A description of the anti-Fock representation of a set of *-algebras connected with piecewise linear-fractional unimodal mappings

We give a method for describing the anti-Fock representation of the relation <$E X X sup *~=~f (X sup * X)> for a unimodal mapping f. Examples of a description of the anti-Fock representation for sets of piecewise linear-fractional mappings are given.

A generalized Malliavin derivative connected with the Poisson- and Gamma-measures

We introduce and study a generalized Malliavin derivative on the spaces of functions that are square integrable with respect to the Poisson measure and the Gamma-measure, and on the Kondratiev generalized functions spaces that are connected with these measures.

A Hilbert space of functions of infinitely many variables

We investigate Hilbert spaces of functions defined on infinite-dimensional Banach spaces. In particular, we study Hardy classes of analytic functions that are weakly uniformly continuous on the unit balls of Banach spaces.

A theorem on zeros of an entire function and its applications

Let <$E F> be an entire sine-type function of exponential type <$E sigma~>>~0>, which is real-valued on <$E bold roman R> and has only real zeros <$E left { lambda sub k right }:~lambda sub k~<<~lambda sub k+1>, <$E k~<174>~bold roman Z>. We consider entire functions of exponential type <$E symbol Г~sigma>, which are real-valued on <$E bold roman R>, satisfying <$E f(x)~=~omicron (|x|)>, <$E x~symbol О~symbol С inf> and <$E (-1) sup k f( lambda sub k )~symbol У~0>, <$E k~<174>~bold roman Z>. It is proved that the zeros of such functions are real and simple except for points <$E left { lambda sub k right }>, each of which can be zeros of multiplicity not greater than 2. These results are applied to concrete classes of functions and to the problem of stability of entire functions. We also improve and complement some results due to Polya.

Boundary triples and Weyl functions for singular perturbations of self-adjoint operators

Given the symmetric operator <$E A sub N> obtained by restricting the self-adjoint operator <$E A> to <$E N>, a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint <$E A sub N sup *> and the corresponding Weyl function. These objects provide us with the self-adjoint extensions of <$E A sub N> and their resolvents.

Centered one-parameter semigroups

We introduce a class of centered one-parameter semigroups which are continuous analogue of a single centered operator, and study their properties. In particular, we prove the Wold decomposition for such semigroups and give a complete description of one-parameter centered semigroups of partial isometries. Also we show that the double commutator relation <$E [a,~[a,~b]]~=~0> is a natural "infinitesimal version" of the condition for the semigroup to be centered.

Continuations of Hermitian indefinite functions and corresponding canonical systems: an example

M. G. Krein established a close connection between the continuation problem of positive definite functions from a finite interval to the real axis and the inverse spectral problem for differential operators. In this note we study such a connection for the function <$E f(t)~=~1~-~|t|,~t~<174>~ bold roman R>, which is not positive definite on <$E bold roman R>: its restrictions <$E f sub a ~:=~f| sub {(-2a,2a)}> are positive definite if <$E a~symbol Г~1> and have one negative square if <$E a~>>~1>. We show that with f a canonical differential equation or a Sturm-Liouville equation can be associated which have a singularity.

Correlation functionals for Gibbs measures and Ruelle bounds

It is proven that Gibbs measures for continuous systems with pair interaction are in one to one correspondence via Ruelle bound to correlation functionals, which fulfill the Kirkwood-Salsburg equations. The same bound we use for an existence proof for infinite volume correlation functionals. As state space X we consider Riemannian manifolds or homogeneous spaces. The results are extended to marked Gibbs measures. Furthermore, unitary representations of a group of diffeomorphisms on X are constructed.

Deterministic viscous hydrodynamics via stochastic analysis on groups of diffeomorphisms

The Navier-Stokes equation is derived as an Euler type equation in the "algebra" of the group of diffeomorphisms of flat n-dimensional torus. It is generated by a certain diffusion process on the group, governed by a stochastic analogue of the second Newton's law with a mechanical constraint that guarantees incompressibility. The diffusion term of the process is connected with viscosity coefficient of the fluid. The constraint is given in invariant geometric terms, the Newton's law is formulated in terms of Nelson's mean backward derivatives. The construction is translated into the finite-dimensional language of processes on the torus.

Double trivalent diagrams and n-hyperbolic knots

An effect of insertions in a knot via special tangle maps the double pure braid <$E n>-commutators, with respect to the Vassiliev knot invariants of order <$E n>, is expressed in terms of trivalent diagrams. The class of "geometric" <$E n>-trivial knots consisting of the <$E n>-hyperbolic knots, introduced by Kalfagianni and Lin, is considered. Using insertions of double pure braid commutators in diagrams for a trivial knot, we show that for each odd integer <$E n~symbol У~3> there is an (<$E n~-~2>)-hyperbolic knot which is not <$E n>-trivial, answering a question raised by Kalfagianni and Lin (1998).

Entropy of Bogoliubov automorphisms on II1-representations of the group <$E bold U (inf)>

The authors define analogies of Bogoliubov automorphisms on II1 factor representations of the group <$E U (inf)> and obtain formulas for its CNT-entropy.

Estimates for periodic eigenvalues of the differential operator <$E bold {(-1) sup m d sup 2m "/" dx sup 2m~+~V}> with <$E bold V>-distribution

The periodic eigenvalue problem for the differential operator <$E (-1) sup m d sup 2m "/" dx sup 2m~+~V> is studied for a complex-valued distribution <$E V> in the Sobolev space <$E H sub roman per sup {-m alpha} [-1,~1]> (<$E m~<174>~roman N,~0~symbol Г~alpha~<<~1>). We prove that the periodic spectrum consists of a sequence <$E ( lambda sub k ) sub {k symbol У 0}> complex eigenvalues satisfying the asymptotics <$E lambda sub 2n-1 ,~lambda sub 2n~=~n sup 2m pi sup 2m~+~V Hat (0)~symbol С~sqrt {V Hat (-2n) V Hat (2n)}~+~omicron (n sup {m(2 alpha -1+ epsilon )})> for any <$E epsilon~>>~0>, where <$E V Hat (k)> denotes the Fourier coefficients of <$E V>. The case <$E m> = 1 was investigated using the same method.

Fine structure of the singular continuous spectrum

A refined version of the well-known Lebesgue decomposition theorem for positive measures is proven. The singular continuous part <$E mu sub roman sc> is shown to admit a unique decomposition into three mutually singular components, <$E mu sub roman sc~=~roman mu sub sc sup C~+~roman mu sub sc sup P~+~roman mu sub sc sup S>, of Cantor, Pratsiovytyi, and Salem types, respectively. This leads to a corresponding unique decomposition <$E sigma sub roman sc (A)~=~roman sigma sub sc sup C~union~roman sigma sub sc sup S~union~roman sigma sub sc sup P> of the singular continuous spectrum <$E sigma sub roman sc (A)> of a self-adjoint operator <$E A> on a Hilbert space <$E H>. Correspondingly, the singular continuous subspace <$E H sub roman sc (A)> of <$E A> admits the orthogonal decomposition <$E H sub roman sc (A)~=~H sub roman sc sup roman C~simbol e~H sub roman sc sup roman P~simbol e~H sub roman sc sup roman S> such that <$E A> is reduced by each of these subspaces, and the spectral measure <$E mu sub psi> associated with a vector <$E psi~symbol <174>~H sub roman sc sup roman C (A)> has only trivial components of P and S types, whereas the spectral measure <$E mu sub psi> associated with a vector <$E psi~symbol <174>~H sub roman sc sup roman P (A)> has only a trivial component of S-type.

Generalized Jacobi fields

We generalize the notion of a Jacobi field which is a family of operators with certain properties acting in the Fock space. Namely, we introduce a notion of generalized Jacobi field acting in a space more general than the Fock space. We discuss a particular case where this spectral measure is the Gamma measure and the operators of the field act in the so-called extended Fock space.

Iterated limit theorem for inductive-projective topologies and application

We investigate the spaces, topology of which is given by iterated inductive-projective limit. The simple conditions for permutation of passages to the limit are proved. The applications to the distribution theory are considered.

Large deviation for a stochastic Cahn - Hilliard equation

We establish the large deviation principle for a sequence of stochastic PDEs defined on rescaled lattices. They are a special class of the Ginzburg-Landau models where the total order parameters is conserved. As the lattice mesh size and the magnitude of randomness go to zero, the large deviation result gives us information about convergence from the stochastic dynamic to the most probable deterministic trajectory, as well as the rate of deviations of the dynamic frorn atypical deterministic trajectories. We prove such large deviation result by using Hamilton - Jacobi equation techniques. A technical core in such a method is the comparison between sub and super (viscosity) solutions for a class of first order Hamilton - Jacobi equations. The state space for these equations is a Hilbert space, and the Hamiltonian admits a special form to which available comparison results cannot be directly applied. We obtain the large deviation result by clarifying different notions of viscosity solutions, and by extending existing comparison techniques. It is the hope that the techniques and connections revealed here will prove to be useful in further investigations of the interactions between infinite dimensional viscosity solution method and the large deviation for stochastic PDEs and interacting particle systems.

On a class of generators of a one-parameter <$E bold C sub 0>-semigroup of operators

On a separable Hilbert space, we consider a class of closed, densely defined operators, inverse to which are <$E n>-dimensional perturbations of Volterra dissipative operators. We find conditions such that operators of this class generate a <$E C sub 0>-semigroup.

On a two-dimensional trigonometric moment problem

Positive definite functions defined on a subset of the lattice <$E bold roman Z sup 2> do not admit, generally speaking, positive definite extension on the whole lattice. We give new and elementary proof of the following theorem: Any positive definite function defined on a rectangular of <$E bold roman Z sup 2> with one side from -1 to 1 always admits representation as trigonometric moments of a positive measure on two-dimensional torus, and, therefore, can be extended on the whole lattice.

On categorical properties of the functor of order-preserving functionals

For a Tychonoff space X, by O(X) we denote the space of order-preserving functional on <$E C sub b (X)>, where <$E C sub b (X)> is the ring of all continuous bounded functions <$E f~:~X~symbol О~R>. It is shown that the functor O in the category Tych does not preserve preimages and weights of Tychonoff spaces. We proof that the remaining conditions of normality hold with slight modifications.

On connection between <$E bold back 60 +>-decomposability and wildness for algebras generated by idempotents

We introduce a natural semigroup structure on the set of <$E k>-algebras <$E Q sub k (n,~alpha )~=~k~<<~1,~e sub 1 , ... , e sub n~|~e sub 1 sup 2~=~e sub 1 ,..., e sub n sup 2~=~e sub n ,~e sub 1 ...~+~e sub n~=~ alpha~symbol <174>~k~>> > (k is a field), having at least one finite-dimensional representation and, for an algebraically closed field of characteristic 0, obtain a connection between decomposability (in the semigroup) and wildness of such algebras.