Characterizing Triviality of the Exponent Lattice of A Polynomial through Galois and Galois-Like Groups

The problem of computing \emph{the exponent lattice} which consists of all the multiplicative relations between the roots of a univariate polynomial has drawn much attention in the field of computer algebra. As is known, almost all irreducible polynomials with integer coefficients have only trivial exponent lattices. However, the algorithms in the literature have difficulty in proving such triviality for a generic polynomial. In this paper, the relations between the Galois group (respectively, \emph{the Galois-like groups}) and the triviality of the exponent lattice of a polynomial are investigated. The \bbbq\emph{-trivial} pairs, which are at the heart of the relations between the Galois group and the triviality of the exponent lattice of a polynomial, are characterized. An effective algorithm is developed to recognize these pairs. Based on this, a new algorithm is designed to prove the triviality of the exponent lattice of a generic irreducible polynomial, which considerably improves a state-of-the-art algorithm of the same type when the polynomial degree becomes larger. In addition, the concept of the Galois-like groups of a polynomial is introduced. Some properties of the Galois-like groups are proved and, more importantly, a sufficient and necessary condition is given for a polynomial (which is not necessarily irreducible) to have trivial exponent lattice.Comment: 19 pages,2 figure

Mellin Amplitudes for Dual Conformal Integrals

Motivated by recent work on the utility of Mellin space for representing conformal correlators in $AdS$/CFT, we study its suitability for representing dual conformal integrals of the type which appear in perturbative scattering amplitudes in super-Yang-Mills theory. We discuss Feynman-like rules for writing Mellin amplitudes for a large class of integrals in any dimension, and find explicit representations for several familiar toy integrals. However we show that the power of Mellin space is that it provides simple representations even for fully massive integrals, which except for the single case of the 4-mass box have not yet been computed by any available technology. Mellin space is also useful for exhibiting differential relations between various multi-loop integrals, and we show that certain higher-loop integrals may be written as integral operators acting on the fully massive scalar $n$-gon in $n$ dimensions, whose Mellin amplitude is exactly 1. Our chief example is a very simple formula expressing the 6-mass double box as a single integral of the 6-mass scalar hexagon in 6 dimensions.Comment: 29+7 page

Patterns of primary care and mortality among patients with schizophrenia or diabetes: a cluster analysis approach to the retrospective study of healthcare utilization

Abstract Background Patients with schizophrenia have difficulty managing their medical healthcare needs, possibly resulting in delayed treatment and poor outcomes. We analyzed whether patients reduced primary care use over time, differentially by diagnosis with schizophrenia, diabetes, or both schizophrenia and diabetes. We also assessed whether such patterns of primary care use were a significant predictor of mortality over a 4-year period. Methods The Veterans Healthcare Administration (VA) is the largest integrated healthcare system in the United States. Administrative extracts of the VA's all-electronic medical records were studied. Patients over age 50 and diagnosed with schizophrenia in 2002 were age-matched 1:4 to diabetes patients. All patients were followed through 2005. Cluster analysis explored trajectories of primary care use. Proportional hazards regression modelled the impact of these primary care utilization trajectories on survival, controlling for demographic and clinical covariates. Results Patients comprised three diagnostic groups: diabetes only (n = 188,332), schizophrenia only (n = 40,109), and schizophrenia with diabetes (Scz-DM, n = 13,025). Cluster analysis revealed four distinct trajectories of primary care use: consistent over time, increasing over time, high and decreasing, low and decreasing. Patients with schizophrenia only were likely to have low-decreasing use (73% schizophrenia-only vs 54% Scz-DM vs 52% diabetes). Increasing use was least common among schizophrenia patients (4% vs 8% Scz-DM vs 7% diabetes) and was associated with improved survival. Low-decreasing primary care, compared to consistent use, was associated with shorter survival controlling for demographics and case-mix. The observational study was limited by reliance on administrative data. Conclusion Regular primary care and high levels of primary care were associated with better survival for patients with chronic illness, whether psychiatric or medical. For schizophrenia patients, with or without comorbid diabetes, primary care offers a survival benefit, suggesting that innovations in treatment retention targeting at-risk groups can offer significant promise of improving outcomes.http://deepblue.lib.umich.edu/bitstream/2027.42/78274/1/1472-6963-9-127.xmlhttp://deepblue.lib.umich.edu/bitstream/2027.42/78274/2/1472-6963-9-127.pdfPeer Reviewe

Exactly Marginal Deformations and Global Symmetries

We study the problem of finding exactly marginal deformations of N=1 superconformal field theories in four dimensions. We find that the only way a marginal chiral operator can become not exactly marginal is for it to combine with a conserved current multiplet. Additionally, we find that the space of exactly marginal deformations, also called the "conformal manifold," is the quotient of the space of marginal couplings by the complexified continuous global symmetry group. This fact explains why exactly marginal deformations are ubiquitous in N=1 theories. Our method turns the problem of enumerating exactly marginal operators into a problem in group theory, and substantially extends and simplifies the previous analysis by Leigh and Strassler. We also briefly discuss how to apply our analysis to N=2 theories in three dimensions.Comment: 23 pages, 2 figure

Minimally helicity violating, maximally simple scalar amplitudes in N=4 SYM

In planar N=4 SYM we study a particular class of helicity preserving amplitudes. These are scalar amplitudes whose flavor configuration is chosen in such a way that only a limited number of diagrams is allowed, which exhibit an iterative structure. For such amplitudes we evaluate the tree level and one-loop contributions, providing a general formula valid for any number of particles. The ratio between the one-loop and tree level results is a simple combination of dual conformally invariant box functions with at most two massive legs. Along with the MHV and NMHV series, this constitutes the third known infinite sequence of one-loop amplitudes in N=4 SYM.Comment: 22 pages, 7 figures, published versio

Precision Gauge Unification from Extra Yukawa Couplings

We investigate the impact of extra vector-like GUT multiplets on the predicted value of the strong coupling. We find in particular that Yukawa couplings between such extra multiplets and the MSSM Higgs doublets can resolve the familiar two-loop discrepancy between the SUSY GUT prediction and the measured value of alpha_3. Our analysis highlights the advantages of the holomorphic scheme, where the perturbative running of gauge couplings is saturated at one loop and further corrections are conveniently described in terms of wavefunction renormalization factors. If the gauge couplings as well as the extra Yukawas are of O(1) at the unification scale, the relevant two-loop correction can be obtained analytically. However, the effect persists also in the weakly-coupled domain, where possible non-perturbative corrections at the GUT scale are under better control.Comment: 26 pages, LaTeX. v6: Important early reference adde

The transcriptional repressor protein NsrR senses nitric oxide directly via a [2Fe-2S] cluster

The regulatory protein NsrR, a member of the Rrf2 family of transcription repressors, is specifically dedicated to sensing nitric oxide (NO) in a variety of pathogenic and non-pathogenic bacteria. It has been proposed that NO directly modulates NsrR activity by interacting with a predicted [Fe-S] cluster in the NsrR protein, but no experimental evidence has been published to support this hypothesis. Here we report the purification of NsrR from the obligate aerobe Streptomyces coelicolor. We demonstrate using UV-visible, near UV CD and EPR spectroscopy that the protein contains an NO-sensitive [2Fe-2S] cluster when purified from E. coli. Upon exposure of NsrR to NO, the cluster is nitrosylated, which results in the loss of DNA binding activity as detected by bandshift assays. Removal of the [2Fe-2S] cluster to generate apo-NsrR also resulted in loss of DNA binding activity. This is the first demonstration that NsrR contains an NO-sensitive [2Fe-2S] cluster that is required for DNA binding activity