Differential Calculus on Compact Quantum Group Uθ（2） and its Applications

In this paper, via constructing special matrices, we will show that there exists a differential calculus on Uθ（2）, where θ is an irrational number. Then using the above results, we shall discuss the properties of infinitesimal generators of corepresentations of Uθ（2）. And in the final, we shall discuss its irreducible corepresentations and give the Peter-Weyl theorem explicitly for compact quantum group Uθ（2）.     

On the classification of compact quantum groups U_θ(2)

In this paper,we show that the corepresentation of compact group Uθ(2) on the vector space is determined by its infinitesimal generators B0,B2,A0,A1 and A2,where θ is an irrational number.We also show that B0 and B2 commute with Aj,j = 0,1,2,so B0,A0,A1 and A2 are sl(2,C) loop algebra.Then we exhibit all irreducible representations of Uθ(2),which are different from those of group U(2),and use above results to give the classification of quantum groups Uθ(2),which is analogous to that of irrational algebra Aθ.At the same time,we also give all the forms of automorphisms on quantum group Uθ(2).     

Recognition of the Projective Special Linear Group over GF（3）

Let P be a finite group and denote by w（P） the set of its element orders. P is called k-recognizable by the set of its element orders if for any finte group G with ω（G） =ω（P） there are, up to isomorphism, k finite groups G such that G ≌P. In this paper we will prove that the group Lp（3）, where p 〉 3 is a prime number, is at most 2-recognizable.     

Lp Convergence of Generalized Fourier Series Deduced by Piecewise Linear Spectral Sequences

In the present paper, we discuss some properties of piecewise linear spectral sequences introduced by Liu and Xu. We have a study on the pointwise and almost everywhere convergence of its corresponding series. Also, it is shown that the set G constructed from piecewise linear spectral sequences are bases, but not unconditional bases, for LP（0, 1） where 1 〈 p 〈 ∞, p ≠2.     

OD-characterization of Almost Simple Groups Related to U3（5）

Let G be a finite group with order ｜G｜=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph （or Gruenberg- Kegel graph） denoted .by г（G） （or GK（G））. This graph is constructed as follows： The vertex set of it is π（G） = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge （and we write pi - pj） if and only if G contains an element of order pipj. The degree deg（pi） of a vertex pj ∈π（G） is the number of edges incident on pi. We define D（G） ：= （deg（p1）, deg（p2）,..., deg（pk））, which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that ｜H｜ = ｜G｜ and D（H） = D（G）. Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L ：= U3（5） be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.     

Fuzzy蕴涵代数的素MP-滤子空间(英文)

In this paper,the topological space(PF_MP（X）,T) based on prime MP-filters of a lattice FI-algebra X is constructed firstly and we proved that it is a compact T₀-space if X with condition（P）.Secondly,we restricted T to the set of all maximal MP-filters MF_MP（X） of X and concluded that(PF_MP（X）,T |_PFMP（X） )is a compact T₂ space if X with conditions（P） and（S）.     

On the Clique-Transversal Number in （Claw,K4）-Free 4-Regular Graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G.The clique-transversal number,denoted by τC(G),is the minimum cardinality of a clique-transversal set in G.In this paper,we first present a lower bound on τC(G) and characterize the extremal graphs achieving the lower bound for a connected(claw,K4)-free 4-regular graph G.Furthermore,we show that for any 2-connected(claw,K4)-free 4-regular graph G of order n,its clique-transversal number equals to [n/3].     

Second Order Nonlinear Evolution Inclusions Ⅰ： Existence and Relaxation Results

This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x（t） and x（t）. In this first part we prove existence and relaxation theorems. We consider the case of an usc, convex valued nonlinearity and we show that for this problem the solution set is nonempty and compact in C^1 （T, H）. Also we examine the Isc, nonconvex case and again we prove the existence of solutions. In addition we establish the existence of extremal solutions and by strengthening our hypotheses, we show that the extremal solutions are dense in C^1 （T, H） to the solutions of the original convex problem （strong relaxation）. An example of a nonlinear hyperbolic optimal control problem is also discussed.     

Composition of fractional Orlicz maximal operators and A 1-weights on spaces of homogeneous type

For a Young function θ with 0 ≤α 〈 1, let Mα,θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type （X, d, μ） by Mα,θf（x） = supx∈（B）α ｜｜f｜｜θ,B, where ｜｜f｜｜θ,B is the mean Luxemburg norm of f on a ball B. When α= 0 we simply denote it by Me. In this paper we prove that if Ф and ψare two Young functions, there exists a third Young function θ such that the composition Mα,ψ o MФ is pointwise equivalent to Mα,θ. As a consequence we prove that for some Young functions θ, if Mα,θf 〈∞a.e. and δ ∈（0,1） then （Mα,θf）δ is an A1-weight.     

一个子集树研究问题的解

In this paper, we solve a research problem on trees of subsets posed by F.R. McMorris. If a collection of subsets are chosen at random from the power set of a finite set X,we give the probability that the collection is a tree of subsets of X.     

Improving on the minimum risk equivariant estimator of a location parameter which is constrained to an interval or a half-interval

For location families with densitiesf ₀(x−θ), we study the problem of estimating θ for location invariant lossL(θ,d)=ρ(d−θ), and under a lower-bound constraint of the form θ≥a. We show, that for quite general (f ₀, ρ), the Bayes estimator δ _U with respect to a uniform prior on (a, ∞) is a minimax estimator which dominates the benchmark minimum risk equivariant (MRE) estimator. In extending some previous dominance results due to Katz and Farrell, we make use of Kubokawa'sIERD (Integral Expression of Risk Difference) method, and actually obtain classes of dominating estimators which include, and are characterized in terms of δ _U . Implications are also given and, finally, the above dominance phenomenon is studied and extended to an interval constraint of the form θ∈[a, b]. Research supported by NSERC of Canada.     

Conductors and newforms for U(1,1)

Let F be a non-Archimedean local field whose residue characteristic is odd. In this paper we develop a theory of newforms forU (1, 1)(F), building on previous work onSL ₂(F). This theory is analogous to the results of Casselman forGL ₂(F) and Jacquet, Piatetski-Shapiro, and Shalika forGL _n(F). To a representation π ofU(1, 1)(F), we attach an integer c(π) called the conductor of π, which depends only on theL-packet π containing π. A newform is a vector in π which is essentially fixed by a congruence subgroup of level c(π). We show that our newforms are always test vectors for some standard Whittaker functionals, and, in doing so, we give various explicit formulae for newforms.     

The character table of the group GL 2(Q) when extended by a certain group of order two

LetG denote either of the groupsGL ₂(q) or SL₂(q). Then θ :G →G given by θ(A) = (A ^t)^t, whereA ^t denotes the transpose of the matrixA, is an automorphism ofG. Therefore we may form the groupG.θ> which is the split extension of the groupG by the cyclic group θ of order 2. Our aim in this paper is to find the complex irreducible character table ofG. θ.     

Stability of Solution of a Class of Quasilinear Elliptic Equations

In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.     

The Bellman function,the two-weight Hilbert transform,and embeddings of the model spaces<Emphasis Type="Italic">K</Emphasis><Subscript>θ</Subscript>

This paper is devoted to embedding theorems for the spaceK _θ , where θ is an inner function in the unit disc D. It turns out that the question of embedding ofK _θ into L²(Μ) is virtually equivalent to the boundedness of the two-weight Hilbert transform. This makes the embedding question quite difficult (general boundedness criteria of Hunt-Muckenhoupt-Wheeden type for the twoweight Hilbert transform have yet to be found). Here we are not interested in sufficient conditions for the embedding ofKg into L²(Μ) (equivalent to a certain two-weight problem for the Hilbert transform). Rather, we are interested in the fact that a certain natural set of conditions is not sufficient for the embedding ofK _θ intoL ² (Μ) (equivalently, a certain set of conditions is not sufficient for the boundedness in a two-weight problem for the Hilbert transform). In particular, we answer (negatively) certain questions of W. Cohn about the embedding ofK _θ into L²(Μ). Our technique leads naturally to the conclusion that there can be a uniform embedding of all the reproducing kernels ofK _θ but the embedding of the wholeK _θ intoL ²(Μ) may fail. Moreover, it may happen that the embedding into a potentially larger spaceL ²(μ) fails too. Both authors are supported by the NSF grant DMS 9970395 and joint American-Israeli grant BSF 00030.     

On the asymptotic efficiency of estimators in an autoregressive process

Summary Let {X _t } be defined recursively byX _t =θX _t−1+U _t (t=1,2, ...), whereX ₀=0 and {U _t } is a sequence of independent identically distributed real random variables having a density functionf with mean 0 and varianceσ ². We assume that |θ|<1. In the present paper we obtain the bound of the asymptotic distributions of asymptotically median unbiased (AMU) estimators of θ and the sufficient condition that an AMU estimator be asymptotically efficient in the sense that its distribution attains the above bound. It is also shown that the least squares estimator of θ is asymptotically efficient if and only iff is a normal density function. University of Electro-Communications     

Construct irreducible representations of quantum groups U q (ƒ m (K))

In this paper, we construct families of irreducible representations for a class of quantum groups U _q (ƒ _m (K)). First, we give a natural construction of irreducible weight representations for U _q (ƒ _m (K)) using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of U _q (ƒ _m (K)). As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.      

On a Function Related to Multinomial Coefficients (I)

Let F _p,t (n) denote the number of the coefficients of (x ₁+1x ₂+...+x _t ) ^j , 0 ≤j≤n− 1, which are not divisible by the prime p. Define G _p,t (n) = F _p,t /n ^θ and β(p,t) = lim infF _p,t )(n)/n ^θ, where θ = (log)/(log p). In this paper, we mainly prove that G _p,t can be extended to a continuous function on ℝ⁺, and the function G _p,t is nowhere monotonic. Both the set of differential points of the function G _p,t and the set of non-differential points of the function G _p,t are dense in ℝ⁺. Received February 18, 2000, Accepted December 7, 2000     

On periods of CUSP forms and algebraic cycles forU(3)

In this paper we discuss relations between the following types of conditions on a representationπ in a cuspidalL-packet ofU(3): (1)L(s, π×ξ) has a pole ats=1 for someξ; (2) aperiod ofπ over some algebraic cycle inU(3) (coming from a unitary group in two variables) is non-zero; and (3) π is atheta-series lifting from some unitary group in two variables. As an application of our analysis, we show that the algebraic cycles on theU(3) Shimura variety arenot spanned (over the Hecke algebra) by the modular and Shimura curves coming from unitary subgroups. All three authors are supported by a grant from the U.S.-Israel Binational Science Foundation; the second author is also supported by an NSF Grant.     

Equivalence classes of matrices inGL 2 (Q) andSL 2 (Q)

LetG denote either of the groupsGL ₂ (q) orSL ₂ (q). The mapping θ sending a matrix to its transpose-inverse is an automophism ofG and therefore we can form the groupG ⁺ =G. <θ>. In this paper conjugacy classes of elements inG ⁺ -G are found. These classes are closely related to the congruence classes of invertible matrices inG.