Symmetry Analysis Research Articles

Analysis of shock wave propagation in two-layered blood flow model via Lie symmetry

In this paper, we study the two-layered blood flow model within the framework of Lie symmetry analysis to investigate a range of closed-form solutions. Our investigation involves optimal classification, revealing the existence of six optimal subalgebras within the model. Further, we streamline the model by reducing it into a system of ordinary differential equations (ODEs), thereby deriving several exact solutions. Our analysis establishes the presence of the nonlinear self-adjointness property in the governing equation, leading to the development of various conservation laws. Furthermore, leveraging one of the exact solutions we obtained, we delve into the behaviour of characteristic shocks, C1 waves, and their interactions, providing a comprehensive understanding of their dynamics.

Enhancing Surgical Approach: Breakthrough Markerless Surface Registration With Augmented Reality for Zygomatic Complex Fracture Surgeries.

Innovative technologies with surgical navigation have been used for enhancing surgical accuracies for zygomaticomaxillary complex (ZMC) fractures and offers advantages in precision, accuracy, effectiveness, predictability, and symmetry improvement. Moreover, augmented reality (AR) navigation technology combines virtual reality, 3-dimensional (3D) reconstruction, and real-time interaction, making it ideal for bone tissue operations. Our study explored the usefulness and clinical efficacy of AR technology in intraoperative guidance for reducing ZMC fractures. We retrospectively studied 35 patients with zygomatic complex fractures, comparing outcomes of AR-guided and conventional methods. Furthermore, the AR system provided real-time visualization and guidance. The evaluation included reduction accuracy using root mean square (RMS) value and symmetry analysis using a mirror image of 3D models. Results demonstrated the feasibility and effectiveness of the AR-guided method in improving outcomes and patient satisfaction. In 35 patients (25 males, 10 females), AR-guided (n = 19) and conventional (n = 16) approaches were compared. Age, sex, and fracture type exhibited no significant differences between groups. No complications occurred, and postoperative RMS error significantly decreased ( P < 0.001). The AR group had a lower postoperative RMS error ( P = 0.034). Augmented reality-guided surgery improved accuracy and outcomes in zygomatic complex fractures. Real-time visualization enhanced precision during reduction and fixation. This innovative approach promises enhanced surgical accuracy and patient outcomes in craniofacial surgery.

Longitudinal-transverse splitting and fine structure of Fermi polarons in two-dimensional semiconductors

Interaction of excitons with resident charge carriers in semiconductors gives rise to bound three-particle complexes, trions, whose optical response is conveniently described in the framework of many-body correlated Fermi polaron states. These states are formed as a result of correlation of photocreated trion with the Fermi sea hole and possess the angular momentum component of ±1 depending on the helicity of the photon. We study theoretically the energy spectrum fine structure of Fermi polarons in two-dimensional semiconductors based on transition metal dichalcogenides. We demonstrate both by the symmetry analysis and microscopic calculation that the Fermi polarons with nonzero in-plane wavevector k are split, similarly to the neutral exciton states, into the linearly polarized longitudinal and transverse, with respect to the k, states. The origin of this longitudinal-transverse splitting is the long-range electron-hole exchange interaction that can be also described as the interaction of Fermi polarons with their induced electromagnetic field. The effective Hamiltonian describing the Fermi polaron fine structure is derived, and its parameters are determined from the microscopic model.

Systematic Search for and Study of Short-timescale Flare Structures in BL Lac Object Gamma-Ray Emission

We present here the first systematic search of short-timescale γ-ray flares from 29 high Galactic latitude BL Lac objects over 14 yr of Fermi Large Area Telescope data. Using a combined Bayesian Blocks and HOP algorithm, we identified seven high-quality orbital timescale flare segments from three sources and quantified 24 short-timescale flare structures. We then performed a comprehensive analysis of flare symmetry, power spectral density (PSD) of variability, and flux–photon index relation. The main results are as follows. (1) The flare symmetry parameter A shows a “U-shaped” distribution. Short-timescale flares are symmetric while long-timescale flares are asymmetric. The number of fast-rise, slow-decay and slow-rise, fast-decay type flares are equal. No correlation is found between A and peak/integral flux. No parameter evolution is seen between consecutive flares, either. The observations support a scenario where longer-timescale flares originate from the superposition of short, symmetric subhour flares. (2) PSD from yearly to hourly timescales is modeled using the CARMA process. At lower frequencies, the PSD follows the typical broken power-law form. The high-frequency region of the PSD exhibits a continuous power-law shape, indicating that γ-ray variability originates from a single physical process across all probed timescales. (3) The flux–photon index distribution shows a pattern of “harder-when-brighter” or “softer-when-brighter,” but becomes flat above a certain critical flux, with Γ ≈ 2. This behavior cannot be simply explained by a two-component or blazar sequence model, and we speculate it may be related to complex interplay between electron acceleration and cooling.

Dual Optical Cycling Centers Mounted on an Organic Scaffold: New Insights from Quantum Chemistry Calculations and Symmetry Analysis.

Molecules cooled to ultracold temperatures are desirable for applications in fundamental physics and quantum information science. However, cooling polyatomic molecules with more than six atoms has not yet been achieved. Building on the idea of an optical cycling center (OCC), a moiety supporting a set of localized and isolated electronic states within a polyatomic molecule, molecules with two OCCs (bi-OCCs) may afford better cooling efficiency by doubling the photon scattering rate. By using quantum chemistry calculations, we assess the extent of the coupling of the two OCCs with each other and the molecular scaffold. We show that promising coolable bi-OCC molecules can be proposed by following chemical design principles.

Noether symmetry analysis for fourth-order modified gravity theory: Cosmological solution and evolution

A modified fourth-order gravity theory namely the [Formula: see text] theory is considered in flat FLRW model to investigate both classical and quantum cosmology. The field equations are reduced to second-order differential equations by considering the technique of Lagrange multiplier. Noether symmetry analysis has been used both for obtaining classical solution and the wave function of the Universe in quantum cosmology. Finally the cosmological solutions are examined from observational view point.

Submodels of 2-d model of the motion of fluid or gas in a porous medium with an external non-stationary source or absorption

The motion of fluid or gas in a porous medium is the main object of investigation of underground hydromechanics. The model describing this motion was called “Porous medium”.In our paper, we study a general 2-d model of the motion of fluid or gas in porous medium with an external non-stationary source or absorption. As the main research method, we use the method of group (symmetry) analysis, which is one of the most effective ways to obtain information about the properties of solutions to differential equations that define a physical model. A group classification of the differential equation defining studied physical model is carried out. As a result, all basic models with different symmetry properties are obtained. For the basic model, which admits the widest group a classification of all its invariant submodels is performed and all invariant submodels, defined by invariant solutions of rank 1 are studied. All solutions found have physical meaning.The set of explicitly found solutions forms a 47-parametric family of exact solutions. Each solution in this family depends on an arbitrary function defining an external non-stationary source or absorption. For each solution, for the specific values of the parameters on which it depends and for this specific function, we obtained a graph of the pressure distribution.Other solutions are used to describe the movement of a liquid or gas if either the pressure and its rate of change, or the pressure and one of its gradient components, are given at an initial time at a fixed point. The set of these solutions depends on 13 parameters and a function defining an external non-stationary source or absorption. For specific values of this function and specific values of the parameters on which each solution depends, the existence and uniqueness of a solution to the corresponding boundary value problem under some additional conditions has been established. Each such problem is solved numerically. The results of these decisions we reflected in the graphs.The significance of obtained solutions lies in the fact that we have obtained a database consisting of two multiparameter families of invariant solutions of rank 1 of the equation describing the motion of fluid or gas in a porous medium in the presence of an external non-stationary source or absorption. Every solution depends on an empirically determined function. These solutions describe specific physical processes. They can also be used as test solutions in numerical calculations related to the study of the motion of fluid or gas in a porous medium during fluids filtering, engineering surveys in the construction of building structures and in the extraction of shale oil or gas.

Solving Nonlinear Second-Order ODEs via the Eisenhart Lift and Linearization

The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The research underscores the effectiveness of this geometric approach in the linearization of a class of Newtonian systems that cannot be linearized through symmetry analysis.

Lie Symmetry Analysis, Power Series Solutions and Conservation Laws of (2+1)-Dimensional Time Fractional Modified Bogoyavlenskii–Schiff Equations

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional modified Bogoyavlenskii–Schiff equations, which is an important model in physics. The one-dimensional optimal system composed by the obtained Lie symmetries is utilized to reduce the system of (2+1)-dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to the system of (1+1)-dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative. Then the power series method is applied to derive explicit power series solutions for the reduced system. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

Automatic three-dimensional facial symmetry reference plane construction based on facial planar reflective symmetry net

ObjectivesThree-dimensional (3D) facial symmetry analysis is based on the 3D symmetry reference plane (SRP). Artificial intelligence (AI) is widely used in the dental and oral sciences. This study developed a novel deep learning model called the facial planar reflective symmetry net (FPRS-Net) to automatically construct an SRP and established a method for defining a 3D point-cloud region of interest (ROI) and high-dimensional feature computations suitable for this network model. MethodsOverall, 240 patients were enroled. The deep learning model was trained and predicted using 200 samples, and its clinical suitability was evaluated with 40 samples. Four FPRS-Net models were prepared, each using supervised and unsupervised learning approaches based on full facial and ROI data (FPRS-NetS, FPRS-NetSR, FPRS-NetU, and FPRS-NetUR). These models were trained on 160 3D facial datasets, validated on 20 cases, and tested on another 20 cases. The model predictions were evaluated using an additional 40 clinical 3D facial datasets by comparing the mean square error of the SRP between the parameters predicted by the four FPRS-Net models and the truth plane. The clinical suitability of FPRS-Net models was evaluated by measuring the angle error between the predicted and ground-truth planes; experts evaluated the predicted SRP of the four FPRS-Net models using the visual analogue scales (VAS) method. ResultsThe FPRS-NetSR and FPRS-NetU models achieved an average angle error of 0.84° and 0.99° in predicting 3D facial SRP, respectively, with a VAS value of >8. Using the four FPRS-Net models to create an SRP in 40 cases of 3D facial data required <4 s. ConclusionsOur study demonstrated a new solution for automatically constructing oral clinical 3D facial SRPs. Clinical significanceThis study proposes a novel deep learning algorithm (FPRS-Net) to construct a symmetry reference plane that can reduce workload, shorten the time required for digital design, reduce dependence on expert experience, and improve therapeutic efficiency and effectiveness in dental clinics.

The type-I, III nodal ring, type-I, III quadratic nodal point, and Dirac valley phonons in 2D kagome lattices M2C3 (M = As, Bi, Cd, Hg, P, Sb, Zn)

Topological phases in kagome systems have garnered considerable interest since the introduction of the colloidal kagome lattice. Our study employs first-principle calculations and symmetry analysis to predict the existence of ideal type-I, III nodal rings (NRs), type-I, III quadratic nodal points (QNPs), and Dirac valley phonons (DVPs) in a collection of two-dimensional (2D) kagome lattices M2C3 (M = As, Bi, Cd, Hg, P, Sb, Zn). Specifically, the Dirac valley points (DVPs) can be observed at two inequivalent valleys with Berry phases of +π and −π , connected by edge arcs along the zigzag and armchair directions. Additionally, the QNP is pinned at the Γ point, and two edge states emerge from its projections. Notably, these kagome lattices also exhibit ideal type-I and III nodal rings protected by time inversion and spatial inversion symmetries. Our work examines the various categories of nodal points and nodal ring phonons within the 2D kagome systems and presents a selection of ideal candidates for investigating topological phonons in bosonic systems.

Testing functional anchor groups for the efficient immobilization of molecular catalysts on silver surfaces

Modifications of complexes by attachment of anchor groups are widely used to control molecule-surface interactions. This is of importance for the fabrication of (catalytically active) hybrid systems, viz. of surface immobilized molecular catalysts. In this study, the complex fac-Re(S-Sbpy)(CO)3Cl (S-Sbpy = 3,3′-disulfide-2,2′-bipyridine), a sulfurated derivative of the prominent Re(bpy)(CO)3Cl class of CO2 reduction catalysts, was deposited onto the clean Ag(001) surface at room temperature. The complex is thermostable upon sublimation as supported by infrared absorption and nuclear magnetic resonance spectroscopy. Its anchoring process has been analyzed using scanning tunneling microscopy (STM) and density functional theory (DFT) calculations. The growth behavior was directly contrasted to the one of the parent complex fac-Re(bpy)(CO)3Cl (bpy = 2,2′-bipyridine). The sulfurated complex nucleates as single molecule at different surface sites and at molecule clusters. In contrast, for the parent complex nucleation only occurs in clusters of several molecules at specifically oriented surface steps. While this shows that surface immobilization of the sulfurated complex is more efficient as compared to the parent, symmetry analysis of the STM topographic data supported by DFT calculations indicates that more than 90% of the complexes adsorb in a geometric configuration very similar to the one of the parent complex.

Calculated vibration spectrum of calcium hexahydroxodizincate dihydrate (qatranaite)

On the basis of first-principles electronic structure calculations, crystallographic parameters have been refined for calcium hydroxozincate (Qatranaite mineral), and the vibration properties (frequencies and eigenvectors) calculated. A detailed analysis of vibration modes is done, in the context of comparison with infrared and Raman spectra previously available. Special attention is paid to a posteriori symmetry analysis of vibration modes, discussing the latters’ attribution to four irreducible representations of the P21/c space group, and to identifying stretchings and bendings of particular chemical bonds, pronounced in different vibrations. It turns out that high-frequency (>700 cm−1) vibrations of hydroxyl groups bridging the Ca or Zn cations differ quite considerably for crystallographically distinct hydroxyl positions. It is shown that the vibrations involving hydroxyl groups and crystalline water typically come about in quadruplets at very close frequencies, whereby different irreducible representations reflect different combinations of similar “molecular” vibrations of four identical entities (of each hydroxyl or water) present in the unit cell. However, some vibrations show exceptions from this rule. In addition to interpretation of earlier experimental investigations, our study indicates that the low-frequency (<700 cm−1) vibrations within the cation–hydroxyl connected skeleton are of more “solid-state-like” character and cannot be reasonably interpreted in terms of “molecular” vibrations within ZnO4 or CaO6 units.

Nonstandard Nearly Exact Analysis of the FitzHugh–Nagumo Model

The FitzHugh–Nagumo model has been used empirically to model certain types of neuronal activities. It is also a non-linear dynamical system applicable to chemical kinetics, population dynamics, epidemiology and pattern formation. In the literature, many approaches have been proposed to study its dynamics. In this paper, initially, we have employed cutting-edge tools from discrete dynamics for discretization and fixed points. It has been proven that an exact discrete scheme exists for this paradigm. This project also considers the phase space and integral surfaces of these evolutionary equations. In addition, it carries out a thorough symmetry analysis of this reaction diffusion system to find equivalent systems. Moreover, steady-state solutions are obtained using ansatzes for traveling wave solutions. The existence of infinite traveling wave solutions has also been proven. Yet again, this investigation establishes the potential of symmetry methods to unravel non-linearity. Finally, singular perturbation theory has been employed to obtain analytical approximations and to study stability in different parameter regimes.

Higher-dimensional integrable deformations of the classical Boussinesq–Burgers system* *This work is supported by the National Natural Science Foundation of China (Grant Nos. 11871396, 12271433).

In this paper, the (1+1)-dimensional classical Boussinesq–Burgers (CBB) system is extended to a (4+1)-dimensional CBB system by using its conservation laws and the deformation algorithm. The Lax integrability, symmetry integrability and a large number of reduced systems of the new higher-dimensional system are given. Meanwhile, for illustration, an exact solution of a (1+1)-dimensional reduced system is constructed from the viewpoint of Lie symmetry analysis and the power series method.

Entanglement between Micro-Magnetism, electromagnetism and the Tensor Magnetic Phase Theory (TMPT) – Symmetry, conservation and invariance laws analysis at low frequency

Ferromagnetic materials show magnetic structures with domains and walls. Thanks to decades of research regarding the origin and behaviour of magnetic domains, we now possess a general foundation which has been verified experimentally in single crystals and powders. The governing equations at the microscopic scale were built in the 1960 s when Brown published calculations of the magnetic moments distribution inside domain walls. This micromagnetic theory uses the so called LLG ‘Landau-Lifshitz-Gilbert’ equation and can include the damping effects. The LLG equation requires a coupling to the field derived from energy contributions: exchange, anisotropy, magnetostriction, stray-field … and the anti-eddy field. At the macroscopic scale, such behaviours are lumped in a homogenized magnetization law for the electromagnetic field equations inside larger polycrystals. Therefore, the inhomogeneous magnetic material nature is always ignored. The Tensor Magnetic Phase Theory (TMPT) describes the magnetic structure and the magnetization thanks to a tensor variable at a mesoscopic scale. The material structuring is then explained thanks to an energy balance which will be discussed.This paper presents the results of investigations on entangled relationships between the micromagnetic theory, the electromagnetic theory and what is called the Tensor Magnetic Phase Theory (TMPT), which statistically describes the magnetic structures of soft ferromagnetic materials. It examines a connection between the TMPT and the LLG by deriving the main energy terms. Then, the TMPT must stay compatible and coupled to the Maxwell equations at low frequencies with volume and surface connections. Additionally, this paper investigates the way to derive the domains structuring and magnetization laws through the Lagrange principle and the corresponding conservation laws with invariants linked to the Nœther theorem. Finally, the TMPT must be discussed while checking its coherence and formulation when changing the reference frame.

Novel dynamical group-invariant solutions and conserved vectors of the Gilson–Pickering equation with applications in plasma physics

This work delves into the analysis of the Gilson–Pickering equation which governs the waves propagation in plasma physics by invoking Lie symmetry analysis. We commence by identifying the Lie point symmetries associated with the equation. These symmetries are then leveraged on to compute the commutator table and subsequently the adjoint representation, ultimately leading to the establishment of an optimal system of one-dimensional subalgebras. Each subalgebra within this system is subsequently utilized to perform symmetry reductions. Through these reductions, various forms of nonlinear ordinary differential equations are obtained, which are subsequently solved using the power series method and Kudryashov’s technique. The resulting solutions are given in terms of hyperbolic functions. To gain deeper insights into the behaviour of these solutions, three-dimensional and two-dimensional plots are presented. Furthermore, applying the Ibragimov’s theorem allows us to derive conserved vectors associated with the Gilson–Pickering equation.

Symmetry-based closed-form solutions and conserved quantities of the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation in fluid mechanics

In this investigation, we explore the mathematical intricacies of the novel (2+1)-dimensional Bogoyavlensky-Konopelchenko equation, a model with practical applications in elucidating the dynamics of internal waves within deep water. The equation’s significance spans various scientific domains, including plasma physics, nonlinear optics, and fluid dynamics. Employing a comprehensive analytical approach, specifically Lie symmetry analysis, we aim to unravel the underlying complexities of this equation and obtain new analytical solutions. To further scrutinize the equation, we apply various methods, namely Kudryashov’s method, the (G′/G)-expansion method, the simplest equation technique, and the power series method, all of which have not been applied to the equation before. Through these techniques, we successfully derive solutions in diverse functional forms, encompassing rational, trigonometric, exponential, hyperbolic, and Jacobi elliptic functions. To enhance comprehension, we present our findings visually using three-dimensional and two-dimensional plots density plots via the Mathematica tool. These graphical representations effectively communicate the intricate characteristics and nuances inherent in the solutions. Our visual representations reveal a spectrum of patterns, including periodic, singular periodic, kink-shaped structures. Additionally, our investigation extends to the determination of conserved quantities associated with the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation. This involves the application of the multiplier method and Ibragimov’s theorem, two potent techniques for identifying and understanding the conservation laws governing the model.

On heat transfer in Carreau fluid flow with thermal slip: An artificial intelligence (AI) based decisions integrated with Lie symmetry

It is well accepted that non-Newtonian fluid flow narrating differential systems with thermal slip conditions at the surface plays an important role in instabilities subject to polymer extrusion like stick–slip and gross-melt-fracture instabilities. Therefore, the present article contains the artificial neural networking evaluation of the friction at magnetized porous surface having thermal and velocity slip boundary conditions. The Carrea fluid flow is mathematically formulated at a porous surface. The novelty is enhanced by considering velocity slip boundary condition, thermal slip boundary condition, chemical reaction, magnetic field, and heat generation effects. The flow differential equations are reduced by using Lie symmetry analysis. The reduced equations are solved by using the shooting method. The neural networking model is constructed by engaging 132 values of SFC. 92 (70%) is marked for training, and 20 (15%) each is marked for validation and testing. 10 number of neurons are chosen in the hidden layer. Levenberg-Marquardt algorithm is carried out to train the network. The performance of the constructed neural networking model is evaluated by MSE and R. The developed ANN is the best to predict the friction values at the magnetized porous plate. Owning to predicted values of ANN, SFC shows inciting values towards the magnetic field parameter.

Semimetallic state transition in Two-Dimensional carbon nitride covalent networks and enhanced electrocatalytic activity for nitrate to ammonia conversion

Single-atom catalysts (SACs), due to their maximum atomic utilization rate, show tremendous potential for application in the electrocatalytic synthesis of ammonia from nitrate. Yet, the development of superior supports that preserve the high selectivity, activity, and stability of SACs remains an imperative challenge. In this work, based on first-principles calculations and tight-binding (TB) model analysis, a new two-dimensional (2D) carbon nitride monolayer, C7N6, is proposed. The C7N6 structure exhibits a strong covalent network, with dynamical, thermal, and mechanical stability. Surprisingly, the structural transition from C9N4 to C7N6 corresponds to a semimetallic state transition. Further symmetry analysis unveils that the Dirac states in C7N6 are protected by space–time inversion symmetry, and the physical origin of the Dirac cone was confirmed using the TB model. Additionally, a non-zero Z2 invariant and significant topological edge states demonstrate its topologically nontrivial nature. Considering the excellent structural and topological properties of C7N6, a three-step screening strategy is designed to identify eligible SACs for electrochemical nitrate reduction reaction (NO3RR), and Ti@C7N6 is identified as possessing the best activity, with the last proton-electron coupling step *NH2→*NH3 being the potential-determining step (PDS), for which the limiting potential is 0.48 V. Moreover, a free energy diagram shows that the *NOH reaction pathway is energetically preferred on Ti@C7N6, and ab initio molecular dynamics (AIMD) calculations at 500 K confirm its good thermal stability. Our study not only provides excellent CN-based support material but also offers theoretical guidance for constructing highly active and selective SACs for nitrate reduction.