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Ktheory for operator algebras
 Mathematical Sciences Research Institute Publications
, 1998
"... p. XII line5: since p. 12: I blew this simple formula: should be α = −〈ξ, η〉/〈η, η〉. p. 2 I.1.1.4: The RieszFischer Theorem is often stated this way today, but neither Riesz nor Fischer (who worked independently) phrased it in terms of completeness of the orthogonal system {e int}. If [a, b] is a ..."
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Cited by 559 (0 self)
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p. XII line5: since p. 12: I blew this simple formula: should be α = −〈ξ, η〉/〈η, η〉. p. 2 I.1.1.4: The RieszFischer Theorem is often stated this way today, but neither Riesz nor Fischer (who worked independently) phrased it in terms of completeness of the orthogonal system {e int}. If [a, b] is a bounded interval in R, in modern language the original statement of the theorem was that L 2 ([a, b]) is complete and abstractly isomorphic to l 2. According to [Jah03, p. 385], the name “Hilbert space ” was first used in 1908 by A. Schönflies, apparently to refer to what we today call l 2. Von Neumann used the same name for Hilbert spaces in the modern sense (complete inner product spaces), which he defined in 1928. p. 3 line6: At the end of the line, 2ɛ should be 4ɛ. p. 3 I.1.2.3: The statement that a dense subspace of a Hilbert space H contains an orthonormal basis for H can be false if H is nonseparable. In fact, I. Farah (private communication) has shown that a Hilbert space of dimension 2ℵ0 has a dense subspace which does not contain any uncountable orthonormal set. A similar example was obtained by Dixmier [Dix53]. p. 89 I.2.4.3(i): Some of the statements on p. 9 can be false if the measure space is not σfinite. p. 13: add after I.2.6.16: I.2.6.17. If X is a compact subset of C not containing 0, and k ∈ N, there is in general no bound on the norm of T −1 as T ranges over all operators with ‖T ‖ ≤ k and σ(T) ⊆ X. For example, let Sn ∈ L(l 2) be the truncated shift: Sn(α1, α2,...) = (0, α1, α2,..., αn, 0, 0,...) and let Tn = I − Sn. ‖Sn ‖ = 1, so ‖Tn ‖ ≤ 2 for all n. Since Sn is nilpotent, σ(Sn) = {0}, so σ(Tn) = {1} for all n. Tn is invertible, with T −1 n = I + Sn + ξ1 ‖ = √ n + 1, so ‖T −1
Mesh Optimization
, 1993
"... We present a method for solving the following problem: Given a set of data points scattered in three dimensions and an initial triangular mesh wH, produce a mesh w, of the same topological type as wH, that fits the data well and has a small number of vertices. Our approach is to minimize an energy f ..."
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Cited by 397 (8 self)
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We present a method for solving the following problem: Given a set of data points scattered in three dimensions and an initial triangular mesh wH, produce a mesh w, of the same topological type as wH, that fits the data well and has a small number of vertices. Our approach is to minimize an energy function that explicitly models the competing desires of conciseness of representation and fidelity to the data. We show that mesh optimization can be effectively used in at least two applications: surface reconstruction from unorganized points, and mesh simplification (the reduction of the number of vertices in an initially dense mesh of triangles).
MAPS: Multiresolution Adaptive Parameterization of Surfaces
, 1998
"... We construct smooth parameterizations of irregular connectivity triangulations of arbitrary genus 2manifolds. Our algorithm uses hierarchical simplification to efficiently induce a parameterization of the original mesh over a base domain consisting of a small number of triangles. This initial param ..."
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Cited by 273 (13 self)
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We construct smooth parameterizations of irregular connectivity triangulations of arbitrary genus 2manifolds. Our algorithm uses hierarchical simplification to efficiently induce a parameterization of the original mesh over a base domain consisting of a small number of triangles. This initial parameterization is further improved through a hierarchical smoothing procedure based on Loop subdivision applied in the parameter domain. Our method supports both fully automatic and user constrained operations. In the latter, we accommodate point and edge constraints to force the align # wailee@cs.princeton.edu + wim@belllabs.com # ps@cs.caltech.edu cowsar@belllabs.com dpd@cs.princeton.edu ment of isoparameter lines with desired features. We show how to use the parameterization for fast, hierarchical subdivision connectivity remeshing with guaranteed error bounds. The remeshing algorithm constructs an adaptively subdivided mesh directly without first resorting to uniform subdivision followed by subsequent sparsification. It thus avoids the exponential cost of the latter. Our parameterizations are also useful for texture mapping and morphing applications, among others.
Multiresolution Signal Processing for Meshes
, 1999
"... We generalize basic signal processing tools such as downsampling, upsampling, and filters to irregular connectivity triangle meshes. This is accomplished through the design of a nonuniform relaxation procedure whose weights depend on the geometry and we show its superiority over existing schemes wh ..."
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Cited by 254 (12 self)
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We generalize basic signal processing tools such as downsampling, upsampling, and filters to irregular connectivity triangle meshes. This is accomplished through the design of a nonuniform relaxation procedure whose weights depend on the geometry and we show its superiority over existing schemes whose weights depend only on connectivity. This is combined with known mesh simplification methods to build subdivision and pyramid algorithms. We demonstrate the power of these algorithms through a number of application examples including smoothing, enhancement, editing, and texture mapping.
Conjugation spaces
, 2004
"... There are classical examples of spaces X with an involution τ whose mod2comhomology ring resembles that of their fixed point set X τ: there is a ring isomorphism κ: H 2 ∗ (X) ≈ H ∗ (X τ). Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ..."
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Cited by 192 (2 self)
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There are classical examples of spaces X with an involution τ whose mod2comhomology ring resembles that of their fixed point set X τ: there is a ring isomorphism κ: H 2 ∗ (X) ≈ H ∗ (X τ). Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ring isomorphism κ is part of an interesting structure in equivariant cohomology called an H ∗frame. An H ∗ frame, if it exists, is natural and unique. A space with involution admitting an H ∗ frame is called a conjugation space. Many examples of conjugation spaces are constructed, for instance by successive adjunctions of cells homeomorphic to a disk in C k with the complex conjugation. A compact symplectic manifold, with an antisymplectic involution compatible with a Hamiltonian action of a torus T, is a conjugation space, provided X T is itself a conjugation space. This includes the coadjoint orbits of any semisimple compact Lie group, equipped with the Chevalley involution. We also study conjugateequivariant complex vector bundles (“real bundles ” in the sense of Atiyah) over a conjugation space and show that the isomorphism κ maps the Chern classes onto the StiefelWhitney classes of the fixed bundle.
Shellable nonpure complexes and posets. I
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 1996
"... The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of ..."
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Cited by 185 (8 self)
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The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of the basic properties of the concept of nonpure shellability. Doubly indexed fvectors and hvectors are introduced, and the latter are shown to be nonnegative in the shellable case. Shellable complexes have the homotopy type of a wedge of spheres of various dimensions, and their StanleyReisner rings admit a combinatorially induced direct sum decomposition. The technique of lexicographic shellability for posets is similarly extended from pure posets (all maximal chains of the same length) to the general case. Several examples of nonpure lexicographically shellable posets are given, such as the kequal partition lattice (the intersection lattice of the kequal subspace arrangement) and the Tamari lattices of binary trees. This leads to simplified computation of Betti numbers for the kequal arrangement. It also determines the homotopy type of intervals in a Tamari lattice and in the lattice of number partitions ordered by dominance, thus strengthening some known Möbius function formulas. The extension to regular CW complexes is briefly discussed and shown to be related to the concept of lexicographic shellability.
Complex reflection groups , Braid groups, Hecke algebras
, 1997
"... Presentations "a la Coxeter" are given for all (irreducible) finite complex reflection groups. They provide presentations for the corresponding generalized braid groups (for all but six cases), which allow us to generalize some of the known properties of finite Coxeter groups and their a ..."
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Cited by 181 (10 self)
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Presentations "a la Coxeter" are given for all (irreducible) finite complex reflection groups. They provide presentations for the corresponding generalized braid groups (for all but six cases), which allow us to generalize some of the known properties of finite Coxeter groups and their associated braid groups, such as the computation of the center of the braid group and the construction of deformations of the finite group algebra (Hecke algebras). We introduce monodromy representations of the braid groups which factorize through the Hecke algebras, extending results of Cherednik, Opdam, Kohno and others.
Progressive Simplicial Complexes
, 1997
"... In this paper, we introduce the progressive simplicial complex (PSC) representation, a new format for storing and transmitting triangulated geometric models. Like the earlier progressive mesh (PM) representation, it captures a given model as a coarse base model together with a sequence of refinement ..."
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Cited by 172 (2 self)
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In this paper, we introduce the progressive simplicial complex (PSC) representation, a new format for storing and transmitting triangulated geometric models. Like the earlier progressive mesh (PM) representation, it captures a given model as a coarse base model together with a sequence of refinement transformations that progressively recover detail. The PSC representation makes use of a more general refinement transformation, allowing the given model to be an arbitrary triangulation (e.g. any dimension, nonorientable, nonmanifold, nonregular), and the base model to always consist of a single vertex. Indeed, the sequence of refinement transformations encodes both the geometry and the topology of the model in a unified multiresolution framework. The PSC representation retains the advantages of PM's. It defines a continuous sequence of approximating models for runtime levelofdetail control, allows smooth transitions between any pair of models in the sequence, supports progressive transmission, and offers a spaceefficient representation. Moreover, by allowing changes to topology, the PSC sequence of approximations achieves better fidelity than the corresponding PM sequence.
The Topological Structure of Asynchronous Computability
 JOURNAL OF THE ACM
, 1996
"... We give necessary and sufficient combinatorial conditions characterizing the tasks that can be solved by asynchronous processes, of which all but one can fail, that communicate by reading and writing a shared memory. We introduce a new formalism for tasks, based on notions from classical algebra ..."
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Cited by 157 (12 self)
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We give necessary and sufficient combinatorial conditions characterizing the tasks that can be solved by asynchronous processes, of which all but one can fail, that communicate by reading and writing a shared memory. We introduce a new formalism for tasks, based on notions from classical algebraic and combinatorial topology, in which a task's possible input and output values are each associated with highdimensional geometric structures called simplicial complexes. We characterize computability in terms of the topological properties of these complexes. This characterization has a surprising geometric interpretation: a task is solvable if and only if the complex representing the task's allowable inputs can be mapped to the complex representing the task's allowable outputs by a function satisfying certain simple regularity properties. Our formalism thus replaces the "operational" notion of a waitfree decision task, expressed in terms of interleaved computations unfolding ...