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.     

Valuations and polarity

Given a collection of convex polytopes, let() denote the set of all convex transversals of. If and are two such collections, of finite cardinality, then there is a simple, arithmetical condition which holds precisely when ()=(). Another such condition, involving what we call the Sallee-Shephard mapping, characterizes those pairs and for which (())=().As these results are established, several distributive lattices involving convex sets are introduced, and relationships between their valuation modules are determined. In particular, it is proven that the Sallee-Shephard mapping is an isomorphism of the additive, abelian group of simple functions generated by the characteristic functions of the open, convex sets and that generated by those of the closed, convex sets.     

A short proof of the nonuniform Ray-Chaudhuri-Wilson inequality

LetL be a set ofs nonnegative integers and a family of subsets of ann-element setX. Suppose that for any two distinct membersA,B we have¦A B¦ L. Assuming in addition that, is uniform, i.e. each member of has the same cardinality, a celebrated theorem of D. K. Ray-Chaudhuri and R. M. Wilson asserts that ¦¦ P. Frankl and R. M. Wilson proved that without the uniformity assumption, we have.We give a short proof of this latter result.     

Some intersection theorems on two-valued functions

Let be a family of two-valued functions defined on ann-element set in which each pair of functions in satisfy a given intersection condition. For certain intersection conditions we determine the maximal value of .     

A generalized inverse ∈-algorithm for constructing intersection projection matrices,with applications

Givenk linear manifolds ₁, ..., _k and corresponding perpendicular projection matricesP ₁, ...,P _k , a closed formula is derived for the perpendicular projection matrix with range. The derivation uses results taken from the theory of generalized inverses together with an application ofWynn's -Algorithm to a convergent sequence of matrices. A variant of this formula is then used in solving arbitrary complex linear systems by iteration and in computing generalized inverses; the latter application provides a solution to least squares linear regression problems.A preliminary version of this paper, MRC Technical Summary Report 604, was sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-11-022-ORD-20$9.     

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.     

The asymptotic numbers of regular tournaments,Eulerian digraphs and Eulerian oriented graphs

LetRT(n), ED(n) andEOG(n) be the number of labelled regular tournaments, labelled loop-free simple Eulerian digraphs, and labelled Eulerian oriented simple graphs, respectively, onn vertices. Then, asn,, for any>0. The last two families of graphs are also enumerated by their numbers of edges. The proofs use the saddle point method applied to appropriaten-dimensional integrals.     

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     

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.     

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.     

The classification of distance-regular graphs of type IIB

The distance-regular graphs of type IIB in Bannai and Ito [1] have intersection numbers of the form whered is the diameter of , andh, x, andt are complex constants. In this paper we show a graph of type IIB and diameterd (3d) is either the antipodal quotient of the Hamming graphH(2d+1,2), or has the same intersection numbers as the antipodal quotient ofH(2d, 2).     

On the relation between the Jackson and Jung constants of the spacesL p

For any infinitely metrizable compact Abelian groupG; 1pq<,n , the following relations are proved: whereK _pq(G, n, G) is the largest Jackson constant in the approximation of the system of characters by polynomials of ordern, d _pq(G, n, G) is the best Jackson constant,J(L _p(G), L_q(G)) is the Jung constant of the pair of real spaces (L _p(G), L_q(G)), and.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 828–836, December, 1995.This work was supported by the Russian Foundation for Basic Research under grant No. 95-01-00657.     

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.     

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.     

Inductive definitions,models of comprehension and invariant definability

The connections between inductive definability and models of comprehension are studied. Let = 〈A, R _l, ...,R _n 〉 be an infinite structure and letI _φ be a set inductively defined by a formulaφ of the second order language . We prove that if is a model of Δ ₁ ¹ -Comprehension relativized toφ, andφ is -absolute, then for everyη smaller than the height of (h( )),I _φ is in . If is aβ-structure which satisfies Σ ₁ ¹ -Comprehension relativized toφ and WF(X), and φ is -absolute, thenI _φ is in and ‖φ| <h ( ). These results imply that Barwise-Grilliot theorem is false in the case of uncountable acceptable structures. We also study the notion of invariant definability over models₁ of Δ ₁ ¹ -Comprehension. This paper is registered as Report ZW 69/76 of the Mathematical Centre.     

Multipliers on Besov spaces

It is proved in this paper that the characteristic function of the half-space is not a multiplier for the pair (B _pq ^1/p , B _p ^1/p ), 1<p<, 1<q . In addition, necessary and sufficient conditions are found for the validity of the inclusion.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 36–50, 1984.     

Formal language theory and the geometry of 3-manifolds

Automatic groups were introduced in connection with geometric problems, in particular with the study of fundamental groups of 3-manifolds. In this article the class of automatic groups is extended to include the fundamental group of every compact 3-manifold which satisfies Thurston's geometrization conjecture. Toward this end, the class of asynchronously groups is introduced and studied, where is an arbitrary full abstract family of languages. For example may be the family of regular languagesReg, context-free languagesCF, or indexed languagesInd. The class consists of precisely those groups which are asynchronously automatic. It is proved that contains all of the above fundamental groups, but that does not. Indeed a virtually nilpotent group belongs to if and only if it is virtually abelian. The first author was partially supported by NSF grant DMS-9203500 and FNRS (Suisse). He also wishes to thank the University of Geneva for its hospitality while this paper was being written. The second author thanks the Institute for Advanced Study for its hospitality while this paper was being written.     

Interpolation in Logiken monotoner systeme

A monotone structure ( ;μ) consists of a structure and a monotone systemμ over the domain of .L(Q ⁿ ) is , enlarged by a newn-ary quantifierQ ⁿ . says in ( ;μ) that there isU ∈μ such thatϕ[ā] is valid in ( ;μ) for allā ∈U ⁿ . If is a class of monotone structures, means thatϕ is valid in all expansions of monotone structures in . We show for the class of all ultrafilters that interpolation with respect to holds forL(Q ⁿ ) exactly in casen=1. Then we prove for a large class of (e.g. the class of topological groups) thatL(Q ⁿ ) satisfies interpolation with respect to for alln ≧ 1. Counterexamples indicate that the class of is sharp in some sense. Finally the results are carried over to certain topological structures and the interior quantifiersI ⁿ instead ofQ ⁿ , thereby generalizing results of Makowsky/Ziegler and Sgro, and to a multidimensional type of monotone structures including uniform spaces.     

A technique for calculating the γ-matrix structures of the diagrams of a total four-fermion interaction with infinite number of vertices ind=2+ε dimensional regularization

It is known [1] that ind=2+ dimensional regularization any four-fermion interaction generates an infinite number of counterterms of the form     

Betweeness for real vectors and lines,III alternative characterizations of betweennesses

Summary This paper introduces terminology enabling the discussion of centerrelated betweennesses defined with respect to centers other than the origin. Thus, it is then proved that the familiar -betweenness in which is origin-centered, is equivalent to a non-origin-centered -betweenness in one lower dimension. A new betweenness inK, denoted by , (r>0), is defined and studied, and it is shown that its restriction to is precisely -betweenness for a certain ν. Finally, by a method elaborated in the earlier papers, a new betweeness, , is induced on the upper open-hemisphere-plus-a-point in 3-space, and a characterization of it is obtained which is expected to facilitate later investigations of betweenness is complex spaces.