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It may be assumed that the planes I' and II' are drawn where the images of the planes I and II are formed by rays near the axis by the ordinary Gaussian rules; and by an extension of these rules, not, however, corresponding to reality, the Gauss image point 0', with co-ordinates 'o, of the point 0 at some distance from the axis could be constructed.

10If all three constants of reproduction be achromatized, then the Gaussian image for all distances of objects is the same for the two colours, and the system is said to be in " stable achromatism."

10The Gaussian theory is only an approximation; monochromatic or spherical aberrations still occur, which will be different for different colours; and should they be compensated for one colour, the image of another colour would prove disturbing.

10Gaussian logarithms are intended to facilitate the finding of the logarithms of the sum and difference of two numbers whose logarithms are known, the numbers themselves being unknown; and on this account they are frequently called addition and subtraction logarithms. The object of the table is in fact to give log (a =b) by only one entry when log a and log b are given.

00The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis (the symmetrical axis of the system) are infinitely small, i.e.

00Consequently the Gaussian theory only supplies a convenient method of approximating to reality; and no constructor would attempt to realize this unattainable ideal.

00If, in the first place, monochromatic aberrations be neglected - in other words, the Gaussian theory be accepted - then every reproduction is determined by the positions of the focal planes, and the magnitude of the focal lengths, or if the focal lengths, as ordinarily happens, be equal, by three constants of reproduction.

00The above approximation for the error bars rests on a gaussian approximation which assumes that the Hessian A is positive definite.

00First of all the image is smoothed by Gaussian convolution.

00convolve a density image is constructed by convolving the image data with a Gaussian density kernel.

00calculate the inverse covariance of the full Gaussian, and use the partitioned inverse equations to obtain the covariance for the known data.

00Support for diagonal and full covariance Gaussian mixture HMMs.

00Seed surface patches are formed by grouping neighboring pixels whose mean and Gaussian curvature have the same sign.

00This is unlike approaches such as discriminatively trained Gaussian mixture models or other discriminative classifiers that discriminate at the frame-level only.

00The models for atmospheric dispersion are based on the Gaussian plume model.

00We recommend using randomize for this, as the above steps are very unlikely to generate nice Gaussian distributions in the data.

00calculate a smooth histogram by convolving the raw data set with a gaussian kernel.

00We use a gaussian kernel with a covariance matrix equal to that of the original data set,, ie.

00The approach is based on a stopping time argument that produces a normalizing transformation facilitating the use of a Gaussian likelihood.

00These maps are assessed for significantly modulated voxels using a Gaussian Mixture Model for the distribution of intensity values.

00Consider a data set { x } modeled as coming from a Gaussian distribution of mean mu and s.d. sigma.

00The surfaces ' points have been corrupted with a Gaussian noise of 2 mm variance.

00Projection steps in the EP iteration that cannot be done analytically are done using Gaussian quadrature.

00Gaussian quadrature is used to obtain a 2-D expression covering all points on the tooth face.

00Solution method: The program solves the Cauchy principal value integrals numerically using adaptive Gaussian quadrature.

00We report equal error rates on the PolyVar database that are 34% lower than a baseline Gaussian mixture model likelihood ratio approach.

00GSIGM = _REAL (Write) The value found for the gaussian sigma on exit from the program.

00skew gaussian from the mlla predictions at the 1% level can be seen.

00skew Gaussian, Cauchy function, double Gaussian, multiple Gaussian.

00With the same probability, Gaussian noise is added to the tolerance value (mean 0, standard deviation 0.01 ).

00Abstract Within inverted optical tweezers we measure both the lateral and axial trapping efficiency obtained with Gaussian and high-order Laguerre-Gaussian beams.

00Gaussian logarithms are intended to facilitate the finding of the logarithms of the sum and difference of two numbers whose logarithms are known, the numbers themselves being unknown; and on this account they are frequently called addition and subtraction logarithms. The object of the table is in fact to give log (a =b) by only one entry when log a and log b are given.

00The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis (the symmetrical axis of the system) are infinitely small, i.e.

00Consequently the Gaussian theory only supplies a convenient method of approximating to reality; and no constructor would attempt to realize this unattainable ideal.

00If the angle u l be very small, O', is the Gaussian image; and 0', 0' 2 is termed the " longitudinal aberration," and 0'1R the " lateral aberration " of the pencils with aperture u 2.

00This ray, named by Abbe a " principal ray " (not to be confused with the " principal rays " of the Gaussian theory), passes through the centre of the entrance pupil before the first refraction, and the centre of the exit pupil after the last refraction.

00It may be assumed that the planes I' and II' are drawn where the images of the planes I and II are formed by rays near the axis by the ordinary Gaussian rules; and by an extension of these rules, not, however, corresponding to reality, the Gauss image point 0', with co-ordinates 'o, of the point 0 at some distance from the axis could be constructed.

00If, in the first place, monochromatic aberrations be neglected - in other words, the Gaussian theory be accepted - then every reproduction is determined by the positions of the focal planes, and the magnitude of the focal lengths, or if the focal lengths, as ordinarily happens, be equal, by three constants of reproduction.

00If all three constants of reproduction be achromatized, then the Gaussian image for all distances of objects is the same for the two colours, and the system is said to be in " stable achromatism."

00The Gaussian theory is only an approximation; monochromatic or spherical aberrations still occur, which will be different for different colours; and should they be compensated for one colour, the image of another colour would prove disturbing.

00Projection steps in the EP iteration that cannot be done analytically are done using Gaussian quadrature.

00Gaussian quadrature is used to obtain a 2-D expression covering all points on the tooth face.

00Solution method: The program solves the Cauchy principal value integrals numerically using adaptive Gaussian quadrature.

00The fitted Gaussian dropped to of the peak value from the center, implying an RMS readout noise of.

00GSIGM = _REAL (Write) The value found for the gaussian sigma on exit from the program.

00In these plots the deviation of the skew gaussian from the mlla predictions at the 1% level can be seen.

00The order of increasing complexity is: single Gaussian, skew Gaussian, Cauchy function, double Gaussian, multiple Gaussian.

00With the same probability, Gaussian noise is added to the tolerance value (mean 0, standard deviation 0.01).

00Abstract Within inverted optical tweezers we measure both the lateral and axial trapping efficiency obtained with Gaussian and high-order Laguerre-Gaussian beams.

00This ray, named by Abbe a " principal ray " (not to be confused with the " principal rays " of the Gaussian theory), passes through the centre of the entrance pupil before the first refraction, and the centre of the exit pupil after the last refraction.

01

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