Dihadron azimuthal correlations in Au+Au collisions at root s(NN)=200 GeV

Azimuthal angle (Delta phi) correlations are presented for a broad range of transverse momentum (0.4 < p(T) < 10 GeV/c) and centrality (0-92%) selections for charged hadrons from dijets in Au+Au collisions at root s(NN) = 200 GeV. With increasing p(T), the away-side Delta phi distribution evolves from a broad and relatively flat shape to a concave shape, then to a convex shape. Comparisons with p + p data suggest that the away-side distribution can be divided into a partially suppressed ""head"" region centered at Delta phi similar to pi, and an enhanced ""shoulder"" region centered at Delta phi similar to pi +/- 1.1. The p(T) spectrum for the associated hadrons in the head region softens toward central collisions. The spectral slope for the shoulder region is independent of centrality and trigger p(T). The properties of the near-side distributions are also modified relative to those in p + p collisions, reflected by the broadening of the jet shape in Delta phi and Delta eta, and an enhancement of the per-trigger yield. However, these modifications seem to be limited to p(T)less than or similar to 4 GeV/c, above which both the hadron pair shape and per-trigger yield become similar to p + p collisions. These observations suggest that both the away- and near-side distributions contain a jet fragmentation component which dominates for p(T) greater than or similar to 5 GeV/c and a medium-induced component which is important for p(T) less than or similar to 4 GeV/c. We also quantify the role of jets at intermediate and low p(T) through the yield of jet-induced pairs in comparison with binary scaled p + p pair yield. The yield of jet-induced pairs is suppressed at high pair proxy energy (sum of the p(T) magnitudes of the two hadrons) and is enhanced at low pair proxy energy. The former is consistent with jet quenching; the latter is consistent with the enhancement of soft hadron pairs due to transport of lost energy to lower p(T).

Quantitative constraints on the transport properties of hot partonic matter from semi-inclusive single high transverse momentum pion suppression in Au plus Au collisions at root S(NN)=200 GeV

The PHENIX experiment has measured the suppression of semi-inclusive single high-transverse-momentum pi(0)'s in Au+Au collisions at root s(NN) = 200 GeV. The present understanding of this suppression is in terms of energy loss of the parent (fragmenting) parton in a dense color-charge medium. We have performed a quantitative comparison between various parton energy-loss models and our experimental data. The statistical point-to-point uncorrelated as well as correlated systematic uncertainties are taken into account in the comparison. We detail this methodology and the resulting constraint on the model parameters, such as the initial color-charge density dN(g)/dy, the medium transport coefficient <(q) over cap >, or the initial energy-loss parameter epsilon(0). We find that high-transverse-momentum pi(0) suppression in Au+Au collisions has sufficient precision to constrain these model-dependent parameters at the +/- 20-25% (one standard deviation) level. These constraints include only the experimental uncertainties, and further studies are needed to compute the corresponding theoretical uncertainties.

Cold nuclear matter effects on J/psi production as constrained by deuteron-gold measurements at root S(NN)=200 GeV

We present a new analysis of J/psi production yields in deuteron-gold collisions at root s(NN) =200 GeV using data taken from the PHENIX experiment in 2003 and previously published in S. S. Adler [Phys. Rev. Lett 96, 012304 (2006)]. The high statistics proton-proton J/psi data taken in 2005 are used to improve the baseline measurement and thus construct updated cold nuclear matter modification factors (R(dAu)). A suppression of J/psi in cold nuclear matter is observed as one goes forward in rapidity (in the deuteron-going direction), corresponding to a region more sensitive to initial-state low-x gluons in the gold nucleus. The measured nuclear modification factors are compared to theoretical calculations of nuclear shadowing to which a J/psi (or precursor) breakup cross section is added. Breakup cross sections of sigma(breakup)=2.8(-1.4)(+1.7) (2.2(-1.5)(+1.6)) mb are obtained by fitting these calculations to the data using two different models of nuclear shadowing. These breakup cross-section values are consistent within large uncertainties with the 4.2 +/- 0.5 mb determined at lower collision energies. Projecting this range of cold nuclear matter effects to copper-copper and gold-gold collisions reveals that the current constraints are not sufficient to firmly quantify the additional hot nuclear matter effect.

Transverse momentum and centrality dependence of dihadron correlations in Au plus Au collisions at root s(NN)=200 GeV: Jet quenching and the response of partonic matter

Azimuthal angle (Delta phi) correlations are presented for charged hadrons from dijets for 0.4 < p(T)< 10 GeV/c in Au+Au collisions at root s(NN)=200 GeV. With increasing p(T), the away-side distribution evolves from a broad and relatively flat shape to a concave shape, then to a convex shape. Comparisons to p+p data suggest that the away-side can be divided into a partially suppressed ""head"" region centered at Delta phi similar to pi and an enhanced ""shoulder"" region centered at Delta phi similar to pi +/- 1.1. The p(T) spectrum for the head region softens toward central collisions, consistent with the onset of jet quenching. The spectral slope for the shoulder region is independent of centrality and trigger p(T), which offers constraints on energy transport mechanisms and suggests that it contains the medium response to energetic jets.

Centrality dependence of charged hadron production in deuteron plus gold and nucleon plus gold collisions at root S(NN)=200 GeV

We present transverse momentum (p(T)) spectra of charged hadrons measured in deuteron-gold and nucleon-gold collisions at root s(NN)=200 GeV for four centrality classes. Nucleon-gold collisions were selected by tagging events in which a spectator nucleon was observed in one of two forward rapidity detectors. The spectra and yields were investigated as a function of the number of binary nucleon-nucleon collisions, nu, suffered by deuteron nucleons. A comparison of charged particle yields to those in p+p collisions show that yield per nucleon-nucleon collision saturates with nu for high momentum particles. We also present the charged hadron to neutral pion ratios as a function of p(T).

Identified charged hadron production in p plus p collisions at root s=200 and 62.4 GeV

Transverse momentum distributions and yields for pi(+/-), K(+/-), p, and (p) over bar in p + p collisions at root s = 200 and 62.4 GeV at midrapidity are measured by the PHENIX experiment at the Relativistic Heavy Ion Collider (RHIC). These data provide important baseline spectra for comparisons with identified particle spectra in heavy ion collisions at RHIC. We present the inverse slope parameter T(inv), mean transverse momentum < p(T)>, and yield per unit rapidity dN/dy at each energy, and compare them to other measurements at different root s in p + p and p + (p) over bar collisions. We also present the scaling properties such as m(T) scaling and x(T) scaling on the p(T) spectra between different energies. To discuss the mechanism of the particle production in p + p collisions, the measured spectra are compared to next-to-leading-order or next-to-leading-logarithmic perturbative quantum chromodynamics calculations.

Recent advances in the pathology of alcoholic myopathy

This article represents the proceedings of a symposium at the 2000 ISBRA Meeting in Yokohama, Japan. The chairs were Victor R. Preedy and Junko Adachi. The presentations were (1) Alcoholic myopathy: Past, present and future, by Timothy J. Peters and Victor R. Preedy; (2) Protein adducts in the type I and II fiber-predominant muscles of the ethanol-fed rat, by Simon Worrall, Seppo Parkkila, and Onni Niemela; (3) Hydroperoxides and changes in alcoholic myopathy, by Junko Adachi, Migiwa Asamo, and Yasuhino Ueno; and (4) A close association between testicular atrophy, muscle atrophy, and the increase in protein catabolism after chronic ethanol administration, by Kunihiko Takeda, Masayoshi Yamauchi, Kazuhiko Sakamoto, Masaru Takagi, Hisato Nakajima, and Gotaro Toda.

CLOCK-controlled polyphonic regulation of circadian rhythms through canonical and noncanonical E-boxes.

In mammalian circadian clockwork, the CLOCK-BMAL1 complex binds to DNA enhancers of target genes and drives circadian oscillation of transcription. Here we identified 7,978 CLOCK-binding sites in mouse liver by chromatin immunoprecipitation-sequencing (ChIP-Seq), and a newly developed bioinformatics method, motif centrality analysis of ChIP-Seq (MOCCS), revealed a genome-wide distribution of previously unappreciated noncanonical E-boxes targeted by CLOCK. In vitro promoter assays showed that CACGNG, CACGTT, and CATG(T/C)G are functional CLOCK-binding motifs. Furthermore, we extensively revealed rhythmically expressed genes by poly(A)-tailed RNA-Seq and identified 1,629 CLOCK target genes within 11,926 genes expressed in the liver. Our analysis also revealed rhythmically expressed genes that have no apparent CLOCK-binding site, indicating the importance of indirect transcriptional and posttranscriptional regulations. Indirect transcriptional regulation is represented by rhythmic expression of CLOCK-regulated transcription factors, such as Krüppel-like factors (KLFs). Indirect posttranscriptional regulation involves rhythmic microRNAs that were identified by small-RNA-Seq. Collectively, CLOCK-dependent direct transactivation through multiple E-boxes and indirect regulations polyphonically orchestrate dynamic circadian outputs.

Acquired antibody responses against Plasmodium vivax infection vary with host genotype for duffy antigen receptor for chemokines (DARC).

BACKGROUND: Polymorphism of the Duffy Antigen Receptor for Chemokines (DARC) is associated with susceptibility to and the severity of Plasmodium vivax malaria in humans. P. vivax uses DARC to invade erythrocytes. Individuals lacking DARC are 'resistant' to P. vivax erythrocytic infection. However, susceptibility to P. vivax in DARC+ individuals is reported to vary between specific DARC genotypes. We hypothesized that the natural acquisition of antibodies to P. vivax blood stages may vary with the host genotype and the level of DARC expression. Furthermore, high parasitemia has been reported to effect the acquisition of immunity against pre-erythrocytic parasites. We investigated the correlation between host DARC genotypes and the frequency and magnitude of antibodies against P. vivax erythrocytic stage antigens. METHODOLOGY/FINDINGS: We assessed the frequencies and magnitudes of antibody responses against P. vivax and P. falciparum sporozoite and erythrocytic antigens in Colombian donors from malaria-endemic regions. The frequency and level of naturally-acquired antibodies against the P. vivax erythrocytic antigens merozoite surface protein 1 (PvMSP1) and Duffy binding protein (PvDBP) varied with the host DARC genotypes. Donors with one negative allele (FY*B/FY*Bnull and FY*A/FY*Bnull) were more likely to have anti-PvMSP1 and anti-PvDBP antibodies than those with two positive alleles (FY*B/FY*B and FY*A/FY*B). The lower IgG3 and IgG1 components of the total IgG response may account for the decreased responses to P. vivax erythrocytic antigens with FY*A/FY*B and FY*B/FY*B genotypes. No such association was detected with P. falciparum erythrocytic antigens, which does not use DARC for erythrocyte invasion. CONCLUSION/SIGNIFICANCE: Individuals with higher DARC expression, which is associated with higher susceptibility to P. vivax infection, exhibited low frequencies and magnitudes of P. vivax blood-stage specific antibody responses. This may indicate that one of the primary mechanisms by which P. vivax evades host immunity is through DARC indirectly down-regulating humoral responses against erythrocytic invasion and development.

The Impact of Cannabis Use on Cognitive Functioning in Patients With Schizophrenia: A Meta-analysis of Existing Findings and New Data in a First-Episode Sample.

Cannabis use is highly prevalent among people with schizophrenia, and coupled with impaired cognition, is thought to heighten the risk of illness onset. However, while heavy cannabis use has been associated with cognitive deficits in long-term users, studies among patients with schizophrenia have been contradictory. This article consists of 2 studies. In Study I, a meta-analysis of 10 studies comprising 572 patients with established schizophrenia (with and without comorbid cannabis use) was conducted. Patients with a history of cannabis use were found to have superior neuropsychological functioning. This finding was largely driven by studies that included patients with a lifetime history of cannabis use rather than current or recent use. In Study II, we examined the neuropsychological performance of 85 patients with first-episode psychosis (FEP) and 43 healthy nonusing controls. Relative to controls, FEP patients with a history of cannabis use (FEP + CANN; n = 59) displayed only selective neuropsychological impairments while those without a history (FEP - CANN; n = 26) displayed generalized deficits. When directly compared, FEP + CANN patients performed better on tests of visual memory, working memory, and executive functioning. Patients with early onset cannabis use had less neuropsychological impairment than patients with later onset use. Together, these findings suggest that patients with schizophrenia or FEP with a history of cannabis use have superior neuropsychological functioning compared with nonusing patients. This association between better cognitive performance and cannabis use in schizophrenia may be driven by a subgroup of "neurocognitively less impaired" patients, who only developed psychosis after a relatively early initiation into cannabis use.

The complete genome sequence of the gram-positive bacterium Bacillus subtilis.

Bacillus subtilis is the best-characterized member of the Gram-positive bacteria. Its genome of 4,214,810 base pairs comprises 4,100 protein-coding genes. Of these protein-coding genes, 53% are represented once, while a quarter of the genome corresponds to several gene families that have been greatly expanded by gene duplication, the largest family containing 77 putative ATP-binding transport proteins. In addition, a large proportion of the genetic capacity is devoted to the utilization of a variety of carbon sources, including many plant-derived molecules. The identification of five signal peptidase genes, as well as several genes for components of the secretion apparatus, is important given the capacity of Bacillus strains to secrete large amounts of industrially important enzymes. Many of the genes are involved in the synthesis of secondary metabolites, including antibiotics, that are more typically associated with Streptomyces species. The genome contains at least ten prophages or remnants of prophages, indicating that bacteriophage infection has played an important evolutionary role in horizontal gene transfer, in particular in the propagation of bacterial pathogenesis.

Polimorfismos del gen Butirilcolinesterasa responsables de reacciones adversas en pacientes consumidores de “cocaína”

La butirilcolinesterasa humana (BChE; EC 3.1.1.8) es una enzima polimórfica sintetizada en el hígado y en el tejido adiposo, ampliamente distribuida en el organismo y encargada de hidrolizar algunos ésteres de colina como la procaína, ésteres alifáticos como el ácido acetilsalicílico, fármacos como la metilprednisolona, el mivacurium y la succinilcolina y drogas de uso y/o abuso como la heroína y la cocaína. Es codificada por el gen BCHE (OMIM 177400), habiéndose identificado más de 100 variantes, algunas no estudiadas plenamente, además de la forma más frecuente, llamada usual o silvestre. Diferentes polimorfismos del gen BCHE se han relacionado con la síntesis de enzimas con niveles variados de actividad catalítica. Las bases moleculares de algunas de esas variantes genéticas han sido reportadas, entre las que se encuentra las variantes Atípica (A), fluoruro-resistente del tipo 1 y 2 (F-1 y F-2), silente (S), Kalow (K), James (J) y Hammersmith (H). En este estudio, en un grupo de pacientes se aplicó el instrumento validado Lifetime Severity Index for Cocaine Use Disorder (LSI-C) para evaluar la gravedad del consumo de “cocaína” a lo largo de la vida. Además, se determinaron Polimorfismos de Nucleótido Simple (SNPs) en el gen BCHE conocidos como responsables de reacciones adversas en pacientes consumidores de “cocaína” mediante secuenciación del gen y se predijo el efecto delos SNPs sobre la función y la estructura de la proteína, mediante el uso de herramientas bio-informáticas. El instrumento LSI-C ofreció resultados en cuatro dimensiones: consumo a lo largo de la vida, consumo reciente, dependencia psicológica e intento de abandono del consumo. Los estudios de análisis molecular permitieron observar dos SNPs codificantes (cSNPs) no sinónimos en el 27.3% de la muestra, c.293A>G (p.Asp98Gly) y c.1699G>A (p.Ala567Thr), localizados en los exones 2 y 4, que corresponden, desde el punto de vista funcional, a la variante Atípica (A) [dbSNP: rs1799807] y a la variante Kalow (K) [dbSNP: rs1803274] de la enzima BChE, respectivamente. Los estudios de predicción In silico establecieron para el SNP p.Asp98Gly un carácter patogénico, mientras que para el SNP p.Ala567Thr, mostraron un comportamiento neutro. El análisis de los resultados permite proponer la existencia de una relación entre polimorfismos o variantes genéticas responsables de una baja actividad catalítica y/o baja concentración plasmática de la enzima BChE y algunas de las reacciones adversas ocurridas en pacientes consumidores de cocaína.

Análisis de la política exterior japonesa en materia económica frente a las dinámicas comerciales de Corea del Sur y Japón. Periodo 2001 – 2011

Este estudio de caso busca analizar la Política Exterior Japonesa en materia económica frente a las dinámicas comerciales tanto de Corea del Sur como del mismo Japón en el Periodo 2001 – 2011, teniendo como objetivo concreto la identificación de la incidencia que tienen dichas dinámicas comerciales, en términos de competitividad por precio y calidad, sobre la Política Exterior Comercial Japonesa (PECJ).

Cambios maculares y de capa de fibras nerviosas ganglionares posterior a la vitrectomía pars plana y sus factores pronósticos en la fundación oftalmológica nacional

OBJETIVOS: Determinar los factores pronóstico, cambios maculares morfológicos y de capa de fibras nerviosas ganglionares posterior a vitrectomía pars plana, en la Fundación Oftalmológica Nacional. MATERIALES Y MÉTODOS: Estudio longitudinal de antes y después (3y6 meses) de la vitrectomía pars plana (VPP) en pacientes con membrana epirretiniana, agujero macular, síndrome de tracción vítreo macular y opacidades vítreas no-inflamatorias. Se realizó seguimiento clínico y con tomografía de coherencia óptica. RESULTADOS: Grupo de 60 pacientes (mujeres 65.0%), edad promedio 65.45+9.49años y tiempo de enfermedad promedio 23+29.79meses. Las indicaciones de VPP (n=60ojos) fueron agujero macular (38.3%) y membrana epirretiniana (36.7%). Se encontró diferencia significativa entre grosor del complejo capa de células ganglionares (CCG)+capa plexiforme interna (CPI) inicial y 3 meses (p=0.039), correlación entre grosor del complejo CCG+CPI al tercer y sexto mes (r=0.704,p<0.001) y grosor del complejo CCG+CPI al tercer mes con grosor foveal central (CFT) al tercer y sexto mes (r=–0.594,p<0.001 y r=–0.595,p=0.001). Mayores de 65años tenían menor grosor de CFNG a 6meses (r=-0.528,p=0.007). El grosor de CFNG promedio y la presencia de la zona elipsoide inicial fueron factores pronósticos de buena agudeza visual al tercer mes de VPP (r2=0.414,p=0.018, y r2=0.414,p=0.010). CONCLUSIÓN: El grosor de CFNG y la presencia de la zona elipsoide inicial tienen alta capacidad predictiva de buena agudeza visual al tercer mes de VPP, y, correlación inversa entre grosor del complejo CCG+CPI con CFT al tercer y sexto mes de VPP.

Modelling, control and supervision for a class of hybrid systems

The aim of this thesis is to narrow the gap between two different control techniques: the continuous control and the discrete event control techniques DES. This gap can be reduced by the study of Hybrid systems, and by interpreting as Hybrid systems the majority of large-scale systems. In particular, when looking deeply into a process, it is often possible to identify interaction between discrete and continuous signals. Hybrid systems are systems that have both continuous, and discrete signals. Continuous signals are generally supposed continuous and differentiable in time, since discrete signals are neither continuous nor differentiable in time due to their abrupt changes in time. Continuous signals often represent the measure of natural physical magnitudes such as temperature, pressure etc. The discrete signals are normally artificial signals, operated by human artefacts as current, voltage, light etc. Typical processes modelled as Hybrid systems are production systems, chemical process, or continuos production when time and continuous measures interacts with the transport, and stock inventory system. Complex systems as manufacturing lines are hybrid in a global sense. They can be decomposed into several subsystems, and their links. Another motivation for the study of Hybrid systems is the tools developed by other research domains. These tools benefit from the use of temporal logic for the analysis of several properties of Hybrid systems model, and use it to design systems and controllers, which satisfies physical or imposed restrictions. This thesis is focused in particular types of systems with discrete and continuous signals in interaction. That can be modelled hard non-linealities, such as hysteresis, jumps in the state, limit cycles, etc. and their possible non-deterministic future behaviour expressed by an interpretable model description. The Hybrid systems treated in this work are systems with several discrete states, always less than thirty states (it can arrive to NP hard problem), and continuous dynamics evolving with expression: with Ki ¡ Rn constant vectors or matrices for X components vector. In several states the continuous evolution can be several of them Ki = 0. In this formulation, the mathematics can express Time invariant linear system. By the use of this expression for a local part, the combination of several local linear models is possible to represent non-linear systems. And with the interaction with discrete events of the system the model can compose non-linear Hybrid systems. Especially multistage processes with high continuous dynamics are well represented by the proposed methodology. Sate vectors with more than two components, as third order models or higher is well approximated by the proposed approximation. Flexible belt transmission, chemical reactions with initial start-up and mobile robots with important friction are several physical systems, which profits from the benefits of proposed methodology (accuracy). The motivation of this thesis is to obtain a solution that can control and drive the Hybrid systems from the origin or starting point to the goal. How to obtain this solution, and which is the best solution in terms of one cost function subject to the physical restrictions and control actions is analysed. Hybrid systems that have several possible states, different ways to drive the system to the goal and different continuous control signals are problems that motivate this research. The requirements of the system on which we work is: a model that can represent the behaviour of the non-linear systems, and that possibilities the prediction of possible future behaviour for the model, in order to apply an supervisor which decides the optimal and secure action to drive the system toward the goal. Specific problems can be determined by the use of this kind of hybrid models are: - The unity of order. - Control the system along a reachable path. - Control the system in a safe path. - Optimise the cost function. - Modularity of control The proposed model solves the specified problems in the switching models problem, the initial condition calculus and the unity of the order models. Continuous and discrete phenomena are represented in Linear hybrid models, defined with defined eighth-tuple parameters to model different types of hybrid phenomena. Applying a transformation over the state vector : for LTI system we obtain from a two-dimensional SS a single parameter, alpha, which still maintains the dynamical information. Combining this parameter with the system output, a complete description of the system is obtained in a form of a graph in polar representation. Using Tagaki-Sugeno type III is a fuzzy model which include linear time invariant LTI models for each local model, the fuzzyfication of different LTI local model gives as a result a non-linear time invariant model. In our case the output and the alpha measure govern the membership function. Hybrid systems control is a huge task, the processes need to be guided from the Starting point to the desired End point, passing a through of different specific states and points in the trajectory. The system can be structured in different levels of abstraction and the control in three layers for the Hybrid systems from planning the process to produce the actions, these are the planning, the process and control layer. In this case the algorithms will be applied to robotics ¡V a domain where improvements are well accepted ¡V it is expected to find a simple repetitive processes for which the extra effort in complexity can be compensated by some cost reductions. It may be also interesting to implement some control optimisation to processes such as fuel injection, DC-DC converters etc. In order to apply the RW theory of discrete event systems on a Hybrid system, we must abstract the continuous signals and to project the events generated for these signals, to obtain new sets of observable and controllable events. Ramadge & Wonham¡¦s theory along with the TCT software give a Controllable Sublanguage of the legal language generated for a Discrete Event System (DES). Continuous abstraction transforms predicates over continuous variables into controllable or uncontrollable events, and modifies the set of uncontrollable, controllable observable and unobservable events. Continuous signals produce into the system virtual events, when this crosses the bound limits. If this event is deterministic, they can be projected. It is necessary to determine the controllability of this event, in order to assign this to the corresponding set, , controllable, uncontrollable, observable and unobservable set of events. Find optimal trajectories in order to minimise some cost function is the goal of the modelling procedure. Mathematical model for the system allows the user to apply mathematical techniques over this expression. These possibilities are, to minimise a specific cost function, to obtain optimal controllers and to approximate a specific trajectory. The combination of the Dynamic Programming with Bellman Principle of optimality, give us the procedure to solve the minimum time trajectory for Hybrid systems. The problem is greater when there exists interaction between adjacent states. In Hybrid systems the problem is to determine the partial set points to be applied at the local models. Optimal controller can be implemented in each local model in order to assure the minimisation of the local costs. The solution of this problem needs to give us the trajectory to follow the system. Trajectory marked by a set of set points to force the system to passing over them. Several ways are possible to drive the system from the Starting point Xi to the End point Xf. Different ways are interesting in: dynamic sense, minimum states, approximation at set points, etc. These ways need to be safe and viable and RchW. And only one of them must to be applied, normally the best, which minimises the proposed cost function. A Reachable Way, this means the controllable way and safe, will be evaluated in order to obtain which one minimises the cost function. Contribution of this work is a complete framework to work with the majority Hybrid systems, the procedures to model, control and supervise are defined and explained and its use is demonstrated. Also explained is the procedure to model the systems to be analysed for automatic verification. Great improvements were obtained by using this methodology in comparison to using other piecewise linear approximations. It is demonstrated in particular cases this methodology can provide best approximation. The most important contribution of this work, is the Alpha approximation for non-linear systems with high dynamics While this kind of process is not typical, but in this case the Alpha approximation is the best linear approximation to use, and give a compact representation.