Commutator-finite orthomodular lattices

The class of orthomodular lattices which have only finitely many commutators is investigated. The following theorems are proved: contains the block-finite orthomodular lattices. Every irreducible element of is simple. Every element of is a direct product of a Boolean algebra and finitely many simple orthomodular lattices. The irreducible elements of which are modular, or are M-symmetric with at least one atom, have height two or less.     

A structure theorem for weak buildings of spherical type

Following earlier work of Tits [8], this paper deals with the structure of buildings which are not necessarily thick; that is, possessing panels (faces of codimension 1) which are contained in two chambers, only. To every building , there is canonically associated a thick building whose Weyl group W( ) can be considered as a reflection subgroup of the Weyl group W() of . One can reconstruct from together with the embedding W( ) W(). Conversely, if is any thick building and W any reflection group containing W( ) as a reflection subgroup, there exists a weak building with Weyl group W and associated thick building .     

Homogeneous algebraic varieties defined by Jordan pairs

Every Jordan pair defines an algebraic varietyX containing as a dense open subset.X is projective (affine) if and only if is separable (radical). The Picard group ofX is generated by the irreducible factors of the generic norm of . If is separable then the automorphism group ofX is the projective group of .     

Rings of regular functions on nilpotent orbits and their covers

Summary LetG be a complex semisimple algebraic group with Lie algebra . Let be a nilpotentG-orbit, its ring of regular functions. We derive a formula for as aG-module and prove some partial results on a cover of . We then relate this formula to various existing multiplicity formulas forK-types in Harish-Chandra bimodules ofG.Supported by National Science Foundation Grant DMS-8505550     

Groupwise density and related cardinals

We prove several theorems about the cardinal associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps from to, all families of size are below all unbounded families. With respect to a natural ordering of filters on, all filters generated by sets are below all non-feeble filters. If then and . (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from by Laflamme are shown to be equivalent to .     

Isometric dilations in nest algebras

LetT be a contraction acting in a separable Hilbert space and leaving invariant a nest of subspaces of . We answer the question: when doesT have an isometric extension to which leaves invariant the nest = {N N :N ;}.     

Exponential Stability of Semigroups Related to Operator Models in Mechanics

In this paper, we consider equations of the form , where is a function with values in the Hilbert space , the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in . The linear operator generating the C ₀-semigroup in the energy space is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.     

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").     

The completions of a commutative lattice group with respect to the intrinsic topologies

In this paper we discuss the completions (, ) of a commutativel-groupG with respect to the intrinsic topologies . We give some conditions under which is the intrinsic topology of the same type on as and give the relations between these completions.     

Quantization of Boltzmann Entropy: Pairs and Correlation Function

We consider the Gibbs statistical ensemble. We introduce the statistical-weight operator , construct the entropy operator of the ensemble using the Boltzmann formula , and define the free-energy operator. We find asymptotic expressions for free-energy eigenvalues with pair correlations taken into account in the limit as the number of systems in the ensemble tends to infinity.     

The lift on a small sphere touching a plane in the presence of a simple shear flow

The contribution from purely viscous forces to the liftL on a sphere of radiusa touching a plane in the presence of a shear flow field of strength is zero. An exact integral expression for the lift to leading order in the Reynolds number is derived using known creeping flow solutions to related problems. The integral is evaluated numerically to obtain the value of the lift orL/F _x 0.287R whereF _x is the lateral viscous force on the sphere.

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Zusammenfassung Der Beitrag der Zähigkeit allein zum AuftriebL an einer Kugel vom Radiusa, welche in einer Scherströmung mit dem Gradienten eine unendliche Ebene berührt ist null. Ein exakter Integralausdruck für den Auftrieb wird in erster Ordnung der Reynoldszahl, , hergeleitet unter Benützung bekannter verwandter Lösungen in schleichender Strömung. Der Wert des Integrals wird numerisch bestimmt und gibt für den Auftrieb oderL/F _x 0.287R; dabei istF _x die seitliche Zähigkeitskraft auf die Kugel.

     

Minimax-robust prediction of discrete time series

Summary We discuss a robust approach for predicting a weakly stationary discrete time series whose spectral density f is not exactly known. We assume that we know that f ), where is a convex set of spectral densities fulfilling some not too stringent conditions. We proof the existence of a most indeterministic density f ⁰ in , and we show that the classical optimal linear predictor for a time series with spectral density f ⁰ is mini-max-robust in the sense that it minimizes the maximal possible prediction error.We investigate some special models , and, in doing so, we illustrate a generally applicable method for determining the robust predictor. In particular, we discuss model sets which are defined by conditions on a finite part of the autocovariance sequence of the corresponding time series. These examples are of particular interest as the most indeterministic density is an autoregressive one, i.e. the robust predictor is finite. We discuss connections between this type of model set and maximum entropy and generalized maximum entropy spectral estimates.     

s-Prime elements in multiplicative lattices

Let be aC-lattice which is strong join principally generated. In this paper, we consider prime elements of for which every semiprimary element is primary. We show, for example, that a compact nonmaximal primep with this property is principal. We also show that if every primepm has this property, then is either a one dimensional domain or a primary lattice. It follows that if every primep satisfies the property, and if there are only a finite number of minimal primes in , then is the finite direct product of one-dimensional domains and primary lattices.     

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 .     

Yangians and $$\mathcal{W} $$ -Algebras

We present a connection between -algebras and Yangians in the case of gl(N) algebras, as well as for twisted Yangians and super-Yangians. We illustrate this connection, which allows constructing an R-matrix for the -algebras and classifying their finite-dimensional irreducible representations, in the framework of the nonlinear Schrödinger equation in 1+1 dimensions.     

Common invariant subspaces for collections of operators

Let be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that is finitely quasinilpotent at a vectorx ₀X whenever for any finite subset of the joint spectral radius of atx ₀ is equal 0. If such collection contains a non-zero compact operator, then and its commutant have a common non-trivial invariant, subspace. If in addition, is a collection of positive operators on a Banach lattice, then has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then has a common non-trivial invariant closed ideal.This work was supported by the Research Ministry of Slovenia.     

A remark concerning the modulus of smoothness introduced by Ditzian and Totik

For each functionf(x) continuous on the segment [–1, 1], we set . We study the relationship between the ordinarykth modulus of continuity of therth derivative of the function and thekth modulus of continuity with weight _r of the rth derivativef ^(r) of the functionf introduced by Ditzian and Totik. Thus, ifr is odd andk is even, we prove that these moduli are equivalent ast0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1627–1638, December, 1995.     

Tensor invariants of the Lie algebra sl2(ℂ[t]) and fundamental representations of the Lie algebras\hat p_{2n}

Generators of the space of tensor invariants of the Lie algebra Sl₂([t]) are constructed. It is proved that the restrictions of a spinor representation of the affine Lie algebra to and form a dual pair. A realization of the fundamental representations of the Lie algebra is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akad. Nauk SSSR, Vol. 172, pp. 137–144, 1989.     

A characterization of the line set of an odd-dimensional Baer subspace

Generalizing a theorem of Beutelspacher and Seeger, we consider line sets inP=PG(2t + 1,q),t IN, with the following properties: (1) any (t + 1)-dimensional subspace ofP contains at least one line of , (2) if a pointx ofP is incident with at least two lines of then the points in the factor geometryP/x which are induced by the lines of throughx form a blocking set of type (t, 1) inP/x, (3) any line of is coplanar with at least one further line of . We will show that the examples of minimal cardinality are exactly the line sets of Baer subspaces ofP.     

Theory of nonequilibrium phenomena based on the BBGKY hierarchy. I. small deviations from equilbrium

The BBGKY hierarchy is expanded in a series with respect to the small parameter , where is the diameter of the particles, and is a characteristic macroscopic length (for example, the diameter of the system). Since neither nor occurs explicitly in the equations of the hierarchy, a preliminary step consists of separation from the distribution functions of short-range components that vary over distances of order and long-range components that vary over distances of order . By a transition to dimensionless variables, terms of zeroth and first order in in the hierarchy are separated, this making it possible to perform the expansion with respect to . It is shown that in the zeroth order in the BBGKY hierarchy determines a state of local equilibrium that for any matter density can be described by a Maxwell distribution with shift. The higher terms of the series in describe the deviations from local equilibrium. At the same time, the long-range correlations that always arise in nonequilibrium systems are described by the balance equations for mass, momentum, and energy, which are also a consequence of the BBGKY hierarchy, whereas the short-range correlations are described by the equations for obtained from the same hierarchy by expanding in a series with respect to .Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 1, pp. 109–122, April, 1995.