The Second-Order Term in the Redshift-Magnitude Relation

1956 ◽  
Vol 68 ◽  
pp. 301 ◽  
Author(s):  
Fred Hoyle ◽  
Allan Sandage
Keyword(s):  
Order Term ◽  
Second Order

Neural computation of motion in the fly visual system: quadratic nonlinearity of responses induced by picrotoxin in the HS and CH cells

1995 ◽  
Vol 74 (6) ◽  
pp. 2665-2684 ◽  
Author(s):  
Y. Kondoh ◽  
Y. Hasegawa ◽  
J. Okuma ◽  
F. Takahashi
Keyword(s):  
Mean Square Error ◽  
Order Term ◽  
Mirror Image ◽  
Second Order ◽  
The Other ◽  
Linear Component ◽  
Mean Square ◽  
Nonlinear Component ◽  
First Order ◽  
Order Kernel

1. A computational model accounting for motion detection in the fly was examined by comparing responses in motion-sensitive horizontal system (HS) and centrifugal horizontal (CH) cells in the fly's lobula plate with a computer simulation implemented on a motion detector of the correlation type, the Reichardt detector. First-order (linear) and second-order (quadratic nonlinear) Wiener kernels from intracellularly recorded responses to moving patterns were computed by cross correlating with the time-dependent position of the stimulus, and were used to characterize response to motion in those cells. 2. When the fly was stimulated with moving vertical stripes with a spatial wavelength of 5-40 degrees, the HS and CH cells showed basically a biphasic first-order kernel, having an initial depolarization that was followed by hyperpolarization. The linear model matched well with the actual response, with a mean square error of 27% at best, indicating that the linear component comprises a major part of responses in these cells. The second-order nonlinearity was insignificant. When stimulated at a spatial wavelength of 2.5 degrees, the first-order kernel showed a significant decrease in amplitude, and was initially hyperpolarized; the second-order kernel was, on the other hand, well defined, having two hyperpolarizing valleys on the diagonal with two off-diagonal peaks. 3. The blockage of inhibitory interactions in the visual system by application of 10-4 M picrotoxin, however, evoked a nonlinear response that could be decomposed into the sum of the first-order (linear) and second-order (quadratic nonlinear) terms with a mean square error of 30-50%. The first-order term, comprising 10-20% of the picrotoxin-evoked response, is characterized by a differentiating first-order kernel. It thus codes the velocity of motion. The second-order term, comprising 30-40% of the response, is defined by a second-order kernel with two depolarizing peaks on the diagonal and two off-diagonal hyperpolarizing valleys, suggesting that the nonlinear component represents the power of motion. 4. Responses in the Reichardt detector, consisting of two mirror-image subunits with spatiotemporal low-pass filters followed by a multiplication stage, were computer simulated and then analyzed by the Wiener kernel method. The simulated responses were linearly related to the pattern velocity (with a mean square error of 13% for the linear model) and matched well with the observed responses in the HS and CH cells. After the multiplication stage, the linear component comprised 15-25% and the quadratic nonlinear component comprised 60-70% of the simulated response, which was similar to the picrotoxin-induced response in the HS cells. The quadratic nonlinear components were balanced between the right and left sides, and could be eliminated completely by their contralateral counterpart via a subtraction process. On the other hand, the linear component on one side was the mirror image of that on the other side, as expected from the kernel configurations. 5. These results suggest that responses to motion in the HS and CH cells depend on the multiplication process in which both the velocity and power components of motion are computed, and that a putative subtraction process selectively eliminates the nonlinear components but amplifies the linear component. The nonlinear component is directionally insensitive because of its quadratic non-linearity. Therefore the subtraction process allows the subsequent cells integrating motion (such as the HS cells) to tune the direction of motion more sharply.

scholarly journals Glacier Sliding Down an Inclined Wavy Bed

1976 ◽  
Vol 17 (77) ◽  
pp. 447-462 ◽  
Author(s):  
L. W. Morland
Keyword(s):  
Order Term ◽  
High Viscosity ◽  
Second Order ◽  
Temperature Fields ◽  
Surface Velocity ◽  
Flow Equations ◽  
Basal Sliding ◽  
Pressure Melting ◽  
The Mean ◽  
Bed Slope

The treatments by Nye and Kamb of glacier sliding over a wavy bed with small slope, which assume the ice to be approximated by a Newtonian fluid of high viscosity, are complemented by the inclusion of the glacier depth and the inclination of the bed to the horizontal. The driving force of the motion, gravity, is therefore present in the flow equations and defines immediately the mean drag on the bed. A geothermal heal flux is also included in order to estimate its possible effect on the flow. A complex variable method is used to determine the velocity and temperature fields to second order in the bed slope. These fields satisfy the zero shear traction and pressure-melting-regelation conditions to the same order on the actual bed profile. It is the balance of the second-order term which determines explicitly the (zero order) basal-sliding velocity and surface velocity in terms of the geometry and physical properties of both ice and bed. An explicit solution is illustrated for a sinusoidal bed. and a simple criterion for the onset of cavitation is obtained.

Second‐order term effect on the dispersion characteristics of a magnetostatic delay line

1988 ◽  
Vol 63 (8) ◽  
pp. 3335-3337 ◽  
Author(s):  
P. De Gasperis ◽  
R. Marcelli ◽  
G. Miccoli
Keyword(s):  
Delay Line ◽  
Order Term ◽  
Second Order ◽  
Term Effect ◽  
Dispersion Characteristics

Higher‐order moveout spectra

Geophysics ◽  
1979 ◽  
Vol 44 (7) ◽  
pp. 1193-1207 ◽  
Author(s):  
Bruce T. May ◽  
Donald K. Straley
Keyword(s):  
Fourth Order ◽  
Seismic Reflection ◽  
Order Term ◽  
Higher Order ◽  
Second Order ◽  
Equation Modeling ◽  
Fourth Order Term ◽  
Higher Order Terms ◽  
Velocity Spectra

Higher‐order terms in the generalized seismic reflection moveout equation are usually neglected, resulting in the familiar second‐order, or hyperbolic, moveout equation. Modeling studies show that the higher‐order terms are often significant, and their neglect produces sizable traveltime residuals after correction for moveout in such cases as kinked‐ray models. Taner and Koehler (1969) introduced velocity spectra for estimating stacking velocity defined on the basis of second‐order moveout. Through the use of orthogonal polynomials, an iterative procedure is defined that permits computation of fourth‐order moveout spectra while simultaneously upgrading the previously computed, second‐order spectra. Emphasis is placed on the fourth‐order term, but the procedure is general and can be expanded to higher orders. When used with synthetic and field recorded common‐midpoint (CMP) trace data, this technique produces significant improvements in moveout determination affecting three areas: (1) resolution and interpretability of moveout spectra, (2) quality of CMP stacked sections, and (3) computation of velocity and depth for inverse modeling.

Ellipsometric Determination of the Spectra of Adsorbed Molecules

1971 ◽  
Vol 49 (7) ◽  
pp. 1115-1132 ◽  
Author(s):  
M. J. Dignam ◽  
B. Rao ◽  
M. Moskovits ◽  
R. W. Stobie
Keyword(s):  
Thin Films ◽  
Film Thickness ◽  
Optical Constants ◽  
Order Term ◽  
Second Order ◽  
Lower Sensitivity ◽  
Model Calculations ◽  
The Media ◽  
Reflecting Surfaces ◽  
Optical Behavior

This paper presents a detailed analysis of the application of ellipsometry to obtaining the optical spectra (principally infrared (i.r.)) of molecules adsorbed on reflecting surfaces. Both external and total internal reflections are considered and the conditions for optimum sensitivity examined. A new empirical quantity, the relative complex optical density, is defined which exhibits thin film properties well, particularly in the case of multiple reflection measurements. An explicit expression is derived for this density function (relating it to the optical constants of the media and other system parameters), which is both reasonably simple and correct to second order terms in the film thickness. It is shown that for thin films, no higher order terms need be included, but that in general the second order term must be retained. Various limiting cases are examined to gain insight into the optical behavior of thin films, and to the same end, model calculations performed for CCl4 physically adsorbed on Ag, Ni, Sb, and Ge. In relation to conventional reflection spectroscopy, ellipsometric spectroscopy is shown to have three major advantages: (1) in general, higher sensitivity to adsorbate properties; (2) very much lower sensitivity to absorption of radiation by the adjacent gas phase; (3) more information, permitting the optical constants and film thickness to be determined. Finally, the practicability of the technique is demonstrated by presenting preliminary results for CH3OH reversibly adsorbed on Ag, showing clearly the C—H stretching bands.

Perturbation Theory and the Triangular Well Potential

1975 ◽  
Vol 53 (1) ◽  
pp. 5-12 ◽  
Author(s):  
W. R. Smith ◽  
D. Henderson ◽  
J. A. Barker
Keyword(s):  
Free Energy ◽  
Distribution Function ◽  
Perturbation Theory ◽  
Radial Distribution Function ◽  
Radial Distribution ◽  
Order Term ◽  
Second Order ◽  
First Order ◽  
Triangular Well Potential

Accurate calculations of the second order term in the free energy and the first order term in the radial distribution function in the Barker–Henderson (BH) perturbation theory are presented for the triangular well potential. The BH theory is found to be fully satisfactory for this system. Thus, the conclusions of Card and Walkley regarding the accuracy of the BH theory are erroneous.

A note on the perturbation theory and the triangular well potential

1977 ◽  
Vol 55 (7-8) ◽  
pp. 632-634 ◽  
Author(s):  
P. C. Wankhede ◽  
K. N. Swamy
Keyword(s):  
Monte Carlo ◽  
Perturbation Theory ◽  
Order Term ◽  
Second Order ◽  
First Order ◽  
Monte Carlo Calculations ◽  
Theory Of Liquids ◽  
Compressibility Factors ◽  
Triangular Well Potential

The integrals which appear in the first order term and the macroscopic compressibility (mc) and the local compressibility (lc) approximation for the second order term in the Barker–Henderson (BH) perturbation theory of liquids are evaluated analytically for the triangular-well potential. The compressibility factors so calculated are compared with the Monte Carlo calculations.

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