A Note on Euler's Approximations

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.     

A Product Formula for Euler's Totient

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.     

A generalization of Euler's hypergeometric transformation

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.

     

A Discrete Analog of Euler's Summation Formula

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 P_n and special polynomials P_n(x) introduced by Korobov in 1996 appear in the formula.     

NURBS曲线的一个插值性质

本介绍了非均匀有理B样条曲线，并给出了非均匀有理B样条曲线的一个插值性质。     

Extension of Euler's beta function

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.     

Statistical solutions of Euler's equations

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.     

Weighted forms of Euler's theorem

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.     

A Fast Augmented Lagrangian Method for Euler's Elastica Models

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.     

A probabilistic approach to some of Euler's number theoretic identities

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.

     

Euler's Function in Residue Classes

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.     

On the iterates of Euler's function

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.     

Criteria for irrationality of Euler's constant

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.