# Invariance

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## invariance

[in′ver·ē·əns]
(mathematics)
(optics)
Any property of a light beam that remains constant when the light is reflected or refracted at one or more surfaces.
(physics)
The property of a physical quantity or physical law of being unchanged by certain transformations or operations, such as reflection of spatial coordinates, time reversal, charge conjugation, rotations, or Lorentz transformations. Also known as symmetry.

## Invariance

in automatic control systems, the independence of some coordinate of a system from external actions applied to it. The independence of one of the controlled coordinates of the system from all external actions or the independence of all coordinates from one particular action is called polyinvariance. Often the conditions of invariance cannot be met precisely; in this case we speak of invariance with a accuracy up to a certain preas-signed magnitude. In order to realize the conditions of invariance the system must have at least two channels for propagation of the action between the point of application of the external action and the coordinate whose invariance must be secured (B. N. Petrov’s principle of two channels). The ideas of invariance are used in automatic control systems for aircraft and ships, in the control of chemical processes, and in the construction of servomechanism and, especially, combined systems in which the principles of deviation and disturbance control are used simultaneously.

### REFERENCES

Kukhtenko, A. I. Problema invariantnosti v anomatike. Kiev, 1963.
Petrov, B. N., and V. Iu. Rutkovskii. “Dvukhkratnaia invariantnost’sistem avtomaticheskogo upravleniia.” Dokl. AN SSSR, 1965, vol. 161, no. 4

V. IU. RUTKOVSKII

## Invariance

invariability, independence of the physical conditions. Invariance is considered most often in a mathematical sense as the invariability of some quantity under certain transformations. For example, if we consider the motion of a material particle in two coordinate systems rotated at some angle to each other, the projections of the velocity of motion will change on transition from one frame of reference to the other, but the square of the velocity and consequently the kinetic energy will remain unchanged, that is, the kinetic energy is invariant with respect to spatial rotations of the frame of reference. The transformations of coordinates and time when switching from one inertial frame of reference to another (Lorentz transformations) are an important case of transformations. Quantities that do not change during such transformations are said to be Lorentz-invariant. An example of such an invariant is the so-called four-dimensional interval whose square is equal to s122 = (x1 − x2)2 + (y1 − y2)2 + (z1 − z2) − c2(t12)2, where x1, y1, z1 and x2, y2, z2 are the coordinates of two points in space at which some event transpires, t1 and t2 are the moments of time at which the event occurs, and c is the velocity of light. Another example: the intensities of an electric field E and a magnetic field H change during Lorentz transformations, but E2H2 and (EH) are Lorentz-invariant. In the general theory of relativity (the theory of gravitation), quantities that are invariant with respect to transformations to arbitrary curvilinear coordinates are considered.

The importance of the concept of invariance is determined by the fact that with its help it is possible to single out quantities independent of the choice of the frame of reference, that is, quantities that characterize the internal properties of the object being investigated.

Invariance is closely connected with the laws of conservation, which are of great importance. The equivalence of all points in space (the homogeneity of space), which is expressed mathematically in the form of the requirement for the invariance of some function defining the equations of motion (Lagrange) with respect to the translations of the origin of coordinates, leads to the law of conservation of momentum. Similarly, the equivalence of all directions in space (the isotropy of space) leads to the law of conservation of the moment of momentum, the equivalence of all moments of time leads to the law of conservation of energy, and so forth (the Noether theorem).

V. I. GRIGOR’EV

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A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research.
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Meeting guidelines by revision, the modified partial scalar invariance model was compared to the metric invariance model.
To perform the invariance analysis we split the sample in two groups according to their age and gender: the "young-old" (60-74 years, n = 292, 53.4% female and 46.6% male) vs.
Because each template has invariance for estimating a parameter of the other one, they complement each other.
The theoretical measurement model for Decisional Balance with two latent Pros and Cons factors (see Figure 1) was assessed for the invariance analyses with this sample.
Therefore, a good image retrieval system should maintain the invariance for the above 5 features.
In this paper, Log-Polar transform of the rotated image will be outlined first, followed by the discussion about extraction of image texture features using NSCF's translational invariance, and the use of Canberra distance to compare the similarities between this image and the image library.
A multi-step protocol is used to assess measurement invariance through a series of two-country multi-group confirmatory factor analyses (MGCFA), comparing the U.S.
The Salk researchers found that neurons that respond to more complicated shapes - like the curve in 5 or in a rock - demonstrated decreased translation invariance. "They need that complicated curve to be in a more restricted range for them to detect it and understand its meaning," Reynolds said.
In MACS, a factor linear model is fit to the data for each group in which the relationship between the item response and latent variable is assumed to be linear; measurement invariance is assessed by comparing the factor loadings (i.e., non-uniform DIF) and item thresholds (i.e., uniform DIF) for each item between two or more groups.

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