Point sets with low <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-discrepancy

In this paper we study the L _p -discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L _p -discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L _p -discrepancy (p an even integer) of order which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L ₂-discrepancy of digitally shifted Hammersley point sets which show that the value of the L ₂-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these. This work is supported by the Austrian Research Fund (FWF), Project P17022-N12 and Project S8305.     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

A Lower Bound of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> Norms in the CLT for Strongly Mixing Random Variables

We derive a lower bound of L _p norms, 1 ⩽ p ⩽ ∞, in the central limit theorem for strongly mixing random variables X ₁,..., X _n with under the boundedness condition ℙ{|X _i | ⩽ M} = 1 with a nonrandom constantM > 0 and condition ∑ _r⩾1 r ²α(r) < ∞, where α(r) are the Rosenblatt strong mixing coefficients. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 587–602, October–December, 2005.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> approximation capability of RBF neural networks

L ^p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R ₊¹ → R ¹ and ∈ L _loc ^p (R ⁿ ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L ^p (K) with any accuracy for any compact set K in R ⁿ , if and only if g(x) is not an even polynomial. Partly supported by the National Natural Science Foundation of China (10471017)     

A new multivariable <Subscript><Emphasis Type="BoldItalic">6</Emphasis></Subscript><Emphasis Type="BoldItalic">ψ</Emphasis><Subscript><Emphasis Type="BoldItalic">6</Emphasis></Subscript> summation formula

By multidimensional matrix inversion, combined with an A _r extension of Jackson’s ₈ φ ₇ summation formula by Milne, a new multivariable ₈ φ ₇ summation is derived. By a polynomial argument this ₈ φ ₇ summation is transformed to another multivariable ₈ φ ₇ summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating ₆ φ ₅ summation. The latter is then extended, by analytic continuation, to a new multivariable extension of Bailey’s very-well-poised ₆ ψ ₆ summation formula. Partly supported by FWF Austrian Science Fund grants P17563-N13, and S9607 (the second is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”).     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-mixed intersection bodies

In this paper the author first introduce a new concept of L _p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L _p intersection body to L _p -mixed intersection body. Inequalities for L _p -dual mixed volumes of L _p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

Multivariate Refinement Equations and Convergence of Cascade Algorithms in Lp（0〈p〈1）Spaces

We consider the solutions of refinement equations written in the form

On the rate of convergence of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> norms in the CLT for poisson and gamma random variables

In the paper, we present upper bounds of L _p norms of order ( X)^-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X− X)/ √ X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09.     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

Geometric dual formulation for first-derivative-based univariate cubic <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> splines

With the objective of generating “shape-preserving” smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based -smooth univariate cubic L ₁ splines. An L ₁ spline minimizes the L ₁ norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an L ₁ spline is a nonsmooth non-linear convex program. Via Fenchel’s conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.     

Upper triangular operators and <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions

For a newform f for Γ₀(N) of even weight k supersingular at a prime p ≥ 5, by using infinite dimensional p-adic analysis, we prove that the p-adic L-function L _p (f,α; χ) has finite order of vanishing at any character of the form [(c)\tilde] _s ( x ) = x^s\tilde \chi _s \left( x \right) = x^s. In particular, under the natural embedding of ℤ _p in the group of ℂ* _p -valued continuous characters of ℤ* _p , the order of vanishing at any point is finite.     

Cyclic Descents and <Emphasis Type="Italic">P</Emphasis>-Partitions

Louis Solomon showed that the group algebra of the symmetric group _n has a subalgebra called the descent algebra, generated by sums of permutations with a given descent set. In fact, he showed that every Coxeter group has something that can be called a descent algebra. There is also a commutative, semisimple subalgebra of Solomon's descent algebra generated by sums of permutations with the same number of descents: an “Eulerian” descent algebra. For any Coxeter group that is also a Weyl group, Paola Cellini proved the existence of a different Eulerian subalgebra based on a modified definition of descent. We derive the existence of Cellini's subalgebra for the case of the symmetric group and of the hyperoctahedral group using a variation on Richard Stanley's theory of P-partitions.     

Localization of <Emphasis Type="Italic">LM</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-algebras

The aim of this paper is to define the localization LM _n -algebra of an LM _n —algebra L with respect to a topology F on L; in Section 5 we prove that the maximal LM _n -algebra of fractions (defined in [3]) and the LM _n -algebra of fractions relative to an Λ—closed system (defined in Section 2) are LM _n -algebras of localization.     

On the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> affine isoperimetric inequalities

We obtain an isoperimetric inequality which estimate the affine invariant p-surface area measure on convex bodies. We also establish the reverse version of L _p -Petty projection inequality and an affine isoperimetric inequality of Γ_− p K.     

On hamiltonicity of <Emphasis Type="Italic">P</Emphasis><Subscript>3</Subscript>-dominated graphs

We introduce a new class of graphs which we call P ₃-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P ₃-dominated graph. We prove that G is hamiltonian if α(G ²) ≤ κ(G), with two exceptions: K _2,3 and K _1,1,3. We also prove that G is hamiltonian, if G is 3-connected and |V(G)| ≤ 5δ(G) − 5. These results extend known results on (quasi-)claw-free graphs. This paper was completed when both authors visited the Center for Combinatorics, Nankai University, Tianjin. They gratefully acknowledge the hospitality and support of the Center for Combinatorics and Nankai University. The work of E.Vumar is sponsored by SRF for ROCS, REM.     

On representation theorem of <Emphasis Type="Italic">G</Emphasis>-expectations and paths of <Emphasis Type="Italic">G</Emphasis>-Brownian motion

We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ ： θ∈θ} representing an important sublinear expectation- G-expectation E[·]. We also give a concrete approximation of a bounded continuous function X（ω） by an increasing sequence of cylinder functions Lip（Ω） in order to prove that Cb（Ω） belongs to the completion of Lip（Ω） under the natural norm E[｜·｜].     

Interpolation of nonlinear operators in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We show that Peetre’s classical interpolation theorem in weighted L _p -spaces is carried over to some classes of nonlinear operators containing in particular the Lipschitz operators and operators close to them in the properties satisfying less restrictive conditions than Lipschitz in each of the spaces of a Banach pair.     

Approximation of sine-shaped functions by constants in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, <Emphasis Type="Italic">p</Emphasis> < 1

We investigate the best approximations of sine-shaped functions by constants in the spaces L_p for p < 1. In particular, we find the best approximation of perfect Euler splines by constants in the spaces L_p for certain p(0,1).Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 745–762, June, 2004.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-convergence of Lagrange interpolation on the semiaxis

In order to approximate functions defined on (0, +∞), the authors consider suitable Lagrange polynomials and show their convergence in weighted L ^p -spaces.      

Rationality of the vertex operator algebra <Emphasis Type="Italic">V</Emphasis><Subscript><Emphasis Type="Italic">L</Emphasis></Subscript>+ for a positive definite even lattice <Emphasis Type="Italic">L</Emphasis>

The lattice vertex operator algebra V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from –1-isometry of L. The fixed point set V_L+ of V_L for the automorphism is naturally a vertex operator algebra. We prove that any ₀-graded weak V_L⁺-module is completely reducible.Supported by JSPS Research Fellowships for Young Scientists.