THE DECOMPOSITION OF THE REPRESENTATIONS Tv AND THE PLANCHEREL FORMULA FOR THE UNIVERSAL COVERING GROUP OF 517(1,1)

The decomposition of the representations T0(v∈R) ore considered here. The Plancherel formula for the universal covering group of SU(1,1) is also deduced.     

SL(n+1,R)/S(GL(1,R)×GL(n,R))上线丛的一个超几何方程（英文）

本文研究了伪黎曼对称空间SL(n+1,R)/S(GL(1,R)×GL(n,R))线丛上的微分方程.利用李代数方法,即Casimir算子得到这个微分算子.这个微分算子是一个超几何方程,这个结论推广了文献[1,3,5]中的微分方程.     

Hamiltonian Gromov–Witten Invariants on C~(n+1) with S~1-action

Consider a Hamiltonian action of S~1 on(C~(n+1), ω_(std)), we shown that the Hamiltonian Gromov–Witten invariants of it are well-defined. After computing the Hamiltonian Gromov–Witten invariants of it, we construct a ring homomorphism from H*_(S~1,CR)(X, R) to the small orbifold quantum cohomology of X //_τ S~1 and obtain a simpler formula of the Gromov–Witten invariants for weighted projective space.     

ABOUT THE INVERSION FORMULA ON THE LIE GROUP SL (2, R)

Harish-Chandra have got a Fourier inversion formula for C_c~-(SL(2, R)). In this paper, we give a discussion on approximation identity kernels on SL(2, R) and get some properties of their Fourier transforms, and then, making use of these properties and Harish-Chandra's result, we prove that the Fourier inversion formula obtained by Harish-Chandra is also valid for C_b~3 (SL(2, R)).     

Generalized wavelets and inversion of the radon transform on the laguerre hypergroup

Let X= Rn+ × R denote the underlying manifold of polyradial functions on the Heisenberg group Hn.We construct a generalized translation on X=Rn+ × R, and establish the Plancherel formula on L2(X,dμ).Using the Gelfand transform we give the condition of generalized wavelets on L2(X,dμ). Moreover, we show the reconstruction formulas for wavelet packet trnasforms and an inversion formula of the Radon transform on X.     

(0,1)-矩阵类u(R,S)的基数函数f(R,S)及其非零集

Let R and S be two vectors with m and n nonnegative integers as conponents respectively. Let u(R, S) be the class consisting of all m×n (0,1) - matrices with row sum vector R and column sum vector S. Suppose that A is the maximal mrixat with row sum vector R. Let S he the column sum vector of A. (of. H. J. Ryser, Combinatorial Mathematics, Carcus Math. Monograph 14 (1963)). Let L(S)={S=(s₁,…,s_m),S-1≥s₂≥…≥s_n}, and let F(R, S) be the cardinal function of u(R,S), i. e.. f(R, S) = |u(R, S) |. Then L(S) is the nonzero-point set of f(R,S). In this paper our principal result is the following.     

SU(n,1)/S(U(n)×U(1))上的中心极限定理

Let G=SU(n,1),K=S(U(n)×u(1)),and for l∈Z,let{η)1∈Z be a one-Dimensional K-type and let E1 be the line bundle over G/K associated to η.In this work we obtain a central limit theorem for the space El.     

ON THE CROSSING NUMBER OF THE COMPLETE TRIPARTITE GRAPH K_(1,8,n)

The well known Zarankiewicz' conjecture is said that the crossing number of the complete bipartite graph Km,n (m≤n) is Z(m,n). where Z(m,n) = [m/2] [(m-1)/2] [n/2] [(n-1)/2](for and real number x, [x] denotes the maximal integer no more than x). Presently, Zarankiewicz' conjecture is proved true only for the case m≤G. In this article, the authors prove that if Zarankiewicz' conjecture holds for m≤9, then the crossing number of the complete tripartite graph K1,8,n is Z(9, n) 12[n/2].     

Automorphisms of a Class of Finite p-groups with a Cyclic Derived Subgroup

Let p be an odd prime,and let k be a nonzero nature number.Suppose that nonabelian group G is a central extension as follows1→G'→G→Z_(p~k)×…×Z_(p~k),where G'≌Z_(p~k),and ζG/G' is a,direct factor of G/G'.Then G is a central product of an extraspecial p~kgroup E and ζG.Let |E|=p~((2n+1)k) and |ζG|=p~((m+1)k).Suppose that the exponents of E and ζG are p~(k+l) and p~(k+r),respectively,where 0≤l,r≤k.Let Aut_(G') G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G',let Aut_(G/ζG,ζG) G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζG and let Aut_(G/ζG,ζG/G') G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on ζG/G'.Then(ⅰ) The group extension 1→Aut G'→Aut G→Aut G'→1 is split.(ⅱ) Aut_(G') G/Aut_(G/ζG,ζG) G≌G_1 × G_2,where Sp(2n-2,Z_(p~k))■H≤G_1≤Sp(2n,Z_(p~k)),H is an extraspecial p~k-group of order p~((2n-1)k) and(GL(m-1,Z_(p~k))■Z_(p~k)~((m-1))■Z_(p~k)~((m))≤G_2≤GL(m,Z_(p~k))■Z_(p~k)~((m)).In particular,G_1=Sp(2n-2,Z~(p~k))■ H if and only if l=k and r=0;G_1=Sp(2n,Z_(p~x)) if and only if l≤r;G_2=(GL(m-1,Z_(p~k))■ Z_(p~k)~((m-1))■ Z_(p~k)~((m)) if and only if r=k;G_2=GL(m,Z_(p~k))■Z_(p~k)((m)) if and only if r=0.(ⅲ) Aut_(G') G/Aut_( G/ζG,ζG/G') G≌G_1 × G_3,where G_1 is defined in(ⅱ);GL(ml,Z_(p~k))■ Z_(p~k)~((m-1))≤G_3 ≤GL(n,Z_(p~k)).In particular,G_3=GL(m-1,Z_(p~k))■ Z_(p~k)~((m-1)) if and only if r=k;G_3=GL(m,Z_(p~k)) if and only if r=0.(ⅳ) Ant_(G/ζG,ζG/G') G≌ Aut_(G/ζG,ζG/G') G■ Z_(p~k)~((m)),If m=0,then Ant_(G/ζG,ζG/G') G=Inn G≌Z_(p~k)~((2n));If m 0,then Ant_(G/ζG,ζG/G') G≌Z_(p~k)~((2nm))×Z_(p~(k-r))~((2n)),and Aut_(G/ζG,ζG) G/Inn G≌Z_(p~k)~((2n(m-1))× Z_(p~(k-r))~((2n)).     

主理想环上子群Gr在线性群中的扩群

Suppose R is a principal ideal ring,R~* is a multiplicative group which is composed of all reversible elements in R,and M_n(R),GL(n,R),SL(n,R) are denoted by, M_n(R)={A=(a_(ij))_(n×n)|a_(ij)∈R,i,j=1,2,…,n},GL(n,R) = {g|g∈M_n(R),detg∈R~*},SL(n,R) = {g∈GL(n,R)|det g=1},SL(n,R)≤G≤GL(n,R)(n≥3),respectively, then basing on these facts,this paper mainly focus on discussing all extended groups of G_r={(AB OD)∈G|A∈GL(r,R),(1≤r相似文献   

ON THE ERROR FUNCTION OF THE SQUARE-FULL INTEGERS

Let L(x) denote the number of square-full integers not exceeding x. It is proved in [1] thatL(x)～(ζ(3/2)/ζ(3))x~(1/2) (ζ(2/3)/ζ(2))x~(1/3) as x→∞,where ζ(s) denotes the Riemann zeta function. Let △(x) denote the error function in the asymptotic formula for L(x). It was shown by D. Suryanaryana~([2]) on the Riemann hypothesis (RH) that1/x integral from n=1 to x |△(t)|dt=O(x~(1/10 s))for every ε>0. In this paper the author proves the following asymptotic formula for the mean-value of △(x) under the assumption of R. H.integral from n=1 to T (△~2(t/t~(6/5))) dt～c log T,where c>0 is a constant.     

AN ASYMPTOTIC ORDER OF FOURIER TRANSFORM ON SL(2,R)

In this paper, a better asymptotic order of Fourier transform on SL(2,R) is obtained by using classical analysis and Lie analysis comparing with that of [5],[6] ,and the Plancherel theorem on C2i(SL(2,R)) is also obtained as an application.     

迭代函数方程$G(x,f(x),\ldots,f^n(x))=F(x)$的$C^1$解

In this paper we consider the iterative equation G(x,f(x),...,f n(x)) = F(x) on R,and give the existence of C 1 solutions near the fixed point of F,which generalize some results on the leading coefficient problem from the form of the polynomial-like iterative equations to the general form.     

NOTES ON THE SPECTRAL PROPERTIES OF THE WEIGHTED MEAN DIFFERENCE OPERATOR G(u,v;?) OVER THE SEQUENCE SPACE ?_1

In the study by Baliarsingh and Dutta [Internat. J.Anal., Vol.2014(2014), Article ID 786437], the authors computed the spectrum and the fine spectrum of the product operator G(u, v; ?) over the sequence space ?_1. The product operator G(u, v; ?) over ?_1 is defined by(G(u,v; ?) x)_k=k∑i=1u_kv_i(x_i-x_(i-1)) with x_k = 0 for all k 0, where x =(x_k) ∈ ?_1,and u and v are either constant or strictly decreasing sequences of positive real numbers satisfying certain conditions. In this article we give some improvements of the computation of the spectrum of the operator G(u v; ?) on the sequence space ?_1.     

SPACELIKE SUBMANIFOLDS IN THE DE SITTER SPACE S_p~(n+p)(c) WITH CONSTANT SCALAR CURVATURE

§ 1　 IntroductionL et Sn+ pp (c) (c>0 ) be an (n+p) - dimensional connected de Sitter space and Mn be aspacelike submanifold isometrically immersed in Sn+ pp (c) . We say Mn is closed if it iscompact and without boundary.Denote by R,H and S the normalized scalar curvature,themean curvature and the square of the length of the second fundamental form of Mn,respectively.By application of the technique of Simons[1 1 ] ,there have been many rigidity results formaximal spacelike submanifolds a…     

THE HOPF BIFURCATION OF QUADRATIC SYSTEM (I)_(n=0)

δlm is the parameter space of quadratic system (I)n=0. A partition of parameters corresponding to the existence and nonexistence of the limit cycle of the system is given in detail. The Hopf bifurcation surfaces of (I)m=0 are obtained, and the sketch of Hopf bifurcation surfaces of (I)n=0 are drawn.     

On inclusion of permutation modules

Let H1,H2 be subgroups of a finite group G. Assume that G=∪i=1mH2yiH1=∪j=1nH1gjH1 and that y1=1,g1=1.Let Di be the set consisting of right cosets of H2 contained in H2yiH1 and let dj(j=1, . . . ,n) be the set consisting of right cosets contained in H1gjH1.We define the n×m matrix Mz(z=1, . . . ,m) whose columns and rows are indexed by Di and dj respectively and the (dk,Dl) entry is |Dzgk∩Dl|. Let M=(M1, . . . ,Mm). Assume that 1H1G and 1H2G are semisimple permutation modules of a finite group G. In this paper, by using the matrix M , we give some sufficient and necessary conditions such that 1H1G is isomorphic to a submodule of 1H2G.As an application, we prove Foulkes' conjecture in special cases.     

ON THE EXISTENCE OF INFINITELY MANY CRITICAL POINTS OF THE EVEN FUNCTIONAL integral from n=Q to (F(x,u,Du)) integral from n=■Q to (G(x,u))

In this paper we deal with the existence of infinitely many critical points of the even functional I(u)=integral from n=Q to (F(x,u,Du)) integral from n=(?)Q to (G(x,u)), u∈W~(1,p)(Ω),where G(x, u)=integral from n=o to u (g(x,t)dt), under the weak structure conditions on F(x, u, q) by the Mountain Pass Lemma.     

Automorphisms of Extensions of Q by a Direct Sum of Finitely Many Copies of Q

Let G be an extension of Q by a direct sum of r copies of Q.(1) If G is abelian, then G is a direct sum of r + 1 copies of Q and Aut G = GL(r + 1, Q);(2) If G is non-abelian, then G is a direct product of an extraspecial Q-group E and m copies of Q, where E/ζ E is a linear space over Q with dimension 2 n and m + 2 n = r. Furthermore, let Aut_G'G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and Aut_(G/ζG),_(ζG)G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G')G→ Aut G→ Aut G'→ 1 is split;(ii)Aut_(G')G/Aut_(G/ζG),_(ζG)G = Sp(2 n, Q) ×(GL(m, Q) Q~(m));(iii) Aut_(G/ζG),ζGG/Inn G= Q~(2 nm).     

Vertex-distinguishing E-total Coloring of Complete Bipartite Graph K_(7,n) when 7≤n≤95

Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex x of G, let C(x) denote the set of colors of vertex x and of the edges incident with x, we call C(x) the color set of x. If C(u)≠ C(v) for any two different vertices u and v of V(G), then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by χ_(vt)~e(G) and is called the VDET chromatic number of G. The VDET coloring of complete bipartite graph K_(7,n)(7 ≤ n ≤ 95) is discussed in this paper and the VDET chromatic number of K_(7,n)(7 ≤ n ≤ 95) has been obtained.