A form of the Euler equation using an impulse formulation is presented. This form is based on a representation of the divergence-free projection operator in terms of a continuous distribution of vortex dipoles which have a finite self-induced velocity. A generalization of the Euler equation is presented as a kinetic equation similar to the Vlasov–Poisson equation. An interesting feature of this...

Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol-Bernoulli and between Apostol-Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.

Finite element formulations based generally on classical beam theories such as Euler-Bernoulli or Timoshenko. Sometimes, these two formulations could be problematic expressed in terms of restrictions of Euler-Bernoulli beam theory, in case of thicker beams due to non-consideration of transverse shear; phenomenon that is known as shear locking characterized the Timoshenko beam theory, in case of...

We propose a new approach to quantize the marginals of the discrete Euler diffusion process. The method is built recursively and involves the conditional distribution of the marginals of the discrete Euler process. Analytically, the method raises several questions like the analysis of the induced quadratic quantization error between the marginals of the Euler process and the proposed quantizati...

We present a new method for stability and modal analysis of shear flows and their acoustic radiation. The Euler equations are modified and solved as a spatial initial value problem in which initial perturbations are specified at the flow inlet and propagated downstream by integration of the equations. The modified equations, which we call one-way Euler equations, differ from the usual Euler equ...

In this paper, the concepts of Euler numbers and Euler polynomials are generalized, and some basic properties are investigated.

If qα is the Euler factor of an odd perfect number N, then we prove that its so-called index σ(N/qα) � qα ≥ 32 × 5× 7 = 315. It follows that for any odd perfect number, the ratio of the non-Euler part to the Euler part is greater than 32 × 5× 7/2.

The purpose of this paper is to construct new q-Euler numbers and polynomials. Finally we will consider the Witt's type formula associated with these q-Euler numbers and polynomials, and construct q-partial zeta functions and p-adic q-l-functions which interpolate new q-Euler numbers and polynomials at negative integers.

The aim oj this article is to study the convergence oj the Euler harmonic series. Firstly, the results concerning the convergence oj the Smaralldache and Erdos harmonic junctions are reviewed Secondly, the Euler harmonic series is proved to be convergent jor a> I, and divergent otherwise. Finally, the slims of the Euler harmonic series are given

Euler diagrams have been used for centuries as a means for conveying ideas in an intuitive, informal way. Recently much research has been conducted to develop formal, diagrammatic reasoning systems based on Euler diagrams. Most of these systems extend Euler diagrams by adding further syntax to increase expressiveness. In this paper we survey such systems and draw comparisons between them.