We obtain computer assisted proofs of several spherical volume inequalities that appear in the analysis of semidefinite programming based approximation algorithms for Boolean constraint satisfaction problems. These inequalities imply, in particular, that the performance ratio achieved by the MAX 3-SAT approximation algorithm of Karloff and Zwick is indeed 7/8, as conjectured by them, and that the performance ratio of the MAX 3-CSP algorithm of the author is indeed ½. Other results are also implied. The computer assisted proofs are obtained using a system called REALSEARCH written by the author. This system uses interval arithmetic to produce rigorous proofs that certain collections of constraints in real variables have no real solution.

Abstract. We present a verified enumeration of tame graphs as defined in Hales ’ proof of the Kepler Conjecture and confirm the completeness of Hales ’ list of all tame graphs while reducing it from 5128 to 2771 graphs. 1

Abstract. This article gives an introduction to a long-term project called Flyspeck, whose purpose is to give a formal verification of the

The body-centered cubic (BCC) grid has received attention in the volume visualization community recently due to its ability to represent the same data with almost 30% fewer samples as compared to the Cartesian cubic (CC) grid. In this paper we present several resampling strategies for raycasting BCC grids. These strategies range from 2D interpolation in planes to piece-wise linear (barycentric) interpolation in a tetrahedral decomposition of the grid to trilinear and sheared trilinear interpolation. We compare them to raycasting with comparable resampling techniques in the commonly used CC grid in terms of computational complexity and visual quality. 1

Abstract—We introduce a family of box splines for efficient, accurate, and smooth reconstruction of volumetric data sampled on the body-centered cubic (BCC) lattice, which is the favorable volumetric sampling pattern due to its optimal spectral sphere packing property. First, we construct a box spline based on the four principal directions of the BCC lattice that allows for a linear C 0 reconstruction. Then, the design is extended for higher degrees of continuity. We derive the explicit piecewise polynomial representations of the C 0 and C 2 box splines that are useful for practical reconstruction applications. We further demonstrate that approximation in the shift-invariant space—generated by BCC-lattice shifts of these box splines—is twice as efficient as using the tensor-product B-spline solutions on the Cartesian lattice (with comparable smoothness and approximation order and with the same sampling density). Practical evidence is provided demonstrating that the BCC lattice not only is generally a more accurate sampling pattern, but also allows for extremely efficient reconstructions that outperform tensor-product Cartesian reconstructions. Index Terms—BCC, box splines, discrete/continuous representations, optimal regular sampling. Ç 1

This paper formalizes the local density inequality approach to getting upper bounds for sphere packing densities in Rn. This approach was first suggested by L. Fejes-Tóth in 1954 as a method to prove the Kepler conjecture that the densest packing of unit spheres in R3 has density π √ , which 18 is attained by the “cannonball packing. ” Local density inequalities give upper bounds for the sphere packing density formulated as an optimization problem of a nonlinear function over a compact set in a finite dimensional Euclidean space. The approaches of L. Fejes-Tóth, of W.-Y. Hsiang, and of T. C. Hales, to the Kepler conjecture are each based on (different) local density inequalities. Recently T. C. Hales, together with S. P. Ferguson, has presented extensive details carrying out a modified version of the Hales approach to prove the Kepler conjecture. We describe the particular local density inequality underlying the Hales and Ferguson approach to prove Kepler’s conjecture and sketch some features of their proof.

Abstract. An abstract lattice of empty set cells is shown to be able to account for a primary substrate in a physical space. Spacetime is represented by ordered sequences of topologically closed Poincaré sections of this primary space. These mappings are constrained to provide homeomorphic structures serving as frames of reference in order to account for the successive positions of any objects present in the system. Mappings from one to the next section involve morphisms of the general structures, standing for a continuous reference frame, and morphisms of objects present in the various parts of this structure. The combination of these morphisms provides spacetime with the features of a nonlinear generalized convolution. Discrete properties of the lattice allow the prediction of scales at which microscopic to cosmic structures should occur. Deformations of primary cells by exchange of empty set cells allow a cell to be mapped into an image cell in the next section as far as mapped cells remain homeomorphic. However, if a deformation involves a fractal transformation to objects, there occurs a change in the dimension of the cell and the homeomorphism is not conserved. Then the fractal kernel stands for a ”particle ” and the reduction

Abstract. Today’s floating point implementations of elementary transcendental functions are usually very accurate. However, with few exceptions, the actual accuracy is not known. In the present paper we describe a rigorous, accurate, fast and portable implementation of the elementary standard functions based on some existing approximate standard functions. The scheme is outlined for IEEE 754, but not difficult to adapt to other floating point formats. A Matlab implementation is available on the net. Accuracy of the proposed algorithms can be rigorously estimated. As an example we prove that the relative accuracy of the exponential function is better than 2.07eps in a slightly reduced argument range (eps denoting the relative rounding error unit). Otherwise, extensive computational tests suggest for all elementary functions and all suitable arguments an accuracy better than about 3eps. 1. A general approach for rigorous standard functions. Todays libraries for the approximation of elementary transcendental functions are very fast and the results are mostly of very high accuracy. For a good introduction and summary of state-of-the-art methods cf. [19]. The achieved accuracy does not exceed one or two ulp for almost all input arguments; however, there is no proof for that. Today computers are more and more used for so-called computer-assisted proofs, where assumptions of mathematical theorems are verified on the computer in order to draw anticipated conclusions. Famous examples are the celebrated Kepler conjecture [9], the enclosure of the Feigenbaum constant [8], bounds for

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This paper presents a user study of the visual quality of an imaging pipeline employing the optimal body-centered cubic (BCC) sampling lattice. We provide perceptual evidence supporting the theoretical expectation that sampling and reconstruction on the BCC lattice offer superior imaging quality over the traditionally popular Cartesian cubic (CC) sampling lattice. We asked 12 participants to choose the better of two images: one image rendered from data sampled on the CC lattice and one image that is rendered from data sampled on the BCC lattice. We used both synthetic and CT volumetric data, and confirm that the theoretical advantages of BCC sampling carry over to the perceived quality of rendered images. Using 25 % to 35 % fewer samples, BCC sampled data result in images that exhibit comparable visual quality to their CC counterparts.