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     

On inductive limits of measure spaces

We introduce a category of (topological) measure spaces in which inductive limitis exist and where the Banach spaces and (1≤p≤+∞) are isometric for arbitrary inductive systems of (topological) measure spaces.     

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.

     

A Schwarz lemma for a domain related to μ-synthesis

We prove a Schwarz lemma for a domain in ℂ³ that arises in connection with a problem in H^∞ control theory. We describe a class of automorphisms of and determine the distinguished boundary of We apply our Schwarz lemma to a special case of the μ-synthesis problem.     

Enumeration ofQ-acyclic simplicial complexes

Let (n, k) be the class of all simplicial complexesC over a fixed set ofn vertices (2≦k≦n) such that: (1)C has a complete (k−1)-skeleton, (2)C has precisely ( _k ⁿ⁻¹ )k-faces, (3)H _k (C)=0. We prove that for ,H _k−1(C) is a finite group, and our main result is: . This formula extends to high dimensions Cayley’s formula for the number of trees onn labelled vertices. Its proof is based on a generalization of the matrix tree theorem.     

A decomposition of 2-weak vertex-packing polytopes

The 2-weak vertex-packing polytope of a loopless graphG withd vertices is the subset of the unitd-cube satisfyingx _i +x _j ≤1 for every edge (i,j) ofG. The dilation by 2 of this polytope is a polytope with integral vertices. We triangulate with lattice simplices of minimal volume and label the maximal simplices with elements of the hyperoctahedral groupB _d . This labeling gives rise to a shelling of the triangulation of , where theh-vector of (and the Ehrharth ^*-vector of can be computed as a descent statistic on a subset ofB _d defined in terms ofG. A recursive way of computing theh-vector of is also given, and a recursive formula for the volume of . This work was partially supported by grants from the Icelandic Council of Science and the Royal Swedish Academy of Sciences, respectively.     

Un teorema sulla rappresentazione degli integrali

Let be a De Possel differentiation basis in a complete measure space , with μ≥0, μ(X)<+∞; letB be a Banach space and . When the F′ exists μ-a.e. onX, we prove that for eachp>0, the μ-integral of‖F′‖ ^p is the lower-bound of a class ofp-variations.     

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.     

Intersections of curve systems and the crossing number ofC 5 ×C 5

It and are two families of pairwise disjoint simple closed curves in the plane such that each curve in intersects each curve in , then the total number of points of intersection in is at least 2(m−1)n, where , and this bound is best possible. We use this to show that the cartesian product of two 5-cycles has crossing number 15.     

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.     

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.     

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     

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 .     

Large sample theory for distributions on the hypersphere with rotational symmetries

Summary A distribution on the unit sphere inℝ ^q with a densityf(‖x _v ‖) is considered where is ans(<q) dimensional subspace andx _v is the part ofx in . For a large sample the estimation of , a test that and a test for rotational symmetry within is given. For several samples with possibly different subspaces but the samef, a test that is given. For all tests power functions for contiguous alternatives are given. For the special density proportional to expk‖x _v ‖ ², additional results are given. Research supported in part by a Contract with the Office of Naval Research N00014-81-K-0146 awarded to Princeton University, Princeton, New Jersey 08544.     

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.     

Generalized quadrangles having a prime parameter

Generalized quadrangles are studied in whichs ort is prime and Aut has rank 3 on points. This research was supported in part by NSF grant GP 37982X.     

The orbit method and Gelfand pairs, associated with nilpotent Lie groups

Let K be a compact Lie group acting by automorphisms on a nilpotent Lie group N. One calls (K, N) a Gelfand pair when the integrable K-invariant functions on N form a commutative algebra under convolution. We prove that in this case the coadjoint orbits for G:= K × N which meet the annihilator of the Lie algebra of K do so in single K-orbits. This generalizes a result of the authors and R. Lipsman concerning Gelfand pairs associated with Heisenberg groups.     

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.     

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.     

Fibrazioni di una classe di near-field regolari di dimensionet+1 sul nucleo

Il contenuto di questo lavoro consiste nel mostrare che le fibrazioni dei near-field regolari di dimensionet+1 sul nucleo , quando , si ottengono da fibrazioni regolari con una operazione di scambio fra alcuni sistemi di matrici, che generalizza il procedimento di sostituzione di regoli con regoli opposti di quadriche rigate nel caso di dimensione due sul nucleo.