On the fractional matching polytope of a hypergraph

For a hypergraph andb:⁺ define Conjecture. There is a matching of such that For uniform andb constant this is the main theorem of [4]. Here we prove the conjecture if is uniform or intersecting, orb is constant.The research was done while the author visited the Department of Mathematics at Rutgers University. Research supported in part by the Hungarian National Science Foundation under grant No. 1812Supported in party by NSF and AFOSR grants and by a Sloan Research Fellowship     

Traces of differential functions on subsets of the euclidean space

In the paper one investigates conditions for the existence and for the equality of the traces in the operator sense and in the sense of strict definition for functions from Sobolev and Besov spaces. One gives a complete solution of the problem on the traces and extensions for the trace operator, in the case when is a countably (_m,m)-rectifiable _m-measurable subset of ⁿ.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 52–66, 1986.     

On the category 𝒪 for rational Cherednik algebras

We study the category of representations of the rational Cherednik algebra A_W attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor: _W-mod, where _W is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between /_tor, the quotient of by the subcategory of A_W-modules supported on the discriminant, and the category of finite-dimensional _W-modules. The standard A_W-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of cells, provided W is a Weyl group and the Hecke algebra _W has equal parameters. We prove that the category is equivalent to the module category over a finite dimensional algebra, a generalized q-Schur algebra associated to W.     

The distribution of sublattices of ℤ m

Lattices , are similar if one can be transformed into the other by an angle-preserving linear map. Similarity classes of lattices of rankn may be parametrized by a fundamental domain of the action ofGL _n () on the generalized upper half-plane _n . Given 1<nm and, letN(D,T) be the number of sublattices of _n which have rankn, similarity class inD, and determinant T. Our most basic result will be thatN(D,T)c ₁(m, n)(D)T ^m asT for suitable setsD, where is the invariant measure on _n . The casen=2 had been dealt with by Roelcke and by Maass using the theory of modular forms.Herrn Professor Hlawka zum achtzigsten Geburtstag gewidmetSupported in part by NSF-DMS-9401426     

Some extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

Studies on the Painlevé equations

Summary In this series of papers, we study birational canonical transformations of the Painlevé system , that is, the Hamiltonian system associated with the Painlevé differential equations. We consider also -function related to and particular solutions of . The present article concerns the sixth Painlevé equation. By giving the explicit forms of the canonical transformations of associated with the affine transformations of the space of parameters of , we obtain the non-linear representation: GG_*, of the affine Weyl group of the exceptional root system of the type F₄ A canonical transformation of G_* can extend to the correspondence of the -functions related to . We show the certain sequence of -functions satisfies the equation of the Toda lattice. Solutions of , which can be written by the use of the hypergeometric functions, are studied in details.     

A free convenient vector space for holomorphic spaces

In this paper we show that there exists a free convenient vector space for the case of holomorphic spaces and holomorphic maps. This means that for every spaceX with a holomorphic structure, there exists an appropriately complete locally convex vector space X and a holomorphic mapl _X:XX, such that for any vector space of the same kind the map (l _X )^*:L(X,E)(X,E) is a bijection. Analogously to the smooth case treated in [2, 5.1.1] the free convenient vector space X can be obtained as the Mackey closure of the linear subspace spanned by the image of the canonical mapX(X).In the second part of the paper we prove that in the case whereX is a Riemann surface, one hasX=(X,).     

On the convergence of Fourier series in the metric ofL 1

L ¹ .      

Simplicial cells in arrangements of hyperplanes

Let be an arrangement of n hyperplanes in P ^d , C() its cell complex, and H any hyperplane of . It is proved: (1) If is not a near pencil then there are at least n–d–1 simplicial d-cells of C(), each having no facet in H. (2) There are at least d+1 simplicial d-cells of C(), each having a facet in H.Material for this paper was taken from the author's doctoral dissertation.     

When Is a Subspace of B(H) Ideal?

Let be a real or complex Hilbert space and let () denote the algebra of all bounded linear operators on . We show that if N is a subspace of (H) and for positive operators P₁, P₂ and every A N, P₁P₂A^* + AP₁P₂ N then N is an ideal. Furthermore if is an infinite dimensional real space then N = (H).AMS Subject Classification (1991): Primary 47B47, 47D25     

Optimal ℋ∞-state feedback control for continuous-time linear systemsfeedback control for continuous-time linear systems

This paper proposes a convex programming method to achieve optimal -state feedback control for continuous-time linear systems. State space conditions, formulated in an appropriate parameter space, define a convex set containing all the stabilizing control gains that guarantee an upper bound on the -norm of the closed-loop transfer function. An optimization problem is then proposed, in order to minimize this upper bound over the previous convex set, furnishing the optimal -control gain as its optimal solution. A limiting bound for the optimum -norm can easily be calculated, and the proposed method will achieve minimum attenuation whenever a feasible state feedback controller exists. Generalizations to decentralized and output feedback control are also investigated. Numerical examples illustrate the theory.This research has been supported in part by grants from Fundação de Amparo à Pesquisa do Estado de São Paulo—FAPESP and Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq—Brazil. The authors are grateful to the anonymous referees for their useful comments on this paper.     

Numerically regular hereditary classes of combinatorial geometries

A hereditary class of combinatorial geometries (or simple matroids) is a collection of geometries closed under minors and direct sums. A geometry G in is extremal if no proper extension of G of the same rank is in . The size function h(n) of is defined by h(n)=max {|G|: G and rank(G)=n}, where |G| is the number of points in G. A hereditary class is numerically regular if for every extremal geometry G in , |G|=h (rank(G)). We determine all the numerically regular hereditary classes for which the set {h(n)–h(n–1): 1n<} of positive integers does not have an upper bound: they are all varieties. We also give several examples of numerically regular hereditary classes which are not varieties.Partially supported by a North Texas State University Faculty Research Grant.     

Familien komplexer räume zu gegebenen infinitesimalen deformationen

Let M be a domain in the complex plane, :XM a flat family of reduced complex spaces, (X_o, _o) the fibre over a point OM, and x_o the sheaf of (1,O)-forms over X_o. The family defines an element (Ext¹ (_Xo, _o))_x for every point xX. We prove: If (X_o, _o) is a normal complex space, x a point in X_o such that (Ext² (_Xo, _o))_x=O, then for each infinitesimal deformation (Ext¹ (_Xo, _o))_x there exists a flat reduced family with =. This statement is analogous to a result of KODAIRA-NIRENBERG-SPENCER in the theory of deformations of compact complex manifolds.     

On positivity in Hecke algebras

Let be the Hecke algebra associated with a Coxeter system (W, R). The structure constants of with respect to various bases are Laurent polynomials, whose coefficients enjoy remarkable positivity properties. We survey these and prove some new ones using the relationship between and the geometry of Schubert varieties.To Professor Jacques Tits on his sixtieth birthday     

Superrigidity of lattices in solvable Lie groups

Summary Let be a closed, cocompact subgroup of a simply connected, solvable Lie groupG, such that Ad _G has the same Zariski closure as AdG. If : GL _n () is any finite-dimensional representation of , we show that virtually extends to a representation ofG. (By combining this with work of Margulis on lattices in semisimple groups, we obtain a similar result for lattices in many groups that are neither solvable nor semisimple.) Furthermore, we show that if is isomorphic to a closed, cocompact subgroup of another simply connected, solvable Lie groupG, then any isomorphism from to extends to a crossed isomorphism fromG toG. In the same vein, we prove a more concrete form of Mostow's theorem that compact solvmanifolds with isomorphic fundamental groups are diffeomorphic.Oblatum 5-VII-1994 & 15-IV-1995     

Une méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique I. Première partie: une démonstration de la conjecture isopérimétrique Pλ2 ≥ πj 0 4 /2 de Pólya et Szegö

Résumé Soient _G la première valeur propre de la membrane à contour fixé sur un domaine simplement connexeG, etP _G la rigidité à la torsion deG. En construisant un cercleK tel queP _K P _G et _{K G}, on démontre la conjecture de Pólya et Szegö [14]; P_{G G} ² j ₀ ² /2. Ce résultat renforce le théorème isopérimétrique classique de Rayleigh [15], Faber [4] et Krahn [10].

---

Summary Let _G be the first eigenvalue of the fixed membrane on a simply connected domainG, and letP _G be the torsional rigidity ofG. We prove Pólya-Szegö's conjecture [14]: P_{G G} ² j ₀ ² /2, by constructing a circleK such thatP _K P _G and _{K G}. This result sharpens the classical isoperimetric theorem of Rayleigh [15], Faber [4] and Krahn [10].

     

Infinitely divisible completely positive mappings

Summary We generalise the theory of infinitely divisible positive definite functions f:G on a group G to a theory of infinite divisibility for completely positive mappings : G() taking values in the algebra of bounded operators on some Hilbert space .We prove a structure theorem for normalised infinitely divisible completely positive mappings which shows that the mapping , its Stinespring representation and its Stinespring isometry are of type S (in the sense of Guichardet [Gui]). Furthermore, we prove that a completely positive mapping is infinitely divisible if and only if it is the exponential (as defined in this paper) of a hermitian conditionally completely positive mapping.     

On the almost everywhere convergence of ergodic averages for power-bounded operators on LP-subspaces

Let X be a closed subspace of L^P(), where is an arbitrary measure and 1A(n) and (n) denote the discrete ergodic averages and Hilbert transform truncates defined by U. We extend to this setting the -a. e. convergence criteria forA(n) and (n) which V. F. Gaposhkin and R. Jajte introduced for unitary operators on L²(). Our methods lift the setting from X to ^p, where classical harmonic analysis and interpolation can be applied to suitable square functions.     

Subgroups of the general symplectic group containing the group of diagonal matrices

There are described the subgroups of the general symplectic group =GSp(2n, R) over a commutative semilocal ring R, containing the group of symplectic diagonal matrices. For each such subgroup P there is uniquely defined a symplectic D-net a such that ()pN(), where () is the net subgroup in corresponding to (cf. RZhMat, 1977, 5A288), and N() is its normalizer. The quotient group N × ()/() is calculated. There are also considered subgroups in Sp(2n, R). Analogous results for subgroups of the general linear group were obtained earlier in RZhMat, 1978, 9A237.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 31–47, 1980.     

Decomposition of large uniform hypergraphs

We determine a minimum cardinality family _{n, k} (resp. _{n, k} ) ofn-uniform,k-edge hypergraphs satisfying the following property: all, except for finitely many,n-uniform hypergraphs satisfying the divisibility condition have an _{n, k} -decomposition (resp. vertex _{n, k} -decomposition).