An Efficient and Continuous Representation for Occupancy Mapping with Random Mapping *

Canonical transformations (Bogoliubov transformations) for fermions with an infinite number of degrees of freedom are studied within a calculus of superanalysis. A continuous representation of the orthogonal group is constructed on a Grassmann module extension of the Fock space. The pull-back of these operators to the Fock space yields a unitary ray representation of the group that implements the Bogoliubov transformations.

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Effect of nonuniformity on the continuous representation of electromechanical dynamics in large power systems

IEEE Power Engineering Review ◽

10.1109/39.666567 ◽

1998 ◽

Vol 18 (5) ◽

pp. 60-61 ◽

Cited By ~ 3

Author(s):

A. Semlyen

Keyword(s):

Power Systems ◽

Continuous Representation ◽

Large Power ◽

Electromechanical Dynamics

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Functionals of random mappings: exact and asymptotic results

Advances in Applied Probability ◽

10.2307/1427616 ◽

1991 ◽

Vol 23 (3) ◽

pp. 437-455 ◽

Cited By ~ 7

Author(s):

P. J. Donnelly ◽

W. J. Ewens ◽

S. Padmadisastra

Keyword(s):

Frequency Spectrum ◽

Exact Results ◽

Asymptotic Results ◽

Number Of Components ◽

Random Mapping ◽

Random Mappings ◽

Simulation Results ◽

Central Tool

A random mapping partitions the set {1, 2, ···, m} into components, where i and j are in the same component if some functional iterate of i equals some functional iterate of j. We consider various functionals of these partitions and of samples from it, including the number of components of ‘small' size and of size O(m) as m → ∞the size of the largest component, the number of components, and various symmetric functionals of the normalized component sizes. In many cases exact results, while available, are uniformative, and we consider various approximations. Numerical and simulation results are also presented. A central tool for many calculations is the ‘frequency spectrum', both exact and asymptotic.

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Continuous Representation of Preferences

SSRN Electronic Journal ◽

10.2139/ssrn.1367774 ◽

1980 ◽

Author(s):

Graciela Chichilnisky

Keyword(s):

Continuous Representation

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Checklist for the Continuous Representation of the CMMI

Interpreting the CMMI (R) ◽

10.1201/9780203504611.axh ◽

2003 ◽

pp. 337-364

Keyword(s):

Continuous Representation

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A 4D continuous representation of myocardial velocity fields from tissue phase mapping magnetic resonance imaging

PLoS ONE ◽

10.1371/journal.pone.0247826 ◽

2021 ◽

Vol 16 (3) ◽

pp. e0247826

Author(s):

Bård A. Bendiksen ◽

Gary McGinley ◽

Ivar Sjaastad ◽

Lili Zhang ◽

Emil K. S. Espe

Keyword(s):

Velocity Field ◽

Velocity Fields ◽

Tissue Phase Mapping ◽

Continuous Representation ◽

Resonance Imaging ◽

Limits Of Agreement ◽

B Splines ◽

End Point ◽

Tissue Phase ◽

Linear B

Myocardial velocities carry important diagnostic information in a range of cardiac diseases, and play an important role in diagnosing and grading left ventricular diastolic dysfunction. Tissue Phase Mapping (TPM) Magnetic Resonance Imaging (MRI) enables discrete sampling of the myocardium’s underlying smooth and continuous velocity field. This paper presents a post-processing framework for constructing a spatially and temporally smooth and continuous representation of the myocardium’s velocity field from TPM data. In the proposed scheme, the velocity field is represented through either linear or cubic B-spline basis functions. The framework facilitates both interpolation and noise reducing approximation. As a proof-of-concept, the framework was evaluated using artificially noisy (i.e., synthetic) velocity fields created by adding different levels of noise to an original TPM data. The framework’s ability to restore the original velocity field was investigated using Bland-Altman statistics. Moreover, we calculated myocardial material point trajectories through temporal integration of the original and synthetic fields. The effect of noise reduction on the calculated trajectories was investigated by assessing the distance between the start and end position of material points after one complete cardiac cycle (end point error). We found that the Bland-Altman limits of agreement between the original and the synthetic velocity fields were reduced after application of the framework. Furthermore, the integrated trajectories exhibited consistently lower end point error. These results suggest that the proposed method generates a realistic continuous representation of myocardial velocity fields from noisy and discrete TPM data. Linear B-splines resulted in narrower limits of agreement between the original and synthetic fields, compared to Cubic B-splines. The end point errors were also consistently lower for Linear B-splines than for cubic. Linear B-splines therefore appear to be more suitable for TPM data.

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