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

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

Dynamic redox conditions control late Ediacaran metazoan ecosystems in the Nama Group, Namibia

The first appearance of skeletal metazoans in the late Ediacaran (~550 million years ago; Ma) has been linked to the widespread development of oxygenated oceanic conditions, but a precise spatial and temporal reconstruction of their evolution has not been resolved. Here we consider the evolution of ocean chemistry from ~550 to ~541. Ma across shelf-to-basin transects in the Zaris and Witputs Sub-Basins of the Nama Group, Namibia. New carbon isotope data capture the final stages of the Shuram/Wonoka deep negative C-isotope excursion, and these are complemented with a reconstruction of water column redox dynamics utilising Fe-S-C systematics and the distribution of skeletal and soft-bodied metazoans. Combined, these inter-basinal datasets provide insight into the potential role of ocean redox chemistry during this pivotal interval of major biological innovation.The strongly negative δ13C values in the lower parts of the sections reflect both a secular, global change in the C-isotopic composition of Ediacaran seawater, as well as the influence of 'local' basinal effects as shown by the most negative δ13C values occurring in the transition from distal to proximal ramp settings. Critical, though, is that the transition to positive δ13C values postdates the appearance of calcified metazoans, indicating that the onset of biomineralization did not occur under post-excursion conditions.Significantly, we find that anoxic and ferruginous deeper water column conditions were prevalent during and after the transition to positive δ13C that marks the end of the Shuram/Wonoka excursion. Thus, if the C isotope trend reflects the transition to global-scale oxygenation in the aftermath of the oxidation of a large-scale, isotopically light organic carbon pool, it was not sufficient to fully oxygenate the deep ocean.Both sub-basins reveal highly dynamic redox structures, where shallow, inner ramp settings experienced transient oxygenation. Anoxic conditions were caused either by episodic upwelling of deeper anoxic waters or higher rates of productivity. These settings supported short-lived and monospecific skeletal metazoan communities. By contrast, microbial (thrombolite) reefs, found in deeper inner- and mid-ramp settings, supported more biodiverse communities with complex ecologies and large skeletal metazoans. These long-lived reef communities, as well as Ediacaran soft-bodied biotas, are found particularly within transgressive systems, where oxygenation was persistent. We suggest that a mid-ramp position enabled physical ventilation mechanisms for shallow water column oxygenation to operate during flooding and transgressive sea-level rise. Our data support a prominent role for oxygen, and for stable oxygenated conditions in particular, in controlling both the distribution and ecology of Ediacaran skeletal metazoan communities

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

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