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34
Quantization
 IEEE TRANS. INFORM. THEORY
, 1998
"... The history of the theory and practice of quantization dates to 1948, although similar ideas had appeared in the literature as long ago as 1898. The fundamental role of quantization in modulation and analogtodigital conversion was first recognized during the early development of pulsecode modula ..."
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Cited by 639 (11 self)
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The history of the theory and practice of quantization dates to 1948, although similar ideas had appeared in the literature as long ago as 1898. The fundamental role of quantization in modulation and analogtodigital conversion was first recognized during the early development of pulsecode modulation systems, especially in the 1948 paper of Oliver, Pierce, and Shannon. Also in 1948, Bennett published the first highresolution analysis of quantization and an exact analysis of quantization noise for Gaussian processes, and Shannon published the beginnings of rate distortion theory, which would provide a theory for quantization as analogtodigital conversion and as data compression. Beginning with these three papers of fifty years ago, we trace the history of quantization from its origins through this decade, and we survey the fundamentals of the theory and many of the popular and promising techniques for quantization.
A proof of the Kepler conjecture
 Math. Intelligencer
, 1994
"... This section describes the structure of the proof of ..."
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Cited by 112 (11 self)
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This section describes the structure of the proof of
Billiard representation for multidimensional cosmology with multicomponent perfect fluid near the singularity, Class. Quantum Grav
, 1995
"... Multidimensional model describing the cosmological evolution of n Einstein spaces in the theory with l scalar fields and forms is considered. When electromagnetic composite pbrane ansatz is adopted, and certain restrictions on the parameters of the model are imposed, the dynamics of the model near ..."
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Cited by 39 (21 self)
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Multidimensional model describing the cosmological evolution of n Einstein spaces in the theory with l scalar fields and forms is considered. When electromagnetic composite pbrane ansatz is adopted, and certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the (N − 1)dimensional Lobachevsky space H N−1, N = n+l. The geometrical criterion for the finiteness of the billiard volume and its compactness is used. This criterion reduces the problem to the problem of illumination of (N −2)dimensional sphere S N−2 by pointlike sources. Some examples with billiards of finite volume and hence oscillating behaviour near the singularity are considered. Among them examples with square and triangle 2dimensional billiards (e.g. that of the BianchiIX model) and a 4dimensional billiard in “truncated” D = 11 supergravity model (without the ChernSimons term) are considered. It is shown that the inclusion of the ChernSimons term destroys the confining of a billiard. PACS number(s): 04.50.+h, 98.80.Hw, 04.60.Kz2 1
Asymptotic Performance of Vector Quantizers with a Perceptual Distortion Measure
 in Proc. IEEE Int. Symp. on Information Theory, p. 55
, 1997
"... Gersho's bounds on the asymptotic performance of vector quantizers are valid for vector distortions which are powers of the Euclidean norm. Yamada, Tazaki and Gray generalized the results to distortion measures that are increasing functions of the norm of their argument. In both cases, the distortio ..."
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Cited by 28 (3 self)
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Gersho's bounds on the asymptotic performance of vector quantizers are valid for vector distortions which are powers of the Euclidean norm. Yamada, Tazaki and Gray generalized the results to distortion measures that are increasing functions of the norm of their argument. In both cases, the distortion is uniquely determined by the vector quantization error, i.e., the Euclidean difference between the original vector and the codeword into which it is quantized. We generalize these asymptotic bounds to inputweighted quadratic distortion measures, a class of distortion measure often used for perceptually meaningful distortion. The generalization involves a more rigorous derivation of a fixed rate result of Gardner and Rao and a new result for variable rate codes. We also consider the problem of source mismatch, where the quantizer is designed using a probability density different from the true source density. The resulting asymptotic performance in terms of distortion increase in dB is shown...
Sphere Packings I
 Discrete Comput. Geom
, 1996
"... : We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is relate ..."
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Cited by 22 (6 self)
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: We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is related to the density of packings, is assigned to each Delaunay star. We conjecture that the score of every Delaunay star is at most the score of the stars in the facecentered cubic and hexagonal close packings. This conjecture implies the Kepler conjecture. To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture. Contents: 1. Introduction, 2. The Program, 3. Quasiregular Tetrahedra, 4. Quadrilaterals, 5. Restrictions, 6. Combinatorics, 7. The Method of Subdivision, 8. Explicit Formulas for Compression, Volume, and Angle, 9. FloatingPoint Calculations. Appendix. D. J. Muder's Proof of Theorem 6.1. Sec...
The OneRound Voronoi Game
, 2002
"... In the oneround Voronoi game, the FRST player places n sites inside a unitsquare Q. Next, the ..."
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Cited by 18 (4 self)
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In the oneround Voronoi game, the FRST player places n sites inside a unitsquare Q. Next, the
Global Optimization In Geometry  Circle Packing Into The Square
"... The present review paper summarizes the research work done mostly by the authors on packing equal circles in the unit square in the last years. 1. ..."
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Cited by 8 (0 self)
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The present review paper summarizes the research work done mostly by the authors on packing equal circles in the unit square in the last years. 1.
An overview of the Kepler conjecture
"... The series of papers in this volume gives a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than π / √ 18 ≈ 0.74048.... This is the oldest problem in discrete geometry and is an important part of Hilbert’s 18th pr ..."
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Cited by 7 (1 self)
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The series of papers in this volume gives a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than π / √ 18 ≈ 0.74048.... This is the oldest problem in discrete geometry and is an important part of Hilbert’s 18th problem. An example of a
Asymptotics of BestPacking on Rectifiable Sets
 PACKING ON RECTIFIABLE SETS 19
"... (Communicated by) Abstract. We investigate the asymptotic behavior, as N grows, of the largest minimal pairwise distance of N points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we ..."
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Cited by 6 (5 self)
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(Communicated by) Abstract. We investigate the asymptotic behavior, as N grows, of the largest minimal pairwise distance of N points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare bestpacking configurations with minimal Riesz senergy configurations and determine the sth root asymptotic behavior (as s → ∞) of the minimal energy constants. We show that the upper and the lower dimension of a set defined through the Riesz energy or bestpacking coincides with the upper and lower Minkowski dimension, respectively. For certain sets in R d of integer Hausdorff dimension, we show that the limiting behavior of the bestpacking distance as well as the minimal senergy for large s is different for different subsequences of the cardinalities of the configurations. 1. Preliminaries.