Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s (d + 1 ) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials. We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions. We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite-Padé approximation in matrix form is given.     

The existence of a weighted mean for almost periodic functions

In a recent paper by Liang et al. (2010) [1], the original question which is that of the existence of a weighted mean for almost periodic functions was raised. In particular, they showed through an example that there exist weights for which a weighted mean for almost periodic functions may or may not exist. In this note we give some sufficient conditions which do guarantee the existence of a weighted mean for almost periodic functions, which will then coincide with the classical Bohr mean. Moreover, we will show that under those conditions, the corresponding weighted Bohr transform exists.     

Fourier–Padé Approximants for Nikishin Systems

We study type I Fourier–Padé approximation for certain systems of functions formed by the Cauchy transform of finite Borel measures supported on bounded intervals of the real line. This construction is similar to type I Hermite–Padé approximation. Instead of power series expansions of the functions in the system, we take their development in a series of orthogonal polynomials. We give the exact rate of convergence of the corresponding approximants. The answer is expressed in terms of the extremal solution of an associated vector-valued equilibrium problem for the logarithmic potential.      

Quadratically convergent algorithms and one-dimensional search schemes

In this paper, the performances of three quadratically convergent algorithms coupled with four one-dimensional search schemes are studied through several nonquadratic examples. The algorithms are the rank-one algorithm (Algorithm I), the projection algorithm (Algorithm II), and the Fletcher-Reeves algorithm (Algorithm III). The search schemes are the exact quadratic search (EQS), the exact cubic search (ECS), the approximate quadratic search (AQS), and the approximate cubic search (ACS). The performances are analyzed in terms of number of iterations and number of equivalent function evaluations for convergence. From the numerical experiments, the following conclusions are found: (a) while the number of iterations generally increases by relaxing the search stopping condition, the number of equivalent function evaluations decreases; therefore, approximate searches should be preferred to exact searches; (b) the numbers of iterations for ACS, ECS, and EQS are about the same; therefore, the use of more sophisticated, higher order search schemes is not called for; the present ACS scheme, modified so that only the function, instead of the gradient, is used in bracketing the minimal point, could prove to be most desirable in terms of the number of equivalent function evaluations; (c) for Algorithm I, ACS and AQS yield almost identical results; it is believed that further improvements in efficiency are possible if one uses a fixed stepsize approach, thus bypassing the one-dimensional search completely; (d) the combination of Algorithm II and ACS exhibits high efficiency in treating functions whose order is higher than two and whose Hessian at the minimal point is singular; and (f) Algorithm III, even with the best search scheme, is inefficient in treating functions with flat bottoms; it is doubtful that the simplicity of its update will compensate for its inefficiency in such pathological cases.This research was supported by the National Science Foundation, Grant No. 32453.     

Sunyer-i-Balaguer’s almost elliptic functions and Yosida’s normal functions

We study the properties of two classes of meromorphic functions in the complex plane. The first one is the class of almost elliptic functions in the sense of Sunyer-i-Balaguer. This is the class of meromorphic functions f such that the family {f(z + h)} _h∈ℂ is normal with respect to the uniform convergence in the whole complex plane. Given two sequences of complex numbers, we provide sufficient conditions for themto be zeros and poles of some almost elliptic function. These conditions enable one to give (for the first time) explicit non-trivial examples of almost elliptic functions. The second class was introduced by K. Yosida, who called it the class of normal functions of the first category. This is the class of meromorphic functions f such that the family {f(z + h)} _h∈ℂ is normal with respect to the uniform convergence on compacta in the complex plane and no limit point of the family is a constant function. We give necessary and sufficient conditions for two sequences of complex numbers to be zeros and poles of some normal function of the first category and obtain a parametric representation for this class in terms of zeros and poles.     

Cambrian lattices

For an arbitrary finite Coxeter group W, we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatorially isomorphic to the generalized associahedron for W. In types A and B we obtain, by means of a fiber-polytope construction, combinatorial realizations of the Cambrian lattices in terms of triangulations and in terms of permutations. Using this combinatorial information, we prove in types A and B that the Cambrian fans are combinatorially isomorphic to the normal fans of the generalized associahedra and that one of the Cambrian fans is linearly isomorphic to Fomin and Zelevinsky's construction of the normal fan as a “cluster fan.” Our construction does not require a crystallographic Coxeter group and therefore suggests a definition, at least on the level of cellular spheres, of a generalized associahedron for any finite Coxeter group. The Tamari lattice is one of the Cambrian lattices of type A, and two “Tamari” lattices in type B are identified and characterized in terms of signed pattern avoidance. We also show that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.     

Löwner chains and a subclass of biholomorphic mappings

In this paper, we give characterization of almost starlike functions of order α (respectively almost starlike mappings of order α) on the unit disc in C (respectively the unit ball in a finite-dimensional complex Banach space) in terms of Löwner chains. Furthermore, using the properties of Löwner chains, we can easily prove that two classes of generalized Roper-Suffridge extension operators preserve almost starlikeness of order α on two important classes of Reinhardt domains in Cn, respectively.     

Orientation-reversing actions on lens spaces and Gaussian integers

A finite group action on a lens space L(p,q) has ‘type OR’ if it reverses orientation and has an invariant Heegaard torus whose sides are interchanged by the orientation-reversing elements. In this paper we enumerate the actions of type OR up to equivalence. This leads to a complete classification of geometric finite group actions on amphicheiral lens spaces L(p,q) with p>2. The family of actions of type OR is partially ordered by lifting actions via covering maps. We show that each connected component of this poset may be described in terms of a subset of the lattice of Gaussian integers ordered by divisibility. This results in a correspondence equating equivalence classes of actions of type OR with pairs of Gaussian integers.     

On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II

The main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz-Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved.     

Asymptotic solutions of inhomogeneous differential equations having a turning point

Asymptotic solutions are derived for inhomogeneous differential equations having a large real or complex parameter and a simple turning point. They involve Scorer functions and three slowly varying analytic coefficient functions. The asymptotic approximations are uniformly valid for unbounded complex values of the argument, and are applied to inhomogeneous Airy equations having polynomial and exponential forcing terms. Error bounds are available for all approximations, including new simple ones for the well-known asymptotic expansions of Scorer functions of large complex argument.     

Reduction of a polling network to a single node

We consider a discrete-time tree network of polling servers where all packets are routed to the same node (called node 0), from which they leave the network. All packets have unit size and arrive from the exterior according to independent batch Bernoulli arrival processes. The service discipline of each node is work-conserving and the service discipline of node 0 has to be HoL-based, which is an additional assumption that is satisfied by, a.o., m _i -limited service, exhaustive service, and priority disciplines. Let a type i packet be a packet that visits queue i of node 0. We establish a distributional relation between the number of type i packets in the network and in a single station system, and we show equality of the mean end-to-end delay of type i packets in the two systems. Essentially this reduces an arbitrary tree network to a much simpler system of one node, while preserving the mean end-to-end delay of type i packets.      

Hyperbolic embeddability of locally complete almost complex submanifolds

In this Note, we generalize to the almost complex setting, a theorem of Zaidenberg (1983) [13] and Thai (1991) [12] by giving a characterization on hyperbolic embeddability of a locally complete and relatively compact almost complex submanifold in terms of extension of pseudo-holomorphic disks from the punctured unit disk and of limit J-complex lines.     

Almost Complex Manifolds and Hyperbolicity

One of the sufficient conditions for a complex manifold to be (complete) hyperbolic (in the sense that its intrinsic pseudo-distance is a (complete) distance) is that it admits a (complete) Hermitian metric with holomorphic sectional curvature bounded above by a negative constant. The concept of hyperbolicity can be readily extended to almost complex manifolds. We will show that the above result for hyperbolicity can be generalized to the almost complex case. As an application, we prove that every point of an almost complex manifold has a complete hyperbolic neighborhood. In real dimension 4, this fact was established by Debalme and Ivashkovich [2] by a completely different method.     

Spectral analysis of the covariance of the almost periodically correlated processes

This paper deals with the spectrum of the almost periodically correlated (APC) processes defined on . It is established that the covariance kernel of such a process admits a Fourier series decomposition, K(s+t, s) , whose coefficient functions b are the Fourier transforms of complex measures m, , which are absolutely continuous with respect to the measure m_o. Considering the APC strongly harmonizable processes, the spectral covariance of the process can be expressed in terms of these complex measures m.

The usual estimators for the second order situation can be modified to provide consistent estimators of the coefficient functions b from a sample of the process. Whenever the measures m are absolutely continuous with respect to the Lebesgue measure, so m(dλ)=f(λ) dλ, the estimation of the corresponding density functions f is considered. Under hypotheses on the covariance kernel K and on the coefficient functions b, we establish rates of convergence in quadratic mean and almost everywhere of these estimators.     

Čech and Steenrod homotopy and the Quigley exact couple in strong shape and proper homotopy theory

Let X be a topological space equipped with the action of a finite group ∏. We may form the twisted group ring of ∏, coefficients being elements of the ring of continuous functions on X with values in the real numbers, complex numbers or quaternions. In this paper we show how the Witt groups of hermitian forms of various kinds over these twisted group rings can be described in terms of the real, complex or quaternionic equivariant K-theory of X.     

Zeroes of Gaussian analytic functions with translation-invariant distribution

We study zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We prove that the horizontal limiting measure of the zeroes exists almost surely, and that it is non-random if and only if the spectral measure is continuous (or degenerate). In this case, the limiting measure is computed in terms of the spectral measure. We compare the behavior with Gaussian analytic functions with symmetry around the real axis. These results extend a work by Norbert Wiener.     

Almost Hermitian structures and quaternionic geometries

Gray and Hervella gave a classification of almost Hermitian structures (g,I) into 16 classes. We systematically study the interaction between these classes when one has an almost hyper-Hermitian structure (g,I,J,K). In general dimension we find at most 167 different almost hyper-Hermitian structures. In particular, we obtain a number of relations that give hyper-Kähler or locally conformal hyper-Kähler structures, thus generalising a result of Hitchin. We also study the types of almost quaternion-Hermitian geometries that arise and tabulate the results.     

VECTOR-VALUED HOLOMORPHIC FUNCTIONS ON THE COMPLEX BALL AND THE ANALYTIC RADON-NIKODYM PROPERTY

1IntroductionAcomplexBanachspacewithvaluesinwhicheveryboundedallalyticfunctionontheopenunitdiskhasradialliniltsatalmostallboundarypointsissaidtohavetheanalyticRa.don-NikodympropertyApaPerofBukhvalovandDanilevich[5lexanlinesthistopic.TheclassofBanachspaceswiththeanalyticRadon-Niliodympropertyisslightlylargerthanthatwiththeffedon-NikodympropertyIndeed,itisshownill[5]thatL1andmoregenerallyallBanachlatticesnotcontainingcoalsoverifytheanalyticffedon-NikodympropertyAftertheirwork,severalchaJra…     

GENERALIZED GEOMETRIC GOPPA CODES

ABSTRACT

Conventional geometric Goppa codes are defined in terms of functions of an algebraic function field associated with a divisor evaluated in places of degree 1. The generalization that will be treated here allows evaluations in places of arbitrary degree. With the appropriate inner product, the dual of the code can be defined and described in terms of Weil differentials similarly to conventional geometric Goppa codes. A decoding algorithm is derived.