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1.
Cubic elliptic functions 总被引:1,自引:1,他引:0
Shaun Cooper 《The Ramanujan Journal》2006,11(3):355-397
The function
occurs in one of Ramanujan’s inversion formulas for elliptic integrals. In this article, a common generalization of the cubic
elliptic functions
is given. The function g1 is the derivative of Ramanujan’s function Φ (after rescaling), and χ3(n) = 0, 1 or −1 according as n≡ 0, 1 or 2 (mod 3), respectively, and |q| < 1. Many properties of the common generalization, as well as the functions g1 and g2, are proved.
2000 Mathematics Subject Classification Primary—33E05; Secondary—11F11, 11F27 相似文献
2.
Fa-en WU~ 《中国科学A辑(英文版)》2007,50(8):1078-1086
Let D be a bounded domain in an n-dimensional Euclidean space Rn. Assume that 0 < λ1 ≤λ2 ≤ … ≤ λκ ≤ … are the eigenvalues of the Dirichlet Laplacian operator with any order l{(-△)lu=λu, in D u=(δ)u/(δ)(→n)=…(δ)l-1u/(δ)(→n)l-1=0,on (δ)D.Then we obtain an upper bound of the (k 1)-th eigenvalue λκ 1 in terms of the first k eigenvalues.k∑i=1(λκ 1-λi) ≤ 1/n[4l(n 2l-2)]1/2{k∑i=1(λκ 1-λi)1/2λil-1/l k∑i=1(λκ 1-λi)1/2λ1/li}1/2.This ineguality is independent of the domain D. Furthermore, for any l ≥ 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang. 相似文献
3.
We determine the minimum length n
q
(k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n
q
(k, d) = g
q
(k, d) + 1 for when k is odd, for when k is even, and for .
This work was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD). (KRF-2005-214-C00175).
This research has been partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science
under Contract Number 17540129. 相似文献
4.
P. Ding 《Journal of Mathematical Sciences》2006,137(2):4645-4653
Let p be a prime number, n be a positive integer, and ƒ(x) = axk + bx. We put
where e(t) = exp(2πit). This special exponential sum has been widely studied in connection with Waring’s problem. We write n in the form n
= Qk + r, where 0 ≤ r ≤ k − 1 and Q ≥ 0. Let α = ord
p(k), β = ord
p(k − 1), and θ = ord
p(b). We define
and J = [ζ]. Moreover, we denote V = min(Q, J). Improving the preceding result, we establish the theorem. Theorem. Let k ≥ 2 and n ≥ 2. If p > 2, then
. An example showing that this result is best possible is given. Bibliography: 15 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 63–75. 相似文献
5.
6.
V. M. Babich 《Journal of Mathematical Sciences》2006,132(1):2-10
The paper is devoted to a detailed consideration of an ansatz known from the seventies:
where
Here the Dp are parabolic-cylinder functions. Analytic expressions in the first approximation for the wave field in the penumbra of the
wave reflected by an impedance or transparent cone are obtained. Bibliography: 11 titles.
Dedicated to P. V. Krauklis on the occasion of his seventieth birthday
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 9–22. 相似文献
7.
G. Kuba 《Archiv der Mathematik》2002,79(6):534-542
Let
(z ∈ ℝ). Further let λ denote a large real parameter. We show that for arbitrary real numbersk and α withk>=2.7013 and 0<α≦1,
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8.
We consider the question of evaluating the normalizing multiplier $$\gamma _{n,k} = \frac{1}{\pi }\int_{ - \pi }^\pi {\left( {\frac{{sin\tfrac{{nt}}{2}}}{{sin\tfrac{t}{2}}}} \right)^{2k} dt} $$ for the generalized Jackson kernel J n,k (t). We obtain the explicit formula $$\gamma _{n,k} = 2\sum\limits_{p = 0}^{\left[ {k - \tfrac{k}{n}} \right]} {( - 1)\left( {\begin{array}{*{20}c} {2k} \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k(n + 1) - np - 1} \\ {k(n - 1) - np} \\ \end{array} } \right)} $$ and the representation $$\gamma _{n,k} = \sqrt {\frac{{24}}{\pi }} \cdot \frac{{(n - 1)^{2k - 1} }}{{\sqrt {2k - 1} }}\left[ {1\frac{1}{8} \cdot \frac{1}{{2k - 1}} + \omega (n,k)} \right],$$ , where $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ . 相似文献
9.
Gensun Fang 《中国科学A辑(英文版)》1998,41(12):1272-1277
The Nikolskii type inequality for cardinal splines
is proved, which is exact in the sense of order, where ∈ ℒ
m,h
, and ℒ
m,k
is the space of cardinal splines with nodes
Project supported by the National Natural Science Foundation of China (Grant No. 19671012), and Doctoral Programme Foundation
of Institution of Higher Education. 相似文献
10.
Piotr Niemiec 《Rendiconti del Circolo Matematico di Palermo》2008,57(3):391-399
The aim of the paper is to prove that every f ∈ L
1([0,1]) is of the form f = , where j
n,k
is the characteristic function of the interval [k- 1 / 2
n
, k / 2
n
) and Σ
n=0∞Σ
k=12n
|a
n,k
| is arbitrarily close to ||f|| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on [0,1], then for any ɛ > 0 there exists a sequence (b
n,k
)
n≧0
k=1,...,2n
of real numbers such that and for each Lipschitz function g: [0,1] → ℝ (Theorem 3).
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