A family of non-oscillatory 6-point interpolatory subdivision schemes

In this paper we propose and analyze a new family of nonlinear subdivision schemes which can be considered non-oscillatory versions of the 6-point Deslauries-Dubuc (DD) interpolatory scheme, just as the Power _p schemes are considered nonlinear non-oscillatory versions of the 4-point DD interpolatory scheme. Their design principle may be related to that of the Power _p schemes and it is based on a weighted analog of the Power _p mean. We prove that the new schemes reproduce exactly polynomials of degree three and stay ’close’ to the 6-point DD scheme in smooth regions. In addition, we prove that the first and second difference schemes are well defined for each member of the family, which allows us to give a simple proof of the uniform convergence of these schemes and also to study their stability as in [19, 22]. However our theoretical study of stability is not conclusive and we perform a series of numerical experiments that seem to point out that only a few members of the new family of schemes are stable. On the other hand, extensive numerical testing reveals that, for smooth data, the approximation order and the regularity of the limit function may be similar to that of the 6-point DD scheme and larger than what is obtained with the Power _p schemes.     

Mesh ratios for best-packing and limits of minimal energy configurations

For N-point best-packing configurations ω _N on a compact metric space (A,ρ), we obtain estimates for the mesh-separation ratio γ(ω _N ,A), which is the quotient of the covering radius of ω _N relative to A and the minimum pairwise distance between points in ω _N . For best-packing configurations ω _N that arise as limits of minimal Riesz s-energy configurations as s→∞, we prove that γ(ω _N ,A)≦1 and this bound can be attained even for the sphere. In the particular case when N=5 on S ² with ρ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid $\omega_{5}^{*}$ , that is the limit (as s→∞) of 5-point s-energy minimizing configurations. Moreover, $\gamma(\omega_{5}^{*},S^{2})=1$ .     

Transitive points via Furstenberg family

Let (X,T) be a topological dynamical system and F be a Furstenberg family (a collection of subsets of Z₊ with hereditary upward property). A point x∈X is called an F-transitive one if {n∈Z₊:Tnx∈U}∈F for every non-empty open subset U of X; the system (X,T) is called F-point transitive if there exists some F-transitive point. In this paper, we aim to classify transitive systems by F-point transitivity. Among other things, it is shown that (X,T) is a weakly mixing E-system (resp. weakly mixing M-system, HY-system) if and only if it is {D-sets}-point transitive (resp. {central sets}-point transitive, {weakly thick sets}-point transitive).It is shown that every weakly mixing system is Fip-point transitive, while we construct an Fip-point transitive system which is not weakly mixing. As applications, we show that every transitive system with dense small periodic sets is disjoint from every totally minimal system and a system is Δ^?(Fwt)-transitive if and only if it is weakly disjoint from every P-system.     

Boundedness and compactness of multidimensional integral operators with homogeneous kernels

We obtain sufficient conditions for the boundedness and compactness of multidimensional integral operators with homogeneous kernels acting from a weighted L _p -space to a weighted L _q -space.     

Some weighted estimates for Littlewood-Paley functions and radial multipliers

We prove some weighted estimates for certain Littlewood-Paley operators on the weighted Hardy spaces H_w^p (0<p?1) and on the weighted L^p spaces. We also prove some weighted estimates for the Bochner-Riesz operators and the spherical means.     

N-point locality for vertex operators: Normal ordered products,operator product expansions,twisted vertex algebras

In this paper we study fields satisfying N-point locality and their properties. We obtain residue formulae for N-point local fields in terms of derivatives of delta functions and Bell polynomials. We introduce the notion of the space of descendants of N-point local fields which includes normal ordered products and coefficients of operator product expansions. We show that examples of N  -point local fields include the vertex operators generating the boson–fermion correspondences of types B, C and D-A. We apply the normal ordered products of these vertex operators to the setting of the representation theory of the double-infinite rank Lie algebras _b∞$b_{\infty }$, _c∞$c_{\infty }$, _d∞$d_{\infty }$. Finally, we show that the field theory generated by N-point local fields and their descendants has a structure of a twisted vertex algebra.     

The realizable extension problem and the weighted graph (K 3,3, l)

This note outlines the realizable extension problem for weighted graphs and provides a detailed analysis of this problem for the weighted graph (K _3,3, l). The main result of this analysis is that the moduli space of planar realizations of (K _3,3, l) can have one, two, four, six or eight connected components and explicit examples of each case are provided. The note culminates with two examples which show that in general, realizability and connectedness results relating to the moduli spaces of weighted cycles which are contained in a larger weighted graph cannot be extended to similar results regarding the moduli space of the larger weighted graph.     

The suitability of the weighted lp-norm in estimating actual road distances

The weighted l_p-norm, suggested as a way to estimate actual road distances, takes time and effort to calculate because it requires two parameters. Compared to the weighted Euclidean norm, a function requiring only one parameter which is relatively easy to obtain, the weighted l_p-norm yields considerably better estimates for some countries and only negligible improvements for others. This paper evaluates whether, on the basis of easy-to-obtain data characterizing the road network of a particular area, one can predict the extent of the improvement in estimate accuracy. Such an ability to predetermine estimate accuracy would help researchers decide when to use the expensive l_p-norm. The investigation shows, however, that no such relationship can be proven. The extent of the improvement in estimate accuracy offered by the weighted l_p-norm cannot be predicted.     

Characterization of the convergence of weighted averages of sequences and functions

The weighted averages of a sequence (c _k ), c _k ?? ?, with respect to the weights (p _k ), p _k ?? 0, with {fx135-1} are defined by {fx135-2} while the weighted average of a measurable function f: ?₊ ?? ? with respect to the weight function p(t) ?? 0 with {fx135-3}. Under mild assumptions on the weights, we give necessary and sufficient conditions under which the finite limit ?? _n ?? L as n ?? ?? or ??(t) ?? L as t ?? ?? exists, respectively. These characterizations may find applications in probability theory.     

Weighted Reed–Muller codes revisited

We consider weighted Reed–Muller codes over point ensemble S ₁ × · · · × S _m where S _i needs not be of the same size as S _j . For m = 2 we determine optimal weights and analyze in detail what is the impact of the ratio |S ₁|/|S ₂| on the minimum distance. In conclusion the weighted Reed–Muller code construction is much better than its reputation. For a class of affine variety codes that contains the weighted Reed–Muller codes we then present two list decoding algorithms. With a small modification one of these algorithms is able to correct up to 31 errors of the [49,11,28] Joyner code.     

Crown multiplications and a higher order Hopf construction

We describe a ternary function (multiplication) from the product of three 12-point crowns into a 4-point crown in the category of partially ordered sets. The ternary map illustrates a version, in the context of posets, of associative multiplication. A higher order analogue of Hopf's construction applied to σ yields a poset model of a certain homotopy class, which we identify, in π₅(S³).     

Norm expansion along a zero variety

The reproducing kernel function of a weighted Bergman space over domains in Cd is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of (complex) codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kernel is thus reduced to the same kind of problem when one of the two entries is on the subvariety. A complete expansion of the reproducing kernel may be achieved in this manner. We carry this out in dimension d=2 for certain classes of weighted Bergman spaces over the bidisk (with the diagonal z₁=z₂ as subvariety) and the ball (with z₂=0 as subvariety), as well as for a weighted Bargmann-Fock space over C² (with the diagonal z₁=z₂ as subvariety).     

Rellich type inequalities related to Grushin type operator and Greiner type operator

We prove some sharp Hardy inequality associated with the gradient ? _?? = (? _x ,|x| ^?? ? _y ) by a direct and simple approach. Moreover, similar method is applied to obtain some weighted sharp Rellich inequality related to the Grushin operator in the setting of L ^p . We also get some weighted Hardy and Rellich type inequalities related to a class of Greiner type operators.     

Multi-variable weighted geometric means of positive definite matrices

We define a family of weighted geometric means {G(t;ω;A)}_t∈[0,1]ⁿ where ω and A vary over all positive probability vectors in Rⁿ and n-tuples of positive definite matrices resp. Each of these weighted geometric means interpolates between the weighted ALM (t=0_n) and BMP (t=1_n) geometric means (ALM and BMP geometric means have been defined by Ando-Li-Mathias and Bini-Meini-Poloni, respectively.) We show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean.     

Cohen’s linearly weighted kappa is a weighted average

An n × n agreement table F?=?{f _ij } with n ?? 3 ordered categories can for fixed m?(2??? m??? n ? 1) be collapsed into ${\binom{n-1}{m-1}}$ distinct m × m tables by combining adjacent categories. It is shown that the components (observed and expected agreement) of Cohen??s weighted kappa with linear weights can be obtained from the m × m subtables. A consequence is that weighted kappa with linear weights can be interpreted as a weighted average of the linearly weighted kappas corresponding to the m?× m tables, where the weights are the denominators of the kappas. Moreover, weighted kappa with linear weights can be interpreted as a weighted average of the linearly weighted kappas corresponding to all nontrivial subtables.     

A Plasticity Principle of Closed Hexahedra in the Three-Dimensional Euclidean Space

We prove a plasticity principle of closed hexahedra in the three dimensional Euclidean space which states that: Suppose that the closed hexahedron A ₁ A ₂?A ₅ has an interior weighted Fermat-Torricelli point A ₀ with respects to the weights B _i and let α _i0j =∠A _i A ₀ A _j . Then these 10 angles are determined completely by 7 of them and considering these five prescribed rays which meet at the weighted Fermat-Torricelli point A ₀, such that their endpoints form a closed hexahedron, a decrease on the weights that correspond to the first, third and fourth ray, causes an increase to the weights that correspond to the second and fifth ray, where the fourth endpoint is upper from the plane which is formed from the first ray and second ray and the third and fifth endpoint is under the plane which is formed from the first ray and second ray. By applying the plasticity principle of closed hexahedra to the n-inverse weighted Fermat-Torricelli problem, we derive some new evolutionary structures of closed polyhedra for n>5. Finally, we derive some evolutionary structures of pentagons in the two dimensional Euclidean space from the plasticity of weighted hexahedra as a limiting case.     

Smoothness and Best Approximation with Respect to Freud-Type Weights,a New Approach

A new measure of smoothness is defined and related to best approximation by polynomials with respect to weighted L _p (R) with Freud-type weights. Other related norms are also discussed. Comparisons with the known measure of smoothness on weighted L _p spaces are obtained. Related K-functionals and realization functionals are introduced. The new measure of smoothness allows us to consider a more general class of function spaces, to achieve Marchaud, Jackson and Bernstein-type inequalities, and to relate it to expressions involving the coefficients of the expansion by orthogonal polynomials with respect to Freud-type weights. Some of the results are new for approximation by Hermite polynomials in the weighted L _p space with the weight \({e^{-x^{2}}}\) .     

A simple heuristic for the p-centre problem

We describe a simple heuristic for determining the p-centre of a finite set of weighted points in an arbitrary metric space. The computational effort is O(np) for an n-point set. We show that the ratio of the objective function value of the heuristic solution to that of the optimum is bounded by min(3, 1 + α), where α is the maximum weight divided by the minimum weight of points in the set.     

Tractability of Multivariate Integration for Periodic Functions

We study multivariate integration in the worst case setting for weighted Korobov spaces of smooth periodic functions of d variables. We wish to reduce the initial error by a factor ε for functions from the unit ball of the weighted Korobov space. Tractability means that the minimal number of function samples needed to solve the problem is polynomial in ε⁻¹ and d. Strong tractability means that we have only a polynomial dependence in ε⁻¹. This problem has been recently studied for quasi-Monte Carlo quadrature rules and for quadrature rules with non-negative coefficients. In this paper we study arbitrary quadrature rules. We show that tractability and strong tractability in the worst case setting hold under the same assumptions on the weights of the Korobov space as for the restricted classes of quadrature rules. More precisely, let γ_j moderate the behavior of functions with respect to the jth variable in the weighted Korobov space. Then strong tractability holds iff ∑^∞_j=1 γ_j<∞, whereas tractability holds iff lim sup_d→∞ ∑^d_j=1 γ_j/ln d<∞. We obtain necessary conditions on tractability and strong tractability by showing that multivariate integration for the weighted Korobov space is no easier than multivariate integration for the corresponding weighted Sobolev space of smooth functions with boundary conditions. For the weighted Sobolev space we apply general results from E. Novak and H. Woźniakowski (J. Complexity17 (2001), 388–441) concerning decomposable kernels.     

Bounds for the weighted Lp discrepancy and tractability of integration

Quite recently Sloan and Woźniakowski (J. Complexity 14 (1998) 1) introduced a new notion of discrepancy, the so-called weighted L^p discrepancy of points in the d-dimensional unit cube for a sequence γ=(γ₁,γ₂,…) of weights. In this paper we prove a nice formula for the weighted L^p discrepancy for even p. We use this formula to derive an upper bound for the average weighted L^p discrepancy. This bound enables us to give conditions on the sequence of weights γ such that there exists N points in [0,1)^d for which the weighted L^p discrepancy is uniformly bounded in d and goes to zero polynomially in N⁻¹.Finally we use these facts to generalize some results from Sloan and Woźniakowski (1998) on (strong) QMC-tractability of integration in weighted Sobolev spaces.