Lie-algebraic interpretation of the maximal superintegrability and exact solvability of the Coulomb–Rosochatius potential inndimensions

An algebraic interpretation of multilayer networks is introduced in relation to conscious experience, brain and body. The discussion is based on a network model for undirected multigraphs with coloured edges whose elements are time-evolving multilayers, representing complex experiential brain-body networks. These layers have the ability to merge by an associative binary operator, accounting for biological composition. As an extension, they can rotate in a formal analogy to how the activity inside layers would dynamically evolve. Under consciousness interpretation, we also studied a mathematical formulation of splitting layers, resulting in a formal analysis for the transition from conscious to non-conscious activity. From this construction, we recover core structures for conscious experience, dynamical content and causal efficacy of conscious interactions, predicting topological network changes after conscious layer interactions. Our approach provides a mathematical account of coupling and splitting layers co-arising with more complex experiences. These concrete results may inspire the use of formal studies of conscious experience not only to describe it, but also to obtain new predictions and future applications of formal mathematical tools.

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Exact solvability of two new 3D and 1D non-relativistic potentials within the TRA framework

Modern Physics Letters A ◽

10.1142/s0217732318501870 ◽

2018 ◽

Vol 33 (32) ◽

pp. 1850187 ◽

Cited By ~ 2

Author(s):

I. A. Assi ◽

H. Bahlouli ◽

A. Hamdan

Keyword(s):

Wave Function ◽

Orthogonal Polynomials ◽

Jacobi Polynomials ◽

Basis Functions ◽

Exact Solvability ◽

Analytical Properties ◽

Expansion Coefficients ◽

Potential Parameters ◽

Square Integrable Basis

This work aims at introducing two new solvable 1D and 3D confined potentials and present their solutions using the Tridiagonal Representation Approach (TRA). The wave function is written as a series in terms of square integrable basis functions which are expressed in terms of Jacobi polynomials. The expansion coefficients are then written in terms of new orthogonal polynomials that were introduced recently by Alhaidari, the analytical properties of which are yet to be derived. Moreover, we have computed the numerical eigenenergies for both potentials by considering specific choices of the potential parameters.

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Extended study on the application of the sextic potential in the frame of X(3)-Sextic

Journal of Physics G Nuclear and Particle Physics ◽

10.1088/1361-6471/ac3a00 ◽

2021 ◽

Author(s):

Mostafa Oulne ◽

Imad Tagdamte

Keyword(s):

Phase Transition ◽

Critical Point ◽

Electromagnetic Properties ◽

Shape Evolution ◽

Isotopic Chain ◽

Oscillator Potential ◽

Extended Study ◽

Deformed Nuclei ◽

Exact Solvability ◽

Shape Phase Transition

Abstract The main aim of the present paper is to study extensively the γ-rigid Bohr Hamiltonian with anharmonic sextic oscillator potential for the variable β and γ = 0. For the corresponding spectral problem, a ﬁnite number of eigenvalues are found explicitly, by algebraic means, so-called Quasi-Exact Solvability (QES). The evolution of the spectral and electromagnetic properties by considering higher exact solvability orders is investigated, especially the approximate degeneracy of the ground and ﬁrst two β bands in the critical point of the shape phase transition from a harmonic to an anharmonic prolate β-soft, also the shape evolution within an isotopic chain. Numerical results are given for 39 nuclei, namely, 98-108Ru, 100-102Mo, 116-130Xe, 180-196Pt, 172Os, 146-150Nd, 132-134Ce, 152-154Gd, 154-156Dy, 150-152Sm, 190Hg and 222Ra. Across this study, it seems that the higher quasi-exact solvability order improves our results by decreasing the rms, mostly for deformed nuclei. The nuclei 100,104Ru, 118,120,126,128Xe, 148Nd and 172Os fall exactly in the critical point.

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Exact solvability of semiclassical quantum gravity in two dimensions and liouville theory

Field Theory, Quantum Gravity and Strings - Lecture Notes in Physics ◽

10.1007/3-540-16452-9_11 ◽

1988 ◽

pp. 180-189

Author(s):

Norma Sanchez

Keyword(s):

Quantum Gravity ◽

Two Dimensions ◽

Liouville Theory ◽

Exact Solvability

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Product of Two Commutators as a Square in a Free Group

Canadian Mathematical Bulletin ◽

10.4153/cmb-1990-032-4 ◽

1990 ◽

Vol 33 (2) ◽

pp. 190-196

Author(s):

Jonell A. Comerford ◽

Y. Lee

Keyword(s):

Free Group ◽

Mapping Class Group ◽

Class Group ◽

Mapping Class ◽

Algebraic Interpretation

AbstractWe show that, if [s,t][u, v] = x2 in a free group, x need not be a commutator. We arrive at our example by use of a result of D. Piollet which characterizes solutions of such equations using an algebraic interpretation of the mapping class group of the corresponding surface.

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