Highest weight representations of a family of Lie algebras of Block type

For an additive subgroup G of a field F of characteristic zero, a Lie algebra B（G） of Block type is defined with basis {Lα,i｜ α∈G, i∈Z＋} and relations [Lα,i, Lβ,j] = （β-α）Lα＋β,i＋j＋（αj-βi）Lα＋β,Lα＋β,i＋j-1.It is proved that an irreducible highest weight B（Z）-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order λ on G and any ∧∈B（G）0^＊（the dual space of B（G）0 = span{L0,i｜i∈Z＋}）, a Verma B（G）-module M（∧,λ） is defined, and the irreducibility of M（A,λ） is completely determined.     

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.     

SPDEs driven by Poisson random measure with non Lipschitz coefficients: existence results

The article deals with SPDEs driven by Poisson random measure with non Lipschitz coefficients. Let A:E→E be a generator of an analytic semigroup on E, E being a certain Banach space. Let be a stochastic basis carrying an E-valued Poisson random measure η with characteristic measure ν and compensator γ. Let 1≤p≤2. Our point of interest is the existence of solutions to SPDE's of e.g.the following type where g:E→L(E,E ₀) is some mapping satisfying ∫ _E |g(x,z)−g(y,z)| ^p ν(dz)≤C|x−y| ^rp , x,y ∈ E, where 0<r<1 satisfy certain condition specified later and is again a certain Banach space. This work was supported by the Austrian Academy of Science, APART 700 and FWF-Project P17273-N12     

A Generalization of Orthogonal Factorizations in Graphs

Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integer-valuated functions defined on V(G) such that g(x) ≤f(x) for all x∈V(G). Then a (g, f)-factor of G is a spanning subgraph H of G such that g(x) ≤d _H (x) ≤f(x) for all x∈V(G). A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let = {F ₁, F ₂, ..., F _m } be a factorization of G and H be a subgraph of G with mr edges. If F _i , 1 ≤i≤m, has exactly r edges in common with H, then is said to be r-orthogonal to H. In this paper it is proved that every (mg + kr, mf−kr)-graph, where m, k and r are positive integers with k < m and g≥r, contains a subgraph R such that R has a (g, f)-factorization which is r-orthogonal to a given subgraph H with kr edges. This research is supported by the National Natural Science Foundation of China (19831080) and RSDP of China     

A simple formula for an analogue of conditional wiener integrals and its applications II

Let C[0, T] denote the space of real-valued continuous functions on the interval [0, T] with an analogue w _ϕ of Wiener measure and for a partition 0 = t ₀ < t ₁ < ... < t _n < t _n+1 = T of [0, T], let X _n : C[0, T] → ℝ ⁿ⁺¹ and X ⁿ⁺¹: C[0, T] → ℝ ⁿ⁺² be given by X _n (x) = (x(t ₀), x(t ₁), ..., x(t _n )) and X _n+1(x) = (x(t ₀), x(t ₁), ..., x(t _n+1)), respectively. In this paper, using a simple formula for the conditional w _ϕ-integral of functions on C[0, T] with the conditioning function X _n+1, we derive a simple formula for the conditional w _ϕ-integral of the functions with the conditioning function X _n . As applications of the formula with the function X _n , we evaluate the conditional w _ϕ-integral of the functions of the form F _m (x) = ∫₀ ^T (x(t)) ^m for x ∈ C[0, T] and for any positive integer m. Moreover, with the conditioning X _n , we evaluate the conditional w _ϕ-integral of the functions in a Banach algebra which is an analogue of the Cameron and Storvick’s Banach algebra . Finally, we derive the conditional analytic Feynman w _ϕ-integrals of the functions in .      

Existence of subgraph with orthogonal (g,f)-factorization

Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F ₁,F ₂,⃛,F ₁| ofG, if |E(H)∩E(F ₁)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G). Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences.     

On <Emphasis Type="Italic">S</Emphasis>-quasinormal and <Emphasis Type="Italic">c</Emphasis>-normal subgroups of a finite group

Let be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are presented: (1) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of H is either c-normal or S-quasinormally embedded in G. (2) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of F*(H), the generalized Fitting subgroup of H, is either c-normal or S-quasinormally embedded in G. (3) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every cyclic subgroup of F*(H) of prime order or order 4 is either c-normal or S-quasinormally embedded in G. Supported by the Natural Science Foundation of China and the Natural Science Foundation of Guangxi Autonomous Region (No. 0249001). Corresponding author. Supported in part by the Natural Science Foundation of China (10571181), NSF of Guangdong Province (06023728) and ARF(GDEI).     

Second cohomology group of extended W-algebras

Let \mathbbF\mathbb{F} be a field of characteristic 0, and let G be an additive subgroup of \mathbbF\mathbb{F}. We define a class of infinite-dimensional Lie algebras \mathbbF\mathbb{F}-basis {L _μ, V _μ, W _μ | μ ∈ G}, which are very closely related to W-algebras. In this paper, the second cohomology group of is determined.     

A New Result on Alspach's Problem

Let G be a simple graph. Let g(x) and f(x) be integer-valued functions defined on V(G) with g(x)≥2 and f(x)≥5 for all x∈V(G). It is proved that if G is an (mg+m−1, mf−m+1)-graph and H is a subgraph of G with m edges, then there exists a (g,f)-factorization of G orthogonal to H. Received: January 19, 1996 Revised: November 11, 1996     

Estimates for multilinear singular integral operators with nonsmooth kernels

Let (, d, μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we consider the behavior on × ⋯ × for the m-linear singular integral operators with nonsmooth kernels which were first introduced by Duong, Grafakos and Yan.     

（g, f）-Factorizations Randomly Orthogonal to a Subgraph in Graphs

Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integervalued functions defined on V(G) such that 2k - 2 ≤g(x)≤f(x) for all x∈V(G). Let H be a subgraph of G with mk edges. In this paper, it is proved that every (mg m-1,mf-m 1)-graph G has (g, f)-factorizations randomly k-orthogonal to H under some special conditions.     

The commutant of analytic Toeplitz operators on Bergman space

In this paper, using the matrix skills and operator theory techniques we characterize the commutant of analytic Toeplitz operators on Bergman space. For f（z） = z^ng（z） （n ≥1）, g（z） = b0 ＋ b1z^p1 ＋b2z^p2 ＋.. , bk ≠ 0 （k = 0, 1, 2,...）, our main result is =A′（Mf） = A′（Mzn）∩A′（Mg） = A′（Mz^s）, where s = g.c.d.（n,p1,p2,...）. In the last section, we study the relation between strongly irreducible curve and the winding number W（f,f（α））, α ∈ D.     

The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs

If G is a connected graph of order n ⩾ 1, then by a hamiltonian coloring of G we mean a mapping c of V (G) into the set of all positive integers such that |c(x) − c(y)| ⩾ n − 1 − D _G (x, y) (where D _G (x, y) denotes the length of a longest x − y path in G) for all distinct x, y ∈ V (G). Let G be a connected graph. By the hamiltonian chromatic number of G we mean

The pressure in operator algebras

We introduce two notions of the pressure in operator algebras, one is the pressure Pα（π, T） for an automorphism α of a unital exact C^＊-algebra A at a self-adjoint element T in A with respect to a faithful unital ＊-representation π the other is the pressure Pτ,α（T） for an automorphism α of a hyperfinite von Neumann algebra M at a self-adjoint element T in M with respect to a faithful normal α-invariant state τ. We give some properties of the pressure, show that it is a conjugate invaxiant, and also prove that the pressure of the implementing inner automorphism of a crossed product A×α Z at a self-adjoint operator T in A equals that of α at T.     

On thek dimensional Piatetski-Shapiro prime number theorem

Thek-dimensional Piatetski-Shapiro prime number problem fork⩾3 is studied. Let π(x ₁ c ₁,⋯,c _k ) denote the number of primesp withp⩽x, , where 1<c ₁<⋯<c _k are fixed constants. It is proved that π(x;c ₁,⋯,c _k ) has an asymptotic formula ifc ₁ ⁻¹ +⋯+c _k ⁻¹ >k−k/(4k ²+2). Project supported by the National Natural Science Foundation of China (Grant No. 19801021) and the Natural Science Foundation of Shandong Province (Grant No.Q98A02110).     

∞-jets of diffeomorphisms preserving orbits of vector fields

Let F be a C ^∞ vector field defined near the origin O ∈ ℝ ⁿ , F(O) = 0, and (F _t ) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝ ⁿ → ℝ ⁿ at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney W ^r topology. Then contains a subset consisting of maps of the form F _α(x)(x), where α: ℝ ⁿ → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = . In this paper we present a class of examples of vector fields with degenerate singularities at O for which formally coincides with , i.e. on the level of ∞-jets at O. We also establish parameter rigidity of linear vector fields and “reduced” Hamiltonian vector fields of real homogeneous polynomials in two variables.      

<Emphasis Type="Italic">K</Emphasis>-Radical classes and product radical classes of <Emphasis Type="Italic">MV</Emphasis>-algebras

For an MV-algebra let J ₀( ) be the system of all closed ideals of ; this system is partially ordered by the set-theoretical inclusion. A radical class X of MV-algebras will be called a K-radical class iff, whenever ∈ X and is an MV-algebra with J ₀( ) ≅ J ₀( ), then ∈ X. An analogous notation for lattice ordered groups was introduced and studied by Conrad. In the present paper we show that there is a one-to-one correspondence between K-radical classes of MV-algebras and K-radical classes of abelian lattice ordered groups. We also prove an analogous result for product radical classes of MV-algebras; product radical classes of lattice ordered groups were studied by Ton. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information, Grant I/2/2005.     

On quantum cluster algebras of finite type

We extend the definition of a quantum analogue of the Caldero-Chapoton map defined by D. Rupel. When Q is a quiver of finite type, we prove that the algebra (Q) generated by all cluster characters is exactly the quantum cluster algebra (Q).     

Cohomology of a flag variety as a Bethe algebra

We interpret the equivariant cohomology H_GLn ^*H_{GL_n }^* (ℱ _λ ,ℂ) of a partial flag variety ℱ _λ parametrizing chains of subspaces 0 = F ₀ ⊂ F ₁ ⊂ … ⊂ F _N = ℂ ⁿ , dimF _i /F _i−1 = λ _i , as the Bethe algebra of the -weight subspace of a [t]-module .     

How Close to Regular Must a Semicomplete Multipartite Digraph Be to Secure Hamiltonicity?

 Let D be a semicomplete multipartite digraph, with partite sets V ₁, V ₂,…, V _c, such that |V ₁|≤|V ₂|≤…≤|V _c|. Define f(D)=|V(D)|−3|V _c|+1 and . We define the irregularity i(D) of D to be max|d ⁺(x)−d ⁻(y)| over all vertices x and y of D (possibly x=y). We define the local irregularity i _l(D) of D to be max|d ⁺(x)−d ⁻(x)| over all vertices x of D and we define the global irregularity of D to be i _g(D)=max{d ⁺(x),d ⁻(x) : x∈V(D)}−min{d ⁺(y),d ⁻(y) : y∈V(D)}. In this paper we show that if i _g(D)≤g(D) or if i _l(D)≤min{f(D), g(D)} then D is Hamiltonian. We furthermore show how this implies a theorem which generalizes two results by Volkmann and solves a stated problem and a conjecture from [6]. Our result also gives support to the conjecture from [6] that all diregular c-partite tournaments (c≥4) are pancyclic, and it is used in [9], which proves this conjecture for all c≥5. Finally we show that our result in some sense is best possible, by giving an infinite class of non-Hamiltonian semicomplete multipartite digraphs, D, with i _g(D)=i(D)=i _l(D)=g(D)+?≤f(D)+1. Revised: September 17, 1998