Representations Of Symmetry Group Research Articles

Structured Dynamics in the Algorithmic Agent.

In the Kolmogorov Theory of Consciousness, algorithmic agents utilize inferred compressive models to track coarse-grained data produced by simplified world models, capturing regularities that structure subjective experience and guide action planning. Here, we study the dynamical aspects of this framework by examining how the requirement of tracking natural data drives the structural and dynamical properties of the agent. We first formalize the notion of a generative model using the language of symmetry from group theory, specifically employing Lie pseudogroups to describe the continuous transformations that characterize invariance in natural data. Then, adopting a generic neural network as a proxy for the agent dynamical system and drawing parallels to Noether's theorem in physics, we demonstrate that data tracking forces the agent to mirror the symmetry properties of the generative world model. This dual constraint on the agent's constitutive parameters and dynamical repertoire enforces a hierarchical organization consistent with the manifold hypothesis in the neural network. Our findings bridge perspectives from algorithmic information theory (Kolmogorov complexity, compressive modeling), symmetry (group theory), and dynamics (conservation laws, reduced manifolds), offering insights into the neural correlates of agenthood and structured experience in natural systems, as well as the design of artificial intelligence and computational models of the brain.

The number of Hamiltonian cycles in groups of Symmetry

Groups are algebraic structures in Abstract Algebra comprised of a set of elements with a binary operation that satisfies closure, associativity, identity and invertibility. Cayley graphs serve as a visualization tool for groups, as they are capable of illustrating certain structures and properties of groups geometrically. In particular, each element in a group is assigned to a vertex in Cayley graphs. By the group action of left-multiplication, distinct elements in the generating sets can act on each element to create varied directed edges (Meier[1]). By contrast, the presence of a Hamiltonian cycles within a graph demonstrates its level of connectivity. In this research paper, utilizing directed Cayley graphs, we present a series of conjectures and theorems regarding the number and existence of Hamiltonian cycles within Dihedral groups, Symmetric groups of Platonic solids and Symmetric groups. By exploring the relationship between Abstract groups and Hamiltonian graphs, this work contributes to the broader field of research pertaining to Groups of Symmetry and Geometric Group Theory.

DFT modeling of electronic and mechanical properties of polytwistane using line symmetry group theory

The electronic and mechanical properties of polytwistane are studied considering helical symmetry for the first time. All presented calculations were carried out using the density functional theory (DFT) and LCAO approach as implemented in the CRYSTAL17 code. It was found that the curve of the formation energy dependence on the rotation angle has only one minimum corresponding to the incommensurate structure (structure without translational symmetry). Symmetry cell (CH) and topological monomer (C6H6) were used for the calculations. It was found that changing the symcell from CH to C6H6 has effect only under extreme torsion. For the polytwistane structure corresponding to the energy minimum, the electronic band gap and mechanical properties were calculated. The results obtained for Young's moduli and band gap indicate a smooth change in these properties in relatively small ranges upon torsion and axial deformation of the structure.

Theory of Radar Polarimetric Interferometry and Its Application to the Retrieval of Sea Ice Elevation in the Western Weddell Sea, Antarctic

AbstractSea ice elevation plays a crucial role in sea ice dynamic processes driven by winds and waves, for which satellite radar remote sensing becomes indispensable to monitor snow‐covered sea ice across the vast polar regions regardless of darkness and clouds. To measure sea ice elevation, a theory of polarimetric interferometry for both monostatic and bistatic radars is developed based on analytic solutions of Maxwell's equations, accounting for realistic and complicated properties of snow, sea ice, and seawater. This analytic method inherently preserves phase information that is imperative for radar polarimetry and interferometry. Among a multitude of complex radar coefficients in the general polarimetric interferometric covariance matrix, the symmetry group theory is utilized to identify and select appropriate terms pertaining to the retrieval of sea ice elevation while avoiding radar parameters that may inadvertently introduce non‐uniqueness and excessive uncertainty. Theoretical calculations compare well with field observations for rough and old sea ice encountered in the Operation‐IceBridge and TanDEM‐X Antarctic Science Campaign over the Western Weddell Sea. The results show that the magnitude of the coefficient of normalized correlation between co‐polarized horizontal and co‐polarized vertical radar returns is inversely related to sea ice elevation, while the associate phase term is nonlinear and noisy and should be excluded. From these analyses, a protocol is set up to measure Antarctic sea ice elevation to be presented in the next companion paper.

Pacman geometries and the Hayward term in JT gravity

We study the Hayward term describing corners in the boundary of the geometry in the context of the Jackiw-Teitelboim gravity. These corners naturally arise in the computation of Hartle-Hawking wave functionals and reduced density matrices, and give origin to AdS spacetimes with conical defects.This set up constitutes a lab to manifestly realize many aspects of the construction recently proposed in [1]. In particular, it can be shown that the Hayward term is required to reproduce the flat spectrum of Rényi entropies in the Fursaev’s derivation, and furthermore, the action with an extra Nambu-Goto term associated to the Dong’s cosmic brane prescription appears naturally.On the other hand, the conical defect coming from Hayward term contribution are subtly different from the defects set as pointlike sources studied previously in the literature. We study and analyze these quantitative differences in the path integral and compare the results. Also study previous proposals on the superselection sectors, and by computing the density operator we obtain the Shannon entropy and some novel results on the symmetry group representations and edge modes. It also makes contact with the so-called defect operator found in [2].Lastly, we obtain the area operator as part of the gravitational modular Hamiltonian, in agreement with the Jafferis-Lewkowycz-Maldacena-Suh proposal.

Progress and prospects in magnetic topological materials.

Magnetic topological materials represent a class of compounds with properties that are strongly influenced by the topology of their electronic wavefunctions coupled with the magnetic spin configuration. Such materials can support chiral electronic channels of perfect conduction, and can be used for an array of applications, from information storage and control to dissipationless spin and charge transport. Here we review the theoretical and experimental progress achieved in the field of magnetic topological materials, beginning with the theoretical prediction of the quantum anomalous Hall effect without Landau levels, and leading to the recent discoveries of magnetic Weyl semimetals and antiferromagnetic topological insulators. We outline recent theoretical progress that has resulted in the tabulation of, for the first time, all magnetic symmetry group representations and topology. We describe several experiments realizing Chern insulators, Weyl and Dirac magnetic semimetals, and an array of axionic and higher-order topological phases of matter, and we survey future perspectives.

RING CURRENT IN ANTHRACENE AND PHENANTHRENE: CORRECTION TO HÜCKEL PARAMETERS

Excessive degeneracy in the ground state of the π-electron energy levels of anthracene was removed using the Hückel method by correcting the Coulomb integral of the four central carbon atoms. A further correction to the resonance integral was proposed based on the ring-current model, which describes the π-ring current flow along the molecule’s perimeter, by introducing a new parameter for the bridge carbon atoms expressed as a fraction of the resonance energy. The solution to the Hamiltonian was obtained based on the symmetry group theory, which provides the advantage of solving the determinant matrix or secular equations in a particular irreducible representation with a relatively small matrix dimension. The results were further analyzed to determine the bond order and bond length. The values of the harmonic oscillator model of aromaticity, GEO, and EN indices of aromaticity and their ratio for the central and outer rings of benzene were used to evaluate the validity of the correction. Applying the same method to phenanthrene as a topological analog of anthracene allows for an interesting comparison of the stabilities of the two molecules.

Ab initio modeling of helically periodic nanostructures using CRYSTAL17: A general algorithm first applied to nanohelicenes

A general algorithm based on the line symmetry group theory is proposed for ab initio modeling of one-dimensional (1D) helical nanoobjects using computer codes designed for periodic structure calculations. The key point of the algorithm is the interpolation between the properties of a sequence of periodic structures to obtain the properties of 1D helical nanoobjects without translational periodicity. A special selection of periodic structures avoids excessively long periods, thereby providing control over computational costs. The developed approach gives an opportunity to simulate the continuous dependences of mechanical and electronic properties on the torsional strain for any helical 1D nanoobject using high-level quantum chemical methods. The superposition of torsional and axial deformations can be used to generate the two-parameter maps of the desired properties.The proposed algorithm is applied to obtain the structure and properties of selected nanohelicenes, which are the helicenes infinitely continued along the helical (screw) axis direction. The CRYSTAL17 computer code based on atomic basis set has been used for this purpose. The main advantage of this program in comparison with other widely available ab initio simulation codes is the built-in accounting for helical symmetry operations in 1D periodic systems. Performed calculations show that nanohelicenes have several local minima of potential energy, differing in the order of the helical axis. The irrational order of the helical axis found at all energy minima indicates the absence of translational symmetry in these structures. Two-parameter electronic band gap maps and the dependences of Young’s moduli on the torsion angle for the selected nanohelicenes are calculated and discussed.

The Elementary Particles of Quantum Fields.

The elementary particles of relativistic quantum field theory are not simple field quanta, as has long been assumed. Rather, they supplement quantum fields, on which they depend on but to which they are not reducible, as shown here with particles defined instead as a unified collection of properties that appear in both physical symmetry group representations and field propagators. This notion of particle provides consistency between the practice of particle physics and its basis in quantum field theory.

3D Printing Technology in Education and Some Examples of 3D Printer Technology Materials Applied in Chemistry Education

Technology affects and changes our lives day by day. The application of technological developments and advances in education is of great importance in order to bring targeted behavior to individuals. One of the technologies we observe in many areas, including education, is three-dimensional (3D) printing technology. Three dimensional printers can be used in the field of education to better visualize complex structures. 3D printers have a truly groundbreaking technology in solid modeling. With this technology, individuals can realize their dreams in a short time and in a concrete way. 3D printing technology also shows tremendous potential in the chemical sciences. This type of technology has begun to enter chemistry education on a wide range of subjects, and chemistry models produced in educational processes such as symmetry and point group theory, unit cell theory, orbital theory and structure-energy relationships contribute to students in terms of vision, touch and detailed examination. In this study, together with general information about the use of 3D printer technology in education, the importance of using this technology in education and information about the materials used in the field of chemistry teaching produced with 3D printer technology are presented.

General Lieb–Schultz–Mattis Type Theorems for Quantum Spin Chains

We develop a general operator algebraic method which focuses on projective representations of symmetry group for proving Lieb-Schultz-Mattis type theorems, i.e., no-go theorems that rule out the existence of a unique gapped ground state (or, more generally, a pure split state), for quantum spin chains with on-site symmetry. We first prove a theorem for translation invariant spin chains that unifies and extends two theorems proved by two of the authors in [OT1]. We then prove a Lieb-Schultz-Mattis type theorem for spin chains that are invariant under the reflection about the origin and not necessarily translation invariant.

The Geometrical Meaning of Spinors Lights the Way to Make Sense of Quantum Mechanics

This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.

Modified MIT bag Models—part I: Thermodynamic consistency, stability windows and symmetry group

In this work we study different variations of the MIT bag model. We start with the so called non-ideal bag model and discuss it in detail. Then we implement a vector interaction in the MIT bag model that simulates a meson exchange interaction and fix the quark-meson coupling constants via symmetry group theory. At the end we propose an original model, inspired by the Boguta-Bodmer models, which allows us to control the repulsion interaction at high densities. For each version of the model we obtain a stability window as predicted by the Bodmer-Witten conjecture and discuss its thermodynamic consistency.

Fluctuating Asymmetry and 'Pincer-Like' Symmetry (PS – Symmetry)

Individual research on the theory of broken symmetry groups in natural, biological and socio-economic systems has brought us closer to the development of basic principles, methods and techniques in understanding of the fundamental essence of symmetry and asymmetry of systems. The main provisions set forth in this article are based on the author’s modest attempt to reveal what is common in the asymmetry of systems, which allows systems to adapt to changing environmental conditions, develop and evolve. In the authors opinions, such a property of systems is the objective fundamental state of any system, conventionally called «pincer-like» symmetry (PS -symmetry). An understanding of the essence of PS-symmetry is presented; particular examples of the presence of similar symmetry among representatives of various forms of existence are given. The presented study opens a new direction in the understanding of the essence of symmetry and its broken groups.

Equivariant PT-symmetric real Chern insulators

It was understood that Chern insulators cannot be realized in the presence of PT symmetry. In this paper, we reveal a new class of PT-symmetric Chern insulators, which has internal degrees of freedom forming real representations of a symmetry group with a complex endomorphism field. As a generalization to the conventional 2n-dimensional Chern insulators with integer n ≥ 1, these PT-symmetric Chern insulators have the n-th complex Chern number as their topological invariant, and have a ℤ classification given by the equivariant orthogonal K theory. Thus, in a fairly different sense, there exist ubiquitously Chern insulators with PT symmetry. By generalizing the Thouless charge pump argument, we find that, for a PT-symmetric Chern insulator with Chern number ν, there are equally many ν flavors of coexisting left- and right-handed chiral modes. Chiral modes with opposite chirality are complex conjugates to each other as complex representations of the internal symmetry group, but are not isomorphic. For the physical dimensionality d = 2, the PT-symmetric Chern insulators may be realized in artificial systems including photonic crystals and periodic mechanical systems.

The Group-Theoretical Analysis of Nonlinear Optimal Control Problems with Hamiltonian Formalism

In this study, we pay attention to novel explicit closed-form solutions of optimal control problems in economic growth models described by Hamiltonian formalism by utilizing mathematical approaches based on the theory of Lie groups. For this analysis, the Hamiltonian functions, which are used to define an optimal control problem, are considered in two different types, namely, the current and present value Hamiltonians. Furthermore, the first-order conditions (FOCs) that deal with Pontrygain maximum principle satisfying both Hamiltonian functions are considered. FOCs for optimal control in the problem are studied here to deal with the first-order coupled systems. This study mainly focuses on the analysis of these systems concerning for to the theory of symmetry groups and related analytical approaches. First, Lie point symmetries of the first-order coupled systems are derived, and then by using the relationships between symmetries and Jacobi last multiplier method, the first integrals and corresponding invariant solutions for two different economic models are investigated. Additionally, the solutions of initial-value problems based on the transversality conditions in the optimal control theory of economic growth models are analyzed.

Invariant solutions and submodels in two-phase fluid mechanics generated by 3-dimensional subalgebras: Barochronous flows

A model of two-phase isothermal ideal fluid is investigated by methods of the symmetry group theory. Different types of submodels are obtained for this model with respect to 3-dimensional subalgebras: invariant and partially invariant submodels. All invariant submodels are integrated. Ten subalgebras of the symmetry algebra are considered. Only these subalgebras generate barochronous motions of two-phase fluid. Many new exact solutions of two-phase fluid mechanics are found. They describe different motions of two-phase fluid, such as homogeneous motions, Galileo invariant motions and the motions with linear field of velocities. Two applications of the solutions are presented.

The Symbolic Meaning of Redesign Batik Motives Kawung Solo based on Applied Mathematics Geometry Transformation and Village Promotion Efforts Kampung Matematika Karanglo, Karanganyar

Indonesian Batik by UNESCO was designated as the Masterpieces of the Oral and Intangible Heritage of Humanity since October 2009. The famous Indonesian batik motif from Solo is the Kawung batik motif. One type of batik is stamped batik. In making batik, there is always a clear repetition, so that the image appears repetitive in the same shape. The problem with making stamp batik is the price of expensive batik stamp. Generally, one stamp of batik can only produce one motif. Therefore, the main problem that was solved in this study was how to develop a Solo-Kawung batik motif design that has a uniqueness from one motif in the stamp can produce many patterns of batik without losing the local identity. The values contained in kawung batik are simplicity, regularity, consistency, and mutual cooperation reflected in community life. This problem will be solved by applying the concept of Geometry Analysis with the Symmetry Group Theory which will then be used Geometry Transformation. The Energy Security Concept is Every aspect of development must be environmentally sound while still striving for effectiveness and efficiency. This research was developed and applied in Karanglo Village, Karanganyar Regency as a Batik Mathematics Village. The development of creative industries applied in Karanglo is an industry that originates from the utilization of individual talents to create welfare and employment through the creation and utilization of these creative and creative powers. There are three directions for developing creative industries, namely industries based on creative cultural industries, creative industries and copyright industries with environmental based.

Non-Kolmogorov dissipation in a turbulent planar jet

Adopting Lie symmetry group theory both Kolmogorov and non-Kolmogorov scaling and dissipation laws are explored theoretically for a turbulent planar jet. We find that the jet entrainment coefficient varies with streamwise distance when non-Kolmogorov scaling laws hold.

Improved Variational Nodal Method Based on Symmetry Group Theory

To reduce the calculation effort and memory requirement for high-order PN expansion calculation in the Variational Nodal Method (VNM), the surficial irreducible basis functions based on the symmetry group theory have been employed to block-diagonalize one of the four nodal response matrices. Its effectiveness encourages our further investigation on the application of the symmetry group theory to volumetric expansion to block-diagonalize the remaining three of the nodal response matrices in this paper. By using the symmetry group theory, the neutron transport problem for each node can be decoupled into several independent subproblems as long as both the geometry and the material distribution of the node are symmetric. Each of these subproblems can be solved by using variational principles as in the traditional VNM, providing their nodal response matrices as the diagonal blocks of the corresponding entire ones. For hexagonal-z node, each nodal response matrix can be reduced into 16 diagonal blocks, among which only 12 have to be calculated due to the properly selected irreducible basis functions. In addition, it is also proved that the response matrices with anisotropic scattering can also be block-diagonalized as the same. Calculation results based on typical problems demonstrate that the new method reduces the time cost for the response matrice calculation by one order of magnitude compared with our previous work. For the total computing time, the speedup ratio is about 2 for P3 calculation and 4 for P5 calculation. Furthermore, almost 40% of the memory requirement can be saved.