Weighted lattice point sums in lattice polytopes, unifying Dehn-Sommerville and Ehrhart-Macdonald

Let $V$ be a real vector space of dimension $n$ and let $M\subset V$ be a lattice. Let $P\subset V$ be an $n$-dimensional polytope with vertices in $M$, and let \varphi\colon V\rightarrow \CC be a homogeneous polynomial function of degree $d$ (i.e., an element of \Sym^{d} (V^{*})). For q\in \ZZ_{>0} and any face $F$ of $P$, let $D_{\varphi ,F} (q)$ be the sum of $\varphi$ over the lattice points in the dilate $qF$. We define a generating function G_{\varphi}(q,y) \in \QQ [q] [y] packaging together the various $D_{\varphi ,F} (q)$, and show that it satisfies a functional equation that simultaneously generalizes Ehrhart--Macdonald reciprocity and the Dehn--Sommerville relations. When $P$ is a simple lattice polytope (i.e., each vertex meets $n$ edges), we show how $G_{\varphi}$ can be computed using an analogue of Brion--Vergne's Euler--Maclaurin summation formula

Revisiting Knowledge, Skills, and Abilities Needed for Development and Delivery Project Staff

This paper is grounded on the proposition that quality and timeliness of provisioning business information system solutions can be advanced by staffing development projects with personnel based on appropriate task related Knowledge, Skills, Abilities and Personal Characteristics (KSA-P). Defining a standard repeatable process for such staffing decisions requires a consistent classification scheme for the KSA-Ps, which this paper develops through a meta-analysis of the relevant literature. A nominal group of CIOs and consulting principals provide additional support for the validity of the classification scheme. The role of general and specific experience in skill and ability development is explored. Implications and future directions of the research are discussed

On Witten multiple zeta-functions associated with semisimple Lie algebras IV

In our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types $A_2$, $A_3$, $B_2$, $B_3$ and $C_3$. In this paper, we consider the case of $G_2$-type. We define certain analogues of Bernoulli polynomials of $G_2$-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of $G_2$-type. Next we consider the meromorphic continuation of the zeta-function of $G_2$-type and determine its possible singularities. Finally, by using our previous method, we give explicit functional relations for them which include Witten's volume formulas.Comment: 22 pag

Hecke operators and Hilbert modular forms Hecke operators and Hilbert modular forms

Abstract. Let F be a real quadratic field with ring of integers Ø and with class number 1. Let Γ be a congruence subgroup of GL2(Ø). We describe a technique to compute the action of the Hecke operators on the cohomology H 3 (Γ ; C). For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way to compute the Hecke action on these Hilbert modular forms