关于亚直接不可约完全左对称Г-环

A left ideal L of a Г-ring M is called left symmetric if aabβb∈L implies aacβb∈L, where a,b, c∈ M, a ,β∈Γ. A Γ-ring M is called to be a Γ-division ring if it has the left unity, and if left oprator ring R is a division ring.In this paper, the following results are pvoved:Suppose that every left ideal of a Γ-ring M is left symmetiic.If A is subdirectly irreducible with more than one element and with no nonzero nilpotents then it is a Γ-division ring.If M is subdirectly irreducible and if D≠M then M/D is a Γ-division ring where D=(a∈M|aΓb=0,0≠b∈M).     

关于Γ-环的稠密性定理

Let U be an Γ-ring and M be an irreducible UΓ-moduie, for α∈U, α∈Γ, we define T_[a,a]: M→M by m T_[a,a] = maa for all m∈M. Let End(M) be the ring of all endomorphisms of the additive group of M. We define as usual End_[Γ,U](M)=={ψ∈End(M)|T_Σ[ai,ai]ψ=ψT_Σ[ai,ai] all a_i∈U,a_i∈Γ}. In this paper the following results are obtained.     

Boundedness and convergence for the non-Liénard type differential equation

In this article, the author studies the boundedness and convergence for the {(x) = a(y) - f(x),(y) = b(y)β(x) - g(x) e(t),where a(y), b(y), f(x),g(x),β(x) are real continuous functions in y ∈ R or x ∈ R,β(x) ≥ 0 for all x and e(t) is a real continuous function on R = {t: t ≥ 0} such that the equation has a unique solution for the initial value problem. The necessary and sufficient conditions are obtained and some of the results in the literatures are improved and extended.     

Operator Equations and Duality Mappings in Sobolev Spaces with Variable Exponents

After studying in a previous work the smoothness of the space UΓ0={u∈W1,p(·)(Ω);u=0 on Γ0 Γ=Ω},where dΓ-measΓ0>0,with p(·)∈C(Ω)and p(x)>1 for all x∈Ω,the authors study in this paper the strict and uniform convexity as well as some special properties of duality mappings defined on the same space.The results obtained in this direction are used for proving existence results for operator equations having the form Ju=Nfu,where J is a duality mapping on UΓ0 corresponding to the gauge function,and Nf is the Nemytskij operator generated by a Carath′eodory function f satisfying an appropriate growth condition ensuring that Nf may be viewed as acting from UΓ0 into its dual.     

BOUNDEDNESS AND CONVERGENCE FOR THE NON-LINARD TYPE DIFFERENTIAL EQUATION

In this article, the author studies the boundedness and convergence for the non-Lienard type differential equation (x|·)=a(y)-f(x) (y|·)=b(y)β(x)-g(x) e(t) where a(y),b(y),f(x),g(x),β(x) are real continuous functions in y∈R or x∈R,β(x)≥0 for all x and e(t) is a real continuous function on R = {t: t≥0} such that the equation has a unique solution for the initial value problem. The necessary and sufficient conditions are obtained and some of the results in the literatures are improved and extended.     

THE GENERALIZED SMASH PRODUCT AND COPRODUCT

91. PreliminariesThroughout the paper all spaces are over a fixed ground field K. If C is a coalgebra,then we always denote the comultiplication and counit by b and E respectively. Set b(c) Z of @ cd for c E C. If M is a right (left) C -comodule, then we use p for the structure mapof M, and set p(m) = ac(o) @ m(1) (p(m) ~ ac(--1) @ m(o)) for m E M. Let H be abialgebra. Denote left H-module category by HM and left H-comodule category by HM.If M and N are in HM, then M @ N is also a l…     

Existence of multiple solutions for a fractional p-Laplacian system with concave-convex term

In this article, we study the existence of multiple solutions for the following system driven by a nonlocal integro-differential operator with zero Dirichlet boundary conditions{(-?)_p~su = a(x)|u|~(q-2) u +2α/α + βc(x)|u|~(α-2) u|v|~β, in ?,(-?)_p~sv = b(x)|v|~(q-2) v +2β/α + βc(x)|u|α|v|~(β-2) v, in ?,u = v = 0, in Rn\?,(0.1) where Ω is a smooth bounded domain in Rn, n ps with s ∈(0,1) fixed, a(x), b(x), c(x) ≥ 0 and a(x),b(x),c(x) ∈L∞(Ω), 1 q p and α,β 1 satisfy pα + βp*,p* =np/n-ps.By Nehari manifold and fibering maps with proper conditions, we obtain the multiplicity of solutions to problem(0.1).?????     

新题征展(52)

A　题组新编1.(1)满足条件 { 1,2 } M { 1,2 ,3,4 ,5 }的集合 M共有个 ;(2 )满足条件 M∪ { a,b,c} ={ a,b,c,d,e}的集合 M共有个 ;(3) M { 1,2 ,3,4 ,5 } ,且满足条件 :若 a∈ M,则 6 - a∈ M,这样的非空集合 M共有个 ;(4 ) A∪ B ={ a,b}的集合 A、B共有对 ;(5 ) A∪ B ={ a,b,c}的集合 A、B共有对 .2 .(1)若 f (x) =x1 x,则 f(1) f(2 ) f(3) … f(2 0 0 4 ) f(12 ) f(13) f(14 ) … f(12 0 0 4 ) =;(2 )若 f(x) =x21 x2 ,则 f (1) f(2 ) f(3) … f(2 0 0 4 ) f(12 ) f(13) f(14 ) … f(12 0 0 4 ) =;(3)若 f(x…     

新题征展(52)

A 题组新编 1.(1)满足条件{1,2}M{1,2,3,4,5}的集合M共有个;(2)满足条件M∪{a,b,c}={a,b,c,d,e}的集合M共有个;(3)M{1,2,3,4,5},且满足条件若a∈M,则6-a∈M,这样的非空集合M共有个;(4)A∪B {a,b}的集合A、B共有对;(5)A∪B={a,b,c}的集合A、B共有对.     

Fixed Point Theorem of {a,b,c} Contraction and Nonexpansive Type Mappings in Weakly Cauchy Normed Spaces

Let C be a closed convex weakly Cauchy subset of a normed space X. Then we define a new {a,b,c} type nonexpansive and {a,b,c} type contraction mapping T from C into C. These types of mappings will be denoted respectively by {a,b,c}-ntype and {a,b,c}-ctype. We proved the following： 1. If T is {a,b,c}-ntype mapping, then inf{ ｜｜ T（x）-x｜｜ ：x C C} =0, accordingly T has a unique fixed point. Moreover, any sequence {Xn}n∈NN in C with limn→∞｜｜T（xn） - Xn｜｜ = 0 has a subsequence strongly convergent to the unique fixed point of T. 2. If T is {a,b,c}-ctype mapping, then T has a unique fixed point. Moreover, for any x∈C the sequence of iterates {Tn （x）}n∈N has subsequence strongly convergent to the unique fixed point of T. This paper extends and generalizes some of the results given in [2,4, 7] and [13].     

全纯函数族的正规定则

Let f be a holomorphic function on a domain D Lontaiu in C, and let a be a finite complex number. We denote by -↑Ef/(a) = {z ∈ D : f(z) = a, ignoring multiplicity} the set of all distinct α-points of f. Let F be a family of holomorphic functions on D. If there exist three finite values a, b(≠ O,α) and c(≠ O) such that for every f ∈F, -↑Ef′/(0) Lontain in -↑Ef(α） and -↑Ef′ (b) Lontain in -↑Ef(c), then F is a normal family on D.     

左模的张量积及其线性映射

引言 设K={O,1_k,a,b,c,…}为有单位元1_k的可换环,R={O,|σ∈∑}、S={O,1_s,S_r|τ∈Γ}分别为有单位元1_R、1_s的K环,当然1_k1_R=1_R=1_R1_k,1_k1_s=1_s=1_s1_k,下面在不至于混淆的情况下,1_k、1_R、1_s均用1表示。M={x_λ|λ∈∧}、M＇={u_i|i∈I}为左R酉模,N={y_u|μ∈Ω}、N＇={U_i|i∈J}为左S酉模。我们用H_R(M,M＇)表示R模M到R模M＇的所有R同态所形成的可换群。文[1]将R模M与S模N的张量积定义为一个左R S模,本文就在此基础上讨论M N作为R S模的一些性质及其线性映射。如果不特别声明,本文中所有的环都有单位元,所有的模都指左酉模。     

Some Normality Criteria for Families of Meromorphic Functions

Let κ be a positive integer and F be a family of meromorphic functions in a domain D such that for each f ∈ F, all poles of f are of multiplicity at least 2,and all zeros of f are of multiplicity at least κ + 1. Let α and b be two distinct finite complex numbers. If for each f ∈ F, all zeros of f~(κ)-α are of multiplicity at least 2,and for each pair of functions f, g ∈ F, f~(κ)and g~(κ) share b in D, then F is normal in D.     

On quasi-reduced quadratic forms

With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 ＋ bXY ＋ cY2 ： a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant △. As a matter of fact, we show that for each reduced quadratic form f = aX2 ＋ bXY ＋ cY2 = （a, b, c） of discriminant △〉0（and of sign σ（f） equal to the sign of a）, the quadratic forms associated with f and defined by {〈a＋bu＋cu2,b＋2cu.c〉},with 1≤σ（f）u≤b/2｜c｜ （whenever they exist）, 〈c,-b-2cu,a＋bu＋cu2〉 with b/2｜c｜≤σ（f）u≤[w（f）]=[b＋√△/2｜c｜], are all different from one another and build a set I（f） whose cardinality is #I（f）={1＋[ω（f）],when（2c）｜b,[ω（f）],when （2c）｜b. If f and g are two different reduced quadratic forms, we show that I（f） ∩ I（g） = Ф. Our main result is that the set Q△ is given by the disjoint union of all I（f） with f running through the set of reduced quadratic forms of discriminant △〉0. This allows us to deduce a formula for #（Q△） involving sums of partial quotients of certain continued fractions.     

Best Simultaneous Approximation for Dunham-Type by an N-Dimensional Subspace on a Set of N+1 Points

Let X={x_0,x_1…,x_n}and let c(X)be the set of all continuous real functions on X with the Chebyshev norm. Let G=span{g_1,g_2,…,g_n}be an n-dimensional subspace of c(X).Let T={(f~+,f~-):f~+≥f~-and f~+,f~-∈c(X)}.If there exists a P∈G such that max{||f~+-P||, ||f~--P||}=inf{max{||f~+-Q||, ||f~--Q||}:Q∈G},(1) then P is called a best simultaneous approximation to(f~+,f~-)from G.     

Short-time Asymptotics of the Heat Kernel on Bounded Domain with Piecewise Smooth Boundary Conditions and Its Applications to an Ideal Gas

The asymptotic expansion of the heat kernel Θ(t)=sum from ∞to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R~n(n=2 or 3) is studied for short-time t for a generalbounded domain Ωwith a smooth boundary (?)Ω.In this paper,we consider the case of a finite number of theDirichlet conditions φ=0 on Γ_i (i=1,...,J) and the Neumann conditions (?)=0 on Γ_i (i=J 1,...,k) andthe Robin conditions ((?) γ_i)φ=0 on Γ_i (i=k 1,...,m) where γ_i are piecewise smooth positive impedancefunctions,such that (?)Ωconsists of a finite number of piecewise smooth components Γ_i(i=1,...,m) where(?)Ω=(?)Γ_i.We construct the required asymptotics in the form of a power series over t.The senior coefficients inthis series are specified as functionals of the geometric shape of the domain Ω.This result is applied to calculatethe one-particle partition function of a“special ideal gas”,i.e.,the set of non-interacting particles set up in abox with Dirichlet,Neumann and Robin boundary conditions for the appropriate wave function.Calculationof the thermodynamic quantities for the ideal gas such as the internal energy,pressure and specific heat revealsthat these quantities alone are incapable of distinguishing between two different shapes of the domain.Thisconclusion seems to be intuitively clear because it is based on a limited information given by a one-particlepartition function;nevertheless,its formal theoretical motivation is of some interest.     

Periodic points and normal families concerning multiplicity

In 1992,Yang Lo posed the following problem:let F be a family of entire functions,let D be a domain in C,and let k 2 be a positive integer.If,for every f∈F,both f and its iteration f~khave no fixed points in D,is F normal in D?This problem was solved by Ess′en and Wu in 1998,and then solved for meromorphic functions by Chang and Fang in 2005.In this paper,we study the problem in which f and f~(k ) have fixed points.We give positive answers for holomorphic and meromorphic functions.(I)Let F be a family of holomorphic functions in a domain D and let k 2 be a positive integer.If,for each f∈F,all zeros of f(z)-z are multiple and f~khas at most k distinct fixed points in D,then F is normal in D.Examples show that the conditions"all zeros of f(z)-z are multiple"and"f~k having at most k distinct fixed points in D"are the best possible.(II)Let F be a family of meromorphic functions in a domain D,and let k 2 and l be two positive integers satisfying l 4 for k=2 and l 3 for k 3.If,for each f∈F,all zeros of f(z)-z have a multiplicity at least l and f~khas at most one fixed point in D,then F is normal in D.Examples show that the conditions"l 3for k 3"and"f~k having at most one fixed point in D"are the best possible.     

The Negative Spectrum of Schr?dinger Operators with Fractal Potentials

Let Γ?R~2 be a regular anisotropic fractal. We discuss the problem of the negative spectrum for the Schr?dinger operators associated with the formal expression H_β=id-?+βtr_b~Γ,β∈R,acting in the anisotropic Sobolev space W_2~(1,α)(R~2), where ? is the Dirichlet Laplanian in R~2 and tr_b~Γ is a fractal potential(distribution) supported by Γ.     

Generalized Inverse Analysis on the Domain Ω(A,A+) in B(E,F)

Let B(E,F) be the set of all bounded linear operators from a Banach space E into another Banach space F,B~+(E, F) the set of all double splitting operators in B(E, F)and GI(A) the set of generalized inverses of A ∈ B~+(E, F). In this paper we introduce an unbounded domain ?(A, A~+) in B(E, F) for A ∈ B~+(E, F) and A~+∈GI(A), and provide a necessary and sufficient condition for T ∈ ?(A, A~+). Then several conditions equivalent to the following property are proved: B = A+(IF+(T-A)A~+)~(-1) is the generalized inverse of T with R(B)=R(A~+) and N(B)=N(A~+), for T∈?(A, A~+), where IF is the identity on F. Also we obtain the smooth(C~∞) diffeomorphism M_A(A~+,T) from ?(A,A~+) onto itself with the fixed point A. Let S = {T ∈ ?(A, A~+) : R(T)∩ N(A~+) ={0}}, M(X) = {T ∈ B(E,F) : TN(X) ? R(X)} for X ∈ B(E,F)}, and F = {M(X) : ?X ∈B(E, F)}. Using the diffeomorphism M_A(A~+,T) we prove the following theorem: S is a smooth submanifold in B(E,F) and tangent to M(X) at any X ∈ S. The theorem expands the smooth integrability of F at A from a local neighborhoold at A to the global unbounded domain ?(A, A~+). It seems to be useful for developing global analysis and geomatrical method in differential equations.     

曲线族的模不等式

研究了曲线族的模 ,得到了 :1 )设Γ是 Rn 中连结不相交的曲线α1 与α2 的曲线族 ,若d(α1 ,α2 )≥ r,minj=1 ,2 dia(αj)≤ s,则 M(Γ )≤ 1 +srnΩn. 2 )设Γ是连结 Rn 中的闭连集 F1 与 F2 的曲线族 ,若minj=1 ,2 dia( Fj)≥ ad( F1 ,F2 ) ,则 M( Γ)≥ C( n,a) . 3 )设 R=R( C,C0 )是 R2 中的环 ,D表示 R2 \C中含 R的一个分支 ,α,β是 C上两条不相交的子曲线 .若 ΓR,ΓD 分别是 R与 D中连结 α和 β的曲线族 ,则 M( ΓR)≤ M( ΓD)≤φ( mod R) M(ΓR)