Faces of matrix models

Partition functions of eigenvalue matrix models possess a number of very different descriptions: as matrix integrals, as solutions to linear and non-linear equations, as tau-functions of integrable hierarchies and as special-geometry prepotentials, as result of the action of W-operators and of various recursions on elementary input data, as gluing of certain elementary building blocks. All this explains the central role of such matrix models in modern mathematical physics: they provide the basic "special functions" to express the answers and relations between them, and they serve as a dream model of what one should try to achieve in any other field.Comment: 10 page

Glassy Random Matrix Models

This paper discusses Random Matrix Models which exhibit the unusual phenomena of having multiple solutions at the same point in phase space. These matrix models have gaps in their spectrum or density of eigenvalues. The free energy and certain correlation functions of these models show differences for the different solutions. Here I present evidence for the presence of multiple solutions both analytically and numerically. As an example I discuss the double well matrix model with potential $V(M)= -{\mu \over 2}M^2+{g \over 4}M^4$ where $M$ is a random $N\times N$ matrix (the $M^4$ matrix model) as well as the Gaussian Penner model with $V(M)={\mu\over 2}M^2-t \ln M$. First I study what these multiple solutions are in the large $N$ limit using the recurrence coefficient of the orthogonal polynomials. Second I discuss these solutions at the non-perturbative level to bring out some differences between the multiple solutions. I also present the two-point density-density correlation functions which further characterizes these models in a new university class. A motivation for this work is that variants of these models have been conjectured to be models of certain structural glasses in the high temperature phase.Comment: 25 pages, Latex, 7 Figures, to appear in PR

Two-Dimensional QCD in the Wu-Mandelstam-Leibbrandt Prescription

We find the exact non-perturbative expression for a simple Wilson loop of arbitrary shape for U(N) and SU(N) Euclidean or Minkowskian two-dimensional Yang-Mills theory regulated by the Wu-Mandelstam-Leibbrandt gauge prescription. The result differs from the standard pure exponential area-law of YM_2, but still exhibits confinement as well as invariance under area-preserving diffeomorphisms and generalized axial gauge transformations. We show that the large N limit is NOT a good approximation to the model at finite N and conclude that Wu's N=infinity Bethe-Salpeter equation for QCD_2 should have no bound state solutions. The main significance of our results derives from the importance of the Wu-Mandelstam-Leibbrandt prescription in higher-dimensional perturbative gauge theory.Comment: 7 pages, LaTeX, REVTE

Moments of vicious walkers and M\"obius graph expansions

A system of Brownian motions in one-dimension all started from the origin and conditioned never to collide with each other in a given finite time-interval $(0, T]$ is studied. The spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue-statistics of random Hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian orthogonal ensemble (GOE) statistics as the time $t$ is going on from 0 to $T$. In the present paper, we characterize this GUE-to-GOE transition by presenting the graphical expansion formula for the moments of positions of vicious walkers. In the GUE limit $t \to 0$, only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus. Following the time evolution of the vicious walkers, however, the graphs with twisted ribbons, called M\"obius graphs, increase their contribution to our expansion formula, and we have to deal with the topology of non-orientable surfaces. Application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the M\"obius expansion using the Stirling numbers of the first kind.Comment: REVTeX4, 11 pages, 1 figure. v.2: calculations of the Green function and references added. v.3: minor additions and corrections made for publication in Phys.Rev.

Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices

Spatially and temporally inhomogeneous evolution of one-dimensional vicious walkers with wall restriction is studied. We show that its continuum version is equivalent with a noncolliding system of stochastic processes called Brownian meanders. Here the Brownian meander is a temporally inhomogeneous process introduced by Yor as a transform of the Bessel process that is a motion of radial coordinate of the three-dimensional Brownian motion represented in the spherical coordinates. It is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue-statistics of Gaussian ensembles of Bogoliubov-deGennes Hamiltonians of the mean-field theory of superconductivity, which have the particle-hole symmetry. We report that the time evolution of the present stochastic process is fully characterized by the change of symmetry classes from the type $C$ to the type $C$I in the nonstandard classes of random matrix theory of Altland and Zirnbauer. The relation between the non-colliding systems of the generalized meanders of Yor, which are associated with the even-dimensional Bessel processes, and the chiral random matrix theory is also clarified.Comment: REVTeX4, 16 pages, 4 figures. v2: some additions and correction

Exploring the Genetic Cipher of Chickpea (Cicer arietinum L.) Through Identification and Multi-environment Validation of Resistant Sources Against Fusarium Wilt (Fusarium oxysporum f. sp. ciceris)

Fusarium wilt (Fusarium oxysporum f. sp. ciceris) of chickpea is the major limitation to chickpea production worldwide. As the nature of the pathogen is soil borne, exploitation of host plant resistance is the most suitable and economical way to manage this disease. Present study was therefore conducted with an aim to find new, stable and durable sources of resistance of chickpea against Fusarium wilt through multi-environment and multi-year screening. During 2007/2008 crop season, 130 promising genotypes having <10% wilt incidence were selected from initial evaluation of 893 chickpea genotypes in wilt sick plot at ICRISAT, Patancheru. Of them 61 highly resistant lines were selected through further evaluation in 2008/2009 and 2009/2010 crop season. Finally, a set of 31 genotypes were selected to constitute a Chickpea Wilt Nursery (CWN) and tested at 10 locations in India for three cropping seasons (2010/2011, 2011/2012 and 2012/2013) coordinated through Indian Council of Agricultural Research (ICAR) and ICRISAT collaboration. The genotype and genotype × environment interaction (GGE) indicated significant variations (p ≤ 0.001) due to genotype × environment (G × E) interaction. Most of genotypes were resistant at two locations, ICRISAT (Patancheru) and Badnapur. On the contrary most of them were susceptible at Dholi and Kanpur indicating the variability in pathogen. GGE biplot analyses allowed the selection six genotypes ICCVs 98505, 07105, 07111, 07305, 08113, and 93706 with high resistance and stability across most of the locations and eight moderately resistant (<20% mean incidence) genotypes viz., ICCVs 08123, 08125, 96858, 07118, 08124, 04514, 08323, and 08117. As chickpea is grown in diverse agro-ecological zones and environments; these stable/durable sources can be used in future resistance breeding program to develop Fusarium wilt resistant cultivars

Achievements and prospects of genomics-assisted breeding in three legume crops of the semi-arid tropics

Advances in next-generation sequencing and genotyping technologies have enabled generation of large-scale genomic resources such as molecular markers, transcript reads and BAC-end sequences (BESs) in chickpea, pigeonpea and groundnut, three major legume crops of the semi-arid tropics. Comprehensive transcriptome assemblies and genome sequences have either been developed or underway in these crops. Based on these resources, dense genetic maps, QTL maps as well as physical maps for these legume species have also been developed. As a result, these crops have graduated from ‘orphan’ or ‘less-studied’ crops to ‘genomic resources rich’ crops. This article summarizes above mentioned advances in genomics and genomics-assisted breeding applications in the form of marker-assisted selection (MAS) for hybrid purity assessment in pigeonpea; marker-assisted backcrossing (MABC) for introgressing QTL region for drought-tolerance related traits, Fusarium wilt (FW) resistance and Ascochyta blight (AB) resistance in chickpea; late leaf spot (LLS), leaf rust and nematode resistance in groundnut. We critically present the case of use of other modern breeding approaches like marker-assisted recurrent selection (MARS) and genomic selection (GS) to utilize the full potential of genomics-assisted breeding for developing superior cultivars with enhanced tolerance to various environmental stresses. In addition, this article recommends the use of advanced- backcross (AB-backcross) breeding and development of specialized populations such as multi-parents advanced generation intercross (MAGIC) for creating new variations that will help developing superior lines with broadened genetic base. In summary, we propose the use of integrated genomics and breeding approach in these legume crops to enhance crop productivity in marginal environments ensuring food security in developing countries

Neurodevelopmental toxicity of prenatal polychlorinated biphenyls (PCBs) by chemical structure and activity: a birth cohort study

Abstract Background Polychlorinated biphenyls (PCBs) are ubiquitous environmental toxins. Although there is growing evidence to support an association between PCBs and deficits of neurodevelopment, the specific mechanisms are not well understood. The potentially different roles of specific PCB groups defined by chemical structures or hormonal activities e.g., dioxin-like, non-dioxin like, or anti-estrogenic PCBs, remain unclear. Our objective was to examine the association between prenatal exposure to defined subsets of PCBs and neurodevelopment in a cohort of infants in eastern Slovakia enrolled at birth in 2002-2004. Methods Maternal and cord serum samples were collected at delivery, and analyzed for PCBs using high-resolution gas chromatography. The Bayley Scales of Infant Development -II (BSID) were administered at 16 months of age to over 750 children who also had prenatal PCB measurements. Results Based on final multivariate-adjusted linear regression model, maternal mono-ortho-substituted PCBs were significantly associated with lower scores on both the psychomotor (PDI) and mental development indices (MDI). Also a significant association between cord mono-ortho-substituted PCBs and reduced PDI was observed, but the association with MDI was marginal (p = 0.05). Anti-estrogenic and di-ortho-substituted PCBs did not show any statistically significant association with cognitive scores, but a suggestive association between di-ortho-substituted PCBs measured in cord serum and poorer PDI was observed. Conclusion Children with higher prenatal mono-ortho-substituted PCB exposures performed more poorly on the Bayley Scales. Evidence from this and other studies suggests that prenatal dioxin-like PCB exposure, including mono-ortho congeners, may interfere with brain development in utero. Non-dioxin-like di-ortho-substituted PCBs require further investigation

Guidelines for Genome-Scale Analysis of Biological Rhythms

Genome biology approaches have made enormous contributions to our understanding of biological rhythms, particularly in identifying outputs of the clock, including RNAs, proteins, and metabolites, whose abundance oscillates throughout the day. These methods hold significant promise for future discovery, particularly when combined with computational modeling. However, genome-scale experiments are costly and laborious, yielding “big data” that are conceptually and statistically difficult to analyze. There is no obvious consensus regarding design or analysis. Here we discuss the relevant technical considerations to generate reproducible, statistically sound, and broadly useful genome-scale data. Rather than suggest a set of rigid rules, we aim to codify principles by which investigators, reviewers, and readers of the primary literature can evaluate the suitability of different experimental designs for measuring different aspects of biological rhythms. We introduce CircaInSilico, a web-based application for generating synthetic genome biology data to benchmark statistical methods for studying biological rhythms. Finally, we discuss several unmet analytical needs, including applications to clinical medicine, and suggest productive avenues to address them