Multiple Integrals Research Articles

Analytical properties and related inequalities derived from multiplicative Hadamard [formula omitted]-fractional integrals

The present article is intended to address the properties and associated inequalities of multiplicative Hadamard k-fractional integrals. The core concept lies in introducing the multiplicative Hadamard k-fractional integrals. In this framework, various analytical characteristics they possess, such as ∗integrability, continuity, commutativity, semigroup property, boundedness, and others, are examined herein. Subsequently, the Hermite–Hadamard-analogous inequalities are formulated for the novelly constructed operators. Meanwhile, an identity is inferred within multiplicative Hadamard k-fractional integrals, based on which a series of Bullen-type inequalities are derived in this article, where the function Λ∗ is GG-convex and the function (lnΛ∗)s is GA-convex for s>1, with a particular focus on discussing the case when 0<s≤1. To facilitate a more profound understanding of the outcomes, we offer illustrative examples together with numerical simulations to confirm the consistency of the theoretical results. Finally, applications of the proposed results in multiplicative differential equations, quadrature formulas, and special means for real numbers are investigated as well.

Product formulas for multiple stochastic integrals associated with Lévy processes

AbstractIn the present paper, we obtain an explicit product formula for products of multiple integrals w.r.t. a random measure associated with a Lévy process. As a building block, we use a representation formula for products of martingales from a compensated-covariation stable family. This enables us to consider Lévy processes with both jump and Gaussian part. It is well known that for multiple integrals w.r.t. the Brownian motion such product formulas exist without further integrability conditions on the kernels. However, if a jump part is present, this is, in general, false. Therefore, we provide here sufficient conditions on the kernels which allow us to establish product formulas. As an application, we obtain explicit expressions for the expectation of products of iterated integrals, as well as for the moments and the cumulants for stochastic integrals w.r.t. the random measure. Based on these expressions, we show a central limit theorem for the long time behaviour of a class of stochastic integrals. Finally, we provide methods to calculate the number of summands in the product formula.

Methodological insights into soil elemental nickel in typical Karst areas: comprehensive analysis of geochemical characteristics, source determination, and influencing factors.

Excessive levels of Nickel in the soil can compromise the security of agricultural products, posing a threat to health of human beings; therefore, the repair and treatment of Nickel exceeding the standard levels in soil are particularly critical. Although it is crucial that the potential restoration of Nickel in ensuring the security of both soil and farm produce within karst regions., few studies have been conducted on the potential restoration of large-scale Nickel-contaminated soils. In this study, the soil in Wuming, Guangxi, a typical karst area, was comprehensively studied. 12,547 surface soil samples, 134 deep soil samples and 60 soil profiles were collected systematically. The results showed that the Nickel background value of the surface soil was 34.9mg/kg, indicating strong background characteristics and high variability. Principal component analysis showed that soil Nickel was primarily derived from natural sources in the geological background and partly derived from agricultural sources. Analysis of variance showed that the Nickel content of the soil was affected by the parent rock, soil type, soil use type, and topography. In addition, the distribution of Nickel in the soil profile increased exponentially with depth. Therefore, the exponential model and multiple integrals were used to derive the formula for the Nickel potential restoration amount at different depth ranges, and the potential restoration amount of soil Nickel was calculated based on different parent material, soil, and land use types. The formula is reasonable and representative and can provide a theoretical basis for the remediation and treatment of Nickel-polluted soil in karst areas.

Higher order $\mathcal{S}^{p}$-differentiability: The unitary case

Consider the set of unitary operators on a complex separable Hilbert space \mathcal{H} , denoted as \mathcal{U}(\mathcal{H}) . Consider 1&lt;p&lt;\infty . We establish that a function f defined on the unit circle \mathbb{T} is n times continuously Fréchet \mathcal{S}^{p} -differentiable at every point in \mathcal{U}(\mathcal{H}) if and only if f\in C^{n}(\mathbb{T}) . Take a function U \colon\R\rightarrow\mathcal{U}(\mathcal{H}) such that the function t\in\R\mapsto U(t)-U(0) takes values in \mathcal{S}^{p} and is n times continuously \mathcal{S}^{p} -differentiable on \R . Consequently, for f\in C^{n}(\mathbb{T}) , we prove that f is n times continuously Gâteaux \mathcal{S}^{p} -differentiable at U(t) . We provide explicit expressions for both types of derivatives of f in terms of multiple operator integrals. In the domain of unitary operators, these results closely follow the n -th order successes for self-adjoint operators achieved by the second author, Le Merdy, Skripka, and Sukochev. Furthermore, as for application, we derive a formula and \mathcal{S}^{p} -estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev, and Tomskova.

The Influence of Dynamic Representations in Mobile Applications on Students’ Learning Achievements in Solving Multiple Integrals

In this paper, we describe the influence of the implementation of methodology for teaching and learning multiple integrals based on the constructivist approach and the interconnectedness of multiple representations (with an emphasis on dynamic representations) for different mathematical objects using GeoGebra’s applications Graphing Calculator and 3D Calculator. The empirical research was conducted at the University of Kragujevac, Serbia, with two groups (the experimental and the control groups) of second-year students. During the learning process of the experimental group, students created digital materials to visualize the domain of the integration for the concrete tasks, to set the integration bounds, to recognize in which situation they needed to introduce the switch of variables, and all that to solve double or triple integral problems. Students in the control group, on the other hand, didn't use technology of any kind, including the programs Graphing Calculator and 3D Calculator. The findings of the empirical study showed that students' use of these tools to create digital materials improved their academic achievement and their practical and theoretical knowledge of multiple integrals.

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.

Coupled Aeropropulsive Design Optimization of an Over-wing Nacelle Configuration

The over-wing nacelle (OWN) configuration has the potential to improve on the conventional under-wing nacelle (UWN) configuration by enabling higher bypass ratios and increasing noise shielding. To explore this potential, we perform coupled aeropropulsive design optimization to study the coupled analysis and the design tradeoffs between aerodynamics and propulsion. We find that the variations in wing shape are not significant when the OWN configuration is optimized for different fan pressure ratios. Optimizing the OWN configuration using the proposed approach results in 3% less shaft power than optimizing the wing and propulsor separately. The OWN has marginally superior performance over the UWN for the same lower fan pressure ratio. However, this low fan pressure ratio may be unattainable for the UWN configuration because of ground clearance constraints, whereas the OWN configuration is not subject to such constraints. We study the optimal OWN placement and show that placing it aft of the trailing edge and away from the wing root results in lower required shaft power. In this study, we perform single-point optimizations to understand the fundamentals of the OWN configuration in terms of aerodynamics and propulsion. However, multiple operating conditions, structural integrity, and inclusion of the pylon geometry should be considered to assess the net benefit of the OWN configuration. Nevertheless, these advancements in aeropropulsive optimization are critical to OWN configuration design and more sustainable aircraft.

Estimation of Lifetime Performance Index for Generalized Inverse Lindley Distribution Under Adaptive Progressive Type-II Censored Lifetime Test

The lifetime performance index (LPI) is an important metric for evaluating product quality, and research on the statistical inference of the LPI is of great significance. This paper discusses both the classical and Bayesian estimations of the LPI under an adaptive progressive type-II censored lifetime test, assuming that the product’s lifetime follows a generalized inverse Lindley distribution. At first, the maximum likelihood estimator of the LPI is derived, and the Newton–Raphson iterative method is adopted to solve the numerical solution due to the log-likelihood equations having no analytical solutions. If the exact distribution of the LPI is not available, then the asymptotic confidence interval and bootstrap confidence interval of the LPI are constructed. For the Bayesian estimation, the Bayesian estimators of the LPI are derived under three different loss functions. Due to the complex multiple integrals involved in these estimators, the MCMC method is used to draw samples and further construct the HPD credible interval of the LPI. Finally, Monte Carlo simulations are used to observe the performance of these estimators in terms of the average bias and mean squared error, and two practical examples are used to illustrate the application of the proposed estimation method.

An Order Statistics Perspective for System Reliability

ABSTRACTStructural reliability integrates the design variables over the safety region characterized by a positive limit state function when the reliability of the entire system is of concern. Calculating the structure function can be challenging for high‐dimensional systems or intricate system architectures. In order to enhance the efficiency of time‐dependent system reliability assessment, we emulate the integral formulation in structural reliability. To elaborate further, we treat each individual unit's lifetime variable as a design variable and subsequently perform calculations involving multiple integrals. Given the ordered nature of unit failure times, we leverage the order statistics distribution to simplify the multiple integrals into a double integral, and then multiply this result by the survival signature to obtain reliability. A two‐terminal nine‐unit network system configuration is illustrated to assess the performance and effectiveness of the proposed method.

Hierarchical Stability Conditions for Two Types of Time-Varying Delay Generalized Neural Networks.

In this article, the stability analysis for generalized neural networks (GNNs) with a time-varying delay is investigated. About the delay, the differential has only an upper boundary or cannot be obtained. For the both two types of delayed GNNs, up to now, the second-order integral inequalities have been the highest-order integral inequalities utilized to derive the stability conditions. To establish the stability conditions on the basis of the high-order integral inequalities, two challenging issues are required to be resolved. One is the formulation of the Lyapunov-Krasovskii functional (LKF), the other is the high-degree polynomial negative conditions (NCs). By transforming the integrals in N-order generalized free-matrix-based integral inequalities (GFIIs) into the multiple integrals, the hierarchical LKFs are constructed by adopting these multiple integrals. Then, the novel modified matrix polynomial NCs are presented for the 2N-1 degree delay polynomials in the LKF differentials. Thus, the hierarchical linear matrix inequalities (LMIs) are set up and the nonlinear problems caused by the GFIIs are solved at the same time. Eventually, the superiority of the provided hierarchical stability criteria is demonstrated by several numeric examples.

Simplified approach to estimate Lorenz number using experimental Seebeck coefficient for non-parabolic band

Reducing lattice thermal conductivity (κL) is one of the most effective ways for improving thermoelectric properties. However, the extraction of κL from the total measured thermal conductivity can be misleading if the Lorenz (L) number is not estimated correctly. κL is obtained using the Wiedemann–Franz law, which estimates the electronic part of thermal conductivity κe = L σT, where σ and T are electrical conductivity and temperature, respectively. κL is then estimated as κL = κT − L σT. For metallic systems, the Lorenz number has a universal value of 2.44 × 10 −8 WΩ K−2 (degenerate limit), but for non-degenerate semiconductors, the value can deviate significantly for acoustic phonon scattering, the most common scattering mechanism for thermoelectric materials above room temperature. Up until now, L is estimated by solving a series of equations derived from Boltzmann transport equations. For the single parabolic band (SPB) model, an equation was proposed to estimate L directly from the experimental Seebeck coefficient. However, using the SPB model will lead to an overestimation of L in the case of low bandgap semiconductors, which results in an underestimation of κL, sometimes even negative κL. In this article, we propose a simpler equation to estimate L for a non-parabolic band. The experimental Seebeck coefficient, bandgap (Eg), and temperature (T) are the main inputs to the equation, which nearly eliminates the need for solving multiple Fermi integrals besides giving accurate values of L.

Hermite–Hadamard-type inequalities for multiplicative harmonic s -convex functions

UDC 517.5 We study the concept of multiplicative harmonic s -convex functions and establish Hermite–Hadamard integral inequalities for this class of functions. Furthermore, we derive a set of Hermite–Hadamard-type inequalities applicable to the product and quotient of multiplicative harmonic s -convex functions. In addition, we deduce new inequalities involving multiplicative integrals for the product and quotient of harmonic convex and multiplicative harmonic s -convex functions. Some results for the class of multiplicative harmonic convex functions are obtained as special cases of our results. The obtained results are verified by providing examples with included graphs.

A nexus of multiple integrations and business performance through supply chain agility and supply flexibility: a dynamic capability view

Purpose Manufacturing capability is a crucial component of every nation’s economy and pharmaceuticals are frequently a significant part of the manufacturing sector. Pharmaceutical supply chains are essential to health-care systems, contributing to living quality and shorter hospital stays. This study aims to examine the role of multiple integrations on business performance (BP) through supply chain flexibility (SCF) and supply chain agility (SCA). Design/methodology/approach Data was collected from 198 supply chain professionals in the pharmaceutical sector of the developing economy of Pakistan. The sample was collected based on a nonprobability purposive sampling approach. A five-point Likert-scale survey was used and analyzed with the PLS-SEM technique using SmartPLS 4. Findings This study found that process integration (PI) does not affect SCA, whereas relationship integration and measurement integration positively affect SCA. SCA positively impacts BP. In contrast, all integrations significantly influenced supply flexibility and BP except for PI. Finally, SCF significantly mediates the relationship between all integrations and BP. Originality/value This study examined the relationships of multiple integrations on BP, directly and indirectly, through SCF and agility. The theory of dynamic capabilities has been applied and extended to increase the comprehensiveness of the findings. A developing economy’s pharmaceutical industry supply chain was examined, producing empirical evidence of the results.

Enhanced framework for solving general energy equations based on metropolis-hasting Markov chain Monte Carlo

Due to the widespread presence of heat and mass transfer phenomena in industrial applications, numerous studies have been devoted to the accurate solution of energy equations, providing a foundation for the analysis of heat and mass transfer processes in practical applications. In this study, a Green's Function Markov Superposition Monte Carlo (GMSMC) for solving general energy equations has been developed based on probability and statistical principles owing to its advantageous features of insensitivity towards dimension and geometric complexity as well as the capability to handle multiple integrals in complex domains. The energy equation is first decomposed, and corresponding probability models are established for each component, considering their interrelationships. Subsequently, a solution framework for solving the general energy equation is constructed by integrating these probability models based on a Markov chain structure. The mathematical principles and formulae of the proposed method are derived in detail. The performance of the proposed method is validated by several heat transfer systems with different combinations of boundary conditions and features, which mainly include the distribution of the internal heat source and whether the convection or transient term is included. Results of the validation show that the temperatures obtained by the proposed method are in good agreement with the FEM based on a fine grid, no matter whether the calculation is for a single point or a distribution.

Stability and passivity analysis of delayed neural networks via an improved matrix-valued polynomial inequality

The stability and passivity of delayed neural networks are addressed in this paper. A novel Lyapunov–Krasovskii functional (LKF) without multiple integrals is constructed. By using an improved matrix-valued polynomial inequality (MVPI), the previous constraint involving skew-symmetric matrices within the MVPI is removed. Then, the stability and passivity criteria for delayed neural networks that are less conservative than the existing ones are proposed. Finally, three examples are employed to demonstrate the meliority and feasibility of the obtained results.

Libepa — A C++/Python library for calculations of cross sections of ultraperipheral collisions

The library provides a set of C++/Python functions for computing cross sections of ultraperipheral collisions of high energy particles under the equivalent photons approximation. Cross sections are represented through multiple integrals over the phase space. The integrals are calculated through recurrent application of algorithms for one dimensional integration. The paper contains an introduction to the theory of ultraperipheral collisions, discusses the library approach and provides a few examples of calculations.PROGRAM SUMMARY•Program title:libepa.•CPC Library link to program files:https://doi.org/10.17632/6h5fjpg7p9.1•Developer's repository link:https://github.com/jini-zh/libepa.•Licensing provisions: GNU General Public License 3•Programming Language: C++, Python.•Supplementary material: attached.•Nature of problem: Ultraperipheral collisions of ultrarelativistic particles are a source of high energy photon-photon collisions. Ultraperipheral collisions occur at particle accelerators, in particular the Large Hadron Collider. They can be used to discover new electrically charged particles or other particles that can be produced in photons fusion (e.g., axions), and to make precision tests of the Standard Model of particle physics. libepa is a tool to calculate cross sections of ultraperipheral collisions with restrictions on the phase space of the produced particles typical to high energy experiments.•Solution method: Cross sections are expressed in terms of multiple integrals over the phase space parameters and numerically calculated through recurrent application of algorithms for one-dimensional integration. Functional programming approach is used to simplify the interface and optimize the calculations.

Mussel-Inspired Polymer-Based Composites for Efficient Thermal Management in Dry and Underwater Environments.

To address the escalating power consumption of processors in data centers and the growing emphasis on environmental sustainability, the prospective shift from traditional air-cooling to immersion liquid cooling necessitates multiple functional integrations in polymer-based thermal conductive materials. Here, drawing inspiration from mussels, we showed a copolymer, poly(dimethylsiloxane-co-dopamine methacrylate) (PDMS-DMA), with a variety of reversible molecular interactions and simply combined with liquid metal (EGaIn) can yield a flexible, waterproof, and electrically insulating thermal conductive composite. The obtained PDMS-DMA/EGaIn composites demonstrate a harmonious blend of attributes, including a low modulus (75.8 kPa), high thermal conductivity of 6.9 W m-1 K-1, and rapid room-temperature self-healing capabilities, capable of complete repair within 20 min, even under water. Based on its electrically insulating and water resistance properties, PDMS-DMA/EGaIn emerges as a promising candidate for efficient and stable heat transfer in both air and underwater thermal management. Consequently, this water-resistant polymer-based composite holds significance for application in thermal protective layers for future immersion liquid cooling systems.

The Exact Density of the Eigenvalues of the Wishart and Matrix-Variate Gamma and Beta Random Variables

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and jth largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.

Multiple Mellin-Barnes integrals and triangulations of point configurations

Mellin-Barnes (MB) integrals are a well-known type of integrals appearing in diverse areas of mathematics and physics, such as in the theory of hypergeometric functions, asymptotics, quantum field theory, solid-state physics, etc. Although MB integrals have been studied for more than a century, it is only recently that, due to a remarkable connection found with conic hulls, N-fold MB integrals can be computed analytically for N&gt;2 in a systematic way. In this article, we present an alternative novel technique by unveiling a new connection between triangulations of point configurations and MB integrals, to compute the latter. To make it ready to use, we have implemented our new method in the package oniculls.wl, an already existing software dedicated to the analytic evaluation of MB integrals using conic hulls. The triangulation method is remarkably faster than the conic hull approach and can thus be used for the calculation of higher-fold MB integrals, as we show here by testing our code on the case of the off-shell massless scalar one-loop N-point Feynman integral up to N=15, for which the MB representation has 104 folds. Among other examples of applications, we present new simpler solutions for the off-shell one-loop massless conformal hexagon and two-loop double-box Feynman integrals, as well as for some complicated 8-fold MB integrals contributing to the hard diagram of the two-loop hexagon Wilson loop in general kinematics. Published by the American Physical Society 2024

Empirical limit theorems for Wiener chaos

We consider empirical measures in a triangular array setup with underlying distributions varying as sample size grows. We study asymptotic properties of multiple integrals with respect to normalized empirical measures. Limit theorems involving series of multiple Wiener–Itô integrals are established.