Estimating scale-invariant directed dependence of bivariate distributions

The best matching procedure described in Chapter 4 is equivalent to the introduction of a principal fibre bundle in configuration space. Essentially one introduces a one-dimensional gauge connection on the time axis, which is a representation of the Euclidean group of rotations and translations (or, possibly, the similarity group which includes dilatations). To accommodate temporal relationalism, the variational principle needs to be invariant under reparametrizations. The simplest way to realize this in point–particle mechanics is to use Jacobi’s reformulation of Mapertuis’ principle. The chapter concludes with the relational reformulation of the Newtonian N-body problem (and its scale-invariant variant).

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Singularities

10.1093/oso/9780198805021.003.0012 ◽

2018 ◽

Author(s):

S. G. Rajeev

Keyword(s):

Stokes Equations ◽

Negative Answer ◽

Three Dimensions ◽

Picard Iteration ◽

Navier Stokes ◽

Scale Invariant ◽

Navier Stokes Equations ◽

L2 Norm ◽

Two Dimensional Flow ◽

Physical Consequences

The initial value problem of the incompressible Navier–Stokes equations is explained. Leray’s classic study of it (using Picard iteration) is simplified and described in the language of physics. The ideas of Lebesgue and Sobolev norms are explained. The L2 norm being the energy, cannot increase. This gives sufficient control to establish existence, regularity and uniqueness in two-dimensional flow. The L3 norm is not guaranteed to decrease, so this strategy fails in three dimensions. Leray’s proof of regularity for a finite time is outlined. His attempts to construct a scale-invariant singular solution, and modern work showing this is impossible, are then explained. The physical consequences of a negative answer to the regularity of Navier–Stokes solutions are explained. This chapter is meant as an introduction, for physicists, to a difficult field of analysis.

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Scaling of entanglement in 2 + 1-dimensional scale-invariant field theories

Journal of Statistical Mechanics Theory and Experiment ◽

10.1088/1742-5468/2015/02/p02010 ◽

2015 ◽

Vol 2015 (2) ◽

pp. P02010 ◽

Cited By ~ 14

Author(s):

Xiao Chen ◽

Gil Young Cho ◽

Thomas Faulkner ◽

Eduardo Fradkin

Keyword(s):

Field Theories ◽

Scale Invariant

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Real-time FPGA-based implementation of the AKAZE algorithm with nonlinear scale space generation using image partitioning

Journal of Real-Time Image Processing ◽

10.1007/s11554-021-01089-9 ◽

2021 ◽

Author(s):

Parastoo Soleimani ◽

David W. Capson ◽

Kin Fun Li

Keyword(s):

Real Time ◽

Memory Management ◽

Scale Space ◽

Image Resolution ◽

Hardware Architecture ◽

Frame Rate ◽

Time Sharing ◽

Scale Invariant ◽

Nonlinear Scale Space ◽

Nonlinear Scale

AbstractThe first step in a scale invariant image matching system is scale space generation. Nonlinear scale space generation algorithms such as AKAZE, reduce noise and distortion in different scales while retaining the borders and key-points of the image. An FPGA-based hardware architecture for AKAZE nonlinear scale space generation is proposed to speed up this algorithm for real-time applications. The three contributions of this work are (1) mapping the two passes of the AKAZE algorithm onto a hardware architecture that realizes parallel processing of multiple sections, (2) multi-scale line buffers which can be used for different scales, and (3) a time-sharing mechanism in the memory management unit to process multiple sections of the image in parallel. We propose a time-sharing mechanism for memory management to prevent artifacts as a result of separating the process of image partitioning. We also use approximations in the algorithm to make hardware implementation more efficient while maintaining the repeatability of the detection. A frame rate of 304 frames per second for a $$1280 \times 768$$ 1280 × 768 image resolution is achieved which is favorably faster in comparison with other work.

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