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Partial differential equations(Involution systems)

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ALGEBRES TOPOLOGIQUES --- ANNEAUX A INVOLUTION --- ALGEBRES DE BANACH

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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.

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Breastfeeding is the preferred method of feeding in early life. It is also one of the most cost-effective childhood survival interventions. Breastfeeding practices are important for preventing child mortality and morbidity, as well as ensuring the optimal growth, health, and development of infants. The public health benefits of breastfeeding have been well documented in the medical literature, and include the following: associations with decreased risk for early-life diseases such as otitis media, respiratory tract infection, diarrhoea, and early childhood obesity (to name but a few). This Special Issue book includes a collection of studies on the use of novel methods to improve breastfeeding rates, and research exploring the short- and long-term benefits of breastfeeding for both the infant and mother, including technology-based approaches.

practice --- milk bank --- galactagogues --- infant --- twins --- children --- perinatal --- circadian rhythm --- lactoferrin --- Aboriginal --- lipidome --- infant crying --- AA --- risk factors --- infants --- EPIC --- pregnancy --- Africa --- ECOWAS --- involution --- cortisol --- educational status --- low milk supply --- lactating mammary gland --- milk flow --- DHA --- EWAS --- breast feeding --- pregnancy outcomes --- NTR --- premature birth --- omega-3 --- DNA methylation --- LC-PUFA --- omega-6 --- culturally and linguistically diverse (CALD) --- vitamin A --- human milk carbohydrates --- childhood --- milk intake --- neonate --- fenugreek --- United Arab Emirates --- prognosis --- sensitivity and specificity --- weaning --- breastfeeding frequency --- birth cohort --- maternal behavior --- temperament --- multiple pregnancy --- mortality --- breastfeeding --- Australia --- hospitalizations --- cortisone --- maternal age --- oligosaccharides --- milk composition --- initiation of breastfeeding --- preterm infant --- antenatal care --- infant mortality --- body composition --- growth trajectory --- maternal protein restriction --- self-efficacy --- infections --- gestational age --- exclusive breastfeeding --- daily intake --- ALSPAC --- maternal stress --- retinoic acid --- breast milk metabolome --- lactation --- antibiotic use --- free amino acid --- maternal anxiety --- early life nutrition --- child nutrition --- parity --- human milk --- lactose --- litter size --- caesarean section --- skin-to-skin --- prolonged lactation --- vitamin A deficiency --- fatty acids --- plasma metabolic parameters --- breast milk --- growth --- glycome --- caries

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Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or "form" of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a "residually pseudo-split" Bruhat-Tits building. The book concludes with a display of the Tits indices associated with each of these exceptional forms.This is the third and final volume of a trilogy that began with Richard Weiss' The Structure of Spherical Buildings and The Structure of Affine Buildings.

Buildings (Group theory) --- Combinatorial geometry. --- Geometric combinatorics --- Geometrical combinatorics --- Combinatorial analysis --- Discrete geometry --- Theory of buildings (Group theory) --- Tits's theory of buildings (Group theory) --- Linear algebraic groups --- Bruhat-Tits building. --- Clifford invariant. --- Coxeter diagram. --- Coxeter group. --- Coxeter system. --- Euclidean plane. --- Fundamental Theorem of Descent. --- Moufang building. --- Moufang condition. --- Moufang polygon. --- Moufang quadrangle. --- Moufang set. --- Moufang structure. --- Pfister form. --- Structure Theorem. --- Tits index. --- abelian group. --- absolute Coxeter diagram. --- absolute Coxeter system. --- absolute rank. --- affine building. --- algebraic group. --- anisotropic pseudo-quadratic space. --- anisotropic quadratic space. --- anti-isomorphism. --- apartment. --- arctic region. --- automorphism. --- bilinear form. --- biquaternion division algebra. --- building. --- canonical isomorphism. --- chamber. --- compatible representation. --- descent group. --- descent. --- discrete valuation. --- exceptional Moufang quadrangle. --- exceptional quadrangle. --- finite dimension. --- fixed point building. --- fixed point theory. --- gem. --- generalized quadrangle. --- hyperbolic plane. --- hyperbolic quadratic module. --- hyperbolic quadratic space. --- involutory set. --- isomorphism. --- isotropic quadratic space. --- length function. --- non-abelian group. --- parallel residues. --- polar space. --- projection map. --- proper indifferent set. --- proper involutory set. --- pseudo-quadratic space. --- pseudo-split building. --- quadratic form. --- quadratic module. --- quadratic space. --- quaternion division algebra. --- ramified quadrangle. --- ramified quaternion division algebra. --- ramified separable quadratic extension. --- relative Coxeter diagram. --- relative Coxeter group. --- relative Coxeter system. --- relative rank. --- residual quadratic spaces. --- residue. --- root group sequence. --- root. --- round quadratic space. --- scalar multiplication. --- semi-ramified quadrangle. --- separable quadratic extension. --- simplicial complex. --- special vertex. --- spherical building. --- split quadratic space. --- standard involution. --- subbuilding of split type. --- subbuilding. --- tamely ramified division algebra. --- thick building. --- thin T-building. --- trace map. --- trace. --- unramified quadrangle. --- unramified quadratic space. --- unramified quaternion division algebra. --- unramified separable quadratic extension. --- vector space. --- vertex. --- weak isomorphism. --- wild quadratic space.