Lp-Approximable Sequences of Vectors and Limit Distribution of Quadratic Forms of Random Variables

The properties of L₂-approximable sequences established here form a complete toolkit for statistical results concerning weighted sums of random variables, where the weights are nonstochastic sequences approximated in some sense by square-integrable functions and the random variables are “two-wing” averages of martingale differences. The results constitute the first significant advancement in the theory of L₂-approximable sequences since 1976 when Moussatat introduced a narrower notion of L₂-generated sequences. The method relies on a study of certain linear operators in the spaces L_p and l_p. A criterion of L_p-approximability is given. The results are new even when the weight generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method.     

Weak Equality and Exact Sequences in H v -modules

The largest class of multivalued systems satisfying the module-like axioms is the H_v-module. H_v-modules first were introduced by Vougiouklis. In this paper we define weak equality between two subsets of an H_v-module and introduced the notion of exact sequences of H_v-modules. Also some results on the weak equality and exact sequences are given.     

The L p -curvature images of convex bodies and L p -projection bodies

Associated with the L _p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L _p -projection bodies are established. As a natural extension of a result due to Lutwak, an L _p -type affine isoperimetric inequality, whose special cases are L _p -Busemann-Petty centroid inequality and L _p -affine projection inequality, respectively, is established. Some L _p -mixed volume inequalities involving L _p -projection bodies are also established.     

Rational Surfaces with Many Nodes

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain n b ₂ –2 disjoint smooth rational curves with self-intersection –2, where b ₂ is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with p _g = 0 and K² = 8, 9 which admit an automorphism of order 2.     

<Emphasis Type="Italic">∂</Emphasis>-operations and <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">V</Emphasis></Subscript>-fields

The hyperoperations, called theta-operations （δ）, are motivated from the usual property, which the derivative has on the derivation of a product of functions. Using any map on a set, one can define δ-operations. In this paper, we continue our study on the δ-operations on groupoids, rings, fields and vector spaces or on the corresponding hyperstructures. Using δ-operations one obtains, mainly, Hwstructures, which form the largest class of the hyperstructures. For representation theory of hyperstructures, by hypermatrices, one needs special Hv-rings or Hy-fields, so these hyperstructures can be used. Moreover, we study the relation of these δ-structures with other classes of hyperstructures, especially with the Hv-structures.     

Existence and Construction of H 1-Splines of Class C k on a Three Directional Mesh

Let be the uniform triangulation generated by the usual three directional mesh of the plane and let H ₁ be the regular hexagon formed by the six triangles of surrounding the origin. We study the space of piecewise polynomial functions in C ^k (R ²) with support H ₁ having a sufficiently high degree n, which are invariant with respect to the group of symmetries of H ₁ and whose sum of integer translates is constant. Such splines are called H ₁-splines. We first compute the dimension of this space in function of n and k. Then we prove the existence of a unique H ₁-spline of minimal degree for any fixed k0. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of this spline.     

On weighted L p -spaces of vector-valued entire analytic functions

The weighted L _p -spaces of entire analytic functions are generalized to the vector-valued setting. In particular, it is shown that the dual of the space is isomorphic to when the function χ _K is an L _p,ρ (E)-Fourier multiplier. This result allows us to give some new characterizations of the so-called UMD-property and to represent several ultradistribution spaces by means of spaces of vector sequences. J. Motos is partially supported by DGI (Spain), Grant BFM 2002-04013 and Grant MTEM 2005-08350-C03-03.     

On potentially K 5-H-graphic sequences

Let K _m -H be the graph obtained from K _m by removing the edges set E(H) of H where H is a subgraph of K _m . In this paper, we characterize the potentially K ₅-P ₄ and K ₅-Y ₄-graphic sequences where Y ₄ is a tree on 5 vertices and 3 leaves. Research was supported by NNSF of China (10271105) and by NSF of Fujian (Z0511034), Fujian Provincial Training Foundation for “Bai-Quan-Wan Talents Engineering”, Project of Fujian Education Department, Project of Zhangzhou Teachers College and by NSERC.     

Some complete metrics on spaces of fuzzy subsets

The classes of the L_p,∞- and L_p-metrics play an important role to develop a probability theory in fuzzy sample spaces. All of these metrics are known to be separable, but not complete. The classes are closely related as for each L_p,∞-metric there exists some L_p-metric which induces the same topology. This paper deals with the completion of the L_p,∞- and L_p-metrics. We can also show that the relationship between the classes of L_p,∞- and L_p-metrics still holds for the obtained respective classes of their completions.     

Lp-Lq DECAY ESTIMATES FOR HYPERBOLIC EQUATIONS WITH OSCILLATIONS IN COEFFICIENTS

This work is concerned with the proof of Lp -Lq decay estimates for solutions of the Cauchy problem for utt -λ2(t)b2(t)/Δu =0. The coefficient consists of an increasing smooth function λ and an oscillating smooth and bounded function b which are uniformly separated from zero. The authors‘ main interest is devoted to the critical case where one has an interesting interplay between the growing and the oscillating part.     

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.     

Holomorphic Functions of Exponential Growth on Abelian Coverings of a Projective Manifold

Let M be a projective manifold, p: M _G M a regular covering over M with a free Abelian transformation group G. We describe the holomorphic functions on M _G of an exponential growth with respect to the distance defined by a metric pulled back from M. As a corollary, we obtain Cartwright and Liouville-type theorems for such functions. Our approach brings together the L ₂ cohomology technique for holomorphic vector bundles on complete Kähler manifolds and the geometric properties of projective manifolds.     

<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)     

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”).     

A simple formula for an analogue of conditional wiener integrals and its applications II

Let C[0, T] denote the space of real-valued continuous functions on the interval [0, T] with an analogue w _ϕ of Wiener measure and for a partition 0 = t ₀ < t ₁ < ... < t _n < t _n+1 = T of [0, T], let X _n : C[0, T] → ℝ ⁿ⁺¹ and X ⁿ⁺¹: C[0, T] → ℝ ⁿ⁺² be given by X _n (x) = (x(t ₀), x(t ₁), ..., x(t _n )) and X _n+1(x) = (x(t ₀), x(t ₁), ..., x(t _n+1)), respectively. In this paper, using a simple formula for the conditional w _ϕ-integral of functions on C[0, T] with the conditioning function X _n+1, we derive a simple formula for the conditional w _ϕ-integral of the functions with the conditioning function X _n . As applications of the formula with the function X _n , we evaluate the conditional w _ϕ-integral of the functions of the form F _m (x) = ∫₀ ^T (x(t)) ^m for x ∈ C[0, T] and for any positive integer m. Moreover, with the conditioning X _n , we evaluate the conditional w _ϕ-integral of the functions in a Banach algebra which is an analogue of the Cameron and Storvick’s Banach algebra . Finally, we derive the conditional analytic Feynman w _ϕ-integrals of the functions in .      

Properties of Distortion Risk Measures

The current literature does not reach a consensus on which risk measures should be used in practice. Our objective is to give at least a partial solution to this problem. We study properties that a risk measure must satisfy to avoid inadequate portfolio selections. The properties that we propose for risk measures can help avoid the problems observed with popular measures, like Value at Risk (VaR _α ) or Conditional VaR _α (CVaR _α ). This leads to the definition of two new families: complete and adapted risk measures. Our focus is on risk measures generated by distortion functions. Two new properties are put forward for these: completeness, ensuring that the distortion risk measure uses all the information of the loss distribution, and adaptability, forcing the measure to use this information adequately. This research was partially funded by ^1,3 Welzia Management, SGIIC SA, RD Sistemas SA, Comunidad Autónoma de Madrid Grant s-0505/tic/000230, and MEyC Grant BEC2000-1388-C04-03 and by ² the Natural Sciences and Engineering Research Council of Canada (NSERC) Grant 36860-06.     

Embeddings of the classesH ω

In this paper we study functions belonging to the classesV _ε and ΛBV, which are encountered in the theory of Fourier trigonometric series. Necessary and sufficient conditions for the embedding of the classesH _ω in the classesV _ϕ and ABV are obtained. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 713–719, November, 1998. This research was supported by the program “Leading Scientific Schools” under grant No. 96/97-15-96073.     

A Variation of an Extremal Theorem Due to Woodall

We consider a variation of an extremal theorem due to Woodall [12, or 1, Chapter 3] as follows: Determine the smallest even integer (₃C₁,n), such that every n-term graphic sequence = (d₁, d₂,..., d_n) with term sum () = d₁ + d₂ + ... + d_n (₃C₁,n) has a realization G containing a cycle of length r for each r = 3,4,...,l. In this paper, the values of (₃C_l,n) are determined for l = 2m – 1,n 3m – 4 and for l = 2m,n 5m – 7, where m 4.AMS Mathematics subject classification (1991) 05C35Project supported by the National Natural Science Foundation of China (Grant No. 19971086) and the Doctoral Program Foundation of National Education Department of China     

Equivalent Norms in Spaces of Functions of Fractional Smoothness on Arbitrary Domains

In this paper, we study the spaces B _pq ^s (G) and L _pq ^s (G) of functions f with positive exponent of smoothness s > 0 given on a domain . The norms on these spaces are defined via integral norms of the difference of the function f of order m > s treated as a function of the point of the domain and of the difference increment. For an arbitrary domain , we characterize these spaces in terms of the local approximations of the function by polynomials of degree m – 1.     

On Weighted α-Besov Spaces and α-Bloch Spaces of Quaternion-Valued Functions

In this paper, we give the definitions of weighted α-Besov-type spaces and α-Bloch spaces of quaternion-valued functions, then we obtain characterizations of these quaternion α-Bloch spaces by quaternion α-Besov-type spaces. Relations between Q _p norms and weighted α-Besov norms are also considered. The role of ρ?^α sequences in securing non-Bloch functions is highlighted in quaternion sense.