Convex mappings on some Reinhardt domains

In this paper, we consider the following Reinhardt domains. Let M = （M1, M2,..., Mn） ： [0,1] → [0,1]^n be a C2-function and Mj（0） = 0, Mj（1） = 1, Mj″ 〉 0, C1jr^pj-1 〈 Mj′（r） 〈 C2jr^pj-1, r∈ （0, 1）, pj 〉 2, 1 ≤ j ≤ n, 0 〈 C1j 〈 C2j be constants. Define
DM={z=（z1,z2,…,Zn）^T∈C^n：n∑j=1 Mj（｜zj｜）〈1}
Then DM C^n is a convex Reinhardt domain. We give an extension theorem for a normalized biholomorphic convex mapping f ： DM -→ C^n.     

Hölder Regularity for Solutions of Ultraparabolic Equations in Divergence Form

We consider the second order differential equation , where (x,t) ^N+1, 0<m ₀N, the coefficients a _i,j belong to a suitable space of vanishing mean oscillation functions VMO _L and B=(b _i,j ) is a constant real matrix. The aim of this paper is to study interior regularity for weak solutions to the above equation assuming that F _j belong to a function space of Morrey type.     

Asymptotic Distribution of Quadratic Forms and Applications

We consider the quadratic formsQ X _j X _k+ (X _j ² -E X _j ² )where X _j are i.i.d. random variables with finite sixth moment. For a large class of matrices (a _jk ) the distribution of Q can be approximated by the distribution of a second order polynomial in Gaussian random variables. We provide optimal bounds for the Kolmogorov distance between these distributions, extending previous results for matrices with zero diagonals to the general case. Furthermore, applications to quadratic forms of ARMA-processes, goodness-of-fit as well as spacing statistics are included.     

Tame Sets in the Complement of Algebraic Variety

Let A?? ^N be an algebraic variety with dim?A≤N?2. Given discrete sequences {a _j },{b _j }?? ^N \ A with slow growth ( $\sum_{j}{1\over|a_{j}|^{2}}<\infty,\sum_{j}{1\over |b_{j}|^{2}}<\inftyLet A⊂ℂ ^N be an algebraic variety with dim A≤N−2. Given discrete sequences {a _j },{b _j }⊂ℂ ^N \ A with slow growth ( ?_j[1/(|a_j|²)] < ￥,?_j[1/(|b_j|²)] < ￥\sum_{j}{1\over|a_{j}|^{2}}<\infty,\sum_{j}{1\over |b_{j}|^{2}}<\infty ) we construct a holomorphic automorphism F with F(z)=z for all z∈A and F(a _j )=b _j for all j∈ℕ. Additional approximation of a given automorphism on a compact polynomially convex set, fixing A, is also possible. Given unbounded analytic variety A there is a tame set E such that F(E)≠{(j,0 ^N−1):j∈ℕ} for all automorphisms F with F| _A =id. As an application we obtain an embedding of a Stein manifold into the complement of an algebraic variety in ℂ ^N with interpolation on a given discrete set.     

A less formal approach to kaluza-klein formalism

The action integrals (a) and , corresponding respectively to gravitational and gravitational-electromagnetic phenomena, are shown to be related under continuous groups of null translations. This relation motivates a modified Kaluza—Klein formalism for which the classical cylindrical metric preserving transformations (c)y ⁵ = =x ⁵ +f ⁵(x ^j ),y ⁱ =f ⁱ (x ^j ) fori = 1, 2, 3, 4 are replaced by (d)y ⁵ =x ⁵,y ⁱ =f ⁱ (x ^j ,x ⁵). The cylindrical metric of V⁵ is nevertheless preserved under (d), since it is assumed thatV ⁵ admits a metric of the form (corresponding to (a)) and that (d) defines a continuous group of null translations in theV ⁴ metric defined byg _ij whenx ⁵ is considered the group parameter. Application of (d) leads to the cylindrical metric corresponding to (b). The resulting electromagnetic fieldsF _ij =B _i,j –B _j,i are then related to the curvatures of theV ⁴ corresponding tog _ij andh _ij ; in particular it is shown that and . When it is shown thatF _ij is a null electromagnetic field which is generally non-trivial. Some physical and geometric interpretations of the mathematical results are also presented.Dedicated to Professor A. Ostrowski on the occasion of his 75th birthday     

An algorithm for a linear diophantine equation and a problem of Frobenius

Summary We solve the diophantine equation for nonnegative variablesx _j , wherea _j andL are positive integers. We characterize both the values ofL that lead to solutions and those that do not lead to solutions. We solve the Frobenius problem of finding the largest value ofL for which no solution exists.     

The structure of Banach algebras of bounded continuous functions on the open disk that contain H∞, the Hoffman algebra,and nontangential limits

LetH _B ^G be an algebra of bounded continuous functions on an open disk representable in the formH _B G, where andH _B is a closed subalgebra in C(D) consisting of the functions that have nontangential limits almost everywhere on {ie1023-06}, and these limits belong to the Douglas algebraB. In this paper we describe the spaceM(H _B ^G ) of maximal ideals of the algebraH _B ^G and prove thatM(H _B ^G ) =M(B) M(H _B ^G and prove thatM(H ₀ ^G ), whereH ₀ ^G is a closed ideal inG consisting of functions having nontangential limits equal to zero almost everywhere on {ie1023-12}. Moreover, it is established that on the disk. The Chang-Marshall theorem is generalized for the Banach algebrasH _B ^G . We also prove that for any Douglas algebraB, whereI B = {u } _B are inner functions such that on.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 924–931, July, 1993.     

The dirichlet problem for sub-elliptic second order equations

The C-regularity up to the boundary of solutions to the Dirichlet problem: is proved, using a comparison principle of L with a Hörmander's type operator X _j ^* X_j, where is a smooth bounded open subset of Rⁿ, and is a second-order degenerate elliptic operator with smooth coefficients, satisfying the so-called Fefferman-Phong's condition.     

Determinants of Matrices Associated with Incidence Functions on Posets

Let S = x ₁,...,x _n} be a finite subset of a partially ordered set P. Let f be an incidence function of P. Let denote the n × n matrix having f evaluated at the meet of x _i and x _j as its i, j-entry and denote the n × n matrix having f evaluated at the join of x _i and x _j as its i, j-entry. The set S is said to be meet-closed if for all 1 i, j n. In this paper we get explicit combinatorial formulas for the determinants of matrices and on any meet-closed set S. We also obtain necessary and sufficient conditions for the matrices and on any meet-closed set S to be nonsingular. Finally, we give some number-theoretic applications.     

A Weighted Goodness-of-Fit Test for GARCH(1,1) Specification

Suppose that (j) is the lag-j autocorrelation of the squared residuals computed from a realization of length n under the assumption that the observations follow a GARCH(1,1) model. We study the asymptotic distribution of the statistics of the form , where the _j are nonnegative summable weights and the matrix , can be estimated from the data. We show that, under weak assumptions on model errors, the statistic Q _n converges in distribution to , where the N _i are iid standard normal. We discuss choices of the weights _j for which the distribution of Q is tabulated. Our results lead to and provide a rigorous justification for Portmanteau goodness-of-fit tests for GARCH(1,1) specification.     

An Intersection Theorem on an Unbounded Set and Its Application to the Fair Allocation Problem

We prove the following theorem. Let m and n be any positive integers with mn, and let be a subset of the n-dimensional Euclidean space ⁿ . For each i=1, . . . , m, there is a class of subsets M _i ^j of ^Tn . Assume that for each i=1, . . . , m, that M _i ^j is nonempty and closed for all i, j, and that there exists a real number B(i, j) such that and its jth component _{xjB(i, j)} imply . Then, there exists a partition of {1, . . . , n} such that for all i and We prove this theorem based upon a generalization of a well-known theorem of Birkhoff and von Neumann. Moreover, we apply this theorem to the fair allocation problem of indivisible objects with money and obtain an existence theorem.     

Schur rings and cyclic association schemes of class three

LetG be a group of finite order andD ₀ = {e},D ₁,...,D _d be a partition ofG. Suppose{d ^–1|d D _i } =D _{i, i {0, 1,..., d}}, for eachi {0, 1,..., d}; and for alli, j where . Then the subalgebra spanned by is called a Schur ring overG. It is known that such a partitionD ₀,D ₁,...,D _d can be used to construct an association scheme of classd. In this paper, we obtain a complete classification for the case whenG is cyclic andd = 3. The result corresponds to a complete classification of cyclic association schemes of class three.     

Heegaard Genus Formula for Haken Manifolds

Suppose M is a compact orientable 3-manifold and a properly embedded orientable boundary incompressible essential surface. Denote the completions of the components of M – Q with respect to the path metric by M ¹, ...,M ^k . Denote the smallest possible genus of a Heegaard splitting of M, or M ^j respectively, for which ∂M, or ∂M ^j respectively, is contained in one compression body by g(M, ∂M), or g(M ^j , ∂M ^j ) respectively. Denote the maximal number of non-parallel essential annuli that can be simultaneously embedded in M ^j by n _j . Then

A class of Banach algebras whose duals have the Radon-Nikodym property

Let A be a complex, commutative Banach algebra and let M_A be the structure space of A. Assume that there exists a continuous homomorphism h : L¹(G) → A with dense range, where L¹(G) is the group algebra of a locally compact abelian group G. The main results of this paper can be summarized as follows: (a) If the dual space A* has the Radon-Nikodym property, then M_A is scattered (i.e., it has no nonempty perfect subset) and . (b) If the algebra A has an identity, then the space A* has the Radon-Nikodym property if and only if . Furthermore, any of these conditions implies that M_A is scattered. Several applications are given. Received: 29 September 2005     

Die asymptotische Verteilung von mehrfachen Koinzidenzen

Zusammenfassung Ist U eine symmetrische Menge des R _m (i. e. -U=U R _m ), so heien zwei Punkte x ₁, x ₂ R _m U-koinzident falls x ₁–x₂U gilt. Dieser Koinzidenzbegriff wurde in [1] eingeführt. In dieser Arbeit wurde folgendes allgemeine Resultat bewiesen:Sind x ₁,...,x _n ER _m gegebene Punkte, ist AR _m beliebig und U=–UR _m , so bezeichne K(x₁,h.,x_n; U, A) die Anzahl der U-koinzidenten Paare unter denjenigen der Punkte x ₁,...,x _n, die in A fallen. Ist eine Folge von unabhÄngigen, m-dimensionalen Zufallsvariablen, mit der allen gemeinsamen, quadratisch integrierbaren Dichte f, so ordne man jeder Folge von symmetrischen Borelmengen des R _m die Folge der stochastischen Prozesse zu. Weiters bezeichne den durch das Ma induzierten Poissonproze.Dann wurde in [1] gezeigt, da unter geeigneten Voraussetzungen über die Folge und für beliebige die Folge der gemeinsamen Verteilungen von (K(x ₁,h.,x_n; U _n , A _j ): j=1(1)r) gegen die gemeinsame Verteilung von (K(A _j ): j=1(1) r) strebt.In der vorliegenden Arbeit wird der Begriff der k-fachen U-Koinzidenz eingeführt und ein analoges Resultat für die asymptotische Verteilung der Anzahlen von k-fachen Koinzidenzen hergeleitet.     

Factorization and reflexivity on Fock spaces

The framework of the paper is that of the full Fock space and the Banach algebraF which can be viewed as non-commutative analogues of the Hardy spacesH ² andH respectively.An inner-outer factorization for any element in as well as characterization of invertible elements inF are obtained. We also give a complete characterization of invariant subspaces for the left creation operatorsS ₁ ,..., S _n of . This enables us to show that every weakly (strongly) closed unital subalgebra of {(S ₁ ,..., S _n ) F } is reflexive, extending in this way the classical result of Sarason [S]. Some properties of inner and outer functions and many examples are also considered.The first author was supported in part by NSF DMS 93-21369 1991Mathematics Subject Classification. Primary 47D25, Secondary 32A35, 47A67.     

Equivariant K-Homology and Restriction to Finite Cyclic Subgroups

For a discrete group G, we prove that a G-map between proper G–CW-complexes induces an isomorphism in G-equivariant K-homology if it induces an isomorphism in C-equivariant K-homology for every finite cyclic subgroup C of G. As an application, we show that the source of the Baum–Connes assembly map, namely K _* ^G (E(G, in)), is isomorphic to K _* ^G (E(G, )), where E(G, ) denotes the classifying space for the family of finite cyclic subgroups of G. Letting be the family of virtually cyclic subgroups of G, we also establish that and related results.     

Weak convergence of an empirical process indexed by the closed convex subsets ofI 2

Summary LetI ² be the unit cube of andX _i be independentI ²-valued random variables that are distributed according to Lebesgue-measure. IfS is the set of closed convex subsets ofI ² we consider the process _n (A) _AS,where .It is proved that this process suitably normalized converges in a suitable weak sense to a Gaussian process.     

The injection and the projection theorem for spectral sets

For a closed normal subgroupN of a locally compact groupG view a closed subset of Prim_* L ¹ (G/N) as a subsetE of Prim_* L ¹ (G) in the canonical way and writeN for Prim_* L ¹ (G/N) as a subset of Prim_* L ¹ (G); then the injection theorem says: IfE is spectral (i.e. of synthesis), then is so; and if andN are spectral, thenE is too. In case of a group of polynomial growth with symmetricL ¹-algebra, where smallest idealsj (E) with given hulls exist, it is known thatN is always spectral. For a closed,G-invariant subsetF of Prim_* L ¹ (N) define a closed subsetE of Prim_* L ¹ (G) by . Denote by e (I') the ideal generated byC ₀₀ (G)*I', where theG-invariant idealI' ofL ¹ (N) is viewed as a subset of measures onG, then the projection theorem states: IfE is spectral, thenF is so, and ifF is spectral withe (j (F))=j (E) thenE is spectral. All assumptions are fulfilled for instance, ifG andN are of polynomial growth with symmetricL ¹-algebra and eitherSIN-groups or solvable.