Extrapolation of transformations of random processes perturbed by white noise

We consider the problem of linear mean square optimal estimation of transformation of a stationary random process (t) in observations of process (t) + n(t) for t < – 0, where (t) is white noise uncorrelated with (t). We find least favorable spectral densities f₀() D and minimax (robust) spectral characteristics of an optimal estimator of transformation A for various classesD of densities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 216–223, February, 1991.     

Reducibility of lacunary polynomials,VII

Letk>1 and let be non-zero algebraic numbers contained in the field . It is shown that for almost all, in the sense of density integer vectorsn ₁,...,n _k the polynomial becomes irreducible over on dividing by the product of all factorsx–, where is a root of unity.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.     

Über die Riemannsche Einbettungsgeometrie der Veronesemannigfaltigkeiten

Let E be a n-dimensional euclidean vector space. The subset V _k ⁿ ={x ... x | x E} of ^kE is called a Veronesemanifold. The scalar product of E induces a euclidean structure on ^kE. Passing to the corresponding projective space , one may consider as a riemannian submanifold of the space form . In this paper we study properties of the pair of riemannian manifolds.     

Path processes and historical superprocesses

Summary A superprocessX over a Markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles.The historical process for is the process whose state at timet is the path of over time interval [0,t]. The superprocess over the historical superprocess over —reflects not only the particle distribution at any fixed time but also the structure of family trees. The principal property of a historical process is that is a function of for alls<t. Every process with this property is calleda path process. We develop a theory of superprocesses over path processes whose core is the integration with respect to measure-functionals. By applying this theory to historical superprocesses we construct the first hitting distributions and prove a special Markov property for superprocesses.Partially supported by National Science Foundation Grant DMS-8802667     

Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

Order comparison of norms of polynomials in regions of the complex plane

For an arbitrary Jordan region G, inequalities are obtained which relate the norm with the norms and of a polynomial P_n(z) of degree n.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1277–1280, September, 1991.     

Complete Asymptotics of the Deviation of a Class of Differentiable Functions from the Set of Their Harmonic Poisson Integrals

On a class of differentiable functions W ^r and the class of functions conjugate to them, we obtain a complete asymptotic expansion of the upper bounds of deviations of the harmonic Poisson integrals of the functions considered.     

On some properties of the metric subalgebras ofℓ ∞

The objects studied are the subalgebras of which contain c_o. These are isometrically isomorphic to the algebras C( ) where is a compactification of a discrete denumerable set N . It is shown: 1) If is metric then there is a projection of norm 1, P: C( ) C( ) with kernel c_o defined by PF = f o where is a retraction of onto = – N . 2) If is metric, then the group of homeomorphisms of is isomorphic to a complete group of permutations of the natural numbers . 3) The group of homeomorphisms of a compact metric space is the homomorphic image of a complete group of permutations of ("complete" means "no outer automorphisms, trivial center").     

Brownian processes in infinite dimension

LetE be a locally convex space endowed with a centered gaussian measure . We construct a continuousE-valued brownian motionW _t with covariance . The main goal is to solve the SDE of Langevin type dX _t= dW _t–AX _t wherea andA are unbounded operators of the Cameron-Martin space of (E, ). It appears as the unique linear measurable extension of the solution of the classical Cauchy problemv(t)= u–Av(t).     

Multidimensional Lagrange–Yen-Type Interpolation Via Kotel'nikov–Shannon Sampling Formulas

Direct finite interpolation formulas are developed for the Paley–Wiener function spaces and , where contains all bivariate entire functions whose Fourier spectrum is supported by the set = Cl{(u, v) |u| + |v| < ], while in the Fourier spectrum support set of its d-variate entire elements is [–, ] ^d . The multidimensional Kotel'nikov–Shannon sampling formula remains valid when only finitely many sampling knots are deviated from the uniform spacing. By using this interpolation procedure, we truncate a sampling sum to its irregularly sampled part. Upper bounds of the truncation error are obtained in both cases.According to the Sun–Zhou extension of the Kadets -theorem, the magnitudes of deviations are limited coordinatewise to . To avoid this inconvenience, we introduce weighted Kotel'nikov–Shannon sampling sums. For , Lagrange-type direct finite interpolation formulas are given. Finally, convergence-rate questions are discussed.     

Universality of the automorphism group of the real line

We discuss the question, whether each automorphism group (of cardinality at most ) of a linear order is embeddable into the automorphism group of the real line. We show that the answer to this question is independent of the axioms of ZFC: the answer is positive, if we assume < and Souslin's hypothesis; the answer is negative, if we assume orV=L.     

Classes of (ψ, β)-differentiable functions of a complex variable and approximation by linear averages of their Faber series

We introduce the notion of (, )-derivative of a function of one complex variable, and define on the basis of this the classes of (, )-differentiable analytic functions in a bounded domain G. The classes consist of the Cauchy-type integrals whose densities f() are such that the induced functions on the unit circle are periodic functions of classes . We consider approximation of functions by algebraic polynomials constructed from their series expansions in Faber polynomials.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1556–1570, November, 1992.     

Prime ideal decomposition inF({}^m\sqrt \mu )

LetF be an algebraic number field and F such thatx ^m– is irreducible, wherem is an integer. Let be a prime ideal inF with . The prime decomposition of in is explicitly obtained in the following cases. Case 1: , (a,m) = 1 (where means , 0 ). Case 2:m lt, wherel is a prime andl 0 . Case 3:m 0 and every prime that dividesm also dividespf–1. It is not assumed that thev ^th roots of unity are inF for anyv 2.     

The closure diagram for nilpotent orbits of the split real form of E8

Let and be adjoint nilpotent orbits in a real semisimple Lie algebra. Write ≥ if is contained in the closure of . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E ₈. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible components of the boundaries and of the intersections .     

Reflection groups and K-loops

The notion reflection group (, ) was introduced in order to give group theoretical characterizations of absolute planes. Here we consider reflection groups with midpoints and associate to each of them an incidence structure . Then is an incidence space which dimension can assume any value. The motion group together with the set of all reflections in points of a Euclidean or hyperbolic geometry are examples of reflection groups with midpoints. Furthermore the set can be turned into a K-loop. The precise results are summarized in the theorems at the end of the paper.Cordially dedicated to Sibylla Prieß on the occasion of her 60th birthdaySupported by the NATO Scientific Affairs Division grant CRG 900103.     

Quotient Groups of Groups of Certain Classes

For an arbitrary variety of groups and an arbitrary class of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with - and -factors (respectively, is a residually -group) if G possesses an invariant system with - and -factors (respectively, is a residually -group) and N (respectively, N is a maximal invariant -subgroup of the group G).     

SU2ℍ als Kollineationsgruppe von sechzehndimensionalen lokalkompakten Translationsebenen

As a contribution to a classification of all sixteen-dimensional translation planes whose collineation group has dimension at least 38, this paper deals with the case that contains a subgroup (locally) isomorphic to SU₂Spin₅. Under various further assumptions, it is shown that such a plane satisfying dim 38 is necessarily isomorphic to the classical plane over the octonions.The complete classification will reveal that these further assumptions may in fact be omitted, except for the case that even contains a subgroup isomorphic to Spin₇. The latter planes have been explicitly determined in previous papers.

Meinem verehrten Lehrer Helmut Salzmann zum 65. Geburtstag     

Spreads and group divisible designs

By removing the components of at-spread of a finite projective spacePG(d, q) from each hyperplane ofPG(d, q), the blocks of a regular group divisible design are obtained We characterize geometrict-spreads as thoset-spreads which are such that the dual of is also a group divisible design.     

Tensor ideals in the category of tilting modules

We study the tensor category of tilting modules over a quantum groupU _q with divided powers. The setX ₊ of dominant weights is a union of closed alcoves numbered by the elementswW ^f of a certain subset of affine Weyl groupW. G. Lusztig and N. Xi defined a partition ofW ^f into canonical right cells and the right order _R on the set of cells. For a cellAW ^f we consider a full subcategory formed by direct sums of tilting modulesQ() with highest weights . We prove that is a tensor ideal in , generalizing H. Andersen's theorem about the ideal of negligible modules which in our notations is nothing else then . The proof is an application of a recent result by W. Soergel who has computed the characters of tilting modules.This material is based upon work supported by the U.S. Civilian Research and Development Foundation under Award No. RM1-265.