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Functional analysis --- Approximation theory --- Embedding theorems --- Approximation theory. --- Embedding theorems. --- Functional analysis. --- 517.518.8 --- 517.518.8 Approximation of functions by polynomials and their generalizations --- Approximation of functions by polynomials and their generalizations --- Complex analysis --- Besov spaces. --- Besov, Espaces de. --- Spaces of measures --- Espaces de mesures --- Spaces of measures. --- Besov, Espaces de --- Approximation et developpements --- Approximation des fonctions

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Since antibodies tagged with markers have been developed, immunocytochemistry has become the method of choice for identifying tissue substances or for the localisation of nucleic acid in tissue by in situ hybridisation in molecular biology. Resin-embedded tissue is routinely used and new techniques are constantly introduced. Thus, the novice entering these fields has a breathtaking variety of methods open to him. This laboratory book covers the embedding of tissue using less sensitive epoxy resin methods to the more sensitive procedures employing the acrylics. The possibilities are discussed and results are presented so that an understanding of the techniques can be acquired and appropriate choices made. In the first part of the book, the background of the various resins available are discussed and information on inexpensive alternative technologies where they exist is provided. The various steps involved in tissue processing, beginning with fixation, are first described in theory, then detailed protocols are presented for their application, including troubleshooting sections. The second part rationalises the great variety of labelling methods that are commonly used for "on-section" cytochemistry and immunocytochemistry, where colloidal gold is currently overwhelmingly the most popular marker. The principles behind the usage of various markers and their limitations and advantages are discussed.

Immunology. Immunopathology --- Analytical biochemistry --- Biological microscopy --- Immunocytochemistry --- Electron microscopy --- Gums and resins --- Immunohistochemistry --- Microscopy, Electron --- Acrylic Resins --- Epoxy Resins --- Plastic Embedding --- Tissue Preservation --- Technique --- methods --- Gums and resins. --- Acrylic Resins. --- Epoxy Resins. --- Technique. --- methods. --- Acrylic resins. --- Methods. --- Cell biology. --- Immunology. --- Biochemistry. --- Cell Biology. --- Biochemistry, general. --- Biological chemistry --- Chemical composition of organisms --- Organisms --- Physiological chemistry --- Biology --- Chemistry --- Medical sciences --- Immunobiology --- Life sciences --- Serology --- Cell biology --- Cellular biology --- Cells --- Composition --- Immunocytochemistry - Technique --- Electron microscopy - Technique --- Immunohistochemistry - methods --- Microscopy, Electron - methods --- Plastic Embedding - methods --- Tissue Preservation - methods

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This book offers a self-contained account of the 3-manifold invariants arising from the original Jones polynomial. These are the Witten-Reshetikhin-Turaev and the Turaev-Viro invariants. Starting from the Kauffman bracket model for the Jones polynomial and the diagrammatic Temperley-Lieb algebra, higher-order polynomial invariants of links are constructed and combined to form the 3-manifold invariants. The methods in this book are based on a recoupling theory for the Temperley-Lieb algebra. This recoupling theory is a q-deformation of the SU(2) spin networks of Roger Penrose. The recoupling theory is developed in a purely combinatorial and elementary manner. Calculations are based on a reformulation of the Kirillov-Reshetikhin shadow world, leading to expressions for all the invariants in terms of state summations on 2-cell complexes. Extensive tables of the invariants are included. Manifolds in these tables are recognized by surgery presentations and by means of 3-gems (graph encoded 3-manifolds) in an approach pioneered by Sostenes Lins. The appendices include information about gems, examples of distinct manifolds with the same invariants, and applications to the Turaev-Viro invariant and to the Crane-Yetter invariant of 4-manifolds.

Drie-menigvuldigheden (Topologie) --- Knopentheorie --- Knot theory --- Noeuds [Theorie des ] --- Three-manifolds (Topology) --- Trois-variétés (Topologie) --- Knot theory. --- Algebraic topology --- Invariants --- Mathematics --- Invariants (Mathematics) --- Invariants. --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Low-dimensional topology --- Topological manifolds --- Knots (Topology) --- 3-manifold. --- Addition. --- Algorithm. --- Ambient isotopy. --- Axiom. --- Backslash. --- Barycentric subdivision. --- Bijection. --- Bipartite graph. --- Borromean rings. --- Boundary parallel. --- Bracket polynomial. --- Calculation. --- Canonical form. --- Cartesian product. --- Cobordism. --- Coefficient. --- Combination. --- Commutator. --- Complex conjugate. --- Computation. --- Connected component (graph theory). --- Connected sum. --- Cubic graph. --- Diagram (category theory). --- Dimension. --- Disjoint sets. --- Disjoint union. --- Elaboration. --- Embedding. --- Equation. --- Equivalence class. --- Explicit formula. --- Explicit formulae (L-function). --- Factorial. --- Fundamental group. --- Graph (discrete mathematics). --- Graph embedding. --- Handlebody. --- Homeomorphism. --- Homology (mathematics). --- Identity element. --- Intersection form (4-manifold). --- Inverse function. --- Jones polynomial. --- Kirby calculus. --- Line segment. --- Linear independence. --- Matching (graph theory). --- Mathematical physics. --- Mathematical proof. --- Mathematics. --- Maxima and minima. --- Monograph. --- Natural number. --- Network theory. --- Notation. --- Numerical analysis. --- Orientability. --- Orthogonality. --- Pairing. --- Pairwise. --- Parametrization. --- Parity (mathematics). --- Partition function (mathematics). --- Permutation. --- Poincaré conjecture. --- Polyhedron. --- Quantum group. --- Quantum invariant. --- Recoupling. --- Recursion. --- Reidemeister move. --- Result. --- Roger Penrose. --- Root of unity. --- Scientific notation. --- Sequence. --- Significant figures. --- Simultaneous equations. --- Smoothing. --- Special case. --- Sphere. --- Spin network. --- Summation. --- Symmetric group. --- Tetrahedron. --- The Geometry Center. --- Theorem. --- Theory. --- Three-dimensional space (mathematics). --- Time complexity. --- Tubular neighborhood. --- Two-dimensional space. --- Vector field. --- Vector space. --- Vertex (graph theory). --- Winding number. --- Writhe.

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A survey, thorough and timely, of the singularities of two-dimensional normal complex analytic varieties, the volume summarizes the results obtained since Hirzebruch's thesis (1953) and presents new contributions. First, the singularity is resolved and shown to be classified by its resolution; then, resolutions are classed by the use of spaces with nilpotents; finally, the spaces with nilpotents are determined by means of the local ring structure of the singularity.

Algebraic geometry --- Analytic spaces --- SINGULARITIES (Mathematics) --- 512.76 --- Singularities (Mathematics) --- Geometry, Algebraic --- Spaces, Analytic --- Analytic functions --- Functions of several complex variables --- Birational geometry. Mappings etc. --- Analytic spaces. --- Singularities (Mathematics). --- 512.76 Birational geometry. Mappings etc. --- Birational geometry. Mappings etc --- Analytic function. --- Analytic set. --- Analytic space. --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Calculation. --- Chern class. --- Codimension. --- Coefficient. --- Cohomology. --- Compact Riemann surface. --- Complex manifold. --- Computation. --- Connected component (graph theory). --- Continuous function. --- Contradiction. --- Coordinate system. --- Corollary. --- Covering space. --- Dimension. --- Disjoint union. --- Divisor. --- Dual graph. --- Elliptic curve. --- Elliptic function. --- Embedding. --- Existential quantification. --- Factorization. --- Fiber bundle. --- Finite set. --- Formal power series. --- Hausdorff space. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Intersection (set theory). --- Intersection number (graph theory). --- Inverse limit. --- Irreducible component. --- Isolated singularity. --- Iteration. --- Lattice (group). --- Line bundle. --- Linear combination. --- Line–line intersection. --- Local coordinates. --- Local ring. --- Mathematical induction. --- Maximal ideal. --- Meromorphic function. --- Monic polynomial. --- Nilpotent. --- Normal bundle. --- Open set. --- Parameter. --- Plane curve. --- Pole (complex analysis). --- Power series. --- Presheaf (category theory). --- Projective line. --- Quadratic transformation. --- Quantity. --- Riemann surface. --- Riemann–Roch theorem. --- Several complex variables. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tensor algebra. --- Theorem. --- Topological space. --- Transition function. --- Two-dimensional space. --- Variable (mathematics). --- Zero divisor. --- Zero of a function. --- Zero set. --- Variétés complexes --- Espaces analytiques

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The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds.In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers.Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.

Algebraic topology --- Characteristic classes --- Classes caractéristiques --- 515.16 --- #WWIS:d.d. Prof. L. Bouckaert/ALTO --- Classes, Characteristic --- Differential topology --- Topology of manifolds --- Characteristic classes. --- 515.16 Topology of manifolds --- Classes caractéristiques --- Additive group. --- Axiom. --- Basis (linear algebra). --- Boundary (topology). --- Bundle map. --- CW complex. --- Canonical map. --- Cap product. --- Cartesian product. --- Characteristic class. --- Charles Ehresmann. --- Chern class. --- Classifying space. --- Coefficient. --- Cohomology ring. --- Cohomology. --- Compact space. --- Complex dimension. --- Complex manifold. --- Complex vector bundle. --- Complexification. --- Computation. --- Conformal geometry. --- Continuous function. --- Coordinate space. --- Cross product. --- De Rham cohomology. --- Diffeomorphism. --- Differentiable manifold. --- Differential form. --- Differential operator. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Directional derivative. --- Eilenberg–Steenrod axioms. --- Embedding. --- Equivalence class. --- Euler class. --- Euler number. --- Existence theorem. --- Existential quantification. --- Exterior (topology). --- Fiber bundle. --- Fundamental class. --- Fundamental group. --- General linear group. --- Grassmannian. --- Gysin sequence. --- Hausdorff space. --- Homeomorphism. --- Homology (mathematics). --- Homotopy. --- Identity element. --- Integer. --- Interior (topology). --- Isomorphism class. --- J-homomorphism. --- K-theory. --- Leibniz integral rule. --- Levi-Civita connection. --- Limit of a sequence. --- Linear map. --- Metric space. --- Natural number. --- Natural topology. --- Neighbourhood (mathematics). --- Normal bundle. --- Open set. --- Orthogonal complement. --- Orthogonal group. --- Orthonormal basis. --- Partition of unity. --- Permutation. --- Polynomial. --- Power series. --- Principal ideal domain. --- Projection (mathematics). --- Representation ring. --- Riemannian manifold. --- Sequence. --- Singular homology. --- Smoothness. --- Special case. --- Steenrod algebra. --- Stiefel–Whitney class. --- Subgroup. --- Subset. --- Symmetric function. --- Tangent bundle. --- Tensor product. --- Theorem. --- Thom space. --- Topological space. --- Topology. --- Unit disk. --- Unit vector. --- Variable (mathematics). --- Vector bundle. --- Vector space. --- Topologie differentielle --- Classes caracteristiques --- Classes et nombres caracteristiques