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.     

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.     

Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

Let _i(L), _i(L^*) denote the successive minima of a latticeL and its reciprocal latticeL ^*, and let [b₁,..., b _n ] be a basis ofL that is reduced in the sense of Korkin and Zolotarev. We prove that and, where and _j denotes Hermite's constant. As a consequence the inequalities are obtained forn7. Given a basisB of a latticeL in ^m of rankn andx ^m , we define polynomial time computable quantities(B) and(x,B) that are lower bounds for ₁(L) and(x,L), where(x,L) is the Euclidean distance fromx to the closest vector inL. If in additionB is reciprocal to a Korkin-Zolotarev basis ofL ^*, then ₁(L) _n ^* (B) and.The research of the second author was supported by NSF contract DMS 87-06176. The research of the third author was performed at the University of California, Berkeley, with support from NSF grant 21823, and at AT&T Bell Laboratories.     

Maße für zyklische symmetrien ebener konvexer körper

Let

denote a finite group of rotations of

A realvalued functional f defined on a subfamily

of the family of all convex bodies in

is called a measure of symmetry with respect to

if (1) 0?(K)?1 for all

; (2) f(K)=1 iff K is symmetric with respect to

; (3)

for all

and for all orientation preserving similarties τ of

; (4) f:

is continuous.

A special measure of symmetry with respect to

,

, is introduced. The following theorems hold: Let

denote the cyclic group of proper rotations of

generated by the rotation about 2π/k with the origin as center, k?3 an integer, and let

denote the family of all plane centrally symmetric convex bodies containing at least two points.

1. 1.

   for k?3, where c_k is a positive real number which is explicitly calculated.

2. 2.

   iff K is a segment. For K=3,4 better results can be obtained. One tool for these improvements is:

3. 3.

   About every plane convex body with interior points an equilateral triangle can be circumscribed such that its center of gravity lies in the interior of K.

     

Isoperimetric inequalities for infinite hyperplane systems

Let be an infinite discrete system ofk-dimensional flats inn-dimensional Euclidean space. If the totalk-dimensional volume of the flats in intersected with the ball of center 0 and radiusr, divided by the volume of that ball, tends to a limit forr→∞, then this limit is called thedensity of . We consider isoperimetric problems of the following kind. If is a hyperplane system of positive density, find sharp upper bounds for the density of the system ofk-flats (k∈{0, ...,n−2}) that are generated as intersections of hyperplanes in . Ideas from the theory of uniform distribution of sequences are employed to define a large class of hyperplane systems, calleduniform, for which all necessary densities exist, isperimetric inequalities can be proved, and systems with extremal intersection densities can be characterized.     

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     

Sull’unione di classi filtrali

Riassunto In questo lavoro si cercano condizioni perchè una classe di algebre simili filtrali sia essa stessa filtrale. Si trova che se è una famiglia finita di algebre simili e filtrali, che siano indipendenti (cfr.A. L. Foster [2]), allora anche è filtrale. Introdotto poi il concetto disemi-indipendenza, si arriva a un risultato analogo per le algebre semifiltrali e a una condizione sufficiente per la semifiltralità di un'algebra finita. Infine si osserva che i risultati trovati possono essere estesi alle classi ideali.

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Summary This paper refers to the concept offiltrale class of similar algebras, which has been introduced byR. Magari in [6]. First I observe that a class of similarfiltrale algebras is not generallyfiltrale, not even if is a finite family of finite algebras. However I find that if is a finite family of similarfiltrale algebras, which are also independent (using this term in the sense ofA. L. Foster [2]), then also isfiltrale. Then introducing the notion of independent varieties, one arrives at a generalization of the above mentioned result. Given then a family of similar algebras , we say that these aresemi-indipendenti if for everyJ⊂I there exists an algebraic functionf _J defined in II _i so thatf _J(x, y)=z wherez _i=x_i ifi∈J andz _i=y_i ifi∉J for every . I prove that if is a finite family ofsemifiltrali andsemi-indipendenti algebras, then also issemifiltrale. I also find that a finite algebra, which is simple andsemi-indipendente of itself (in the sense that there exists in a binary algebraic functionf withf(x, y)=<x ₀,y ₁>) issemifiltrale. Then I give the notion ofsemi-indipendente variety, which leads to some further results. Finally I observe that all these results can be extended to ideal classes of similar algebras.

     

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.     

Multiplicity One for Representations Corresponding to Spherical Distribution Vectors of Class ρ

In this paper one considers a unimodular second countable locally compact group G and the homogeneous space X:=H\G, where H is a closed unimodular subgroup of G. Over X complex vector bundles are considered such that H acts on the fibers by a unitary representation with closed image. The natural action of G on the space of square integrable sections is unitary and possesses an integral decomposition in so-called spherical distributions of class . The uniqueness of this decomposition can be characterized by a number of equivalent properties. Uniqueness is shown to hold for a class of semidirect products. In the case that H is compact, the multiplicity free decomposition is shown to be equivalent with the commutativity of a suitable convolution algebra. As an example, one takes for X a symmetric k-variety , with k a locally compact field of characteristic not equal to two, and for a character of _k, whose square is trivial. Here is a reductive algebraic group defined over k and is the fixed point group of an involution of defined over k. It is shown then that the natural representation of G_k on the Hilbert space is multiplicity free if is anisotropic. Next a criterion is derived that leads to multiplicity one also in the noncompact situation. Finally, in the non-Archimedean case, a general procedure is given that might lead to showing that a pair is a generalized Gelfand pair. Here and are suitable algebraic groups defined over k.Mathematics Subject Classifications (2000) 20G15, 20G20, 22E15, 22E46.Aloysius G. Helminck: Partially supported by N.S.F. Grant DMS-9977392.     

Finding minimum areak-gons

Given a setP ofn points in the plane and a numberk, we want to find a polygon with vertices inP of minimum area that satisfies one of the following properties: (1) is a convexk-gon, (2) is an empty convexk-gon, or (3) is the convex hull of exactlyk points ofP. We give algorithms for solving each of these three problems in timeO(kn ³). The space complexity isO(n) fork=4 andO(kn ²) fork5. The algorithms are based on a dynamic programming approach. We generalize this approach to polygons with minimum perimeter, polygons with maximum perimeter or area, polygons containing the maximum or minimum number of points, polygons with minimum weight (for some weights added to vertices), etc., in similar time bounds.This paper includes work done while David Eppstein was at Columbia University, Department of Computer Science, and while Günter Rote and Gerhard Woeginger were at the Freie Universität Berlin, Fachbereich Mathematik, Institut für Informatik. Research was partially supported by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM).     

Sull'apertura di luoghi negli spettri affine e projettivo di un dominio graduato noetheriano

Riassunto Sia una proprietà degli anelli discende per fedele piattezza, ascende quando le fibre sono buone, di carattere locale e per cui vale il criterio di Nagata. SeS è un dominio graduato noetheriano valgono alcune implicazioni per l'apertura del in SpecS, ProjS e altri schemi collegati adS.

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Summary Let be a property meaningful for rings which descends by faithful flatness, ascends when there are good fibers, with local character and for which Nagata's criterion holds. IfS is any graded noetherian domain some implications hold for the openness of the in SpecS, ProjS and other schemes related toS.

     

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.     

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 .     

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.     

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.     

Symmetric centres of braided monoidal categories

This paper introduces the concept of ‘symmetric centres’ of braided monoidal categories. LetH be a Hopf algebra with bijective antipode over a fieldk. We address the symmetric centre of the Yetter-Drinfel’d module category: and show that a left Yetter-Drinfel’d moduleM belongs to the symmetric centre of and only ifM is trivial. We also study the symmetric centres of categories of representations of quasitriangular Hopf algebras and give a sufficient and necessary condition for the braid of, _Hℳ to induce the braid of , or equivalently, the braid of , whereA is a quantum commutativeH-module algebra     

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.     

Structures of polyhedra determined by submodular functions on crossing families

The present paper shows that for any submodular functionf on a crossing family with , if the polyhedron is nonempty, then there exist a unique distributive lattice with and a unique submodular function with such thatB(f) coincides with the base polyhedron associated with the submodular system . Here, iff is integer-valued, thenf ₁ is also integer-valued. Based on this fact, we also show the relationship between the independent-flow problem considered by the author and the minimum cost flow problem considered by J. Edmonds and R. Giles.     

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.