Let $L_p=L_p(X,\mu)$, $1\leq p\leq\infty$, be the usual Banach Spaces of real valued functions on a complete non-atomic probability space. Let $(T_1,\ldots,T_{K})$ be $L_2$-contractions. Let $0<\varepsilon < \delta\leq1$. Call a function $f$ a $\delta$-spanning function if $\|f\|_2 = 1$ and if $\|T_kf-Q_{k-1}T_kf\|_2\geq\delta$ for each $k=1,\ldots,K$, where $Q_0=0$ and $Q_k$ is the orthogonal projection on the subspace spanned by $(T_1f,\ldots,T_kf)$. Call a function $h$ a $(\delta,\varepsilon)$-sweeping function if $\|h\|_\infty\leq1$, $\|h\|_1<\varepsilon$, and if $\max_{1\leq k\leq K}|T_kh|>\delta-\varepsilon$ on a set of measure greater than $1-\varepsilon$. The following is the main technical result, which is obtained by elementary estimates. There is an integer $K=K(\varepsilon,\delta)\geq1$ such that if $f$ is a $\delta$-spanning function, and if the joint distribution of $(f,T_1f,\ldots,T_Kf)$ is normal, then $h=\bigl((f\wedge M)\vee(-M)\bigr)/M$ is a $(\delta,\varepsilon)$-sweeping function, for some $M>0$. Furthermore, if $T_k$s are the averages of operators induced by the iterates of a measure preserving ergodic transformation, then a similar result is true without requiring that the joint distribution is normal. This gives the following theorem on a sequence $(T_i)$ of these averages. Assume that for each $K\geq1$ there is a subsequence $(T_{i_1},\ldots,T_{i_K})$ of length $K$, and a $\delta$-spanning function $f_K$ for this subsequence. Then for each $\varepsilon>0$ there is a function $h$, $0\leq h\leq1$, $\|h\|_1<\varepsilon$, such that $\limsup_iT_ih\geq\delta$ a.e.. Another application of the main result gives a refinement of a part of Bourgain's ``Entropy Theorem'', resulting in a different, self contained proof of that theorem.

We present a subordination theory for spatial branching processes. This theory is developed in three different settings, first for branching Markov processes, then for superprocesses and finally for the path-valued process called the {\it Brownian snake}. As a common feature of these three situations, subordination can be used to generate new branching mechanisms. As an application, we investigate the compact support property for superprocesses with a general branching mechanism.

For a meromorphic (or harmonic) function $f$, let us call the dilation of $f$ at $z$ the ratio of the (spherical) metric at $f(z)$ and the (hyperbolic) metric at $z$. Inequalities are known which estimate the $\sup$ norm of the dilation in terms of its $L^p$ norm, for $p>2$, while capitalizing on the symmetries of $f$. In the present paper we weaken the hypothesis by showing that such estimates persist even if the $L^p$ norms are taken only over the set of $z$ on which $f$ takes values in a fixed spherical disk. Naturally, the bigger the disk, the better the estimate. Also, We give estimates for holomorphic functions without zeros and for harmonic functions in the case that $p=2$.

Positive, copositive, onesided and intertwining (co-onesided) polynomial and spline approximations of functions $f\in\Wp^k\mll$ are considered. Both uniform and pointwise estimates, which are exact in some sense, are obtained.

This paper studies closed subspaces $L$ of the Hardy spaces $H^p$ which are $g$-invariant ({\it i.e.}, $g\cdot L \subseteq L)$ where $g$ is inner, $g\neq 1$. If $p=2$, the Wold decomposition theorem implies that there is a countable ``$g$-basis'' $f_1, f_2,\ldots$ of $L$ in the sense that $L$ is a direct sum of spaces $f_j\cdot H^2[g]$ where $H^2[g] = \{f\circ g \mid f\in H^2\}$. The basis elements $f_j$ satisfy the additional property that $\int_T |f_j|^2 g^k=0$, $k=1,2,\ldots\,.$ We call such functions $g$-$2$-inner. It also follows that any $f\in H^2$ can be factored $f=h_{f,2}\cdot (F_2\circ g)$ where $h_{f,2}$ is $g$-$2$-inner and $F$ is outer, generalizing the classical Riesz factorization. Using $L^p$ estimates for the canonical decomposition of $H^2$, we find a factorization $f=h_{f,p} \cdot (F_p \circ g)$ for $f\in H^p$. If $p\geq 1$ and $g$ is a finite Blaschke product we obtain, for any $g$-invariant $L\subseteq H^p$, a finite $g$-basis of $g$-$p$-inner functions.

No abstract.

Exterior powers of the adjoint representation of a complex semisimple Lie algebra are decomposed into irreducible representations, to varying degrees of satisfaction.

We have previously shown how to construct a deformation quantization of any locally compact space on which a vector group acts. Within this framework we show here that, for a natural class of Hamiltonians, the quantum evolutions will have the classical evolution as their classical limit.

Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions $|x_1|^{\alpha_1}\cdots |x_d|^{\alpha_d}$ on the unit sphere $S^{d-1}$ in $\RR^d$. The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.

In the present paper, we face the problem of classifying classes of orientable PL $5$-manifolds $M^5$ with $h \geq 1$ boundary components, by making use of a combinatorial invariant called {\it regular genus} ${\cal G}(M^5)$. In particular, a complete classification up to regular genus five is obtained: $${\cal G}(M^5) = \gG \leq 5 \Longrightarrow M^5 \cong \#_{\varrho - \gbG}(\bdo) \# \smo_{\gbG},$$ where $\gbG = {\cal G}(\partial M^5)$ denotes the regular genus of the boundary $\partial M^5$ and $\smo_{\gbG}$ denotes the connected sum of $h\geq 1$ orientable $5$-dimensional handlebodies $\cmo_{\alpha_i}$ of genus $\alpha_i\geq 0$ ($i=1,\ldots, h$), so that $\sum_{i=1}^h \alpha_i = \gbG.$ \par Moreover, we give the following characterizations of orientable PL $5$-manifolds $M^5$ with boundary satisfying particular conditions related to the ``gap'' between ${\cal G}(M^5)$ and either ${\cal G}(\partial M^5)$ or the rank of their fundamental group $\rk\bigl(\pi_1(M^5)\bigr)$: $$\displaylines{{\cal G}(\partial M^5)= {\cal G}(M^5) = \varrho \Longleftrightarrow M^5 \cong \smo_{\gG}\cr {\cal G}(\partial M^5)= \gbG = {\cal G}(M^5)-1 \Longleftrightarrow M^5 \cong (\bdo) \# \smo_{\gbG}\cr {\cal G}(\partial M^5)= \gbG = {\cal G}(M^5)-2 \Longleftrightarrow M^5 \cong \#_2 (\bdo) \# \smo_{\gbG}\cr {\cal G}(M^5) = \rk\bigl(\pi_1(M^5)\bigr)= \varrho \Longleftrightarrow M^5 \cong \#_{\gG - \gbG}(\bdo) \# \smo_{\gbG}.\cr}$$ \par Further, the paper explains how the above results (together with other known properties of regular genus of PL manifolds) may lead to a combinatorial approach to $3$-dimensional Poincar\'e Conjecture.

In this paper we prove, that under certain hypotheses, the planar differential equation: $\dot x=X_1(x,y)+X_2(x,y)$, $\dot y=Y_1(x,y)+Y_2(x,y)$, where $(X_i,Y_i)$, $i=1$, $2$, are quasi-homogeneous vector fields, has at most two limit cycles. The main tools used in the proof are the generalized polar coordinates, introduced by Lyapunov to study the stability of degenerate critical points, and the analysis of the derivatives of the Poincar\'e return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.

The spectral theory for the Neumann Laplacian on planar domains with symmetric, horn-like ends is studied. For a large class of such domains, it is proven that the Neumann Laplacian has no singular continuous spectrum, and that the pure point spectrum consists of eigenvalues of finite multiplicity which can accumulate only at $0$ or $\infty$. The proof uses Mourre theory.

Symplectic and orthogonal Schur functions can be defined combinatorially in a manner similar to the classical Schur functions. This paper demonstrates that they can also be expressed as determinants. These determinants are generated using planar decompositions of tableaux into strips and the equivalence of these determinants to symplectic or orthogonal Schur functions is established by Gessel-Viennot lattice path techniques. Results for rational (also called {\it composite}) Schur functions are also obtained.

A formula is obtained for the rank of the $2$-Sylow subgroup of the ideal class group of imaginary bicyclic biquadratic fields. This formula involves the number of primes that ramify in the field, the ranks of the $2$-Sylow subgroups of the ideal class groups of the quadratic subfields and the rank of a $Z_2$-matrix determined by Legendre symbols involving pairs of ramified primes. As applications, all subfields with both $2$-class and class group $Z_2 \times Z_2$ are determined. The final results assume the completeness of D.~A.~Buell's list of imaginary fields with small class numbers.

We give several new characterizations of Riordan Arrays, the most important of which is: if $\{d_{n,k}\}_{n,k \in {\bf N}}$ is a lower triangular array whose generic element $d_{n,k}$ linearly depends on the elements in a well-defined though large area of the array, then $\{d_{n,k}\}_{n,k \in {\bf N}}$ is Riordan. We also provide some applications of these characterizations to the lattice path theory.

In this paper we prove a complete convergence theorem for attractive, reversible, super-critical nearest particle systems satisfying a natural regularity condition. In particular this implies that under these conditions there exist precisely two extremal invariant measures. The result we prove is relevant to question seven of Liggett (1985), Chapter~VII.

In this paper, we study the local bifurcations of critical periods in the neighborhood of a nondegenerate centre of the reduced Kukles system. We find at the same time the isochronous systems. We show that at most three local critical periods bifurcate from the Christopher-Lloyd centres of finite order, at most two from the linear isochrone and at most one critical period from the nonlinear isochrone. Moreover, in all cases, there exist perturbations which lead to the maximum number of critical periods. We determine the isochrones, using the method of Darboux: the linearizing transformation of an isochrone is derived from the expression of the first integral. Our approach is a combination of computational algebraic techniques (Gr\"obner bases, theory of the resultant, Sturm's algorithm), the theory of ideals of noetherian rings and the transversality theory of algebraic curves.

In \cite{Anker}, Jean-Philippe Anker conjectures an upper bound for the heat kernel of a symmetric space of noncompact type. We show in this paper that his prediction is verified for the space of positive definite $n\times n$ real matrices.

Limit transitions will be derived between the five parameter family of Askey-Wilson polynomials, the four parameter family of big $q$-Jacobi polynomials and the three parameter family of little $q$-Jacobi polynomials in $n$ variables associated with root system $\BC$. These limit transitions generalize the known hierarchy structure between these families in the one variable case. Furthermore it will be proved that these three families are $q$-analogues of the three parameter family of $\BC$ type Jacobi polynomials in $n$ variables. The limit transitions will be derived by taking limits of $q$-difference operators which have these polynomials as eigenfunctions.

Explicit bounds for the Hurwitz constants for general cofinite Fuchsian groups have been found. It is shown that the bounds obtained are exact for the Hecke groups and triangular groups with signature $(0:2,p,q)$.

We give explicit combinatorial expresssions for the characteristic cycles associated to certain canonical sheaves on Schubert varieties $X$ in the classical Hermitian symmetric spaces: namely the intersection homology sheaves $IH_X$ and the constant sheaves $\Bbb C_X$. The three main cases of interest are the Hermitian symmetric spaces for groups of type $A_n$ (the standard Grassmannian), $C_n$ (the Lagrangian Grassmannian) and $D_n$. In particular we find that $CC(IH_X)$ is irreducible for all Schubert varieties $X$ if and only if the associated Dynkin diagram is simply laced. The result for Schubert varieties in the standard Grassmannian had been established earlier by Bressler, Finkelberg and Lunts, while the computations in the $C_n$ and $D_n$ cases are new. Our approach is to compute $CC(\Bbb C_X)$ by a direct geometric method, then to use the combinatorics of the Kazhdan-Lusztig polynomials (simplified for Hermitian symmetric spaces) to compute $CC(IH_X)$. The geometric method is based on the fundamental formula $$CC(\Bbb C_X) = \lim_{r\downarrow 0} CC(\Bbb C_{X_r}),$$ where the $X_r \downarrow X$ constitute a family of tubes around the variety $X$. This formula leads at once to an expression for the coefficients of $CC(\Bbb C_X)$ as the degrees of certain singular maps between spheres.

Using Feferman-Vaught techniques we show a certain property of the fine spectrum of an admissible class of structures leads to a first-order law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first-order law. We present three conditions for verifying that the above property actually holds. The first condition is that the count function of an admissible class has regular variation with a certain uniformity of convergence. This applies to a wide range of admissible classes, including those satisfying Knopfmacher's Axiom A, and those satisfying Bateman and Diamond's condition. The second condition is similar to the first condition, but designed to handle the discrete case, {\it i.e.}, when the sizes of the structures in an admissible class $K$ are all powers of a single integer. It applies when either the class of indecomposables or the whole class satisfies Knopfmacher's Axiom A$^\#$. The third condition is also for the discrete case, when there is a uniform bound on the number of $K$-indecomposables of any given size.

The abstract Witt rings which are Gorenstein have been classified when the dimension is one and the classification problem for those of dimension zero has been reduced to the case of socle degree three. Here we classifiy the Gorenstein Witt rings of fields with dimension zero and socle degree three. They are of elementary type.

We show that the Meixner, Pollaczek, Meixner-Pollaczek, the continuous $q$-ultraspherical polynomials and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures. We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials.

We introduce a double sum extension of a very well-poised series and extend to this the transformations of Bailey and Sears as well as the ${}_6\f_5$ summation formula of F.~H.~Jackson and the $q$-Dixon sum. We also give $q$-integral representations of the double sum. Generalizations of the Nassrallah-Rahman integral are also found.

For ${n \over {n-2}}\leq p<\infty$ we show that the conditions $C_{2,q}(G\setminus \dox)=C_{2,q}(G \setminus X)$ for all open sets $G$, $C_{2,q}$ denoting Bessel capacity, are not sufficient to characterize the compact sets $X$ with the property that each function harmonic on $\dox$ and in $L^p(X)$ is the limit in the $L^p$ norm of a sequence of functions which are harmonic on neighbourhoods of $X$.

In this work we study the global geometry of planar quadratic Hamiltonian systems with a center and we sum up the dynamics of these systems in geometrical terms. For this we use the algebro-geometric concept of multiplicity of intersection $I_p(P,Q)$ of two complex projective curves $P(x,y,z) = 0$, $Q(x,y,z) = 0$ at a point $p$ of the plane. This is a convenient concept when studying polynomial systems and it could be applied for the analysis of other classes of nonlinear systems.

In group theory Schreier's technique provides a basis for a subgroup of a free group. In this paper an analogue is developed for free Lie algebras. It hinges on the idea of cutting a Hall set into two parts. Using it, we show that proper subalgebras of finite codimension are not finitely generated and, following M.~Hall, that a finitely generated subalgebra is a free factor of a subalgebra of finite codimension.

In the genus polynomial of the graph $G$, the coefficient of $x^k$ is the number of distinct embeddings of the graph $G$ on the oriented surface of genus $k$. It is shown that for several infinite families of graphs all the zeros of the genus polynomial are real and negative. This implies that their coefficients, which constitute the genus distribution of the graph, are log concave and therefore also unimodal. The geometric distribution of the zeros of some of these polynomials is also investigated and some new genus polynomials are presented.

Using Feferman-Vaught techniques a condition on the fine spectrum of an admissible class of structures is found which leads to a first-order 0--1 law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first-order 0--1 law. If the condition is satisfied (and hence we have a first-order %% 0--1 law)

In this paper we study the $q$-Carleson measures for a space $h_\alpha^p$ of ${\cal M}$-harmonic potentials in the unit ball of $\C^n$, when $q<p$. We obtain some computable sufficient conditions, and study the relations among them.

We make precise the structure of the first two reduction morphisms associated with codimension two non-singular subvarieties of non-singular quadrics $\Q^n$, $n\geq 5$. We give a coarse classification of the same class of subvarieties when they are assumed not to be of log-general-type.}

In this work we study the geodesic flow on $n$-dimensional ideal polyhedra and establish classical (for manifolds of negative curvature) results concerning the distribution of closed orbits of the flow.

For a truncated matrix moment problem, we describe in detail the set of positive definite matrices of measures $\mu$ in $V_{2n}$ (this is the set of solutions of the problem of degree $2n$) for which the polynomials up to degree $n$ are dense in the corresponding space ${\cal L}^2(\mu)$. These matrices of measures are exactly the extremal measures of the set $V_n$.

Let $L/K$ be a finite Galois extension of local fields which are finite extensions of $\bQ_p$, the field of $p$-adic numbers. Let $\Gal (L/K)=G$, and $\euO_L$ and $\bZ_p$ be the rings of integers in $L$ and $\bQ_p$, respectively. And let $\euP_L$ denote the maximal ideal of $\euO_L$. We determine, explicitly in terms of specific indecomposable $\bZ_p[G]$-modules, the $\bZ_p[G]$-module structure of $\euO_L$ and $\euP_L$, for $L$, a composite of two arithmetically disjoint, ramified cyclic extensions of $K$, one of which is only weakly ramified in the sense of Erez \cite{erez}.

It is proved in this note, that a strongly continuous semigroup of (sub)positive contractions acting on an $L^p$-space, for $1<p<\infty$ $p \not= 2$, can be dilated by a strongly continuous group of (sub)positive isometries in a manner analogous to the dilation M.~A.~Ak\c{c}oglu and L.~Sucheston constructed for a discrete semigroup of (sub)positive contractions. From this an improvement of a von Neumann type estimation, due to R.~R.~Coifman and G.~Weiss, on the transfer map belonging to the semigroup is deduced.

Following a method outlined by Greenberg, root number computations give a conjectural lower bound for the ranks of certain Mordell-Weil groups of elliptic curves. More specifically, for $\PQ_{n}$ a \pgl{{\bf Z}/p^{n}{\bf Z}}-extension of ${\bf Q}$ and $E$ an elliptic curve over {\bf Q}, define the motive $E \otimes \rho$, where $\rho$ is any irreducible representation of $\Gal (\PQ_{n}/{\bf Q})$. Under some restrictions, the root number in the conjectural functional equation for the $L$-function of $E \otimes \rho$ is easily computible, and a `$-1$' implies, by the Birch and Swinnerton-Dyer conjecture, that $\rho$ is found in $E(\PQ_{n}) \otimes {\bf C}$. Summing the dimensions of such $\rho$ gives a conjectural lower bound of $$ p^{2n} - p^{2n - 1} - p - 1 $$ for the rank of $E(\PQ_{n})$.

All finite dimensional indecomposable representations of $U_t (\rmsl (2))$ at roots of $1$ are determined.

We prove the existence of trace functions in the rings of fractions of polycyclic-by-finite group rings or their homomorphic images. In particular a trace function exists in the ring of fractions of $KH$, where $H$ is a polycyclic-by-finite group and $\char K > N$, where $N$ is a constant depending on $H$.

In this paper, we consider the parabolic equation with anisotropic growth conditions, and obtain some criteria on boundedness of solutions, which generalize the corresponding results for the isotropic case.

Rational functions orthogonal on the unit circle with prescribed poles lying outside the unit circle are studied. We use the potential theory to discuss the zeros distribution for the orthogonal rational functions.

Une des questions fondamentales de la th\'eorie des groupes de Lie de dimension infinie concerne l'int\'egrabilit\'e des sous-alg\`ebres de Lie topologiques $\cal H$ de l'alg\`ebre de Lie $\cal G$ d'un groupe de Lie $G$ de dimension infinie au sens de Milnor. Par contraste avec ce qui se passe en th\'eorie classique il peut exister des sous-alg\`ebres de Lie ferm\'ees $\cal H$ de $\cal G$ non-int\'egrables en un sous-groupe de Lie. C'est le cas des alg\`ebres de Lie de champs de vecteurs $C^{\infty}$ d'une vari\'et\'e compacte qui ne d\'efinissent pas un feuilletage de Stefan. Heureusement cette ``imperfection" de la th\'eorie n'est pas partag\'ee par tous les groupes de Lie int\'eressants. C'est ce que montre cet article en exhibant une tr\`es large classe de groupes de Lie de dimension infinie exempte de cette imperfection. Cela permet de traiter compl\`etement le second probl\`eme fondamental de Sophus Lie pour les groupes de jauge de la physique-math\'ematique et les groupes formels de diff\'eomorphismes lisses de $\R^n$ qui fixent l'origine.

Non-hermitian solutions of algebraic matrix Riccati equations (of the continuous and discrete types) are studied. Existence is proved of non-hermitian solutions with given upper bounds of the ranks of the skew-hermitian parts, under the sign controllability hypothesis.

Let $M(X,Y)$ denote the space of all continous functions between $X$ and $Y$ and $M_f(X,Y)$ the path component corresponding to a given map $f: X\rightarrow Y.$ When $X$ and $Y$ are classical flag manifolds, we prove the components of $M(X,Y)$ corresponding to ``simple'' maps $f$ are classified up to rational homotopy type by the dimension of the kernel of $f$ in degree two cohomology. In fact, these components are themselves all products of flag manifolds and odd spheres.

The genus series for maps is the generating series for the number of rooted maps with a given number of vertices and faces of each degree, and a given number of edges. It captures topological information about surfaces, and appears in questions arising in statistical mechanics, topology, group rings, and certain aspects of free probability theory. An expression has been given previously for the genus series for maps in locally orientable surfaces in terms of zonal polynomials. The purpose of this paper is to derive an integral representation for the genus series. We then show how this can be used in conjunction with integration techniques to determine the genus series for monopoles in locally orientable surfaces. This complements the analogous result for monopoles in orientable surfaces previously obtained by Harer and Zagier. A conjecture, subsequently proved by Okounkov, is given for the evaluation of an expectation operator acting on the Jack symmetric function. It specialises to known results for Schur functions and zonal polynomials.

We prove an integration formula involving Jack polynomials conjectured by I.~P.~Goulden and D.~M.~Jackson in connection with enumeration of maps in surfaces.

For a fixed algebraic number $\alpha$ we discuss how closely $\alpha$ can be approximated by a root of a $\{0,+1,-1\}$ polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, $k$, of the polynomial at $\alpha$. In particular we obtain the following. Let ${\cal B}_{N}$ denote the set of roots of all $\{0,+1,-1\}$ polynomials of degree at most $N$ and ${\cal B}_{N}(\alpha,k)$ the roots of those polynomials that have a root of order at most $k$ at $\alpha$. For a Pisot number $\alpha$ in $(1,2]$ we show that \[ \min_{\beta \in {\cal B}_{N}\setminus \{ \alpha \}} |\alpha -\beta| \asymp \frac{1}{\alpha^{N}}, \] and for a root of unity $\alpha$ that \[ \min_{\beta \in {\cal B}_{N}(\alpha,k)\setminus \{\alpha\}} |\alpha -\beta|\asymp \frac{1}{N^{(k+1) \left\lceil \frac{1}{2}\phi (d)\right\rceil +1}}. \] We study in detail the case of $\alpha=1$, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When $k=0$ or 1 we can describe the extremal polynomials explicitly.

We give a new geometric model for the quantization of the 4-dimensional conical (nilpotent) adjoint orbit $\OR$ of $\SL(3,\R)$. The space of quantization is the space of holomorphic functions on ${\C}^2-\{0\})$ which are square integrable with respect to a signed measure defined by a Meijer $G$-function. We construct the quantization out a non-flat Kaehler structure on ${\C}^2-\{0\})$ (the universal cover of $\OR$) with Kaehler potential $\rho=|z|^4$.

We consider the shift-invariant space, $\bbbs(\Phi)$, generated by a set $\Phi=\{\phi_1,\ldots,\phi_r\}$ of compactly supported distributions on $\RR$ when the vector of distributions $\phi:=(\phi_1,\ldots,\phi_r)^T$ satisfies a system of refinement equations expressed in matrix form as $$ \phi=\sum_{\alpha\in\ZZ}a(\alpha)\phi(2\,\cdot - \,\alpha) $$ where $a$ is a finitely supported sequence of $r\times r$ matrices of complex numbers. Such {\it multiple refinable functions} occur naturally in the study of multiple wavelets. The purpose of the present paper is to characterize the {\it accuracy} of $\Phi$, the order of the polynomial space contained in $\bbbs(\Phi)$, strictly in terms of the refinement mask $a$. The accuracy determines the $L_p$-approximation order of $\bbbs(\Phi)$ when the functions in $\Phi$ belong to $L_p(\RR)$ (see Jia~[10]). The characterization is achieved in terms of the eigenvalues and eigenvectors of the subdivision operator associated with the mask $a$. In particular, they extend and improve the results of Heil, Strang and Strela~[7], and of Plonka~[16]. In addition, a counterexample is given to the statement of Strang and Strela~[20] that the eigenvalues of the subdivision operator determine the accuracy. The results do not require the linear independence of the shifts of $\phi$.

Let $A$ be a simple $C^*$-algebra with real rank zero, stable rank one and weakly unperforated $K_0(A)$ of countable rank. We show that a monomorphism $\phi\colon C(S^2) \to A$ can be approximated pointwise by homomorphisms from $C(S^2)$ into $A$ with finite dimensional range if and only if certain index vanishes. In particular, we show that every homomorphism $\phi$ from $C(S^2)$ into a UHF-algebra can be approximated pointwise by homomorphisms from $C(S^2)$ into the UHF-algebra with finite dimensional range. As an application, we show that if $A$ is a simple $C^*$-algebra of real rank zero and is an inductive limit of matrices over $C(S^2)$ then $A$ is an AF-algebra. Similar results for tori are also obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$ for lower dimensional spaces is also studied.

In this paper we give a characterization of the pairs of weights $(\w,v)$ such that $T$ maps $L^p(v)$ into $L^q(\w)$, where $T$ is a general one-sided operator that includes as a particular case the Weyl fractional integral. As an application we solve the following problem: given a weight $v$, when is there a nontrivial weight $\w$ such that $T$ maps $L^p(v)$ into $L^q(\w )$?

The convergence behavior of best uniform rational approximations $r^\ast_{mn}$ with numerator degree~$m$ and denominator degree~$n$ to the function $|x|^\alpha$, $\alpha>0$, on $[-1,1]$ is investigated. It is assumed that the indices $(m,n)$ progress along a ray sequence in the lower triangle of the Walsh table, {\it i.e.} the sequence of indices $\{ (m,n)\}$ satisfies $$ {m\over n}\rightarrow c\in [1, \infty)\quad\hbox{as } m+ n\rightarrow\infty. $$ In addition to the convergence behavior, the asymptotic distribution of poles and zeros of the approximants and the distribution of the extreme points of the error function $|x|^\alpha - r^\ast_{mn} (x)$ on $[-1,1]$ will be studied. The results will be compared with those for paradiagonal sequences $(m=n+2[\alpha/2])$ and for sequences of best polynomial approximants.

We consider the multiparameter nonlinear Sturm-Liouville problem $$\displaylines{ u''(x) - \sum_{k=1}^m\mu_k u(x)^{p_k} + \sum_{k=m+1}^n \mu_ku(x)^{p_k} = \lambda u(x)^q, \quad x \in I := (-1,1), \cr u(x) > 0, \quad x \in I, \cr u(-1) = u(1) = 0,\cr}$$ where $\mu = (\mu_1, \mu_2, \ldots, \mu_m, \mu_{m+1}, \ldots \mu_n) \in \bar{R}_+^m \times R_+^{n-m} \bigl(R_+ := (0, \infty)\bigr)$ and $\lambda \in R$ are parameters. We assume that $$1 \le q \le p_1 < p_2 < \cdots < p_n < 2q + 3.$$ We shall establish an asymptotic formula of variational eigenvalue $\lambda = \lambda(\mu,\alpha)$ obtained by using Ljusternik-Schnirelman theory on general level set $N_{\mu,\alpha} (\alpha > 0:$ parameter of level set). Furthermore, we shall give the optimal condition of $\{(\mu, \alpha)\}$, under which $\mu_i (m + 1 \le i \le n: \hbox{\rm fixed})$ dominates the asymptotic behavior of $\lambda(\mu,\alpha)$.

We study sets on which measurable real-valued functions on a measurable space with negligibles are determined by their range.

Let $\VN(G)$ be the von Neumann algebra of a locally compact group $G$. We denote by $\mu$ the initial ordinal with $\abs{\mu}$ equal to the smallest cardinality of an open basis at the unit of $G$ and $X= \{\alpha; \alpha < \mu \}$. We show that if $G$ is nondiscrete then there exist an isometric $*$-isomorphism $\kappa$ of $l^{\infty}(X)$ into $\VN(G)$ and a positive linear mapping $\pi$ of $\VN(G)$ onto $l^{\infty}(X)$ such that $\pi\circ\kappa = \id_{l^{\infty}(X)}$ and $\kappa$ and $\pi$ have certain additional properties. Let $\UCB (\hat{G})$ be the $C^{*}$-algebra generated by operators in $\VN(G)$ with compact support and $F(\hat{G})$ the space of all $T \in \VN(G)$ such that all topologically invariant means on $\VN(G)$ attain the same value at $T$. The construction of the mapping $\pi$ leads to the conclusion that the quotient space $\UCB (\hat{G})/F(\hat{G})\cap \UCB(\hat{G})$ has $l^{\infty}(X)$ as a continuous linear image if $G$ is nondiscrete. When $G$ is further assumed to be non-metrizable, it is shown that $\UCB(\hat{G})/F (\hat{G})\cap \UCB(\hat{G})$ contains a linear isomorphic copy of $l^{\infty}(X)$. Similar results are also obtained for other quotient spaces.

Soient $A$, $B$ et $C$ trois entiers non nuls premiers entre eux deux \`a deux, et $p$ un nombre premier. Comme cons\'equence des travaux de A. Wiles et F. Diamond sur la conjecture de Taniyama-Weil, on explicite une constante $f(A,B,C)$ telle que, sous certaines conditions portant sur $A$, $B$ et $C$, l'\'equation $Ax^p+By^p+Cz^p=0$ n'a aucune solution non triviale dans $\Z$, si $p$ est $>f(A,B,C)$. On d\'emontre par ailleurs, sans condition suppl\'ementaire portant sur $A$, $B$ et $C$, que cette \'equation n'a aucune solution non triviale dans $\Z$, si $p$ divise $xyz$, et si $p$ est $>f(A,B,C)$.

In this paper we introduce the concepts of hyperbolic and elliptic areas and prove uncountably many new geometric isoperimetric inequalities on the surfaces of constant curvature.

In this paper we consider the following problem: Given a unital \cs\ $A$ and a collection of elements $S$ in the identity component of the invertible group of $A$, denoted \ino, characterize the group of finite products of elements of $S$. The particular $C^*$-algebras studied in this paper are either unital purely infinite simple or of the form \tenp, where $A$ is any \cs\ and $K$ is the compact operators on an infinite dimensional separable Hilbert space. The types of elements used in the factorizations are unipotents ($1+$ nilpotent), positive invertibles and symmetries ($s^2=1$). First we determine the groups of finite products for each collection of elements in \tenp. Then we give upper bounds on the number of factors needed in these cases. The main result, which uses results for \tenp, is that for $A$ unital purely infinite and simple, \ino\ is generated by each of these collections of elements.

Let $g$ be a semisimple Lie algebra. Quantum analogs of the enveloping algebra of the fixed Lie subalgebra are introduced for involutions corresponding to the negative of a diagram automorphism. These subalgebras of the quantized enveloping algebra specialize to their classical counterparts. They are used to form an Iwasawa type decompostition and begin a study of quantum Harish-Chandra modules.

We give the irreducible decomposition of the tensor product of an analytic continuation of the holomorphic discrete series of $\SU(2, 2)$ with its conjugate.

We prove that if $X$ is a complex strictly monotone sequence space with $1$-un\-con\-di\-tion\-al basis, $Y \subseteq X$ has no bands isometric to $\ell_2^2$ and $Y$ is the range of norm-one projection from $X$, then $Y$ is a closed linear span a family of mutually disjoint vectors in $X$. We completely characterize $1$-complemented subspaces and norm-one projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q < \infty$. Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are $1$-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space $X$ is not isomorphic to $\ell_p$ for some $1 \leq p < \infty$ then the only subspaces of $X$ which are $1$-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients.

Let $G$ be a finite group. To a set of subgroups of order two we associate a $\mod 2$ Hecke algebra and construct a homomorphism, $\psi$, from its units to the class-group of ${\bf Z}[G]$. We show that this homomorphism takes values in the subgroup, $D({\bf Z}[G])$. Alternative constructions of Chinburg invariants arising from the Galois module structure of higher-dimensional algebraic $K$-groups of rings of algebraic integers often differ by elements in the image of $\psi$. As an application we show that two such constructions coincide.

Pieri's formula describes the intersection product of a Schubert cycle by a special Schubert cycle on a Grassmannian. We present a new geometric proof, exhibiting an explicit chain of rational equivalences from a suitable sum of distinct Schubert cycles to the intersection of a Schubert cycle with a special Schubert cycle. The geometry of these rational equivalences indicates a link to a combinatorial proof of Pieri's formula using Schensted insertion.

In this paper, we solve the $\dbar$-Neumann problem on $(0,q)$ forms, $0\leq q \leq n$, in the strictly pseudoconvex non-isotropic Siegel domain: \[ \cU=\left\{ \begin{array}{clc} &\bz=(z_1,\ldots,z_n) \in \C^{n},\\ (\bz,z_{n+1}):&&\Im (z_{n+1}) > \sum_{j=1}^{n}a_j |z_j|^2 \\ &z_{n+1}\in \C; \end{array} \right\}, \] where $a_j> 0$ for $j=1,2,\ldots, n$. The metric we use is invariant under the action of the Heisenberg group on the domain. The fundamental solution of the related differential equation is derived via the Laguerre calculus. We obtain an explicit formula for the kernel of the Neumann operator. We also construct the solution of the corresponding heat equation and the fundamental solution of the Laplacian operator on the Heisenberg group.

In the first paper with the same title the authors were able to determine all partially oriented flag manifolds that are stably parallelizable or parallelizable, apart from four infinite families that were undecided. Here, using more delicate techniques (mainly K-theory), we settle these previously undecided families and show that none of the manifolds in them is stably parallelizable, apart from one 30-dimensional manifold which still remains undecided.

No abstract.