A Note on the Embedding of Subspaces of Lp into ℓNr, 0 < r < 1, r ≦ p < 2

In this paper we extend a result by Bourgain-Lindenstrauss-Milman (see [1]). We prove: Let 0 < ? < 1/2, 0< r < 1, r< p < 2. There exists a constant C = C(r,p,?) such that if X is any n-dimensional subspace of L_p(0, l), then there exists Y ? ?^N_r with d(X, Y) ≦ 1 + ?, whenever N > Cn. As an application, we obtain the following partial result: Let 0 < r < 1. There exist constants C = C(r) and C' = C' (r) such that if X is any n-dimensional subspace of L_r(0,1), then there exists Y ? ^N_r with d(X, Y) ≦ C (logn)^l/^r, whenever N ≧ C'n.     

Completely 𝒲-Resolved Complexes

Let R be a ring and 𝒲 a self-orthogonal class of left R-modules which is closed under finite direct sums and direct summands. A complex C of left R-modules is called a 𝒲-complex if it is exact with each cycle Z ⁿ (C) ∈ 𝒲. The class of such complexes is denoted by 𝒞_𝒲. A complex C is called completely 𝒲-resolved if there exists an exact sequence of complexes D _· = … → D _?1 → D ₀ → D ₁ → … with each term D _i in 𝒞_𝒲 such that C = ker(D ₀ → D ₁) and D _· is both Hom(𝒞_𝒲, ?) and Hom(?, 𝒞_𝒲) exact. In this article, we show that C = … → C ^?1 → C ⁰ → C ¹ → … is a completely 𝒲-resolved complex if and only if C ⁿ is a completely 𝒲-resolved module for all n ∈ ?. Some known results are obtained as corollaries.     

Holomorphic liftings and Bergman kernel estimates for 𝒟ℱ𝒩‐domains

Let E be a 𝒟ℱ𝒩‐space and let U ⊂ E be open. By applying the nuclearity of the Fréchet space ℋ︁(U) of holomorphic functions on U we show that there are finite measures μ on U leading to Bergman spaces of μ ‐square integrable holomorphic functions. We give an explicit construction for μ by using infinite dimensional Gaussian measures. Moreover, we prove boundary estimates for the corresponding Bergman kernels K_μ on the diagonal and we give an application of our results to liftings of μ ‐square integrable Banach space valued holomorphic functions over U. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

𝒱-Nilpotent Groups and 5-Term Exact Sequence

In this paper, we prove that any nontrivial normal subgroup N of 𝒱-nilpotent group G, intersects nontrivially with the marginal subgroup of G with respect to the variety 𝒱. This is a generalization of a well-known theorem by Philip Hall which refers to the case where 𝒱 is the variety of abelian groups. Moreover, we give a generalized version of a 5-term exact sequence of homology groups and establish some isomorphisms of the factors of the lower marginal series.     

Lie Superalgebras Over 𝔤𝔩2

Whereas Holm proved that the ring of differential operators on a generic hyperplane arrangement is finitely generated as an algebra, the problem of its Noetherian properties is still open. In this article, after proving that the ring of differential operators on a central arrangement is right Noetherian if and only if it is left Noetherian, we prove that the ring of differential operators on a central 2-arrangement is Noetherian. In addition, we prove that its graded ring associated to the order filtration is not Noetherian when the number of the consistuent hyperplanes is greater than 1.     

The wi‐club filter on 𝒫κ λ

We develop the theory of C_{κ, λ}ⁱ, a strongly normal filter over ??_κ λ for Mahlo κ. We prove a minimality result, showing that any strongly normal filter containing {x ∈ ??_κ λ: |x | = |x ∩ κ | and |x | is inaccessible} also contains C_{κ, λ}ⁱ. We also show that functions can be used to obtain a basis for C_{κ, λ}ⁱ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Density Property of P𝒮C Fields

Let 𝒮 be a finite set of finite and real local primes of a field K. We prove two results. First, a PAC field over K is already PAC over each nonempty 𝒮‐open subset of K; if K is global, this implies that it is also PAC over each separable Hilbert subset of K. Second, each P𝒮C field inside K_tot,𝒮 has the 𝒮‐density property.     

Factorization of operator valued Lp for 0 ≤ p < 1

In [5], it is proved that a bounded linear operator u, from a Banach space Y into an L_p(S, ν) factors through L_p1 (S, ν) for some p₁ > 1, if Y* is of finite cotype; (S, ν) is a probability space for p = 0, and any measure space for 0 < p < 1. In this paper, we generalize this result to uv, where u : Y → L_p(S, ν) and v : X → Y are linear operators such that v* is of finite Ka?in cotype. This result gives also a new proof of Grothendieck's theorem. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

ω‐saturated quasi‐minimal models of Th(ℚω, +, σ, 0)

We show that (ℚ^ω, +, σ, 0) is a quasi-minimal torsion-free divisible abelian group. After discussing the axiomatization of the theory of this structure, we present its ω-saturated quasi-minimal model. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lp – Lq Decay Estimates fo the Solutions of Strictly Hyperbolic Equations of Second Order with Increasing in Time Coefficients

One of the most important questions in the theory of nonlinear wave equations is that for global existence of solutions. An essential tool is the Strichartz inequality for special solutions of the wave equation.In the last time different results were proved generalizing the classical one of Strichartz. In the present paper L_p – L_q estimates are proved for the solutions of strictly hyperbolic equations of second order with time dependent coefficients where these are unbounded at infinity. In the first step the WKB method is applied to the construction of a fundamental system of solutions for ordinary differential equations depending on a parameter. In a second step the method of stationary phase yields the asymptotical behaviour of Fourier multipliers with nonstandard phase functions depending on a parameter.     

Order plus size of τ‐critical graphs

Let $G=\left(V,E\right)$ be a $\tau$‐critical graph with $\tau \left(G\right)=t$. Erd?s and Gallai proved that $\mid V\mid \le 2t$ and the bound $\mid E\mid \le \left(\genfrac{}{}{0ex}{}{t+1}{2}\right)$ was obtained by Erd?s, Hajnal, and Moon. We give here the sharp combined bound $\mid E\mid +\mid V\mid \le \left(\genfrac{}{}{0ex}{}{t+2}{2}\right)$ and find all graphs with equality.     

On the self‐dual 𝔽5‐codes constructed from Hadamard matrices of order 24

There are exactly 60 inequivalent Hadamard matrices of order 24. In this note, we give a classification of the self‐dual ??₅‐codes of length 48 constructed from the Hadamard matrices of order 24. © 2004 Wiley Periodicals, Inc.     

Asymptotic Property of the 𝕀-Invariant of the Associated Graded Modules

Abstract

The present article clarifies the stability of the 𝕀-invariant of the associated graded modules G(𝔞 ⁿ ; E) when n is increasing. It will be shown that the 𝕀-invariant 𝕀(G(𝔞 ⁿ ; E)) becomes a constant if n is sufficiently large.     

Admissible perturbations of α‐ψ‐pseudocontractive operators: convergence theorems

We prove some convergence theorems for α‐ψ‐pseudocontractive operators in real Hilbert spaces, by using the concept of admissible perturbation. Our results extend and complement some theorems in the existing literature. Copyright © 2016 John Wiley & Sons, Ltd.     

τ-Functions for a two-point version of the KP-hierarchy

In this paper a multipoint version of the linearization of the KP-hierarchy is described. Solutions of this system in the form of wave functions are constructed starting from a suitable Grassmann manifold and a group of commuting flows corresponding to the configuration of n points in . The failure of equivariance at lifting these flows to the determinant bundle over this Grassmann manifold is measured by the determinants of certain Fredholm operators, the so-called τ-functions. In the case of two points an explicit relation between these τ-functions and the wave functions is derived.