THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON RN WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

In this paper, we prove the existence of at least one positive solution pair (u, v) ∈ H 1 (R N ) × H 1 (R N ) to the following semilinear elliptic system{-u + u = f(x, v), x ∈RN ,-v + v = g(x,u), x ∈ R N ,(0.1) by using a linking theorem and the concentration-compactness principle. The main con-ditions we imposed on the nonnegative functions f, g ∈ C 0 (R N × R 1 ) are that, f (x, t) and g(x, t) are superlinear at t = 0 as well as at t = +∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245(2008), 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem{-u + u = f(x, u), x ∈Ω,u ∈H10(Ω)where ΩRN is bounded and a result of Li and Yang [G. Li and J. Yang: Communications in P.D.E. Vol. 29(2004) Nos.5 6.pp.925–954, 2004] concerning (0.1) when f and g are asymptotically linear.     

不定酉不变线性映射的刻戈及相关结果

Let A and B be unital C*-algebras, and let J ∈ A, L ∈ B be Hermitian invertible elements. For every T ∈ A and S ∈ B,define TJ(?)=J-1T*J and SL(?) =L-1S*L. Then in such a way we endow the C*-algebras A and B with indefinite structures. We characterize firstly the Jordan (J, L)-(?)-homomorphisms on C*-algebras. As applications, we further classify the bounded linear maps ?:A→B preserving (J, L)-unitary elements. When A = B(H) and B = B(K), where H and K are infinite dimensional and complete indefinite inner product spaces on real or complex fields, we prove that indefinite-unitary preserving bounded linear surjections are of the form T →UVTV-1((?)T ∈ B(H)) or T→UVT(?)V-1 ((?)T ∈ B(H)), where U ∈ B(K) is indefinite unitary and, V : H→K is generalized indefinite unitary in the first form and generalized indefinite anti-unitary in the second one. Some results on indefinite orthogonality preserving additive maps are also given.     

Inverse limits in representations of a restricted Lie algebra

Let(g,[p]) be a restricted Lie algebra over an algebraically closed field of characteristic p 0.Then the inverse limits of "higher" reduced enveloping algebras {uχs(g)|s∈N} with χ running over g* make representations of g split into different "blocks".In this paper,we study such an infinitedimensional algebra Aχ(g):= ■Uχs(g) for a given χ∈g*.A module category equivalence is built between subcategories of U(g)-mod and Aχ(g)-mod.In the case of reductive Lie algebras,(quasi) generalized baby Verma modules and their properties are described.Furthermore,the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized χ-reduced module category are precisely determined,and a higher reciprocity in the case of regular nilpotent is obtained,generalizing the ordinary reciprocity.     

Cotorsion Pairs and Stable Categories over Triangular Matrix Rings北大核心CSCD

In general, the properties of modules over a triangular matrix ring T = (0ABU) are studied via modules over diagonal “small rings” A and B. However, we use model structures on the category of T-modules to characterize the stable categories GP(A), GP(B) of Gorenstein projective modules over A and B. To this end, we introduce two subcategories of Gorenstein T-modules, and obtain two corresponding complete cotorsion pairs. Moreover, cotorsion pairs of modules are lifted to T-complexes, and the equivalences and recollements of homotopy categories of complexes are studied. © 2022 Chinese Academy of Sciences. All rights reserved.     

On modules with <Emphasis Type="Italic">d</Emphasis>-Koszul-type submodules

The so-called weakly d-Koszul-type module is introduced and it turns out that each weakly d-Koszul-type module contains a d-Koszul-type submodule. It is proved that, M ∈ W H J^d（A） if and only if M admits a filtration of submodules： 0 belong to U0 belong to U1 belong to ... belong to Up = M such that all Ui/Ui-1 are d-Koszul-type modules, from which we obtain that the finitistic dimension conjecture holds in W H J^d（A） in a special case. Let M ∈ W H J^d（A）. It is proved that the Koszul dual E（M） is Noetherian, Hopfian, of finite dimension in special cases, and E（M） ∈ gr0（E（A））. In particular, we show that M ∈ W H J^d（A） if and only if E（G（M）） ∈ gr0（E（A））, where G is the associated graded functor.     

POINCAR SERIES AND AN APPLICATION TO WEYL ALGEBRAS

Let A n be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over A n . By further exploring the relationship between the Poincar′e series and the dimension and the multiplicity of M , we are able to prove that the tensor product of two finitely generated modules over A n has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of A n .     

Poincaré series and an application to Weyl algebras

Let A n be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over A n . By further exploring the relationship between the Poincar′e series and the dimension and the multiplicity of M , we are able to prove that the tensor product of two finitely generated modules over A n has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of A n .     

The existence of nontrivial solutions to a semilinear elliptic system on without the ambrosetti-rabinowitz condition

In this paper, we prove the existence of at least one positive solution pair （u, v）∈ H1（RN） × H1（RN） to the following semilinear elliptic system {-△u＋u=f（x,v）,x∈RN,-△u＋u=g（x,v）,x∈RN （0.1）,by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈C0（RN× R1） are that, f（x, t） and g（x, t） are superlinear at t = 0 as well as at t =＋∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245（2008）, 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem {-△u＋u=f（x,u）,x∈Ω,u∈H0^1（Ω） where Ω ∩→RN is bounded and a result of Li and Yang [G. Li and J. Yang： Communications in P.D.E. Vol. 29（2004） Nos.5＆ 6.pp.925-954, 2004] concerning （0.1） when f and g are asymptotically linear.     

映射的横截性定理

Hirsch conjectured: M、N、A are differential manifolds, g∈C^∞(A,N), then the set T = {f∈C^∞(M,N)|f∈g} is dense in C∞(M,N)and open if g is proper.In this paper, we prove the transversality theorem of map in the Jet bundle.Theorem 1 Let M, N. A be differential manifolds, g∈C^∞(A,J^τ(M、N)), then the set T{f∈C^∞(M,N)|f^τf∈g } is residual in C^∞(M,N) and open if g is proper.Theorem 1 contains Thom's transversal ity theorem as a special case. We can obtain Hirsch's conjecture by using theorem 1.     

Preservation of quasi-isomorphisms of complexes

We consider the preservation property of the homomorphism and tensor product functors for quasi-isomorphisms and equivalences of complexes.Let X and Y be two classes of R-modules with Ext ≥1(X,Y) = 0 for each object X ∈X and each object Y ∈Y.We show that if A,B ∈C■(R) are X-complexes and U,V ∈ C■(R) are Y-complexes,then U■V■Hom(A,U)■Hom(A,V);A■B■Hom(B,U)■Hom(A,U).As an application,we give a sufficient condition for the Hom evaluation morphism being invertible.     

ON 3-CHOOSABIL ITY OF PL ANE GRAPHSON3 -CHOOSABIL ITY OF PL ANE GRAPHS WITHOUT 6-,7-AND 9-CYCLES

§ 1　 IntroductionAll graphs considered in this paper are finite,simple plane graphs.G=(V,E,F)denotes a plane graph,with V,E and F being the set of vertices,edges and faces of G,respectively.Two vertices u and v are adjacent,denoted by uv∈E,if there is an edge in Ejoining them.A vertex u is incident with an edge e if u is an endvertex of e.Two faces aresaid to be adjacent if they share a common edge.We use b(f) to denote the boundary of aface f.A face is incident with all vertices and e…     

GROUND STATES FOR FRACTIONAL SCHR?DINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS AND CRITICAL GROWTH

In this article, we study the following fractional Schr?dinger equation with electromagnetic fields and critical growth (-?)_A~su + V(x)u = |u|~(2_s~*-2) u + λf(x, |u|~2)u, x ∈ R~N,where(-?)_A~s is the fractional magnetic operator with 0 s 1, N 2s, λ 0, 2_s~*=2N/(N-2s),f is a continuous function, V ∈ C(R~N, R) and A ∈ C(R~N, R~N) are the electric and magnetic potentials, respectively. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by Nehari method.     

相对半投射和直投射模

In this paper, the ＂relative＂ version of semi-projectivity is considered. Let M and N be modules. N is called M-semi-projective, if any homomorphism from N to an M-eyclie submodule f（M） of N can be factored through to a homomorphism from N to M and f, where f∈[M, N]. Some properties of relative semi-projectivity are obtained. Next, we consider a wider class of ＂elatively semi-projective＂ modules such as ＂elatively direct-projective＂ modules which were introduced by Nicholson and Zhou. Several properties of their homomorphisms are also obtained.     

推出-拉回图上的覆盖性同态

In a pushout-pullback diagram,which consists of four morphisms f : A → B,g : A → C,α : C → D and β : B → D,we give some relations among the covers of these four modules.If kerf ∈ I(L ),then g : A → C is L -covering if and only if β : B → D is L -covering.If every module has an L -precover and kerf ∈ I(L ),then A and C have isomorphic L -precovers if and only if B and D have isomorphic L -precovers.     

Approximately Linear Mappings in Banach Modules over a C^＊-algebra

Let X and Y be vector spaces. The authors show that a mapping f ： X →Y satisfies the functional equation 2d f（∑^2d j=1（-1）^j＋1xj/2d）=∑^2dj=1（-1）^j＋1f（xj） with f（0） = 0 if and only if the mapping f ： X→ Y is Cauchy additive, and prove the stability of the functional equation （≠） in Banach modules over a unital C^＊-algebra, and in Poisson Banach modules over a unital Poisson C＊-algebra. Let A and B be unital C^＊-algebras, Poisson C^＊-algebras or Poisson JC^＊- algebras. As an application, the authors show that every almost homomorphism h ： A →B of A into is a homomorphism when h（（2d-1）^nuy） =- h（（2d-1）^nu）h（y） or h（（2d-1）^nuoy） = h（（2d-1）^nu）oh（y） for all unitaries u ∈A, all y ∈ A, n = 0, 1, 2,.... Moreover, the authors prove the stability of homomorphisms in C^＊-algebras, Poisson C^＊-algebras or Poisson JC^＊-algebras.     

GLOBAL ASYMPTOTICS TOWARD THE REREFACTION WAVES FOR A PARABOLIC-ELLIPTIC SYSTEM RELATED TO THE CAMASSA-HOLM SHALLOW WATER EQUATION

This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut＋（u^2/2）x＋px=εuxx, t〉0,x∈R, -αPxx＋P=f（u）＋α/2ux^2-1/2u^2, t〉0,x∈R, （E） with the initial data u（0,x）=u0（x）→u±, as x→±∞ （I） Here, u_ 〈 u＋ are two constants and f（u） is a sufficiently smooth function satisfying f＂ （u） 〉 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u_ 〈 u＋, the above Riemann problem admits a unique global entropy solution u^R（x/t） u^R（x/t）={u_,（f′）^-1（x/t）,u＋, x≤f′（u_）t, f′（u_）t≤x≤f′（u＋）t, x≥f′（u＋）t. Let U（t, x） be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u0（x） - U（0,x） ∈ H^1（R） and u_ 〈 u＋, the above Cauchy problem （E） and （I） admits a unique global classical solution u（t, x） which tends to the rarefaction wave u^R（x/t） as → ＋∞ in the maximum norm. The proof is given by an elementary energy method.     

Strongly Irreducible Submodules of Modules

Strongly irreducible submodules of modules are defined as follows： A submodule N of an Rmodule M is said to be strongly irreducible if for submodules L and K of M, the inclusion L ∩ K ∈ N implies that either L ∈ N or K ∈ N. The relationship among the families of irreducible, strongly irreducible, prime and primary submodules of an R-module M is considered, and a characterization of Noetherian modules which contain a non-prime strongly irreducible submodule is given.     

A Note on a Problem of M. S. Berger

In this paper, we discuss the problem concerning global and local structure of solutions of an operator equation posed by M. S. Berger. Let f : U (?)E→ F be a C1 map, where E and F are Banach spaces and U is open in E. We show that the solution set of the equation f(x)=y for a fixed generalized regular value y of f is represented as a union of disjoint connected C1 Banach submanifolds of U, each of which has a dimension and its tangent space is given. In particular, a characterization of the isolated solutions of the equation f(x) = y is obtained.     

有关M.S.Berger问题的注记

In this paper, we discuss the problem concerning global and local structure of solutions of an operator equation posed by M. S. Berger. Let f : U 真包含 E → F be a C1 map, where E and F are Banach spaces and U is open in E. We show that the solution set of the equation f(x) = y for a fixed generalized regular value y of f is represented as a union of disjoint connected C1 Banach submanifolds of U, each of which has a dimension and its tangent space is given. In particular, a characterization of the isolated solutions of the equation f(x) = y is obtained.     

3$\times$3阶上三角型算子矩阵的可能谱

Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For given A ∈ B（H1）, B ∈ B（H2） and C ∈ B（H3）, the sets ∪D,E,F^σp（M（D,E,F））,∪D,E,F ^σr（M（D,E,F））,∪D,E,F ^σc（M（D,E,F）） and ∪D,E,F σ（M（D,E,F）） are characterized, where D ∈ B（H2,H1）, E ∈B（H3, H1）, F ∈ B（H3,H2） and σ（·）, σp（·）, σr（·）, σc（·） denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.