Relations for Bernoulli--Barnes Numbers and Barnes Zeta Functions

The \emph{Barnes $\zeta$-function} is \zeta_n (z, x; \a) := \sum_{\m \in \Z_{\ge 0}^n} \frac{1}{\left(x + m_1 a_1 + \dots + m_n a_n \right)^z} defined for $\Re(x) > 0$ and $\Re(z) > n$ and continued meromorphically to \C. Specialized at negative integers $-k$, the Barnes $\zeta$-function gives \zeta_n (-k, x; \a) = \frac{(-1)^n k!}{(k+n)!} \, B_{k+n} (x; \a) where B_k(x; \a) is a \emph{Bernoulli--Barnes polynomial}, which can be also defined through a generating function that has a slightly more general form than that for Bernoulli polynomials. Specializing B_k(0; \a) gives the \emph{Bernoulli--Barnes numbers}. We exhibit relations among Barnes $\zeta$-functions, Bernoulli--Barnes numbers and polynomials, which generalize various identities of Agoh, Apostol, Dilcher, and Euler.Comment: 11 page

A Philip Glass Retrospective: Paul Barnes, piano, December 3, 2016

This is the concert program of the A Philip Glass Retrospective: Paul Barnes, piano performance on Saturday, December 3, 2016 at 4:30 p.m., at the Marshall Room, 855 Commonwealth Avenue. Works performed were the following by Philip Glass: Etudes 6, 8, 11, 16, 18 and 20 from "The Complete Piano Etudes," II. Conclusion from "Satyagraha" from "Trilogy Sonata" (arranged by Paul Barnes), II. Orphée's Bedroom and IV. Orphée and the Princess from "Orphée Suite for Piano" (arranged by P. Barnes), and III. The Land from Piano Concerto No. 2 (After Lewis and Clark) (arranged by P. Barnes). Digitization for Boston University Concert Programs was supported by the Boston University Humanities Library Endowed Fund

A Study of Spiders (Araneae) on Maple Trees (Acer Spp.)

We began this study to determine whether spider species occur randomly on maple species or whether they are selective in picking either their host species or their position on that host. Several papers have been published on habitat selection of spiders in relation to structural components or microclimate (Barnes, 1953; Barnes & ,Barnes, 1954, 1955; Cherrett, 1964; Duffey, 1962a. 1962b, 1966; Hackmann, 1957; Kuenzler, 1958; Norgaard, 1951). Duffey (1956) and Kuenzler (1958) also discussed the influence of microclimate on the activities of spiders. None of the above, however, dealt with arboreal spiders with the exception of Duffey (1956) who discussed aerial dispersal rather than habitat selection

Being Metaphysically Unsettled: Barnes and Williams on Metaphysical Indeterminacy and Vagueness

This chapter discusses the defence of metaphysical indeterminacy by Elizabeth Barnes and Robert Williams and discusses a classical and bivalent theory of such indeterminacy. Even if metaphysical indeterminacy arguably is intelligible, Barnes and Williams argue in favour of it being so and this faces important problems. As for classical logic and bivalence, the chapter problematizes what exactly is at issue in this debate. Can reality not be adequately described using different languages, some classical and some not? Moreover, it is argued that the classical and bivalent theory of Barnes and Williams does not avoid the problems that arise for rival theories

q-Analogues of the Barnes multiple zeta functions

In this paper, we introduce $q$-analogues of the Barnes multiple zeta functions. We show that these functions can be extended meromorphically to the whole plane, and moreover, tend to the Barnes multiple zeta functions when $q\uparrow 1$ for all complex numbers.Comment: 13 page

New proofs for the two Barnes lemmas and an additional lemma

Mellin-Barnes (MB) representations have become a widely used tool for the evaluation of Feynman loop integrals appearing in perturbative calculations of quantum field theory. Some of the MB integrals may be solved analytically in closed form with the help of the two Barnes lemmas which have been known in mathematics already for one century. The original proofs of these lemmas solve the integrals by taking infinite series of residues and summing these up via hypergeometric functions. This paper presents new, elegant proofs for the Barnes lemmas which only rely on the well-known basic identity of MB representations, avoiding any series summations. They are particularly useful for presenting and proving the Barnes lemmas to students of quantum field theory without requiring knowledge on hypergeometric functions. The paper also introduces and proves an additional lemma for a MB integral \int dz involving a phase factor exp(+-i pi z).Comment: 6 page

Barnes Hospital Bulletin

https://digitalcommons.wustl.edu/bjc_barnes_bulletin/1071/thumbnail.jp