共查询到20条相似文献,搜索用时 10 毫秒
1.
2.
A Note on Euler's Approximations 总被引:1,自引:0,他引:1
We prove that Euler's approximations for stochastic differential equations on domains of
d
converge almost surely if the drift satisfies the monotonicity condition and the diffusion coefficient is Lipschitz continuous. 相似文献
3.
We review and generalize certain known facts about the intersectionsor projections of a given integral lattice with or into rationalsubspaces, and apply them to get a product formula for a numericalinvariant associated to subspaces of Rn which are orthogonalto ( 1, 1,..., 1) Rn. 相似文献
4.
Robert S. Maier 《Transactions of the American Mathematical Society》2006,358(1):39-57
Euler's transformation formula for the Gauss hypergeometric function is extended to hypergeometric functions of higher order. Unusually, the generalized transformation constrains the hypergeometric function parameters algebraically but not linearly. Its consequences for hypergeometric summation are explored. It has as a corollary a summation formula of Slater. From this formula new one-term evaluations of and are derived by applying transformations in the Thomae group. Their parameters are also constrained nonlinearly. Several new one-term evaluations of with linearly constrained parameters are derived as well.
5.
6.
7.
8.
In this paper, we prove a discrete analog of Euler's summation formula. The difference from the classical Euler formula is in that the derivatives are replaced by finite differences and the integrals by finite sums. Instead of Bernoulli numbers and Bernoulli polynomials, special numbers Pn and special polynomials Pn(x) introduced by Korobov in 1996 appear in the formula. 相似文献
9.
10.
A. V. Ustinov 《Mathematical Notes》2006,79(1-2):146-147
11.
12.
《Journal of Computational and Applied Mathematics》1997,78(1):19-32
An extension of Euler's beta function, analogous to the recent generalization of Euler's gamma function and Riemann's zeta function, for which the usual properties and representation are naturally and simply extended, is introduced. It is proved that the extension is connected to the Macdonald, error and Whittaker functions. In addition, the extended beta distribution is introduced. 相似文献
13.
N. B. Maslova 《Journal of Mathematical Sciences》1993,64(6):1240-1247
Solutions of nonstationary Euler equations on a torus in R3
are investigated. For a broad class of random initial velocity fields a measure in L2(× [0, T]) is constructed, the moments of which for any finite T are solutions of the corresponding chain of moment equations.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 27–37, 1990. 相似文献
14.
William Y.C. Chen 《Journal of Combinatorial Theory, Series A》2007,114(2):360-372
In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan's “lost” notebook, we obtain weighted forms of Euler's theorem on partitions with odd parts and distinct parts. This work is inspired by the insight of Andrews on the connection between Ramanujan's identities and Euler's theorem. Our combinatorial formulations of Ramanujan's identities rely on the notion of rooted partitions. Pak's iterated Dyson's map and Sylvester's fish-hook bijection are the main ingredients in the weighted forms of Euler's theorem. 相似文献
15.
In this paper, a fast algorithm for Euler's elastica functional is proposed,
in which the Euler's elastica functional is reformulated as a constrained
minimization problem. Combining the augmented Lagrangian method and operator
splitting techniques, the resulting saddle-point problem is solved by a serial
of subproblems. To tackle the nonlinear constraints arising in the model, a novel
fixed-point-based approach is proposed so that all the subproblems either is a
linear problem or has a closed-form solution. We show the good performance of our
approach in terms of speed and reliability using numerous numerical examples on
synthetic, real-world and medical images for image denoising, image inpainting
and image zooming problems. 相似文献
16.
Don Rawlings 《Transactions of the American Mathematical Society》1998,350(7):2939-2951
Probabilistic proofs and interpretations are given for the -binomial theorem, -binomial series, two of Euler's fundamental partition identities, and for -analogs of product expansions for the Riemann zeta and Euler phi functions. The underlying processes involve Bernoulli trials with variable probabilities. Also presented are several variations on the classical derangement problem inherent in the distributions considered.
17.
A. S. Fainleib 《Mathematical Notes》1967,1(6):428-432
18.
We discuss the distribution of integers n with (n) in a particular residue class, showing that if a residue class contains a multiple of 4, then it must contain infinitely many numbers (n). We get asymptotic formulae for the distribution of (n) in the various residue classes modulo 12. 相似文献
19.
On the iterates of Euler's function 总被引:1,自引:0,他引:1
R. Warlimont 《Archiv der Mathematik》2001,76(5):345-349
Asymptotic representations are given for the three sums ?n £ x j(n)/j(j(n))\textstyle\sum\limits \limits _{n\le x} \varphi (n)/\varphi \bigl (\varphi (n)\bigr ), ?n £ x j(n)/j(j(n))\textstyle\sum\limits \limits _{n\le x} \varphi (n)/\varphi \bigl (\varphi (n)\bigr ), ?n £ x log j(n)/j(j(n)) ; j\textstyle\sum\limits \limits _{n\le x}\ \log \, \varphi (n)/\varphi \bigl (\varphi (n)\bigr )\ ; \ \varphi is Euler's function. 相似文献
20.
Jonathan Sondow 《Proceedings of the American Mathematical Society》2003,131(11):3335-3344
By modifying Beukers' proof of Apéry's theorem that is irrational, we derive criteria for irrationality of Euler's constant, . For 0$">, we define a double integral and a positive integer , and prove that with the following are equivalent:
1. The fractional part of is given by for some .
2. The formula holds for all sufficiently large .
3. Euler's constant is a rational number.
A corollary is that if infinitely often, then is irrational. Indeed, if the inequality holds for a given (we present numerical evidence for and is rational, then its denominator does not divide . We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact . A by-product is a rapidly converging asymptotic formula for , used by P. Sebah to compute correct to 18063 decimals.