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464
Spherical Averages and Applications to Spherical Splines and Interpolation
 ACM Transactions on Graphics
, 2001
"... This paper introduces a method for computing weighted averages on spheres based on least squares minimization that respects spherical distance. We prove existence and uniqueness properties of the weighted averages, and give fast iterative algorithms with linear and quadratic convergence rates. O ..."
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Cited by 99 (1 self)
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This paper introduces a method for computing weighted averages on spheres based on least squares minimization that respects spherical distance. We prove existence and uniqueness properties of the weighted averages, and give fast iterative algorithms with linear and quadratic convergence rates. Our methods are appropriate to problems involving averages of spherical data in meteorological, geophysical and astronomical applications. One simple application is a method for smooth averaging of quaternions, which generalizes Shoemake's spherical linear interpolation.
Continuation and Path Following
, 1992
"... CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful ..."
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Cited by 95 (6 self)
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CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful theoretical tools in modern mathematics. Their use can be traced back at least to such venerated works as those of Poincar'e (18811886), Klein (1882 1883) and Bernstein (1910). Leray and Schauder (1934) refined the tool and presented it as a global result in topology, viz., the homotopy invariance of degree. The use of deformations to solve nonlinear systems of equations Partially supported by the National Science Foundation via grant # DMS9104058 y Preprint, Colorado State University, August 2 E. Allgower and K. Georg may be traced back at least to Lahaye (1934). The classical embedding methods were the
An IV Model of Quantile Treatment Effects
 Econometrica
, 2001
"... Headnote.The ability of quantile regression models to characterize the heterogeneous impact of variables on different points of an outcome distribution makes them appealing in many economic applications. However, in observational studies, the variables of interest (e.g. education, prices) are ofte ..."
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Cited by 82 (4 self)
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Headnote.The ability of quantile regression models to characterize the heterogeneous impact of variables on different points of an outcome distribution makes them appealing in many economic applications. However, in observational studies, the variables of interest (e.g. education, prices) are often endogenous, making conventional quantile regression inconsistent and hence inappropriate for recovering the causal effects of these variables on the quantiles of economic outcomes. In order to address this problem, we develop a model of quantile treatment effects (QTE) in the presence of endogeneity and obtain conditions for identification of the QTE without functional form assumptions. The principal feature of the model is the imposition of conditions which restrict the evolution of ranks across treatment states. This feature allows us to overcome the endogeneity problem and recover the true QTE through the use of instrumental variables. The proposed model can also be equivalently viewed as a structural simultaneous equation model with nonadditive errors, where QTE can be interpreted as the structural quantile effects (SQE). Key Words: endogeneity, quantile regression, simultaneous equations, instrumental regression, identification, nonlinear model, monotone likelihood ratio, bounded completeness,
Spin Geometry and Seiberg–Witten Invariants
, 1996
"... Over the last year remarkable new developments have no less than revolutionized the subject of 4manifold topology. When Seiberg and Witten discovered their monopole equations in October 1994 it was soon realized by Kronheimer, Mrowka, Taubes, and others that these new invariants led to remarkably s ..."
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Cited by 80 (13 self)
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Over the last year remarkable new developments have no less than revolutionized the subject of 4manifold topology. When Seiberg and Witten discovered their monopole equations in October 1994 it was soon realized by Kronheimer, Mrowka, Taubes, and others that these new invariants led to remarkably simpler proofs of many of Donaldson’s theorems and gave rise to new interconnections between Riemannian geometry, 4manifolds, and symplectic topology. For example, manifolds with nontrivial invariants do not admit metrics of positive scalar curvature, Kronheimer and Mrowka finally settled the Thom conjecture, and Taubes proved that symplectic 4manifolds have nontrivial invariants, thus settling a longstanding conjecture related to the existence of symplectic structures. One of the deepest and most striking new results in this circle of ideas is Taubes ’ theorem about the relation between the SeibergWitten and the Gromov invariants in the symplectic case. This can be interpreted as an existence theorem for Jholomorphic curves and it gave rise to a number of new theorems about
Generic Global Rigidity
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2004
"... Suppose a finite configuration of labeled points p = (p1,...,pn) in Ed is given along with certain pairs of those points determined by a graph G such that the coordinates of the points of p are generic, i.e., algebraically independent over the integers. If another corresponding configuration q = (q ..."
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Cited by 72 (7 self)
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Suppose a finite configuration of labeled points p = (p1,...,pn) in Ed is given along with certain pairs of those points determined by a graph G such that the coordinates of the points of p are generic, i.e., algebraically independent over the integers. If another corresponding configuration q = (q1,...,qn) in Ed is given such that the corresponding edges of G for p and q have the same length, we provide a sufficient condition to ensure that p and q are congruent in Ed. This condition, together with recent results of Jackson and Jordán [JJ], give necessary and sufficient conditions for a graph being generically globally rigid in the plane.
The Maslov Index for Paths
 Topology
, 1992
"... Maslov’s famous index for a loop of Lagrangian subspaces was interpreted by Arnold [1] as an intersection number with an algebraic variety known as the Maslov cycle. Arnold’s general position arguments apply equally well to the case of a path of Lagrangian subspaces whose endpoints lie in the comple ..."
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Cited by 71 (6 self)
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Maslov’s famous index for a loop of Lagrangian subspaces was interpreted by Arnold [1] as an intersection number with an algebraic variety known as the Maslov cycle. Arnold’s general position arguments apply equally well to the case of a path of Lagrangian subspaces whose endpoints lie in the complement of the Maslov cycle. Our aim in this paper is to define a Maslov index for any path regardless of where its endpoints lie. Our index is invariant under homotopy with fixed endpoints and is additive for catenations. Duistermaat [4] has proposed a Maslov index for paths which is not additive for catenations but is independent of the choice of the Lagrangian subspace used to define the Maslov cycle. By contrast our Maslov index depends on this choice. We have been motivated by two applications in [10] and [12] as well as the index introduced by Conley and Zehnder in [2] and [3]. In [12] we show how to define a signature for a certain class of one dimensional first order differential operators whose index and coindex are infinite. In [10] we relate
Visualizing Nonlinear Vector Field Topology
 IEEE TVCG
, 1999
"... Abstract — We present our results on the visualization of nonlinear vector field topology. The underlying mathematics is done in Clifford algebra, a system describing geometry by extending the usual vector space by a multiplication of vectors. We started with the observation that all known algorith ..."
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Cited by 70 (9 self)
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Abstract — We present our results on the visualization of nonlinear vector field topology. The underlying mathematics is done in Clifford algebra, a system describing geometry by extending the usual vector space by a multiplication of vectors. We started with the observation that all known algorithms for vector field topology are based on piecewise linear or bilinear approximation and that these methods destroy the local topology if nonlinear behavior is present. Our algorithm looks for such situations, chooses an appropriate polynomial approximation in these areas and finally visualizes the topology. This overcomes the problem and the algorithm is still very fast because we are using linear approximation outside these small but important areas. The paper contains a detailed description of the algorithm and a basic introduction to Clifford algebra.
Minimal surfaces in pseudohermitian geometry and the Bernstein problem in the Heisenberg group
, 2004
"... We develop a surface theory in pseudohermitian geometry. We define a notion of (p)mean curvature and the associated (p)minimal surfaces. As a differential equation, the pminimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate the go through theo ..."
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Cited by 59 (10 self)
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We develop a surface theory in pseudohermitian geometry. We define a notion of (p)mean curvature and the associated (p)minimal surfaces. As a differential equation, the pminimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate the go through theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and hence solves the analogue of the Bernstein problem in the Heisenberg group H1. In H1, identified with the Euclidean space R 3, the pminimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set. We interpret the pmean curvature: as the curvature of a characteristic curve, as the tangential sublaplacian of a defining function, and as a quantity in terms of calibration geometry. We also show that there are no closed, connected, C 2 smoothly embedded constant pmean curvature or pminimal surfaces of genus greater than one in the standard S 3. This fact