The chiral optical Tamm state (COTS) is a special localized state at the interface of a handedness-preserving mirror and a structurally chiral medium such as a cholesteric liquid crystal or a chiral sculptured thin film. The spectral behavior of COTS, observed as reflection resonances, is described by the temporal coupled-mode theory. Mode coupling is different for two circular light polarizations because COTS has a helix structure replicating that of the cholesteric. The mode coupling for co-handed circularly polarized light exponentially attenuates with the cholesteric layer thickness since the COTS frequency falls into the stop band. Cross-handed circularly polarized light freely goes through the cholesteric layer and can excite COTS when reflected from the handedness-preserving mirror. The coupling in this case is proportional to anisotropy of the cholesteric and theoretically it is only anisotropy of magnetic permittivity that can ultimately cancel this coupling. These two couplings being equal results in a polarization crossover (the Kopp--Genack effect) for which a linear polarization is optimal to excite COTS. The corresponding cholesteric thickness and scattering matrix for COTS are generally described by simple expressions.

Hundreds of minerals are chiral, that is, they appear in nature in two forms – left-handed and right-handed. Yet except for quartz, that key structural property remained, by-and-large in the shadow of the world of minerals in research, in museum displays, and for collectors. This review is devoted for providing a full picture of chiral minerals in nature. It starts with a general outline of the crystallographic background needed for the characterization of chiral minerals, continues with a detailed description of the many chemical and physical processes leading to their formation, follows with their chemical reactivities and transformations, with their physical properties, and with the ways to analyze and identify them. Many tables with listings of the various types of chiral minerals, are provided. It is hoped that this review will spark interest in this aspect nature’s crystals.

Helical (frontier) molecular orbitals [Chem. Sci. 2013, 4, 4278] have been reported almost a decade ago. Yet, helical molecular spin densities [Mol. Phys. 2022, e2157774] have been observed by us only very recently. In this work, we have observed that some chiral boron clusters (B–16, B–20, B–24, and B–28) can simultaneously have helical molecular orbitals and helical spin densities. To our best knowledge, it is the first time we have discovered that inorganic boron clusters assume such an unprecedented property. We have unambiguously unraveled that it is chiral Jahn-Teller distortion of quasi-planar boron clusters that drives the formation of helical molecular spin densities. More interestingly, we have shown that elongation/enhancement of helical molecular orbitals can be achieved by simply adding more building blocks via a linker. Aromaticity properties of these boron clusters are also discussed. Implication of this work is straightforward that boron clusters may find potential applications in spintronics, such as molecular magnets.

Chirality depends on particular symmetries. For crystal structures it describes the absence of mirror planes and inversion centers, and in addition to translations, only rotations are allowed as symmetry elements. However, chiral space groups have additional restrictions on the allowed screw rotations as a symmetry element, because they always appear in enantiomorphous pairs. This study classifies and distinguishes the chiral structures and space groups. The chirality is quantified using Hausdorff distances and continuous chirality measures and selected crystal structures are reported. The chirality is discussed for bulk solids and their surfaces. Moreover, the band structure, and thus, the density of states, is found to be affected by the same crystal parameters as the chirality. However, it is independent of handedness. The Berry curvature, as a topological measure of the electronic structure, depends on the handedness, but is not a proof for the chirality because it responds on the inversion of a structure. For molecules, optical circular dichroism is one of the most important measures for the chirality. Thus, it is proposed in this study that the circular dichroism in the angular distribution of photoelectrons in high symmetry configurations can be used to distinguish the handedness of chiral solids and their surfaces.

We compare Chiral Perturbation Theory (ChPT) and the Linear Sigma Model (LSM) as realizations of low energy QCD for light mesons in a chirally imbalanced medium. The relations between the low-energy constants of the Chiral Lagrangian and the corresponding constants of the Linear Sigma Model are established as well as the expressions for the decay constant of $\pi $-meson in the medium and for the mass of the $a_0$. In the large $N_c$ count taken from QCD the correspondence of ChPT and LSM is remarkably good and give a solid ground for search of chiral imbalance manifestation in pion physics. A possible experimental detection of chiral imbalance (and therefore a phase with Local Parity Breaking) is outlined in the charged pion decays inside the fireball.

In the field of quality analysis of food and flavoring products, gas chromatography – quadrupole mass spectrometry – ion mobility spectrometry (GC-QMS-IMS) is a powerful technique for the simultaneous detection of volatile organic compounds (VOC) by both QMS and IMS. GC is an established technique for the separation of complex VOC-rich foods such as beverages, dairy and fruit products as well as flavorings in general. While the subsequent detection by IMS features soft ionization of fragile compounds (e.g., terpenes) with characteristic drift times, MS provides the analytes’ m/z-values for database substance identification. A limitation of the prominently used static headspace-based GC-QMS-IMS systems is the substantially higher sensitivity of IMS in comparison to full scan QMS, which results in the detection of relevant VOC only in IMS, but not in the QMS analyzer. The present study describes a new prototypic trapped-headspace (THS)-GC-QMS-IMS setup, with optimized active flow regulation, able to gain valuable data of both detectors. This ultimately allows the combination of a soft ionization with m/z values obtained from database-searchable electron ionization (EI) spectra and as such, offers a substantial advantage in fields of quality control. The new setup features aligned retention times for IMS and MS and sufficient signal intensities for QMS and IMS. The analytical setup was evaluated with a set of mango puree samples from various provenances and the data was evaluated by suitable machine learning approaches. The results demonstrate that THS-GC-QMS-IMS allows for the classification of mango purees from different cultivars and can be a promising alternative method for authenticity control of food, flavors, and beverages.

A comprehensive review on symmetry and conditional symmetry is made from the core conception of symmetry and conditional symmetry. For a dynamical system, the structure of symmetry means its robustness against the polarity change of some of the system variables. Symmetric systems typically show symmetrical dynamics and even when the symmetry is broken, symmetric pairs of coexisting attractors are born annotating the symmetry in another way. The polarity balance can be recovered by the combinations of the polarity reverse of system variables, and furthermore, it can also be restored by the offset boosting of some of the system variables if the variables lead to the polarity reverse from their functions. In this case, conditional symmetry is constructed giving a chance for a dynamical system outputting coexisting attractors. Symmetric strange attractors typically represent the flexible polarity reverse of some of the system variables, which brings more alternatives of chaotic signal and more convenience for chaos application. Symmetric and conditionally symmetric coexisting attractors can also be found in memristive systems and circuits. Therefore, symmetric chaotic system and those of conditional symmetry provide sufficient system options for chao-based applications.

There is a deep link existing between symmetry and nature that may be best apprehended by using group theory. In physics, giving a symmetry group is entirely equivalent to fix natural laws (Noether’s theorem). Within such a framework, it follows that all fundamental mechanical laws may be given by defining a group of symmetry operations belonging to the Gal(3,1) symmetry group having ten generators (3 + 3 + 3 + 1 = 10). The move to special relativity amounts simply to substitute another group ISO(3,1) to Gal(3,1) and having the same number of generators. In both cases, one immediately perceives that we live in universe with 3 dimensions of space and 1 dimension of time. A further progress could be made by looking for the symmetry group that leaves Maxwell’s equations invariant. The answer to this fundamental question is the group ISO(4,2)⊗U(2)⊗U(2) characterized by 23 generators : 15 = 10 + 5 for the conformal group ISO(4,2), 4 for U(2) and hence 8 = 4 + 4 for the compact part U(2)⊗U(2) that could be related to the existence of quantum physics besides relativity. As ISO(3,1) is a sub-group of ISO(4,2), the two numbers 4 and 2 suggests that, taking into account the existence of light besides matter, we live in an universe having 4 spacelike dimensions and 2 time-like dimensions. Here, it is proposed that, as music is characterized by a scale-invariant symmetry operation (transposition by octaves) and that as time may be perceived differently in shamanic trance for instance, the two “new” dimensions of our universe may be related to music and consciousness. Related to the fifth dimension of music, the notion of diapason linking physical frequencies to a musical note is revisited. It is thus proposed to adopt a new A4 = 429.62 Hz value based on universal physical constants of our universe and on the mass of the water molecule H₂O, the most abundant molecule in the universe. Demonstration is done that this new diapason is superior to other diapasons based on frequencies of 440 Hz or even 432 Hz

Differences between the quantum mechanical and relativistic concepts of time observed in GPS satellites are accounted for by applying the equivalence principle to the transitioning electron of an atomic clock. This allows a differential equation of motion to be derived for the electron in the Minkowski space between electron shells. By applying Hamilton’s principle we transform the differential equation into a relativistically correct integral equation of motion, the time integral of a Lagrangian. This newly derived equation accounts for the abstract mathematics of the more familiar non-relativistic model by means of a physical interpretation. Abstract rotations in Hilbert space are replaced by real rotations of particle field geometries in Minkowki space that describe the electron transition and emission of a photon. Because the properties of energy are universal the same integral equations are able to be used to describe the time evolution of galaxies despite vast differences in lifetime. The comparison of gravitational and electromagnetic energy exchanges by means of radial and transverse fields leads to an alternative to the standard model that is based on a field model. It explains the observed properties of galaxies as a superposition of neutrinos localized within the horizon of a black hole with spins aligned. The particle properties of free neutrinos are accounted by means of a classical field geometry; specifically, a spin that is oriented in space with rotational inertia due to energy. A test of the standard model is proposed based on differing views for flavor oscillation.

The first diastereoselective synthesis of (–)-1-epi-lentiginosine stars from a common chiral trans-epoxyamide derived from 2-pyridincarbaldehyde is reported. This methodology involves a sequential oxirane ring opening and intramolecular 5-exo-tet cyclization of tosylate tras-epoxyalcohol to afford a diastereomeric mixture of indolizinium salts in a one-pot fashion, followed by regio- and diastereospecific pyridinium ring reduction.

With the development of the volume of information exchange and perception increases, the demands for intelligent, miniaturized and integrated optical devices for information acquisition are also increasing. As the core component of optical networks for transmitting information, how to further optimize its structural characteristics to generate richer optical characteristics and apply them to information exchange and optical field control has become a key research hotspot. The introduction of chiral twist characteristics has led to new phenomena and applications in optical field transmission and transformation for traditional optical fibers or microstructured optical fibers (MOF). Therefore, this review mainly starts from the principle of chiral optical fibers, introduces their preparation and latest application scenarios, and finally looks forward to their potential future development prospects.

Dark energy and dark matter are described as different manifestations of vacuum energy in a new cyclic universe framework of Extra Dimensional Symmetry (EDS) that doubles large spatial dimensions with microscopic partners. In this testable framework, vacuum energy, constrained by a Planck density constraint, exists in a gravitationally dark state where actual gravitational constant G_o = 0G. While a small repulsive component is constrained by an asymptotically evolving asymmetry to the active state where G_o = 2G, some components in the inert state are enabled to gravitationally attract by neutrino induced gravitational constant with light and shear activated heavy components. Observational and experimental tests are briefly discussed.

In this brief review, the concepts of quantum tunneling and parity violation are introduced in the context of chiral molecules. A particle moving in a double well potential provides a good model to study the behavior of chiral molecules, where the left well and right well represent the L and R enantiomers, respectively. If the model considers the quantum behavior of matter, the concept of quantum tunneling emerges, giving place to a stereomutation dynamics between left- and right-handed chiral molecules. Parity-violating interactions, like the electroweak one, can be also considered, making possible the existence of an energy difference between the L and R enantiomers, the so-called parity-violating energy difference (PVED). Here we provide a brief account of some theoretical methods usually employed to calculate this PVED, also commenting on relevant experiments devoted to experimentally detect the aforementioned PVED in chiral molecules.

Recently Palle has investigated the chiral vorticity and Cartan torsion in neutrino asymmetries. In his case he addressed this problem in Goedel s like anisotropic Einstein-Catan cosmology. In this paper we discusse how these ideas applied to sheared Bianchi types I Einstein-Cartan (EC) neutrino amisotropic cosmology, affect the handness of neutrinos in the universe. Actually here a novel concept of the chiral metric is introduced where metric functions also possess two distinct signs as in neutrino flipping or helicity. The axial anomaly equation for neutrinos in the presence of torsion and metric chirality is shown to produce left-handed neutrinos from right-handed torsion. Metric chirality is shown to be able to define how the metric would behave far away of neutrino density. Chiral flipping of the chiral neutrinos in the presence of torsion is also investigated. It is shown that when the chiral torsion is left-handed the chemical chiral potential vanishes as the universe expands.

Actuators are essential components for intelligent machines that can fulfill certain tasks in response to environmental stimuli. In recent years, actuators that can transform from a 2D ribbon shape to a 3D helical configuration under certain external stimuli have attracted significant attention due to the potential applications of the targeted helical structures in springs, propulsion generation, and artificial muscles. Inspired by the chiral opening of Bauhinia variegate‘s seedpods and the coiling of the Towel Gourd tendril with perversions, researchers have made significant breakthroughs in synthesizing state-of-the-art actuators capable of mimicking helical transformations. In this review, we give a brief overview of the shape evolution mechanisms of these two plant structures and then review recent progress in the fabrication of biomimetic helical actuators. These structures are categorized by the stimuli-responsive materials involved, including hydrogels, liquid crystal networks/elastomers, shape memory polymers, and multiwall carbon nanotubes. By providing this survey on important recent advances along with our perspectives, we hope to solicit new inspirations and insights on the development and fabrication of smart actuators, as well as the future development of interdisciplinary research at the interface of physics, engineering, and biology.

This review deals with the functionalization of double bonds carried out in the presence of a chiral catalyst exploiting the intramolecular attack to haliranium ions by nucleophilic nitrogen of amides or carbamates prepared from achiral aminoalkenes, and the C-N bonds formation leads to highly enantioenriched non-aromatic heterocycles. A range of protocols are reported, emphasizing the synthesis of many natural and biologically active products of pharmacological interest prepared according to this methodology.

This study introduces a data obfuscation technique, leveraging the exponential map associated with the generators of Lie groups. Originating from quantum machine learning frameworks, our method illustrates the practical application of quantum mechanics principles in data processing. Specifically, it employs the exponential map of a generator algebra to introduce controlled noise into the data, achieving obfuscated data while preserving its utility for machine learning tasks. This strategy is shown to safeguard privacy in sensitive datasets, such as discussed medical records, and to enhance dataset volume and diversity through augmentation. Our empirical analysis, benchmarked against standard machine learning approaches, demonstrates that our method can maintain or even improve the predictive accuracy of the original data. This research highlights the potential of Lie group theory for advancing data privacy in medicine, marking a significant contribution to machine learning methodologies by offering the dual benefits of data obfuscation and enrichment. Through this synthesis of algebraic structures and machine learning, we propose new pathways for the secure and effective use of data in sensitive areas.

Formal Calculation uses an auxiliary form to calculate various nested sums and provides results in three forms. It is also a powerful tool for analysis. This article studies the symmetry of the coefficients in Formal Calculation. Three types of extended numbers were introduced, and many formulas for symmetric functions were obtained.

Either stereo reactants or stereo catalysis from achiral or chiral molecules are prerequisite to obtain pure enantiomeric lipid derivatives. We reviewed a few plausible organic syntheses of phospholipids under prebiotic conditions with a special attention to the starting materials as pro-chiral dihydroxyacetone and dihydroxyacetone phosphate (DHAP), which are the key molecules to break symmetry in phospholipids. The advantages of homochiral membranes compared to those of heterochiral membranes were analysed in term of specific recognition, optimal functions of enzymes, membrane fluidity and topological packing. All biological membranes contain enantiomeric lipids in modern bacteria, eukarya and archaea. The contemporary archaea, comprising of methanogens, halobacteria and thermoacidophiles are living under extreme conditions reminiscent of primitive environment and may indicate the origin of one ancient evolution path of lipid biosynthesis. The analysis of lipid metabolism reveals that all modern cells including archaea synthetize enantiomeric lipid precursors from prochiral DHAP. sn-glycerol-1-phosphate dehydrogenase (G1PDH), usually found in archaea, catalyses the formation of sn-glycerol-1-phosphate (G1P), while sn-glycerol-3-phosphate dehydrogenase (G3PDH) catalyses the formation of sn-glycerol-3-phosphate (G3P) in bacteria and eukarya. The selective enzymatic activity seems to be the main strategy that evolution retained to obtain enantiomeric pure lipids. The occurrence of two genes encoding for G1PDH and G3PDH, served to build up an evolution tree and the basis of our review focusing on the evolution of these two genes. Gene encoding for G3PDH in Eukarya may originate from G3PDH gene found in rare archaea indicating that archaea appeared earlier in the evolution tree than eukarya. Archaea and bacteria evolved probably separately, due to their distinct respective genes coding for G1PDH and G3PDH. The suggested hypothesis is that catalysis of homochiral G1P or G3P from DHAP are more efficient than those leading to racemic G1P and G3P, since there are no enzymes able to synthesize racemic G1P and G3P from DHAP. We propose that G1PDH or G3DPH, which are not “image mirror enzymes” but belonging to distinct family of proteins, emerged separately during evolution. They were probably selected for their efficient catalytic activities during evolution from large libraries of vesicles containing various biopolymers, including amino acids, carbohydrates, nucleic acids, lipids, and meteorite components to induce chemical imbalance.

Based on detection of elliptic particle tracks, $\simeq 137^2 n^2$ bigger than Bohr-Sommerfeld electron orbits, indicating the possible detection of superluminal electrons masquerading as magnetic monopoles, a new structure emerges leading to: (i) a new set of seven elementary lengths; (ii) replacement of the usual motion (Lorentz) transformations by scale transformations between $v^2<c^2$ and $v^2>c^2$ frames; (iii) equivalence of charge between $v^2<c^2$ and $v^2>c^2$ frames based on the Dirac-Schwinger quantization condition; (iv) a relativistic foundation for the Dirac-Schwinger quantization condition; (v) a possible cause of charge quantization; and (vi) the prospect of symmetry in Maxwell's equations. If the elliptic particle tracks are viewed as a magnification of electron orbits, the effect is suggestive of a spacetime distortion such as those predicted in general relativistic theories.

Time evolution of the symmetry of symbols constituting alphabets rooted in the Phoenician script was investigated. A diversity of quantitative measures of symmetry of graphemes appearing in Phoenician, Western Greek, Etruscan from Marsiliana, Archaic Etruscan, Neo-Etruscan, Euclidian Greek, Archaic and Classical Latin, and Proto-Hebrew scripts, constituting the Phoenician script family, were calculated. The same measures were calculated for the Hebrew/Ashurit and English scripts. The Shannon-like measures of symmetry were calculated. The Shannon diversity index was calculated. The Shannon diversity index increased with time in a monotonic way for the scripts emerging from the Phoenician one. The diversity of symmetry groups inherent for alphabets emerging from the Phoenician one grows with time. We also introduced the symmetry factor of the alphabet. The symmetry factor quantifies the averaged level of symmetrization of the alphabet and the possible parsimony of graphical information necessary for the drawing of the entire set of graphemes, constituting the alphabet. The symmetry factor is decreased with time for the alphabets rooted in the Phoenician one. This means that the average level of symmetrization of the studied alphabet is increased with time. The parsimony of graphical information necessary for writing graphemes is increased with time. The values of the symmetry factor calculated for the addressed scripts are close one to another, with the pronounced exception of the Hebrew/Ashurit script. Our study supplies the arguments for the point of view, according to which the modern Hebrew/Ashurit script does not emerge from the Phoenician one.

This paper addresses the fate of extended space-time symmetries, in particular conformal symmetry and supersymmetry, in two-dimensional Rindler space-time appropriate to a uniformly accelerated non-inertial frame in flat 1+1-dimensional space-time. Generically, in addition to a conformal co-ordinate transformation, the transformation of fields from Minkowski to Rindler space is accompanied by local conformal and Lorentz transformations of the components, which also affect the Bogoliubov transformations between the associated Fock spaces. I construct these transformations for massless scalars and spinors, as well as for the ghost and super-ghost fields necessary in theories with local conformal and supersymmetries, as arising from coupling to 2-D gravity or supergravity. Cancellation of the anomalies in Minkowski and in Rindler space requires theories with the well-known critical spectrum of particles arising in string theory in the limit of infinite strings, and is relevant for the equivalence of Minkowski and Rindler frame theories.

Entropy is usually understood as the quantitative measure of “chaos” or “disorder”. However, the notions of “chaos” and “disorder” are definitely obscure. This leads to numerous misinterpretations of entropy. We propose to see the disorder as an absence of symmetry and to identify “ordering” with symmetrizing of a physical system; in other words, introducing the elements of symmetry into initially disordered physical system. We demonstrate with the binary system of elementary magnets that introducing elements of symmetry necessarily diminishes its entropy. This is true for 1D and 2D systems of elementary magnets. Imposing symmetry does not influence the Landauer Principle valid for the addressed systems. Imposing the symmetry restrictions onto the system built of particles contained within the chamber divided by the permeable partition also diminishes its entropy.

We specify basic definitions of the Chern-Simons current in cohomology theory, then we calculate its value by using a model of quantum machine learning, the so-called supersymmetric support Dirac machine. The supercurrent is generated from the coupling between three states of a quantum flux of modified Wilson loop of the Cooper pairs. We use Holo-Hilbert spectrum in frequency modulation to visualize the network as a coupling behavior of convolutional neuron network in superstatistical theory with the application to the theory of a superconductor. We also calculate the number of carbon atoms in the stable support spinor network structure of a graphene wormhole to produce a supercurrent of Cooper pairs as graviphoton states by using the Holo-Hilbert spectral analysis.

This paper presents an optimized feature-centered reflection symmetry axis detection and localization framework for object perception. The proposed framework is formed to obtain an improved reflection symmetry axis based on the salient symmetry feature. It starts with a refined Multi-scale Saliency Symmetry Model (MSSM), which is realized by applying isotropic symmetry operator on salient points in scale-space rather than all pixels. In each scale, salient points are initially extracted as local extremal from an image, and they are further refined by a multi-scale implementation for generating salient symmetry feature maps. A Symmetric Transformation Matrix is then computed using the optimal feature matching pairs, which can be explicitly used as an abstract representation of the constraint regions of symmetry objects in an image to optimize the performance of the potential symmetry axis detection. The framework has been investigated experimentally both on the classical dataset from a symmetry detection challenge and the latest dataset. It has shown that the framework can get a better or comparative result and also can be further adapted into terminated human--computer equipment for reflection symmetry object perception and tracking.

Carbon nanotubes (CNTs) possess unique mechanical, physical, electrical and absorbability properties coupled with their nanometer dimensional scale that renders them extremely valuable for applications in many fields including nanotechnology and chromatographic separation. The aim of this review is to provide an updated overview about the applications of CNTs in chiral and achiral separations of pharmaceuticals, biologics and chemicals. Chiral SWCNTs and MWCNTs have been directly applied for the enantioseparation of pharmaceuticals and biologicals by using them as stationary or pseudostationary phases in chromatographic separation techniques such as high-performance liquid chromatography (HPLC), capillary electrophoresis (CE) and gas chromatography (GC). Achiral MWCNTs have been used for achiral separations as efficient sorbent objects in solid-phase extraction techniques of biochemicals and drugs. Achiral SWCNTs have been applied in achiral separation of biological samples. Achiral SWCNTs and MWCNTs have been also successfully used to separate achiral mixtures of pharmaceuticals and chemicals. Collectively, functionalized CNTs have been indirectly applied in separation science by enhancing the enantioseparation of different chiral selectors whereas, non-functionalized CNTs have shown efficient capabilities for chiral separations by using techniques such as encapsulation or immobilization in polymer monolithic columns.

Chiral hydroxamic acid (HA) and bis-hydroxamic acid (BHA) ligands have made significant contributions to the field of asymmetric synthesis, particularly in the synthesis of natural products. These ligands possess unique molecular structures that allow for exceptional stereochemical control, leading to their widespread use in catalytic systems. This review highlights the advancements made in asymmetric synthesis using chiral hydroxamic acid and bis-hydroxamic acid ligands and their impact on the synthesis of complex natural products. The discussion encompasses their role in enantioselective C-C bond formation, the functionalization of C-H bonds, asymmetric transformations involving heteroatoms, and their application in the total synthesis of natural products. The versatility and efficiency of chiral hydroxamic acid ligands and bis-hydroxamic acid ligands make them invaluable tools for synthetic chemists working towards the efficient and selective synthesis of natural products. This review provides a comprehensive overview of their contributions, showcasing their potential to expand the boundaries of chemical synthesis and access the diverse array of natural product scaffolds.

We propose an experimental method for the unidirectional excitation of spin waves. By structuring Au nanowires arrays within a coplanar waveguide onto a thin yttrium iron garnet (YIG) film, we observe a chiral coupling between the excitation field geometry of the nanowires grating and several well-resolved propagating magnon modes. We report a propagating spin wave spectroscopy study with unprecedented spectral definition, wavelengths down to 130 nm and attenuation lengths well above the 100 μm over the 20 GHz frequency band. The proposed experiment paves the way for future non-reciprocal magnonic devices.

Hybrid organic–inorganic perovskites (HOIPs) are promising materials in several fields related to electronics, offering long carrier-diffusion lengths, high absorption coefficients, tunable band gaps, and long spin lifetimes. Recently, chiral perovskites have attracted a huge interest thanks to the possibility of further widening the applications of HOIPs. Chiral materials, being intrinsically non-centrosymmetric, display several attractive physico-chemical properties, including circular dichroism, circularly polarized photoluminescence, nonlinear optics, ferroelectricity, and spin-related effects. Recent studies have shown that chirality can be transferred from the chiral organic ligands into the inorganic perovskite framework, resulting in materials combining the advantages of both chirality and perovskite superior optoelectronic characteristics. Thus, chiral HOIPs have emerged as promising materials for chiroptoelectronic, spintronic, and ferroelectric applications. Here, we report a summary of the latest advances in the field of chiral HOIPs, with a specific focus on lead-free materials and on the first attempts to understand the correlation between the crystal structure features and the chirality-induced functional properties

We examine here characteristics of electromagnetic waves that propagate through an unbounded space filled with a homogeneous isotropic chiral medium. Resulting characters are compared to those of the electromagnetic waves propagating through an achiral free space. To this goal, we form energy conservation laws for key bilinear parameters in a chiral case. Due to a nonzero medium chirality, conservation laws turn out to contain extra terms that are linked to the spin-orbit coupling, which is absent for an achiral case. As an example, we take a plane wave for achiral case to evaluate those bilinear parameters. Resultantly, the conservation laws for a chiral case are found to reveal inconsistencies among several bilinear parameters that constitute the con-servation laws , thereby prompting us to establish partial remedies for formulating proper wave-propagation problems.

Can physical objects be in time-symmetry? Physical objects can only exist in a medium that has physical attributes, which means this medium is a type of energy. Is time energy? This article will show that time is not energy and there is no possibility that physical objects could be in time-symmetry. Physical objects only can be in symmetry in the time-invariant space in which they exist. In this perspective time measured with clocks is the result of the observer’s measurement in the time-invariant space. The time-symmetry model is flawed.

Proper identification of oriented knots and 2-component links requires a precise link nomenclature. Motivated by questions arising in DNA topology, this study aims to produce a nomenclature unambiguous with respect to link symmetries. For knots, this involves distinguishing a knot type from its mirror image. In the case of 2-component links, there are up to sixteen possible symmetry types for each topology. The study revisits the methods previously used to disambiguate chiral knots and extends them to oriented 2-component links with up to nine crossings. Monte Carlo simulations are used to report on writhe, a geometric indicator of chirality. There are ninety-two prime 2-component links with up to nine crossings. Guided by geometrical data, linking number and the symmetry groups of 2-component links, a canonical link diagram for each link type is proposed. All diagrams but six were unambiguously chosen ( $8_1^{2}5$ , $9_5^2$ , $9_3^{2}4$ , $9_3^{2}5$ , $9_3^{2}9$ , and $9_4^{2}1$ ). We include complete tables for prime knots with up to ten crossings and prime links with up to nine crossings. We also prove a result on the behavior of the writhe under local lattice moves.

A numerical application of linear-molecule symmetry properties, described by the D_∞h point group, is formulated in terms of lower-order symmetry groups D_nh with finite n. Character tables and irreducible representation transformation matrices are presented for D_nh groups with arbitrary n-values. These groups are subsequently used in the construction of symmetry-adapted ro-vibrational basis functions for solving the Schrödinger equations of linear molecules as part of the variational nuclear motion program TROVE. The TROVE symmetrisation procedure is based on a set of "reduced" vibrational eigenvalue problems with simplified Hamiltonians. The solutions of these eigenvalue problems have now been extended to include the classification of basis-set functions using ℓ, the eigenvalue (in units of ℏ) of the vibrational angular momentum operator ${\widehat{L}}_z$ . This facilitates the symmetry adaptation of the basis set functions in terms of the irreducible representations of D_nh. ¹²C₂H₂ is used as an example of a linear molecule of D_∞h point group symmetry to illustrate the symmetrisation procedure.

The general theory of relativity (GR) is known to be invariant under smooth coordinate transformations (diffeomorphism). This group has a subgroup known as the Lorentz group of symmetry which is manifested in the weak field approximation to GR. The dominant operator in the weak field equation of GR is thus the d'Alembert (wave) operator which has a retarded potential solution. Galaxies are huge physical systems having dimensions of many tens of thousands of light years. Thus any change at the galactic center will be noticed at the rim only tens of thousands of years later. Those retardation effects are neglected in present day galactic modelling used to calculate rotational velocities of matter in the rims of the galaxy and surrounding gas. The significant differences between the predictions of Newtonian instantaneous action at a distance and observed velocities are usually explained by either assuming dark matter or by modifying the laws of gravity (MOND). In this paper we will show that taking general relativity seriously without neglecting retardation effects one can explain the radial velocities of galactic matter in the M33 galaxy without postulating dark matter.

Evolution and geometry generate complexity in similar ways. Evolution drives natural selection while geometry may capture the logic of this selection and express it visually, in terms of specific generic properties representing some kind of advantage. Geometry is ideally suited for expressing the logic of evolutionary selection for symmetry, which is found in the shape curves of vein systems and other natural objects such as leaves, cell membranes, or tunnel systems built by ants. The topology and geometry of symmetry is controlled by numerical parameters, which act in analogy with a biological organism's DNA. The introductory part of this paper reviews findings from experiments illustrating the critical role of two-dimensional design parameters and shape symmetry for visual or tactile shape sensation, and for perception-based decision making in populations of experts and non-experts. Thereafter, results from a pilot study on the effects of fractal symmetry, referred to herein as the symmetry of things in a thing, on aesthetic judgments and visual preference are presented. In a first experiment (psychophysical scaling procedure), non-expert observers had to rate (scale from 0 to 10) the perceived beauty of a random series of 2D fractal trees with varying degrees of fractal symmetry. In a second experiment (two-alternative forced choice procedure), they had to express their preference for one of two shapes from the series. The shape pairs were presented successively in random order. Results show that the smallest possible fractal deviation from "symmetry of things in a thing" significantly reduces the perceived attractiveness of such shapes. The potential of future studies where different levels of complexity of fractal patterns are weighed against different degrees of symmetry is pointed out in the conclusion.

A topological representation theory of chiral spin liquid is proposed to unify the integral and fractional quantum Hall fluid based on hyperbolic geometry, knot lattice and Thurston's train track theory. The integral and fractional Hall resistivity obeys an exact unified equation, $R_{xy} = (h/e^2)B_{e}$, $B_{e} = (n+\nu)^{-1}$, with respect to the bulk energy gap ${\Delta}E = {\hbar{e}B_{e}}/{Mc}$. The Wen-Zee matrix formulation of topological fluid for quantum Hall effect is constructed by topological surgery and train track. The chirality of spin liquid is spontaneously distinguished by Jones polynomial. The integral Chern number for Hall conductance is fractionalized by generalizing Thurston's train track to folding laminations in three dimensions, which spontaneously bring fractional quantum Hall fluid and anyons into three dimensions and provided an theoretical explanation on the competing fractional quantum Hall states. The many body system of interacting fermions or bosons are mapped into many interacting liquid droplets that interwind into knot lattice, which provide an effective topological representation for superfluid and superconductor, the strange metal state in superconductor is explained by topological dislocation and topological surgery. The Kauffman decomposition rule in knot theory is extended to describe the topological relations between different two dimensional manifolds, which further reveals the non-trivial relationship between the partition functions of pairing fermions and free electrons. The minimal energy state is the most stable topological state, even thought it maybe shares the same topological number with the excited states. The topological representation theory predicts a circular motion of electrons in hyperbolic space, that is implementable and testable by inhomogeneous magnetic field in three dimensions. The folding lamination representation of topological quantum fluid provides an promising implementation of string theory in high energy physics.

A general method is presented that permits enantiomers, originally present in a low enantiomeric excess (%ee1), to accumulate in a high %ee (%ee2) without a chiral source present. An equation was derived that demonstrates the final %ee2 of any unreacted chiral starting material will become higher when that starting material has reacted to generate an additional chiral center, hence forming two sets of diastereomeric enantiomeric products. Importantly, chiral amplification factor (multiples), %ee2/%ee1, depends on the yield of this reaction (p), which exhibits a nonlinear increase in amplification as p approaches 100%. This process is inevitable. It represents a possible route by which small initial %ee1 values of prebiotic molecules could have been raised to high %ee2 contents during the pre-evolution of life period on Earth, without the need for an added chiral source to be present. This predicted behavior was then experimentally verified in Pictet-Spengler reactions of L-tryptophane methyl ester (L-TME) samples present in low enantiomeric excess (%ee1) with biacetyl or oxaldehyde (model reactions) run to various yields. Each reaction generates stereoisomer products, and chiral amplification of the L-TME’s initial %ee1 values to higher %ee2 values occurs in the unreacted (recovered) L-TME. Repeated cycles of these reactions create very high L-TME %ee values. Self-cyclodimerization of low %ee1 L-amino acids with their D-enantiomers to generate L,L-, D,D- and L,D-cyclic products were computationally studied, leading to the same conclusion. Small %ee1 values are enhanced to larger %ee2 in the unreacted starting materials.

We analyze the transformation of a very broad class of metrics that can be put in terms of static coordinates. Starting from a general ansatz, we obtain a relation for the parameters in which one can impose further symmetries, or restrictions. One of the simplest restrictions leads to FLRW cases, while transforming from the initial static to other static-type coordinates can lead to near horizon coordinates, Wheeler-Regge, isotropic coordinates, among others. As less restrictive cases, we show an indirect route for obtaining Kruskal-Szekeres within this approach, as well as Lema\^{\i}tre coordinates. We use Schwarzschild spacetime as prototype for testing the procedure in individual cases. However, application to other spacetimes, such as de-Sitter, Reissner-Nordstr\"{o}m or Schwarzschild de Sitter, can be readily generalized.

We discuss three key points: defining mathematically the notions of phase, symmetry breaking, topological defects and their stability in a formal language of homotopy theory and fiber bundle formalism; postulating and proving two theorems for successive symmetry breaking scenarios in condensed matter physics and cosmology, and demonstrating application of theorems to superfluid Helium 3 and Kibble-Lazarides-Shafi (KLS) walls, supporting and mathematically justifying the results of [Volovik, 2020].

We report on recent experimental results on β decay into self-conjugate (N=Z) nuclei with mass number 58≤A≤70. Super-allowed β decays from the Jπ=0+ ground state of a Z=N+2 parent nucleus are to the isobaric analogue state, through so-called Fermi transitions, and to Jπ=1+ states, by way of Gamow–Teller (GT) transitions. The operator of the latter decay is a generator of Wigner’s SU(4) algebra and as a consequence GT transitions obey selection rules associated with this symmetry. Since SU(4) is progressively broken with increasing A, mainly as a consequence of the spin–orbit interaction, this symmetry is not relevant for the nuclei considered here. We argue, however, that the pseudo-spin–orbit splitting can be small in nuclei with 58≤A≤70, in which case nuclear states exhibit an approximate pseudo-SU(4) symmetry. To test this conjecture, GT decay strength is calculated with use of a schematic Hamiltonian with pseudo-SU(4) symmetry. Some generic features of the GT β decay due to pseudo-SU(4) symmetry are pointed out. The experimentally observed GT strength indicates a restoration of pseudo-SU(4) symmetry for A=70.

A review is here provided on the thermal effects on the optical chirality. To this goal, chiral objects dispersed in an embedding fluid are examined for their magnetoelectric coupling. Archetypal twisted-Omega particles are examined with respect to electron transport, phonon dynamics, and temperature effects. Continuum-mechanical aspect of thermo-elasticity is reviewed along with transverse deformations. A transition temperature delineating a sign flip in the chirality parameter is identified as well.

In this paper, Lie symmetry and Painlevé Techniques are applied to the SIRD (Susceptible, Infected, Recovered and Dead) model. A demonstration of the integrability of the model is provided to present an exact solution. The study revealed that nonlinear system passes Painlevé test and does not possesses complex chaotic behaviour. However, the system fails to pass the Painlevé test while constraints reach values equivalent to the corresponding complex chaotic behaviour. The two-dimensional Lie symmetry algebra and the commutator table of the infinitesimal generators are obtained. Lie symmetry analysis serves to linearize the nonlinear system and find the corresponding invariant solution.

We report positive correlations between letter frequency and symmetry (more symmetric symbols are more frequent) in English and Russian languages and in four types of numerals (Western, Roman, Arabic, and Chinese) with the coefficients of determination of R2=0.899 (English), R2=0.693 (Russian) and R2=0.740 (numerals). For comparison, we studied "symbol-like" 2D colloidal clusters and found negative correlations between frequency and symmetry (R2=0.814 and R2=0.994). While in human-made systems the frequency of using of a symbol is the driving force which leads to symbols simplification and symmetrizing, in natural system, symmetry leads to lower frequency of occurrence because there are fewer ways of building symmetric clusters than asymmetric ones. We interpret these results as a correlation between the symbol as a type and as a token and we attribute this correlation to the inherent iconicity of symbols

It is shown that the Lambda component in the cosmological Lambda-CDM model can be conceived as vacuum energy, consisting of gravitational particles subject to Heisenberg’s energy-time uncertainty. These particles can be modelled as elementary polarisable Dirac-type dipoles (“darks”) in a fluidal space at thermodynamic equilibrium, with spins that are subject to the Bekenstein-Hawking entropy. Around the baryonic kernels, uniformly distributed in the universe, the spins are polarized, thereby invoking an increase of the effective gravitational strength of the kernels. It explains the dark matter effect to the extent that the numerical value of Milgrom’s acceleration constant can be assessed by theory. Non-polarized vacuum particles beyond the baryonic kernels compose the dark energy. The result is a quantum mechanical interpretation of gravity in terms of quantitatively established shares in baryonic matter, dark matter and dark energy, which correspond with the values of the Lambda-CDM model..

We reelaborate on the basic properties of PT symmetry from a geometrical Perspective. The transfer matrix associated with these systems induces a Möbius transformation in the complex plane. The trace of this matrix classifies the actions into three types that represent rotations, translations, and parallel displacements. We find that a PT invariant system can be pictured as a complex conjugation followed by an inversion in a circle. We elucidate the physical meaning of these geometrical operations and link them with measurable properties of the system.

Bijective Analysis of 4D Minkowski space-time model confirms that 4D space-time where time would be 4th dimension of space does not exist. In 4D manifold, time is the duration of a physical event which runs in space. An event does not have the duration in time, duration itself is time. It means that any given physical phenomena can only be symmetric in space; not in time.

A review is here provided about the thermal effects on the optical chirality. To this goal, chiral objects dispersed in an embedding fluid are examined for their magnetoelectric coupling. Thermal effects on several chiral meta-atoms and their ensembles are examined. To this goal, DNA-like helical structures are examined in detail. Mechanical aspect of thermo-elasticity is reviewed along with transverse deformations, while drawing analogies from condensed-matter physics. In this respect, chirality-induced spin selection is reviewed along with the temperature-mediated elec-tron-phonon interactions. A wide range of materials such as polymers and biological cells are also examined for temperature effects. A transition temperature delineating a sign flip in the chirality parameter is identified as well. Chirality-associated functionalities such as ratchet motions, switching, and modulations are investigated for their respective thermal effects. Issues of fabri-cating chiral meta-atoms are also discussed.

Voronoi entropy for the random patterns and patterns demonstrating various elements of symmetry are calculated. The symmetric patterns are characterized by the values of the Voronoi entropy very close to those inherent to random ones. This contradicts the idea that the Voronoi entropy quantifies the ordering of the seed points, constituting the pattern. The extension of the Shannon-like formula embracing symmetric patterns is suggested. Analysis of Voronoi diagrams enables revealing of the elements of symmetry of the pattern.

The grand antiprism A is an outlier among the uniform 4-polytopes, since it is not obtainable from Wythoff’s construction. Its symmetry group G(A) has been incorrectly described as [[10,2+,10]] or even as an `ionic diminished Coxeter group’. In fact, G(A) is another group of order 400, namely the group ±[D10×D10]·2, in the notation of Conway and Smith.

In this paper, we consider particles together with the surrounding space as an indivisible system, and then investigate the maximum symmetry of the system at the quantum level. We introduce a new quantity to describe the evolution of this system before and after measurement. The effective mass of gravity is estimated. We also show that a free quantum particle as the source of energy and momentum do not produce gravity.

A connection between the phenomenon of wafer bifurcation and that of spontaneous symmetry breaking (SSB) has been established. Indeed, by developing an analytical approximation of the elastic energy of a wafer coated with a thin layer (e.g. a metal), it is shown as the elastic potential energy, interpreted as a quantity contributing to the thermodynamic free energy, can be investigated within the framework of the Landau theory of the second order phase transitions. The elastic energy of a bifurcated wafer is a complex function of the stress and the curvatures in the two perpendicular directions. In this work, it is shown as a translation of the coordinate system of the curvatures allows to gain a potential which has a “Mexican hat” shape. This is a distinctive trait of the phenomenon of spontaneous symmetry breaking (SSB). Moreover, it is shown as the values of the coordinates at the minimum of the SSB potential agrees with those provided by the theory. Bifurcation is hence a phenomenon that can also be interpreted as a spontaneous symmetry breaking where the rotational symmetry of a disc shaped wafer is broken. It occurs because of a lowering of the wafer energy in the SSB broken symmetry configuration.

An asymmetric [3+2] cycloaddition of azomethine ylides generated from the corresponding imino esters to trans-β-nitrostyrene catalyzed by chiral aziridine-containing phosphines and phosphine oxides is described. Of the sixteen stereoisomers that could be formed as a result of the title reac-tion, three were formed, two of which were obtained in an enantiomerically enriched or pure form, and one in a racemic form. One of the products underwent epimerization under basic reaction conditions.

Motivated by the intriguing discovery of $X(6900)$ by LHCb Collaboration, we undertake a comprehensive study of the $s\bar{s}s\bar{s}$ tetraquark system in positive parity, employing the Gaussian expansion within the chiral quark model method. We consider two structures, diquark-antidiquark ($ss$-$\bar{s}\bar{s}$) structure and meson-meson ($s\bar{s}$-$s\bar{s}$)strcuture, encompassing all conceivable color and spin configurations. Despite the absence of bound states in our calculations, we have identified potential resonant states with $J^P=0^{+}$, namely $R(0,2150)$ and $R(0,2915)$, as well as a resonant state with $J^P=1^{+}$ denoted as $R(1,2950)$, and a resonant state with $J^P=2^{+}$ denoted as $R(2,2850)$, utilizing the real-scaling method. Through a comparison of their energies and widths, we propose that $R(0,2915)$ and $R(1,2950)$ may share characteristics with $X(6900)$, while $R(0,2150)$ could be a promising experimental candidate for $f_0(2100)$. We strongly advocate for experimental investigations to shed light on the existence and properties of these resonant states.

Informational (Shannon) measures of symmetry are introduced and analyzed for the patterns built of 1D and 2D shapes. The informational measure of symmetry Hsym (G) characterizes the an averaged uncertainty in the presence of symmetry elements from the group G in a given pattern; whereas the Shannon-like measure of symmetry Ωsym (G) quantifies averaged uncertainty of appearance of shapes possessing in total n elements of symmetry belonging to group G in a given pattern. Hsym(G1)=Ωsym(G1)=0 for the patterns built of irregular, non-symmetric shapes. Both of informational measures of symmetry are intensive parameters of the pattern and do not depend on the number of shapes, their size and area of the pattern. They are also insensitive to the long-range order inherent for the pattern. Informational measures of symmetry of fractal patterns are addressed. The mixed patterns including curves and shapes are considered. Time evolution of the Shannon measures of symmetry is treated. The close-packed and dispersed 2D patterns are analyzed.

When discussing the gauge symmetries of any theory, the Henneaux-Teitelboim transformations are often underappreciated or even completely ignored, due to their on-shell triviality. Nevertheless, these gauge transformations play an important role in understanding the structure of the full gauge symmetry group of any theory, especially regarding the subgroup of diffeomorphisms. We give a review of the Henneaux-Teitelboim transformations and the resulting gauge group in the general case, and then discuss its role in the applications to the class of topological theories called nBF models, relevant for the constructions of higher gauge theories and quantum gravity.

Here we investigate how the local properties of particles in a thermal bath may influence the thermodynamics of the bath, and consequently alter the statistical mechanics of subsystems that comprise the bath. We are guided by the theory of small-system thermodynamics, which is based on two primary postulates: that small systems can be treated self-consistently by coupling them to an ensemble of similarly small systems, and that a large ensemble of small systems forms its own thermodynamic bath. We adapt this “nanothermodynamics” to investigate how a large system may subdivide into an ensemble of smaller subsystems, causing internal heterogeneity across multiple size scales. For the semi-classical ideal gas, maximum entropy favors subdividing a large system of “atoms” into an ensemble of “regions” of variable size. The mechanism of region formation could come from quantum exchange symmetry that makes atoms in each region indistinguishable, while decoherence between regions allows atoms in separate regions to be distinguishable by their distinct locations. Combining regions reduces the total entropy, as expected when distinguishable particles become indistinguishable, and as required by a theorem in quantum mechanics for sub-additive entropy. Combining large volumes of small regions gives the usual entropy of mixing for a semi-classical ideal gas, resolving Gibbs paradox without invoking quantum symmetry for atoms that may be meters apart. Other models presented here are based on Ising-like spins, which are solved analytically in one dimension. Focusing on the bonds between the Ising-like spins we find similarity in the equilibrium properties of a two-state model in the nanocanonical ensemble and a three-state model in the canonical ensemble. Thus, emergent phenomena may alter the thermal behavior of microscopic models, and the correct ensemble is necessary for fully-accurate predictions. Another result using Ising-like spins involves simulations that include a nonlinear correction to Boltzmann’s factor, which mimics the statistics of indistinguishable states by imitating the dynamics of spin exchange on intermediate lengths. These simulations exhibit 1/f-like noise at low frequencies (f), and white noise at higher f, similar to equilibrium thermal fluctuations found in many materials.

Cortical actomyosin flows, among other mechanisms, scale up spontaneous symmetry breaking and thus play pivotal roles in cell differentiation, division, and motility. According to many model systems, myosin motor-induced local contractions of initially isotropic actomyosin cortices are nucleation points for generating cortical flows. However, the positive feedback mechanisms by which spontaneous contractions can be amplified towards large-scale directed flows remain mostly speculative. To investigate such a process on spherical surfaces, we reconstituted and confined initially isotropic minimal actomyosin cortices to the interfaces of emulsion droplets. The presence of ATP leads to myosin-induced local contractions that self-organize and amplify into directed, large-scale actomyosin flows. By combining our experiments with theory, we found that the feedback mechanism leading to a coordinated, directional motion of actomyosin clusters can be described as asymmetric cluster vibrations, caused by intrinsic non-isotropic ATP consumption, in conjunction with spatial confinement. By tracking individual actomyosin clusters, we identified fingerprints of vibrational states as the basis of directed motions. These vibrations may represent a generic key driver of directed actomyosin flows under spatial confinement in vitro and in living systems.

Abstract Using the Pauli algebra version of the Dirac equation six symmetric but non-equivalent copies of the QED Lagrangian are constructed. The group of symmetries is the group of outer automorphisms of the finite quaternion group, isomorphic to the group of permutations on three letters. Solutions to corresponding Dirac equations form six sectors of fermion fields. The intersections between different sectors can only contain massless fields, implying that the sectors are distinct. If the symmetry could be extended to the full Standard Model Lagrangian it would predict the ratio of dark matter to ordinary matter of 5 : 1 which is close to the ratio of 5.2 : 1 according to current observations.

In this article, we give a brief review of the origin of the neutrino mass in some interesting non-linear supersymmetric models with R-symmetry. These models are able to address and solve the most important problems of the particle physics and provide mechanisms for neutrino mass generation and their mixing parameters in agreement with the current experimental data. Their prediction could be experimentally tested in the near future by collider experiments.

The nucleon-pair shell model (NPSM) with broken symmetry basis is studied. The results demonstrate the validity of NPSM with symmetry broken basis, which leads to a significant reduction in the dimensionality of the multi-pair configuration space. We have observed that the axial-deformed basis provides a satisfactory description of the low-lying spectrum, even without the need for the projection procedure. In the case of the triaxial-deformed basis, we have investigated the variation after angular momentum projection, indicating the potential for NPSM to find exact solutions in the half-closed system.

Quantum effects arising from manifestly broken time-reversal symmetry are investigated using time-dependent perturbation theory in a simple model. The forward time and the backward time Hamiltonians are taken to be different and hence the forward and backward amplitudes become unsymmetrical and are not complex conjugates of each other. The effects vanish when the symmetry breaking term is absent and ordinary quantum mechanical results such as Fermi Golden rule are recovered.

In this paper we consider Changhee polynomials of type two, which are motivated from the recent work of Kim and Kim. We investigate some symmetry identities for the Changhee polynomials of type two which are derived from the properties of symmtery for the fermionic p-adic integral on Zp.

The circular photogalvanic effect (CPGE) is the photocurrent generated in an optically active material in response to an applied ac electric field, and it changes sign depending on the chirality of the incident circularly polarized light. It is a non-linear dc current as it is second-order in the applied electric field, and for a certain range of low frequencies, takes on a quantized value proportional to the topological charge for a system which is a source of nonzero Berry flux. We show that for a non-interacting double-Weyl node, the CPGE is proportional to two quanta of Berry flux. On examining the effect of short-ranged Hubbard interactions upto first-order corrections, we find that this quantization is destroyed. This implies that unlike the quantum Hall effect in gapped phases or the chiral anomaly in field theories, the quantization of the CPGE in topological semimetals is not protected.

We elucidate the versatility of intriguing symmetry and asymmetry phenomena outside and inside black holes. For description of the test particle (planet and photon) motion it is used the Kerr-Newman metric of the rotating and electrically charged black hole. In particular, it is demonstrated the symmetry and asymmetry of the one-way Einstein-Rosen bridge inside black hole toward and through the plethora of endless and infinite universes.

We establish a new Gagliardo-Nirenberg inequality characterized by radial symmetry and involving potentials exhibiting pure power polynomial behaviour. As an application of our result, we investigate the existence of extremals for this inequality, which also correspond to stationary solutions for the nonlinear Schrödinger equation with inhomogeneous nonlinearity, competing with Hs-subcritical nonlinearities, either of local or non-local nature.

In physics, symmetry is mostly about the symmetry between elementary particles, between matter and antimatter. In the universe, we have two fundamental types of energy: energy of matter and energy of space. These two energies have a symmetry that is not geometry type, not electric charge type, it is energy type. Every physical object with mass m symmetrically diminishes the energy density of space in its center exactly for the amount of its mass. This symmetry of matter and space energies is the physical source of gravity that is also valid in the centre of black holes.

In this paper, we investigate the behaviour of a relativistic quantum oscillator under the effects of Lorentz symmetry violation determined by a tensor (KF)µναβ out of the Standard Model Extension. We analyze the quantum system under a Coulomb-type radial electric field and a uniform magnetic induced by Lorentz symmetry breaking effects under a Cornell-type potential, and obtain the bound states solution by solving the Klein-Gordon oscillator. We see a quantum effect due to the dependence of the angular frequency of the oscillator on the quantum numbers of the system, and the energy eigenvalues and the wave-function of the oscillator field get modified by the Lorentz symmetry breaking parameters as well as due to the presence of Cornell-type potential.

Magneto-electrochemistry (MEC) is a unique paradigm in science, where electrochemical experiments are carried out as a function of an applied magnetic field, creating a new horizon of potential scientific and technological applications. Over the time, detailed understanding of this research domain was developed to identify and rationalize the possible effects exerted by a magnetic field on the various microscopic processes occurring in an electrochemical system, such as: electrolyte properties governed by charge-transfer process (electric conductivity, viscosity, and diffusivity), mass transfer, electrochemical kinetics and on the structure/quality of products formed either at the working electrode or in the electrochemical cell. Particularly, magnetic field controlled chiral architecture obtained from deposited metal, alloys and catalyst and their excellent enantio-recognition in experimental frame is highly appealing. Interestingly, Hall effect was also demonstrated in electrolytic medium via an impressive experimental technique which is being employed for further theoretical understanding in the field of magneto-electrochemical science. Later, a highly reproducible local temperature variation was observed in electrochemical electrolytes exposed to perpendicular magnetic and electric fields. However, until recent studies, none of the above mentioned reports considered the possibility of a spin-dependent related charge-transfer process. Recent experimental and theoretical studies reveal that electron’s transmission through chiral molecules is spin-selective and this effect has been referred to as chiral-induced spin-selectivity (CISS) effect. The CISS effect pave the way for the building up of a system characterized by a net magnetic moment exploiting the spin-filtering ability of chiral molecules. This interplay between chirality and magnetism may shed light on fundamental scientific aspects underlying the enantio-recognition and highly efficient electron-transfer that occurs in biological process.

The current report contributes to the understanding of the stereoselectivity of the chiral dirhodium(II) carboxylate catalysts derived from N-protected amino acid ligands. Investigating the possible effect of ligand stereo-purity on catalyst structure and enantioselectivity was carried out. This was justified through a new X-ray crystal structure for Rh2(S,S,S,R-PTTL)4 diastereomer.

We demonstrate that the black hole evaporation can be modelled as a process where one symmetry of the system is spontaneously broken continuously. We then identify three free-parameters of the system. The sign of one of the free-parameters, governs whether the particles emitted by the black-hole are fermions or bosons. The present model explains why the Black Hole evaporation process is so universal. Interestingly, this universality emerges naturally inside certain modifications of gravity.

To understand the mystery of final unification, in our earlier publications, we proposed that, 1) There exist three atomic gravitational constants associated with electroweak, strong and electromagnetic interactions; and 2) There exists a strong interaction elementary charge (e_s) in such a way that, it's squared ratio with normal elementary charge is close to inverse of the strong coupling constant. In this context, starting from lepton rest masses to stellar masses, we have developed many interesting and workable relations. We noticed that, electroweak field seems to be operated by a primordial massive fermion of rest energy 585 GeV. It can be considered as the zygote of all elementary particles and galactic dark matter. Proceeding further, with a characteristic fermion-boson mass ratio of 2.27, quarks can be classified into quark fermions and quark bosons. Considering strong charge conservation and electromagnetic charge conservation, fractional charge quark fermions and quark bosons can be understood. Quark fermions that generate observable massive baryons can be called as Fluons. Quark bosons that generate observable mesons can be called as Bluons. By considering a new hadronic fermion of rest energy 103.4 GeV, rest masses of fluons and bluons can be estimated and there by baryon masses and meson masses can be estimated.

Recent observations of the dark energy density demonstrates the fine-tuning problem and challenges in theoretical modelling. In this study, we apply the self-similar symmetry (SSS) model, describing the hierarchical structure of the universe based on the Dirac large numbers hypothesis, to Einstein's cosmological term. We introduce a new similarity dimension, D_B, in the SSS model. Using the D_B SSS model, the cosmological constant, vacuum energy density, and Hubble parameter can be simply expressed as a function of the cosmic microwave background (CMB) temperature. We show that the initial value of the vacuum energy density at the creation of the universe is ρ₀ = 1/8πα_f⁶, where α_f is the fine structure constant. The results indicate that the CMB is the primary factor for the evolution of the universe, providing a unified understanding of the problems of naturalness.

Lysine plays a crucial role in promoting development, enhancing immune function, and improving the function of central nervous system tissues. The two configurational isomers of amino acids have significantly different effects. Currently, methods for chiral recognition of lysine all have drawbacks such as expensive equipment and complicated detection processes, whereas fluorescence analysis boasts high sensitivity, strong selectivity, and simple operation. In this study, we synthesized four novel Binaphthyl-Amine (BINAM)-based fluorescent probes capable of specifically identifying the L-configuration of lysine among the twenty amino acids constituting human proteins. The enantiomeric fluorescence enhancement ratio (ef or ΔIL/ΔID) reached up to 15.29, demonstrating high enantioselectivity. We also evaluated the probe's recognition ability under different pH, reaction times, and metal ion conditions, as well as the Limit of Detection of the probe. Our findings indicate that this probe is a highly stable tool for detecting chiral lysine.

Results are presented concerning the influence on the water splitting process of enantiopure tartaric acid present in bulk solution. Stainless steel and electrodeposited nickel are used as working electrode (WE) surface. The latter is obtained by electrodeposition on the two poles of a magnet. The influence and role played by the chiral compound in solution has been assessed by comparing the current values, in cyclic voltammetry (CV) experiments, recorded in the potential range at which oxygen evolution reaction (OER) occurs. In the case of tartaric acid and nickel WE a spin polarization of about 4 % is found. The use of the chiral environment (bulk solution) and ferromagnetic chiral Ni electrode allows for observing the OER at a more favourable potential: about 50 mV (i.e. a cathodic, less positive, shift of the potential at which the oxygen evolution is observed).

The realization of a BRST cohomology of the 4D topological gauge-affine gravity is established in terms of a superconnection formalism. Identification of fields in the quantized theory is occurred directly as usual in terms of superconnection and its supercurvature components with the double covering of the general affine group GA¯(4,R). Then, by means of an appropriate decomposition of the metalinear double-covering group SL¯(5,R) with respect to the general linear double-covering group GL¯(4,R) one can easily obtain the enlargements of the fields while remaining consistent with the BRST algebra. This leads to the descent equations allowing us to build the observables of the theory by means of the BRST algebra constructed using a sa¯(5,R) algebra-valued superconnection. In particular, we discuss the construction of topological invariants with torsion.

In this paper, a relativistic scalar particle under Lorentz symmetry breaking effects in the presence of a scalar potential is investigated. We introduce the scalar potential by modifying the mass via transformation M → M+S(r) in the wave equation and analyze the behaviour of a scalar particle. We see that the analytical solution to the KleinGordon equation can be achieved, and the energy eigenvalues and the wave function depends on the Lorentz symmetry breaking parameters as well as potential

A black hole in a Schwarzschild spacetime is considered. A transformation is proposed that describes the relationship between the coordinate systems exterior and interior to an event horizon. Application of this transformation permits considerations of the (a)symmetry of a range of phenomena taking place on both sides of the event horizon. The paper investigates two distinct problems of a uniformly accelerated particle. In one of these, although the equations of motion are the same in the regions on both sides, the solutions turn out to be very different. This manifests the differences of the properties of these two ranges.

The synthesis of a new class of epoxide derivatives from prochiral vinylidene bisphosphonate (VBP) precursors is reported using hydrogen peroxide as the terminal oxidant. The reaction is carried out using a series of possible organic catalysts having a basic character, the best results being observed using quinine and sparteine. These catalysts not only provide from good to excellent epoxide yields with a large variety of VBPs but also interesting enantioselectivities in the 67-96% ee range at least in the case of the Ph and m-MeO-Ph VBP derivatives opening the way to a number of chiral anti-osteoporosis potentially active pharmaceutical ingredients.

The search for fundamental determinants of bio-molecular chirality is a hot topic in biology, clarifying the meaning of evolution and the enigma of life's origin. Non-biological and biological entities obey the universal laws of nature grounded on space-time symmetry (STS) transformations. The fabric of STS is the immense subject of our review. The symmetry, encompassing behavior of elementary particles and galaxy structure, imposes its fundamental laws on all hierarchical levels of the biological world. Objects across spatial scales may be classified as chiral or achiral concerning a specific space-related symmetry transformation - mirror reflection. Chirality as a symmetry element reflecting the property of asymmetry (or dissymmetry) is the critical structural feature of natural systems, including sub-atomic particles and living matter. The inheritance of molecular symmetry from the symmetry of elementary particles points to the bi-directional causal pathway of prevalent bio-chirality. We assume that both extremities of spatial dimension naturally influence this transitional state of biological matter. The current review promotes an integrative holistic approach to the new experimental results in fast-developing and divergent branches of STS and biological chirality. The generalized view on biological chirality is a necessary and promising avenue for adequately understanding the link between protein folding, cell morphology, neuronal signaling, and laterality of cognitive functions.

Within the spectrum of a natural image, the amplitude of modulation decreases with spatial frequency. The speed of such an amplitude decrease, or the amplitude spectrum slope, of an image affects the perceived aesthetic value. Additionally, a human observer would consider a symmetric image more appealing than they do an asymmetric one. We investigated how these two factors jointly affect aesthetic preferences by manipulating both the amplitude spectrum slope and the symmetric level of images to assess their effects on aesthetic preference on a 6-point Likert scale. Our results showed that the preference ratings increased with the symmetry level but had an inverted U-shape relation to amplitude spectrum slope. In addition, a strong interaction existed between symmetry level and amplitude spectrum slope on preference rating, in that symmetry can amplify the amplitude spectrum slope’s effects. Such effects can be described by a quadratic function of the spectrum slope. That is, preference is an inverted U-shape function of spectrum slope whose intercept is determined by the number of symmetry axis. In addition, the interaction between the two factors is manifested as the modulation depth of the quadratic function.

Quantum coherence is an important physical resource in quantum information science, and also as one of the most fundamental and striking features in quantum physics. In this paper, we obtain a symmetry-like relation of relative entropy measure $C_r(\rho)$ of coherence for $n$-partite quantum states $\rho$, which gives lower and upper bounds for $C_r(\rho)$. Meanwhile, we discuss the conjecture about the validity of the inequality $C_r(\rho)\leq C_{\ell_1}(\rho)$ for any state $\rho$. We observe that every mixture $\eta$ of a state $\rho$ satisfying $C_r(\rho)\leq C_{\ell_1}(\rho)$ and any incoherent state $\sigma$ also satisfies the conjecture. We also note that if the von Neumann entropy is defined by the natural logarithm $\ln$ instead of $\log_2$, then the reduced relative entropy measure of coherence $\bar{C}_r(\rho)=-\rho_{\rm{diag}}\ln \rho_{\rm{diag}}+\rho\ln \rho$ satisfies the inequality ${\bar{C}}_r(\rho)\leq C_{\ell_1}(\rho)$ for any mixed state $\rho$.

Using a nonclassical symmetry of nonlinear reaction-diffusion equations, some exact multi-dimensional time-dependent solutions are constructed for a fourth-order Allen-Cahn-Hilliard equation. This models a phase field that gives a phenomenological description of a two-phase system near the critical temperature. Solutions are given for the changing phase of a cylindrical or spherical inclusion, allowing for a 'mushy zone' with mixed state that is controlled by imposing a pure state at the boundary. The diffusion coefficients for transport of one phase through the mixture, depend on the phase field value, since the physical structure of the mixture depends on the relative proportions of the two phases. A source term promotes stability of both of the pure phases but this tendency may be controlled or even reversed through the boundary conditions.

Introduction: The purpose of this study was to determine the normative values of muscles’ contractile properties, lateral symmetry, and the impact of aging on muscle contractility in gymnastics. Materials and Methods: A survey of 81 athletes from different disciplines was undertaken: MAG (n = 26), WAG (n = 28), and RG (n = 27). The athletes’ average age was 15.41 ± 5.03 years. We try to establish the normative values for contraction time (muscle pairs) in gymnastics for Slovenian athletes for ten skeletal muscles. Results: Athletes’ age affects contractility in a negative way. Differences between the duration of contractility and age were found in all disciplines: m. BF (r = 0.48, p < 0.001), m. TB (r = 0.37, p < 0.013), m. ES (r = 0.17, p < 0.025), m. VL (r = 0.36, p < 0.00) and m. VM (r = 0.40, p < 0.000) at a statistical significance of p < 0.05. Conclusions: A comparison between the left and right sides shows little asymmetry in WAG and that the occurrence of injuries is in the middle range. In RG we see a bigger deviation, which could trigger the emergence of pain or injury in m. BF (13%) and m. GL (14.5%), but in MAG the deviation is greater in m. BB (12%), m. BF (11%) and m. GM (13%).

Various optically active 2-hydroxyamide derivatives are produced based on the kinetic resolution of racemic 2-hydroxyamides with a diphenylacetyl component and (R)-benzotetramisole [(R)-BTM], a chiral acyl-transfer catalyst, via asymmetric esterification and acylation. It was revealed that a tertiary amide can be used with this novel protocol to achieve high selectivity [22 examples; s-value reaching over 250]. The resulting chiral compounds could be transformed into other useful structures while maintaining their chirality.

Particles and photons appear to be total opposites; the former has rest mass which requires space to exist; the latter has kinetic energy which requires time to occur (oscillate). But they do share certain properties (e.g., quantization) that remain invariant when one is transformed (swapped) for the other. This gauge invariance is developed in some detail.The symmetry between particle and photon turns out to be one of inversion. It is the equalities of special relativity that support this inversion and the accompanying invariances: mass transformed to energy; space transformed to time. The great advantage of these symmetries (inversions) is that they provide guidance for an object little understood (the photon) based upon an object well understood (the particle). On this basis, progress can be made in the understanding of some long-standing issues: wave-particle duality, time’s arrow, the constant speed of light and nonlocality.

Analytical modeling has symmetries and aesthetic-mathematical characteristics that distinguish it from numerical computation. Nanoscience plays an extremely important role in unification efforts that also include holistic aspects of reality. In this paper I present new discovered results about the complete analytical quantum relativistic form of the mean square deviation of position false related to a lately appeared Drude-Lorentz-like model (DS model), already performed at classical, quantum and relativistic level. The function false gives precise information about the distance crossed by carriers (electrons, ions, etc.) inside a nanostructure, considering both quantum effects and relativistic velocities. The model has a wide scale range of applicability; the nanoscale is considered in this paper, but it holds from sub-pico-level to macro-level because of the existence of a gauge factor, making it applicable to every oscillating processes in Nature. Examples of application and suggestions supplement the paper, as well as interesting developments to be studied related to the model and to one of the basic elements of a current unified holistic approach based on the vacuum energy.

Optinalysis, as a method of symmetry detection, is a new algorithm that intrametrically (within elements or variables) or intermetrically (between elements or variables) computes and compares two or more univariate or multi-clustered or multivariate sequences as a mirror-like reflection of each other (optics-like manner), hence the name is driven. Optinalysis is based by the principles of reflection and moment about a symmetrical line which detects symmetry that reflects a similarity measurement. This proposed methodology was validated in comparison with Pearson method of skewness detection, and also with some algorithms for pairewise alignment and comparison of genomic sequences (Needle, Stretcher, Water, Matcher) on EMBL-EBI website. A results comparison shows that optinalysis is more advance, more sensitive, more inferential and simple alternative approach of skewness detection and pairewise sequence comparison.

The notion of the informational measure of symmetry is introduced according to: HsymG=-i=1kPGilnPGi, where PGi is the probability of appearance of the symmetry operation Gi within the given 2D pattern. HsymG is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as HsymD3=1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. Informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. Informational measure of symmetry does not correlate neither with the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns.

Physical roots, exemplifications and consequences of periodic and aperiodic ordering (represented by Fibonacci series) in biological systems are discussed. The role and physical and biological roots of symmetry and asymmetry appearing in biological patterns is addressed. Generalization of the Curie-Neumann Principle as applied to biological objects is presented, briefly summarized as: “asymmetry is what creates a biological phenomenon”. The “up-bottom approach” and “bottom up” approaches to the explanation of symmetry in organisms are presented in detail. The “up-bottom approach”, implies that the symmetry of the biological structure follows the symmetry of media in which this structure is functioning; the “bottom-up” approach, in turn, adopt that the symmetry of biological structures emerges from the symmetry of molecules constituting the structure. A diversity of mathematical measures applicable for quantification of ordering in biological patterns is introduced. The continuous, Shannon and Voronoi measures of symmetry/ordering and their application to biology objects are addressed. The fine structure of the notion of “ordering” is discussed. Informational/algorithmic roots of ordering inherent for the biological systems are considered. Ordered/symmetrical patterns provide economy of biological information, necessary for algorithmic description of a biological entity. Application of the Landauer principle bridging physics and theory of information to the biological systems is discussed.

Symmetry-breaking is a phenomenon that is observed in various contexts, from the early universe to complex organisms, and is considered a key puzzle in understanding the emergence of life. The presence of enantiomeric amino acids and proteins highlights its critical role. However, the origin of symmetry-breaking has yet to be comprehensively explained, particularly from an energetic standpoint. This article explores a novel approach by considering energy dissipation, specifically lost free energy, as a crucial factor in elucidating symmetry-breaking. Through a thorough thermodynamic analysis applicable to enantiomeric systems at various scales, from elementary particles to aggregated matter like crystals, we present experimental findings that establish a direct correlation between non-equilibrium free energy and energy dissipation during formation processes. Results emphasize the pivotal role of energy dissipation not only as an outcome but as the trigger for symmetry-breaking. This insight suggests that understanding the origins of complex systems, from cells to living beings and the universe itself, requires a lens focused on non-equilibrium processes, ultimately facilitating the formation of non-racemic systems.

Symmetry in biological and physical systems is a product of self-organization driven by evolutionary processes, or mechanical systems under constraints. Symmetry-based feature extraction or representation by neural networks may unravel the most informative contents in large image databases. Despite significant achievements of artificial intelligence in recognition and classification of regular patterns, the problem of uncertainty remains a major challenge in ambiguous data. In this study, we present an artificial neural network that detects symmetry uncertainty states in human observers. To this end, we exploit a neural network metric in the output of a biologically inspired Self-Organizing Map, the Quantization Error (SOM-QE). Shape pairs with perfect geometric mirror symmetry but a non-homogenous appearance, caused by local variations in hue, saturation, or lightness within and/or across the shapes in a given pair produce, as shown here, longer choice RT for ‘yes’ responses relative to symmetry. These data are consistently mirrored by the variations in the SOM-QE from unsupervised neural network analysis of the same stimulus images. The neural network metric is thus capable of detecting and scaling human symmetry uncertainty in response to patterns. Such capacity is tightly linked to the metric’s proven selectivity to local contrast and color variations in large and highly complex image data.

This study investigates via Lie symmetry analysis the Hunter-Saxton equation, an equation relevant to the theoretical analysis of nematic liquid crystals. We employ the multiplier method to obtain conservation laws of the equation derived from first-order multipliers. Conservation laws of the Hunter-Saxton equation, in combination with the Lie point symmetries of the equation, enable us to perform symmetry reductions through employment of the double reduction method. The method exploits the relationship between Lie point symmetries and conservation laws to effectively reduce both the number of variables and the order of the equation. Five nontrivial conservation laws of the Hunter-Saxton equation are derived, four of which are found to have associated Lie point symmetries. Applying the double reduction method to the Hunter-Saxton equation results in a set of first-order ordinary differential equations, the solutions of which represent invariant solutions for the equation.

We study the Gaudin models with gl(1|1) symmetry that are twisted by a diagonal matrix and defined on tensor products of polynomial evaluation gl(1|1)[t]-modules, extending all the results of to the twisted case. Namely, we give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation gl(1|1)[t]-modules and show that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also give dimensions of the generalized eigenspaces.

A new method of line shape calculations of hydrogen-like atoms in magnetized plasmas is presented. This algorithm makes it possible to solve two fundamental problems in the broadening theory: the analytical description of the radiation transition array between excited atomic states and account of a thermal ion motion effect on the line shapes formation. The solution to the first problem is based on the semiclassical approach to dipole matrix elements calculations and the usage of the specific symmetry properises of the Coulomb field. The second one is considered in terms of the kinetic treatment of the frequency fluctuation model (FFM). As the result one has a universal description of line shapes under the action of the dynamic of ion’s microfield. The final line shape is obtained by the convolution of the ionic line shape with the Voigt electron-Doppler profile. The method is applicable formally for large values of principle quantum numbers. However, it is demonstrated the efficiency of the results even for well known first members of the hydrogen Balmer series Dalpha and Dbeta. The comparison of obtained results with accurate quantum calculations is presented. The new method may be of interest for investigations of spectral line shapes of hydrogen-like ions presented in different kinds of hot ionized environments with the presence of a magnetic field, including SoL and divertor tokamak plasmas.

Abstract: The second part of the paper develops the approach, suggested in the Entropy 2020, 22(1), 11; https://doi.org/10.3390/e22010011 , which relates ordering in physical systems to their symmetrizing. Entropy is frequently interpreted as a quantitative measure of “chaos” or “disorder”. However, the notions of “chaos” and “disorder” are vague and subjective to a much extent. This leads to numerous misinterpretations of entropy. We propose to see the disorder as an absence of symmetry and to identify “ordering” with symmetrizing of a physical system; in other words, introducing the elements of symmetry into an initially disordered physical system. We explore the initially disordered system of elementary magnets exerted to the external magnetic field H ⃗. Imposing symmetry restrictions diminishes the entropy of the system and decreases its temperature. The general case of the system of elementary magnets demonstrating the j-fold symmetry is treated. The interrelation T_j=T/j takes place, where T and T_j are the temperatures of non-symmetrized and j-fold-symmetrized systems of the magnets correspondingly.

A comprehensive theory is developed for the chiral optical response of two configurations of the N-oscillator Born-Kuhn model (NOBK): the helically stacked and the corner stacked models. In the helical NOBK model, there is always a chiral response regardless of the value of N, whereas in the corner NOBK, only configurations with even N demonstrate a chiral response. Generally, the magnitudes of optical rotatory dispersion (ORD) and circular dichroism (CD) increase with N when the parameters of each oscillator are fixed. In cases of weak coupling, the spectral shapes of ORD and CD remain invariant, while strong coupling significantly alters the spectral shapes. For large damping, the spectral amplitude becomes smaller, and the spectral features become broader. In the presence of small damping, strong coupling introduces degeneracy in the coupled oscillator system, leading to multiple spectral features in both ORD and CD across the entire spectral region. This simple model not only can help in the design of tunable chiral metamaterials but also enhance our understanding of chiro-optical responses in structures with different configurations.

We show that the compound samarium hexaboride is a strong topological insulator using the eigenvalues of the space inversion operator in the low-energy limit of the periodic Anderson model. Additionally, we assume the presence of the ferromagnetic exchange interaction (M). A Dirac cone like feature in the surface state energy spectra is observed for M equals zero in a certain parameter range. For M not equal to zero, there is no Kramers degeneracy. We have been able to show that this phase corresponds to the quantum anomalous Hall state by calculating Berry curvature and the Chern number. Using Floquet theory, we further show that the access to a novel state with broken time reversal symmetry is possible due to the normal incidence of circularly polarized optical field on the surface of the compound despite M being absent.

In light of Liu \emph{at el.}'s original works, this paper revisits the solution of Burgers's nonlinear equation $u_t=a(u_x)^2+bu_{xx} $. The study found two exact and explicit solutions for groups $G_4$ and $G_6$, as well as a general solution. A numerical simulation is carried out. In the appendix a Maple code is provided

In this work, we investigate the behaviour of relativistic quantum oscillator under the effects of Lorentz symmetry violation determined by a tensor $(K_F)_{\mu\nu\alpha\beta}$ out of the Standard Model Extension. We analyze this relativistic system under an inverse radial electric field and a constant magnetic field induced by Lorentz symmetry violation. We see that the presence of Lorentz symmetry breaking terms modified the energy spectrum of the system, and a quantum effect arise due to the dependence of the linear charge density on the quantum numbers of the system

The construction of molecular robotic-like objects that imitate living things is an important challenge for current chemists. Such molecular devices are expected to perform their duties robustly to carry out mechanical motion, process information, and make independent decisions. Dissipative self-organization plays an essential role in meeting these purposes. To produce a micro-robot that can perform the above tasks autonomously as a single entity, a function generator is required. Although many elegant review articles featuring chemical devices that mimic biological mechanical functions have been published recently, the dissipative structure, which is the minimum requirement, has not been sufficiently discussed. This article aims to show clearly that dissipative self-organization is a phenomenon involving autonomy, robustness, mechanical functions, and energy transformation. Moreover, the author details the recent experimental results of an autonomous light-driven molecular device that achieves all of these features. In addition, a chemical model of cell-amplification is also discussed to focus on the generation of hierarchical movement by dissipative self-organization. By reviewing this research, it may be perceived that mainstream approaches to synthetic chemistry have not always been appropriate. In summary, the author proposes that the integration of catalytic functions is a key issue for the creation of autonomous microarchitecture.