Some Applications of the Bounded Convergence Theorem for an Introductory Course in Analysis

The Arzela bounded convergence theorem is the special case of the Lebesgue dominated convergence theorem in which the functions are assumed to be Riemann integrable

XTE J2123-058: A New Neutron Star X-Ray Transient

We report on optical and RXTE observations of a new high-latitude bursting X-ray transient, XTE J2123-058. We identified the optical counterpart, measured the 5.9573 hr orbital period and constrained the binary inclination and the source distance. The distance lower limit indicates that the source is at least 2.6 kpc from the Galactic plane, which is unusual for an LMXB. RXTE observations were made between June and August 1998 during the first detected outburst from this source. We describe correlations between X-ray timing and spectral properties and discuss the possibility that the propeller mechanism turned on during the decay of the outburst. During one of the RXTE observations, we detect a pair of high frequency QPOs at 847.1 +/- 5.5 Hz and 1102 +/- 13 Hz simultaneously. According to the beat frequency model, the QPO separation implies a neutron star spin period near 3.9 ms. As the X-ray flux decreases at the end of the outburst, the amplitude of the optical modulation increases significantly. This behavior can be explained if the size of the accretion disk decreases during the decay of the outburst.Comment: 24 pages, 9 figures, accepted by Ap

Thirty-two Goldbach Variations

We give thirty-two diverse proofs of a small mathematical gem--the fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also discuss various generalizations for multiple harmonic (Euler) sums and some of their many connections, thereby illustrating both the wide variety of techniques fruitfully used to study such sums and the attraction of their study.Comment: v1: 34 pages AMSLaTeX. v2: 41 pages AMSLaTeX. New introductory material added and material on inequalities, Hilbert matrix and Witten zeta functions. Errors in the second section on Complex Line Integrals are corrected. To appear in International Journal of Number Theory. Title change

Elliptic integral evaluations of Bessel moments

We record what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. More generally, we develop formulae for integrals of products of six or fewer Bessel functions. In consequence, we are able to discover and prove closed forms for $c_{n,k}:=\int_0^\infty t^k K_0^n(t) {\rm d}t$ with integers $n=1,2,3,4$ and $k\ge0$, obtaining new results for the even moments $c_{3,2k}$ and $c_{4,2k}$. We also derive new closed forms for the odd moments $s_{n,2k+1}:=\int_0^\infty t^{2k+1}I_0^{}(t) K_0^{n-1}(t) {\rm d}t$ with $n=3,4$ and for $t_{n,2k+1}:=\int_0^\infty t^{2k+1}I_0^2(t) K_0^{n-2}(t) {\rm d}t$ with $n=5$, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of $s_{5,2k+1}$, make substantial progress on the evaluation of $c_{5,2k+1}$, $s_{6,2k+1}$ and $t_{6,2k+1}$ and report more limited progress regarding $c_{5,2k}$, $c_{6,2k+1}$ and $c_{6,2k}$. In the process, we obtain 8 conjectural evaluations, each of which has been checked to 1200 decimal places. One of these lies deep in 4- dimensional quantum field theory and two are probably provable by delicate combinatorics. There remains a hard core of five conjectures whose proofs would be most instructive, to mathematicians and physicists alike.Comment: 51 pages, 1 Postscript figure, uses amsmath.sty, added reference

Earth BioGenome Project: Sequencing life for the future of life.

Increasing our understanding of Earth's biodiversity and responsibly stewarding its resources are among the most crucial scientific and social challenges of the new millennium. These challenges require fundamental new knowledge of the organization, evolution, functions, and interactions among millions of the planet's organisms. Herein, we present a perspective on the Earth BioGenome Project (EBP), a moonshot for biology that aims to sequence, catalog, and characterize the genomes of all of Earth's eukaryotic biodiversity over a period of 10 years. The outcomes of the EBP will inform a broad range of major issues facing humanity, such as the impact of climate change on biodiversity, the conservation of endangered species and ecosystems, and the preservation and enhancement of ecosystem services. We describe hurdles that the project faces, including data-sharing policies that ensure a permanent, freely available resource for future scientific discovery while respecting access and benefit sharing guidelines of the Nagoya Protocol. We also describe scientific and organizational challenges in executing such an ambitious project, and the structure proposed to achieve the project's goals. The far-reaching potential benefits of creating an open digital repository of genomic information for life on Earth can be realized only by a coordinated international effort

Why sequence all eukaryotes?

Life on Earth has evolved from initial simplicity to the astounding complexity we experience today. Bacteria and archaea have largely excelled in metabolic diversification, but eukaryotes additionally display abundant morphological innovation. How have these innovations come about and what constraints are there on the origins of novelty and the continuing maintenance of biodiversity on Earth? The history of life and the code for the working parts of cells and systems are written in the genome. The Earth BioGenome Project has proposed that the genomes of all extant, named eukaryotes-about 2 million species-should be sequenced to high quality to produce a digital library of life on Earth, beginning with strategic phylogenetic, ecological, and high-impact priorities. Here we discuss why we should sequence all eukaryotic species, not just a representative few scattered across the many branches of the tree of life. We suggest that many questions of evolutionary and ecological significance will only be addressable when whole-genome data representing divergences at all of the branchings in the tree of life or all species in natural ecosystems are available. We envisage that a genomic tree of life will foster understanding of the ongoing processes of speciation, adaptation, and organismal dependencies within entire ecosystems. These explorations will resolve long-standing problems in phylogenetics, evolution, ecology, conservation, agriculture, bioindustry, and medicine

Standards recommendations for the Earth BioGenome Project

A global international initiative, such as the Earth BioGenome Project (EBP), requires both agreement and coordination on standards to ensure that the collective effort generates rapid progress toward its goals. To this end, the EBP initiated five technical standards committees comprising volunteer members from the global genomics scientific community: Sample Collection and Processing, Sequencing and Assembly, Annotation, Analysis, and IT and Informatics. The current versions of the resulting standards documents are available on the EBP website, with the recognition that opportunities, technologies, and challenges may improve or change in the future, requiring flexibility for the EBP to meet its goals. Here, we describe some highlights from the proposed standards, and areas where additional challenges will need to be met