Additivity of Lie multiplicative maps on triangular algebras

Let 𝒜 and ? be unital algebras over a commutative ring ?, and ? be a (𝒜,??)-bimodule, which is faithful as a left 𝒜-module and also as a right ?-module. Let 𝒰?=?Tri(𝒜,??,??) be the triangular algebra and 𝒱 any algebra over ?. Assume that Φ?:?𝒰?→?𝒱 is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST???TS)?=?Φ(S)Φ(T)???Φ(T)Φ(S) for all S, T?∈?𝒰. Then Φ(S?+?T)?=?Φ(S)?+?Φ(T)?+?Z _S,T for all S, T?∈?𝒰, where Z _S,T is an element in the centre 𝒵(𝒱) of 𝒱 depending on S and T.     

For most large underdetermined systems of linear equations the minimal 𝓁1‐norm solution is also the sparsest solution

We consider linear equations y = Φx where y is a given vector in ?ⁿ and Φ is a given n × m matrix with n < m ≤ τn, and we wish to solve for x ∈ ?^m. We suppose that the columns of Φ are normalized to the unit ??₂‐norm, and we place uniform measure on such Φ. We prove the existence of ρ = ρ(τ) > 0 so that for large n and for all Φ's except a negligible fraction, the following property holds: For every y having a representation y = Φx₀ by a coefficient vector x₀ ∈ ?^m with fewer than ρ · n nonzeros, the solution x₁ of the ??₁‐minimization problem is unique and equal to x₀. In contrast, heuristic attempts to sparsely solve such systems—greedy algorithms and thresholding—perform poorly in this challenging setting. The techniques include the use of random proportional embeddings and almost‐spherical sections in Banach space theory, and deviation bounds for the eigenvalues of random Wishart matrices. © 2006 Wiley Periodicals, Inc.     

Maps preserving product X*Y + YX* on factor von Neumann algebras

Let 𝒜 and ? be two factor von Neumann algebras. In this article, we prove that a nonlinear bijective map Φ?:?𝒜?→?? satisfies Φ(X*?Y?+?YX*)?=Φ(X)*Φ(Y)?+?Φ(Y)Φ(X)* (?X,?Y?∈?𝒜), if and only if Φ is a *-ring isomorphism. In particular, if 𝒜 and ? are type I factors, then Φ is a unitary isomorphism or conjugate unitary isomorphism.     

Nonlinear Strong Commutativity Preserving Maps on Prime Rings

Let 𝒜 be a unital prime ring containing a nontrivial idempotent P. Assume that Φ: 𝒜 → 𝒜 is a nonlinear surjective map. It is shown that Φ preserves strong commutativity if and only if Φ has the form Φ(A) = αA + f(A) for all A ∈ 𝒜, where α ∈ {1, ?1} and f is a map from 𝒜 into 𝒵(𝒜). As an application, a characterization of nonlinear surjective strong commutativity preserving maps on factor von Neumann algebras is obtained.     

The Mal'cev Rank for Modules over Pseudo-Valuation Domains

Let R be a commutative ring and A be an R-module. The Mal'cev rank μ(A) of A is the sup of genN, where N ranges over the finitely generated submodules of A, and genN is the minimum number of generators of N. We prove that μ is both sub-additive and pre-additive as an invariant of Mod(R). Our main goal is to investigate μ for modules over pseudo-valuation domains. Specifically, we establish which pseudo-valuation domains R satisfy the property that an R-module of finite Mal'cev rank must be finitely generated. We split the class 𝒞 of pseudo-valuation domains as a union 𝒞 = 𝒞₁ ∪ 𝒞₂ ∪ 𝒞₃ ∪ 𝒞₄ of suitably defined subclasses, and prove that the property holds if and only if R ∈ 𝒞₃ ∪ 𝒞₄. In that case we can describe the R-modules A where μ(A) < ∞. We also show that, for R ∈ 𝒞₄, there exist indecomposable R-modules of arbitrarily large finite Mal'cev rank.     

Another lattice of varieties of completely regular semigroups

Completely regular semigroups S are taken here with the unary operation of inversion within the maximal subgroups of S. As such they form a variety 𝒞? whose lattice of subvarieties is denoted by ?(𝒞?). The relation on ?(𝒞?) which identifies two varieties if they contain the same bands is denoted by B^∧. The upper ends of B^∧-classes which are neither equal to 𝒞? nor contained in the variety 𝒞𝒮 of completely simple semigroups are generated by two countably infinite ascending chains called canonical varieties. In a previous publication, we constructed the sublattice Σ of ?(𝒞?) generated by 𝒞𝒮 and the first four canonical varieties. Here we extend Σ to the sublattice Ψ of ?(𝒞?) generated by 𝒞𝒮 and the first six canonical varieties. For each of the varieties in Ψ?Σ, we construct the ladder and a basis of its identities.     

Generalized Skew Derivations with Nilpotent Values in Prime Rings

Let ? be a prime ring, 𝒞 the extended centroid of ?, ? a Lie ideal of ?, F be a nonzero generalized skew derivation of ? with associated automorphism α, and n ≥ 1 be a fixed integer. If (F(xy) ? yx) ⁿ  = 0 for all x, y ∈ ?, then ? is commutative and one of the following statements holds:

(1) Either ? is central;

(2) Or ? ? M ₂(𝒞), the 2 × 2 matrix ring over 𝒞, with char(𝒞) = 2.     

On Some Problems of P. Turán Concerning Lm Extremal Polynomials and Quadrature Formulas

The L_m extremal polynomials in an explicit form with respect to the weights (1−x)^−1/2 (1+x)^(m−1)/2 and (1−x)^(m−1)/2 (1+x)^−1/2 for even m are given. Also, an explicit representation for the Cotes numbers of the corresponding Turán quadrature formulas and their asymptotic behavior is provided.     

Characterizations of Jordan derivations and Jordan homomorphisms

Let 𝒜 be a unital Banach algebra and ? be a unital 𝒜-bimodule. We show that if δ is a linear mapping from 𝒜 into ? satisfying δ(ST)?=?δ(S)T?+Sδ(T) for any S, T?∈?𝒜 with ST?=?W, where W is a left or right separating point of ?, then δ is a Jordan derivation. Also, it is shown that every linear mapping h from 𝒜 into a unital Banach algebra ? which satisfies h(S)h(T)?=?h(ST) for any S,?T?∈?𝒜 with ST?=?W is a Jordan homomorphism if h(W) is a separating point of ?.     

Some results on (k, μ)′-almost Kenmotsu manifolds

In this paper, we study the Weyl conformal curvature tensor 𝒲 and the concircular curvature tensor 𝒞 on a (k, μ)′-almost Kenmotsu manifold M²ⁿ⁺¹ of dimension greater than 3. We obtain that if M²ⁿ⁺¹ satisfies either R · 𝒲 = 0 or 𝒞 · 𝒞 = 0, then it is locally isometric to either the hyperbolic space ?²ⁿ⁺¹ (?1) or the Riemannian product ?ⁿ⁺¹(?4) × ?ⁿ.     

For most large underdetermined systems of equations,the minimal 𝓁1‐norm near‐solution approximates the sparsest near‐solution

We consider inexact linear equations y ≈ Φx where y is a given vector in ?ⁿ, Φ is a given n × m matrix, and we wish to find x_0,? as sparse as possible while obeying ‖y ? Φx_0,?‖₂ ≤ ?. In general, this requires combinatorial optimization and so is considered intractable. On the other hand, the ??₁‐minimization problem is convex and is considered tractable. We show that for most Φ, if the optimally sparse approximation x_0,? is sufficiently sparse, then the solution x_1,? of the ??₁‐minimization problem is a good approximation to x_0,?. We suppose that the columns of Φ are normalized to the unit ??₂‐norm, and we place uniform measure on such Φ. We study the underdetermined case where m ～ τn and τ > 1, and prove the existence of ρ = ρ(τ) > 0 and C = C(ρ, τ) so that for large n and for all Φ's except a negligible fraction, the following approximate sparse solution property of Φ holds: for every y having an approximation ‖y ? Φx₀‖₂ ≤ ? by a coefficient vector x₀ ∈ ?^m with fewer than ρ · n nonzeros, This has two implications. First, for most Φ, whenever the combinatorial optimization result x_0,? would be very sparse, x_1,? is a good approximation to x_0,?. Second, suppose we are given noisy data obeying y = Φx₀ + z where the unknown x₀ is known to be sparse and the noise ‖z‖₂ ≤ ?. For most Φ, noise‐tolerant ??₁‐minimization will stably recover x₀ from y in the presence of noise z. We also study the barely determined case m = n and reach parallel conclusions by slightly different arguments. Proof techniques include the use of almost‐spherical sections in Banach space theory and concentration of measure for eigenvalues of random matrices. © 2006 Wiley Periodicals, Inc.     

Linear spaces and preservers of symmetric matrices of bounded rank-two

Let 𝔽 be a field of characteristic two. Let S _n (𝔽) denote the vector space of all n?×?n symmetric matrices over 𝔽. We characterize i. subspaces of S _n (𝔽) all whose elements have rank at most two where n???3,

ii. linear maps from S _m (𝔽) to S _n (𝔽) that sends matrices of rank at most two into matrices of rank at most two where m, n???3 and |𝔽|?≠?2.

     

Cluster-Tilted Algebras of Type D n

Let H be a hereditary algebra of Dynkin type D _n over a field k and 𝒞 _H be the cluster category of H. Assume that n ≥ 5 and that T and T′ are tilting objects in 𝒞 _H . We prove that the cluster-tilted algebra Γ = End_{𝒞 H} (T)^op is isomorphic to Γ′ = End_{𝒞 H} (T′)^op if and only if T = τ ⁱ T′ or T = στ ^j T′ for some integers i and j, where τ is the Auslander–Reiten translation and σ is the automorphism of 𝒞 _H defined in Section 4.     

Qualitative Properties of Saddle-Shaped Solutions to Bistable Diffusion Equations

We consider the elliptic equation ? Δu = f(u) in the whole ?^2m , where f is of bistable type. It is known that there exists a saddle-shaped solution in ?^2m . This is a solution which changes sign in ?^2m and vanishes only on the Simons cone 𝒞 = {(x ¹, x ²) ∈ ? ^m × ? ^m : |x ¹| = |x ²|}. It is also known that these solutions are unstable in dimensions 2 and 4.

In this article we establish that when 2m = 6 every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution.

These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.     

Idempotence preserving maps between matrix spaces

Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽₃. Let M _n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n idempotent matrices and GL _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n invertible matrices. Let Φ_𝔽(n,?m) denote the set of all maps from M _n (𝔽) to M _m (𝔽) satisfying A???λB?∈?P _n (𝔽)???φ(A)???λφ(B)?∈?P _m (𝔽) for every A,?B?∈?M _n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ_𝔽(n,?m), then there exists T?∈?GL _m (𝔽) such that φ(A)?=?T?[A???I _p ?⊕?A ^t ???I _q ?⊕?0]T?^?1 for every A?∈?M _n (𝔽), where I ₀?=?0. This improves the results of some related references.     

The Poincaré series of the algebras of simultaneous invariants and covariants of two binary forms

Let ? _{d 1, d 2} and 𝒞 _{d 1, d 2} be the algebras of simultaneous invariants and simultaneous covariants of the two binary forms of degrees d ₁ and d ₂. Formulas for computation of the Poincaré series 𝒫? _{d 1, d 2} (z), 𝒫𝒞 _{d 1, d 2} (z) of the algebras are found. By using these formulas, we have computed the series for d ₁, d ₂ ≤ 20.     

Invariant Constructions of Simple and Maximal Sets

The main results of the present paper are the following theorems: 1. There is no e ∈ ω such that for any A, B ? ω, S^A = W is simple in A, and if A′ ?_T B′, then S^A =* S^B. 2 There is an e ∈ ω such that for any A, B ? ω, M^A = W_e is incomplete maximal in A, and if A =* B, then M^A ?_T M^B.     

A generalization of Everett's result on the Collatz 3x + 1 problem

It is a well-known conjecture that given m ε N, the set of natural numbers, the sequence {m_n}^∞_n−0, defined by the iterative formula m₀ = m,

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has some iterate m_j = 1. It is shown in this paper that for any k ε N, “almost every” natural number m greater than unity has k iterates less than m.