The Radon–Nikodym Property for Tensor Products of Banach Lattices

Let 1 ≤ p < ∞. We show that , the Fremlin projective tensor product of ℓ_p with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property; and that , the Wittstock injective tensor product of ℓ_p with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property and each positive operator from ℓ_p' to X is compact, where 1/p +1/p'= 1 and let ℓ_p' = c₀ if p = 1. The author gratefully acknowledges support from the Office of Naval Research Grant # N00014-03-1-0621     

Differentiability of Lipschitz Maps from Metric Measure Spaces to Banach Spaces with the Radon–Nikodym Property

We prove the differentiability of Lipschitz maps X → V, where X denotes a PI space, i.e. a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon–Nikodym Property (RNP). As a consequence, we obtain a bi-Lipschitz nonembedding theorem for RNP targets. The differentiation theorem depends on a new specification of the differentiable structure for PI spaces involving directional derivatives in the direction of velocity vectors to rectifiable curves. We give two different proofs of this, the second of which relies on a new characterization of the minimal upper gradient. There are strong implications for the infinitesimal structure of PI spaces which will be discussed elsewhere.     

On the connection between the Hilger and Radon–Nikodym derivatives

We show that the Hilger derivative on time scales is a special case of the Radon–Nikodym derivative with respect to the natural measure associated with every time scale. Moreover, we show that the concept of delta absolute continuity agrees with the one from measure theory in this context.     

The limiting form of the Radon–Nikodym property is true for all Fréchet spaces

In this paper, we propose a new limiting form of the Radon–Nikodym property for the Bochner integral. We prove that the limiting form holds for an arbitrary Fr′echet space as opposed to an ordinary Radon–Nikodym property. We consider some applications in linear and nonlinear analysis.     

A Radon–Nikodym type theorem for forms

The Lebesgue decomposition theorem and the Radon–Nikodym theorem are the cornerstones of the classical measure theory. These theorems were generalized in several settings and several ways. Hassi, Sebestyén, and de Snoo recently proved a Lebesgue type decomposition theorem for nonnegative sesquilinear forms defined on complex linear spaces. The main purpose of this paper is to formulate and prove also a Radon–Nikodym type result in this setting. As an application, we present a Lebesgue type decomposition theorem and solve a special case of the infimum problem for densely defined (not necessarily bounded) positive operators.     

The Radon–Kipriyanov transform of the generalized spherical mean of a function

A formula relating the Radon transform of functions of spherical symmetries to the Radon–Kipriyanov transform K_γ for a naturalmulti-index γ is given. For an arbitrary multi-index γ, formulas for the representation of the K_γ-transform of a radial function as fractional integrals of Erdelyi–Kober integral type and of Riemann–Liouville integral type are proved. The corresponding inversion formulas are obtained. These results are used to study the inversion of the Radon–Kipriyanov transform of the generalized (generated by a generalized shift) spherical mean values of functions that belong to a weighted Lebesgue space and are even with respect to each of the weight variables.     

Radon–Nikodym Theorems for Nonnegative Forms,Measures and Representable Functionals

The aim of this paper is to establish two Radon–Nikodym-type theorems for nonnegative Hermitian forms defined on a real or complex vector space and to apply these results to provide some known Radon–Nikodym-type theorems of the theory of representable positive functionals, \(\sigma \)-additive and finitely additive measures.     

A New Characterization of the Analytic Radon-Nikodym Property for Bounded S

ＡＮｅｗＣｈａｒａｃｔｅｒｉｚａｔｉｏｎｏｆｔｈｅＡｎａｌｙｔｉｃＲａｄｏｎ－ＮｉｋｏｄｙｍＰｒｏｐｅｒｔｙｆｏｒＢｏｕｎｄｅｄＳｕｂｓｅｔｓＢｕＳｈａｎｇｑｕａｎ（步尚全）（ＤｅｐａｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，Ｔｓｉｎｇｈ...     

On the Average of Exponents

ＯｎｔｈｅＡｖｅｒａｇｅｏｆＥｘｐｏｎｅｎｔｓＣａｏＨｕｉｚｈｏｎｇ（曹惠中）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａｎｄｏｎｇＵｎｉｖｅｒｓｉｔｙ，Ｊｉｎａｎ２５０１００）Ａｂｓｔｒａｃｔ：Ｌｅｔｎ＞１ａｎｄｂｅｔｈｅｐｒｉｍｅｆａ...     

The Infinite Krull–Schmidt Property in the Case 2

We study regularity in the infinite direct sum decomposition in cocomplete categories. In particular we investigate and characterize a weak form of the Krull–Schmidt–Azumaya Theorem,in which the uniqueness of the direct sum decomposition is granted up to not one but two bijections,providing an abstract setting for a behaviour observed in the category of serial modules and other categories.     

Quasiunconditional Basis Property of the Faber–Schauder System

Ukrainian Mathematical Journal - We prove that, for any 0 &lt; ?? &lt; 1, there exists a measurable set E?? ? [0, 1], mes (E??) &gt; 1...     

On the Szegő–Radon projection of monogenic functions

We define a version of the Radon transform for monogenic functions which is based on Szegő kernels. The corresponding Szegő–Radon projection is abstractly defined as the orthogonal projection of a Hilbert module of left monogenic functions onto a suitable closed submodule of functions depending only on two variables. We also establish the inversion formula based on the dual transform.     

The Effect of Corners on the Complexity of Approximate Range Searching

Given an n-element point set in ℝ ^d , the range searching problem involves preprocessing these points so that the total weight, or for our purposes the semigroup sum, of the points lying within a given query range η can be determined quickly. In ε-approximate range searching we assume that η is bounded, and the sum is required to include all the points that lie within η and may additionally include any of the points lying within distance ε⋅diam(η) of η’s boundary. In this paper we contrast the complexity of approximate range searching based on properties of the semigroup and range space. A semigroup (S,+) is idempotent if x+x=x for all x∈S, and it is integral if for all k≥2, the k-fold sum x+⋅⋅⋅+x is not equal to x. Recent research has shown that the computational complexity of approximate spherical range searching is significantly lower for idempotent semigroups than it is for integral semigroups in terms of the dependencies on ε. In this paper we consider whether these results can be generalized to other sorts of ranges. We show that, as with integrality, allowing sharp corners on ranges has an adverse effect on the complexity of the problem. In particular, we establish lower bounds on the worst-case complexity of approximate range searching in the semigroup arithmetic model for ranges consisting of d-dimensional unit hypercubes under rigid motions. We show that for arbitrary (including idempotent) semigroups and linear space, the query time is at least . In the case of integral semigroups we prove a tighter lower bound of Ω(1/ε ^d−2). These lower bounds nearly match existing upper bounds for arbitrary semigroups. In contrast, we show that the improvements offered by idempotence do apply to smooth convex ranges. We say that a range is smooth if at every boundary point there is an incident Euclidean sphere that lies entirely within the range whose radius is proportional to the range’s diameter. We show that for smooth ranges and idempotent semigroups, ε-approximate range queries can be answered in O(log n+(1/ε)^(d−1)/2log (1/ε)) time using O(n/ε) space. We show that this is nearly tight by presenting a lower bound of Ω(log n+(1/ε)^(d−1)/2). This bound is in the decision-tree model and holds irrespective of space. A preliminary version of this paper appeared in the Proc. 22nd Annu. ACM Sympos. Comput. Geom., pp. 11–20, 2006. The research of S. Arya was supported by the Research Grants Council, Hong Kong, China under project number HKUST6184/04E. The research of D.M. Mount was partially supported by the National Science Foundation under grant CCR-0635099 and the Office of Naval Research under grant N00014-08-1-1015.     

The Range Set of Meromorphic Derivatives

ＴｈｅＲａｎｇｅＳｅｔｏｆＭｅｒｏｍｏｒｐｈｉｃＤｅｒｉｖａｔｉｖｅｓＩｎｄｒａｊｉｔＬａｈｉｒｉ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＪａｄａｖｐｕｒＵｎｉｖｅｒｓｉｔｙ，Ｃａｌｃｕｔａ－７０００３２，Ｉｎｄｉａ）ＡｂｓｔｒａｃｔＩｎｔ...     

Probabilistic Extensions of the Erdős–Ko–Rado Property

The classical Erdős–Ko–Rado (EKR) Theorem states that if we choose a family of subsets, each of size k, from a fixed set of size , then the largest possible pairwise intersecting family has size . We consider the probability that a randomly selected family of size t=t _n has the EKR property (pairwise nonempty intersection) as n and k=k _n tend to infinity, the latter at a specific rate. As t gets large, the EKR property is less likely to occur, while as t gets smaller, the EKR property is satisfied with high probability. We derive the threshold value for t using Janson’s inequality. Using the Stein–Chen method we show that the distribution of X ₀, defined as the number of disjoint pairs of subsets in our family, can be approximated by a Poisson distribution. We extend our results to yield similar conclusions for X _i , the number of pairs of subsets that overlap in exactly i elements. Finally, we show that the joint distribution (X ₀, X ₁, ..., X _b ) can be approximated by a multidimensional Poisson vector with independent components.      

The Property of Bair in r-Space

ＴｈｅＰｒｏｐｅｒｔｙｏｆＢａｉｒｉｎｒ－ＳｐａｃｅＳｏｎｇＺｈｅｎｍｉｎｇ（宋振明）（ＨｅｎａｎＵｎｉｖｅｒｓｉｔｙ）Ａｂｓｔｒａｃｔ：ＩｎｔｈｉｓＰａｐｅｒ，ｗｅＳｔｕｄｙｐｒｏｐｅｒｔｉｅｓｏｆＢａｉｒｂｙｓｙｍｍｅｔｒｉｃｄｉｆｆｅｒｅｎｃｅ...     

A Space-Time Characterization of the Kerr–Newman Metric

In the present paper, the characterization of the Kerr metric found by Marc Mars is extended to the Kerr–Newman family. A simultaneous alignment of the Maxwell field, the Ernst two-form of the pseudo-stationary Killing vector field, and the Weyl curvature of the metric is shown to imply that the space-time is locally isometric to domains in the Kerr–Newman metric. The paper also presents an extension of Ionescu and Klainerman’s null tetrad formalism to explicitly include Ricci curvature terms. Submitted: November 16, 2008. Accepted: February 9, 2009.     

The Average Eccentricity of Sierpiński Graphs

We determine the eccentricity of an arbitrary vertex, the average eccentricity and its standard deviation for all Sierpiński graphs ${S_p^n}$ . Special cases are the graphs ${S_2^{n}}$ , which are isomorphic to the state graphs of the Chinese Rings puzzle with n rings and the graphs ${S_3^{n}}$ isomorphic to the Hanoi graphs ${H_3^{n}}$ representing the Tower of Hanoi puzzle with 3 pegs and n discs.