A class of identities satisfied by both Bernoulli and Euler polynomials is established. Recurrence relations for Bernoulli and Euler numbers are derived.

An Euler equation is a difference or differential equation that is an intertemporal first-order condition for a dynamic choice problem. It describes the evolution of economic variables along an optimal path. It is a necessary but not sufficient condition for a candidate optimal path, and so is useful for partially characterizing the theoretical implications of a range of models for dynamic beha...

By always mapping data from lower dimensional space into higher or even infinite dimensional space, kernel k-means is able to organize data into groups when data of different clusters are not linearly separable. However, kernel k-means incurs the large scale computation due to the representation theorem, i.e. keeping an extremely large kernel matrix in memory when using popular Gaussian and spa...

Abstract. The purpose of this paper is to construct of λ-Euler numbers and polynomials by using fermionic expression of p-adic q-integral at q = −1. From these λ-Euler polynomials, we derive the harmonic sums of higher order. Finally, we investigate several interesting properties and relationships involving the classical as well as the generalized Euler numbers and polynomials. As an applicatio...

We consider the symplectic Euler method for Hamiltonian systems with holonomic constraints and its generalizations to a class of overdetermined differential-algebraic equations (ODAEs) which are nonlinear in the algebraic variables. It is shown that a straightforward generalization of the symplectic Euler method is inconsistent. A different non-trivial generalization of the symplectic Euler met...

The governing equation of turbulence, that we are interested in, is the incompressible 2D Navier– Stokes equation under periodic boundary conditions. We are particularly interested in investigating the dynamics of 2D Navier–Stokes equation in the infinite Reynolds number limit and of 2D Euler equation. Our approach is different from many other studies on 2D Navier–Stokes equation in which one s...

*Correspondence: [email protected] 3Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Republic of Korea Full list of author information is available at the end of the article Abstract We use the p-adic q-integral and group action to count the number of the generating functions of modified q-Euler polynomials in a prescribed set. Some generating function yield...

This note studies the behavior of Euler characteristics and of intersection homology Euler characterstics under proper morphisms of algebraic (or analytic) varieties. The methods also yield, for algebraic (or analytic) varieties, formulae comparing these two kinds of Euler characteristics. The main results are direct consequences of the calculus of constructible functions and Grothendieck group...