Derivations with power central values on Lie ideals in prime rings

Let R be a prime ring of char R ≠ 2 with a nonzero derivation d and let U be its noncentral Lie ideal. If for some fixed integers n ₁ ⩾ 0, n ₂ ⩾ 0, n ₃ ⩾ 0, (u ⁿ¹ [d(u), u]u ⁿ²) ⁿ³ ∈ Z(R) for all u ∈ U, then R satisfies S ₄, the standard identity in four variables.     

Power Values of Derivations with Annihilator Conditions on Lie Ideals in Prime Rings

Let R be a prime ring of char R ≠ 2, d a nonzero derivation of R, U a noncentral Lie ideal of R, and a ∈ R. If au ^{n ₁} d(u) ^{n ₂} u ^{n ₃} d(u) ^{n ₄} u ^{n ₅} … d(u) ^{n _k?1} u ^{n _k}  = 0 for all u ∈ U, where n ₁, n ₂,…,n _k are fixed non-negative integers not all zero, then a = 0 and if a(u ^s d(u)u ^t ) ⁿ  ∈ Z(R) for all u ∈ U, where s ≥ 0, t ≥ 0, n ≥ 1 are some fixed integers, then either a = 0 or R satisfies S ₄, the standard identity in four variables.     

Power-Centralizing Automorphisms of Lie Ideals in Prime Rings

Let R be a prime ring with center Z, L a noncentral Lie ideal of R, and σ a nontrivial automorphism of R such that [u ^σ,u] ⁿ  ∈ Z for all u ∈ L. If either char(R) > n or char(R) = 0, then R satisfies s ₄, the standard identity in 4 variables.     

Certain additive maps on m-power closed Lie ideals

Let R be a prime ring with extended centroid C and m a fixed positive integer >?1. A Lie ideal L of R is called m-power closed if ${u^m \in L}$ for all ${u \in L}$ . We prove that if char R = 0 or a prime p?>?m, then every non-central, m-power closed Lie ideal L of R contains a nonzero ideal of R except when dim _C RC?=?4, m is odd, and ${u^{m-1} \in C}$ for all ${u \in L}$ . Moreover, the additive maps d : L ?? R satisfying d(u ^m )?=?mu ^m-1 d(u) (resp. d(u ^m )?=?u ^m-1 d(u)) for all ${u \in L}$ are completely characterized if char R = 0 or a prime p?>?2(m ? 1).     

Annihilators of derivations with Engel conditions on lie ideals

LetR a prime ring,L a noncentral Lie ideal ofR anda ∈R. Suppose thatd is a nonzero derivation ofR such thata[d(u),u] _k =0 for allu ∈L, wherek is a fixed positive integer. Thena=0 except when charR=2 and dim _C RC=4.     

Generalized derivations as Jordan homomorphisms on lie ideals and right ideals

Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the standard identity s4（x1, x2, x3, x4）, L is commutative and u2 ∈ Z（R）, for any u C L. We also examine some consequences of this result related to generalized derivations which act as Jordan homomorphisms on the set [I, I], where I is a non-zero right ideal of R.     

Stochastic wave equation with critical nonlinearities: Temporal regularity and uniqueness

We consider a nonlinear wave equation on Rd driven by a spatially homogeneous Wiener process W with a finite spectral measure and with nonlinear terms f, g of critical growth. We study pathwise uniqueness and norm continuity of paths of (u,ut) in H¹(Rd)⊕L²(Rd) under the hypothesis that there exists a local solution u such that (u,ut) has weakly continuous paths in H¹(Rd)⊕L²(Rd).     

On Lie ideals with derivations as homomorphisms and anti-homomorphisms

In [2, Theorem 3], Bell and Kappe proved that if d is a derivation of a prime ring R which acts as a homomorphism or an anti-homomorphism on a nonzero right ideal I of R, then d = 0 on R. In the present paper our objective is to extend this result to Lie ideals. The following result is proved: Let R be a 2-torsion free prime ring and U a nonzero Lie ideal of R such that u ² ∈ U, for all u ∈ U. If d is a derivation of R which acts as a homomorphism or an anti-homomorphism on U, then either d=0 or U ?Z(R).     

Annihilator Properties of Skew Monoid Rings

Let R be a ring with an endomorphism σ and F ∪ {0} the free monoid generated by U = {u ₁,…, u _t } with 0 added, and M a factor of F setting certain monomial in U to 0, enough so that, for some n, M ⁿ  = 0. In this article, we study various annihilator properties and a variety of conditions and related properties that the skew monoid ring R[M; σ] is inherited from R.     

A Note on Strongly Clean Matrix Rings

Let R be an associative ring with identity. An element a ∈ R is called strongly clean if a = e + u with e ² = e ∈ R, u a unit of R, and eu = ue. A ring R is called strongly clean if every element of R is strongly clean. Strongly clean rings were introduced by Nicholson [7 Nicholson , W. K. ( 1999 ). Strongly clean rings and Fitting's lemma . Comm. Algebra 27 : 3583 – 3592 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. It is unknown yet when a matrix ring over a strongly clean ring is strongly clean. Several articles discussed this topic when R is local or strongly π-regular. In this note, necessary conditions for the matrix ring 𝕄 _n (R) (n > 1) over an arbitrary ring R to be strongly clean are given, and the strongly clean property of 𝕄₂(RC ₂) over the group ring RC ₂ with R local is obtained.     

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.

     

Wave front set for solutions to Schrödinger equations

We consider solutions to Schrödinger equation on Rd with variable coefficients. Let H be the Schrödinger operator and let u(t)=e−itHu₀ be the solution to the Schrödinger equation with the initial condition u₀∈L²(Rd). We show that the wave front set of u(t) in the nontrapping region can be characterized by the wave front set of e−itH₀u₀, where H₀ is the free Schrödinger operator. The characterization of the wave front set is given by the wave operator for the corresponding classical mechanical scattering (or equivalently, by the asymptotics of the geodesic flow).     

About the p-adic Yosida equation inside a disk

Let K be a complete ultrametric algebraically closed field and let ?(d(0, R^?)) be the field of meromorphic functions inside the disk d(0,R⁻) = {x ∈ K ∣ ∣x∣ < R}. Let ?_b(d(0, R^?)) be the subfield of bounded meromorphic functions inside d(0,R⁻) and let ?_u(d(0, R^?)) = ?(d(0, R^?)) ? ?_b(d(0, R^?)) be the subset of unbounded meromorphic functions inside d(0,R⁻). Initially, we consider the Yosida Equation: , where m ∈ ?^* and F(X) is a rational function of degree d with coefficients in ?_b(d(0, R^?)). We show that, if d ≥ 2m + 1, this equation has no solution in ?_u(d(0, R^?)).Next, we examine solutions of the above equation when F(X) is apolynomial with constant coefficients and show that it has no unbounded analytic functions in d(0,R⁻). Further, we list the only cases when the equation may eventually admit solutions in ?_u(d(0, R^?)). Particularly, the elliptic equation may not.     

The Dichotomy Theorem for evolution bi-families

We prove that the operator G, the closure of the first-order differential operator −d/dt+D(t) on L₂(R,X), is Fredholm if and only if the not well-posed equation u^′(t)=D(t)u(t), t∈R, has exponential dichotomies on R₊ and R₋ and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t)=A+B(t), A is the generator of a bi-semigroup, B(⋅) is a bounded piecewise strongly continuous operator-valued function. Also, we prove some perturbations results and consider various examples of not well-posed problems.     

An identity with generalized derivations on lie ideals, right ideals and Banach algebras

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, F a non-zero generalized derivation of R. Suppose that [F(u), u]F(u) = 0 for all u ε L, then one of the following holds:

1. there exists α ε C such that F(x) = α x for all x ε R
2. R satisfies the standard identity s ₄ and there exist a ε U and α ε C such that F(x) = ax + xa + αx for all x ε R.

We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.     

Commuting Values of Generalized Derivations on Multilinear Polynomials

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, U the right Utumi quotient ring of R, f(x ₁,…, x _n ) a noncentral multilinear polynomial over K, and G a nonzero generalized derivation of R. Denote f(R) the set of all evaluations of the polynomial f(x ₁,…, x _n ) in R. If [G(u)u, G(v)v] = 0, for any u, v ∈ f(R), we prove that there exists c ∈ U such that G(x) = cx, for all x ∈ R and one of the following holds: 1. f(x ₁,…, x _n )² is central valued on R;

2. R satisfies s ₄, the standard identity of degree 4.

     

The Cohen–Macaulay and Gorenstein Properties of Rings Associated to Filtrations

Let (R, m) be a Cohen–Macaulay local ring, and let ? = {F _i } _i∈? be an F ₁-good filtration of ideals in R. If F ₁ is m-primary we obtain sufficient conditions in order that the associated graded ring G(?) be Cohen–Macaulay. In the case where R is Gorenstein, we use the Cohen–Macaulay result to establish necessary and sufficient conditions for G(?) to be Gorenstein. We apply this result to the integral closure filtration ? associated to a monomial parameter ideal of a polynomial ring to give necessary and sufficient conditions for G(?) to be Gorenstein. Let (R, m) be a Gorenstein local ring, and let F ₁ be an ideal with ht(F ₁) = g > 0. If there exists a reduction J of ? with μ(J) = g and reduction number u: = r _J (?), we prove that the extended Rees algebra R′(?) is quasi-Gorenstein with a-invariant b if and only if J ⁿ : F _u  = F _n+b?u+g?1 for every n ∈ ?. Furthermore, if G(?) is Cohen–Macaulay, then the maximal degree of a homogeneous minimal generator of the canonical module ω _G(?) is at most g and that of the canonical module ω _R′(?) is at most g ? 1; moreover, R′(?) is Gorenstein if and only if J ^u : F _u  = F _u . We illustrate with various examples cases where G(?) is or is not Gorenstein.     

Cocentralizing and vanishing derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2 and extended centroid C and let f(x1,..., x _n ) be a multilinear polynomial over C not central-valued on R, while δ is a nonzero derivation of R. Suppose that d and g are derivations of R such that

$\delta (d(f(r_1 , \ldots ,r_n ))f(r_1 , \ldots ,r_n ) - f(r_1 , \ldots ,r_n )g(f(r_1 , \ldots ,r_n ))) = 0$

for all r1,..., r _n ∈ R. Then d and g are both inner derivations on R and one of the following holds: (1) d = g = 0; (2) d = ?g and f(x ₁,..., x _n )² is central-valued on R.

     

Existenz und Eindeutigkeit für Lösungen des Neumann—Tricomi-Problems im ℝ2

In a bounded simply-connected domainG \( \subseteq \) ?² a boundary value problem for a linear partial differential equation of second orderLu=f is studied. The operatorL is elliptic inG?{y>0}, parabolic forG?{y=0} and hyperbolic inG?{y<0}. The boundary value problem consists in findingu satisfyingLu=f inG, d _n u=φ on the elliptic part of the boundary ofG, u=ψ on the noncharacteristic part (which is not empty) of the hyperbolic part of the boundary ofG.d _n u denotes the conormal (with respect toL) derivative ofu. It is proved that the problem has a generalized solution in anL ²-weight space. Uniqueness is otained in the class of quasiregular solutions. In order to get the results apriori estimates are proved; theorems from functional analysis are used.     

Control problems governed by Sobolev partial differential equations

Let G be a bounded subset of Rⁿ with a smooth boundary and Q = G × (0, T]. We consider a control problem governed by the Sobolev initial-value problem My_t(u) + Ly(u) = u in L²(Q), y(·, 0; u) = 0 in L₂(G), where M = M(x) and L = L(x) are symmetric uniformly strongly elliptic operators of orders 2m and 2l, respectively. The problem is to find the control u₀ of L²(Q)-norm at most b that steers to within a prescribed tolerance ? of a given function Z in L²(G) and that minimizes a certain energy functional. Our main results establish regularity properties of u₀. We also give results concerning the existence and uniqueness of the optimal control, the controllability of Sobolev initial-value problems, and properties of the Lagrange multipliers associated with the problem constraints.