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21.
Generators of some Ramanujan formulas 总被引:2,自引:0,他引:2
Jesús Guillera 《The Ramanujan Journal》2006,11(1):41-48
In this paper we prove some Ramanujan type formulas for 1/π but without using the theory of modular forms. Instead we use
the WZ—method created by H. Wilf and D. Zeilberger and find some hypergeometric functions in two variables which are second
components of WZ—pairs than can be certified using Zeilberger's EKHAD package. These certificates have an additional property
which allows us to get generalized Ramanujan's type series which are routinely proven by computer. We call these second hypergeometric
components of the WZ—pairs generators. Finding generators seems a hard task but using a kind of experimental research (explained
below), we have succeeded in finding some of them. Unfortunately we have not found yet generators for the most impressive
Ramanujan's formulas. We also prove some interesting binomial sums for the constant 1/π2. Finally we rewrite many of the obtained series using pochhammer symbols and study the rate of convergence.
2000 Mathematics Subject Classification Primary—33C20 相似文献
22.
R. López-Martens J. Mauritsson P. Johnsson K. Varjú A. L’Huillier W. Kornelis J. Biegert U. Keller M. Gaarde K. Schafer 《Applied physics. B, Lasers and optics》2004,78(7-8):835-840
We characterize the temporal structure of high-order harmonic radiation on both the femtosecond and attosecond time scales. The harmonic emission is characterized by mixed-color two-photon ionization with an infrared femtosecond laser using a Mach–Zehnder interferometer where both pump and probe arms travel completely separate paths. In a first experiment, we measure the duration and chirp of individual harmonics. In a second experiment, we resolve, for the first time with this type of setup, the attosecond beating of several harmonics generated under conditions similar to the first experiment. We suggest that the results of both measurements can be combined to determine the full attosecond time structure of the harmonic emission. PACS 32.80.Rm; 42.65.Ky 相似文献
23.
In this work we study semifield planes of orderq
n(q=p
r
,p prime) with a collineation whose order is ap-primitive divisor ofq
n–1.Research supported in part by NSF Grant No. DMS-9107372Research supported in part by NSF grants RII-9014056, component IV of the EPSCoR of Puerto Rico grant and ARO grant for Cornell MSI. 相似文献
24.
Francisco Jesús Castro-Jiménez José Marı́a Ucha-Enrı́quez 《Comptes Rendus Mathematique》2004,338(6):461-466
We prove a duality theorem for some logarithmic -modules associated with a class of divisors. We also give some results for the locally quasi-homogeneous case. To cite this article: F.J. Castro-Jiménez, J.M. Ucha-Enr??quez, C. R. Acad. Sci. Paris, Ser. I 338 (2004). 相似文献
25.
Leibniz agebras are a generalization of Lie algebras, where no symmetry properties of the bracket are required. In this Letter we introduce a notion of R-matrices for this structure and the related Yang–Baxter equations, and discuss some of their basic properties. 相似文献
26.
We present here an improved version of the method introduced by the first author to derive
pointwise gradient estimates for the solutions of one-dimensional parabolic problems. After considering
a general qualinear equation in divergence form we apply the method to the case of a nonlinear
diffusion-convection equation. The conclusions are stated first for classical solutions and then for
generalized and mild solutions. In the case of unbounded initial datum we obtain several regularizing
effects for t > 0. Some unilateral pointwise gradient
estimates are also obtained. The case of
the Dirichlet problem is also considered. Finally, we collect, in the last section, several comments
showing the connections among these estimates and the study of the free boundaries
associated to the solutions of the diffusion-convection equation. 相似文献
27.
28.
29.
We evaluate the financial performance of most of the branch offices of a large European savings bank for a recent accounting
period. We employ a complementary pair of nonparametric techniques to evaluate their financial performance, in terms of their
ability to conserve on the expenses they incur in building their customer bases and providing customer services. We find variation
in the ability of branch offices to perform this task, and agreement on the identity of the laggard branches. We then employ
parametric techniques to determine that the list of indicators on which their financial performance is evaluated can be reduced
without statistically significant loss of information to bank management. Both findings suggest ways in which the bank can
increase the profitability of its branch network.
A previous version of this paper was presented at CORE, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, where
we received many helpful comments from the audience. We are grateful to three referees for their helpful comments as well. 相似文献
30.