Symmetry

Counting polynomials find their way into chemical graph theory through quantum chemistry in two ways: as approximate solutions to the Schrödinger equation or by storing information in a mathematical form and trying to find a pattern in the roots of these expressions. Coefficients count how many times a property occurs, and exponents express the extent of the property. They help understand the origin of regularities in the chemistry of specific classes of compounds. Our objective is to accelerate the research of newcomers into chemical graph theory. One problem in understanding these concepts is in the different approaches and notations of each research study; some researchers provide online tools for computing these mathematical concepts, but these need to be maintained for functionality. We take advantage of similar mathematical aspects of 14 such polynomials that merge theoretical chemistry and pure mathematics; give examples, differences, and similarities; and relate them to recent research. Full article

In this paper, we establish certain $L^p$ bounds for several classes of rough Marcinkiewicz integrals over surfaces of revolution on product spaces. By using these bounds and using an extrapolation argument, we obtain the $L^p$ boundedness of these Marcinkiewicz integrals under very weak conditions on the kernel functions. Our results represent natural extensions and improvements of several known results on Marcinkiewicz integrals. Full article

Mathematical modeling in epidemiology, biology, and life sciences requires the use of stochastic models. In this paper, we derive a competitive two-strain stochastic SIR epidemic model by considering the change in state of the epidemic process due to an event. Based on the density-dependent process theory, we construct a six-dimensional deterministic model that can be used to describe the diffusion limit of the stochastic epidemic on a heterogeneous network. Furthermore, we show the explicit expressions for the variances of infectious individuals with strain 1 and strain 2 when the level of infection is increasing exponentially. In particular, we find that the expressions of the variances are symmetric. Finally, simulations for epidemics spreading on networks are performed to confirm our analytical results. We find a close agreement between the simulations and theoretical predictions. Full article

This paper explores the novel concept of discontinuous unpredictable and Poisson-stable motions within impulsive inertial neural networks. The primary focus is on a specific neural network architecture where impulses mimic the structure of the original model, that is, continuous and discrete parts are symmetrical. This unique modeling decision aligns with the real-world behavior of systems, where voltage typically remains smooth and continuous but may exhibit sudden changes due to various factors such as switches, sudden loads, or faults. The paper introduces the representation of these abrupt voltage transitions as discontinuous derivatives, providing a more accurate depiction of real-world scenarios. Thus, the focus of the research is a model, exceptional in its generality. To study Poisson stability, the method of included intervals is extended for discontinuous functions and B-topology. The theoretical findings are substantiated with numerical examples, demonstrating the practical feasibility of the proposed model. Full article

The numerical investigation of bioconvective nanofluid (NF) flow, which involves gyrotactic microbes and heat and mass transmission analysis above an inclined extending axisymmetric cylinder, is presented in this study. The study aims to investigate the bioconvection flow of nanofluid under the influence of heat sources/sinks. Through proper transformation, all partial differential equations are transformed into a non-linear ODE scheme. A new set of variables is presented in the directive to get the first-order convectional equations and then solved numerically using bvp4c MATLAB, embedded in the function. The proposed model is validated after calculating the error estimation and obtaining the residual error. The influence of various factors on the velocity, energy, concentration, and density of motile microorganisms is examined and studied. The analysis describes and addresses all physical measures of concentration such as Skin Friction (SF), Sherwood number, the density of motile microorganisms, and Nusselt number. To validate the present study, a comparison is conducted with previous studies, and excellent correspondence is found. In addition, the ND-Solve approach is utilized to confirm the bvp4c. The mathematical model is confirmed through error analysis. This study provides the platform for industrial applications such as cooling capacity polymers, heat exchange, and chemical production sectors. Full article

In this paper, we study soft sets of the first and second Baire categories. The soft sets of the first Baire category are examined to be small soft sets from the point of view of soft topology, while the soft sets of the second Baire category are examined to be large. The family of soft sets of the first Baire category in a soft topological space forms a soft $\sigma$-ideal. This contributes to the development of the theory of soft ideal topology. The main properties of these classes of soft sets are discussed. The concepts of soft points where soft sets are of the first or second Baire category are introduced. These types of soft points are subclasses of non-cluster and cluster soft sets. Then, various results on the first and second Baire category soft points are obtained. Among others, the set of all soft points at which a soft set is of the second Baire category is soft regular closed. Moreover, we show that there is symmetry between a soft set that is of the first Baire category and a soft set in which each of its soft points is of the first Baire category. This is equivalent to saying that the union of any collection of soft open sets of the first Baire category is again a soft set of the first Baire category. The last assertion can be regarded as a generalized version of one of the fundamental theorems in topology known as the Banach Category Theorem. Furthermore, it is shown that any soft set can be represented as a disjoint soft union of two soft sets, one of the first Baire category and the other not of the first Baire category at each of its soft points. Full article

Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards. Full article

In rough set theory, there are many covering approximation spaces, so how to classify covering approximation spaces has become a hot issue. In this paper, we propose the concepts of a covering approximation $T_1$-space, F-symmetry, covering rough continuous mapping, and covering rough homeomorphism mapping, and we obtain some interesting results. We have used the above definitions and results to classify covering approximation spaces. Finally, we find a new method for constructing topologies, obtain some properties, and provide an example to illustrate our method’s similarities and differences with other construction methods. Full article

Due to the complexity and uncertainty of decision-making, probabilistic linguistic term sets (PLTSs) are currently important tools for qualitative evaluation of decision-makers. The asymmetry of evaluation information can easily lead to the loss of subjective preference information for decision-makers, and the existing operation of decision-maker evaluation information fusion operators is difficult to solve this problem. To solve such problems, this paper proposes some new operational methods for PLTSs based on Dombi T-conorm and T-norm. Considering the interrelationships between the input independent variables of PLTSs, the probabilistic linguistic weighted Dombi Bonferroni mean Power average (PLWDBMPA) operators are extended and the properties of these aggregation operators are proposed. Secondly, the PLWDBMPA operator is used to fuse the evaluation information of decision-makers, avoiding the loss of decision information as much as possible. This paper uses social media platforms and web crawler technology to obtain online comments from users on decision-making to obtain the public’s attitude towards decision events. TF-IDF and Word2Vec are used to calculate the weight of alternatives on each attribute. Under traditional group decision-making methods and integrating the wisdom of the public, a novel multi-attribute group decision-making method based on TODIM method is proposed. Finally, the case study of Turkey earthquake shelter selection proves this method is scientific and effective. Meanwhile, the superiority of this method was further verified through comparisons with the PL-TOPSIS, PLWA, SPOTIS and PROMETHEE method. Full article

We review several variants of three-dimensional quantum electrodynamics (QED₃) with N_f fermion (or boson) flavors, including fermionic (or spinorial) QED3, bosonic (or scalar) QED₃, N = 1 supersymmetric QED and also models of reduced QED (supersymmetric or not). We begin with an introduction to these models and their flow to a stable infra-red fixed point in the large-Nf limit. We then present detailed state-of-the-art computations of the critical exponents of these models within the dimensional regularization (and reduction) scheme(s), at the next-to-leading order in the 1/N_f expansion and in an arbitrary covariant gauge. We finally discuss dynamical (matter) mass generation and the current status of our understanding of the phase structure of these models. Full article

In this study, we first generalize the Lorentzian inner and vector products, and then we define the generalized split quaternions by means of the generalized Lorentzian inner and vector products. Next, on any hyperboloid of one or two sheets, which is a generalized Lorentzian sphere, non-parabolic conical rotations with nonnull axes are expressed using the generalized split quaternions with supporting numerical examples. Full article

Following the discovery of an artificial protein cage with a paradoxical geometry, we extend the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages, for which all the faces are equivalent, and define bi-homogeneous symmetric polyhedral cages made of two different types of faces, where all the faces of a given type are equivalent. We parametrise the possible connectivity configurations for such cages, analytically derive p-cages that are regular, and numerically compute near-symmetric p-cages made of polygons with 6 to 18 edges and with deformation not exceeding 10%. Full article

Covalent Organic Frameworks (COFs) are a newly emerged class of porous materials consisting of organic building blocks linked by strong covalent bonds. The physical and chemical properties of COFs, i.e., modularity, porosity, well-developed specific surface area, crystallinity, and chemical-thermal stability, make them a good application material, especially in the aspects of adsorption and gas separation. The organic compositions of their building blocks also render them with biocompatible properties; therefore, they also have potential in biomedical applications. Depending on the symmetry of the building blocks, COF materials form two-dimensional (2D COF) or three-dimensional (3D COF) crystal structures. 3D COF structures have a higher specific surface area, they are much lighter due to their low density, and they have a larger volume than 2D COF crystals, but, unlike the latter, 3D COF crystals are less frequently obtained and studied. Selecting and obtaining suitable building blocks to form a stable 3D COF crystal structure is challenging and therefore of interest to the chemical community. Triptycene, due to its 3D structure, is a versatile building block for the synthesis of 3D COFs. Polymeric materials containing triptycene fragments show good thermal stability parameters and have a very well-developed surface area. They often tend to be characterized by more than one type of porosity and exhibit impressive gas adsorption properties. The introduction of a triptycene backbone into the structure of 3D COFs is a relatively new procedure, the results of which only began to be published in 2020. Triptycene-based 3D COFs show interesting physicochemical properties, i.e., high physical stability and high specific surface area. In addition, they have variable porosities with different pore diameters, capable of adsorbing both gases and large biological molecules. These promising parameters, guaranteed by the addition of a triptycene backbone to the 3D structure of COFs, may create new opportunities for the application of such materials in many industrial and biomedical areas. This review aims to draw attention to the symmetry of the building blocks used for COF synthesis. In particular, we discussed triptycene as a building block for the synthesis of 3D COFs and we present the latest results in this area. Full article

Supercritical CO₂ phase change fracturing technology has been widely used in rock engineering, with the advantages of low disturbance and no pollution. However, the phase change shock wave inevitably affects the surrounding environment, and the influence range is still unclear. In this paper, we present a computational model for the symmetric generation, propagation, and attenuation of supercritical CO₂ phase transition shock waves, with the center of the borehole as the origin, based on the C–J theory. The attenuation of the shock wave in the rock medium under the influence of the type of fracturing tube, the thickness of the shear sheet, and the rock performance parameters are further analyzed. The results show that the rock stress under the action of the phase change shock wave attenuates logarithmically with the propagation distance, which correlates with the magnitude of the incident rock stress at the borehole wall. The incident rock stress decreases with the increase in the initial density of CO₂ in the fracturing tube, increases linearly with the thickness of the shear sheet, and correlates with the rock wave impedance. Full article

Differentiated production and supply chain management (SCM) areas benefit from the IoT, Big Data, and the data-management capabilities of the AI paradigm. Many businesses have wondered how the arrival of AI will affect planning, organization, optimization, and logistics in the context of SCM. Information symmetry is very important here, as maintaining consistency between output and the supply chain is aided by processing and drawing insights from big data. We consider continuous (production) and discontinuous (supply chain) data to satisfy delivery needs to solve the shortage problem. Despite a surplus of output, this article addresses the voluptuous deficiency problem in supply chain administration. This research serves as an overview of AI for SCM practitioners. The report then moves into an in-depth analysis of the most recent studies on and applications of AI in the supply chain industry. This work introduces a novel approach, Incessant Data Processing (IDP), for handling harmonized data on both ends, which should reduce the risk of incorrect results. This processing technique detects shifts in the data stream and uses them to predict future suppressions of demand. Federated learning gathers and analyzes information at several points in the supply chain and is used to spot the shifts. The learning model is educated to forecast further supply chain actions in response to spikes and dips in demand. The entire procedure is simulated using IoT calculations and collected data. An improved prediction accuracy of 9.93%, a reduced analysis time of 9.19%, a reduced data error of 9.77%, and increased alterations of 10.62% are the results of the suggested method. Full article

The influence of Ar + SiH₄ + O₂ plasma formulation on the phase composition and optical properties of amorphous SiO_x films with silicon nanoclusters obtained using PECVD with DC discharge modulation was studied. Using a unique technique of ultrasoft X-ray emission spectroscopy, it was found that at a 0.15 mol.% plasma oxygen content, amorphous silicon a-Si films are formed. At a high oxygen content (≥21.5 mol.%), nanocomposite films based on SiO_x silicon suboxide containing silicon nanoclusters ncl-Si are formed. It was found that the suboxide matrix consists of a mixture of SiO_1.3 and SiO₂ phases, and the average oxidation state x in the SiO_x suboxide matrix is ~1.5. An increase in the concentration of O₂ in the reactor atmosphere from 21.5 to 23 mol.% leads to a decrease in ncl-Si content from 40 to 15% and an increase in the average oxidation state x of SiO_x from 1.5 to 1.9. In this case, the suboxide matrix consists of two phases of silicon dioxide SiO₂ and non-stoichiometric silicon oxide SiO_1.7. Thus, according to the experimental data obtained using USXES, the phase composition of these films in pure form differs in their representation in both random coupling and random mixture models. A decrease in the ncl-Si content of SiO_x films is accompanied by a decrease in their sizes from ~3 to ~2 nm and a shift in the photoluminescence band from 1.9 eV to 2.3 eV, respectively. Full article

Let $G\left(R\right)$ be the unit graph associated with a ring R. Let p be a prime number and let R be a finite ring of order p or $p^2$ and be one of the rings ${\mathbb{Z}}_p,{\mathbb{Z}}_{p^2},GF\left(p^2\right),{\mathbb{Z}}_p(+){\mathbb{Z}}_p$ or ${\mathbb{Z}}_p\times {\mathbb{Z}}_p$. We determine the geodetic number $g\left(G\right(R\left)\right)$ associated with each such ring. Full article

The fundamental unit for comprehending the physicochemical properties of water, the Zundel cation configuration H₅O₂⁺, has yet to be exhaustively evaluated in terms of its interaction with terahertz (THz) electromagnetic waves, characterized by sub-picosecond oscillation periods or pulse widths. In this study, we embark on an investigation of the broad resonance and high-field nonresonant effects of intense THz radiation (ITR) on Zundel cations, utilizing a multifaceted methodological approach that includes density functional theory (DFT) calculations, finite difference time domain (FDTD) algorithm of the Schrödinger equation, and ab initio molecular dynamics (AIMD) simulations. Our analysis reveals that the proton potential energy surface (PES) varies in response to the external electric (E) field, suggesting that the interaction frequency of the central proton with the electromagnetic wave encompasses the THz band. This resonance effect is associated with proton behavior that may oscillate or demonstrate periodic tunneling. Moreover, our work uncovers the high-field nonresonant effects of ITR on Zundel cations, manifesting in proton transfer and vibrational excitation of the system. Our findings contribute to the understanding of the interaction between Zundel species and electromagnetic waves by presenting a microscopic view of proton transfer as informed by wavefunction evolution. Full article

This paper discusses some properties of complex-valued fuzzy metric spaces and introduces the $\alpha$-admissible mappings in the setting of complex-valued fuzzy metric spaces. We establish fixed point theorems for mappings satisfying symmetric contractive conditions with control functions. The results of this paper generalize, extend, and improve several results from metric, fuzzy metric, and complex-valued fuzzy metric spaces. Several examples are presented that verify and illustrate the new concepts, claims, and results. Full article