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.     

On the Cotorsion Pair (𝒫1, 𝒟)

Abstract

We classify the C ₂ × C ₂-gradings on M ₂(k), k an arbitrary field. If char(k) ≠ 2 our approach relies on the duality between gradings and actions for finite abelian groups, and if k is algebraically closed we find precisely one isomorphism type of grading which is not isomorphic to a grading with all the matrix units being homogeneous elements. If char(k) = 2 we use a computational approach and we find that any C ₂ × C ₂-grading is induced by a C ₂-grading.     

On the Nonseparable Subspaces of J(η) and C([1, η])

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 𝓁₂(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c₀(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω₁) (resp. C([1, ω₁])) contains a complemented copy of 𝓁₂(ω₁) (resp.c₀(ω₁)). As consequence, we give examples (J(ω₁), C([1, ω₁]), C(V), V being the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i.e., for every subspace Y ⊆ X we have that Dens (Y) = w*–Dens (Y*)), in spite of these spaces are not weakly Lindelof determined (WLD).     

Symmetric Powers of Nat 𝔰𝔩2(𝕂)

We identify the spaces of homogeneous polynomials in two variables 𝕂[Y^k, XY^k?1, ?, X^k] among representations of the Lie ring 𝔰𝔩₂(𝕂). This amounts to constructing a compatible 𝕂-linear structure on some abstract 𝔰𝔩₂(𝕂)-modules, where 𝔰𝔩₂(𝕂) is viewed as a Lie ring.     

Embedding Subspaces of Lp into 𝓁Np, 0 < p < 1

It is shown that if X is an n – dimensional subspace of L_p, 0 < p < 1, then there exists a subspace Y of 𝓁^N_p such that d(X, Y) ≤ 1 + ε and N ≤ C(ε, p)n(log n)³.     

On a Class of p- Valently α-Convex Functions

The object of the present is to prove some properties of a class M_p(α) of p-valently α-convex Functions in the unit disk. Also, an integral representation for functions belonging to the class M_p(α) is shown.     

Infinitely many solutions for nonlocal elliptic systems of (p1(x),⋯,pn(x))‐Kirchhoff type

In this paper, we obtain the existence of infinitely solutions for a class of nonlocal elliptic systems of (p₁(x),?,p_n(x))‐Kirchhoff type. Our main results are new. Our approach are based on general variational principle because of B. Ricceri and the theory of the variable exponent Sobolev spaces. Copyright © 2015 John Wiley & Sons, Ltd.     

Quasi-Deformations of 𝔰𝔩2(𝔽) Using Twisted Derivations

In this article we apply a method devised in Hartwig, Larsson, and Silvestrov (2006 Hartwig , J. T. , Larsson , D. , Silvestrov , S. D. ( 2006 ). Deformations of Lie algebras using σ-derivations . J. Algebra 295 : 314 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]) and Larsson and Silvestrov (2005a Larsson , D. , Silvestrov , S. D. (2005a). Quasi-hom-Lie algebras, Central extensions and 2-cocycle-like identities. J. Algebra 288:321–344.[Crossref], [Web of Science ®] , [Google Scholar]) to the simple 3-dimensional Lie algebra 𝔰𝔩₂(𝔽). One of the main points of this deformation method is that the deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present article that when our deformation scheme is applied to 𝔰𝔩₂(𝔽) we can, by choosing parameters suitably, deform 𝔰𝔩₂(𝔽) into the Heisenberg Lie algebra and some other 3-dimensional Lie algebras in addition to more exotic types of algebras, this being in stark contrast to the classical deformation schemes where 𝔰𝔩₂(𝔽) is rigid.     

A classification result on weighted {δ (p3 + 1), δ; 3, p3}‐minihypers

We classify all {δ (p³ + 1), δ; 3, p³}‐minihypers, , for a prime number p₀ ≥ 7, with excess e ≤ p³. Such a minihyper is a sum of lines and of possibly one projected subgeometry PG(5, p), or a sum of lines and a minihyper which is a projected subgeometry PG(5, p) minus one line. When p is a square, also (possibly projected) Baer subgeometries PG(3, p^3/2) can occur. © 2004 Wiley Periodicals, Inc.     

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)     

A note about stabilization in Aℝ(𝔻)

It is shown that for A_?(??) functions f₁ and f₂ with and f₁ being positive on real zeros of f₂ then there exists A_?(??) functions g₂ and g₁, g₁^–1 with and This result is connected to the computation of the stable rank of the algebra A_?(??) and to Control Theory (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-recursive functions,knots “with thick ropes,” and self-clenching “thick” hyperspheres

We introduce an approach to certain geometric variational problems based on the use of the algorithmic unrecognizability of the n-dimensional sphere for n ≥ 5. Sometimes this approach allows one to prove the existence of infinitely many solutions of a considered variational problem. This recursion-theoretic approach is applied in this paper to a class of functionals on the space of C^1.1-smooth hypersurfaces diffeomorphic to Sⁿ in Rⁿ⁺¹, where n is any fixed number ≥ 5. The simplest of these functionals k_v is defined by the formula k_v(Σⁿ) = (vol(Σⁿ))^1/n/r(Σⁿ), where r(Σⁿ) denotes the radius of injectivity of the normal exponential map for Σⁿ ? Rⁿ+l. We prove the existence of an infinite set of distinct locally minimal values of k_v on the space of C^1.1-smooth topological hyperspheres in Rⁿ⁺¹ for any n ≥ 5. The functional k_v naturally arises when one attempts to generalize knot theory in order to deal with embeddings and isotopies of “thick” circles and, more generally, “thick” spheres into Euclidean spaces. We introduce the notion of knot “with thick rope” types. The theory of knot “with thick rope” types turns out to be quite different from the classical knot theory because of the following result: There exists an infinite set of non-trivial knot “with thick rope” types in codimension one for every dimension greater than or equal to five.     

On –δu = K(x)u5 in ℝ3

In this paper, we study the equation –Δu = K(x)u⁵ in ?³ and provide a large class of positive functions K(x) for which we obtain infinitely many positive solutions which decay at infinity at the rate of |x|^?1. © 1993 John Wiley & Sons, Inc.     

C1,α domains and unique continuation at the boundary

It is shown that the square of a nonconstant harmonic function u that either vanishes continuously on an open subset V contained in the boundary of a Dini domain or whose normal derivative vanishes on an open subset V in the boundary of a C^1,1 domain in ℝ^d satisfies the doubling property with respect to balls centered at points Q ∈ V. Under any of the above conditions, the module of the gradient of u is a B₂(dσ)-weight when restricted to V, and the Hausdorff dimension of the set of points {Q ∈ V : ∇u(Q) = 0} is less than or equal to d−2. These results are generalized to solutions to elliptic operators with Lipschitz second-order coefficients and bounded coefficients in the lower-order terms. © 1997 John Wiley & Sons, Inc.     

𝔏-Rickart Modules

It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M _R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M _R is a finitely generated 𝔏-Rickart module, we prove that End _R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End _R (M) is a left extending ring iff M is a Baer module and End _R (M) is left cononsingular.     

The central limit theorem for weighted empirical processes indexed by sets

Sufficient conditions are found for the weak convergence of a weighted empirical process {(ν_n(C)/q(P(C))) 1 _{[P(C) λn]}: C }, indexed by a class of sets and weighted by a function q of the size of each set. We find those functions q which allow weak convergence to a sample-continuous Gaussian process, and, given q, determine the fastest rate at which one may allow λ_n → 0.