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RADIAL DENSITY IN APOLLONIAN PACKINGS
"... Abstract. Given P, an Apollonian Circle Packing, and a circle C0 = ∂B(z0, r0) in P, color the set of disks in P tangent to C0 red. What proportion of the concentric circle C = ∂B(z0, r0 + ) is red, and what is the behavior of this quantity as → 0? Using equidistribution of closed horocycles on t ..."
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on the modular surface H2/SL(2,Z), we show that the answer is 3 pi = 0.9549... We also describe an observation due to Alex Kontorovich connecting the rate of this convergence in the FareyFord packing to the Riemann Hypothesis. For the analogous problem for Soddy Sphere packings, we find that the limiting radial
BEAM WITH A ROUNDED RADIAL DENSITY PROFILE
, 1982
"... 19. KEY WORDS (Continue on reverse aide it neceeary end identi.y by block nmber) Selfpinched Beam Radial Density Profile 20. AESTRACT (Continue an reverse aide If neceeeary and Identity by block number) The axisymmetric perturbations (sausage and hollowing modes) of an intense relativistic selfpin ..."
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19. KEY WORDS (Continue on reverse aide it neceeary end identi.y by block nmber) Selfpinched Beam Radial Density Profile 20. AESTRACT (Continue an reverse aide If neceeeary and Identity by block number) The axisymmetric perturbations (sausage and hollowing modes) of an intense relativistic self
RADIAL DENSITY STATISTICS OF THE GALAXY DISTRIBUTION AND THE LUMINOSITY FUNCTION
, 2014
"... ar ..."
Radial density profiles of timedelay lensing galaxies
, 704
"... We present nonparametric radial mass profiles for ten QSO strong lensing galaxies. Five of the galaxies have profiles close to ρ(r) ∝ r−2, while the rest are closer to r−1, consistent with an NFW profile. The former are all relatively isolated earlytypes and dominated by their stellar light. The ..."
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Cited by 1 (0 self)
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We present nonparametric radial mass profiles for ten QSO strong lensing galaxies. Five of the galaxies have profiles close to ρ(r) ∝ r−2, while the rest are closer to r−1, consistent with an NFW profile. The former are all relatively isolated earlytypes and dominated by their stellar light
Probabilistic Visual Learning for Object Representation
, 1996
"... We present an unsupervised technique for visual learning which is based on density estimation in highdimensional spaces using an eigenspace decomposition. Two types of density estimates are derived for modeling the training data: a multivariate Gaussian (for unimodal distributions) and a Mixtureof ..."
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Cited by 705 (15 self)
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We present an unsupervised technique for visual learning which is based on density estimation in highdimensional spaces using an eigenspace decomposition. Two types of density estimates are derived for modeling the training data: a multivariate Gaussian (for unimodal distributions) and a Mixture
On the Four Vertex Theorem on planes with radial density e ϕ(r)
, 2008
"... It is showed that on a plane with a radial density the Four Vertex Theorem holds for the class of all simple closed curves if and only if the density is constant. But for the class of simple closed curves that are invariant under a rotation about the origin, the Four Vertex Theorem holds for every r ..."
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It is showed that on a plane with a radial density the Four Vertex Theorem holds for the class of all simple closed curves if and only if the density is constant. But for the class of simple closed curves that are invariant under a rotation about the origin, the Four Vertex Theorem holds for every
Training Support Vector Machines: an Application to Face Detection
, 1997
"... We investigate the application of Support Vector Machines (SVMs) in computer vision. SVM is a learning technique developed by V. Vapnik and his team (AT&T Bell Labs.) that can be seen as a new method for training polynomial, neural network, or Radial Basis Functions classifiers. The decision sur ..."
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Cited by 728 (1 self)
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We investigate the application of Support Vector Machines (SVMs) in computer vision. SVM is a learning technique developed by V. Vapnik and his team (AT&T Bell Labs.) that can be seen as a new method for training polynomial, neural network, or Radial Basis Functions classifiers. The decision
Bayesian Interpolation
 Neural Computation
, 1991
"... Although Bayesian analysis has been in use since Laplace, the Bayesian method of modelcomparison has only recently been developed in depth. In this paper, the Bayesian approach to regularisation and modelcomparison is demonstrated by studying the inference problem of interpolating noisy data. T ..."
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Cited by 721 (17 self)
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Although Bayesian analysis has been in use since Laplace, the Bayesian method of modelcomparison has only recently been developed in depth. In this paper, the Bayesian approach to regularisation and modelcomparison is demonstrated by studying the inference problem of interpolating noisy data. The concepts and methods described are quite general and can be applied to many other problems. Regularising constants are set by examining their posterior probability distribution. Alternative regularisers (priors) and alternative basis sets are objectively compared by evaluating the evidence for them. `Occam's razor' is automatically embodied by this framework. The way in which Bayes infers the values of regularising constants and noise levels has an elegant interpretation in terms of the effective number of parameters determined by the data set. This framework is due to Gull and Skilling. 1 Data modelling and Occam's razor In science, a central task is to develop and compare models to a...
Shiftable Multiscale Transforms
, 1992
"... Orthogonal wavelet transforms have recently become a popular representation for multiscale signal and image analysis. One of the major drawbacks of these representations is their lack of translation invariance: the content of wavelet subbands is unstable under translations of the input signal. Wavel ..."
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Cited by 557 (36 self)
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Orthogonal wavelet transforms have recently become a popular representation for multiscale signal and image analysis. One of the major drawbacks of these representations is their lack of translation invariance: the content of wavelet subbands is unstable under translations of the input signal. Wavelet transforms are also unstable with respect to dilations of the input signal, and in two dimensions, rotations of the input signal. We formalize these problems by defining a type of translation invariance that we call "shiftability". In the spatial domain, shiftability corresponds to a lack of aliasing; thus, the conditions under which the property holds are specified by the sampling theorem. Shiftability may also be considered in the context of other domains, particularly orientation and scale. We explore "jointly shiftable" transforms that are simultaneously shiftable in more than one domain. Two examples of jointly shiftable transforms are designed and implemented: a onedimensional tran...
Macroscopic strings as heavy quarks in large N gauge theory and Antide Sitter supergravity
 PHYS. J. C22
"... Maldacena has put forward large N correspondence between superconformal field theories on the brane and antide Sitter supergravity in spacetime. We study some aspects of the correspondence between N = 4 superconformal gauge theory on D3brane and maximal supergravity on adS5 × S5 by introducing mac ..."
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Cited by 510 (1 self)
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Maldacena has put forward large N correspondence between superconformal field theories on the brane and antide Sitter supergravity in spacetime. We study some aspects of the correspondence between N = 4 superconformal gauge theory on D3brane and maximal supergravity on adS5 × S5 by introducing macroscopic strings as heavy (anti)quark probes. The macroscopic strings are semiinfinite Type IIB strings ending on D3brane worldvolume. We first study deformation and fluctuation of D3brane when a macroscopic BPS string is attached. We find that both dynamics and boundary conditions agree with those for macroscopic string in antide Sitter supergravity. As a byproduct we clarify how Polchinski’s Dirichlet and Neumann open string boundary conditions arise. We then study nonBPS macroscopic string antistring pair configuration as physical realization of heavy quark Wilson loop. We obtain Q ¯ Q static potential and again find agreements between the gauge theory and the supergravity results. By turning on RamondRamond zeroform potential, we also study θ vacuum angle dependence of the static potential. We finally discuss dynamical realization of loop equation via turning on local electric field and recoil of heavy quark thereof, and of heavy baryon via of Nprong string junction. Throughout comparisons of the correspondence, we emphasize crucial role played by ‘geometric duality ’ between coordinates perpendicular to D3brane and parallel ones, hinting a possible explanation of emergence of the entire adS5 spacetime.
Results 1  10
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