E-special Ordered Regular Semigroups

An ordered regular semigroup S is E-special if for every x ∈ S there is a biggest x ⁺ ∈ S such that both xx ⁺ and x ⁺ x are idempotent. Every regular strong Dubreil–Jacotin semigroup is E-special, as is every ordered completely simple semigroup with biggest inverses. In an E-special ordered regular semigroup S in which the unary operation x → x ⁺ is antitone the subset P of perfect elements is a regular ideal, the biggest inverses in which form an inverse transversal of P if and only if S has a biggest idempotent. If S ⁺ is a subsemigroup and S does not have a biggest idempotent, then P contains a copy of the crown bootlace semigroup.     

Minimizing the L ∞ Norm of the Gradient with an Energy Constraint

ABSTRACT

The variational problem in L ^∞ considered is to minimize F(u) = ‖Du‖ _{L ^∞(Ω)} subject to ∈ t _Ω |Du|² dx ≤ E for given E > 0. It is proven that a constrained minimizer exists and satisfies an Aronsson-Euler equation in the viscosity sense which depends on a parameter Λ_∞ ≥ 0. This parameter splits Ω into two parts. In one part the minimizer satisfies the infinity laplace equation and in the remaining part the minimizer is the solution of the elasto-plastic torsion problem with constraint ‖Du‖ _{L ^∞}  ≤ Λ_∞.     

An Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f(x ₁,…, x _n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r ₁,…, r _n  ∈ R, a[F ²(f(r ₁,…, r _n )), f(r ₁,…, r _n )] = 0, then one of the following statements hold: 1. a = 0;

2. There exists λ ∈C such that F(x) = λx, for all x ∈ R;

3. There exists c ∈ U such that F(x) = cx, for all x ∈ R, with c ² ∈ C;

4. There exists c ∈ U such that F(x) = xc, for all x ∈ R, with c ² ∈ C.

     

Generalized Derivations with Engel Conditions on One-Sided Ideals

Let R be a noncommutative prime ring and I a nonzero left ideal of R. Let g be a generalized derivation of R such that [g(r ^k ), r ^k ] _n  = 0 for all r ∈ I, where k, n are fixed positive integers. Then there exists c ∈ U, the left Utumi quotient ring of R, such that g(x) = xc and I(c ? α) = 0 for a suitable α ∈ C. In particular we have that g(x) = α x, for all x ∈ I.     

A result concerning nilpotent values with generalized skew derivations on Lie ideals

ABSTRACT

Let n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F(x)F(y)?yx)ⁿ = 0 for all x,y∈L, then char(R) = 2 and R?M₂(C), the ring of 2×2 matrices over C.     

Subsets with Generalized Derivations Having Nilpotent Values on Lie Ideals

Let R be a prime ring, with no nonzero nil right ideal, Q the two-sided Martindale quotient ring of R, F a generalized derivation of R, L a noncommutative Lie ideal of R, and b ∈ Q. If, for any u, w ∈ L, there exists n = n(u, w) ≥1 such that (F(uw) ? bwu)ⁿ = 0, then one of the following statements holds:

1. F = 0 and b = 0;

2. R ? M₂(K), the ring of 2 × 2 matrices over a field K, b² = 0, and F(x) = ?bx, for all x ∈ R.

     

Symmetric colorings of the dihedral group

Let G be a finite group and let r∈?. An r-coloring of G is any mapping χ:G→{1,…,r}. Colorings χ and ψ are equivalent if there exists g∈G such that χ(xg^?1) = ψ(x) for every x∈G. A coloring χ is symmetric if there exists g∈G such that χ(gx^?1g) = χ(x) for every x∈G. Let S_r(G) denote the number of symmetric r-colorings of G and s_r(G) the number of equivalence classes of symmetric r-colorings of G. We count S_r(G) and s_r(G) in the case where G is the dihedral group D_n.     

Decompositions of Quotient Rings and m-Power Commuting Maps

Let R be a semiprime ring with symmetric Martindale quotient ring Q, n ≥ 2 and let f(X) = X ⁿ h(X), where h(X) is a polynomial over the ring of integers with h(0) = ±1. Then there is a ring decomposition Q = Q ₁ ⊕ Q ₂ ⊕ Q ₃ such that Q ₁ is a ring satisfying S _2n?2, the standard identity of degree 2n ? 2, Q ₂ ? M _n (E) for some commutative regular self-injective ring E such that, for some fixed q > 1, x ^q  = x for all x ∈ E, and Q ₃ is a both faithful S _2n?2-free and faithful f-free ring. Applying the theorem, we characterize m-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring.     

Generalized Skew Derivations with Power Central Values on Lie Ideals

Let R be a prime ring with center Z and L a noncommutative Lie ideal of R. Suppose that f is a right generalized β-derivation of R associated with a β-derivation δ such that f(x) ⁿ  ∈ Z for all x ∈ L, where n is a fixed positive integer. Then f = 0 unless dim  _C RC = 4.     

On Behavior of Solvable Ideals of Lie Algebras Under Outer Derivations

Let L be a finite dimensional Lie algebra over a field F. It is well known that the solvable radical S(L) of the algebra L is a characteristic ideal of L if char F = 0, and there are counterexamples to this statement in case char F = p > 0. We prove that the sum S(L) of all solvable ideals of a Lie algebra L (not necessarily finite dimensional) is a characteristic ideal of L in the following cases: 1) char F = 0; 2) S(L) is solvable and its derived length is less than log₂ p.     

Remotal Sets in Orlicz Spaces

Let X be a Banach space, (I, μ) be a finite measure space. By L ^Φ(I, X), let us denote the space of all X-valued Bochner Orlicz integrable functions on the unit interval I equipped with the Luxemburg norm. A closed bounded subset G of X is called remotal if for any x ∈ X, there exists g ∈ G such that ‖x ? g‖ = ρ(x, G) = sup {‖x ? y‖: y ∈ G}. In this article, we show that for a separable remotal set G ? X, the set of Bochner integrable functions, L ^Φ(I, G) is remotal in L ^Φ(I, X). Some other results are presented.     

Multipliers of classes of cauchy transforms

The analytic map g on the unit disk D is said to induce a multiplication operator L from the Banach space X to the Banach space Y if L(f)=f·g∈Y for all f∈X. For z ∈ D and α>0 the families of weighted Cauchy transforms F_α are defined by ?(z) = ∫_T K_x ^α (z)dμ(x) where μ(x) is complex Borel measures, x belongs to the unit circle T and the kernel K_x (z) = (1- xz)^?1. In this article we will explore the relationship between the compactness of the multiplication operator L acting on F ₁ and the complex Borel measures μ(x). We also give an estimate for the essential norm of L     

Supercentralizing Superderivations on Prime Superalgebras

ABSTRACT

Let A = A ₀ ⊕ A ₁ be an associative superalgebra over a commutative associative ring F, and let Z _s (A) be its supercenter. An F-mapping f of A into itself is called supercentralizing on a subset S of A if [x, f(x)] _s  ∈ Z _s (A) for all x ∈ S. In this article, we prove a version of Posner's theorem for supercentralizing superderivations on prime superalgebras.     

Vanishing Derivations and Co-Centralizing Generalized Derivations on Multilinear Polynomials in Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x₁,…, x_n) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) ? f(r)G(f(r))) = 0 for all r = (r₁,…, r_n) ∈ Rⁿ, then one of the following holds:

1. There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a ? q] = 0 and f(x₁,…, x_n)² is central valued on R;

2. There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;

3. There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa ? bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C;

4. R satisfies s₄ and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa ? bx and G(x) = ax + xb for all x ∈ R;

5. There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb ? δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x₁,…, x_n)² is central valued on R.

     

On Classes of E-Inversive Semigroups and Semigroups Whose Idempotents Form a Subsemigroup

Abstract

A semigroup S is called E-inversive if for every a ∈ S there is an x ∈ S such that ax is idempotent. The purpose of this paper is the investigation of E-inversive semigroups and semigroups whose idempotents form a subsemigroup. Basic properties are analysed and, in particular, semigroups whose idempotents form a semilattice or a rectangular band are considered. To provide examples and characterizations, the construction methods of generalized Rees matrix semigroups and semidirect products are employed.     

Solvability of a Commutative Algebra which Satisfies (x 2)2 = 0

We studied the solvability of the algebra which satisfies the polynomial identity (x ²)² = 0. We believe that, if A is a finite dimensional commutative algebra over a field F of characteristic not 2 which satisfies (x ²)² = 0 for all x ∈ A, then A is solvable. In this article we proved this when dim  _F A ≤ 7.     

Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds

Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x ₀)⊂E→F be C¹ nonlinear map, whereU (x ₀) is an open set containing pointx ₀∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x ₀) near x₀,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C¹ Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.     

On Engel Elements of a Group Relative to Certain Subgroup

An element g in a group G is called a left Engel element of G, if for each x ∈ G, there is a positive integer n = n(g, x) such that [x, _n g] = 1. In this article, we will study a generalization of the left Engel elements and its connections with the generalized Hirsch–Plotkin and Baer radical.