共查询到20条相似文献,搜索用时 15 毫秒
1.
Akos Magyar 《Monatshefte für Mathematik》2002,136(4):287-296
Optimal lower bounds are given for the discrepancy of point distributions w.r.t. geodesic balls on spheres and hyperbolic
spaces. The mean discrepancy is estimated below by using a non-commutative version of the Fourier transform method developed
by Beck for Euclidean spaces.
Received March 2, 2000; in revised form February 28, 2002 Published online August 19, 2002 相似文献
2.
Yossi Moshe 《Journal of Number Theory》2007,123(1):224-240
Let H(x) be a monic polynomial over a finite field F=GF(q). Denote by Na(n) the number of coefficients in Hn which are equal to an element a∈F, and by G the set of elements a∈F× such that Na(n)>0 for some n. We study the relationship between the numbers (Na(n))a∈G and the patterns in the base q representation of n. This enables us to prove that for “most” n's we have Na(n)≈Nb(n), a,b∈G. Considering the case H=x+1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind. 相似文献
3.
4.
The number of visible (primitive) lattice points in the sphere of radius R is well approximated by . We consider an integral expression involving the error term E∗(R), which leads to E∗(R)=Ω(R(logR)1/2). This is comparable to what is known in the sphere problem. We can avoid the use of the second power moment (which is in this case unknown) by employing an auxiliary trigonometric series correlated to E∗(R). This approach to prove Ω-results seems to be new and could be useful in other problems. 相似文献
5.
Shabnam Akhtari 《Journal of Number Theory》2008,128(4):884-894
In this article, we study the cyclotomic polynomials of degree N−1 with coefficients restricted to the set {+1,−1}. By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p(x) with coefficients ±1 of even degree N−1 is cyclotomic if and only if p(x)=±Φp1(±x)Φp2(±xp1)?Φpr(±xp1p2?pr−1), where N=p1p2?pr and the pi are primes, not necessarily distinct. Here is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree α2pβ−1 with odd prime p or separable polynomials of any odd degree. 相似文献
6.
Werner Georg Nowak 《Archiv der Mathematik》2008,90(2):181-192
This article gives an asymptotic result for the lattice point discrepancy of a large body of rotation in , whose boundary is piecewise smooth and contains points of vanishing Gaussian curvature.
Received: 2 November 2006 相似文献
7.
Let X be a compact abelian group. A subgroup H of X is called characterized if there exists a sequence u=(un) of characters of X such that H=su(X), where su(X):={x∈X:(un,x)→0 in T}. Every characterized subgroup is an Fσδ-subgroup of X . We show that every Gδ-subgroup of X is characterized. On the other hand, X has non-characterized Fσ-subgroups. 相似文献
8.
Michael Aristidou 《Bulletin des Sciences Mathématiques》2006,130(3):246
Let Ω be a symmetric cone and V the corresponding simple Euclidean Jordan algebra. In our previous papers (some with G. Zhang) we considered the family of generalized Laguerre functions on Ω that generalize the classical Laguerre functions on R+. This family forms an orthogonal basis for the subspace of L-invariant functions in L2(Ω,dμν), where dμν is a certain measure on the cone and where L is the group of linear transformations on V that leave the cone Ω invariant and fix the identity in Ω. The space L2(Ω,dμν) supports a highest weight representation of the group G of holomorphic diffeomorphisms that act on the tube domain T(Ω)=Ω+iV. In this article we give an explicit formula for the action of the Lie algebra of G and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on R+. 相似文献
9.
In this paper, we obtain a generalization of an identity due to Carlitz on Bernoulli polynomials. Then we use this generalized formula to derive two symmetric identities which reduce to some known identities on Bernoulli polynomials and Bernoulli numbers, including the Miki identity. 相似文献
10.
In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich–Dickinson–Velani [6] and Vaughan–Velani [22]. Furthermore, we complete the Lebesgue theory of Diophantine approximation on weakly non-degenerate planar curves that was initially developed by Beresnevich–Zorin [5] in the divergence case. 相似文献
11.
Anthony To-Ming Lau Hiromichi Miyake Wataru Takahashi 《Nonlinear Analysis: Theory, Methods & Applications》2007
The purpose of this paper is to study iterative schemes of Browder and Halpern types for a semigroup of nonexpansive mappings on a compact convex subset of a smooth (and strictly convex) Banach space with respect to a sequence of strongly asymptotic invariant means defined on an appropriate space of bounded real valued functions of the semigroup. Various applications to the additive semigroup of nonnegative real numbers and commuting pairs of nonexpansive mappings are also presented. 相似文献
12.
M. Drmota 《Discrete Mathematics》2008,308(7):1191-1208
Let tj=(-1)s(j) be the Thue-Morse sequence with s(j) denoting the sum of the digits in the binary expansion of j. A well-known result of Newman [On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721] says that t0+t3+t6+?+t3k>0 for all k?0.In the first part of the paper we show that t1+t4+t7+?+t3k+1<0 and t2+t5+t8+?+t3k+2?0 for k?0, where equality is characterized by means of an automaton. This sharpens results given by Dumont [Discrépance des progressions arithmétiques dans la suite de Morse, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983) 145-148]. In the second part we study more general settings. For a,g?2 let ωa=exp(2πi/a) and , where sg(j) denotes the sum of digits in the g-ary digit expansion of j. We observe trivial Newman-like phenomena whenever a|(g-1). Furthermore, we show that the case a=2 inherits many Newman-like phenomena for every even g?2 and large classes of arithmetic progressions of indices. This, in particular, extends results by Drmota and Ska?ba [Rarified sums of the Thue-Morse sequence, Trans. Amer. Math. Soc. 352 (2000) 609-642] to the general g-case. 相似文献
13.
This paper describes a new approach to the problem of computing spherical expansions of zonal functions on Euclidean spheres.
We derive an explicit formula for the coefficients of the expansion expressing them in terms of the Taylor coefficients of
the profile function rather than (as done usually) in terms of its integrals against Gegenbauer polynomials. Our proof of
this result is based on a polynomial identity equivalent to the canonical decomposition of homogeneous polynomials and uses
only basic properties of this decomposition together with simple facts concerning zonal harmonic polynomials. As corollaries,
we obtain direct and apparently new derivations of the so-called plane wave expansion and of the expansion of the Poisson
kernel for the unit ball.
Received: 26 January 2007 相似文献
14.
In Schweiger (2003) [1], Fritz Schweiger introduced the algorithm of the generalized continued fraction (GCF), and in Zhong (2008) [2], T. Zhong studied some basic metric properties of the GCF. In this paper, under the restriction of −1<ε(k)?1, the “0-1” law and the central limit theorem of quotients in the GCF expansions are studied. 相似文献
15.
Let G(p,n) and G(q,n) be the affine Grassmann manifolds of p- and q-planes in Rn, respectively, and let be the Radon transform from smooth functions on G(p,n) to smooth functions on G(q,n) arising from the inclusion incidence relation. When p<q and dimG(p,n)=dimG(p,n), we present a range characterization theorem for via moment conditions. We then use this range result to prove a support theorem for . This complements a previous range characterization theorem for via differential equations when dimG(p,n)<dimG(p,n). We also present a support theorem in this latter case. 相似文献
16.
Bei Zhang 《Journal of Number Theory》2011,131(3):419-439
In this paper, I discuss the construction of the p-adic L-function attached to a Hilbert modular form f, supersingular or ordinary, which turns out to be the non-archimedean Mellin transform of an h-admissible measure. And h is explicitly given. As a special case, when the Fourier coefficient of f at p|p is zero, plus/minus p-adic L-functions are furthermore defined as bounded functions, and they interpolate special values of L(f,χ,s) for cyclotomic characters χ. This can be used to formulate Iwasawa main conjecture for supersingular elliptic curve defined over a totally real field. 相似文献
17.
Text
Extending recent work of others, we provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato-Tate law. We present two methods of proof. Both use the framework of Murty and Sinha (2009) [MS]; the first involves only knowledge of the moments of the Fourier coefficients of the L-functions and combinatorics, and saves a logarithm, while the second requires a Sato-Tate law. Our purpose is to illustrate how the caliber of the result depends on the error terms of the inputs and what combinatorics must be done.Video
For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=faW2iDpe5IE. 相似文献18.
Let f1,…,fd be an orthogonal basis for the space of cusp forms of even weight 2k on Γ0(N). Let L(fi,s) and L(fi,χ,s) denote the L-function of fi and its twist by a Dirichlet character χ, respectively. In this note, we obtain a “trace formula” for the values at integers m and n with 0<m,n<2k and proper parity. In the case N=1 or N=2, the formula gives us a convenient way to evaluate precisely the value of the ratio L(f,χ,m)/L(f,n) for a Hecke eigenform f. 相似文献
19.
Martijn de Vries 《Topology and its Applications》2009,156(3):652-657
Let q>1 be a real number and let m=m(q) be the largest integer smaller than q. It is well known that each number can be written as with integer coefficients 0?ci<q. If q is a non-integer, then almost every x∈Jq has continuum many expansions of this form. In this note we consider some properties of the set Uq consisting of numbers x∈Jq having a unique representation of this form. More specifically, we compare the size of the sets Uq and Ur for values q and r satisfying 1<q<r and m(q)=m(r). 相似文献
20.
Hao Pan 《Journal of Number Theory》2008,128(6):1646-1654
Let e?1 and b?2 be integers. For a positive integer with 0?aj<b, define