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73
Complexity of Bezout's theorem V: Polynomial time
 Theoretical Computer Science
, 1994
"... this paper is to show that the problem of finding approximately a zero of a polynomial system of equations can be solved in polynomial time, on the average. The number of arithmetic operations is bounded by cN ..."
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Cited by 52 (5 self)
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this paper is to show that the problem of finding approximately a zero of a polynomial system of equations can be solved in polynomial time, on the average. The number of arithmetic operations is bounded by cN
Special Lagrangian submanifolds with isolated conical singularities
 V. Survey and applications
"... Special Lagrangian mfolds (SL mfolds) are a distinguished class of real mdimensional minimal submanifolds which may be defined in C m, or in Calabi– ..."
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Cited by 24 (8 self)
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Special Lagrangian mfolds (SL mfolds) are a distinguished class of real mdimensional minimal submanifolds which may be defined in C m, or in Calabi–
A Discrete Global Minimization Algorithm for Continuous Variational Problems
, 2004
"... In this paper, we apply the ideas from combinatorial optimization to find globally optimal solutions to continuous variational problems. At the heart of our method is an algorithm to solve for globally optimal discrete minimal surfaces. This discrete surface problem is a natural generalization of ..."
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Cited by 23 (0 self)
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In this paper, we apply the ideas from combinatorial optimization to find globally optimal solutions to continuous variational problems. At the heart of our method is an algorithm to solve for globally optimal discrete minimal surfaces. This discrete surface problem is a natural generalization of the planargraph shortest path problem.
Singularities of special Lagrangian fibrations and the SYZ conjecture
, 2000
"... In 1996, Strominger, Yau and Zaslow [22] suggested a geometrical interpretation of Mirror Symmetry between Calabi–Yau 3folds M, ˆ M in terms of dual fibrations ..."
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Cited by 20 (4 self)
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In 1996, Strominger, Yau and Zaslow [22] suggested a geometrical interpretation of Mirror Symmetry between Calabi–Yau 3folds M, ˆ M in terms of dual fibrations
Rectifiability of flat chain
, 1999
"... We prove (without using Federer’s structure theorem) that a finitemass flat chain over any coefficient group is rectifiable if and only if almost all of its 0dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a sim ..."
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Cited by 20 (0 self)
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We prove (without using Federer’s structure theorem) that a finitemass flat chain over any coefficient group is rectifiable if and only if almost all of its 0dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and sufficient condition on the coefficient group in order for every finitemass flat chain to be rectifiable.
C.: Revisiting histograms and isosurface statistics
 IEEE Transactions on Visualization and Computer Graphics
"... Abstract—Recent results have shown a link between geometric properties of isosurfaces and statistical properties of the underlying sampled data. However, this has two defects: not all of the properties described converge to the same solution, and the statistics computed are not always invariant unde ..."
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Cited by 16 (2 self)
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Abstract—Recent results have shown a link between geometric properties of isosurfaces and statistical properties of the underlying sampled data. However, this has two defects: not all of the properties described converge to the same solution, and the statistics computed are not always invariant under isosurfacepreserving transformations. We apply Federer’s Coarea Formula from geometric measure theory to explain these discrepancies. We describe an improved substitute for histograms based on weighting with the inverse gradient magnitude, develop a statistical model that is invariant under isosurfacepreserving transformations, and argue that this provides a consistent method for algorithm evaluation across multiple datasets based on histogram equalization. We use our corrected formulation to reevaluate recent results on average isosurface complexity, and show evidence that noise is one cause of the discrepancy between the expected figure and the observed one. 1
Regularity of Horizons and The Area Theorem
"... We prove that the area of sections of future event horizons in spacetimes satisfying the null energy condition is nondecreasing towards the future under the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperboli ..."
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Cited by 16 (12 self)
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We prove that the area of sections of future event horizons in spacetimes satisfying the null energy condition is nondecreasing towards the future under the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperbolic spacetime and there exists a conformal completion with a "regular" I + ; 3) the horizon is a black hole event horizon in a spacetime which has a globally hyperbolic conformal completion. (Some related results under less restrictive hypotheses are also established.) This extends ...
Lower Bound Gradient Estimates for FirstOrder HamiltonJacobi Equations and Applications to the Regularity of Propagating Fronts.
 Adv. Differential Equations
, 2001
"... This paper is concerned with firstorder timedependent HamiltonJacobi Equations. Exploiting some ideas of Barron and Jensen [9], we derive lower bound estimates for the gradient of a locally Lipschitz continuous viscosity solution u of equations with a convex Hamiltonian. Using these estimates in ..."
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Cited by 14 (9 self)
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This paper is concerned with firstorder timedependent HamiltonJacobi Equations. Exploiting some ideas of Barron and Jensen [9], we derive lower bound estimates for the gradient of a locally Lipschitz continuous viscosity solution u of equations with a convex Hamiltonian. Using these estimates in the context of the levelset approach to front propagation, we investigate the regularity properties of the propagating front of u, namely \Gamma t = fx 2 IR n : u(x; t) = 0g for t 0. We show that, contrary to the smooth case, such estimates do not guarantee, in general, any expected regularity for \Gamma t even if u is semiconcave. 1 Introduction We consider the firstorder timedependent HamiltonJacobi Equation @! @t +H(x; t; D x !) = 0 in IR n \Theta (0; T ) (1) with the initial condition !(x; 0) = u 0 (x) in IR n (2) where H 2 C(IR n \Theta [0; T ] \Theta IR n ), u 0 2 C(IR n ) and the solution ! is a realvalued function. Under suitable assumptions on the Hamiltonian H...