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Conic tangency equations and Apollonius problems in biochemistry and pharmacology
"... circles in the plane, find or construct a circle tangent to all three.This was generalized by replacing some circles with straight lines.Vieta [19] (or Viète) solved the problem using circle inversions before 1580.Two generations later, Descartes considered a special case in which all four circles ..."
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Cited by 7 (2 self)
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in these generalized cases.Doing so is quite a challenge for the best computer algebra systems.We report below some comparative times using various computer algebra systems on some of these problems.We also consider conic tangency equations for general conics in two and three dimensions. Apollonius problems
Conic tangency equations and Apollonius problems in biochemistry and pharmacology
, 2002
"... The Apollonius Circle Problem dates to Greek antiquity, circa 250 b.c. Given three circles in the plane, find or construct a circle tangent to all three. This was generalized by replacing some circles with straight lines. Viéte [Canon mathematicus seu Ad triangula cum adpendicibus, Lutetiae: Apud Io ..."
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for the best computer algebra systems. We report later some comparative times using various computer algebra systems on some of these problems. We also consider conic tangency equations for general conics in two and three dimensions. Apollonius problems are of interest in their own right. However
Tangency of Conics and Quadrics
"... Abstract: Our paper discusses a simple way of determining tangency of conics using the concept of pencils of conics and the polepolar relationship. We discuss the method, analyze the different situations of tangency for conics, and extend it to find the tangency of quadrics in 3d space. Although t ..."
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Abstract: Our paper discusses a simple way of determining tangency of conics using the concept of pencils of conics and the polepolar relationship. We discuss the method, analyze the different situations of tangency for conics, and extend it to find the tangency of quadrics in 3d space. Although
Superconformal field theory on threebranes at a CalabiYau singularity
 Nucl. Phys. B
, 1998
"... Just as parallel threebranes on a smooth manifold are related to string theory on AdS5 × S 5, parallel threebranes near a conical singularity are related to string theory on AdS5 × X5, for a suitable X5. For the example of the conifold singularity, for which X5 = (SU(2) × SU(2))/U(1), we argue that ..."
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Cited by 690 (37 self)
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Just as parallel threebranes on a smooth manifold are related to string theory on AdS5 × S 5, parallel threebranes near a conical singularity are related to string theory on AdS5 × X5, for a suitable X5. For the example of the conifold singularity, for which X5 = (SU(2) × SU(2))/U(1), we argue
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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Cited by 800 (26 self)
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likelihoods, marginal probabilities and most probable configurations. We describe how a wide varietyof algorithms — among them sumproduct, cluster variational methods, expectationpropagation, mean field methods, maxproduct and linear programming relaxation, as well as conic programming relaxations — can
Convex Analysis
, 1970
"... In this book we aim to present, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems. The title Variational Analysis reflects this breadth. For a lo ..."
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Cited by 5350 (67 self)
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progressed also to the study of socalled stationary points, critical points, and other indications of singularity that a point might have relative to its neighbors, especially in association with existence theorems for differential equations.
Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition
 JOURNAL OF POLITICAL ECONOMY
, 1974
"... ..."
Liquidity Risk and Expected Stock Returns
, 2002
"... This study investigates whether marketwide liquidity is a state variable important for asset pricing. We find that expected stock returns are related crosssectionally to the sensitivities of returns to fluctuations in aggregate liquidity. Our monthly liquidity measure, an average of individualsto ..."
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Cited by 590 (4 self)
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This study investigates whether marketwide liquidity is a state variable important for asset pricing. We find that expected stock returns are related crosssectionally to the sensitivities of returns to fluctuations in aggregate liquidity. Our monthly liquidity measure, an average of individualstock measures estimated with daily data, relies on the principle that order flow induces greater return reversals when liquidity is lower. Over a 34year period, the average return on stocks with high sensitivities to liquidity exceeds that for stocks with low sensitivities by 7.5 % annually, adjusted for exposures to the market return as well as size, value, and momentum factors.
Sketchpad: A manmachine graphical communication system
, 2003
"... The Sketchpad system uses drawing as a novel communication medium for a computer. The system contains input, output, and computation programs which enable it to interpret information drawn directly on a computer display. It has been used to draw electrical, mechanical, scientific, mathematical, and ..."
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Cited by 702 (6 self)
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The Sketchpad system uses drawing as a novel communication medium for a computer. The system contains input, output, and computation programs which enable it to interpret information drawn directly on a computer display. It has been used to draw electrical, mechanical, scientific, mathematical, and animated drawings; it is a general purpose system. Sketchpad has shown the most usefulness as an aid to the understanding of processes, such as the notion of linkages, which can be described with pictures. Sketchpad also makes it easy to draw highly repetitive or highly accurate drawings and to change drawings previously drawn with it. The many drawings in this thesis were all made with Sketchpad.
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