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In Remarks on the Foundations of Mathematics, Wittgenstein notes that he has 'not yet made the role of miscalculating clear' and that 'the role of the proposition: "I must have miscalculated"...is really the key to an understanding of the "foundations" of mathematics.' In this paper, I hope to get clear on how this is the case. First, I will explain Wittgenstein's understanding of a 'foundation' for mathematics. Then, by showing how the proposition 'I must have miscalculated' differentiates mathematics from the physical sciences, we will see how this proposition is the key to understanding the foundations of mathematics. [ABSTRACT FROM AUTHOR]

Shape memory polymers (SMPs) are one of the intriguing functional materials and have been widely and intensively studied. In order to apply these new polymers to load bearing engineering structures and devices, developing physics-based thermomechanical constitutive models is mandatory. The aim of this Tutorial is to demonstrate how to establish a thermomechanical constitutive model for SMPs. It begins with classifications of SMPs, followed by a discussion on the underlying physics for different SMPs. After that, three classical SMP thermomechanical modeling frameworks are introduced, which include the visco-elasto-plastic based rheological framework, the storage strain-based phase transition framework, and the representative unit cell based multi-branch framework. Next, three commonly adopted new model establishment methods are presented within these frameworks with detailed examples. Finally, future perspectives on this research direction are discussed. We hope that this Tutorial will help readers understand the roadmap from physics to mathematical modeling of SMPs. [ABSTRACT FROM AUTHOR]

This article provides a qualitative analysis of simple harmonic motion (SHM) using elementary knowledge in kinematics, Newton's second law, Hooke's law, and mechanical energy conservation. The analysis focuses on sketching the velocity vs. time curve, which is largely determined by the fact that the tangent line slope on this curve is the instantaneous acceleration. While the obtained curve is not proven to be sinusoidal, it visually appears to be. The qualitative approach complements rigorous mathematical approaches and provides rich physics content. The article also discusses the preparation and details of curve sketching for the velocity vs. time curve. Overall, this qualitative approach can be used to introduce SHM to students and help them understand the motion without relying on complex mathematics. [Extracted from the article]

The article is an editorial introducing a special issue of the scientific journal Axioms titled "Mathematical Models and Simulations." The issue contains 13 papers that cover various mathematical models and simulations in fields such as physics, biology, finance, and engineering. The papers explore different types of mathematical models, including deterministic and stochastic models, and discuss their applications and numerical simulations. The authors provide a brief overview of each paper, highlighting their main findings and methodologies. The article aims to provide library patrons with a concise summary of the special issue, allowing them to decide if they want to explore the individual papers further. [Extracted from the article]

In this study, we present and examine a fractional integral operator with an I$$ I $$‐function in its kernel. This operator is used to solve several fractional differential equations (FDEs). FDE has a set of particular cases whose solutions represent different physical phenomena. Much mathematical physics, biology, engineering, and chemistry problems are identified and solved using FDE. We first solve the FDE and the integral operator for the incomplete I$$ I $$‐function (I I$$ I $$F) for the generalized composite fractional derivative (GCFD). This is followed by the discovery and investigation of several important exceptional cases. The significant finding of this study is a first‐order integer‐differential equation of the Volterra type that clearly describes the unsaturated nature of free‐electron lasers. [ABSTRACT FROM AUTHOR]

According to classical physics, particles are basic constituents of the physical world. Quantum theory is much less friendly to particles; in particular, relativistic quantum field theory (RQFT) creates serious obstacles for the idea that particles are fundamental. Apparently, when moving from the domain of RQFT to that of classical mechanics (CM), particles have to emerge at some stage. It is standard to assume that this emergence has been completed at the level of quantum mechanics, halfway between RQFT and CM, even though particles of the same kind in many-particle quantum mechanics have the curious feature of being 'entities without identity'. Against this 'Received View' about the nature of quantum particles we outline and defend an Alternative View (AV), in which the emergent character of particles is emphasized. According to this AV, the step to a particle theory has not yet been made in quantum mechanics: conditions have still to be satisfied in order to make the particle concept applicable. If these conditions are met, the quantum particles that emerge are distinguishable individuals possessing physically defined identities, in stark contrast to what the Received View asserts. We will compare and contrast the two Views, both from a physical and a logical/conceptual point of view. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'. [ABSTRACT FROM AUTHOR]

Identical quantum subsystems can possess a property which does not have any classical counterpart: indistinguishability. As a long-debated phenomenon, identical particles' indistinguishability has been shown to be at the heart of various fundamental physical results. When concerned with the spatial degree of freedom, identical constituents can be made indistinguishable by overlapping their spatial wave functions via appropriately defined spatial deformations. By the laws of quantum mechanics, any measurement designed to resolve a quantity which depends on the spatial degree of freedom only and performed on the regions of overlap is not able to assign the measured outcome to one specific particle within the system. The result is an entangled state where the measured property is shared between the identical constituents. In this work, we present a coherent formalization of the concept of deformation in a general N -particle scenario, together with a suitable measure of the degree of indistinguishability. We highlight the basic differences with non-identical particles scenarios and discuss the inherent role of spatial deformations as entanglement activators within the spatially localized operations and classical communication operational framework. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'. [ABSTRACT FROM AUTHOR]

There is an extensive philosophical literature on the interrelated issues of identity, individuality and distinguishability in quantum systems. A key consideration is whether quantum systems are subject to a strong form of individuality termed 'haecceity' (from the Latin for 'this-ness'). I argue that the traditional, strong form of haecceity does not apply at the quantum level, but that in order to properly account for the need for symmetrization in quantum systems, a weaker kind of haecceity must be involved, which I call quantum haecceity. In the process, I also question some generally accepted tenets of the current debate, such as the idea that symmetrization of states for identical quanta must be postulated and reflects permutation invariance. Instead, I note that a perturbative Hamiltonian is required for exchange effects, which suggests that the need for symmetrization arises from specific physical conditions. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'. [ABSTRACT FROM AUTHOR]

The study of entanglement in systems composed of identical particles raises interesting challenges with far-reaching implications in both, our fundamental understanding of the physics of composite quantum systems, and our capability of exploiting quantum indistinguishability as a resource in quantum information theory. Impressive theoretical and experimental advances have been made in the last decades that bring us closer to a deeper comprehension and to a better control of entanglement. Yet, when it involves composites of indistinguishable quantum systems, the very meaning of entanglement, and hence its characterization, still finds controversy and lacks a widely accepted definition. The aim of the present paper is to introduce, within an accessible and self-contained exposition, the basic ideas behind one of the approaches advanced towards the construction of a coherent definition of entanglement in systems of indistinguishable particles, with focus on fermionic systems. We also inquire whether the corresponding tools developed for studying entanglement in identical-fermion systems can be exploited when analysing correlations in distinguishable-party systems, in which the complete information of the individual parts is not available. Further, we open the discussion on the broader problem of constructing a suitable framework that accommodates entanglement in the presence of generalized statistics. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'. [ABSTRACT FROM AUTHOR]

This article follows an unorthodox approach to the individuation of quantum particles of the same type. According to this approach, individuation of the components of a composite system of identical particles is done not with the help of unphysical labels (indices) but physically meaningful projection operators. This unorthodox conception requires a modification of the standard notion of entanglement, in order to exclude states whose non-factorizability arises merely from the (anti)symmetrization of a product state. I will report several facts regarding the connections of the modified concept of entanglement with the issue of discernibility. I will also discuss recent experiments involving measurement-induced entanglement, and I will point out that they do not threaten the cogency of the new concept of entanglement. The statistical correlations observed in these experiments are explainable not by the entanglement of the initial state but by the creation of a new, genuinely entangled state by means of a pre-measurement selection. Finally, I will identify and discuss a genuine difference between non-entangled but non-factorizable states of identical particles and ordinary product states, which is that the former but not the latter admits an infinity of alternative and incompatible individuations by single-particle properties. This phenomenon can be accounted for using the concept of emergent particles proposed by Dennis Dieks. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'. [ABSTRACT FROM AUTHOR]

In this brief survey, we discuss some of the scientific and philosophical problems and debates that underlie the notions of identity, individuality and indistinguishability in physics and mathematics. We critically analyse the different positions for or against the existence of indistinguishable objects in different scientific theories, notably quantum mechanics and gauge theories in physics and homotopy type theory in mathematics. We argue that the different forms of indistinguishability that occur in many areas of physics and mathematics—far from being a problem to be eradicated—exhibit a rich formal structure that plays a key role in the corresponding theories that needs to be properly understood. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'. [ABSTRACT FROM AUTHOR]

We present what Aristotle wrote on identity in a leisurely manner, which is much more than is generally known, save among the cognoscenti (Aristotle scholars), and mutatis mutandis about the introduction of the identity-symbol (=). We add two codas, one on the so-called Leibniz' Law, which is different (but resembles) what passes for it in logic and philosophy, and one on the status of identity, as accepted by mathematicians and logicians, in physics. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'. [ABSTRACT FROM AUTHOR]

While entanglement in the case of distinguishable particles is clearly understood, in the case of indistinguishable particles, there is still a wide debate regarding basic conceptual issues. Here, I will address two such issues. First, the debate almost always rests on a differentiated treatment of the cases of distinguishability and indistinguishability, even with respect to the definition of the concept of entanglement. Second, the fact that symmetrized and antisymmetrized states of indistinguishable particles are non-factorizable leads to certain perplexities about the very nature of entanglement. In the present work, both issues will be addressed from the perspective of an ontology of properties, which finds its natural expression in the algebraic formalism of quantum mechanics. The final goal will be to show that the proposed ontology allows a unified treatment of entanglement in the distinguishability and indistinguishability cases and offers a straightforward way out to perplexities. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'. [ABSTRACT FROM AUTHOR]

We shall propose a conceptual-oriented discussion of the so-called Univalent Foundations Program, that is, of Martin–Löf type theory enriched with a homotopic interpretation, together with the univalence axiom proposed by Voevodsky. We shall argue that the type-theoretic notion of propositional equality encodes the notion of indiscernibility, we shall address the homotopic interpretation of Martin–Löf type theory, and we shall analyse whether Leibniz's principle of the identity of indiscernibles holds or not in Univalent Foundations. We shall finally argue that univalence can be understood as a particular implementation of a constructive notion of abstraction that resolves Fregean abstraction. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'. [ABSTRACT FROM AUTHOR]

In this research, we investigate the integrability properties of the Schamel–Korteweg–de Vries (S-KdV) equation, which is important for understanding the effect of electron trapping in the nonlinear interaction of ion-acoustic waves. Using the optimal system, we come over reduced ordinary differential equations (ODEs). To deal with reduced ODEs for this problem, Lie symmetry analysis is combined with the modified auxiliary equation (MAE) procedure and the generalized Jacobi elliptic function expansion (JEF) method. The analytical solutions reported here are novel and have a wide range of applications in mathematical physics. [ABSTRACT FROM AUTHOR]

The fractional coupled Konno-Onno model, which is frequently used in numerous fields of scientific and engineering disciplines, is being investigated in the current study in order to gain an understanding of complex phenomena and systems. The two main goals of this study are to be accomplished. Firstly, the research aims to identify novel solitons for the fractional coupled Konno-Onno model using the unified technique, which is currently absent from the literature. Secondly, a novel strategy that hasn't been previously investigated is phase portrait analysis for both perturbed and non-perturbed dynamical systems. The current study uses appropriate parametric values in phase plane analysis, 2D, 3D, and density plots to ensure the results are physically compatible. The results validate the claim that the technique used in this research to produce complete and uniform responses is not only simple to use and effective, but also substantially faster in computing. The technique is useful for resolving more complex phenomena that arise in engineering and mathematical physics. [ABSTRACT FROM AUTHOR]

Chemical graph theory is a well-established discipline within chemistry that employs discrete mathematics to represent the physical and biological characteristics of chemical substances. In the realm of chemical compounds, graph theory-based topological indices are commonly employed to depict their geometric structure. The main aim of this paper is to investigate the degree-based topological indices of dominating David derived networks (DDDN) and assess their effectiveness. DDDNs are widely used in analyzing the structural and functional characteristics of complex networks in various fields such as biology, social sciences, and computer science. We considered the FN*, M 2 ∗ , and HM N topological indices for DDDNs. Our computations' findings provide a clear understanding of the topology of networks that have received limited study. These computed indices exhibit a high level of accuracy when applied to the investigation of QSPRs and QSARs, as they demonstrate the strongest correlation with the acentric factor and entropy. [ABSTRACT FROM AUTHOR]

A generalized version of the abcd -Boussinesq class of systems is derived to accommodate variable bottom topography in two-dimensional space. This extension allows for the conservation of suitable energy functionals in some cases and enables the description of water waves in closed basins with well-justified slip–wall boundary conditions. The derived systems possess a form that ensures their solutions adhere to important principles of physics and mathematics. By demonstrating their consistency with the Euler equations and estimating their approximation error, we establish the validity of these new systems. Their derivation is based on the assumption of small bathymetric variations. With practical applications in mind, we assess the effectiveness of some of these new systems through comparisons with standard benchmarks. The results indicate that the assumptions made during the derivation are not overly restrictive. The applications of the new systems encompass a wide range of scenarios, including the study of tsunamis, tidal waves and waves in ports and lakes. [ABSTRACT FROM AUTHOR]

In recent years. Electrorheological Fluids have been paid more attention because of their smart nature. This article presents the mathematical investigation of electrorheological fluid and its applications for study of macroscopic structure of the electric particles. Electrorheological fluid is a special viscous fluid, consisting of solid electric particles and that shows significant changes. This article provides insight into the relationship between the macroscopic properties of an electric particles and microscopic structural changes. [ABSTRACT FROM AUTHOR]

The fractional coupled nonlinear Schrödinger model (FCNLSM) is widely utilized in various fields such as nonlinear optics, optical communication systems, plasmas and mathematical physics. In this study, we aim to achieve three primary objectives. Firstly, we seek to obtain novel soliton solutions for the FCNLSM, which have not been previously reported in the literature. Secondly, we employ the Sardar sub-equation method and the improved generalized tanh-function method to effectively analyze the dynamics of solutions and solve the studied model. These methods provide valuable insights into the behavior of the system. Lastly, we conduct bifurcation and stability analyses to explore the dynamical properties of the model. To ensure the physical validity of our findings, we present 2-dimensional, 3-dimensional and contour plots using carefully selected parameter values. The obtained results demonstrate the feasibility, efficiency and computational speed of the employed techniques in obtaining comprehensive and reliable solutions. The study represents a novel and significant contribution to the field by expanding the understanding of soliton solutions in the FCNLSM, introducing new techniques for their investigation and conducting a comprehensive analysis of bifurcation and stability properties. The findings of this research open new avenues for exploration and application in the areas of nonlinear optics, optical communication systems and other related fields. [ABSTRACT FROM AUTHOR]

This article examines how shocks and three-dimensional nonlinear dust-ion-acoustic waves propagate across uniform magnetized electron–positron–ion plasmas. The two-variable  (G ′ / G , 1 / G) -expansion and generalized exp( − ϕ (ξ) )-expansion techniques are presented to construct the ion-acoustic wave results of a (3+1)-dimensional extended Zakharov–Kuznetsov (eZK) model. As a result, the novel soliton and other wave solutions in a variety of forms, including kink- and anti-kink-type breather waves, dark and bright solitons, kink solitons, and multi-peak solitons, etc., are attained. With the help of software, the solitary wave results (that signify the electrostatic potential field), electric and magnetic fields, and quantum statistical pressures are also constructed. These solutions have numerous applications in various areas of physics and other areas of applied sciences. Graphical representations of some of the obtained results, and the electric and magnetic fields as well as the electrostatic field potential are also presented. These results demonstrate the effectiveness of the presented techniques, which will also be useful in solving many other nonlinear models that arise in mathematical physics and several other applied sciences fields. [ABSTRACT FROM AUTHOR]

The Potts model with invisible states was introduced to explain discrepancies between theoretical predictions and experimental observations of phase transitions in some systems where Z q symmetry is spontaneously broken. It differs from the ordinary q-state Potts model in that each spin, besides the usual q visible states, can be also in any of r so-called invisible states. Spins in an invisible state do not interact with their neighbours, but they do contribute to the entropy of the system. As a consequence, an increase in r may cause a phase transition to change from second to first order. Potts models with invisible states describe a number of systems of interest in physics and beyond and have been treated by various tools of statistical and mathematical physics. In this paper, we aim to give a review of this fundamental topic. [ABSTRACT FROM AUTHOR]

Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-Kähler surfaces of Kodaira dimension κ ≥ 0. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized Kähler structures on these spaces. [ABSTRACT FROM AUTHOR]

The Extended Jacobi Elliptic Function Technique (EJET) is a powerful technique for finding the solutions for traveling waves form coming from Non-Linear Waveguides (NLWs). As a result, solitary and shock-wave profiles are obtained simultaneously with corresponding amplitudes and speeds by this method for three types of nonlinear wave equations. A class of nonlinear wave equations of particular interest in mathematical physics have been used to investigate the legality and credibility of this technique. A short script is considered a symbolic software package that calculates traveling wave solutions in exact form. [ABSTRACT FROM AUTHOR]

The paper aims to construct optical solitons and travelling wave solutions to two birefringent nonlinear models which consist of two-component form of vector solitons in optical fibre: the Biswas–Arshed model with Kerr-type nonlinearity and without four-wave mixing terms and the nonlinear Schrödinger equation with quadratic-cubic law of refractive index along with four-wave mixing terms. These nonlinear Schrödinger equations are applied in many physical and engineering fields. Optical solitons are considered in the context of photonic crystal fibres, couplers, polarisation-preserving fibres, metamaterials, birefringent fibres, and so on. Two reliable integration architectures, namely, the extended simplest equation method and the generalised sub-ODE approach, are adopted. As a result, bright soliton, kink and dark soliton, singular soliton, hyperbolic wave, a periodic wave, elliptic function solutions of Weierstrass and Jacobian types, and other travelling wave solutions, such as breather solutions and optical rogons, are derived, together with the existence conditions. In addition, the amplitude and intensity diagrams are portrayed by taking appropriate values for a few selected solutions. Furthermore, based on linear stability analysis, the modulation instability was explored for the obtained steady-state solutions. The reported results of this paper can enrich the dynamical behaviours of the two considered nonlinear models and can be useful in many scientific fields, such as mathematical physics, mathematical biology, telecommunications, engineering and optical fibres. This study confirms that the proposed approaches are sufficiently effective in extracting a variety of analytical solutions to other nonlinear models in both engineering and science. [ABSTRACT FROM AUTHOR]

In this paper, we propose two efficient methods for solving the fractional-order Schrödinger–KdV system. The first method is the Laplace residual power series method (LRPSM), which involves expressing the solution as a power series and using residual correction to improve the accuracy of the solution. The second method is a new iterative method (NIM) that simplifies the problem and obtains a recursive formula for the solution. Both methods are applied to the Schrödinger–KdV system with fractional derivatives, which arises in many physical applications. Numerical experiments are performed to compare the accuracy and efficiency of the two methods. The results show that both methods can produce highly accurate solutions for the fractional Schrödinger–KdV system. However, the new iterative method is more efficient in terms of computational time and memory usage. Overall, our study demonstrates the effectiveness of the residual power series method and the new iterative method in solving fractional-order Schrödinger–KdV systems and provides a valuable tool for researchers and practitioners in applied mathematics and physics. [ABSTRACT FROM AUTHOR]

In addition, this Special Issue addressed other qualitative properties of linear and nonlinear equations and systems of mathematical physics, such as scattering theory, inverse problems, variational methods, and variational calculus. One paper, [[11]], deals with the problem of determining the Hill equation (or the one-dimensional Schrödinger equation) based on its spectrum and deriving the Hill equation from the specific properties of its discriminant. The main approach to studying the problem under consideration was based on the spectral theory of differential operators as well as on the properties of the spectrum HT ht of the one-dimensional Schrödinger operator HT ht with periodic coefficients HT ht and HT ht . B Differential Equations of Mathematical Physics b : In the section on Differential Equations of Mathematical Physics, questions related to the solvability, regularity, stability, and asymptotic behavior of solutions to the equations of mathematical physics and PDE, including the hydrodynamic (Stokes equations) and Helmgotz equations, were proposed for consideration. [Extracted from the article]

The Fermi–Pasta–Ulam–Tsingou (FPUT) problem addresses fundamental questions in statistical physics, and attempts to understand the origin of recurrences in the system have led to many great advances in nonlinear dynamics and mathematical physics. In this work, we revisit the problem and study quasiperiodic recurrences in the weakly nonlinear α -FPUT system in more detail. We aim to reconstruct the quasiperiodic behavior observed in the original paper from the canonical transformation used to remove the three-wave interactions, which is necessary before applying the wave turbulence formalism. We expect the construction to match the observed quasiperiodicity if we are in the weakly nonlinear regime. Surprisingly, in our work, we find that this is not always the case and in particular, the recurrences observed in the original paper cannot be constructed by our method. We attribute this disagreement to the presence of small denominators in the canonical transformation used to remove the three-wave interactions before arriving at the starting point of wave turbulence. We also show that these small denominators are present even in the weakly nonlinear regime, and they become more significant as the system size is increased. We also discuss our results in the context of the problem of equilibration in the α -FPUT system and point out some mathematical challenges when the wave turbulence formalism is applied to explain thermalization in the α -FPUT problem. We argue that certain aspects of the α -FPUT system such as thermalization in the thermodynamic limit and the cause of quasiperiodicity are not clear, and that they require further mathematical and numerical studies. [ABSTRACT FROM AUTHOR]

Up until now, inspired by the formulation of a local form of the classical Second Law of Thermodynamics, quantum dynamic versions of the Second Law of Thermodynamics have been developed [[13], [26], [28], [30], [32], [34], [36]] using the Von Neumann quantum entropy I S i SB VN sb applied to quantum open thermodynamic systems. The aim of this Special Issue is to collect articles on new trends, ideas, and perspectives about unsolved and recent debated aspects of the Second Law of Thermodynamics in its classical, local, quantum, relativistic, and quantum relativistic formulations and about its related subjects. The Second Law of Thermodynamics represents a milestone in the history of not only physics but also chemistry, engineering, and, more generally, life and natural sciences. If recurring to the proper coordinates and to a local observer, the Second Law of Thermodynamics can be written as a four-dimensional expression in terms of entropy density, not only in special relativity but also in general relativity with, in the latter case, the additional replacement of the ordinary divergence operator with the contracted covariant derivative [[24]]. [Extracted from the article]

A huge number of mathematical models are called Boussinesq-type equations; therefore, a wide range of sixth-order Boussinesq-type equations attracts a lot of attention from outside researchers around the world. The paper studies the classical solution of one nonlinear inverse boundary value problem for linearized Boussinesq equation of the sixth order with an additional integral condition. One method is based on the application of the Fourier method. The other method is the application of the method of contracted mappings. The essence of the problem is that it is required, together with the solution, to determine the unknown kernel. The problem is considered in a rectangular area. When solving the original inverse boundary value problem, a transition from the original inverse problem to some auxiliary inverse problem is carried out. The existence and uniqueness of a solution to an auxiliary problem are proved with the help of contracted mappings. Then a transition to the original inverse problem is made again; as a result, a conclusion is made about the solvability of the original inverse problem. The proposed methods for finding solutions to the inverse problem can be used in the study of solvability for various problems of mathematical physics. [ABSTRACT FROM AUTHOR]

In various fields of science and engineering, such as financial mathematics, mathematical physics models, and radiation transfer, stochastic integral equations are important and practical tools for modeling and describing problems. Due to the existence of random factors, we face a fundamental problem in solving stochastic integral equations, and that is the lack of analytical solutions or the great complexity of these solutions. Therefore, finding an efficient numerical solution is essential. In this paper, we intend to propose and study a new method based on the Floater-Hormann interpolation and the spectral collocation method for linear and nonlinear stochastic Itô-Volterra integral equations (SVIEs). The Floater-Hormann interpolation offers an approximation regardless of the distribution of the points. Therefore, this method can be mentioned as a meshless method. The presented method reduces SVIEs under consideration into a system of algebraic equations that can be solved by the appropriate method. We presented an error bound to be sure of the convergence and reliability of the method. Finally, the efficiency and the applicability of the present scheme are investigated through some numerical experiments. [ABSTRACT FROM AUTHOR]

In this work, we will try to find lump solutions, interaction between lump wave and solitary wave solutions, kink-solitary wave solutions and shock wave-type solutions to (3 + 1) -dimensional generalized nonlinear evolution equation arising in the shallow water waves. The lump solutions, the interaction between lump wave and solitary wave solutions and kink-solitary wave solutions are derived with symbolic computation based on a logarithmic derivative transform which is derived by the help of Hirota's simple method. The shallow water waves in this equation are associated with some natural problems such as tides, storms, atmospheric currents and tsunamis. For the physical presentation of the solutions, we draw 3D and counter graphics by giving the suitable values to include the free parameters. We believe that disciplines such as mathematical physics, nonlinear dynamics, fluid mechanics and engineering sciences can benefit from this study. [ABSTRACT FROM AUTHOR]

Context. To directly image rocky exoplanets in reflected (polarized) light, future space- and ground-based high-contrast imagers and telescopes aim to reach extreme contrasts at close separations from the star. However, the achievable contrast will be limited by reflection-induced polarization aberrations. While polarization aberrations can be modeled with numerical codes, these computations provide little insight into the full range of effects, their origin and characteristics, and possible ways to mitigate them. Aims. We aim to understand polarization aberrations produced by reflection off flat metallic mirrors at the fundamental level. Methods. We used polarization ray tracing to numerically compute polarization aberrations and interpret the results in terms of the polarization-dependent spatial and angular Goos-Hänchen and Imbert-Federov shifts of the beam of light as described with closed-form mathematical expressions in the physics literature. Results. We find that all four beam shifts are fully reproduced by polarization ray tracing. We study the origin and characteristics of the shifts as well as the dependence of their size and direction on the beam intensity profile, incident polarization state, angle of incidence, mirror material, and wavelength. Of the four beam shifts, only the spatial Goos-Hänchen and Imbert-Federov shifts are relevant for high-contrast imagers and telescopes because these shifts are visible in the focal plane and create a polarization structure in the point-spread function that reduces the performance of coronagraphs and the polarimetric speckle suppression close to the star. Conclusions. Our study provides a fundamental understanding of the polarization aberrations resulting from reflection off flat metallic mirrors in terms of beam shifts and lays out the analytical and numerical tools to describe these shifts. The beam shifts in an optical system can be mitigated by keeping the f-numbers large and angles of incidence small. Most importantly, mirror coatings should not be optimized for maximum reflectivity, but should be designed to have a retardance close to 180°. The insights from our study can be applied to improve the performance of SPHERE-ZIMPOL at the VLT and future telescopes and instruments such as the Roman Space Telescope, the Habitable Worlds Observatory, GMagAO-X at the GMT, PSI at the TMT, and PCS (or EPICS) at the ELT. [ABSTRACT FROM AUTHOR]

A comprehensive framework has been developed to apply the monomiality principle from mathematical physics to various mathematical concepts from special functions. This paper presents research on a novel family of multivariate Hermite polynomials associated with Apostol-type Frobenius–Euler polynomials. The study derives the generating expression, operational rule, differential equation, and other defining characteristics for these polynomials. Additionally, the monomiality principle for these polynomials is verified. Moreover, the research establishes series representations, summation formulae, and operational and symmetric identities, as well as recurrence relations satisfied by these polynomials. [ABSTRACT FROM AUTHOR]

This paper deals with the study of the existence and multiplicity of solutions for the class of nonlocal problems that have arised in recent developments in the mathematical physics of string theory and cosmology given by (P) { − Δ e − c Δ u + u = λ P (x) (u + f (x , u)) , in R N lim | x | → ∞ ⁡ u (x) = 0 , u ∈ H c , ∞ (R N) , where N ≥ 3 , c > 0 , λ > 0 , P : R N → R is a positive continuous function, f : R N × R → R is C 1 -function, e − c Δ is defined via a power series and H c , ∞ (R N) is a Hilbert space as introduced in [11]. The main tools used here are: the Minimax Theorems and a bifurcation result via variational methods due to Rabinowitz. [ABSTRACT FROM AUTHOR]

This paper addresses discontinuities in the solutions of mathematical physics that describe actual processes and are not observed in experiments. The appearance of discontinuities is associated in this paper with the classical differential calculus based on the analysis of infinitesimal quantities. Nonlocal functions and nonlocal derivatives, which are not specified, in contrast to the traditional approach to a point, but are the results of averaging over small but finite intervals of the independent variable are introduced. Classical equations of mathematical physics preserve the traditional form but include nonlocal functions. These equations are supplemented with additional equations that link nonlocal and traditional functions. The proposed approach results in continuous solutions of the classical singular problems of mathematical physics. The problems of a string and a circular membrane loaded with concentrated forces are used to demonstrate the procedure. Analytical results are supported with experimental data. [ABSTRACT FROM AUTHOR]

HT ht I is i B diophantine b I if for some i I > 0 i I and i I > 0 i I it belongs to (see Fig. So, we can consider it as a I submanifold i HT ht . One can smoothly embed every manifold I P i into some vector space HT ht . Obviously, the flow on HT ht equals SB I t i sb ( I i ) = I i + I t i (mod 2 I i ). [Extracted from the article]

Canonical equations play a pivotal role in various sub-fields of physics and mathematics. However, for complex systems and systems without first principles, deriving canonical equations analytically is quite laborious or might even be impossible. This work is devoted to automatedly distilling the canonical equations solely from random state data. The random state data are collected from stochastically excited, dissipative dynamical systems either experimentally or numerically, while other information, such as the system characterization itself and the excitations, is not needed. The identification procedure comes down to a nested optimization problem, and the explicit expressions of the momentum (density) functions and energy (density) functions are identified simultaneously. Three representative examples are investigated to illustrate its high accuracy of identification, the small requirement for data amount, and high robustness to excitations and dissipation. The identification procedure serves as a filter, filtering out nonconservative information while retaining conservative information, which is especially suitable for systems with unobtainable excitations. [ABSTRACT FROM AUTHOR]

In the present paper, we define the three cases of the geometric phase equations associated with a monochromatic linearly polarized light wave traveling along an optical fiber in three-dimensional Walker manifold (M, ge f). Walker manifolds have many applications in mathematics and theoretical physics. We are working in the context of a pseudo-Riemannian manifold (i.e. a manifold equipped with a non-degenerate arbitrary signature metric tensor). That is, we generalize the motion of the light wave in the optical fiber and the associated electromagnetic curves that describe the motion of a charged particle under the influence of an electromagnetic field over a Walker space defined as a pseudo-Riemannian manifold with a light-like distribution, parallel to the Levi-Civita junction. These manifolds (especially Lorentzian) are important in physics because of their applications in general relativity. Then, we obtain the Rytov curves related to the cases of geometric phase models. Moreover, we give some examples and visualize the evolution of the electric field along the optical fiber in (M, ge f) via MAPLE program. [ABSTRACT FROM AUTHOR]

This expository paper advocates an approach to physics in which "typicality" is identified with a suitable form of algorithmic randomness. To this end various theorems from mathematics and physics are reviewed. Their original versions state that some property Φ (x) holds for P-almost all x ∈ X , where P is a probability measure on some space X. Their more refined (and typically more recent) formulations show that Φ (x) holds for all P-random x ∈ X . The computational notion of P-randomness used here generalizes the one introduced by Martin-Löf in 1966 in a way now standard in algorithmic randomness. Examples come from probability theory, analysis, dynamical systems/ergodic theory, statistical mechanics, and quantum mechanics (especially hidden variable theories). An underlying philosophical theme, inherited from von Mises and Kolmogorov, is the interplay between probability and randomness, especially: which comes first? [ABSTRACT FROM AUTHOR]

This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using the widely debated results of surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. The relevance of these operators and the possibility to be involved in the modeling of an important class of real phenomena is once again revealed by their applications. The use of certain variational methods facilitates the complete solution of the problem using appropriate numerical methods and computational algorithms. Some theoretical results are involved to complete the solutions for a sequence of models issued from real phenomena drawing. [ABSTRACT FROM AUTHOR]

In this paper, we succeed at discovering the new exact wave solutions to the truncated M-fractional complex three coupled Maccari's system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many applications in nonlinear optics, fiber optics, deep water-waves, plasma physics, mathematical physics, fluid mechanics, hydrodynamics and engineering, where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots with the help of Maple software. The gained solutions are helpful for the further development of a concerned model. Finally, this technique is simple, fruitful and reliable to handle nonlinear fractional partial differential equations (NLFPDEs). [ABSTRACT FROM AUTHOR]

In this article, we discuss the findings of new developments in a class of new triangular functions that blend the quantity functions of the traditional triangular. Considering the significant role played by the triangular functions in applied mathematics, physics, and engineering, it is conceivable to predict that the theory of new triangular functions will provide us with additional interpretations and discoveries in mathematics and physics. The solutions which consider variable separation based on arbitrary functions are constructed to the (3+1)-dimensional Burgers model by presenting the Fibonacci Riccati technique and the linearly independent variable separation approach. This technique's fundamental concept is to describe the solution of the Burgers model as a polynomial in the Riccati Equation solution that satisfies the symmetrical hyperbolic and triangular Fibonacci functions. Depending on the choice of suitable functions for variable separation, an abundance of new localized solutions were obtained. Moreover, examples such as embedded solitons, rectangle-solitons, plateau-type ring solitons, taper-like solitons, and their interactions with each other, following the symmetrical hyperbolic and triangular Fibonacci functions, as well as the golden mean, could be explored. [ABSTRACT FROM AUTHOR]