q-ENGEL SERIES EXPANSIONS AND SLATER'S IDENTITIES

Abstract

Dedicated to the memory of John Knopfmacher (1937–1999)

We describe the q-Engel series expansion for Laurent series discovered by John Knopfmacher and use this algorithm to shed new light on partition identities related to two entries from Slater's list. In our study Al-Salam/Ismail and Santos polynomials play a crucial r?ole.     

Degree of the divisor of solutions of a differential equation on a projective variety

Using the data schemes from [1] we give a rigorous definition of algebraic differential equations on the complex projective space Pⁿ. For an algebraic subvariety S?Pⁿ, we present an explicit formula for the degree of the divisor of solutions of a differential equation on S and give some examples of applications. We extend the technique and result to the real case.     

Gauss inequalities on ordered linear spaces

Markov inequalities on ordered linear spaces are tightened through the α-unimodality of the corresponding measures. Modality indices are studied for various induced measures, including the singular values of a random matrix and the periodogram of a time series. These tools support a detailed study of linear inference and the ordering of random matrices, to include fixed and random designs and probability bounds on their comparative efficiencies. Other applications include probability bounds on quadratic forms and of order statistics on Rⁿ, on periodograms in the analysis of time series, and on run-length distributions in multivariate statistical process control. Connections to other topics in applied probability and statistics are noted.     

Gauss-Tchebycheff quadrature formulas

Summary It is well known that the Tchebycheff weight function (1-x ²)^–1/2 is the only weight function (up to a linear transformation) for which then point Gauss quadrature formula has equal weights for alln. In this paper we describe explicitly all weight functions which have the property that then _k-point Gauss quadrature formula has equal weights for allk, where (n _k),n ₁<n ₂<..., is an arbitrary subsequence of . Furthermore results on the possibility of Tchebycheff quadrature on several intervals are given.     

Compact Operators on Bergman Spaces

We prove that a bounded operator S on L _a ^p for p > 1 is compact if and only if the Berezin transform of S vanishes on the boundary of the unit disk if S satisfies some integrable conditions. Some estimates about the norm and essential norm of Toeplitz operators with symbols in BT are obtained.     

Remarks on the length of conjugacy classes of finite groups

Let G be a finite group. The question of how certain arithmetical conditions on the lengths of the conjugacy classes of G influence the group structure has been studied by several authors. In this paper we study restrictions on the structure of a finite group in which the lengths of conjugacy classes are not divisible by p ² for some prime p. We generalise and provide simplified proofs for some earlier results.     

On nilpotent ideals in the cohomology ring of a finite group

In this paper we find upper bounds for the nilpotency degree of some ideals in the cohomology ring of a finite group by studying fixed point free actions of the group on suitable spaces. The ideals we study are the kernels of restriction maps to certain collections of proper subgroups. We recover the Quillen-Venkov lemma and the Quillen F-injectivity theorem as corollaries, and discuss some generalizations and further applications.We then consider the essential cohomology conjecture, and show that it is related to group actions on connected graphs. We discuss an obstruction for constructing a fixed point free action of a group on a connected graph with zero “k-invariant” and study the class related to this obstruction. It turns out that this class is a “universal essential class” for the group and controls many questions about the groups essential cohomology and transfers from proper subgroups.     

Equidistribution quantitative des points de petite hauteur sur la droite projective

We introduce a new class of adelic heights on the projective line. We estimate their essential minimum and prove a result of equidistribution (at every place) for points of small height with estimates on the speed of convergence. To each rational function R in one variable and defined over a number field K, is associated a normalized height on the algebraic closure of K. We show that these dynamically defined heights are adelic in our sense, and deduce from this equidistribution results for preimages of points under R at every place of K. Our approach follows that of Bilu, and relies on potential theory in the complex plane, as well as in the Berkovich space associated to the projective line over , for each prime p. Le premier auteur tient à remercier chaleureusement le project MECESUP UCN0202, ainsi que l'ACI ``Systèmes Dynamiques Polynomiaux' qui ont permis son séjour à l'Université Catholique d'Antofagasta. Le deuxième auteur est partiellement soutenu par le projet FONDECYT N 1040683. Enfin, nous remercions le rapporteur pour sa lecture détaillée de l'article.     

Nonexistence of codimension one Anosov flows on compact manifolds with boundary

A flow is Anosov if it exhibits contracting and expanding directions forming with the flow a continuous tangent bundle decomposition. An Anosov flow is codimension one if its contracting or expanding direction is one-dimensional. Examples of codimension one Anosov flows on compact boundaryless manifolds can be exhibited in any dimension ?3. In this paper, we prove that there are no codimension one Anosov flows on compact manifolds with boundary. The proof uses an extension to flows of some results in Hirsch [On Invariant Subsets of Hyperbolic Sets, Essays on Topology and Related Topics, Memoires dédiés à Georges de Rham, 1970, pp. 126-135] related to Question 10(b) in Palis and Pugh [Fifty problems in dynamical systems, in: J. Palis, C.C. Pugh (Eds.), Dynamical Systems-Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E.C. Zeeman on his fiftieth birthday), Lecture Notes in Mathematics, vol. 468, Springer, Berlin, 1975, pp. 345-353].     

Free actions on handlebodies

The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable three-dimensional handlebody of genus g?1 can be enumerated in terms of sets of generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g−1)/|G|, under Nielsen equivalence (or weak Nielsen equivalence). For Abelian and dihedral G, this allows a complete determination of the equivalence and weak equivalence classes of actions for all genera. Additional information is obtained for other classes of groups. For all G, there is only one equivalence class of actions on the genus g handlebody if g is at least 1+?(G)|G|, where ?(G) is the maximal length of a chain of subgroups of G. There is a stabilization process that sends an equivalence class of actions to an equivalence class of actions on a higher genus, and some results about its effects are obtained.     

Structure of commutative cancellative integral residuated lattices on (0, 1]

ΠMTL-algebras were introduced as an algebraic counterpart of the cancellative extension of monoidal t-norm based logic. It was shown that they form a variety generated by ΠMTL-chains on the real interval [0, 1]. In this paper the structure of these generators is investigated. The results illuminate the structure of cancellative integral commutative residuated chains, because every such algebra belongs to the quasivariety generated by the zero-free subreducts on (0, 1] of all ΠMTL-chains on [0, 1]. The work of the author was partly supported by the grant No. A100300503 of the Grant Agency of the Academy of Sciences of the Czech Republic and partly by the Institutional Research Plan AV0Z10300504.     

Sufficient conditions for ε-minimum

The paper contains sufficient conditions for a point x ∈ U, function f X→ R ∪ ∪ {-∞,∞}, and ε≧ 0 to hold the inequality f(x) ≦ inf f=ε on the subset U of a topological vector space X.     

Bass-Tits minimization of automata, quotients of trees and diameters

Let X be a tree and let G=Aut(X), Bass and Tits have given an algorithm to construct the ‘ultimate quotient’ of X by G starting with any quotient of X, an ‘edge-indexed’ graph. Using a sequence of integers that we compute at consecutive steps of the Bass-Tits (BT) algorithm, we give a lower bound on the diameter of the ultimate quotient of a tree by its automorphism group. For a tree X with finite quotient, this gives a lower bound on the minimum number of generators of a uniform X-lattice whose quotient graph coincides with G?X. This also gives a criterion to determine if the ultimate quotient of a tree is infinite. We construct an edge-indexed graph (A,i) for a deterministic finite state automaton and show that the BT algorithm for computing the ultimate quotient of (A,i) coincides with state minimizing algorithm for finite state automata. We obtain a lower bound on the minimum number of states of the minimized automaton. This gives a new proof that language for the word problem in a finitely generated group is regular if and only if the group is finite, and a new proof that the language of the membership problem for a subgroup is regular if and only if the subgroup has finite index.     

Onp-generator fully symmetric quadrature rules

Summary We consider fully symmetric quadrature rules for fully symmetricn-dimensional integration regions. When the region is a product region it is well known that product Gaussian rules exist. These obtain an approximation of polynomial degree 4p+1 based on (2p+1) ⁿ function values arranged on a rectangular grid. We term rules using such a grid,p-generator rules. In this paper we determine the necessary conditions on the region of integration forp-generator rules of degree 4p+1 to exist. Regions with this property are termed PropertyQ regions and besides product spaces, this class includes the hypersphere and other related regions.Work performed under the auspices of the U.S. Energy Research and Development Administration     

On homogeneous connections with exotic holonomy

In Proc. Symp. Pure Math. 53 (1991), 33–88, Bryant gave examples of torsion free connections on four-manifolds whose holonomy is exotic, i.e. is not contained on Berger's classical list of irreducible holonomy representations. The holonomy in Bryant's examples is the irreducible four-dimensional representation of S1(2, #x211D;) (G1(2, #x211D;) resp.) and these connections are called H ₃-connections (G ₃-connections resp.).In this paper, we give a complete classification of homogeneous G ₃-connections. The moduli space of these connections is four-dimensional, and the generic homogeneous G ₃-connection is shown to be locally equivalent to a left-invariant connection on U(2). Thus, we prove the existence of compact manifolds with G ₃-connections. This contrasts a result in by Schwachhöfer (Trans. Amer. Math. Soc. 345 (1994), 293–321) which states that there are no compact manifolds with an H ₃-connection.     

Galerkin methods with splines for singular integral equations over (0, 1)

Summary In this paper a convergence analysis of Galerkin methods with splines for strongly elliptic singular integral equations over the interval (0, 1) is given. As trial functions we utilize smoothest polynomial splines on arbitrary meshes and continuous splines on special nonuniform partitions, multiplied by a weight function. Using inequalities of Gårding type for singular integral operators in weightedL ² spaces and the complete asymptotics of solutions at the endpoints, we provide error estimates in certain Sobolev norms.     

Euler solutions of pseudodifferential equations

We consider a homogeneous pseudodifferential equation on a cylinderC=×X over a smooth compact closed manifoldX whose symbol extends to a meromorphic function on the complex plane with values in the algebra of pseudodifferential operators overX. When assuming the symbol to be independent on the variablet , we show an explicit formula for solutions of the equation. Namely, to each non-bijectivity point of the symbol in the complex plane there corresponds a finite-dimensional space of solutions, every solution being the residue of a meromorphic form manufactured from the inverse symbol. In particular, for differential equations we recover Euler's theorem on the exponential solutions. Our setting is model for the analysis on manifolds with conical points sinceC can be thought of as a stretched manifold with conical points att=– andt=. When compared with the general theory, our approach is constructive while highlighting all the features of this latter.     

Prescribed Q-curvature problem on closed 4-Riemannian manifolds in the null case

The main objective of this short note is to give a sufficient condition for a non constant function k to be Q curvature candidate for a conformal metric on a closed Riemannian manifold with the null Q-curvature. In contrast to the prescribed scalar curvature on the two-dimensional flat tori, the condition we provided is not necessary as some examples show. The second author would like to thank Department of Mathematics, University of Paris XII for their invitation, hospitality and financial support during his visit. His research is partially supported by NUS research grant: R-146-000-077-112.     

Canonical height functions for affine plane automorphisms

Let be a polynomial automorphism of dynamical degree δ≥2 over a number field K. We construct height functions defined on that transform well relative to f, which we call canonical height functions for f. These functions satisfy the Northcott finiteness property, and a -valued point on is f-periodic if and only if its height is zero. As an application, we give an estimate on the number of points with bounded height in an infinite f-orbit.