Dirac operators on the quantum group SU(2) and the quantum sphere

The problem of constructing the Dirac operators on the quantum groupSU(2) and the quantum sphere S_qμ ² are discussed. In both cases, the constructions presented have the sameSU _q(2)-invariant form and are directly connected with the corresponding Laplace operators. Bibliography:16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 49–65. Translated by P. N. Bibikov.     

Pitt's theorem for the Lorentz and Orlicz sequence spaces

LetL(X, Y) be the Banach space of all continuous linear operators fromX toY, and letK(X, Y) be the subspace of compact operators. Some versions of the classical Pitt theorem (ifp>q, thenK(l _p, l_q)=L(l_p, l_q)) for subspaces of Lorentz and Orlicz sequence spaces are established. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 18–25, January, 1997. Translated by V. N. Dubrovsky     

Summation rational operators of the Jackson type

Summation rational positive operatorsD _4n−n(x; f) of the Jackson type are constructed on the real axis. The corresponding approximations of continuous functionsf onℝwith coinciding finite limits lim_x→−∞ f(x) and lim_x→+∞ f(x) are estimated. Translated fromMatematischeskie Zametki, Vol. 61, No. 2, pp. 270–277, February, 1997. Translated by N. K. Kulman     

Approximation of functions of boundedp-variation by polynomials in terms of the faber-schauder system

In this paper the best polynomial approximation in terms of the system of Faber-Schauder functions in the spaceC _p [0, 1] is studied. The constant in the estimate of Jackson’s inequality for the best approximation in the metric ofC _p [0, 1] and the estimate of the modulus of continuity ω_1−1/p are refined. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 363–371, September, 1997. Translated by N. K. Kulman     

Symmetric solutions for the first and second Painlevé equations

For the equations P₁ and P₂, particular solutions of special form are given, for which the Cauchy problem is effectively solved, i.e., for the parameters describing their behavior near zero and infinity the connection formulas are obtained. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 187, pp. 129–138, 1990. Translated by N. Yu. Netsvetaev.     

Approximation to the Sobolev classesW q r of functions of several variables by bilinear forms inL p for 2≤q≤p≤∞

We study the approximation of functions of several variables by bilinear forms that are the pairwise products of functions of fewer variables. The order of approximation of Sobolev classesW _q ^r by bilinear forms inL _p for 2≤q≤p≤∞ is found. Translated by N. K. Kulman Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 18–34, July, 1997.     

Comparison of various generalizations of continued fractions

We use the Euler, Jacobi, Poincaré, and Brun matrix algorithms as well as two new algorithms to evaluate the continued fraction expansions of two vectorsL related to two Davenport cubic formsg ₁ andg ₂. The Klein polyhedra ofg ₁ andg ₂ were calculated in another paper. Here the integer convergentsP _k given by the cited algorithms are considered with respect to the Klein polyhedra. We also study the periods of these expansions. It turns out that only the Jacobi and Bryuno algorithms can be regarded as satisfactory. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 339–348, March, 1997. Translated by V. E. Nazaikinskii     

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     

Curvature identities for normal manifolds of killing type

We present two curvature identities and study the corresponding classesR ₁ andR ₂ of normal manifolds of Killing type. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 351–362, September, 1997. Translated by S. S. Anisov     

On the interpolation of some generalized function spaces of different anisotropy

In the paper, the interpolation properties of the spacesH _p ^s (v; ℝ _n ) of Sobolev-Liouville type and the spacesB _{p, q} ^s (μ ℝ _n ) of Nikol'skii-Besov type generated by functions of polynomial growth that are infinitely differentiable outside of the origin are studied. Interpolation formulas for the pairs {H(v _o ),H(v ₁)} and {B(μ₀),B(μ₁)} of spaces of the above types for which the anisotropies of the interpolated spaces do not depend on each other are proved. The investigated spaces, for certain specification of the generating functions, coincide with the classical (isotropic and anisotropic) Sobolev-Liouville and Nikol'skii-Besov spaces. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 666–672, November, 1997. Translated by A. I. Shtern     

Analogs of the jackson-nikol'skii inequalities for trigonometric polynomials in spaces with nonsymmetric norm

The paper is concerned with the evaluation of where ▮·▮p ₁, p₂ is a nonsymmetric norm. The order of this number is obtained. Lower bounds involve new polynomials whose properties are studied in detail. In the casep ₁=p ₂,q ₁=q ₂, the estimate obtained is reduced to the well-known Jackson-Nikol'skii inequality. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 687–699, May, 1997. Translated by N. K. Kulman     

Quantum algebras,Kravchuk q-polynomials,and Kravchuk-Meixner q-functions

Addition and multiplication theorems are proved for Kravchuk q-polynomials by means of methods from the theory of representations of the quantum algebras U _q (su ₂ ) and U _q (su _1,1 ), and Kravchuk-Meixner q-functions are introduced and are shown to be orthogonal on the set of integers.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 7, pp. 983–986, July, 1992.     

An application of the neyman-pearson lemma to gaussian processes

Letξ _i (t),i=1,2,t ∈ [0, 1], be Gaussian zero mean processes with continous sample paths. Bounds for the probabilitiesβ _i =P{ξ _i -α _i ∈B},i=1,2, are given, where a_iε C0, 1., and B is a Borel subset of C[0, 1] Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 207, pp. 5–12, 1993. Translated by V. Sudakov.     

The problem of compensation

Let W₁, W₂ be stationary connected Wiener processes. We investigate the possibility of regular representation for an element of the linear space generated by the processes W₁, W₂. Biliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 144–152. Translated by V. Sudakov.     

Functions of two variables continuous along straigt lines

For a functionf(x, y), the setsJ _a of all its discontinuity points with a jump ofa or more (that is, such that the oscillation of the function in the neighborhood of any point fromJ _a is not smaller thana) are studied. Two cases are considered: (1)f is continuous along any straight line; (2)f is continuous along lines parallel to thex- andy-axes. In the first case, conditions that must be met by the setJ _a are given. In the second case, it is shown that a (closed) setF can be the setJ _a for a certain function if and only if the projections ofF on the coordinate axes nowhere dense. Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 306–311, August, 1997. Translated by V. N. Dubrovsky     

Implicit functions determined by equations with unbounded operators and the solvability of operator equations

For implicit functions determined by the equation T_x+F(λ,x)=0, where T is an unbounded operator, existence and differentiability conditions are established. As an application, the solvability conditions for equations of the form Tx+f(x)=0 are derived. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 216–224. Translated by L. Yu. Kolotilina.     

The sum of coefficients of bounded univalent functions

We solve the maximal value problem for the functional in the class of functionsf(z)=z+a ₂z²+… that are holomorphic and univalent in the unit disk and satisfy the inequality |f(z)|<M. We prove that the Pick functions are extremal for this problem for sufficiently largeM whenever the set of indicesk ₁,…,k_m contains an even number. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 728–733, May, 1997. Translated by S. S. Anisov     

An algorithm for determining wave fields in multilayer elastic media

An algorithm for computing wave fields propagating in elastic media with plane-parallel interfaces is proposed. This algorithm is based on summation of kinematically equivalent waves. All kinds of multiple waves of the exchange type arriving to an observer at the fixed interval of time (t₁, t₂) are taken into account. Dynamic analogues are determined within each kinematic group. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 154–171, 1990. Translated by N. S. Zabavnikova.     

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.     

The multipole green functions in the Kondo-lattice theory

A method of taking account of the strong Coulomb correlations for the Kondo-lattice model is suggested. The elementary excitations and phase transitions in this model for the CuO₂ layers in the superconducting oxides are investigated by this method. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 121–136. Translated by B. M. Bekker.