Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction

The $p\times p$ matrix version of the $r$-KdV hierarchy has been recently treated as the reduced system arising in a Drinfeld-Sokolov type Hamiltonian symmetry reduction applied to a Poisson submanifold in the dual of the Lie algebra $\widehat{gl}_{pr}\otimes {\Complex}[\lambda, \lambda^{-1}]$. Here a series of extensions of this matrix Gelfand-Dickey system is derived by means of a generalized Drinfeld-Sokolov reduction defined for the Lie algebra $\widehat{gl}_{pr+s}\otimes {\Complex}[\lambda,\lambda^{-1}]$ using the natural embedding $gl_{pr}\subset gl_{pr+s}$ for $s$ any positive integer. The hierarchies obtained admit a description in terms of a $p\times p$ matrix pseudo-differential operator comprising an $r$-KdV type positive part and a non-trivial negative part. This system has been investigated previously in the $p=1$ case as a constrained KP system. In this paper the previous results are considerably extended and a systematic study is presented on the basis of the Drinfeld-Sokolov approach that has the advantage that it leads to local Poisson brackets and makes clear the conformal ($\cal W$-algebra) structures related to the KdV type hierarchies. Discrete reductions and modified versions of the extended $r$-KdV hierarchies are also discussed.Comment: 60 pages, plain TE

Yangians, Integrable Quantum Systems and Dorey's rule

We study tensor products of fundamental representations of Yangians and show that the fundamental quotients of such tensor products are given by Dorey's rule.Comment: We have made corrections to the results for the Yangians associated to the non--simply laced algebra

Solitons and Vertex Operators in Twisted Affine Toda Field Theories

Affine Toda field theories in two dimensions constitute families of integrable, relativistically invariant field theories in correspondence with the affine Kac-Moody algebras. The particles which are the quantum excitations of the fields display interesting patterns in their masses and coupling and which have recently been shown to extend to the classical soliton solutions arising when the couplings are imaginary. Here these results are extended from the untwisted to the twisted algebras. The new soliton solutions and their masses are found by a folding procedure which can be applied to the affine Kac-Moody algebras themselves to provide new insights into their structures. The relevant foldings are related to inner automorphisms of the associated finite dimensional Lie group which are calculated explicitly and related to what is known as the twisted Coxeter element. The fact that the twisted affine Kac-Moody algebras possess vertex operator constructions emerges naturally and is relevant to the soliton solutions.Comment: 27 pages (harvmac) + 3 figures (LaTex) at the end of the file, Swansea SWAT/93-94/1

Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I

We define the partition and $n$-point correlation functions for a vertex operator superalgebra on a genus two Riemann surface formed by sewing two tori together. For the free fermion vertex operator superalgebra we obtain a closed formula for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from torus Szeg\"o kernels. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. Using the bosonized formalism, a new genus two Jacobi product identity is described for the Riemann theta series. We compute and discuss the modular properties of the generating function for all $n$-point functions in terms of a genus two Szeg\"o kernel determinant. We also show that the Virasoro vector one point function satisfies a genus two Ward identity.Comment: A number of typos have been corrected, 39 pages. To appear in Commun. Math. Phy

Factors Predicting Detrimental Change in Declarative Memory Among Women With HIV: A Study of Heterogeneity in Cognition

Objective: Statistical techniques used to study cognitive function in HIV typically yield normative estimates and can mask the heterogeneity in cognitive trajectories over time. We applied a novel statistical approach to identify clusters of individuals with distinct patterns of change in declarative memory in HIV-seropositive (HIV+) and HIV-seronegative (HIV−) women. Methods: 1731 women from the Women’s Interagency HIV Study, a multi-center, prospective cohort study, completed the Hopkins Verbal Learning Test-Revised (HLVT-R) at >2 visits. To derive subgroups with similar patterns of decline by HIV-serostatus, we used a mixed-effects framework that modeled the trajectory of multiple declarative memory outcomes over time, while simultaneously clustering individuals. Results: Of the 1731 participants, 1149 were HIV+ (70% Black/African American [AA]; 30% White/Other [W/O]) and 582 were HIV− (68% AA; 32% W/O). Race stratification was necessary to optimize clustering. Among HIV+AA’s, four subgroups emerged: a subgroup with minimal decline, two with accelerated decline, and one with stable but low performance. In HIV− AA, three subgroups emerged: one with minimal decline and two with accelerated decline. In multivariable-adjusted models among HIV+, individuals with accelerated decline were less educated (P < 0.001) and more likely to have a history of depression (P < 0.001) versus those with minimal decline. Similar subgroups were identified in W/O HIV+ and W/O HIV− participants. Conclusion: We identified clinically meaningful subgroups of women with distinct phenotypes of declarative memory decline, which depend on race and HIV-serostatus using a data driven approach. Identification of underlying mechanisms and risk factors contributing to the observed differences are warranted. More broadly our modeling approach could be other populations to identify risk factors for accelerated cognitive decline and to personalize interventions

Global extent and drivers of mammal population declines in protected areas under illegal hunting pressure

Illegal hunting is a persistent problem in many protected areas, but an overview of the extent of this problem and its impact on wildlife is lacking. We reviewed 40 years (1980–2020) of global research to examine the spatial distribution of research and socio-ecological factors influencing population decline within protected areas under illegal hunting pressure. From 81 papers reporting 988 species/site combinations, 294 mammal species were reported to have been illegally hunted from 155 protected areas across 48 countries. Research in illegal hunting has increased substantially during the review period and showed biases towards strictly protected areas and the African continent. Population declines were most frequent in countries with a low human development index, particularly in strict protected areas and for species with a body mass over 100 kg. Our results provide evidence that illegal hunting is most likely to cause declines of large-bodied species in protected areas of resource-poor countries regardless of protected area conservation status. Given the growing pressures of illegal hunting, increased investments in people’s development and additional conservation efforts such as improving anti-poaching strategies and conservation resources in terms of improving funding and personnel directed at this problem are a growing priority

Author Correction: Widespread long-range cis-regulatory elements in the maize genome.

An amendment to this paper has been published and can be accessed via a link at the top of the paper

Widespread long-range cis-regulatory elements in the maize genome.

Genetic mapping studies on crops suggest that agronomic traits can be controlled by gene–distal intergenic loci. Despite the biological importance and the potential agronomic utility of these loci, they remain virtually uncharacterized in all crop species to date. Here, we provide genetic, epigenomic and functional molecular evidence to support the widespread existence of gene–distal (hereafter, distal) loci that act as long-range transcriptional cis-regulatory elements (CREs) in the maize genome. Such loci are enriched for euchromatic features that suggest their regulatory functions. Chromatin loops link together putative CREs with genes and recapitulate genetic interactions. Putative CREs also display elevated transcriptional enhancer activities, as measured by self-transcribing active regulatory region sequencing. These results provide functional support for the widespread existence of CREs that act over large genomic distances to control gene expression