共查询到20条相似文献,搜索用时 31 毫秒
1.
Let R be a prime ring and set [x, y]1 = [x, y] = xy ? yx for ${x,y\in R}$ and inductively [x, y] k = [[x, y] k-1, y] for k > 1. We apply the theory of generalized polynomial identities with automorphisms and skew derivations to obtain the following result: If δ is a nonzero σ-derivation of R and L is a noncommutative Lie ideal of R so that [δ(x), x] k = 0 for all ${x \in L}$ , where k is a fixed positive integer, then charR = 2 and ${R\subseteq M_{2}(F)}$ for some field F. This result generalizes the case of derivations by Lanski and also the case of automorphisms by Mayne. 相似文献
2.
Let R be a semiprime ring and F be a generalized derivation of R and n??? 1 a fixed integer. In this paper we prove the following: (1) If (F(xy) ? yx) n is either zero or invertible for all ${x,y\in R}$ , then there exists a division ring D such that either R?=?D or R?=?M 2(D), the 2?× 2 matrix ring. (2) If R is a prime ring and I is a nonzero right ideal of R such that (F(xy) ? yx) n ?=?0 for all ${x,y \in I}$ , then [I, I]I?=?0, F(x)?=?ax?+?xb for ${a,b\in R}$ and there exist ${\alpha, \beta \in C}$ , the extended centroid of R, such that (a ? ??)I?=?0 and (b ? ??)I?=?0, moreover ((a?+?b)x ? x)I?=?0 for all ${x\in I}$ . 相似文献
3.
Let R be a ring with center Z(R). An additive mapping ${F : R \longrightarrow R}$ is said to be a generalized derivation on R if there exists a derivation ${d : R \longrightarrow R}$ such that F(xy) = F(x)y + xd(y), for all ${x, y \in R}$ (the map d is called the derivation associated with F). Let R be a semiprime ring and U be a nonzero left ideal of R. In the present note we prove that if R admits a generalized derivation F, d is the derivation associated with F such that d(U) ≠ (0) then R contains some nonzero central ideal, if one of the following conditions holds: (1) R is 2-torsion free and ${F(xy) \in Z(R)}$ , for all ${x, y \in U}$ , unless F(U)U = UF(U) = Ud(U) = (0); (2) ${F(xy) \mp yx \in Z(R)}$ , for all ${x,y \in U}$ ; (3) ${F(xy) \mp [x,y] \in Z(R)}$ , for all ${x,y \in U}$ ; (4) F ≠ 0 and F([x,y]) = 0, for all ${x, y \in U}$ , unless Ud(U) = (0); (5) F ≠ 0 and ${F([x, y]) \in Z(R)}$ , for all ${x, y \in U}$ , unless either d(Z(R))U = (0) or Ud(U) = (0)n. 相似文献
4.
Let R be a prime ring with center Z(R). For a fixed positive integer n, a permuting n-additive map ${\Delta : R^n \to R}$ is known to be permuting n-derivation if ${\Delta(x_1, x_2, \ldots, x_i x'_{i},\ldots, x_n) = \Delta(x_1, x_2, \ldots, x_i, \ldots, x_n)x'_i + x_i \Delta(x_1, x_2, \ldots, x'_i, \ldots, x_n)}$ holds for all ${x_i, x'_i \in R}$ . A mapping ${\delta : R \to R}$ defined by δ(x) = Δ(x, x, . . . ,x) for all ${x \in R}$ is said to be the trace of Δ. In the present paper, we have proved that a ring R is commutative if there exists a permuting n-additive map ${\Delta : R^n \to R}$ such that ${xy + \delta(xy) = yx + \delta(yx), xy- \delta(xy) = yx - \delta(yx), xy - yx = \delta(x) \pm \delta(y)}$ and ${xy + yx = \delta(x) \pm \delta(y)}$ holds for all ${x, y \in R}$ . Further, we have proved that if R is a prime ring with suitable torsion restriction then R is commutative if there exist non-zero permuting n-derivations Δ1 and Δ2 from ${R^n \to R}$ such that Δ1(δ 2(x), x, . . . ,x) = 0 for all ${x \in R,}$ where δ 2 is the trace of Δ2. Finally, it is shown that in a prime ring R of suitable torsion restriction, if ${\Delta_1, \Delta_2 : R^n \longrightarrow R}$ are non-zero permuting n-derivations with traces δ 1, δ 2, respectively, and ${B : R^n \longrightarrow R}$ is a permuting n-additive map with trace f such that δ 1 δ 2(x) = f(x) holds for all ${x \in R}$ , then R is commutative. 相似文献
5.
Basudeb Dhara Vincenzo De Filippis Giovanni Scudo 《Mediterranean Journal of Mathematics》2013,10(1):123-135
Let R be a prime ring, H a nonzero generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that there exists ${0 \neq a \in R}$ such that a(u s H(u)u t ) n = 0 for all ${u \in L}$ , where s ≥ 0, t ≥ 0, n ≥ 1 are fixed integers. Then s = 0, H(x) = bx for all ${x \in R}$ with ab = 0, unless R satisfies s 4, the standard identity in four variables. We also describe completely this last case. 相似文献
6.
Let R be a prime, locally matrix ring of characteristic not 2 and let Q ms (R) be the maximal symmetric ring of quotients of R. Suppose that ${\delta}\colon R\to Q_{ms}(R)$ is a Jordan τ-derivation, where τ is an anti-automorphism of R. Then there exists a?∈?Q ms (R) such that δ(x)?=?xa???aτ(x) for all x?∈?R. Let X be a Banach space over the field ${\mathbb F}$ of real or complex numbers and let ${\mathcal B}(X)$ be the algebra of all bounded linear operators on X. We prove that $Q_{ms}({\mathcal B}(X))={\mathcal B}(X)$ , which provides the viewpoint of ring theory for some results concerning derivations on the algebra ${\mathcal B}(X)$ . In particular, all Jordan τ-derivations of ${\mathcal B}(X)$ are inner if $\text{dim}_{\mathbb F}X>1$ . 相似文献
7.
Kaoru Yoneda 《Analysis Mathematica》1981,7(4):297-302
Пусть?(x) — ограниченн ая функция на отрезке [0,1] и ее функция распределен ияΦ(t) удовлетворяет услов ию $$\Phi \left( t \right) + \Phi \left( { - t} \right) = 1.$$ Еслиf(x) — конечная поч ти всюду функция, то дл яF n (t) — функции распределе ния произведенияf(x)?(nx) — вы полнены соотношения и В частности, еслиf(x) — и нтегрируемая функци я, то из (1) следует, что $$\mathop {\lim }\limits_{n \to \infty } \mathop \smallint \limits_0^1 f\left( x \right)\varphi \left( {nx} \right)dx = 0 $$ 相似文献
8.
Let R be a prime ring of characteristic different from 2, with right Utumi quotient ring U and extended centroid C, and let ${f(x_1, \ldots, x_n)}$ be a multilinear polynomial over C, not central valued on R. Suppose that d is a derivation of R and G is a generalized derivation of R such that $$G(f(r_1, \ldots, r_n))d(f(r_1, \ldots, r_n)) + d(f(r_1, \ldots, r_n))G(f(r_1, \ldots, r_n)) = 0$$ for all ${r_1, \ldots, r_n \in R}$ . Then either d = 0 or G = 0, unless when d is an inner derivation of R, there exists ${\lambda \in C}$ such that G(x) = λ x, for all ${x \in R}$ and ${f(x_1, \ldots, x_n)^2}$ is central valued on R. 相似文献
9.
Vincenzo de Filippis Giovanni Scudo Mohammad S. Tammam El-Sayiad 《Czechoslovak Mathematical Journal》2012,62(2):453-468
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:
- there exists α ε C such that F(x) = α x for all x ε R
- 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.
10.
Letq be a regular quadratic form on a vector space (V,F) and letf be the bilinear form associated withq. Then, \(\dot V: = \{ z \in V|q(z) \ne 0\} \) is the set of non-singular vectors ofV, and forx, y ∈ \(\dot V\) , ?(x, y) ?f(x, y) 2/(q(x) · q(y)) is theq-measure of (x, y), where ?(x,y)=0 means thatx, y are orthogonal. For an arbitrary mapping \(\sigma :\dot V \to \dot V\) we consider the functional equations $$\begin{gathered} (I)\sphericalangle (x,y) = 0 \Leftrightarrow \sphericalangle (x^\sigma ,y^\sigma ) = 0\forall x,y \in \dot V, \hfill \\ (II)\sphericalangle (x,y) = \sphericalangle (x^\sigma ,y^\sigma )\forall x,y \in \dot V, \hfill \\ (III)f(x,y)^2 = f(x^\sigma ,y^\sigma )^2 \forall x,y \in \dot V, \hfill \\ \end{gathered} $$ and we state conditions on (V,F,q) such thatσ is induced by a mapping of a well-known type. In case of dimV ∈N?{0, 1, 2} ∧ ∣F∣ > 3, each of the assumptions (I), (II), (III) implies that there exist aρ-linear injectionξ :V →V and a fixed λ ∈F?{0} such thatF x σ =F x ξ ?x ∈ \(\dot V\) andf(x ξ,y ξ)=λ · (f(x, y))ρ ?x, y ∈V. Moreover, (II) implies ρ =id F ∧q(x ξ) = λ ·q(x) ?x ∈ \(\dot V\) , and (III) implies ρ=id F ∧ λ ∈ {1,?1} ∧x σ ∈ {x ξ, ?x ξ} ?x ∈ \(\dot V\) . Other results obtained in this paper include the cases dimV = 2 resp. dimV ?N resp. ∣F∣ = 3. 相似文献
11.
Moshe Marcus Laurent Veron 《Calculus of Variations and Partial Differential Equations》2013,48(1-2):131-183
We prove that any positive solution of ${\partial_tu-\Delta u+u^q=0 (q > 1)}$ in ${\mathbb{R}^N \times (0, \infty)}$ with initial trace (F, 0), where F is a closed subset of ${\mathbb{R}^{N}}$ can be represented, up to two universal multiplicative constants, by a series involving the Bessel capacity ${C_{2/q, q^{\prime}}}$ . As a consequence we prove that there exists a unique positive solution of the equation with such an initial trace. We also characterize the blow-up set of u(x, t) when ${t \downarrow 0}$ , by using the “density” of F expressed in terms of the ${C_{2/q, q^{\prime}}}$ -Bessel capacity. 相似文献
12.
L. Socha 《Journal of Optimization Theory and Applications》1981,33(3):393-399
Let a quasilinear control system having the state space \(\bar X \subseteq R^n \) be governed by the vector differential equation $$\dot x = G(u(t))x,$$ wherex(0) =x 0 andU is the family of all bounded measurable functions from [0,T] intoU, a compact and convex subset ofR m.LetG:U ?R be a bounded measurable nonlinear function, such thatG(U) is compact and convex.G ?1 can be convex onG(U) or concave. The main results of the paper establish the existence of a controlu ∈U which minimizes the cost functional $$I(u) = \int_0^T {L(u(t))x(t)dt,} $$ whereL(·) is convex. A practical example of application for chemical reactions is worked out in detail. 相似文献
13.
A. A. Arkhipova 《Journal of Mathematical Sciences》1997,87(2):3284-3303
Properties of generalized solutions of model nonlinear elliptic systems of second order are studied in the semiball $B_1^ + = B_1 (0) \cap \{ x_n > 0\} \subset $ ? n , with the oblique derivative type boundary condition on $\Gamma _1 = B_1 (0) \cap \{ x_n = 0\} $ . For solutionsu∈H 1(B 1 + ) of systems of the form $\frac{d}{{dx_\alpha }}a_\alpha ^k (u_x ) = 0, k \leqslant {\rm N}$ , it is proved that the derivatives ux are Hölder in $B_1^ + \cup \Gamma _1 )\backslash \Sigma $ , where Hn?p(σ)=0,p>2. It is shown for continuous solutions u from H1(B1/+) of systems $\frac{d}{{dx_\alpha }}a_\alpha ^k (u,u_x ) = 0$ that the derivatives ux are Hölder on the set $(B_1^ + \cup \Gamma _1 )\backslash \Sigma , dim_\kappa \Sigma \leqslant n - 2$ . Bibliography: 13 titles. 相似文献
14.
W. Kratz 《Rendiconti del Circolo Matematico di Palermo》1987,36(3):457-473
The following limit theorem on Hamiltonian systems (resp. corresponding Riccati matrix equations) is shown: Given(N, N)-matrices,A, B, C andn ∈ {1,…, N} with the following properties:A and kemelB(x) are constant, rank(I, A, …, A n?1) B(x)≠N,B(x)∈C n(R), andB(x)(A T)j-1 C(x)∈C n-j(R) forj=1, …, n. Then \(\mathop {\lim }\limits_{x \to x_0 } \eta _1^T \left( x \right)V\left( x \right)U^{ - 1} \left( x \right)\eta _2 \left( x \right) = d_1^T \left( {x_0 } \right)U\left( {x_0 } \right)d_2 \) forx 0∈R, whenever the matricesU(x), V(x) are a conjoined basis of the differential systemU′=AU + BV, V′=CU?A TV, and whenever ηi(x)∈R N satisfy ηi(x 0)=U(x 0)d i ∈ imageU(x 0) η′i-Aηni(x) ∈ imageB(x),B(x)(η′i(x)-Aηi(x)) ∈C n-1 R fori=1,2. 相似文献
15.
K. L. Olifirov 《Mathematical Notes》1975,18(5):1050-1053
For an arbitrary element x with spectrum sp(x) in a Banach algebra with identity e ≠ 0 we define the upper (lower) spectral abscissa \(\mathop {\sigma + (x)}\limits_{( - )} = \mathop {\max }\limits_{(\min )} \operatorname{Re} \lambda ,\lambda \in sp(x)\) . With the aid of the spectral radius \(\rho (x) = \mathop {\max }\limits_{\lambda \in sp(x)} \left| \lambda \right| = \mathop {\lim }\limits_{n \to + \infty } \parallel x^n {{1 - } \mathord{\left/ {\vphantom {{1 - } n}} \right. \kern-0em} n}\) we prove the following bounds: γ?(x)?σ?(x)?Γ?(x)?+(x)?σ+(x)?γ+(x), Γ(±)(x)=(2δ(±))?1 (ρ δ 2 )(±)?δ (±) 2 ?ρ 0 2 )(δ(±)≠0), γ(±)(x)= (±)ρδ(±)?δ(±), δ+?0, δ??0 ρ (±) δ = ρ(x+eδ(±)). We mention a case where equality is achieved, some corollaries,and discuss the sharpness of the bounds: for every ? > 0 there is a δ: ¦δ¦ ≥ρ 0 2 /2?, such that Δ: = ¦γ(±) x?Γ(±) x¦?ε and conversely, if the bounds are computed for some δ ≠ 0, then △ ≤ρ 0 2 /2 ¦δ¦. An example is considered. 相似文献
16.
I. I. Sharapudinov 《Mathematical Notes》2013,94(1-2):281-293
We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x 2)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S n ?1 (f) = S n ?1 (f, x) is the projection onto the subspace of algebraic polynomials p n = p n (x) of degree at most n, i.e., S n (p n ) = p n ; in addition, S n ?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S n ?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ n (x) of the partial sums S n ?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ . 相似文献
17.
In this paper we establish existence of solutions of singular boundary value problem ?(p(x)y ′(x))′=q(x)f(x,y,py′) for 0<x≤b and $\lim_{x\rightarrow0^{+}}p(x)y^{\prime}(x)=0$ , α 1 y(b)+β 1 p(b)y ′(b)=γ 1 with p(0)=0 and q(x) is allowed to have integrable discontinuity at x=0. So the problem may be doubly singular. Here we consider $\lim_{x\rightarrow0^{+}}\frac{q(x)}{p'(x)}\neq0$ therefore $\lim_{x\rightarrow0^{+}}p(x)y'(x)=0$ does not imply y′(0)=0 unless $\lim_{x\rightarrow0^{+}}f(x,y(x),p(x)y'(x))=0$ . 相似文献
18.
19.
S. A. Iskhokov 《Differential Equations》2008,44(2):241-255
Let Ω be an arbitrary open set in R n , and let σ(x) and g i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W p r (Ω, σ, $ \vec g $ ), where r ∈ N, p ≥ 1, and $ \vec g $ (x) = (g 1(x), g 2(x), ..., g n (x)), with the norm $$ \left\| {u;W_p^r (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\left\| {u;L_{p,r}^r (\Omega ;\sigma ,\vec g)} \right\|^p + \left\| {u;L_{p,r}^0 (\Omega ;\sigma ,\vec g)} \right\|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where $$ \left\| {u;L_{p,r}^m (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\sum\limits_{\left| k \right| = m} {\int\limits_\Omega {(\sigma (x)g_1^{k_1 - r} (x)g_2^{k_2 - r} (x) \cdots g_n^{k_n - r} (x)\left| {u^{(k)} (x)} \right|)^p dx} } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W p r (Ω, σ, $ \vec g $ ) for the functional $$ \Phi (u) = \sum\limits_{\left| k \right| \leqslant r} {\frac{1} {{p_k }}\int\limits_\Omega {a_k (x)} \left| {u^{(k)} (x)} \right|^{p_k } } dx - \left\langle {F,u} \right\rangle , $$ where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation. 相似文献
20.
Let R be a prime ring with extended centroid F and let δ be an F-algebraic continuous derivation of R with the associated inner derivation ad(b). Factorize the minimal polynomial μ(λ) of b over F into distinct irreducible factors ${\mu(\lambda)=\prod_i\pi_i(\lambda)^{n_i}}$ . Set ? to be the maximum of n i . Let ${R^{(\delta)}{\mathop{=}\limits^{{\rm def.}}}\{x\in R\mid \delta(x)=0\}}$ be the subring of constants of δ on R. Denote the prime radical of a ring A by ${{\mathcal{P}}(A)}$ . It is shown among other things that $${\mathcal{P}}(R^{(\delta)})^{2^\ell-1}=0\quad\text{and}\quad{\mathcal{P}}(R^{(\delta)})=R^{(\delta)}\cap {\mathcal{P}}(C_R(b))$$ . 相似文献