Lie Transformation Group Theory Research Articles

Soliton solutions of BLMP equation by Lie symmetry approach

Shallow water wave equations are usually described by Korteweg–de Vries (KdV)-type equations. In this paper, we have used Lie transformation group theory to solve (2+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation. We have obtained some exact solutions of BLMP equation in the explicit form through similarity reduction. All the reported results are expressed in closed form and analysed physically through their evolution profiles. The physical analysis reveals that the nature of solutions is parabolic, quasi-periodic, multisoliton and asymptotic.

An integrable SIS model

We provide a demonstration of the integrability of a classical model of an infectious disease which neither kills nor induces autoimmunity by means of the Painlevé analysis and use the Lie theory of transformation groups to present an explicit solution.

Invariance Signatures: Characterizing Contours by Their Departures from Invariance

In this paper, a new invariant feature of two-dimensional contours is reported: the invariance signature. The invariance signature is a measure of the degree to which a contour is invariant under a variety of transformations, derived from the theory of Lie transformation groups. It is shown that the invariance signature is itself invariant under shift, rotation, and scaling of the contour. Since it is derived from local properties of the contour, it is well-suited to a neural network implementation. It is shown that a model-based neural network (MBNN) can be constructed which computes the invariance signature of a contour and classifies patterns on this basis. Experiments demonstrate that invariance signature networks can be employed successfully for shift-, rotation-, and scale-invariant optical character recognition.

Some Remarks on Symmetries and Transformation Groups in General Relativity

The motivation for this paper is the recent interest in the study of symme tries in general relativity and its purpose is to discuss the mathematical foundations required for such a study. The general (formal and informal) ideas of what constitutes a symmetry of space-time are discussed and developed and the idea of a Lie algebra of symmetry vector fields is studied in detail. The relationship between such Lie algebras and the ideas of Lie transformation group theory (Palais' theorems) is stated and a general theorem regarding the orbits of such symmetries is given. Finally some specific symmetries in general relativity are explored and some of their similarities and differences noted.

Lie symmetries and multiple solutions in reaction-diffusion systems

Lie theory of transformation groups is applied to the study of ! reaction-diffusion systems in two-dimensional media. Our study proves that they are invariant with respect to a five-parameter symmetry group. Multiple types of invariant solutions with physical interest are possible, and some of them can be found in the literature applied to particular models.

Pattern-specific contrast threshold elevation

Visual channels are defined psychophysically; stimuli that interact share information in the same channel, and those that do not interact are processed in different channels. Channels are often investigated by means of adaptation to one stimulus, testing contrast threshold elevation with one (or more) others. Much recent work has tested the tuning of channels for orientation and spatial frequency, using simple line gratings. This study examined the pattern-specificity of such adaptation, testing the hypothesis that the fundamental operators of the Lie Transformation Group Theory of Neuropsychology (LTG/NP) define psychophysical channels. In Experiment I the three basic pattern pairs of LTG/NP were used as adaptation and test stimuli in a conventional contrast threshold-elevation experiment. Threshold elevation was pattern-specific, thus supporting the hypothesis. In subsequent experiments various 'fractured' patterns, and patterns generated by combinations of Lie operators were used both for adaptation and test. The results were mixed; some supported the original hypothesis, but many did not. Relations between local contour orientations in adaptation and test patterns could explain some results, but not all. The hypothesis that adaptation occurs to the oriented spatial-frequency components of the test patterns, on the other hand, gave a good fit to the data. It is concluded that there is pattern-specificity in contrast threshold elevation, but it is a form of specificity that can be explained without recourse to a model of geometrical pattern processing, at least for the simple patterns used here.

A functional theory of the McCollough effect.

Pattern-contingent color aftereffects, or McCollough effects (MEs), are used to probe the visual brain's operations psychophysically. Their neural substrate is unknown, however, and theories about them are weak. Our theory proposes a strong functional role for MEs and a neuropsychological basis that accounts for "top-down" (global) constraints ignored by other theories. The functional aspect of the theory is based on the concept of contingent adaptation level, following Helson (1964), and on the "error-correcting device" of Andrews (1964), which tracks and adjusts internal representation to external-world contingencies. The neuropsychological part of the theory postulates that global properties are the result of MEs being generated not at the individual detector level but in vectorfields of which the detectors are elements. It is an implementation of Lie transformation group theory (Hoffman, 1966). Evidence for this model is assessed.

Pattern-contingent color aftereffects on noninduced patterns.

In a series of experiments, we found that in addition to expected reports of color aftereffects on patterns viewed during induction, reliable and predictable reports of color were given by subjects to patterns they did not view during induction. These reports to noninduced patterns were generally to patterns that were orthogonal to the patterns seen during induction. Induction with, for example, a red vertical grating led to appropriate aftereffects (i.e., green) on that vertical pattern and to the complementary aftereffect (i.e., pink) on a horizontal grating. We suggest that such color aftereffects on noninduced patterns are based on a shift in the activity of orientation coding mechanisms as a result of viewing the inducing patterns. We further propose that the results are consistent with the Lie transformation group theory of neuropsychology and that they add to a growing body of research demonstrating the applicability of this theory to the understanding of pattern-contingent color aftereffects.

The roles of pattern orthogonality and color contrast in the generation of pattern-contingent color aftereffects.

A previous report (Emerson, Humphrey, & Dodwell, 1985) presented evidence that several pattern-contingent color aftereffects could be established simultaneously in individual observers, using as induction patterns the three basic orthogonal pattern pairs of Hoffman’s (1966) Lie transformation group theory of neuropsychology. Contingent aftereffects (CAEs) generated with these patterns show little interference between pattern pairs. The present experiments explored the induction and test conditions that are sufficient and]or necessary for generating single and multiple CAEs. The data indicated that, for CAE generation to occur, it is necessary that two orthogonal inducing patterns not be viewed in light of the same color. It is sufficient to induce with two distinct patterns in different colors, but more CAEs are reported if these two patterns are orthogonal than if they are nonorthogonal. The scheduling of the inducing conditions is relatively unimportant. In testing, the probability of obtaining a CAE report is greatly increased by presenting the subject with a test pattern designed to elicit two CAEs of different hues; orthogonality of test pattern quadrants has little influence. The results support the view that pattern “channels” bounded by orthogonal pairmates exist in the human visual system and that operations in one channel are relatively independent of operations in another.

A theory of extended lie transformation groups

The theory of Lie transformation groups is extended to a theory of extended Lie transformation groups by extending the group parameters to functions of coordinates in the base manifolds. The result is global both in the group manifolds (the O. Schreier's fundamental theorems being not taken into account) as well as in the base differentiable manifolds owing to the introduction of the author'sII-geodesic parallel coordinates. The Lie's fundamental theorems are extremely simplified.

A global formulation of the Lie theory of transformation groups

A global formulation of the Lie theory of transformation groups