Symmetry Research Articles

An Efficient Solution of Multiplicative Differential Equations through Laguerre Polynomials

The field of multiplicative analysis has recently garnered significant attention, particularly in the context of solving multiplicative differential equations (MDEs). The symmetry concept in MDEs facilitates the determination of invariant solutions and the reduction of these equations by leveraging their intrinsic symmetrical properties. This study is motivated by the need for efficient methods to address MDEs, which are critical in various applications. Our novel contribution involves leveraging the fundamental properties of orthogonal polynomials, specifically Laguerre polynomials, to derive new solutions for MDEs. We introduce the definitions of Laguerre multiplicative differential equations and multiplicative Laguerre polynomials. By applying the power series method, we construct these multiplicative Laguerre polynomials and rigorously prove their basic properties. The effectiveness of our proposed solution is validated through illustrative examples, demonstrating its practical applicability and potential for advancing the field of multiplicative analysis.

Strain-induced long-range charge-density wave order in the optimally doped Bi2Sr2−xLaxCuO6 superconductor

The mechanism of high-temperature superconductivity in copper oxides (cuprate) remains elusive, with the pseudogap phase considered a potential factor. Recent attention has focused on a long-range symmetry-broken charge-density wave (CDW) order in the underdoped regime, induced by strong magnetic fields. Here by 63,65Cu-nuclear magnetic resonance, we report the discovery of a long-range CDW order in the optimally doped Bi2Sr2−xLaxCuO6 superconductor, induced by in-plane strain exceeding ∣ε∣ = 0.15 %, which deliberately breaks the crystal symmetry of the CuO2 plane. We find that compressive/tensile strains reduce superconductivity but enhance CDW, leaving superconductivity to coexist with CDW. The findings show that a long-range CDW order is an underlying hidden order in the pseudogap state, not limited to the underdoped regime, becoming apparent under strain. Our result sheds light on the intertwining of various orders in the cuprates.

Systematic study of flow vector fluctuations in sNN=5.02 TeV Pb-Pb collisions

Measurements of the pT-dependent flow vector fluctuations in Pb–Pb collisions at sNN=5.02TeV using azimuthal correlations with the ALICE experiment at the Large Hadron Collider are presented. A four-particle correlation approach [ALICE Collaboration, ] is used to quantify the effects of flow angle and magnitude fluctuations separately. This paper extends previous studies to additional centrality intervals and provides measurements of the pT-dependent flow vector fluctuations at sNN=5.02TeV with two-particle correlations. Significant pT-dependent fluctuations of the V⃗2 flow vector in Pb–Pb collisions are found across different centrality ranges, with the largest fluctuations of up to ∼15% being present in the 5% most central collisions. In parallel, no evidence of significant pT-dependent fluctuations of V⃗3 or V⃗4 is found. Additionally, evidence of flow angle and magnitude fluctuations is observed with more than 5σ significance in central collisions. These observations in Pb–Pb collisions indicate where the classical picture of hydrodynamic modeling with a common symmetry plane breaks down. This has implications for hard probes at high pT, which might be biased by pT-dependent flow angle fluctuations of at least 23% in central collisions. Given the presented results, existing theoretical models should be reexamined to improve our understanding of initial conditions, quark–gluon plasma properties, and the dynamic evolution of the created system. ©2024 CERN, for the ALICE Collaboration 2024 CERN

On Foci and Asymptotes of Conics in the Isotropic Plane

The paper shows that every conic with foci in the isotropic plane can be represented by the equation of the form $y^2=\epsilon x^2+x$, where $\epsilon \in \{-1, 0, 1\}$ for an ellipse, a parabola and a hyperbola with foci respectively. Using this equation some important properties of the foci are proved. According to duality the properties of asymptotes of the hyperbola in the isotropic plane are valid as well. 2000 Mathematics Subject Classification. 51N25

Curvature of the Focal Conic in the Isotropic Plane

It is shown in \cite{beba} that every focal conic $\mathcal{C}$ in the isotropic plane can be represented by the equation $y^2=\epsilon x^2+x$, $\epsilon \in \{-1,0,1\}$ and a parametrization. This paper gives the equation of the circle of curvature at the point $T$ of the focal conic $\mathcal{C}$. The radius of curvature $\rho$ at the point $T$ of the focal conic $\mathcal{C}$ is given as well as its relation to the span $\delta $ from the center of $\mathcal{C}$ to the tangent $\lijepot$ at the point $T$ and to the length of the half diameter of $\mathcal{C}$ on the diameter parallel to the $\textup{tangent}$. 2000 Mathematics Subject Classification. 51N25

Dual Feuerbach Theorem in an Isotropic Plane

The dual Feuerbach theorem for an allowable triangle in an isotropic plane is proved analytically by means of the so-called standard triangle. A number of statements about relationships between some concepts of the triangle and their dual concepts are also proved. 2000 Mathematics Subject Classification. 51N25

Investigating Fryette's mechanics in computed tomography scans: an analysis of vertebrae spinal physiology using open-sourced datasets and three-dimensional vertebral orientation.

Fryette's mechanics is taught as a simplistic model of coupled vertebral movement, fundamental in osteopathic practice. This study seeks to better understand the likelihood of Fryette's model by calculating vertebral orientation in computed tomography (CT) scans. Given previous findings oflow angular coupled movements during overall spinal motion, static calculations provide a unique perspective on the likelihood of Fryette's mechanics. This analysis aims to evaluate the efficacy of Fryette's principles in predicting vertebral positioning in CT scans by comparing their 3-dimensional (3D) orientation to movements described by Fryette. 3D models of 953 thoracic and lumbar vertebrae were obtained from 82 CT scans within the VerSe`20 open-source dataset. A stepwise algorithm generated three unique symmetry planes for each vertebra, offering 3D angular orientation with respect to the vertebral level below. A total of 422 vertebrae were omitted from the analysis due to the presence of pathologies significant enough to affect their motion, inaccurate symmetry plane calculations, or absence of vertebral level below. The remaining 531 vertebra were analyzed to compare quantitative coupled positioning against expected coupled spinal movements in line with Fryette's mechanics. One-sample proportional z-scoring was implemented for all vertebral levels with an ∝=0.05 and a null hypothesis of Fryette's primed positioning occurring by chance of 50 %. Further analysis was performed with individual z-scoring for each individual level to see which levels were statistically significant. Data from the VerSe`20 dataset revealed that 56.9 % of successfully analyzed vertebrae demonstrated positions compatible with Fryette's mechanics (p=0.0014, power=89 %). The 302 vertebral levels that did display coupled positioning consisted of Type I (166 vertebrae) and Type II (136 vertebrae) compatible with Fryette's mechanics. Levels that demonstrated statistical significance consisted of T5 (p=0, power=99 %), T6 (p=0.0023, power=77 %), T7 (p=0.041, power=46 %), and T10 (p=0.017, power=60 %). Our analysis suggests that the static positions of vertebrae in CT scans may align with Fryette's descriptions, although not very often. Notably, vertebral levels T5-T7 and T10 exhibit strong evidence of their static positions aligning with expected movements, warranting further investigation into the Fryette phenomenon at these levels. Future studies should explore the dynamic implications of these findings to enhance our understanding of spinal biomechanics.

Electromechanically Reconfigurable Terahertz Stereo Metasurfaces.

Dynamic terahertz devices are vital for the next generation of wireless communication, sensing, and non-destructive imaging technologies. Metasurfaces have emerged as a paradigm-shifting platform, offering varied functionalities, miniaturization, and simplified fabrication compared to their 3D counterparts. However, the presence of in-plane mirror symmetry and reduced degree of freedom impose fundamental limitations on achieving advanced chiral response, beamforming, and reconfiguration capabilities. In this work, a platform composed of electrically actuated resonators that can be colossally reconfigured between planar and 3D geometries is demonstrated. To illustrate the platform, metadevices with 3D Split Ring Resonators are fabricated, wherein two counteracting driving forces are combined: i) folding induced by stress mismatch, which enables non-volatile state design and ii) unfolding triggered by the strain associated with insulator-to-metal transition in VO2, which facilitates volatile structural reconfiguration. This large structural reconfiguration space allows for resonance mode switching, widely tunable magnetic and electric polarizabilities, and increased frequency agility. Moreover, the unique properties of VO2, such as the hysteretic nature of its phase transition is harnessed to demonstrate a multi-state memory. Therefore, these VO2 integrated metadevices are highly attractive for the realization of 6G communication devices such as reconfigurable intelligent surfaces, holographic beam formers, and spatial light modulators.

Hypersonic turbulent boundary layer over the windward side of a lifting body

In the present study, we performed direct numerical simulations for a hypersonic turbulent boundary layer over the windward side of a lifting body, the HyTRV model, at Mach number $6$ and attack angle 2 $^{\circ }$ to investigate the global and local turbulent features, and evaluate its difference from canonical turbulent boundary layers. By scrutinizing the instantaneous and averaged flow fields, we found that the transverse curvature on the windward side of the HyTRV model induces the transverse opposing pressure gradients that push the flow on both sides towards the windward symmetry plane, yielding significant effects of the azimuthal inhomogeneity and large-scale cross-stream circulations, moderate and azimuthal independent influences of adverse pressure gradient, and negligible impact of the mean flow three-dimensionality. Further inspecting the local turbulent statistics, we identified that the mean and fluctuating velocity become increasingly similar to the highly decelerated turbulent boundary layers over flat plates in that the mean velocity deficit is enhanced, and the outer layer Reynolds stresses are amplified as it approaches the windward symmetry plane, and prove to be self-similar under the scaling of Wei & Knopp (J. Fluid Mech., vol. 958, 2023, A9) for adverse-pressure-gradient turbulent boundary layers. Conditionally averaged Reynolds stresses based on strong sweeping and ejection events demonstrated that the Kelvin–Helmholtz instability of the strong embedded shear layer induced by the large-scale cross-stream circulations is responsible for the turbulence amplification in the outer layer. The strong Reynolds analogy that relates the mean velocity and temperature was refined to incorporate the non-canonical effects, showing considerable improvements in the accuracy of such a formula. On the other hand, the temperature fluctuations are still transported passively, as indicated by their resemblance to the velocity. The conclusions obtained in the present study provide potentially profitable information for turbulent modelling modification for the accurate predictions of skin friction and wall heat transfer.

Bound state in the continuum (BIC) excited by the metasurface breaking in-plane symmetry and quasi-BIC for terahertz sensing

Bound state in the continuum (BIC) excited by the metasurface breaking in-plane symmetry and quasi-BIC for terahertz sensing

Percolation thresholds of disks with random nonoverlapping patches on four regular two-dimensional lattices.

In percolation of patchy disks on lattices, each site is occupied by a disk, and neighboring disks are regarded as connected when their patches contact. Clusters of connected disks become larger as the patchy coverage of each disk χ increases. At the percolation threshold χ_{c}, an incipient cluster begins to span the whole lattice. For systems of disks with n symmetric patches on Archimedean lattices, a recent work [Wang et al., Phys. Rev. E 105, 034118 (2022)2470-004510.1103/PhysRevE.105.034118] found symmetric properties of χ_{c}(n), which are due to the coupling of the patches' symmetry and the lattice geometry. How does χ_{c} behave with increasing n if the patches are randomly distributed on the disks? We consider two typical random distributions of the patches, i.e., the equilibrium distribution and a distribution from random sequential adsorption. Combining Monte Carlo simulations and the critical polynomial method, we numerically determine χ_{c} for 106 models of different n on the square, honeycomb, triangular, and kagome lattices. The rules governing χ_{c}(n) are investigated in detail. They are quite different from those for disks with symmetric patches and could be useful for understanding similar systems.

Constraints on Bianchi-I type universe with SH0ES anchored Pantheon+ SNIa data

We study the Bianchi-I cosmological model motivated by signals of statistical isotropy violation seen in cosmic microwave background (CMB) observations and others. To that end, we consider various kinds of anisotropic matter that source anisotropy in our model, specifically Cosmic strings, Magnetic fields, Domain walls and Lorentz violation generated magnetic fields. These anisotropic matter sources, taking one at a time, are studied for their co-evolution with standard model (isotropic) sources viz., dust-like (dark/normal) matter, and dark energy modelled as cosmological constant. We constrain the Hubble parameter, density fractions of anisotropic matter, cold dark matter (CDM), and dark energy (Λ) in a Bianchi-I universe with planar symmetry i.e., which has a global ellipsoidal geometry, and try to find signatures of a cosmic preferred axis if any.The latest compilation of Type Ia Supernova (SNIa) data from Pantheon+SH0ES collaboration is used in our analysis to obtain constraints on cosmological parameters and any preferred axis for our universe.In our analysis, we found mild evidence for a cosmic preferred axis. It is interesting to note that this preferred axis lies broadly in the vicinity of other prominent cosmic anisotropy axes reported in the literature from diverse data sets.Also we find some evidence for non-zero (negative) cosmic shear and eccentricity that characterize different expansion rates in different directions and deviation from an isotropic scale factor respectively.The energy density fractions of two of the sources considered are found to be non-zero at a2σ confidence level.To be more conclusive, we require more SNIa host galaxy data for tighter constraints on distance and absolute magnitude calibration which are expected to be available from the future JWST observations and others.

Modeling the Nonmonotonic Immune Response in a Tumor–Immune System Interaction

Tumor–immune system interactions are very complicated, being highly nonlinear and not well understood. A large number of tumors can potentially weaken the immune system through various mechanisms such as secreting cytokines that suppress the immune response. In this paper, we propose a tumor–immune system interaction model with a nonmonotonic immune response function and adoptive cellular immunotherapy (ACI). The model has a tumor-free equilibrium and at most three tumor-presence equilibria (low, moderate and high ones). The stability of all equilibria is studied by analyzing their characteristic equations. The consideration of nonmonotonic immune response results in a series of bifurcations such as the saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. In addition, numerical simulation results show the coexistence of periodic orbits and homoclinic orbits. Interestingly, along with various bifurcations, we also found two bistable scenarios: the coexistence of a stable tumor-free as well as a high-tumor-presence equilibrium and the coexistence of a stable-low as well as a high-tumor-presence equilibrium, which can show symmetric and antisymmetric properties in a range of model parameters and initial cell concentrations. The new findings indicate that under ACI, patients can possibly reach either a stable tumor-free state or a low-tumor-presence state in the presence of nonmonotonic immune response once the immune system is activated.

In-plane magnetization orientation driven topological phase transition in OsCl3 monolayer

The quantum anomalous Hall effect resulting from the in-plane magnetization in the OsCl3 monolayer is shown to exhibit different electronic topological phases determined by the crystal symmetries and magnetism. In this Chern insulator, the Os-atoms form a two dimensional planar honeycomb structure with an easy-plane ferromagnetic configuration and the required non-adiabatic paths to tune the topology of electronic structure exist for specific magnetic orientations based on mirror symmetries of the system. Using density functional theory (DFT) calculations, these tunable phases are identified by changing the orientation of the magnetic moments. We argue that in contrast to the buckled system, here the Cl-ligands bring non-trivial topology into the system by breaking the in-plane mirror symmetry. The interplay between the magnetic anisotropy and electronic band-topology changes the Chern number and hence the topological phases. Our DFT study is corroborated with comprehensive analysis of relevant symmetries as well as a detailed explanation of topological phase transitions using a generic tight binding model.

Mechanical characterization and constitutive modeling of additively-manufactured polymeric materials and lattice structures

Additively manufactured polymeric lattice structures are being extensively studied, primarily because their mechanical properties can be tailored by controlling the unit cell geometry, giving them higher designability than stochastic materials. However, the inherent layer-wise additive manufacturing process affects the base material properties related to the printing direction, which in turn affects the macroscopic responses of the entire lattice materials. A robust understanding and modeling of lattice structures' elastic and plastic yield behavior in a homogenized approach are essential to enhance their design and analysis efficiency in engineering applications. In pursuit of this goal, a unified printing angle-dependent constitutive model of base materials is proposed in line with the tensile experimental data. The elastic material properties (elastic modulus, shear modulus, and Poisson's ratio), obtained through numerical simulations of one unit-cell with periodic boundary conditions, exhibit anisotropic properties, with the degree of anisotropy determined by the angle of the constituent members and base materials. Furthermore, both experimental and numerical results of lattices demonstrate anisotropic mechanical response under horizontal and vertical compression. Virtual multiaxial experiments are conducted through multi-cell numerical simulations, enabling the determination of initial yielding points of two different lattice structures (Kelvin and Simple cubic and body-centered cubic hybrid structures) under various loading conditions using a dissipation energy-based criterion. Overall, the multiaxial yield surface of the investigated lattices under various stress states, except for the isotropic principal stress plane, can be properly depicted by the Extended-Hill anisotropic yield criterion.

On evolutes of curves in the isotropic plane

On evolutes of curves in the isotropic plane

High‐Energy Nitrogen Rings Stabilized by Superatomic Properties

AbstractHow to stabilize nitrogen‐rich high‐energy‐density molecules under conventional conditions is particularly important for the energy storage and conversion of such systems and has attracted extensive attention. In this work, the theoretical study shows for the first time that the stabilization mechanism of the nitrogen ring conforms to the superatomic properties at the atomic level. This result occurs because the stabilized anionic nitrogen rings generally exhibit planar high symmetry and the injected electrons occupy the superatomic molecular orbitals (SAMOs) of the nitrogen rings. According to these results, the typical stabilized anionic nitrogen ring structures N64−, N5−, and N42− are identified, which possess delocalized molecular orbitals with the same symmetry as the atomic orbitals and electron arrangement consistent with the electron shell arrangement of the atom. On this basis, a pathway is further designed to stabilize nitrogen rings by introducing metal atoms as electron donors to form neutral ThN6, LiN5, and MgN4 structures, thereby replacing the anionization of systems. This study highlights the importance of developing nitrogen‐rich energetic materials from the perspective of superatoms.

Damped kink motions in a system of two solar coronal tubes with elliptic cross sections

Aims. This study is motivated by observations of coordinated transverse displacements in neighboring solar active region loops, addressing specifically how the behavior of kink motions in straight two-tube equilibria is impacted by tube interactions and tube cross-sectional shapes. Methods. We worked with linear, ideal, pressureless magnetohydrodynamics. Axially standing kink motions were examined as an initial value problem for transversely structured equilibria involving two identical, field-aligned, density-enhanced tubes with elliptic cross sections (elliptic tubes). Continuously nonuniform layers were implemented around both tube boundaries. We numerically followed the system response to external velocity drivers, largely focusing on the quasi-mode stage of internal flows to derive the pertinent periods and damping times. Results. The periods and damping times that we derive for two-circular-tube setups justify the available modal results found with the T-matrix approach. Regardless of cross-sectional shapes, our nonuniform layers feature the development of small-scale shears and energy accumulation around Alfvén resonances, indicative of resonant absorption and phase mixing. As with two-circular-tube systems, our configurational symmetries still make it possible to classify lower-order kink motions by the polarization and symmetric properties of the internal flows; hence, such motions are labeled as S​x and Ax. However, the periods and damping times for two-elliptic-tube setups further depend on cross-sectional aspect ratios, with Ax motions occasionally damped less rapidly than S​x motions. We find uncertainties up to ∼20% (∼50%) for the axial Alfvén time (the inhomogeneity lengthscale) if the periods (damping times) computed for two-elliptic-tube setups are seismologically inverted with canonical theories for isolated circular tubes. Conclusions. The effects of loop interactions and cross-sectional shapes need to be considered when the periods, and in particular the damping times, are seismologically exploited for coordinated transverse displacements in adjacent coronal loops.

Symmetry of polychromatic beams and transverse energy flows

The article considers the mechanisms of the formation of transverse energy flows in polychromatic fields with different structures. It is shown that the magnitude of the Poynting vector transverse component and the angular momentum for symmetric polychromatic beams are the same as in coherent monochromatic fields. Changes in the characteristics of energy flows due to the transformation and destruction of beam symmetry are demonstrated. The data of computer simulation are presented.

The sign of bistable wave speed for a plankton model

The sign of the speed of bistable traveling wave solution is an important focus in the field of population dynamics, since it can be used to predict which species will win the competition and ultimately survive. In this paper, the sign of bistable wave speed of a plankton model with cubic nonlinearity is studied. By a delicate analysis, a sufficient condition for bistable wave to be stationary is first obtained. Symmetric and multiplication properties for stationary bistable wave are established. By upper-lower solution method and comparison principle, two sufficient conditions are found so that the bistable wave speed is negative or positive. Moreover, numerical simulations are performed to illustrate our theoretic conclusions.