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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Special Lagrangian m-folds (SL m-folds) are a distinguished class of real m-dimensional minimal submanifolds which may be defined in C m, or in Calabi–

In 1996, Strominger, Yau and Zaslow [22] suggested a geometrical interpretation of Mirror Symmetry between Calabi–Yau 3-folds M, ˆ M in terms of dual fibrations

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 planar-graph shortest path problem.

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 isosurface-preserving 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 isosurface-preserving 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

We prove that the area of sections of future event horizons in space-times satisfying the null energy condition is non-decreasing 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 space-time and there exists a conformal completion with a "regular" I + ; 3) the horizon is a black hole event horizon in a space-time which has a globally hyperbolic conformal completion. (Some related results under less restrictive hypotheses are also established.) This extends ...

In 1997 the most prestigious high school science fair in the United States—the Westinghouse Science Competition [Berger 94] —was won by Adam Cohen, then a senior at Hunter High School in New York City. Cohen's project, "Near-Field Photolithography", involved the construction of a home-built scanning tunneling microscope (or STM—a high-resolution

Contents 1. Introduction 2 2. The diffuse interface model 7 3. Existence for the diffuse interface system 12 3.1. The gradient flow structure 12 3.2. Assumptions 15 3.3. Weak solutions 16 3.4. The implicit time discretisation 17 3.5. Uniform estimates 21 3.6. Proof of the existence theorem 25 3.7. Uniqueness for homogeneous linear elasticity 26 4. Logarithmic free energy 29 4.1. A regularised problem 32 4.2. Higher integrability for the strain tensor 36 4.3. Higher integrability for the logarithmic free energy 42 4.4. Proof of the existence theorem 45 5. The sharp interface limit 46 5.1. The \Gamma--limit of the elastic Ginzburg--Landau energies 52 5.2. Euler--Lagrange equation for the sharp interface functional 60 6. The Gibbs--Thomson equation as a singular limit in the scalar case 70 7. Discussion 79 8. Appendix 81 9. Notation 86 References 90 1 1. Introduction We study a mathematical model describing phase separation in multi-- component alloy