An Euler brick is a cuboid with integer side dimensions such that the face diagonals are integers. Already in 1740, families of Euler bricks have been found. Euler himself constructed more families. If the space diagonal of an Euler brick is an integer too, an Euler brick is called a perfect Euler brick. Nobody has found one. There might be none. Nevertheless, it is an entertaining sport to go ...

Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant KO -homology of M. The universal equivariant Euler characteristic of M, which lives in a group UG(M), counts the equivariant cells of M , taking the component structure of the var...

1 Localized Euler classes on topological orbifolds 5 1.1 Banach orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Localized Euler classes in finite dimensions . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Abstract transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Localized Euler classes on Banach orbifolds . . ...

Recently, Choi et al. 2008 have studied the q-extensions of the Apostol-Bernoulli and the ApostolEuler polynomials of order n and multiple Hurwitz zeta function. In this paper, we define Apostol’s type q-Euler numbers En,q,ξ and q-Euler polynomials En,q,ξ x . We obtain the generating functions of En,q,ξ and En,q,ξ x , respectively. We also have the distribution relation for Apostol’s type q-Eul...

This paper introduces a new graph problem to find an Optimal Euler Circuit (OEC) in an Euler graph. OEC is defined as the Euler circuit that maximizes the sum of contiguous costs along it, where the contiguous cost is assigned for each of the two contiguous edges incident to a vertex. We prove that the OEC problem is NP-complete. A polynomial time algorithm will be presented for the case of a g...

We investigate the large time behavior of an axisymmetric model for the 3D Euler equations. In [22], Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier-Stokes equations with swirl. This model shares many properties of the 3D incompressible Euler and Navier-Stokes equations. The main difference between the 3D model of Hou and Lei and the reformulated 3D Euler an...

Generating functions for the number of commuting m-tuples in the symmetric groups are obtained. We define a natural sequence of “orbifold Euler characteristics” for a finite group G acting on a manifold X . Our definition generalizes the ordinary Euler characteristic of X/G and the string-theoretic orbifold Euler characteristic. Our formulae for commuting m-tuples underlie formulas that general...

Copyright q 2012 J. Yang and Z. Zhuang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We rigorously justify a singular Euler-Poisson approximation of the incompressible Euler equations in the quasi-neutral regime for plasma phys...