SOME SUBGROUPS OF THE EXTENDED HECKE GROUPS (H)(λq)

The authors consider the extended Hecke groups H(λq) generated by T(z) =-1 / z, S(z) = -1/(z+λq) and R(z) = 1 / -z with λq = 2cos(π/q) for q ≥ 3 an integer. Inthis paper, the even subgroup He(λq), the second commutator subgroup H"(λq) and theprincipal congruence subgroups Hp(λq) of the extended Hecke groups H(λq) are studied.Also, relations between them are given.     

Congruence subgroups of Hecke groups

Hecke groups are an important tool in subgroups of Hecke groups play an important rule investigating functional equations, and congruence in research of the solutions of the Dirichlet series. When q, m are two primes, congruence subgroups and the principal congruence subgroups of level m of the Hecke group H（√q） have been investigated in many papers. In this paper, we generalize these results to the case where q is a positive integer with q ≥ 5, √q ￠ Z and m is a power of an odd prime.     

CERTAIN ASYMPTOTIC EXPANSIONS FOR LAGUERRE POLYNOMIALS AND CHARLIER POLYNOMIALS

Here proposed are certain asymptotic expansion formulas for Ln(w-1)(λz) and Cn(ω)(λz) in whichO(λ) and n = 0(λ1/2 )(λ→∞) , z being x complex number. Also presented are certain estimates for the remainders(error bounds) of the asymptotic expansions within the regions D1( - ∞相似文献   

Mordell-Weil groups and Selmer groups of twin- prime elliptic curves

Let E = Eσ : y2 = x(x + σp)(x + σq) be elliptic curves, where σ = ±1, p and q are primenumbers with p+2 = q. (i) Selmer groups S(2)(E/Q), S(φ)(E/Q), and S(φ)(E/Q) are explicitly determined,e.g. S(2)(E+1/Q)= (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p ≡ 5, 1 (or 3), and 7(mod 8), respectively. (ii)When p ≡ 5 (3, 5 for σ = -1) (mod 8), it is proved that the Mordell-Weil group E(Q) ≌ Z/2Z Z/2Z,symbol, the torsion subgroup E(K)tors for any number field K, etc. are also obtained.     

THE VALUE DISTRIBUTION AND UNIQUENESS OF ONE CERTAIN TYPE OF DIFFERENTIAL-DIFFERENCE POLYNOMIALS

In this article, we investigate the distribution of the zeros and uniqueness of differential-difference polynomials G(z) =(fn(fm(z)- 1)dΠj=1f(z + cj)vj)(k)- α(z),H(z) =(fn(f(z)- 1)mdΠj=1f(z + cj)vj)(k)- α(z),where f is transcendental entire function of finite order, cj(j = 1, 2, ···, d), n, m, d, and vj(j = 1, 2, ···, d) are integers, and obtain some theorems, which extended and improved many previous results.     

一类解析函数的Fekete-Szeg不等式(英文)

The Fekete-Szego inequality for a subclass H(α,λ,A,B) of the class H of normalized analytic functions is studied.For each f(z)=z+~∞∑_(n=2)αnz~n ∈ H(α,λ,A,B),the sharp upper bounds of |α_3-α_2~2|for any complex parameter u are obtained by using the fundamental inequalities of analytic functions and analytical techniques and the applications of the inequality of functions defined with Hadaniard product are proved.     

扭曲退化同宿分支

§ 1　 HypothesesConsider the following system:z.=f(z) , (1 .1 )and its perturbed systemz.=f(z) +g(z,μ) (1 .2 )where z∈ Rm+n,μ∈ Rk,k≥ 3,0≤ |μ| 1 ,f,g∈ Cr,r≥ 4 ,g(z,0 ) =0 .For simplicity,we sup-pose thatf(p) =0 ,g(p,μ) =0 .Moreover,for(1 .1 ) we assume(H1 ) The stable manifold Wspand the unstable manifold Wupof z=p are m-dimension-al and n-dimensional,respectively.The linearization Df(p) atthe equilibrium z=p has realmultiple-2 eigenvaluesλ1 and -ρ1 ,such thatany remaining eige…     

具有负系数的单叶函数的推广

Let R_A,B(α,β,q) denote the class of functions f(z) = z(?) which are analytic and nnivalent in < 1, and satisfy the condition. Also let R_A,B(α,β,q,t) denote the class of functions (1 - t)z + tf(z), where f(z) ∈R_A,B(α,β, q) and t ∈(0, 1].Sharp results concerning representation, distortion theorems and radius of convexityfor the class R_A,B(α,β, q, t) are determined. It is shown that the class R_A,B(α,β,q, t)is closed under linear combinations. Furthermore extreme points and support points forthe class R_A,B(α,β,q, t) are also determined.     

EXISTENCE OF SOLUTIONS IN THE FUTURE FOR A CLASS OF NONLINEAR DIFFERENTIAL SYSTEMS

1 IntroductionThe problem cited in the title above is an old problem, and has been studiedby many authors. Some good results can be found in [1-5] and referencesquoted therein.In this paper we are concerned with this problem for the nonlinear differential systemJ 1 -- ':)>.--,,<,> ~ F(x), (1 l)where E,P, F, q and g are continuous real functions defined on R. Clearly, ifputting E(z) = z, p(y) = y and q(y) H 1, then (1.1) gives the well-knownLi6nard system{ 1:,.:<">, (l 2)Therefore, (1.1) is…     

Sign changes of Fourier coefficients of Maass eigenforms

Let f be a Maass eigenform that is a new form of level N on Γ0(N),with Laplace eigenvalue 1/4 + νf2 . Then,all its Fourier coefficients {λf(n)}∞n=1 are real,and we may normalize so that λf (1) = 1. It is proved,in this paper,that the first sign change in the sequence {λf(n)}n∞=1 occurs at some n satisfying n《{(3+|νf|2)N }1/2-δ and (n,N)=1. This generalizes the previous results for holomorphic Hecke eigenforms.     

关于有限群两个超可解子群之积的问题

In this paper, we give some sufficient conditions for products of two supersolvable sub-groups to be supersolvable groups. Our results generalize some known results.Theorem 1 Let G = HK,(|H|,|K|) = 1, Where H and K are two supersolvable sub-groups. If H is commutative with every maximal subgroup of K, and K is commutative with every maximal subgroup of H, then G is supersolvable.Theorem 2 Let G = HK, H ∩ K = 1, H G, and K be quasinormal in H. If H, K are supersolvable, the G is supersolvable.Theorem 3 Let G= HK,(|H|,|K|) = 1,H,K be two supersolvable subgroups. If H is commutative with any Sylow subgroup of K and any maximal subgroup of every sylow subgroup of K, and K is commutative with any sylow subgroup of H and any maximal subgroup of every sylow subgroup of H, then G is supersolvable. Theorem 4 If H,K are two supersolvable subgroups of G, G= HK, G′is nilpotent, H is quasi normal K, and K is quasi normal in H,then G is supersolvable. Theorem 5 If H,K are two supersolvable subgroups of G, G= HK, H′? G,[H,K]? G,[H,K] is nilpotent, H is quasi normal in K, and K is quasi normal in H,then G is supersolvable.     

On Φ-τ-Supplement Subgroups of Finite Groups

Let τ be a subgroup functor and H a p-subgroup of a finite group G. Let G= G/H_G and H= H/H_G. We say that H is Φ-τ-supplement in G if G has a subnormal subgroup T and a τ-subgroup S contained in H such that G=H T and H∩T≤SΦ(H). In this paper,some new characterizations of hypercyclically embedability and p-nilpotency of a finite group are obtained based on the assumption that some primary subgroups are Φ-τ-supplement in G.     

On s-semipermutable subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. We say that H is ssemipermutable in G if HGv = GpH for any Sylow p-subgroup Gp of G with （p, ｜H｜） = 1. We investigate the influence of s-semipermutable subgroups on the structure of finite groups. Some recent results are generalized and unified.     

QUASI-PERMUTATION REPRESENTATIONS OF ALTERNATING AND SYMMETRIC GROUPS

The quantities c(G), q(G) and p(G) for finite groups were defined by H. Behravesh. In this article, these quantities for the alternating group An and the symmetric group Sn are calculated. It is shown that c(G) = q(G)=p(G) = n, when G = An or Sn.     

THE PLANCHEREL FORMULA FOR THE LINE BUNDLES ON SL(n + 1, R)/S(GL(1, R) × GL(n,R))

In this paper we obtain the Plancherel formula for the spaces of L~2-sections of the line bundles over the pseudo-Riemannian symmetric space G/H where G = SL(n + 1, R)and H = S(GL(1, R) × GL(n, R)). The Plancherel formula is given in an explicit form by means of spherical distributions associated with the character χ——λof the subgroup H. We follow the method of Faraut, Kosters and van Dijk.     

On s-quasinormal and c-normal subgroups of a finite group

Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown： （1） G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; （2） G ∈F if and only if there is a soluble normal subgroup H such that G/H∈F and every maximal subgroup of all Sylow subgroups of F（H）, the Fitting subgroup of H, is either e-normally or s-quasinormally embedded in G.     

SUBGROUPS OF CLASS GROUPS OF ALGEBRAIC QUADRATIC FUNCTION FIELDS

Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessary and sufficient condition is given for the class group H(K) to contain a cyclic subgroup of any order n, which holds true for both real and imaginary fields K. Then several series of function fields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, are given, for which the class groups H(K) are proved to contain cyclic subgroups of order n.     

MEROMORPHIC SOLUTIONS OF A TYPE OF SYSTEM OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

Applying Nevanlinna theory of the value distribution of meromorphic functions,we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form∑nj=1aj(z)f1(λj1)(z+cj) = R2(z, f2(z)),∑nj=1βj(z)f2(λj2)(z+cj)=R1(Z,F1(z)).(*)where λij(j = 1, 2, ···, n; i = 1, 2) are finite non-negative integers, and cj(j = 1, 2, ···, n)are distinct, nonzero complex numbers, αj(z), βj(z)(j = 1, 2, ···, n) are small functions relative to fi(z)(i = 1, 2) respectively, Ri(z, f(z))(i = 1, 2) are rational in fi(z)(i = 1, 2)with coefficients which are small functions of fi(z)(i = 1, 2) respectively.     

Normal Families and Uniqueness of Entire Functions and Their Derivatives

Let f be a nonconstant entire function; let k ≥ 2 be a positive integer; and let a be a nonzero complex number. If f（z） = a→f′（z） = a, and f′（z） = a →f^（k）（z） = a, then either f = Ce^λz ＋ a or f = Ce^λz ＋ a（λ - 1）/）λ, where C and ）, are nonzero constants with λ^k-1 = 1. The proof is based on the Wiman-Vlairon theory and the theory of normal families in an essential way.     

High-dimensional D. H. Lehmer problem over short intervals

Letk be a positive integer and n a nonnegative integer,0 λ1,...,λk+1 ≤ 1 be real numbers and w =(λ1,λ2,...,λk+1).Let q ≥ max{[1/λi ]:1 ≤ i ≤ k + 1} be a positive integer,and a an integer coprime to q.Denote by N(a,k,w,q,n) the 2n-th moment of(b1··· bk c) with b1··· bk c ≡ a(mod q),1 ≤ bi≤λiq(i = 1,...,k),1 ≤ c ≤λk+1 q and 2(b1+ ··· + bk + c).We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N(a,k,w,q,n),which generalized the result of Zhang.Then we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N(a,k,w,q,n) in the case ofw =(1/2,1/2,...,1/2) and n = 0.In order to show our result is close to the best possible,the mean-square value of N(a,k,q) φk(q)/2k+2and the mean value weighted by the high-dimensional Cochrane sum are studied too.