Fredholm Determinants, Differential Equations and Matrix Models

Orthogonal polynomial random matrix models of NxN hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (phi(x) psi(y) - psi(x) phi(y))/x-y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is a union of open intervals. The emphasis is on the determinants thought of as functions of the end-points of these intervals. We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as phi and psi satisfy a certain type of differentiation formula. There is also an exponential variant of this analysis which includes the circular ensembles of NxN unitary matrices.Comment: 34 pages, LaTeX using RevTeX 3.0 macros; last version changes only the abstract and decreases length of typeset versio

Dark Energy from structure: a status report

The effective evolution of an inhomogeneous universe model in any theory of gravitation may be described in terms of spatially averaged variables. In Einstein's theory, restricting attention to scalar variables, this evolution can be modeled by solutions of a set of Friedmann equations for an effective volume scale factor, with matter and backreaction source terms. The latter can be represented by an effective scalar field (`morphon field') modeling Dark Energy. The present work provides an overview over the Dark Energy debate in connection with the impact of inhomogeneities, and formulates strategies for a comprehensive quantitative evaluation of backreaction effects both in theoretical and observational cosmology. We recall the basic steps of a description of backreaction effects in relativistic cosmology that lead to refurnishing the standard cosmological equations, but also lay down a number of challenges and unresolved issues in connection with their observational interpretation. The present status of this subject is intermediate: we have a good qualitative understanding of backreaction effects pointing to a global instability of the standard model of cosmology; exact solutions and perturbative results modeling this instability lie in the right sector to explain Dark Energy from inhomogeneities. It is fair to say that, even if backreaction effects turn out to be less important than anticipated by some researchers, the concordance high-precision cosmology, the architecture of current N-body simulations, as well as standard perturbative approaches may all fall short in correctly describing the Late Universe.Comment: Invited Review for a special Gen. Rel. Grav. issue on Dark Energy, 59 pages, 2 figures; matches published versio

Guidebook for Rangitoto Island AQUA field trip, Auckland, 2016

Rangitoto is arguably Auckland’s most beautiful and omnipresent landscape feature. It is a symmetrical, ~6-km wide, basaltic shield volcano that last erupted ~550‒500 cal. yr BP (c. 1400‒1450 AD), not long after the arrival and settlement of Polynesians in the Auckland region (c. 1280 AD). It is by far the largest, and the youngest, volcano in the Auckland Volcanic Field (AVF). The AVF consists of approximately 53 individual eruptive centres, all of which are within the boundaries of the Auckland urban area. Recent research on cryptotephras (defined below) in sediments from Lake Pupuke on North Shore and in wetlands on Motutapu Island, and on a 150-m-long drill core obtained in February, 2014, has revealed that Rangitoto has a much more complex history that previously thought, and may be better viewed as a ‘volcanic complex with multiple episodes of eruptions’ (Linnell et al. 2016). In summary, (1) activity commenced c. 6000 cal. yr BP involving minor effusive and pyroclastic volcanism; (2) a voluminous shield-building phase took place from c. 650550 cal. yr BP (c. 13001400 AD), forming the main island ediface; and (3) the final phase of activity, from c. 550500 cal. yr BP (c. 14001450 AD), was explosive and less voluminous, producing scoria cones at the summit. The flora on Rangitoto is unique among the islands situated in the Hauraki Gulf because of the island’s young age, and the fact that technically Rangitoto is an ‘oceanic’ island. Its flora and fauna are derived entirely from long distance dispersal. The island contains some 582 vascular plant taxa of which 228 (39%) are indigenous. Various other special ecological features, and studies on plant succession and their drivers, make the island a truly fascinating place to visit. At this time of year, we should see many pohutukawa (Metrosideros excelsa) trees in flower

Fredholm determinants and pole-free solutions to the noncommutative Painleve' II equation

We extend the formalism of integrable operators a' la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi-infinite interval and to matrix integral operators with a kernel of the form E_1^T(x) E_2(y)/(x+y) thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painleve' II (recently introduced by Retakh and Rubtsov), a related noncommutative equation of Painleve' type. We construct a particular family of solutions of the noncommutative Painleve' II that are pole-free (for real values of the variables) and hence analogous to the Hastings-McLeod solution of (commutative) Painleve' II. Such a solution plays the same role as its commutative counterpart relative to the Tracy-Widom theorem, but for the computation of the Fredholm determinant of a matrix version of the Airy kernel.Comment: 46 pages, no figures (oddly

Clinical spectrum of SIX3-associated mutations in holoprosencephaly: correlation between genotype, phenotype and function

BACKGROUND: Holoprosencephaly (HPE) is the most common structural malformation of the human forebrain. There are several important HPE mutational target genes, including the transcription factor SIX3, which encodes an early regulator of Shh, Wnt, Bmp and Nodal signalling expressed in the developing forebrain and eyes of all vertebrates. OBJECTIVE: To characterise genetic and clinical findings in patients with SIX3 mutations. METHODS: Patients with HPE and their family members were tested for mutations in HPE-associated genes and the genetic and clinical findings, including those for additional cases found in the literature, were analysed. The results were correlated with a mutation-specific functional assay in zebrafish. RESULTS: In a cohort of patients (n = 800) with HPE, SIX3 mutations were found in 4.7% of probands and additional cases were found through testing of relatives. In total, 138 cases of HPE were identified, 59 of whom had not previously been clinically presented. Mutations in SIX3 result in more severe HPE than in other cases of non-chromosomal, non-syndromic HPE. An over-representation of severe HPE was found in patients whose mutations confer greater loss of function, as measured by the functional zebrafish assay. The gender ratio in this combined set of patients was 1.5:1 (F:M) and maternal inheritance was almost twice as common as paternal. About 14% of SIX3 mutations in probands occur de novo. There is a wide intrafamilial clinical range of features and classical penetrance is estimated to be at least 62%. CONCLUSIONS: Our data suggest that SIX3 mutations result in relatively severe HPE and that there is a genotype-phenotype correlation, as shown by functional studies using animal models

The Dynamics of Brane-World Cosmological Models

Brane-world cosmology is motivated by recent developments in string/M-theory and offers a new perspective on the hierarchy problem. In the brane-world scenario, our Universe is a four-dimensional subspace or {\em brane} embedded in a higher-dimensional {\em bulk} spacetime. Ordinary matter fields are confined to the brane while the gravitational field can also propagate in the bulk, leading to modifications of Einstein's theory of general relativity at high energies. In particular, the Randall-Sundrum-type models are self-consistent and simple and allow for an investigation of the essential non-linear gravitational dynamics. The governing field equations induced on the brane differ from the general relativistic equations in that there are nonlocal effects from the free gravitational field in the bulk, transmitted via the projection of the bulk Weyl tensor, and the local quadratic energy-momentum corrections, which are significant in the high-energy regime close to the initial singularity. In this review we discuss the asymptotic dynamical evolution of spatially homogeneous brane-world cosmological models containing both a perfect fluid and a scalar field close to the initial singularity. Using dynamical systems techniques it is found that, for models with a physically relevant equation of state, an isotropic singularity is a past-attractor in all orthogonal spatially homogeneous models (including Bianchi type IX models). In addition, we describe the dynamics in a class of inhomogeneous brane-world models, and show that these models also have an isotropic initial singularity. These results provide support for the conjecture that typically the initial cosmological singularity is isotropic in brane-world cosmology.Comment: Einstein Centennial Review Article: to appear in CJ

Efeito do alumínio sobre a absorção, o acúmulo e o fracionamento do fósforo em sorgo

O trabalho teve como objetivo estudar o efeito do Al sobre a absorção, o acúmulo e o fracionamento do P em duas cultivares de sorgo. As plantas foram expostas a níveis tóxicos de Al durante dez dias e, então, colhidas e determinados o crescimento em tamanho e produção de massa seca, os teores de Al e de P total e as diversas formas de P nas duas partes das plantas. Avaliou-se, também, o efeito do Al sobre a absorção de P pelas raízes de plantas intactas. O Al reduziu o crescimento da raiz seminal e a produção de matéria seca de raízes e parte aérea nas duas cultivares, especialmente na sensível. Os teores de Al e de P total aumentaram nas raízes, mas não foram modificados na parte aérea nas duas cultivares. A absorção de P, entretanto, decresceu na presença de Al nas duas cultivares, principalmente na sensível. O Al, de modo geral, modificou as concentrações das várias formas de P solúvel (Pi e Porg) e insolúvel (P RNA e Presidual), exceto a da forma P LIP. Algumas dessas modificações parecem ser importantes e podem estar relacionadas com o mecanismo de tolerância ao Al em sorgo.The objective of this work was to evaluate Al effect on uptake, accumulation and fractionation of P in two sorghum cultivars. Plants were treated with toxic levels of Al during ten days and then they were harvested and growth, dry matter yield, Al and total P contents and concentrations of the various P forms in the two parts of the plants were determined. Aluminum effect on P uptake was also evaluated in intact plants. Aluminum reduced the growth of the seminal root and dry matter yield in roots and tops of both cultivars, especially in the sensitive one. Aluminum and P contents increased in roots but did not change in the top of both cultivars. Phosphate uptake by roots, however, decreased in the presence of Al in both cultivars, especially in the sensitive one. Aluminum, in general, changed concentrations of all soluble (Pi e Porg) and insoluble P forms (P RNA e Presidual), except of the P LIP form. Some of these modifications seem to be important and may be related to Al tolerance mechanism in sorghum

Basic Methods for Computing Special Functions

This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website

Phylogenomics and the rise of the angiosperms

Angiosperms are the cornerstone of most terrestrial ecosystems and human livelihoods1,2. A robust understanding of angiosperm evolution is required to explain their rise to ecological dominance. So far, the angiosperm tree of life has been determined primarily by means of analyses of the plastid genome3,4. Many studies have drawn on this foundational work, such as classification and first insights into angiosperm diversification since their Mesozoic origins5,6,7. However, the limited and biased sampling of both taxa and genomes undermines confidence in the tree and its implications. Here, we build the tree of life for almost 8,000 (about 60%) angiosperm genera using a standardized set of 353 nuclear genes8. This 15-fold increase in genus-level sampling relative to comparable nuclear studies9 provides a critical test of earlier results and brings notable change to key groups, especially in rosids, while substantiating many previously predicted relationships. Scaling this tree to time using 200 fossils, we discovered that early angiosperm evolution was characterized by high gene tree conflict and explosive diversification, giving rise to more than 80% of extant angiosperm orders. Steady diversification ensued through the remaining Mesozoic Era until rates resurged in the Cenozoic Era, concurrent with decreasing global temperatures and tightly linked with gene tree conflict. Taken together, our extensive sampling combined with advanced phylogenomic methods shows the deep history and full complexity in the evolution of a megadiverse clade