Theory Of Equations Research Articles

Time-dependent magneto-optical transmission maximization and determination of critical concentration of the delocalization-localization transition

This study leads novel approaches to overcome cluster induced magnetic field formation and the associated reduced light transmission beyond a certain critical magnetic field utilizing the ring morphology and time-varying magnetic field. This hypothesis is challenged by the fabrication of large size ferromagnetic, hematite nanoring and nanodisk samples to increase the coupling constant and the interparticle interactions. The nanoring sample transmission outperformed the nanodisk one by 40–50 % under 0.5 mT, 1–5 kHz time-varying magnetic fields, 12 % in the uniform 0–6.5 mT field, and 27 % in the 40x greater gradient field amplitude by Neodymium magnet. Furthermore, the dominant literature view of the dynamics of optical transmission through ferrofluids is corrected. Moreover, the decrease in the optical transmission in the 350–450 nm band is justified in terms of localization, which is found to occur beyond a certain critical concentration, described by the proposed, experimentally verified power law utilizing the results of the renormalization group equation and scaling theory, for the first time. Therefore, challenging limitation of restriction to their qualitative utilization and non-observable quantities besides the previously doubted matching between the method of invariant imbedding and the renormalization group equation have been all resolved, which is crucial for the entire field of complex media.

Understanding the Negative Apparent Activation Energy for Cu2O and CoO Oxidation Kinetics at High Temperature near Equilibrium

The pairs of Cu2O/CuO and CoO/Co3O4 as the carriers of transferring oxygen and storing heat are essential for the recently emerged high-temperature thermochemical energy storage (TCES) system. Reported research results of Cu2O and CoO oxidation kinetics show that the reaction rate near equilibrium decreases with the temperature, which leads to the negative activation energy obtained using the Arrhenius equation and apparent kinetics models. This study develops a first-principle-based theoretical model to analyze the Cu2O and CoO oxidation kinetics. In this model, the density functional theory (DFT) is adopted to determine the reaction pathways and to obtain the energy barriers of elementary reactions; then, the DFT results are introduced into the transition state theory (TST) to calculate the reaction rate constants; finally, a rate equation is developed to describe both the surface elemental reactions and the lattice oxygen concentration in a grain. The reaction mechanism obtained from DFT and kinetic rate constants obtained from TST are directly implemented into the rate equation to predict the oxidation kinetics of Cu2O without fitting experimental data. The accuracy of the developed theory is validated by experimental data obtained from the thermogravimetric analyzer (TGA). Comparing the developed theory with the traditional apparent models, the reasons why the latter cannot appropriately predict the true oxidation characteristics are explained. The reaction rate is jointly controlled by thermodynamics (reaction driving force) and kinetics (reaction rate constant). Without considering the effect of the reaction driving force, the negative apparent activation energy of Cu2O oxidation is obtained. However, for CoO oxidation, the negative apparent activation energy is still obtained although the effect of the reaction driving force is considered. According to the DFT results, the activation energy of the overall CoO oxidation reaction is negative, but the energy barriers of the elementary reactions are positive. Moreover, according to the first-principle-based rate equation theory, the pre-exponential factor in the kinetic model is dependent on the partition function ratio and decreases with the temperature for the Cu2O and CoO oxidation near equilibrium, which results in the apparent activation energy being slightly lower than the actual value.

An Enhanced Darbo-Type Fixed Point Theorems and Application to Integral Equations

This manuscript introduces a generalized operator and presents new Darbo-type fixed point theorems pivotal in the existence theory of integral and differential equations. The significance of these theorems lies in their ability to provide conditions under which solutions to complex mathematical problems can be guaranteed. We establish our results by employing the measure of noncompactness within the context of Banach spaces, a framework that allows for a comprehensive analysis of functional equations. Our findings extend existing Darbo-type fixed point theorems and offer a deeper understanding of the underlying mathematical structures. By generalizing these results, we contribute to the broader field of fixed-point theory, enhancing its applicability to various mathematical disciplines. The implications of our work are substantial, as they facilitate the development of new methods for solving integral and differential equations that arise in both theoretical and applied contexts.

The effect of time-varying characteristics of shallow-sea waveguides on low-frequency acoustic signal transmission

The time-varying characteristic of shallow sea channel plays one of the most important role in the exploration of ocean. However, there is still lack of estimation model to describe the time-varying characteristics of shallow water low-frequency acoustic channels. This paper proposes a low-frequency channel model based on wave equation and ray theory to estimate the time-varying characteristics of shallow sea. Firstly, for the actual characteristics of shallow sea, suitable boundary conditions are designed to solve the wave equation. Secondly, the constraint conditions of shallow sea wave equation are optimized by combining ray theory. Then, based on the physical characteristics of shallow sea, time-delay variation and phase variation are solved by shallow wave equation. Combining bellhop propagation model with ocean disturbance model, the simulation is designed to verify the time-varying characteristic model of low-frequency shallow sea in this paper. Finally, with different propagation distance measurement experiment of sea, estimation value of time-delay variation and phase variation are confirmed, which is significant to shallow sea channel.

Medium-Range Structural Order as the Driver of Activated Dynamics and Complexity Reduction in Glass-Forming Liquids.

We analyze in depth the Elastically Collective Nonlinear Langevin Equation theory of activated dynamics in metastable liquids to establish that the predicted inter-relationships between the alpha relaxation time, local cage and collective elastic barriers, dynamic localization length, and shear modulus are causally related within the theory to the medium range order (MRO) static correlation length. The latter grows exponentially with density for metastable hard sphere fluids and as a nonuniversal inverse power law with temperature for supercooled liquids under isobaric conditions. The physical origin of predicted connections between the alpha time and other metrics of cage order and the thermodynamic inverse dimensionless compressibility is fully established. It is discovered that although kinetic constraints from the real space first coordination shell are important for the alpha time, they are of secondary importance compared to the consequences of the more universal MRO correlations in both the modestly and deeply metastable regimes. This understanding sheds new light on the theoretical basis for, and prior successes of, the predictive mapping of chemically complex thermal liquids to effective hard sphere fluids based on matching their dimensionless compressibilities, a scheme we call "complexity reduction". In essence, the latter is equivalent to the physical requirement that the thermal liquid MRO correlation equals that of its effective hard sphere analog. The mapping alone is shown to provide a remarkable level of quantitative predictive power for the glass transition temperature Tg of 21 molecular and polymer liquids. Predictions for the chemically specific absolute magnitude and growth with cooling of the MRO correlation length are obtained and lie in the window of 2-6 nm at Tg. Dynamic heterogeneity, elastic facilitation, and beyond pair structure issues are briefly discussed. Future opportunities to theoretically analyze the equilibrated deep glass regime are outlined.

Solvent‐Controlled Separation of Integratively Self‐Sorted Pd2LA2LB2 Coordination Cages

AbstractThe integrative implementation of multiple different components into metallosupramolecular self‐assemblies requires sophisticated strategies to avoid the formation of statistical mixtures. Previously, the key focus was set on thermodynamically driven reactions of simple homoleptic into complex heteroleptic structures. Using Pd2LA2LB2‐type coordination cages, we herein show that integrative self‐sorting can be reversed by a change of solvent (from DMSO to MeCN) to favor narcissistic re‐segregation into coexisting homoleptic species Pd2LA4 and Pd3LB6. Full separation (“unsorting”) back to a mixture of the homoleptic precursors was finally achieved by selective precipitation of Pd3LB6 with anionic guest G1 from MeCN, keeping pure Pd2LA4 in solution. When a mixture of homoleptic Pd3LB6 and heteroleptic Pd2LA2LB2 is exposed to a combination of two different di‐anions (G1 and G2) in DMSO, selective guest uptake gives rise to two defined coexisting host–guest complexes. A joint experimental and deep theoretical investigation via liquid‐state integral equation theory of the reaction thermodynamics on a molecular level accompanied by solvent distribution analysis hints at solvent expulsion from Pd2LA4 to favor the formation of Pd2LA2LB2 in DMSO as the key entropic factor for determining the solvent‐specific modulation of the cage conversion equilibrium.

Solvent-Controlled Separation of Integratively Self-Sorted Pd2LA 2LB 2 Coordination Cages.

The integrative implementation of multiple different components into metallosupramolecular self-assemblies requires sophisticated strategies to avoid the formation of statistical mixtures. Previously, the key focus was set on thermodynamically driven reactions of simple homoleptic into complex heteroleptic structures. Using Pd2LA 2LB 2-type coordination cages, we herein show that integrative self-sorting can be reversed by a change of solvent (from DMSO to MeCN) to favor narcissistic re-segregation into coexisting homoleptic species Pd2LA 4 and Pd3LB 6. Full separation ("unsorting") back to a mixture of the homoleptic precursors was finally achieved by selective precipitation of Pd3LB 6 with anionic guest G1 from MeCN, keeping pure Pd2LA 4 in solution. When a mixture of homoleptic Pd3LB 6 and heteroleptic Pd2LA 2LB 2 is exposed to a combination of two different di-anions (G1 and G2) in DMSO, selective guest uptake gives rise to two defined coexisting host-guest complexes. A joint experimental and deep theoretical investigation via liquid-state integral equation theory of the reaction thermodynamics on a molecular level accompanied by solvent distribution analysis hints at solvent expulsion from Pd2LA 4 to favor the formation of Pd2LA 2LB 2 in DMSO as the key entropic factor for determining the solvent-specific modulation of the cage conversion equilibrium.

Improvements on the Leighton‐type oscillation criteria for impulsive differential equations and extension to non‐canonical case

We investigate the oscillatory behavior of solutions of second‐order linear differential equations with impulses. These equations can be categorized into two types: canonical and non‐canonical. This study examines the canonical and non‐canonical impulsive differential equations in connection with the famous Leighton‐type oscillation theorem. The well‐known theorem states that every solution of where and are continuous with , is oscillatory if and This pioneering work of Leighton has received considerable attention since its inception, and hence, it has been extended to various types of equations, including delay differential equations, dynamic equations, and impulsive differential equations. There are also studies generalizing the Leighton oscillation theorem when the condition fails, that is, when the equation is of non‐canonical type. Our work focuses on refining the Leighton oscillation theorem to treat the canonical and non‐canonical cases. By doing so, we correct, supplement, and enhance the current literature on the oscillation theory of differential equations with impulses. Examples are also given to illustrate the significance of the obtained results.

Solving Partial Differential Equations Using Fourier Series

This research paper aims to study the methods of solving partial differential equations using Fourier series. Partial differential equations are a powerful and practical tool in mathematics and physics to describe many phenomena and processes that involve continuous change at the level of equations and functions. The basic concepts related to the Fourier series and how to represent functions using it in the field of mathematics will be reviewed. Application examples of partial differential equations, such as waves, heat and diffusion, and how to use the Fourier series to solve them will also be presented. The steps of solving the partial differential equation using the Fourier series will be explained. The importance of Fourier series in the theory of partial differential equations is that periodic functions f(x) defined on (-∞,∞) or functions defined on a finite interval can be represented by an infinite interval of sines and cosines. In this research, the solution of the partial differential equation using the Fourier series was studied with the solution of an example.

Topological Anderson insulators by homogenization theory

. A central property of (Chern) topological insulators is the presence of robust asymmetric transport along interfaces separating two-dimensional insulating materials in different topological phases. A topological Anderson Insulator is an insulator whose topological phase is induced by spatial fluctuations. This article proposes a mathematical model of perturbed Dirac equations and shows that for sufficiently large and highly oscillatory perturbations, the system is in a different topological phase than the unperturbed model. In particular, a robust asymmetric transport indeed appears at an interface separating perturbed and unperturbed phases. The theoretical results are based on careful estimates of resolvent operators in the homogenization theory of Dirac equations and on the characterization of topological phases by the index of an appropriate Fredholm operator.

Nonlinear combined resonance of magneto-electro-elastic plates

The existing articles on the nonlinear resonances of magneto-electro-elastic (MEE) structures are mainly devoted to the parametric resonance and primary resonance, neglecting the coupling effect between parametric and forced excitations. To reveal this issue, the coupled longitudinal and transverse excitations are considered to investigate the combined resonance of MEE plates, in which the simply supported ends is considered. The expressions of magnetic-, electric- and displacement-fields are determined using the Maxwell's equation and first order shear deformation theory (FSDT). Applying the Galerkin method, the nonlinear ordinary differential equations are derived. The combined resonance problem of MEE plates with simply supported ends is solved by developing the method of varying amplitude (MVA). And the resonance trajectory is depicted by amplitude-frequency diagram in the follow-up analysis, in which the complicated dynamical phenomena with jump, hysteresis and multi-stable solution can be observed. To reveal the dynamic mechanism, comprehensive numerical analyses including electric potential, magnetic potential, excitations, temperature variation and other factors are conducted, the bifurcation and chaotic dynamic behaviors are also analyzed. The results elucidate that these parameters under discussion play significant roles.

Challenges in Mathematics Arising from Theoretical and Computational Chemistry

The combination of theoretical chemistry and mathematics represents a significant advancement in science since it has yielded insights into the enigmas surrounding atoms and molecules. The advancement of technology has greatly benefited drug development, leading to significant advancements in both medicine and materials research. This summary discusses innovative methodologies and computational challenges in the expansive domain of theoretical and computational chemistry. Despite the computational challenges associated with the Schrodinger equation and Density Functional Theory (DFT), quantum mechanics offers a basic understanding of the behaviour of molecules and atoms. While MD simulations may not handle long-range interactions and complex configuration spaces, they nonetheless excel at capturing various time scales of molecular behaviour. Nevertheless, the process of transforming abstract notions into computer models for the purpose of solving MD simulations continues to be difficult; statistical mechanics offers a theoretical framework for understanding MD simulations. Data-driven methodologies and machine learning have significantly transformed the perspective of computational chemistry by enabling the observation and comprehension of very complex chemical interactions. We must also address concerns about interpretability, reliability in limited data streams, and system transferability. These challenges expedite the advancement of novel materials, product designs, and pharmaceuticals by leveraging machine learning and integrating it with specialised domain expertise. We used available data from many reputable databases, spanning the time period from 2010 to 2024. It is imperative to prioritise the advancement of multiscale modelling techniques and hybrid quantum-classical tools in order to achieve sufficient mastery over chemical phenomena. Another potential benefit is the use of novel mathematical models to uncover previously unknown patterns and connections in genetic data. The most effective way to tackle mathematical problems is by employing a straightforward approach and engaging in collaboration with experts from other fields. By adopting this approach, we can effectively address significant societal issues and propel advancements in chemistry. Keywords: mathematics, computational, chemistry, theoretical, challenges, quantum mechanics

Ion-Specific Surface Tension of Aqueous Electrolyte Solutions: Analytical Insights from a Restricted Primitive Model.

The ion-specific surface tension of aqueous electrolyte solutions is of fundamental importance in physical chemistry, solution chemistry, electrochemistry, and biochemistry, yet it remains a challenge to be qualitatively predicted. In this work, an analytical theory of ion-specific surface tension is developed. By modeling the solution to a restricted primitive model (RPM), the surface tension increment of the electrolyte solution reduces to the surface tension of the RPM, which is further determined analytically by using the integral equation theory and the morphological thermodynamics theory. According to our formula, the surface tension increment of the electrolyte solution consists of a dominant and positive hard sphere contribution, which depends on the salt concentration, and a secondary electrostatic contribution, which depends on the inverse Debye length and the salt concentration. Our theory is applied to 1:1, 2:2, 1:2, 2:1, 3:1, and 3:2 electrolyte solutions with typical salt concentrations up to several mol/L and compared with experimental data. Without introducing any adjustable parameters, our theory leads to a good prediction of the surface tension increment of more than 50 kinds of aqueous solutions. Such a good agreement demonstrates the great potential of our theory for a fundamental understanding of specific ion effects in a variety of electrolyte solutions.

Application of Digital Technology for Oil and Gas Fields

The paper investigates the problem of isothermal filtration using an approximate method for solving the theory of partial differential equations. The mathematical model under consideration is nonlinear and does not lend itself to analytical methods of solution. The results obtained indicate the need for wide application in the development of oil and gas fields in the Republic of Kazakhstan. In particular, the results of the study make it possible to solve the problems of adapting mathematical models and evaluating changes in technological indicators, which are necessary attributes in the digital technology ""Information System for the Analysis of oil and Gas Field Development"" (ISAR). Many problems and mathematical problems of filtration theory arose while working on specific oil and gas fields in the western region of the Republic of Kazakhstan. The above approximate solution methods have found applications not only in filtration theory, but also in other problems (geophysics, ecology, etc.)"

Relativistic theory of the viscosity of fluids across the entire energy spectrum.

The shear viscosity is a fundamental transport property of matter. Here we derive a general theory of the viscosity of gases based on the relativistic Langevin equation(deduced from a relativistic Lagrangian) and nonaffine linear response theory. The proposed relativistic theory is able to recover the viscosity of nonrelativistic classical gases, with all its key dependencies on mass, temperature, particle diameter, and Boltzmann constant, in the limit of Lorentz factor γ=1. It also unveils the relativistic enhancement mechanism of viscosity. In the limit of ultrarelativistic fluids, the theory provides an analytical formula which reproduces the cubic increase of viscosity with temperature in agreement with various estimates for hot dense matter and the quark-gluon-plasma-type fluid.

Dynamic pricing and inventory model for perishable products with reference price and reference quality

Reference price (quality) is a benchmark point used by consumers to make price (quality) judgements. Combining reference price and reference quality with dynamic pricing and inventory management, this paper applies differential equations theory to construct a optimal control model for perishable products when the quality of products declines exponentially. The demand of products depends on price, quality, reference price and reference quality, among which reference price and reference quality are influenced by consumers’ past memory. The aim is to maximize the retailer’s profit during the period. The conclusions are as follows. Firstly, a optimal dynamic pricing and inventory model for perishable products with reference price and reference quality is constructed, and the model is extended to dual decision variables, stochastic demand, time-dependent effects, competitive and infinite planing horizon setups. Secondly, through the Pontryagin maximum principle, the analytical expression of the optimal dynamic price is derived. Thirdly, a linear search method to solve the optimal static price is proposed. Finally, the sensitivity of the main parameters is analyzed and the corresponding management enlightenments are given. By comparison of dynamic and static pricing, we find that dynamic pricing can achieve more profits and take a shorter selling period. In addition, for the sales problem of perishable products affected by both reference price and reference quality, retailers should adopt skimming pricing strategy (the optimal price decreases with time). Furthermore, to obtain more profit, retails should strive to increase the sensitivity coefficients of two types of reference, and the reference quality memory coefficient, while to decrease the reference price memory coefficient.

Enhanced Oscillation Criteria for Non-Canonical Second-Order Advanced Dynamic Equations on Time Scales

This study aims to establish novel iterative oscillation criteria for second-order half-linear advanced dynamic equations in non-canonical form. The results extend and enhance recently established criteria for this type of equation by various authors and also encompass the classical criteria for related ordinary differential equations. Our methodology involves transforming the non-canonical equation into its corresponding canonical form. The inherent symmetry of these canonical forms plays a pivotal role in deriving our new criteria. By employing techniques from the theory of symmetric differential equations and utilizing symmetric functions, we establish precise conditions for oscillation. Several illustrative examples highlight the accuracy, applicability, and versatility of our results.

To the Theory of Three-Dimensional Model Integral Volterra-Type Equations with Boundary Singular, Weakly Singular, and Strongly Singular Kernels

To the Theory of Three-Dimensional Model Integral Volterra-Type Equations with Boundary Singular, Weakly Singular, and Strongly Singular Kernels

Asymptotic integrability of nonlinear wave equations.

We introduce the notion of asymptotic integrability into the theory of nonlinear wave equations. It means that the Hamiltonian structure of equations describing propagation of high-frequency wave packets is preserved by hydrodynamic evolution of the large-scale background wave so that these equations have an additional integral of motion. This condition is expressed mathematically as a system of equations for the carrier wave number as a function of the background variables. We show that a solution of this system for a given dispersion relation of linear waves is related to the quasiclassical limit of the Lax pair for the completely integrable equation having the corresponding dispersionless and linear dispersive behavior. We illustrate the theory with several examples.

The solution of sound propagation modeling problems for environment impact assessment by the mode parabolic equations methoda).

The method of sound propagation modeling based on the mode parabolic equations (MPEs) theory is applied to the verification scenarios for environmental impact assessment. The results for selected scenarios from the 2022 Cambridge Joint Industry Programme Acoustic Modelling Workshop and the configuration of the computational programs AMPLE and MPE for these scenarios is discussed. Furthermore, it is revealed how the results for these scenarios change in the case of the bottom slope across and along the propagation path. It is observed that for the cross-slope propagation scenario, the distribution of acoustic energy over decidecade frequency bands does not depend on the slope angle and is practically the same as that for range-independent environment. At the same time, the dependence of energy distribution is noticeable for up- and downslope propagation scenarios, where greater slope angles result in higher propagation loss. It is also shown that MPEs are capable of adequately handling typical sound propagation problems related to the environmental impact assessment for frequencies up to 1000 Hz. A possibility of using frequency-dependent mesh size and number of modes must be implemented in codes based on this approach.