Theory Of Equations Research Articles

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.

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.)"

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.

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.

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.

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.

Towards a probabilistic foundation of relativistic quantum theory: the one-body Born rule in curved spacetime

In this work, we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of relativistic quantum field theory (QFT) such as the ‘problem of infinities’ (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work, we consider the 1-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the 1-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the ‘problem of time’ in this approach and, therefore, serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved. We also provide an example to illustrate how this approach can in principle be employed to model particle creation and annihilation.

Comparison of integral equation theories of the liquid state

Comparison of integral equation theories of the liquid state

Interface-packing analysis of F1-ATPase using integral equation theory and manifold learning

It has been shown that the translational entropy of water plays a key role in biological processes such as protein folding and ligand binding. Under the physiological condition, tightly packed protein conformations like native structures are achieved so that the translational entropy of water is maximized. In this study, we investigate the rotation mechanism of a rotary protein motor, F1-ATPase, by analyzing the packing at the interfaces between the subunits. The packing at the interface between a subunit pair is analyzed using the change in the solvent entropy upon forming subunit pair, ∆S. It is found that as the γ subunit rotates, the ∆S value of a α-β subunit pair decrease because the interface packing becomes loose. However, because the interface packing of another α-β subunit pair becomes tighter upon the rotation, ∆S of this α-β subunit pair increases, leading to a compensation of the decrease in ∆S. Such compensation would be necessary to maximize the solvent entropy of F1-ATPase. In this study, packing at the interfaces between the subunits is also analyzed using a manifold-learning technique, and it is suggested a possibility that a qualitative estimation of the ∆S values of some α-β subunit pairs can be predicted using a manifold-learning technique.

Twisted Mahler Discrete Residues

Abstract Recently we constructed Mahler discrete residues for rational functions and showed they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $g(x^{p})-g(x)$ for some rational function $g(x)$ and an integer $p> 1$. Here we develop a notion of $\lambda $-twisted Mahler discrete residues for $\lambda \in \mathbb{Z}$, and show that they similarly comprise a complete obstruction to the twisted Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $p^{\lambda } g(x^{p})-g(x)$ for some rational function $g(x)$ and an integer $p>1$. We provide some initial applications of twisted Mahler discrete residues to differential creative telescoping problems for Mahler functions and to the differential Galois theory of linear Mahler equations.

Existence and uniqueness of the solution to initial and inverse problems for integro-differential heat equations with fractional load

This work is devoted to the unique solvability of the direct and inverse problems for a multidimensional heat equation with a fractional load in Holder spaces. In the problem under consideration, the loaded term is in the form of a fractional integral operator for the time variable. We prove the existence and uniqueness of the solution to these problems by the contraction mapping theorem and the theory of integral equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/64/abstr.html

Contributions to Generalized Oscillation Theory of Linear Hamiltonian Systems

In this paper we present several new contributions to the oscillation theory of linear differential equations, in particular of linear Hamiltonian systems, when the traditional Legendre condition is absent. Following our recent work (Discrete Contin. Dyn. Syst. 43(12):4139–4173, 2023), we introduce the multiplicity of a generalized right focal point and derive the corresponding local Sturmian separation theorem. We also examine the relation between the existence of finitely many generalized right focal points, or in the special case the nonexistence of generalized right focal points, with the Legendre condition. As the main tools we use new notions of the minimal multiplicities at a given point and the dual comparative index — an object from matrix analysis or differential geometry (Maslov index theory). Furthermore, we study local limit properties of the dual comparative index and the comparative index and apply them for deriving new oscillation results phrased in terms of the generalized right and left focal points. The investigation of the interplay between generalized right and left focal points leads to conditions characterizing the situation, when in the local Sturmian separation theorem the corresponding multiplicities attain the minimal possible value. This also provides a generalization of the concepts of the right and left proper focal point defined by Kratz (Analysis 23(2):163–183, 2003) and Wahrheit (Int. J. Differ. Equ. 2(2):221–244, 2007) to the setting, which does not impose the Legendre condition. The results are new even for completely controllable linear Hamiltonian systems, including the Sturm–Liouville differential equations of arbitrary even order.

Homogenization of a Thermoelastic Bristly Structure Immersed in a Thermofluid

The article considers the mathematical model describing the joint motion of a viscous compressible heat-conducting fluid and a thermoelastic plate with a fine two-level thermoelastic bristly microstructure attached to it. The bristly microstructure consists of a great amount of taller and shorter bristles, which are periodically located on the surface of the plate, and the model under consideration incorporates a small parameter, which is the ratio of the characteristic lengths of the microstructure and the entire plate. Using classical methods in the theory of partial differential equations, we prove that the initial-boundary value problem for the considered model is well-posed. After this, we fulfill the homogenization procedure, i.e., we pass to the limit as the small parameter tends to zero, and, as a result, we derive the effective macroscopic model in which the dynamics of the interaction of the ‘liquid–bristly structure’ is described by equations of two homogeneous thermoviscoelastic layers with memory effects. The homogenization procedure is rigorously justified by means of the Allaire–Briane three-scale convergence method. The developed effective macroscopic model can potentially find application in further mathematical modeling in biotechnology and bionics taking account of heat transfer.

The non-exponential decay characteristics of HTS NI-class coils in sudden discharge experiments

Abstract The decay of the voltage and magnetic field of the high-temperature superconductivity (HTS) no-insulation-class (NI-class) coil in the sudden discharge experiment is usually considered to be exponential. However, our experiment found that it is not accurate to describe the decay characteristics of the NI-class coil with a simple exponential function. Combined with equivalent circuit models of the NI-class coil, we use the theory of ordinary differential equations to explain the non-exponential decay characteristics and point out that it is closer to the matrix exponential characteristics. Based on this finding, we believe that the non-uniform turn-to-turn contact resistivity (TTCR) is an important cause of the non-exponential decay characteristics of the NI-class coil. Reverse thinking leads us to realize that the voltage decay curve of the NI-class coil in the sudden discharge experiment contains the distribution information of its TTCR. Therefore, we propose a method for measuring the TTCR of NI-class coils based on the matrix exponential model. The co-simulation combining the finite element model and the equivalent circuit model shows that the non-uniformly distributed TTCR has a great influence on the transient process of the NI-class coil, which makes the current and current density in the transient process vary greatly in different regions of the NI-class coil. Those findings mean that the simulation study of the NI-class coils and magnets with uniform TTCR usually used in the past may be far from the actual situation.