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The aim of this book is to develop the combinatorics of Young tableaux and to show them in action in the algebra of symmetric functions, representations of the symmetric and general linear groups, and the geometry of flag varieties. The first part of the book is a self-contained presentation of the basic combinatorics of Young tableaux, including the remarkable constructions of 'bumping' and 'sliding', and several interesting correspondences. In Part II these results are used to study representations with geometry on Grassmannians and flag manifolds, including their Schubert subvarieties, and the related Schubert polynomials. Much of this material has never appeared in book form.There are numerous exercises throughout, with hints or answers provided. Researchers in representation theory and algebraic geometry as well as in combinatorics will find Young Tableaux interesting and useful; students will find the intuitive presentation easy to follow.

Young tableaux. --- Tableaux, Young --- Partitions (Mathematics) --- Young tableaux --- Group theory

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The possibilities of astronomical observation have dramatically increased over the last decade. Major satellites, like the Hubble Space Telescope, Chandra and XMM Newton, are complemented by numerous large ground-based observatories, from 8m-10m optical telescopes to sub-mm and radio facilities. As a result, observational astronomy has access to virtually the whole electromagnetic spectrum of galaxies, even at high redshifts. Theoretical models of galaxy formation and cosmological evolution now face a serious challenge to match the plethora of observational data. In October 2003, over 170 astronomers from 15 countries met for a 4-day workshop to extensively illustrate and discuss all major observational projects and ongoing theoretical efforts to model galaxy formation and evolution. This volume contains the complete proceedings of this meeting and is therefore a unique and timely overview of the current state of research in this rapidly evolving field.

Galaxies --- Astronomy --- Astronomie --- Evolution. --- Measurement. --- Charts, diagrams, etc. --- Evolution --- Mesure --- Tableaux, graphiques, etc --- Astrophysics --- Astronomy - General --- Astronomy & Astrophysics --- Physical Sciences & Mathematics --- Measurement --- Charts, diagrams, etc --- Astronomical charts --- Atlases, Astronomical --- Charts, Astronomical --- Extragalactic nebulae --- Nebulae, Extragalactic --- Galactic evolution --- Galaxy evolution --- Physics. --- Astronomy. --- Astrophysics. --- Cosmology. --- Astronomy, Astrophysics and Cosmology. --- Astronomical physics --- Cosmic physics --- Physics

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This publication seeks to provide a global overview of the nature and extent of injury mortality and morbidity in the form of user-friendly tables and charts. It is hoped that the graphical representation of the main patterns of the burden of disease due to injury will raise awareness of the importance of injuries as a public health issue and facilitate the implementation of effective prevention programmes.

Charts, diagrams, etc. --- Cost of medical care --- Coût des soins médicaux --- Forecasting --- Geneeskundige zorgen--Kosten --- Health care costs --- Kosten van de geneeskundige zorgen --- Medical care [Cost of ] --- Medical care--Cost --- Medical service [Cost of ] --- Medicine--Cost of medical care --- Medische zorgen--Kosten --- Prévision --- Soins médicaux--Coût --- Tabellen, grafische voorstellingen, enz. --- Tableaux, graphiques, etc. --- Voorspelling --- Accidents. --- Wounds and injuries. --- Business. --- Health. --- Medicine. --- Social Science. --- Wounds and injuries --- Accidents --- Death --- World health --- Public Health --- Social Sciences --- Vital Statistics --- Health --- Diseases --- Environment and Public Health --- Medicine --- Population Characteristics --- Anthropology, Education, Sociology and Social Phenomena --- Data Collection --- Health Care --- Health Occupations --- Demography --- Epidemiologic Methods --- Epidemiologic Measurements --- Information Science --- Disciplines and Occupations --- Investigative Techniques --- Analytical, Diagnostic and Therapeutic Techniques and Equipment --- Cause of Death --- Epidemiology --- Mortality --- World Health --- Wounds and Injuries --- Cost of Illness --- Business & Economics --- Surgery & Anesthesiology --- Health & Biological Sciences --- Wounds & Injuries --- Causes --- Epidemiology. --- Human beings --- Injuries --- Trauma, Physical --- Wounds --- Surgical emergencies --- Traumatology --- Costs

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Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang-Baxter equation.

Dirichlet series. --- Weyl groups. --- Weyl's groups --- Group theory --- Series, Dirichlet --- Series --- BZL pattern. --- Class I. --- Eisenstein series. --- Euler product. --- Gauss sum. --- Gelfand-Tsetlin pattern. --- Kashiwara operator. --- Kashiwara's crystal. --- Knowability Lemma. --- Kostant partition function. --- Riemann zeta function. --- Schur polynomial. --- Schützenberger involution. --- Snake Lemma. --- Statement A. --- Statement B. --- Statement C. --- Statement D. --- Statement E. --- Statement F. --- Statement G. --- Tokuyama's Theorem. --- Weyl character formula. --- Weyl denominator. --- Weyl group multiple Dirichlet series. --- Weyl vector. --- Whittaker coefficient. --- Whittaker function. --- Yang-Baxter equation. --- Yang–Baxter equation. --- accordion. --- adele group. --- affine linear transformation. --- analytic continuation. --- analytic number theory. --- archimedean place. --- basis vector. --- bijection. --- bookkeeping. --- box-circle duality. --- boxing. --- canonical indexings. --- cardinality. --- cartoon. --- circling. --- class. --- combinatorial identity. --- concurrence. --- critical resonance. --- crystal base. --- crystal graph. --- crystal. --- divisibility condition. --- double sum. --- episode. --- equivalence relation. --- f-packet. --- free abelian group. --- functional equation. --- generating function. --- global field. --- ice-type model. --- inclusion-exclusion. --- indexing. --- involution. --- isomorphism. --- knowability. --- maximality. --- nodal signature. --- nonarchimedean local field. --- noncritical resonance. --- nonzero contribution. --- p-adic group. --- p-adic integral. --- p-adic integration. --- partition function. --- polynomial. --- preaccordion. --- prototype. --- reduced root system. --- representation theory. --- residue class field. --- resonance. --- resotope. --- row sums. --- row transfer matrix. --- short pattern. --- six-vertex model. --- snakes. --- statistical mechanics. --- subsignature. --- tableaux. --- type. --- Γ-equivalence class. --- Γ-swap.