infinitely many simultaneous ψ-approximations, but nocψ-approximations with some constantc>0. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 706–716, May, 1997. Translated by S. K. Lando     

Vacuum solutions in the Neveu-Schwarz field theory of a fermionic string and the zero-curvature representation in graded spaces

We demonstrate that vacuum solutions in the Neveu-Schwarz field theory of a fermionic string incorporating the GSO(−) sector are naturally related to the zero-curvature representation in a graded space of special form. We use the same representation to describe the equivalence of the cubic and nonpolynomial theories with the GSO(−) sector also taken into account. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 131–141, April, 2009.     

Gauges for the cookie-cutter sets

Let E be a cookie-cutter set with dimH E =s. It is known that the Hausdorff s-measure and the packing s-measure of the set E are positive and finite. In this paper, we prove that for a gauge function g the set E has positive and finite Hausdorff g-measure if and only if 0 〈 liminft→0 g（t）/ts 〈 ∞. Also, we prove that for a doubling gauge function g the set E has positive and finite packing g-measure if and only if 0 〈 lim supt→0 g（t）/ts 〈 ∞.     

Integral representations of functions and embeddings of sobolev spaces on cuspidal domains

Some classes of cuspidal domainsG ⊂ ℝ ⁿ are introduced, and embeddings of the formW _p ^(l) (G)↪L_q(G),l ∈ ℕ, for sobolev spaces are established. To this end, estimates of some integral operators are needed. These operators cannot be estimated via Riesz potentials or their anisotropic analogs. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 201–219, February, 1997. Translated by V. E. Nazaikinskii