Loading [MathJax]/jax/output/SVG/jax.js
Research article

On an identity involving generalized derivations and Lie ideals of prime rings

  • Received: 28 December 2019 Accepted: 31 March 2020 Published: 08 April 2020
  • MSC : 16W25, 16N60, 16R50

  • Let R be a prime ring, U the Utumi quotient ring of R, C the extended centroid of R and L a noncentral Lie ideal of R. If Radmits a generalized derivation F associated with a derivation δ of R such that for some fixed integers m,n1, F([u,v])m=[u,v]n for all u,vL, then one of the following holds true: (ⅰ) R satisfies s4, the standard identity in four variables. (ⅱ) there exists λC such that F(x)=λx for all xR. Moreover, if n=1, then λm=1 and if n>1, then F=0.

    Citation: Gurninder Singh Sandhu. On an identity involving generalized derivations and Lie ideals of prime rings[J]. AIMS Mathematics, 2020, 5(4): 3472-3479. doi: 10.3934/math.2020225

    Related Papers:

    [1] Zhongzi Zhao, Meng Yan . Positive radial solutions for the problem with Minkowski-curvature operator on an exterior domain. AIMS Mathematics, 2023, 8(9): 20654-20664. doi: 10.3934/math.20231052
    [2] Wenjia Li, Guanglan Wang, Guoliang Li . The local boundary estimate of weak solutions to fractional p-Laplace equations. AIMS Mathematics, 2025, 10(4): 8002-8021. doi: 10.3934/math.2025367
    [3] Sobajima Motohiro, Wakasugi Yuta . Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain. AIMS Mathematics, 2017, 2(1): 1-15. doi: 10.3934/Math.2017.1.1
    [4] Limin Guo, Jiafa Xu, Donal O'Regan . Positive radial solutions for a boundary value problem associated to a system of elliptic equations with semipositone nonlinearities. AIMS Mathematics, 2023, 8(1): 1072-1089. doi: 10.3934/math.2023053
    [5] Keqiang Li, Shangjiu Wang . Symmetry of positive solutions of a p-Laplace equation with convex nonlinearites. AIMS Mathematics, 2023, 8(6): 13425-13431. doi: 10.3934/math.2023680
    [6] Zhanbing Bai, Wen Lian, Yongfang Wei, Sujing Sun . Solvability for some fourth order two-point boundary value problems. AIMS Mathematics, 2020, 5(5): 4983-4994. doi: 10.3934/math.2020319
    [7] Lin Zhao . Monotonicity and symmetry of positive solution for 1-Laplace equation. AIMS Mathematics, 2021, 6(6): 6255-6277. doi: 10.3934/math.2021367
    [8] Manal Alfulaij, Mohamed Jleli, Bessem Samet . A hyperbolic polyharmonic system in an exterior domain. AIMS Mathematics, 2025, 10(2): 2634-2651. doi: 10.3934/math.2025123
    [9] Maria Alessandra Ragusa, Abdolrahman Razani, Farzaneh Safari . Existence of positive radial solutions for a problem involving the weighted Heisenberg p()-Laplacian operator. AIMS Mathematics, 2023, 8(1): 404-422. doi: 10.3934/math.2023019
    [10] Zhiqian He, Liangying Miao . Multiplicity of positive radial solutions for systems with mean curvature operator in Minkowski space. AIMS Mathematics, 2021, 6(6): 6171-6179. doi: 10.3934/math.2021362
  • Let R be a prime ring, U the Utumi quotient ring of R, C the extended centroid of R and L a noncentral Lie ideal of R. If Radmits a generalized derivation F associated with a derivation δ of R such that for some fixed integers m,n1, F([u,v])m=[u,v]n for all u,vL, then one of the following holds true: (ⅰ) R satisfies s4, the standard identity in four variables. (ⅱ) there exists λC such that F(x)=λx for all xR. Moreover, if n=1, then λm=1 and if n>1, then F=0.


    Boundary value problems with p-Laplace operator Δpu=div(|u|p2u) arise in many different areas of applied mathematics and physics, such as non-Newtonian fluids, reaction-diffusion problems, non-linear elasticity, etc. But little is known about the p-Laplace operator cases (p2) compared to the vast amount of knowledge for the Laplace operator (p=2). In this paper, we discuss the existence of positive radial solution for the p-Laplace boundary value problem (BVP)

    {Δpu=K(|x|)f(u),xΩ,un=0,xΩ,lim|x|u(x)=0, (1.1)

    in the exterior domain Ω={xRN:|x|>r0}, where N2, r0>0, 1<p<N, un is the outward normal derivative of u on Ω, K:[r0,)R+ is a coefficient function, f:R+R is a nonlinear function. Throughout this paper, we assume that the following conditions hold:

    (A1) KC([r0,),R+) and 0<r0rN1K(r)dr<;

    (A2) fC(R+,R+);

    For the special case of p=2, namely the Laplace boundary value problem

    {Δu=K(|x|)f(u),xΩ,un=0,xΩ,lim|x|u(x)=0, (1.2)

    the existence of positive radial solutions has been discussed by many authors, see [1,2,3,4,5,6,7]. The authors of references[1,2,3,4,5,6] obtained some existence results by using upper and lower solutions method, priori estimates technique and fixed point index theory. In [7], the present author built an eigenvalue criteria of existing positive radial solutions. The eigenvalue criterion is related to the principle eigenvalue λ1 of the corresponding radially symmetric Laplace eigenvalue problem (EVP)

    {Δu=λK(|x|)u,xΩ,un=0,xΩ,u=u(|x|),lim|x|u(|x|)=0. (1.3)

    Specifically, if f satisfies one of the following eigenvalue conditions:

    (H1) f0<λ1, f>λ1;

    (H2) f<λ1, f0>λ1,

    the BVP(1.2) has a classical positive radial solution, where

    f0=lim infu0+f(u)u,f0=lim supu0+f(u)u,f=lim infuf(u)u,f=lim supuf(u)u.

    See [7,Theorem 1.1]. This criterion first appeared in a boundary value problem of second-order ordinary differential equations, and built by Zhaoli Liu and Fuyi Li in [8]. Then it was extended to general boundary value problems of ordinary differential equations, See [9,10]. In [11,12], the radially symmetric solutions of the more general Hessian equations are discussed.

    The purpose of this paper is to establish a similar existence result of positive radial solution of BVP (1.1). Our results are related to the principle eigenvalue λp,1 of the radially symmetric p-Laplce eigenvalue problem (EVP)

    {Δpu=λK(|x|)|u|p2u,xΩ,un=0,xΩ,u=u(|x|),lim|x|u(|x|)=0. (1.4)

    Different from EVP (1.3), EVP (1.4) is a nonlinear eigenvalue problem, and the spectral theory of linear operators is not applicable to it. In Section 2 we will prove that EVP (1.4) has a minimum positive real eigenvalue λp,1, see Lemma 2.3. For BVP (1.1), we conjecture that eigenvalue criteria is valid if f0, f0, f and f is replaced respectively by

    fp0=lim infu0+f(u)up1,fp0=lim supu0+f(u)up1,fp=lim infuf(u)up1,fp=lim supuf(u)up1. (1.5)

    But now we can only prove a weaker version of it: In second inequality of (H1) and (H2), λp,1 needs to be replaced by the larger number

    B=[10Ψ(1stp1a(t)dt)ds](p1), (1.6)

    where aC+(0,1] is given by (2.4) and ΨC(R) is given by (2.7). Our result is as follows:

    Theorem 1.1. Suppose that Assumptions (A1) and (A2) hold. If the nonlinear function f satisfies one of the the following conditions:

    (H1) fp0<λp,1, fp>B;

    (H2) fp<λp,1, fp0>B,

    then BVP (1.1) has at least one classical positive radial solution.

    As an example of the application of Theorem 1.1, we consider the following p-Laplace boundary value problem

    {Δpu=K(|x|)|u|γ,xΩ,un=0,xΩ,lim|x|u(x)=0. (1.7)

    Corresponding to BVP (1.1), f(u)=|u|γ. If γ>p1, by (1.5) fp0=0, fp=+, and (H1) holds. If 0<γ<p1, then fp=0, fp0=+, and (H2) holds. Hence, by Theorem 1.1 we have

    Corollary 1. Let K:[r0,)R+ satisfy Assumption (A1), γ>0 and γp1. Then BVP (1.7) has a positive radial solution.

    The proof of Theorem 1.1 is based on the fixed point index theory in cones, which will be given in Section 3. Some preliminaries to discuss BVP (1.1) are presented in Section 2.

    For the radially symmetric solution u=u(|x|) of BVP (1.1), setting r=|x|, since

    Δpu=div(|u|p2u)=(|u(r)|p2u(r))N1r|u(r)|p2u(r),

    BVP (1.1) becomes the ordinary differential equation BVP in [r0,)

    {(|u(r)|p2u(r))N1r|u(r)|p2u(r)=K(r)f(u(r)),r[r0,),u(r0)=0,u()=0, (2.1)

    where u()=limru(r).

    Let q>1 be the constant satisfying 1p+1q=1. To solve BVP (2.1), make the variable transformations

    t=(r0r)(q1)(Np),r=r0t1/(q1)(Np),v(t)=u(r(t)), (2.2)

    Then BVP (2.1) is converted to the ordinary differential equation BVP in (0,1]

    {(|v(t)|p2v(t))=a(t)f(v(t)),t(0,1],v(0)=0,v(1)=0, (2.3)

    where

    a(t)=rq(N1)(t)(q1)p(Np)pr0q(Np)K(r(t)),t(0,1]. (2.4)

    BVP (2.3) is a quasilinear ordinary differential equation boundary value problem with singularity at t=0. A solution v of BVP (2.3) means that vC1[0,1] such that |v|p2vC1(0,1] and it satisfies the Eq (2.3). Clearly, if v is a solution of BVP (2.3), then u(r)=v(t(r)) is a solution of BVP (2.1) and u(|x|) is a classical radial solution of BVP (1.1). We discuss BVP (2.3) to obtain positive radial solutions of BVP (1.1).

    Let I=[0,1] and R+=[0,+). Let C(I) denote the Banach space of all continuous function v(t) on I with norm vC=maxtI|v(t)|, C1(I) denote the Banach space of all continuous differentiable function on I. Let C+(I) be the cone of all nonnegative functions in C(I).

    To discuss BVP (2.3), we first consider the corresponding simple boundary value problem

    {(|v(t)|p2v(t))=a(t)h(t),t(0,1],v(0)=0,v(1)=0, (2.5)

    where hC+(I) is a given function. Let

    Φ(v)=|v|p2v=|v|p1sgnv,vR, (2.6)

    then w=Φ(v) is a strictly monotone increasing continuous function on R and its inverse function

    Φ1(w):=Ψ(w)=|w|q1sgnw,wR, (2.7)

    is also a strictly monotone increasing continuous function.

    Lemma 2.1. For every hC(I), BVP (2.5) has a unique solution v:=ShC1(I). Moreover, the solution operator S:C(I)C(I) is completely continuous and has the homogeneity

    S(νh)=νq1Sh,hC(I),ν0. (2.8)

    Proof. By (2.4) and Assumption (A1), the coefficient a(t)C+(0,1] and satisfies

    10a(t)dt=1[(q1)(Np)]p1r0Npr0rN1K(r)dr<. (2.9)

    Hence aL(I).

    For every hC(I), we verify that

    v(t)=t0Ψ(1sa(τ)h(τ)dτ)ds:=Sh(t),tI (2.10)

    is a unique solution of BVP (2.5). Since the function G(s):=1sa(τ)h(τ)dτC(I), from (2.10) it follows that vC1(I) and

    v(t)=Ψ(1ta(τ)h(τ)dτ),tI. (2.11)

    Hence,

    |v(t)|p2v(t)=Φ(v(t))=1ta(τ)h(τ)dτ,tI.

    This means that (|v(t)|p2v(t)C1(0,1] and

    (|v(t)|p2v(t))=a(t)h(t),t(0,1],

    that is, v is a solution of BVP (2.5).

    Conversely, if v is a solution of BVP (2.5), by the definition of the solution of BVP (2.5), it is easy to show that v can be expressed by (2.10). Hence, BVP (2.5) has a unique solution v=Sh.

    By (2.10) and the continuity of Ψ, the solution operator S:C(I)C(I) is continuous. Let DC(I) be bounded. By (2.10) and (2.11) we can show that S(D) and its derivative set {v|vS(D)} are bounded sets in C(I). By the Ascoli-Arzéla theorem, S(D) is a precompact subset of C(I). Thus, S:C(I)C(I) is completely continuous.

    By the uniqueness of solution of BVP (2.5), we easily verify that the solution operator S satisfies (2.8).

    Lemma 2.2. If hC+(I), then the solution v=Sh of LBVP (2.5) satisfies: vc=v(1), v(t)tvC for every tI.

    Proof. Let hC+(I) and v=Sh. By (2.10) and (2.11), for every tI v(t)0 and v(t)0. Hence, v(t) is a nonnegative monotone increasing function and vC=maxtIv(t)=v(1). From (2.11) and the monotonicity of Ψ, we notice that v(t) is a monotone decreasing function on I. For every t(0,1), by Lagrange's mean value theorem, there exist ξ1(0,t) and ξ2(t,1), such that

    (1t)v(t)=(1t)(v(t)v(0))=v(ξ1)t(1t)v(t)t(1t),tv(t)=tv(1)t(v(1)v(t))=tv(1)tv(ξ2)(1t)tv(1)v(t)t(1t).

    Hence

    v(t)=tv(t)+(1t)v(t)tv(1)=tvC.

    Obviously, when t=0 or 1, this inequality also holds. The proof is completed.

    Consider the radially symmetric p-Laplace eigenvalue problem EVP (1.3). We have

    Lemma 2.3. EVP (1.4) has a minimum positive real eigenvalue λp,1, and λp,1 has a radially symmetric positive eigenfunction.

    Proof. For the radially symmetric eigenvalue problem EVP (1.4), writing r=|x| and making the variable transformations of (2.2), it is converted to the one-dimensional weighted p-Laplace eigenvalue problem (EVP)

    {(|v(t)|p2v(t))=λa(t)|v(t)|p2v(t),t(0,1],v(0)=0,v(1)=0, (2.12)

    where v(t)=u(r(t)). Clearly, λR is an eigenvalue of EVP (1.4) if and only if it is an eigenvalue of EVP (2.12). By (2.4) and (2.9), aC+(0,1]L(I) and 10a(s)ds>0. This guarantees that EVP (2.12) has a minimum positive real eigenvalue λp,1, which given by

    λp,1=inf{10|w(t)|pdt10a(t)wp(t)dt|wC1(I),w(0)=0,w(1)=0,10a(t)wp(t)dt0}. (2.13)

    Moreover, λp,1 is simple and has a positive eigenfunction ϕC+(I)C1(I). See [13, Theorem 5], [14, Theorem 1.1] or [15, Theorem 1.2]. Hence, λp,1 is also the minimum positive real eigenvalue of EVP (1.4), and ϕ((r0/|x|)(q1)(Np)) is corresponding positive eigenfunction.

    Now we consider BVP (2.3). Define a closed convex cone K of C(I) by

    K={vC(I)|v(t)tvC,tI}. (2.14)

    By Lemma 2.2, S(C+(I))K. Let fC(R+,R+), and define a mapping F:KC+(I) by

    F(v)(t):=f(v(t)),tI. (2.15)

    Then F:KC+(I) is continuous and it maps every bounded subset of K into a bounded subset of C+(I). Define the composite mapping by

    A=SF. (2.16)

    Then A:KK is completely continuous by the complete continuity of the operator S:C+(I)K. By the definitions of S and K, the positive solution of BVP (2.3) is equivalent to the nonzero fixed point of A.

    Let E be a Banach space and KE be a closed convex cone in E. Assume D is a bounded open subset of E with boundary D, and KD. Let A:K¯DK be a completely continuous mapping. If Avv for every vKD, then the fixed point index i(A,KD,K) is well defined. One important fact is that if i(A,KD,K)0, then A has a fixed point in KD. In next section, we will use the following two lemmas in [16,17] to find the nonzero fixed point of the mapping A defined by (2.16).

    Lemma 2.4. Let D be a bounded open subset of E with 0D, and A:K¯DK a completely continuous mapping. If μAvv for every vKD and 0<μ1, then i(A,KD,K)=1.

    Lemma 2.5. Let D be a bounded open subset of E with 0D, and A:K¯DK a completely continuous mapping. If Avv and Avv for every vKD, then i(A,KD,K)=0.

    Proof of Theorem 1.1. We only consider the case that (H1)* holds, and the case that (H2)* holds can be proved by a similar way.

    Let KC(I) be the closed convex cone defined by (2.14) and A:KK be the completely continuous mapping defined by (2.16). If vK is a nontrivial fixed point of A, then by the definitions of S and A, v(t) is a positive solution of BVP (2.3) and u=v(r0N2/|x|N2) is a classical positive radial solution of BVP (1.1). Let 0<R1<R2<+ and set

    D1={vC(I):vC<R1},D2={vC(I):vC<R2}. (3.1)

    We prove that A has a fixed point in K(¯D2D1) when R1 is small enough and R2 large enough.

    Since fp0<λp,1, by the definition of fp0, there exist ε(0,λp,1) and δ>0, such that

    f(u)(λp,1ε)up1,0uδ. (3.2)

    Choosing R1(0,δ), we prove that A satisfies the condition of Lemma 2.4 in KD1, namely

    μAvv,vKD1,0<μ1. (3.3)

    In fact, if (3.3) does not hold, there exist v0KD1 and 0<μ01 such that μ0Av0=v0. By the homogeneity of S, v0=μ0S(F(v0))=S(μ0p1F(v0)). By the definition of S, v0 is the unique solution of BVP (2.5) for h=μ0p1F(v0)C+(I). Hence, v0C1(I) satisfies the differential equation

    {(|v0(t)|p2v0(t))=μ0p1a(t)f(v0(t)),t(0,1],v0(0)=0,v0(1)=0. (3.4)

    Since v0KD1, by the definitions of K and D1,

    0v0(t)v0C=R1<δ,tI.

    Hence by (3.2),

    f(v0(t))(λp,1ε)v0p1(t),tI.

    By this inequality and Eq (3.4), we have

    (|v0(t)|p2v0(t))μ0p1(λp,1ε)a(t)v0p1(t),t(0,1].

    Multiplying this inequality by v0(t) and integrating on (0,1], then using integration by parts for the left side, we have

    10|v0(t)|pdtμ0p1(λp,1ε)10a(t)v0p(t)dt(λp,1ε)10a(t)v0p(t)dt. (3.5)

    Since v0KD, by the definition of K,

    10a(t)v0p(t)dtv0Cp10tpa(t)dt=R1p10tpa(t)dt>0.

    Hence, by (2.13) and (3.5) we obtain that

    λp,110|v0(t)|pdt10a(t)v0p(t)dtλp,1ε,

    which is a contradiction. This means that (3.3) holds, namely A satisfies the condition of Lemma 2.4 in KD1. By Lemma 2.4, we have

    i(A,KD1,K)=1. (3.6)

    On the other hand, by the definition (1.6) of B, we have

    B<[1σΨ(1stp1a(t)dt)ds](p1)B(σ0+),σ(0,1). (3.7)

    Since fp>B, by (3.7) there exists σ0(0,1), such that

    B0:=[1σ0Ψ(1stp1a(t)dt)ds](p1)<fp. (3.8)

    By this inequality and the definition of fp, there exists H>0 such that

    f(u)>B0up1,u>H. (3.9)

    Choosing R2>max{δ,H/σ0}, we show that

    AvCvC,vKD2. (3.10)

    For vKD2 and t[σ0,1], by the definitions of K and D2

    v(t)tvCσ0R2>H.

    By this inequality and (3.9),

    f(v(t))>B0vp1(t)B0vp1Ctp1,t[σ0,1]. (3.11)

    Since Av=S(F(v)), by the expression (2.10) of the solution operator S and (3.11), noticing (p1)(q1)=1, we have

    AvCAv(1)=10Ψ(1sa(t)f(v(t))dt)ds1σ0Ψ(1sa(t)f(v(t))dt)ds1σ0Ψ(1sa(t)B0vp1Ctp1dt)ds=Bq10vC1σ0Ψ(1stp1a(t)dt)ds=vC.

    Namely, (3.10) holds. Suppose A has no fixed point on D2. Then by (3.10), A satisfies the condition of Lemma 2.5 in KD2. By Lemma 2.5, we have

    i(A,KD2,K)=0. (3.12)

    By the additivity of fixed point index, (3.6) and (3.11), we have

    i(A,K(D2¯D1),K)=i(A,KD2,K)i(A,KD1,K)=1.

    Hence A has a fixed point in K(D2¯D1).

    The proof of Theorem 1.1 is complete.

    The authors would like to express sincere thanks to the reviewers for their helpful comments and suggestions. This research was supported by National Natural Science Foundations of China (No.12061062, 11661071).

    The authors declare that they have no competing interests.



    [1] M. Ashraf, N. Rehman, On commutativity of rings with derivations, Result. Math., 42 (2002), 3-8. doi: 10.1007/BF03323547
    [2] K. I. Beidar, Rings with generalized identities III, Moscow Univ. Math. Bull., 33 (1978), 53-58.
    [3] K. I. Beidar, W. S. Martindale III, A. V. Mikhalev, Rings with generalized identities, Pure and Applied Math., 196, New York: Marcel Dekker Inc., 1996.
    [4] J. Bergen, I. N. Herstein, J. W. Kerr, Lie ideals and derivations of prime rings, J. Algebra, 71 (1981), 259-267. doi: 10.1016/0021-8693(81)90120-4
    [5] C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103 (1988), 723-728. doi: 10.1090/S0002-9939-1988-0947646-4
    [6] M. N. Daif, H. E. Bell, Remarks on derivations on semiprime rings, International Journal of Mathematics and Mathematical Sciences, 15 (1992), 205-206. doi: 10.1155/S0161171292000255
    [7] T. S. Erickson, W. S. Martindale III, J. M. Osborn, Prime nonassociative algebras, Pacific J. Math., 60 (1975), 49-63. doi: 10.2140/pjm.1975.60.49
    [8] V. De Filippis, S. Huang, Generalized derivations on semiprime rings, Bull. Korean Math. Soc., 48 (2011), 1253-1259. doi: 10.4134/BKMS.2011.48.6.1253
    [9] S. Huang, B. Davvaz, Generalized derivations of rings and Banach algebras, Commun. Algebra, 41 (2013), 1188-1194. doi: 10.1080/00927872.2011.642043
    [10] S. Huang, N. Rehman, Generalized derivations in prime and semiprime rings, Bol. Soc. Paran. Mat., 34 (2016), 29-34. doi: 10.5269/bspm.v34i2.21774
    [11] N. Jacobson, Structure of rings, Amer. Math. Soc., Providence, RI, 1964.
    [12] V. K. Kharchenko, Differential identities of prime rings, Algebra Logic, 17 (1978), 155-168. doi: 10.1007/BF01670115
    [13] T. K. Lee, Generalized derivations of left faithful rings, Commun. Algebra, 27 (1999), 4057-4073. doi: 10.1080/00927879908826682
    [14] C. Lanski, S. Montgomery, Lie structure of prime rings of characteristic 2, Pacific J. Math., 42 (1972), 117-136. doi: 10.2140/pjm.1972.42.117
    [15] C. Lanski, An Engel condition with derivation, Proc. AMer. Math. Soc., 118 (1993), 731-734. doi: 10.1090/S0002-9939-1993-1132851-9
    [16] W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), 576-584. doi: 10.1016/0021-8693(69)90029-5
    [17] M. A. Quadri, M. S. Khan, N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math., 34 (2003), 1393-1396.
    [18] T. L. Wong, Derivations with power values on multilinear polynomials, Algebra Colloq., 3 (1996), 369-378.
  • This article has been cited by:

    1. Bo Yang, Radially Symmetric Positive Solutions of the Dirichlet Problem for the p-Laplace Equation, 2024, 12, 2227-7390, 2351, 10.3390/math12152351
    2. Yongxiang Li, Pengbo Li, Radial solutions of p-Laplace equations with nonlinear gradient terms on exterior domains, 2023, 2023, 1029-242X, 10.1186/s13660-023-03069-y
    3. 旭莹 唐, The Existence of Positive Solutions to Quasilinear Differential Equation on Infinite Intervals, 2023, 13, 2160-7583, 2103, 10.12677/PM.2023.137217
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4102) PDF downloads(338) Cited by(0)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog