Gauge-invariant discrete models of Yang-Mills equations

Two discrete models of Yang-Mill equations are constructed in the space ℝⁿ for some matrix-valued Lie group. A gauge-invariant discrete model is examined. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 742–754, May, 1997. Translated by O. V. Sipacheva     

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.     

Finite groups with 2-nilpotent subgroups of even index

Finite groups with 2-nilpotent subgroups of even index are described. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 700–705, May, 1997. Translated by A. I. Shtern     

Cobordisms of embeddings of oriented five-dimensional manifolds in Euclidean space

The groups EΩ_5,k of cobordism classes of embeddings of oriented five-dimensional manifolds in 5+ k-dimensional Euclidean space are studied. It is proved that EΩ_5,3≅ℤ₂ and EΩ_5,4≅ℤ₂⊕ℤ₂ and specific generators are given. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 148–168. Translated by N. Yu. Netsvetaev.     

Bernstein and Gelfand widths of certain classes of analytic functions. II

The Gelfand widths of the unit ball of H²(ν) (the weighted Hardy space) with respect to the metric of the space L_∞(T_r) are considered (here T_r is the circle of radius r centered at the origin), as well as the Bernstein widths of the unit ball of H^∞ with respect to the metric of the space L₂(T_r, μ). Asymptotic formulas for the widths in question are established for arbitrary measures ν, μ. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 134–140. Translated by S. V. Kislyakov.     

Equivalence of commutation relations

We consider C*-algebras of commutation relations over the fields ℚ _p, p = 2, 3, 5, …, ∞. We describe all the irreducible separable representations of these algebras. We prove that the algebras are not isomorphic at different p. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 406–412, December, 2008.     

Stability properties of cycles and tori of a simplest nonresonant wave-type equation

For a number of meaningful examples, nonresonant wave-type equations are shown to be characterized by periodic dynamics. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 744–750, November, 1997. Translated by O. V. Sipacheva     

Electronic and magnetic states in the two-band Hubbard model

Electronic and magnetic states are considered in the two-band Hubbard model. Bibliography:5 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 240–249. Translated by B. Bekker.     

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     

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.     

Quantitative Whitney-type theorems

The crucial point in the construction of an almost optimal decomposition for the couple (L_q, W _p ^k ) was the covering theorem in which the parameter α (α-capacity) was controlled. Here we give a detailed proof of this theorem for negative α. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 96–113. Translated by S. V. Kislyakov.     

The measure of nonconvexity and the jung constant

For a Banach space X, a new constant G(X)=sup {λ(X) | A ⊂ X, d(A)=1} is introduced. The main result is that G (X) coincides with the Jung constant J (X) (Theorem 1), which yields an estimate for the latter. Some other results concerning J (X) and the measure of nonconvexity λ are given. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 174–181. Translated by O. A. Ivanov.     

Continuation of a linear operator to an involution operator

A bounded linear operatorA:X→X in a linear topological spaceX is called ap-involution operator,p≥2, ifA ^p=I, whereI is the identity operator. In this paper, we describe linearp-involution operators in a linear topological space over the field ℂ and prove that linear operators can be continued to involution operators. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 671–676, May, 1997. Translated by M. A. Shishkova     

Embeddings of the classesH ω

In this paper we study functions belonging to the classesV _ε and ΛBV, which are encountered in the theory of Fourier trigonometric series. Necessary and sufficient conditions for the embedding of the classesH _ω in the classesV _ϕ and ABV are obtained. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 713–719, November, 1998. This research was supported by the program “Leading Scientific Schools” under grant No. 96/97-15-96073.     

Elliptic Operators in a Refined Scale of Functional Spaces

We study the theory of elliptic boundary-value problems in a refined two-sided scale of the Hormander spaces H ^s , ^ϕ, where s ∈ R and ϕ is a functional parameter slowly varying at +∞. For the Sobolev spaces H ^s , the function ϕ(|ξ|) ≡ 1. We establish that the considered operators possess the Fredholm property, and solutions are globally and locally regular. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 689–696, May, 2005.     

Lamb's inverse axially symmetric problem

The problem of restoring the velocity of transversal waves in an elastic half-space by a known wave field on the boundary of a medium is studied. The problem is reduced to a system of “integral” equations similar to the Volterra one. Theorems on existence in the small and on uniqueness of the solution are proved. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 51–67. Translated by T. N. Surkova.     

Justification of a construction of the solution of the problem on a point source ofSH waves in a layered medium with vertical boundary

The solution is determined by the method of separation of variables. The method is justified by using a suitable space of generalized functions. Boundary conditions “are hidden” in this space. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 56–66. Original article This work was supported by the Russian Foundation of Fundamental Research (Grant No. 93-011-16148). Translated by T. N. Surkova.     

Combinatorial properties of rational representations of the groupGL(n, ℂ)

A generalized Robinson-Schensted-Knuth correspondence related to the decomposition of representations of GL(n, ℂ) in mixed tensor spaces is constructed. Bibliography: 5 titles. To Olga Alexandrovna Ladyzhenskaya on the occasion of her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 83–90. Translated by N. B. Lebedinskaya.     

Strongly Symmetric Polyhedra

Strongly symmetric polyhedra in three-space are defined and classified. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 314–325.     

Anisotropic norms of dirichlet kernels and some other trigonometric polynomials

In the paper new estimates of anisotropic norms for trigonometric polynomials are obtained. Translated fromMatematischeskie Zametki, Vol. 67, No. 5, pp. 686–701, May, 2000.