Integral Theorem Research Articles

Feynman-Kac formula for tempered fractional general diffusion equations driven by TFBM

In this article, we investigate the following stochastic tempered fractional general diffusion equation driven by tempered fractional Brownian motion (TFBM) ∂ t β , η u ( t , x ) = L u ( t , x ) + u ( t , x ) B ˙ H , λ ( t ) , ( t , x ) ∈ R + × R d , where ∂ t β , η denotes the Caputo tempered fractional derivative with order β ∈ ( 0 , 1 ) and tempered parameter η > 0, 𝔏 is the infinitesimal generator of some general time-homogeneous symmetric strong Markov process { X t } t ≥ 0 with the corresponding transition semigroup { P t , t ≥ 0 } defined by P t f(x): = 𝔼 x [f(X t )], and { B H , λ ( t ) } t ≥ 0 is a TFBM with Hurst index H ∈ ( 1 2 , 1 ) and tempering parameter λ > 0 which is independent of { X t } t ≥ 0 . First of all, a novel estimate of multiple stochastic integrals with respect to the inverse tempered β-stable subordinator D β, η (t) is presented, which greatly contributes to the stochastic analysis. Then after establishing the Fubini Theorem for the stochastic integrals with respect to TFBM B H, λ (t) and the inverse tempered β-stable subordinator D β, η (t), we show that the Feynman-Kac representation u ( t , x ) = E D [ P D β , η ( t ) f ( x ) e ∫ 0 t ∫ 0 t δ ( s − r ) d D β , η ( s ) d B H , λ ( r ) ] is the weak solution of the stochastic tempered fractional general diffusion equation driven by TFBM with the initial value f by means of the techniques of Malliavin calculus and convergence analysis. From the Feynman-Kac formula, several kinds of continuity of the solutions with respect to time and tempered parameter η of tempered fractional derivative are proved by using the scaling property of the inverse tempered β-stable subordinator D β, η (t) and the novel estimate of stochastic integrals with respect to B H, λ (t) and D β, η (t). In particular for the stochastic fractional general diffusion equation driven by fractional Brownian motion (FBM) but in the time fractional derivative case, we can also obtain the sharp upper and lower bounds for the Hölder regularity and the moments of the solution.

Residue Theorem and Logarithmic Residue Theorem

This passage is written for researching the application of residue theorem and Logarithmic Residue theorem. The residue theorem is a powerful tool for the analysis of complex functions. It also can be used to calculate the integral of the real functions. Residue theorem is the extension of the Cauchy’s integral theorem and Cauchy integral formula. Passage researched two theorems by defining the residue in two ways. Each of them is defined by Laurent series and defined by integral. Then the residue theorem is introduced with its definition and applications. The passage points out two instances for the applications of the residue theorem. After the residue theorem, the Logarithmic Residue theorem is listed with its definition and applications. There are also three instances for its applications. The residue theorem and Logarithmic Residue theorem mainly contribute for the research of the complex functions’ integral calculations. The residue theorem provides powerful mathematical tools in certain special types of real integration problems.

Deductions, Applications and Expansion of the Cauchy’s Residue Theorem

Cauchy’s integral formula is one of the most important discovery in the development history of the complex variable integration. This article focuses on the methods of the integration in mathematics. The author aims to write about how to deduce the residue theorem from Cauchy’s integral formula and how to use the Cauchy’s integral theorem and residue theorem in the integral questions. The following parts of the article include using method of limits and Euler’s formula to deduce roughly the Cauchy’s Integral formula. In addition, the author also uses the differentiation to transfer the integral function. In part three, the author calculates three different types of examples and considers that residue theorem and Cauchy’s integral formula are very useful in the contour integral with singularities. The article is in order to show that the practicability of these theorem and formula. These formulas and theorems are used in a wide range of the subjects and areas.

The First Mean Value Theorem of integral in Perplex Plane

Perplex numbers are an extension of the real numbers, consisting of pairs of real numbers that form a commutative ring. This article considers the partial order structure of perplex numbers and discusses the concept of perplex interval. Based on the definition of integration of perplex functions of perplex variable, this article we investigate the first mean value theorem of integral in perplex plane which can significantly enrich the mathematical toolbox by providing essential methods for estimating and approximating perplex number functions. It also simplifies the solution of complex integral problems involving perplex numbers and fosters theoretical extensions by exploring the behavior of perplex number functions under various conditions. This, in turn, promotes the development of perplex calculus, enhancing the overall theoretical and practical framework of perplex number analysis. In physics, particularly in relativity, studying the first mean value theorem for perplex numbers can provide more precise methods for describing and estimating physical quantities, enhancing the application of hyperbolic numbers in these fields.

EFFECT OF NONUNIFORM POROSITY IN CURVED BREAKWATER ON WATER WAVES

Abstract Assuming linear theory, the phenomenon of scattering of waves by a circular arc shaped barrier with nonuniform porosity is studied. The water region is considered to be of infinite or finite depth. Based on a judicious application of Green’s integral theorem, the corresponding boundary value problem is reduced to a hypersingular integral equation of second kind. The boundary element method and the collocation method are adopted to solve the hypersingular integral equation, and we ensure a good matching of the solutions obtained by the two methods. The reflection coefficient and energy dissipation are evaluated by using the solution of the integral equation which is then studied graphically. Different choices of distributions of pores on the barrier are considered, and we observe that the nonuniform porosity of the barrier has significant effect on the reflected wave and the energy dissipation.

The discrete fractional Karamata theorem and its applications

Abstract We establish a fractional extension of the discrete Karamata integration theorem for two types of fractional sum operators. This result, along with other properties of regularly varying sequences and the tools such as comparison theorems and a fixed point theorem in FK-space, is then used to study asymptotic properties of solutions to a nonlinear fractional difference equation. We also establish and utilize a fractional extension of the Stolz–Cesáro theorem, i.e., of the discrete l’Hospital rule. Both, the discrete fractional Karamata theorem and the discrete fractional l’Hospital rule, are believed to find applications in a broader context within the discrete fractional calculus.

Trace principle for Riesz potentials on Herz-type spaces and applications

We establish trace inequalities for Riesz potentials on Herz-type spaces and examine the optimality of conditions imposed on specific parameters. We also present some applications in the form of Sobolev-type inequalities, including the Gagliardo–Nirenberg–Sobolev inequality and the fractional integration theorem in the Herz space setting. In addition, we obtain a Sobolev embedding theorem for Herz-type Sobolev spaces.

The continuous Choquet integral representation theorems in an abstract setting

The purpose of the paper is to formulate Choquet integral representation theorems for a monotone functional on a collection of functions in such a way that the representing measures are simultaneously inner and outer continuous on appropriate collections of sets such as open, closed, compact, and measurable. This type of theorem is referred to as the continuous Choquet integral representation theorem and will be discussed in a setting general enough for practical use. The benefits of our results are as follows:(i)the representing measures are simultaneously inner and outer continuous,(ii)the collections of sets for which the representing measures are inner and outer continuous are larger than those in previous studies,(iii)the regularity of the representing measures is also considered, and(iv)it is possible to handle not only σ-continuous but also τ-continuous functionals.

Improved modeling for dark photon detection with dish antennas

A vector dark matter candidate, also known as dark photon, would induce an oscillating electric field through kinetic mixing. One detection strategy uses a spherical reflector to focus the induced emission at its center of curvature. First, we investigate the effects of diffraction in this type of experiment from an analytical standpoint, making use of the Kirchhoff integral theorem in the low-curvature dish limit. Next, we estimate the impact of mode matching, in the case of detection by a pyramidal horn antenna. We show that the expected signal intensity can be significantly reduced compared to usual estimates. Our method is applied to the reinterpretation of the SHUKET experiment data, the results of which are shown to be degraded by a factor of ∼50 due to both diffraction and mode matching. The analytical method allows optimizing some experimental parameters to gain sensitivity in future runs. Our results can be applied to any dish antenna experiment using a low-curvature reflector. Published by the American Physical Society 2024

Purity and 2-Calabi–Yau categories

For various 2-Calabi–Yau categories C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathscr{C}$\\end{document} for which the classical stack of objects M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{M}$\\end{document} has a good moduli space p:M→M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p\\colon \\mathfrak{M}\\rightarrow \\mathcal{M}$\\end{document}, we establish purity of the mixed Hodge module complex p!Q_M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p_{!}\\underline{{\\mathbb{Q}}}_{{\\mathfrak {M}}}$\\end{document}. We do this by using formality in 2CY categories, along with étale neighbourhood theorems for stacks, to prove that the morphism p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p$\\end{document} is modelled étale-locally by the semisimplification morphism from the stack of modules of a preprojective algebra. Via the integrality theorem in cohomological Donaldson–Thomas theory we then prove purity of p!Q_M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p_{!}\\underline{{\\mathbb{Q}}}_{{\\mathfrak {M}}}$\\end{document}. It follows that the Beilinson–Bernstein–Deligne–Gabber decomposition theorem for the constant sheaf holds for the morphism p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p$\\end{document}, despite the possibly singular and stacky nature of M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathfrak {M}}$\\end{document}, and the fact that p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p$\\end{document} is not proper. We use this to define cuspidal cohomology for M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathfrak {M}}$\\end{document}, which conjecturally provides a complete space of generators for the BPS algebra associated to C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathscr{C}$\\end{document}. We prove purity of the Borel–Moore homology of the moduli stack M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{M}$\\end{document}, provided its good moduli space ℳ is projective, or admits a suitable contracting C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathbb{C}}^{*}$\\end{document}-action. In particular, when M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{M}$\\end{document} is the moduli stack of Gieseker semistable sheaves on a K3 surface, this proves a conjecture of Halpern-Leistner. We use these results to moreover prove purity for several stacks of coherent sheaves that do not admit a good moduli space. Without the usual assumption that r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r$\\end{document} and d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d$\\end{document} are coprime, we prove that the Borel–Moore homology of the stack of semistable degree d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d$\\end{document} rank r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r$\\end{document} Higgs sheaves is pure and carries a perverse filtration with respect to the Hitchin base, generalising the usual perverse filtration for the Hitchin system to the case of singular stacks of Higgs sheaves.

Exploring the mechanisms behind dual-peak formation in energy harvesting efficiency of semi-passive flapping airfoils with varied spring stiffness

Flapping airfoil energy harvesting has emerged as a promising paradigm inspired by birds and marine organisms. In this investigation, a semi-passive flapping airfoil device with a predetermined pitching motion was examined using a transient numerical simulation method employing an overlapping grid technique. The impact of the spring stiffness on the energy harvesting characteristics of the flapping airfoil was analyzed through the implementation of the integral momentum theorem and dynamic mode decomposition. The results demonstrated that the efficiency and power curves of the flapping airfoil exhibited a distinctive bimodal M shape in response to variations of the spring stiffness. The initial peak efficiency point, which corresponded to the system's natural frequency in proximity to the pitching frequency, was found to be heavily influenced by the presence of leading-edge vortexes, thereby influencing lift generation. By utilizing the integral momentum theorem, it was determined that the Lamb vector term, fluid acceleration term, and total pressure term predominantly contributed to the lift generation. The emergence of the second peak efficiency point was primarily linked to torque variations induced by trailing-edge vortexes.

Noncoercive convex sweeping processes with velocity constraints

In this paper, we investigate noncoercive monotone convex sweeping processes with velocity constraints, a topic not previously investigated. Using some fundamental results on Bochner integration, the Tikhonov regularization method, a solution existence result for coercive convex sweeping processes with velocity constraints and a useful fact on measurable single-valued mappings, we prove the solution existence and properties of the solution set of noncoercive convex sweeping processes with velocity constraints under suitable conditions. These results represent a significant contribution, addressing a specific aspect of an open question posed in our recent work [Adly et al., Convex and nonconvex sweeping processes with velocity constraints: well-posedness and insights. Appl Math Optim. 2023;88(Paper No. 45)]. Additionally, we resolve two other open questions from the same paper concerning the behaviour of the regularized trajectories, relying on the Dominated Convergence Theorem for Bochner integration.

Integral theorems for the gradient of a vector field, with a fluid dynamical application

The familiar divergence and Kelvin–Stokes theorem are generalized by a tensor-valued identity that relates the volume integral of the gradient of a vector field to the integral over the bounding surface of the tensor product of the vector field with the exterior normal. The importance of this long-established yet relatively little-known result is discussed. In flat two-dimensional space, it reduces to a relationship between an integral over an area and that over its bounding curve, combining the two-dimensional divergence and Kelvin–Stokes theorems together with two related theorems involving the strain, as is shown through a decomposition using a suitable tensor basis. A fluid dynamical application to oceanic observations along the trajectory of a moving platform is given. The potential extension of the generalized two-dimensional identity to curved surfaces is considered and is shown not to hold. Finally, the paper includes a substantial background section on tensor analysis, and presents results in both symbolic notation and index notation in order to emphasize the correspondence between these two notational systems.

Rules-reduced fuzzy neural network-based learning control for multiple constraints robots using online identification and compensation methods

This paper presents a novel fuzzy-learning adaptive control approach for uncertain robotic systems with input dead zone and output saturation constraints. To address the input dead zone and output saturation and improve the control stability of the robot, an internal model compensation method was proposed, which utilizes online identification of an arctanh function to approximate the output saturation of actuators. The paper also introduces an adaptive learning control system based on a rules-reduced fuzzy neural network (RRFNN) algorithm, which considers the dead zone and output saturation function to enhance control performance, and an adaptive law driven by approximation mistakes is employed to handle multiple constraints. Furthermore, the controller utilizes the integral Lyapunov stability theorem and RRFNN to design the fuzzy-learning adaptive control law, ensuring global convergence and stability of the control system. Extensive simulations and experiments on a robotic manipulator are conducted to verify the feasibility of the proposed control method. Without the proposed method, the robot’s joint tracking error exceeds 0.5 rad, while with the proposed controller, it is less than 0.05 rad and does not violate output saturation constraints. Thus, the proposed method can realize the desired performance and overcome multiple constraints.

Semi-analytical approach to study wave interaction with a pair of obliquely submerged heterogeneous flexible structures

ABSTRACT Analyzing inclined flexible plates of varying thickness is crucial for optimising and attaining precise control over wave reflection and transmission. This is particularly significant in the context of breakwater construction. In contrast to horizontal breakwaters, inclined barriers can penetrate multiple layers of fluid with varying particle velocities, facilitating interactions and causing wave breaking, which results in the dissipation of wave energy. Moreover, compared to vertical structures, inclined ones demonstrate more effective wave attenuation. In the context of the present study, we investigate the interaction of water waves with a pair of obliquely submerged flexible thin plates characterised by non-uniform thickness. In order to model this complex physical scenario, we employ the principles of linear water wave theory in conjunction with Kirchhoff's thin plate theory. By employing repeated integration and Green's integral theorem, we reduce the boundary value problem to a system of coupled integral equations. Through appropriate approximations, we solve this system and obtain numerical values for various hydrodynamic quantities. This model provides comprehensive results for both horizontal and vertical plates, making it highly versatile. We examine the impact of varying thickness and the inclination of the two flexible plates to understand their roles in the wave scattering process and the associated physical parameters. The results obtained from this study shed light on the intricate dynamics involved in wave-structure interactions and can inform the design and optimisation of breakwater systems.

Integral fluctuation theorems and trace-preserving map.

The detailed fluctuation theorem implies symmetry in the generating function of entropy production probability. The integral fluctuation theorem directly follows from this symmetry and the normalization of the probability. In this paper, we rewrite the generating function by integrating measurements and evolution into a constructed mapping. This mapping is completely positive, and the original integral fluctuation theorem is determined by the trace-preserving property of these constructed maps. We illustrate the convenience of this method by discussing the eigenstate fluctuation theorem and heat exchange between two baths. This set of methods is also applicable to the generating functions of quasiprobability, where we observe the Petz recovery map arising naturally from this approach.

Universal scale laws for colors and patterns in imagery

Distribution of colors and patterns in images is observed through cascades that adjust spatial resolution and dynamics. Cascades of colors reveal the emergent universal property that Fully Colored Images (FCIs) of natural scenes adhere to the debated continuous linear log-scale law (slope −2.00±0.01) (L1). Cascades of discrete 2×2 patterns are derived from pixel square reductions onto the seven unlabeled rotation-free textures (0000, 0001, 0011, 0012, 0101, 0102, 0123). They exhibit an unparalleled universal entropy maximum of 1.74±0.013 at some dynamics regardless of spatial scale (L2). Patterns also adhere to the Integral Fluctuation Theorem (1.00±0.01) (L3), pivotal in studies of chaotic systems. Images with fewer colors exhibit quadratic shift and bias from L1 and L3 but adhere to L2. Randomized Hilbert fractal FCIs better match the laws than basic-to-AI-based simulations. Those results are of interest in Neural Networks, out-of-equilibrium physics, and spectral imagery.

Parameter estimation for the stochastic heat equation with multiplicative noise from local measurements

For the stochastic heat equation with multiplicative noise we consider the problem of estimating the diffusivity parameter in front of the Laplace operator. Based on local observations in space, we first study an estimator, derived in Altmeyer and Reiß (2021) for additive noise. A stable central limit theorem shows that this estimator is consistent and asymptotically mixed normal. By taking into account the quadratic variation, we propose two new estimators. Their limiting distributions exhibit a smaller (conditional) variance and the last estimator also works for vanishing noise levels. The proofs are based on local approximation results to overcome the intricate nonlinearities and on a stable central limit theorem for stochastic integrals with respect to cylindrical Brownian motion. Simulation results illustrate the theoretical findings.

A Generalization of Wayment’s Mean Value Theorem for Integrals

In this short note, we present a generalization of Wayment’s Mean Value Theorem for Integrals.

Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field

In algebra, the sedenions, an extension of the octonion system, form a 16-dimensional noncommutative and nonassociative algebra over the real numbers. It can be expressed as two octonions, and a function and differential operator can be defined to treat the sedenion, expressed as two octonions, as a variable. By configuring elements using the structure of complex numbers, the characteristics of octonions, the stage before expansion, can be utilized. The basis of a sedenion can be simplified and used for calculations. We propose a corresponding Cauchy–Riemann equation by defining a regular function for two octonions with a complex structure. Based on this, the integration theorem of regular functions with a sedenion of the complex structure is given. The relationship between regular functions and holomorphy is presented, presenting the basis of function theory for a sedenion of the complex structure.