Implicit Symmetry Research Articles

Abundant Traveling Wave and Numerical Solutions of Weakly Dispersive Long Waves Model

In this article, plenty of wave solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony ((2 + 1)-D KP-BBM) model are constructed by employing two recent analytical schemes (a modified direct algebraic (MDA) method and modified Kudryashov (MK) method). From the point of view of group theory, the proposed analytical methods in our article are based on symmetry, and effectively solve those problems which actually possess explicit or implicit symmetry. This model is a vital model in shallow water phenomena where it demonstrates the wave surface propagating in both directions. The obtained analytical solutions are explained by plotting them through 3D, 2D, and contour sketches. These solutions’ accuracy is also tested by calculating the absolute error between them and evaluated numerical results by the Adomian decomposition (AD) method and variational iteration (VI) method. The considered numerical schemes were applied based on constructed initial and boundary conditions through the obtained analytical solutions via the MDA, and MK methods which show the synchronization between computational and numerical obtained solutions. This coincidence between the obtained solutions is explained through two-dimensional and distribution plots. The applied methods’ symmetry is shown through comparing their obtained results and showing the matching between both obtained solutions (analytical and numerical).

The multi-variable unified family of generalized Apostol-type polynomials

The aim of this paper is to investigate and give a new family of multi-variable Apostol-type polynomials. This family is related to Apostol-Euler, Apostol-Bernoulli, Apostol-Genocchi and Apostol-laguerre polynomials. Moreover, we derive some implicit summation formulae and general symmetry identities. The new family of polynomials introduced here, gives many interesting special cases.

Decomposing the Toda hierarchy by an implicit symmetry constraint

An implicit symmetry constraint of the famous Toda lattice hierarchy is presented. Using this symmetry constraint, every lattice equation in the Toda hierarchy is decomposed by an integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system.

MoSa: A Modeling and Sentiment Analysis System for Mobile Application Big Data

The development of mobile internet has led to a massive amount of data being generated from mobile devices daily, which has become a source for analyzing human behavior and trends in public sentiment. In this paper, we build a system called MoSa (Mobile Sentiment analysis) to analyze this data. In this system, sentiment analysis is used to analyze news comments on the THAAD (Terminal High Altitude Area Defense) event from Toutiao by employing algorithms to calculate the sentiment value of the comment. This paper is based on HowNet; after the comparison of different sentiment dictionaries, we discover that the method proposed in this paper, which use a mixed sentiment dictionary, has a higher accuracy rate in its analysis of comment sentiment tendency. We then statistically analyze the relevant attributes of the comments and their sentiment values and discover that the standard deviation of the comments’ sentiment value can quickly reflect sentiment changes among the public. Besides that, we also derive some special models from the data that can reflect some specific characteristics. We find that the intrinsic characteristics of situational awareness have implicit symmetry. By using our system, people can obtain some practical results to guide interaction design in applications including mobile Internet, social networks, and blockchain based crowdsourcing.

DECOMPOSING A NEW NONLINEAR DIFFERENTIAL-DIFFERENCE SYSTEM UNDER A BARGMANN IMPLICIT SYMMETRY CONSTRAINT

Firstly, a hierarchy of integrable lattice equations and its bi-Hamilt-onian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.

Symmetries of Light and Emergence of Matter

Objectives: This article will suggest a mathematical model for the strong connection between light and matter. Proposed properties of light will be shown to the birth matter at any level of complexity. Methods/Statistical Analysis: A conceptual analytical model is constructed to elaborate the structure of light and its link with the matter. Consistent with the genre of modern-day theoretical models in the realm of physics this model is exploratory at this stage and proceeds by first creating a symmetrical multi-layered model of light. The implicit quaternary symmetry is modeled by mathematical equations and shown to have a direct relationship on the emergence of matter at the varying scale of complexity. Findings: This article suggests a unification of light and matter. The unifying principle is an essential set of characteristics that is posited to be true of the nature of light that is also found to exist in the perceived reality of matter as it emerges from the field level, through the quantum, atom, and cell levels – hence at subsequent levels of complexity. The mathematical transformations that spawn layers of light are erected so as to allow infinite diversity to emerge from essential unity. Hence the diversity of matter and matter-based organizations is suggested to be an outcome of simple mathematical transformations that emanate from the four fundamental properties of light. A first transformation creates four sets each comprising of up to an infinite number of similar elements. A second transformation creates seeds organized through a combination of the elements from each set. Application/Improvements: Interpretations of Heisenberg’s Uncertainty principle and Schrödinger’s Wave Equation. More detailed mappings of particles, of atoms, of molecular plans to the seed-type equations introduced in this article. Keywords: Emergence of Matter, Symmetries of Light

A new class of generalized Laguerre–Euler polynomials

A variety of polynomials, their extensions and variants have been extensively investigated, due mainly to their potential of applications in diverse research areas. In this sequel, we introduce a new class of generalized Laguerre–Euler polynomials and present certain potentially useful formulas and identities such as implicit summation formulae and symmetry identities. The new class of polynomials introduced and the results presented here, being very general, are shown to be specialized to reduce to various known polynomials and to yield some known and new formulas and identities, respectively.

Two symmetry constraints for a generalized Dirac integrable hierarchy

By a two-by-two matrix spectral problem, a generalized Dirac integrable hierarchy is presented. A Hamiltonian structure of the obtained hierarchy is established by trace identity, and its Liouville integrability is proved. Then, through Bargmann symmetry constraint, spatial part of the Lax pairs and adjoint Lax pairs is nonlinearized as a completely integrable finite-dimensional Hamiltonian system. Next, under an implicit symmetry constraint, both spatial part and temporal parts of the Lax pairs and adjoint Lax pairs are all nonlinearized as completely integrable finite-dimensional Hamiltonian systems. Ultimately, the involutive representation of solution of the generalized Dirac integrable hierarchy is given.

AGT correspondence: Ding–Iohara algebra at roots of unity and Lepowsky–Wilson construction

It was recently conjectured that the AGT correspondence between the —instanton counting on and the two-dimensional field theories with the conformal symmetry algebra can be considered as a root of unity limit of its K-theoretic analogue. From this point of view, the algebra and a special basis in its representation are limits of the Ding–Iohara algebra and the Macdonald polynomials respectively. In this paper we confirm this conjecture for the special case r = 1. We uncover the implicit symmetry in this limit. We also found that the vertex operators in the special basis have factorized AFLT form.

Binary nonlinearization of the nonlinear Schrödinger equation under an implicit symmetry constraint

By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint, we obtain a finite dimensional system from the Lax pair of the nonlinear Schrodinger equation. We show that this system is a completely integrable Hamiltonian system.

A family of Liouville integrable lattice equations with a parameter and its two symmetry constraints

A two-by-two matrix spectral problem with an arbitrary constant is introduced, a family of integrable lattice equations involving a parameter is derived by the technique of Lax pairs. The typical lattice equation in the obtained family is a deformed Toda lattice system. The bi-Hamiltonian structure of the obtained family of integrable lattice equations is established by using the discrete trace identity. And we prove the Liouville integrability of obtained family of integrable lattice equations. Then, Bargmann symmetry constraint and an implicit symmetry constraint of the obtained family are presented by binary nonlinearization method. Under these two symmetry constraints, we obtain two series of integrable symplectic maps and two sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense, and every integrable lattice equation in the obtained family is decomposed by an integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system. As a special case, two integrable decompositions of the deformed Toda lattice equation are given.

Binary nonlinearization of the super AKNS system under an implicit symmetry constraint

For the super AKNS system, an implicit symmetry constraint between the potentials and the eigenfunctions is proposed. After introducing some new variables to explicitly express potentials, the super AKNS system is decomposed into two compatible finite-dimensional super systems (x-part and tn-part). Furthermore, we show that the obtained super systems are integrable super Hamiltonian systems in the supersymmetry manifold .

An implicit symmetry constraint of the modified Korteweg-de Vries (mKdV) equation

In this paper, an implicit symmetry constraint is calculated and its associated binary nonlinearization of the Lax pairs and the adjoint Lax pairs is carried out for the modified Korteweg-de Vries (mKdV) equation. After introducing two new independent variables, we find that under the implicit symmetry constraint, the spatial part and the temporal part of the mKdV equation are decomposed into two finite-dimensional systems. Furthermore we prove that the obtained finite-dimensional systems are Hamiltonian systems and completely integrable in the Liouville sense.

Sex Differences in Neural Mechanisms Underlying Implicit Symmetry Processing

Some species, including humans, prefer symmetrical to asymmetrical patterns. Preferences for symmetrical stimuli could have arisen in both sexes because specialized systems developed to detect and identify symmetrical stimuli, which are more likely to be figure than background. Females, it has been suggested, prefer symmetry because it may provide a cue to biological fitness when choosing a mate. Using functional magnetic resonance imaging, we found sex differences in neural activity when subjects distinguished figure from ground in abstract patterns whose component shapes differed in colour, brightness, and symmetry/asymmetry. Although no participant explicitly reported using symmetry as a decision criterion, women showed more neural activity than men in visual processing areas when the display contained a symmetrical shape. This enhanced activity in occipital cortex (middle and superior occipital gyrus) occurred bilaterally in women irrespective of whether they chose the symmetrical element in the display as the figure. Contrariwise, men showed a significant neural response in right temporal (superior temporal gyrus) and left parietal cortex (inferior parietal cortex/temporo-parietal junction) only when they chose a symmetrical element as the figure. The female brain thus appears to register symmetry automatically as a stimulus attribute, while the differential neurophysiological response of the male brain is dependent upon an explicit behavioural response to symmetry (choosing the symmetrical part of the display as figure), even if the criterion for choice (symmetry) is not reported verbally.

Towards a more comprehensive crystallography

Despite the intrinsic difference from the point of view of structure of various types of crystals, such as commensurate, incommensurately modulated, intergrowth crystals and quasicrystals, a common approach to their symmetry seems to be possible which eventually will lead to a more comprehensive crystallography. The unifying elements become apparent through treatments which, at first, seem to be contradictory with the geometry of the crystal structures involved. Examples are the description of aperiodic crystals in terms of lattice-periodic structures (going beyond three-dimensionality), the investigation of scaling symmetry in quasicrystals by means of a Z-module of translations generating a dense set of translationally equivalent atomic positions (going beyond discreteness) and finally the characterization of Euclidean properties of normal crystals through non-Euclidean symmetries (going beyond Euclidean metric). These changing approaches do not modify, however, the fundamental nature of crystals to be three-dimensional, discrete and Euclidean. They only allow implicit symmetry groups like the superspace groups (unifying the crystallography of incommensurate and commensurate crystals) and the multimetrical space groups (unifying the possible symmetries of quasicrystals and normal crystals) to be made explicit. Aspects of crystal diffraction, morphology and crystal structure are presented from this unifying point of view, without intending to cover the whole crystallography.

On the propagation of pure plane waves in anisotropic media

In this letter it is shown that, for plane longitudinal waves propagating in generally anisotropic media, any direction in which the polarization vector is collinear with the propagation direction is also a direction in which the energy flux vector is collinear with the propagation direction. For transverse waves, the condition that the polarization vector is perpendicular to the propagation direction is shown to imply that the energy flux vector is collinear with the propagation direction only if the anisotropic media satisfies certain implicit symmetry conditions. Possible applications of these results to the determination of the elastic moduli of anisotropic media are discussed.

Some properties of the spectrum of the Sierpinski gasket in a magnetic field

The spectrum of the Sierpinski gasket in a magnetic field is discussed using a synthetic Green's-function technique. This directly relates the spectrum of an ($n+1$)-stage gasket to that of its $n$-stage components and allows effective use of the implicit symmetry. It is found that the ($n+1$)-stage spectrum is nested with three eigenvalues belonging to the three different representations between any two consecutive stage-$n$ eigenvalues. For the special points where the eigenvalues for stage $n$ and $n+1$ coincide we provide proofs for the two Rammal-Toulouse [Phys. Rev. Lett. 49, 1194 (1982)] nesting properties, derive explicit expressions for the evolution of the degeneracies, and construct the eigenfunctions. Some of the implications and remaining problems are also discussed.