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.

By using Euler-Maclaurin's summation formula and the way of real analysis, a more accurate multipleHilbert-type inequality and the equivalent form are given. We also prove that the same constantfactor in the equivalent inequalities is the best possible.

In this paper, I construct noncompact analogs of the Chern classes of equivariant vector bundles over complex reductive groups. Then “Chern classes” of the tangent bundle are used to carry over to the case of an arbitrary reductive group some of the well-known results that hold for a complex torus. One of the results of this paper is a formula for the Chern classes of all regular equivariant co...

In the case of a 2-cell embedding (i.e., in the case where every region of the embedding is homeomorphic to a disk), the'inequality can easily be derived from the Euler Formula, v e + f = 2 2"fwhere e is the number of edges and f is the number of 2-cells of the embedding): the Handshaking Lemma asserts 2e = dv, and applying the Handshaking Lemma to the dual graph yields 2e >. gf; incorporating ...

It is fairly well known that rotation in three dimensions can be expressed as a quadratic in a skew symmetric matrix via the Euler-Rodrigues formula. A generalized Euler-Rodrigues polynomial of degree 2n in a skew symmetric generating matrix is derived for the rotation matrix of tensors of order n. The Euler-Rodrigues formula for rigid body rotation is recovered by n = 1. A Cayley form of the n...

(with DF = 1 for F = P). As explained in [2], essential properties required on the operators DF are ”locality” and ”computability”. At each face F of P, the operator DF should depend only of the translation class modulo Z of the normal cone No(P, F ) to P at a generic point of F (if F is of codimension k, the normal cone is an affine cone of dimension k). In particular if P is an integral polyt...

Abstract. The Euler-MacLaurin summation formula relates a sum of a function to a corresponding integral, with a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Pla...

Let G be a median graph on n vertices and m edges and let k be the number of equivalence classes of the Djoković’s relation Θ defined on the edge-set of G. Then 2n − m − k ≤ 2. Moreover, 2n − m − k = 2 if and only if G is cube-free. A median graph is a connected graph such that, for every triple of vertices u, v, w, there is a unique vertex x lying on a geodesic (i.e. shortest path) between eac...

In this paper, we survey recent results on Gauss-Bonnet-Chern formulae and related issues for closed Riemannian manifolds with variable curvature. Among other things, we address the following problem: “if M is an oriented 2n-dimensional closed manifold with non-positive curvature, then is it true that its Euler number χ(M) satisfies the inequality (−1)χ(M) ≥ 0?” We will present some partial ans...

Dynamical systems can often be decomposed into loosely coupled subsystems. The system of ordinary differential equations (ODEs) modelling such a problem can then be partitioned corresponding to the subsystems, and the loose couplings can be exploited by special integration methods to solve the problem using a parallel computer or just solve the problem more efficiently than by standard methods....