Venkov's reduction theory of positive-quadratic forms

We prove the absolute finiteness of the number of faces (independent of the parameter) of Venkov's reduction domain (Izv. Akad. Nauk SSSR, Ser. Mat.4, 37–52 (1940)) ofn-ary positive quadratic forms. The casen=3 is given special consideration. We study the change of the reduction domain when changes along a line segment in the space of coefficients.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 108–116, 1983.     

Hankel-Schur multipliers and multipliers of the space H1

In this paper there is given a sufficient condition for a Hankel matrix _F to belong to the space of Schur multipliers of all bounded operators in ² (or, what is the same, to the tensor algebra V²). It is shown that ifw is a nonnegative function on T, such that is a sequence of integers, {F_i}_j1 is a sequence of polynomials,) and, then _FV². It follows from this that under these conditions F is a multiplier of the space H¹, i.e.,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 113–119, 1984     

Gauge group and gauge transformation in the continual theory of defects

A representation of the algebra (3)=t(3) S0(3, ) by differential Schaefer's operators is proposed, and an external algebra of (3)-valued differential forms is constructed. The requirement of local gauge invariance is formulated in the model of the (3)-valued field, which enables a group of gauge transformations of the continual theory of defects to be obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 173–184, 1991.I wish to thank V. N. Popov for his interest.     

Transformation of condensers in n-dimensional space

By a condenser in     

Estimate of the norm ∥f(A1, A2)-f(B1, B2)∥ for pairs of self-adjoint commuting operators

In this note we announce the estimate for the difference mentioned in the title, where are the lengths of intervals containing the spectra of the operators A₁, B₂ and A₂, B₂; the function f is from the class Lip 1 (with constant 1). The outline of the proof of this estimate is given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 175–177, 1984.     

Solutions of the cauchy problem for the polyharmonic equation (uniqueness and approximation)

One finds conditions which ensure the possibility of weighted mean-square approximation of a vector-function defined on the boundary of an n-dimensional domain by vector-functions of the form , where u is, the solution of the equation Δ^m _u=0 in while∂/∂v denotes differentiation along the normal. The weight function is continuous and positive everywhere on with the point whose relative neighborhood is contained in some (n-1)-dimensional plane. The solution of this approximation problem is closely related with a certain uniqueness theorem for the solution of the Cauchy problem for the polyharmonic equation, also proved in the paper. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 65, pp. 164–171, 1976.     

A contribution to the theory of Maxwell fluids. II

The local classical unique solvability of the periodic initial-boundary problem and of the cauchy problem is proved for the nonstationary quasilinear system describing the motion of Maxwell fluids.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 131, pp. 106–113, 1983.     

Spectral properties of one-dimensional disperse crystals

The connection between the spectral characteristics of the one-dimensional Schrödinger operator with periodic potential, asa, and the spectral characteristic of the Schrödinger operatorl(y)=–y+q(x)y with a decreasing potentialq(x) is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 197–211, 1984.     

Spherically symmetric solutions of Euclidean Yang-Mills equations

We consider the Euclidean Yang-Mills equations with structure groupSu(2). The action functional and the topological charge are invariant under the transformations, whereg runs over the set of unit length quaternions, andgx denotes the product ofg by the quaternionx=x ₄+ ix₁+jx_g+kx₃. ThisSu(2)-symmetry permits us to apply Coleman's principle. For the potentials A_µ, we obtain the following spherically symmetric Ansatz: while the Yang-Mills equations and the duality equations reduce to ordinary differential equations for the function. We show that every solution of the form (1) of the Yang-Mills equations with finite action and positive (negative) charge satisfies the duality equationsF=* F (respectively,F=–* F) and has charge 1 (respectively, –1). Moreover, we describe explicitly all solutions of form (1) of the duality equations; among them there are the 1-instanton solutions of Belavin, Polyakov, et al.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 126–132, 1984.     

Functions of class II

We consider the class II of operator functions f(z), meromorphic for ¦z¦1, which admit a representation in the form f(z)=f ₂ ^–1 (z)f₁(z), where f₁(z) is an operatorvalued holomorphic function and f₂(z) a scalar-valued bounded holomorphic function, such that the strong limits and coincide a.e. on the unit circle ¦¦=1. We prove that any function of class II can be represented as a block of some J-inner function of class II. We describe all such representations. The results found are applied to the question of realizations of functions of class II as transfer functions of linear systems.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta.im. V. A. Steklova AN SSSR, Vol. 135, pp. 5–30, 1984.     

Orders of torsion points of elliptic curves

We give a simple proof of the following proposition: if the elliptic curve contains a primitiveK -point of order 3 then theK -torsion of the curveT is uniformly bounded. We include application of the theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 47–57, 1983.     

Compact Hopf hypersurfaces of constant mean curvature in complex space forms

We prove that every connected compact Hopf hypersurface of a complex space form, contained in a geodesic ball of radius strictly smaller than the injectivity radius of, having constant mean curvature and with if if < 0 is a geodesic sphere of.Work partially supported by DGICYT Grant No. PB91-0324.     

Stochastic dilations of uniformly continuous completely positive semigroups

For an arbitrary uniformly continuous completely positive semigroup ( _t :t0) on the space of bounded operators on a Hilbert space, we construct a family (U(t)t0) of unitary operators on a Hilbert space and a conditional expectation from to, such that, for arbitraryt0,. The unitary operatorsU(t) satisfy a stochastic differential equation involving a noncommutative generalisation of infinite dimensional Brownian motion. They do not form a semigroup.Part of this work was completed when the first author was visiting research associate at the Center for Relativity, Physics Department, The University of Texas at Austin, Austin, TX 78712, U.S.A., supported in part by NSF PHY 81-01381.     

Grassmann manifold of aJH-algebra

We study the set of rankp idempotents in a topologically simple Hilbert Jordan algebra (JH-algebra for short). To produce the differential geometric structure on, we establish Jordan algebraic results concerning the structure of some two-generator subalgebras. We identify geodesics, the Riemannian distance and the sectional curvature of by using the Jordan algebraic structure.     

Quasiclassical approximation for models of spin-spin interaction on a one-dimensional lattice

The one-dimensional anisotropic Heisenberg model in an external field is considered. For the corresponding energy operator with parameter, the quasiclassical (h0) spectral series are constructed using the method of the canonical operator. In the integrable case (in the absence of the external field) the results obtained coincide with the exact ones.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 63–76, 1984.     

Construction of spaces of local functions on a nonuniform mesh

A general approach is considered to the construction of linear spaces of local functions without regularity conditions. This approach allows one to obtain uniform and integral estimates for the rate of convergence in the case of an essentially nonuniform mesh. As an example, the author considers approximation of functions with a singular point near which polynomial growth of the derivatives is observed. Estimates are given for approximation in the spaces, and also some weighted estimates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 140–163, 1983.     

On bands in which every finitely generated subband is an endomorphic image

A semigroup S is calledE-semigroup ( -semigroup) if every (finitely generated) subsemigroup of S is an endomorphic image of S and Ē-semigroup ( -semigroup) if every subsemigroup of S is an E-semigroup -semigroup. All classes X ε {Ē, , E, } are distinct even in the case of semilattices. It is established when a free band (semilattice) is an X-semigroup. -, - and Ē-chains and E-chains with identity or zero, Ē-and , X-bands with identity and X-semigroups with identity and zero are found.     

Two methods for avoiding hereditary completeness

A family of vectors of a Hubert space H is said to be hereditarily complete if it posses a biorthogonal family {x_n′;n≥1}((x_n,x_k′)=δ_nk) and if any elementx, xε H can be reconstructed in terms of the component of its Fourier series, i.e., x∈V((x,x′_n)x_n:n≥1),∀x∈H. In the paper we indicate two simple methods for constructing nonhereditary complete minimal families having a total biorthogonal family, which just not long ago has caused well-known difficulties (see Ref. Zh. Mat., 1975, 7B802). The first method consists in the fact that a given pair of biorthogonal families Y, Y′ of the space H′,H′⊂H is represented as the projection of the families of the same type but already complete in H.. Clearly, in this case cannot be hereditarily complete. The second method consists in considering linear deformation _n :n⩾1 of the orthogonal bases_n: n⩾1; here A is an unbounded operator of a special type. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 183–188, 1976.     

Convolution of the Fourier coefficients of the Eisenstein-Maass series

The convolution is defined as the sum where for n≠0 and W₀,W₁ are arbitrary smooth functions. Question: how to express these sums in the form of the combinations of the N-th Fourier coefficients of the eigenfunctions of the automorphic Laplacian? The answer is given in terms of the bilinear form of the Hecke series associated with the eigenfunctions of the automorphic Laplacian and with regular cusp forms. The final identity may lead to a new possibility for the solution of the moment problem of the Riemann zeta-function.