Homogenization with corrector for a periodic elliptic operator near an edge of inner gap

For an ellipticoperator with rapidly oscillating coefficients we consider a homogenization procedure near the edge of an interior gap in the spectrum of this operator. At a point close to the edge, we obtain an approximation of the resolvent in the operator L ₂(ℝ)-norm. The first order corrector is taken into account in the approximation. Bibliography: 11 titles. Dedicated to dear Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 41, May 2009, pp. 127–142.     

Two inequalities with an application

Summary An inequality used in Brown and Cohen (1974,Ann. Statist., 2(5), 963–976) and Bhattacharya (1978,Ann. Inst. Statist. Math., A,30, 407–414) is generalized and another useful inequality derived from it. An application of the latter which provides a more elegant approach to and an improvement over a result in Shinozaki (1978,Commun. Statist. Theor. Meth., A,7, 1421–1432), is also presented.     

Estimates for weighted eigenvalues of a fourth-order elliptic operator with variable coefficients

We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in \mathbbRⁿ {\mathbb{R}^n} . We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the (k + 1)th eigenvalue in terms of the first k eigenvalues. Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang–Xia inequality (J. Funct. Anal., 245, No. 1, 334–352 (2007)) for the clamped-plate problem to a fourth-order elliptic operator with variable coefficients.     

On the spectrum of the two-dimensional periodic Dirac operator

We prove the absolute continuity of the Dirac operator spectrum inR ² with the scalar potential V and the vector potential A=(A₁, A₂) being periodic functions (with a common period lattice) such that V, A_j≠L _loc ^q (R ²), q>2. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 3–14, January, 1999.     

A new concept of a generalized solution of operator equations in a Banach space

We define a new notion of a generalized solution of an operator equation with a closed linear operator in a Banach space as an element of the completion of this space with respect to some locally convex topology. We prove a theorem on the existence and uniqueness of the generalized solution. Bibliography: 5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 93–99.     

Boundedness of imaginary powers of the stokes operator in an infinite layer

In this article we prove the existence of bounded purely imaginary powers of the Stokes operator , which is defined on the space of solenoidal vector fields < q < , where is an infinite layer. It is a consequence of a special representation of the resolvent of the Stokes operator in terms of the Stokes operator on , a composition of a trace and a Poisson operator – a singular Green operator – and a negligible part.     

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     

Invariant symmetric restrictions of a self-adjoint operator. II

We give a criterion of invariance and symmetry of the restriction of an arbitrary unbounded self-adjoint operator in the space L ₂(ℝⁿ, dx) by using the introduced notion of support of an arbitrary vector and the notion of capacity of a subspace N ⊂ ℝⁿ. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 781–791, June, 1998. This work was partially supported by the Foundation for Fundamental Research of the Ministry of Science and Technology of the Ukraine (grant No. 1/238 “Operator”).     

Some spectral properties of the streaming operator with general boundary conditions

This paper deals with the spectral study of the streaming operator with general boundary conditions defined by means of a boundary operator K. We study the positivity and the irreducibility of the generated semigroup proved in [M. Boulanouar, L’opérateur d’Advection: existence d’un C ₀-semi-groupe (I), Transp. Theory Stat. Phys. 31, 2002, 153–167], in the case ‖K‖ ⩾ 1. We also give some spectral properties of the streaming operator and we characterize the type of the generated semigroup in terms of the solution of a characteristic equation.     

A modified inexact operator splitting method for monotone variational inequalities

The Douglas–Peaceman–Rachford–Varga operator splitting methods (DPRV methods) are attractive methods for monotone variational inequalities. He et al. [Numer. Math. 94, 715–737 (2003)] proposed an inexact self-adaptive operator splitting method based on DPRV. This paper relaxes the inexactness restriction further. And numerical experiments indicate the improvement of this relaxation.      

Convexity properties for interior operator games

Interior operator games arose by abstracting some properties of several types of cooperative games (for instance: peer group games, big boss games, clan games and information market games). This reason allow us to focus on different problems in the same way. We introduced these games in Bilbao et al. (Ann. Oper. Res. 137:141–160, 2005) by a set system with structure of antimatroid, that determines the feasible coalitions, and a non-negative vector, that represents a payoff distribution over the players. These games, in general, are not convex games. The main goal of this paper is to study under which conditions an interior operator game verifies other convexity properties: 1-convexity, k-convexity (k≥2 ) or semiconvexity. But, we will study these properties over structures more general than antimatroids: the interior operator structures. In every case, several characterizations in terms of the gap function and the initial vector are obtained. We also find the family of interior operator structures (particularly antimatroids) where every interior operator game satisfies one of these properties.     

Parafermion vertex operator algebras

This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module L(k, 0) of positive integer level k for any affine Kac-Moody Lie algebra ĝ, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C ₂-cofiniteness of the parafermion vertex operator algebras are discussed. A proof of the well-known fact that the parafermion vertex operator algebra can be realized as the commutant of a lattice vertex operator algebra in L(k, 0) is also given.     

A smoothing operator polynomial in the Hilbert spaceH λ

A mapping and an operator polynomial are considered as elements of the space H(λ). The polynomial is proved to be smoothing for the mapping in the metric of the space H(λ), that is, to be a solution to the corresponding extremum problem. Bibliography:4 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 77, 1993, pp. 27–34.     

Hyperinvariant subspaces for some <Emphasis Type="Italic">B</Emphasis>–circular operators

We show that if A is a Hilbert–space operator, then the set of all projections onto hyperinvariant subspaces of A, which is contained in the von Neumann algebra υN(A) that is generated by A, is independent of the representation of υ N(A), thought of as an abstract W^*–algebra. We modify a technique of Foias, Ko, Jung and Pearcy to get a method for finding nontrivial hyperinvariant subspaces of certain operators in finite von Neumann algebras. We introduce the B–circular operators as a special case of Speicher's B–Gaussian operators in free probability theory, and we prove several results about a B–circular operator z, including formulas for the B–valued Cauchy– and R–transforms of z^*z. We show that a large class of L^∞([0,1])–circular operators in finite von Neumann algebras have nontrivial hyperinvariant subspaces, and that another large class of them can be embedded in the free group factor L(F₃). These results generalize some of what is known about the quasinilpotent DT–operator. Supported in part by NSF Grant DMS-0300336. with an Appendix by Gabriel Tucci     

Applications of Wigner’s theorem to positive maps preserving norm of operator products

We apply Wigner’s theorem to positive maps on standard operator algebras that preserve norm of operator products or sum of singular values of operator products. It follows that such preservers are of the form ϕ(A) = U AU* with U either a unitary or antiunitary operator. Selected from Journal of Mathematical Study, 2004, 37(1): 11–16     

Local spectral multiplicity of a linear operator with respect to a measure

Let T be a bounded linear operator in a separable Banach space X and let μ be a nonnegative measure in χ with compact support. A function m_T,μ is considered that is defined μ-a.e. and has nonnegative integers or +∞ as values. This function is called the local multiplicity of T with respect to the measure μ. This function has some natural properties, it is invariant under similarity and quasisimilarity; the local spectral multiplicity of a direct sum of operators equals the sum of local multiplicities, and so on. The definition is given in terms of the maximal diagonalization of the operator T. It is shown that this diagonalization is unique in the natural sense. A notion of a system of generalized eigenvectors, dual to the notion of diagonalization, is discussed. Some examples of evaluation of the local spectral multiplicity function are given. Bibliography:10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1995, pp. 293–306.     

Solution of the spectral problem for the curl and Stokes operators with periodic boundary conditions

In this paper, we establish relations between eigenvalues and eigenfunctions of the curl operator and Stokes operator (with periodic boundary conditions). These relations show that the curl operator is the square root of the Stokes operator with ν = 1. The multiplicity of the zero eigenvalue of the curl operator is infinite. The space L ₂(Q, 2π) is decomposed into a direct sum of eigenspaces of the operator curl. For any complex number λ, the equation rot u + λu = f and the Stokes equation −ν(Δv + λ ²v) + ∇p = f, div v = 0, are solved. Bibliography: 15 titles. Dedicated to the memory of Olga Aleksandrovna Ladyzhenskaya __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 246–276.     

Experimental evidence for the occurrence of E 8 in nature and the radii of the Gosset circles

A recent experimental discovery involving the spin structure of electrons in a cold one-dimensional magnet points to a validation of a (1989) Zamolodchikov model involving the exceptional Lie group E ₈. The model predicts 8 particles and predicts the ratio of their masses. The conjectures have now been validated experimentally, at least for the first five masses. The Zamolodchikov model was extended in 1990 to a Kateev–Zamolodchikov model involving E ₆ and E ₇ as well. In a seemingly unrelated matter, the vertices of the 8-dimensional Gosset polytope identifies with the 240 roots of E ₈. Under the famous two-dimensional (Peter McMullen) projection of the polytope, the images of the vertices are arranged in eight concentric circles, hereafter referred to as the Gosset circles. The McMullen projection generalizes to any complex simple Lie algebra (in particular not restricted to A-D-E types) whose rank is greater than 1. The Gosset circles generalize as well, using orbits of the Coxeter element on roots. Applying results in Kostant (Am J Math 81:973–1032, 1959), I found some time ago a very easily defined operator A on a Cartan subalgebra, the ratio of whose eigenvalues is exactly the ratio of squares of the radii r _i of the generalized Gosset circles. The two matters considered above relate to one another in that the ratio of the masses in the E ₆, E ₇, E ₈ Kateev–Zamolodchikov models are exactly equal to the ratios of the radii of the corresponding generalized Gosset circles.     

Recursion operator for the intrinsic generalized sine-Gordon equation

We find the inverse and direct recursion operator for the intrinsic generalized sine-Gordon equation in any number n > 2 of independent variables. Among the flows generated by the direct operator we identify a higher-dimensional analogue of the pmKdV equation. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 12, No. 7, pp. 117–128, 2006.     

A note on slant submanifolds of nearly trans-Sasakian manifolds

The geometry of slant submanifolds of a nearly trans-Sasakian manifold is studied when the tensor field Q is parallel. It is proved that Q is not parallel on the submanifold unless it is anti-invariant and thus the result of [CABRERIZO, J. L.—CARRIAZO, A.—FERNANDEZ, L. M.—FERNANDEZ, M.: Slant submanifolds in Sasakian manifolds, Glasg. Math. J. 42 (2000), 125–138] and [GUPTA, R. S.—KHURSHEED HAIDER, S. M.—SHARFUDIN, A.: Slant submanifolds of a trans-Sasakian manifold, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 47 (2004), 45–57] are generalized.