Integral Function Research Articles

Gradient Descent Provably Escapes Saddle Points in the Training of Shallow ReLU Networks

AbstractDynamical systems theory has recently been applied in optimization to prove that gradient descent algorithms bypass so-called strict saddle points of the loss function. However, in many modern machine learning applications, the required regularity conditions are not satisfied. In this paper, we prove a variant of the relevant dynamical systems result, a center-stable manifold theorem, in which we relax some of the regularity requirements. We explore its relevance for various machine learning tasks, with a particular focus on shallow rectified linear unit (ReLU) and leaky ReLU networks with scalar input. Building on a detailed examination of critical points of the square integral loss function for shallow ReLU and leaky ReLU networks relative to an affine target function, we show that gradient descent circumvents most saddle points. Furthermore, we prove convergence to global minima under favourable initialization conditions, quantified by an explicit threshold on the limiting loss.

Impact Of Innovation and Training & Development on Operational Performance: Mediation of Green Lean Six Sigma

This research tends to discover the influence of innovation and training & development on the operational performance of an organization. Performance is an integral function of the success of any organization. Therefore, this research tends to discover the influence of innovation and training & development on the operational performance of an organization. Green lean six sigma is also an important concept that emerged in the last few years. The motive of this research is to explore the mediation of green lean six sigma among innovation, training & development, and operational performance. The current study uses a quantitative method to explore this current topic. The data utilized in this study was obtained from both customers and employees of textile and apparel manufacturing facilities located in Pakistan. Questionnaires were used to collect data from the respondents of the research. Results of the study indicate innovation and training & development are significantly associated with an organization's operational performance. Green lean six sigma also mediated the association between innovation, training & development, and operational performance. This study also includes limitations, future direction, and practical and managerial implications.

A Robust Parallel Predictive Torque Control Based on a Proportional Integral Cost Function for PMSM

A Robust Parallel Predictive Torque Control Based on a Proportional Integral Cost Function for PMSM

Integral BLF-Based Adaptive Dynamic Event-Triggered Boundary Control for a Flexible Riser System.

For the flexible riser systems modeled with partial differential equations (PDEs), this article explores the boundary control problem in depth for the first time using a dynamic event-triggered mechanism (DETM). Given the intrinsic time-space coupling characteristic inherent in PDE computations, implementing a state-dependent DETM for PDE-based flexible risers presents a significant challenge. To overcome this difficulty, a novel dynamic event-triggered control method is introduced for flexible riser systems, focusing on optimizing available control inputs. In order to save computational costs from the controller to the actuator, a dynamic event-triggered adaptive boundary controller is designed to effectively reduce boundary position vibrations. Additionally, considering external disturbances, an adaptive bounded compensation term is incorporated to counteract the influence of external disturbances on the system. Addressing boundary position constraints, a new integral barrier Lyapunov function (iBLF) tailored specifically for flexible riser systems is introduced, thereby alleviating conservatism in the controller design of flexible risers modeled by PDEs. At last, the validity of the proposed method is demonstrated through a simulation example.

Stereologically adapted Crofton formulae for tensor valuations

The classical Crofton formula explains how intrinsic volumes of a convex body K in n-dimensional Euclidean space can be obtained from integrating a measurement function at sections of K with invariantly moved affine flats. We generalize this idea by constructing stereologically adapted Crofton formulae for translation invariant Minkowski tensors, expressing a prescribed tensor valuation as an invariant integral of a measurement function of section profiles with flats. The measurement functions are weighed sums of powers of the metric tensor times Minkowski valuations. The weights are determined explicitly from known Crofton formulae using Zeilberger's algorithm. The main result is an exhaustive set of measurement functions where the invariant integration is over flats.With the main result at hand, a Blaschke-Petkantschin formula allows us to establish new measurement functions valid when the invariant integration over flats is replaced by an invariant integration over subspaces containing a fixed subspace of lower dimension. Likewise, a stereologically adapted Crofton formula valid in the scheme of vertical sections is constructed. Only some special cases of this result have been stated explicitly before, with even the three-dimensional case yielding a new stereological formula. Here, we obtain new vertical section formulae for the surface tensors of even rank.

Multiplicity and transverse momentum dependence of charge-balance functions in pPb and PbPb collisions at LHC energies

Measurements of the charge-dependent two-particle angular correlation function in proton-lead (pPb) collisions at a nucleon-nucleon center-of-mass energy of sNN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{s_{\ extrm{NN}}} $$\\end{document} = 8.16 TeV and lead-lead (PbPb) collisions at sNN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{s_{\ extrm{NN}}} $$\\end{document} = 5.02 TeV are reported. The pPb and PbPb data sets correspond to integrated luminosities of 186 nb−1 and 0.607 nb−1, respectively, and were collected using the CMS detector at the CERN LHC. The charge-dependent correlations are characterized by balance functions of same- and opposite-sign particle pairs. The balance functions, which contain information about the creation time of charged particle pairs and the development of collectivity, are studied as functions of relative pseudorapidity (∆η) and relative azimuthal angle (∆ϕ), for various multiplicity and transverse momentum (pT) intervals. A multiplicity dependence of the balance function is observed in ∆η and ∆ϕ for both systems. The width of the balance functions decreases towards high-multiplicity collisions in the momentum region < 2 GeV, for pPb and PbPb results. Integrals of the balance functions are presented in both systems, and a mild dependence of the charge-balancing fractions on multiplicity is observed. No multiplicity dependence is observed at higher transverse momentum. The data are compared with hydjet, hijing, and ampt generator predictions, none of which capture completely the multiplicity dependence seen in the data. The comparison of results with different center-of-mass energies suggests that the balance functions become narrower at higher energies, which is consistent with the idea of delayed hadronization and the effect of radial flow.

Adaptive passive fault tolerant control of DFIG-based wind turbine using a self-tuning fractional integral sliding mode control

Controlling variable wind speed turbine (VWT) system based on a doubly-fed induction generator (DFIG) is a challenging task. It requires a control law that is both adaptable and robust enough to handle the complex dynamics of the closed control loop system. Sliding mode control (SMC) is a robust control technology that has shown good performance when employed as a passive fault-tolerant control for wind energy systems. To improve the closed control loop of VWT based on DFIG with the aim of improving energy efficiency, even in presence of nonlinearities and a certain range of bounded parametric uncertainties, whether electrically or mechanically, an adaptive passive fault tolerant control (AP-FTC) based on a self-tuning fractional integral sliding mode control law (ST-FISMC) developed from a novel hyperbolic fractional surface is proposed in this paper. ST-FISMC introduces a nonlinear hyperbolic function into the sliding manifold for self-tuning adaptation of control law, while fractional integral of the control law smooths discontinuous sign function to reduce chattering. Additionally, this work introduces an adaptive observer, developed and proved based on a chosen Lyapunov function. This observer is designed to estimate variations in electrical parameters and stator flux, ensuring sensorless decoupling in indirect field- oriented control (SI-FOC) of DFIG. Lyapunov theory is also used to prove stability of states vectors in closed control loop with presence of bounded parameters uncertainties or external disturbances. Simulation results show that the proposed approach offers better performance in capturing optimal wind energy, as well as the ability to regulate active/reactive power and high resilience in presence of occurring parameter uncertainties or external disturbances.

Can you hear the Planck mass?

For the Laplacian of an n-Riemannian manifold X, the Weyl law states that the k-th eigenvalue is asymptotically proportional to (k/V)2/n, where V is the volume of X. We show that this result can be derived via physical considerations by demanding that the gravitational potential for a compactification on X behaves in the expected (4+n)-dimensional way at short distances. In simple product compactifications, when particle motion on X is ergodic, for large k the eigenfunctions oscillate around a constant, and the argument is relatively straightforward. The Weyl law thus allows to reconstruct the four-dimensional Planck mass from the asymptotics of the masses of the spin 2 Kaluza-Klein modes. For warped compactifications, a puzzle appears: the Weyl law still depends on the ordinary volume V, while the Planck mass famously depends on a weighted volume obtained as an integral of the warping function. We resolve this tension by arguing that in the ergodic case the eigenfunctions oscillate now around a power of the warping function rather than around a constant, a property that we call weighted quantum ergodicity. This has implications for the problem of gravity localization, which we discuss. We show that for spaces with Dp-brane singularities the spectrum is discrete only for p = 6, 7, 8, and for these cases we rigorously prove the Weyl law by applying modern techniques from RCD theory.

Propagation properties of the partially coherent radially polarized twisted beam through optical system in turbulence atmosphere

In this paper, the transmission properties of the partially coherent radially polarized twisted (PCRPT) beam propagating in the turbulence atmosphere are investigated. The analytical formulas for the components of the cross-spectral density matrix for the PCRPT beam passing through the optical system in the turbulent atmosphere are developed using the Collins integral and aperture function. Research results indicate that modifying the variable parameters and dimensions of the optical system can control the near-field and far-field distributions of the beam, while providing a more flexible choice of receivers for the PCRPT beam in the receiving plane. By utilizing the Cassegrain reflector system and adjusting the optical system parameters, it is possible to achieve collimated transmission of the PCRPT beam and significantly enhance the beam transmission efficiency in turbulent atmospheric conditions. The derivation process and the research results presented in this paper can be expanded to analyze the application of optical systems to control high-dimensional beam field variations. The envisioned utilization of the results obtained from this research investigation pertains to the fields of beam shaping and optical communications.

Acceleration Slip Regulation Control Method for Distributed Electric Drive Vehicles under Icy and Snowy Road Conditions

To achieve a rapid and stable dynamic response of the drive anti-slip system for distributed electric vehicles on low-friction surfaces, this paper proposes an adaptive acceleration slip regulation control strategy based on wheel slip rate. An attachment coefficient fusion estimation algorithm based on an improved singular value decomposition unscented Kalman filter is designed. This algorithm combines Sage–Husa with the unscented Kalman filter for adaptive improvement, allowing for the quick and accurate determination of the road friction coefficient and, subsequently, the optimal slip rate. Additionally, a slip rate control strategy based on dynamic adaptive compensation sliding mode control is designed, which introduces a dynamic weight integral function into the control rate to adaptively adjust the integral effect based on errors, with its stability proven. To verify the performance of the road estimator and slip rate controller, a model is built with vehicle simulation software, and simulations are conducted. The results show that under icy and snowy road conditions, the designed estimator can reduce estimation errors and respond rapidly to sudden changes. Compared to traditional equivalent controllers, the designed controller can effectively reduce chattering, decrease overshoot, and shorten response time. Especially during road transitions, the designed controller demonstrates better dynamic performance and stability.

Integral of Continuous Three Variable Functions

Summary In this article we continue our proofs on integrals of continuous functions of three variables in Mizar. In fact, we use similar techniques as in the case of two variables: we deal with projections of continuous function, the continuity of three variable functions in general, aiming at pure real-valued functions (not necessarily extended real-valued functions), concluding with integrability and iterated integrals of continuous functions of three variables.

Determination of the probabilistic content of organic fertilizer fractions after exposure to arch-destroying and crushing agrotechnological devices

Abstract Achieving stable and uniform dosing of organic fertilizers during local application necessitates the crushing of large lumps to a specific size range. The process of breaking down these lumps follows a random distribution. By leveraging the integral function of the normal distribution of random variables intertwined with principles of probability theory, equations are formulated to ascertain the distribution of lump sizes within organic fertilizers. Furthermore, these equations enable the calculation of the likelihood of obtaining lumps of specified sizes post the interaction of the vault-breaker-shredder’s fingers with counter-cutting ribs on the fertilizer. This analytical approach provides a systematic method to optimize the sizing and distribution of organic fertilizer particles for enhanced application efficiency and effectiveness.

Consumer heterogeneity and inefficiency in oligopoly markets

Motivated by several recent studies on oligopolistic markets with consumer heterogeneity, we characterize the geometric properties of a class of inequalities involving the product of two integrals of the probability density quantile function and its reciprocal. Through three applications, we illustrate how our characterization can help enrich our understanding of the existing findings and shed some new insights, regarding the extent to which consumer welfare and social efficiency depend on the properties of the distributions of consumer valuation heterogeneity.

An experimental and theoretical investigation on the hyper-viscoelasticity of polyamide 12 produced by selective laser sintering.

Polyamide 12 (PA12) is vastly utilized in many additive manufacturing methods, such as Selective Laser Sintering (SLS), and a better understanding of its mechanical behaviors promotes available knowledge on the behaviors of 3D-printed parts made from this polymer. In this paper, SLS-produced standard tensile specimens are studied under monotonic and cyclic tension tests, as well as stress relaxation experiments, and the obtained force-displacement responses are shown to be consistent with a hyper-viscoelastic material model. This finding is also observed in typical pantographic structures produced by the same manufacturing parameters. To propose a constitutive model for predicting these behaviors, the convolution integral of a strain-dependent function and a time-dependent function is developed where the material parameters are determined with the use of both short-term and long-term responses of the specimens. Numerical results of the presented model for standard test specimens are shown to be in good agreements with the experimental ones under various loading conditions. To prove the capabilities of the proposed model in studying any SLS-produced part, finite element implementation of the constitutive equations is shown to provide numerical results in agreement with the empirical findings for tensile loading of the 3D-printed pantographic structure.

Observer-based guaranteed-cost bipartite time-varying formation tracking control for multiagent systems subject to communication delays

The paper centers on the guaranteed-cost bipartite time-varying formation tracking control for multiagent systems with nonuniform time-varying communication delays under the framework of a distributed observer. An integral quadratic cost function is presented, founded on formation tracking and observation error. Firstly, distributed state observers, using the relative measurement output between neighbors, are established so that the leader and follower can accurately estimate each agent's unmeasurable state. Subsequently, a bipartite formation tracking protocol is proposed, incorporating estimation information and nonuniform time-varying communication delays. This protocol aims to achieve bipartite time-varying formation tracking for multiagent systems. Next, a new augmented Lyapunov-Krasovskii functional with delay-product and triple integral terms is constructed. By integrating the second-order Bessel-Legendre inequality technique, the delay-dependent reciprocally convex combination inequality method, and the free weight matrix technique, a new guaranteed-cost bipartite time-varying formation tracking criterion is developed, offering less conservatism. Compared to certain existing delay conditions, the stability conditions assure a greater allowable upper bound of delay. Finally, a numerical example is conducted to confirm the validity and superiority of the theoretical findings.

Fixed-time disturbance observer-based funnel control for controllable excitation linear synchronous motor with state constraints

This paper proposes a fixed-time disturbance observer-based funnel control method for a controllable excitation linear synchronous motor (CELSM) with state constraints and disturbances. This approach simultaneously resolves the issues of prescribed tracking performance and state constraints, taking into account disturbances. We employ a fixed-time disturbance observer (DO) to estimate both matched and mismatched disturbances. Additionally, we design a modified funnel variable to confine the tracking error within a prescribed region and the singularity problem is avoided. An integral barrier Lyapunov function (iBLF) is utilized to directly tackle state constraints. Furthermore, we employ a fixed-time differentiator (FTD) to address the “explosion of complexity” issue in backstepping control. Theoretical analysis demonstrates that the tracking error asymptotically converges to the origin within a prescribed region while satisfying state constraints. Finally, simulation results demonstrate the efficacy of the proposed method.

On the Katugampola fractional integral and dimensional analysis of the fractal basin boundary for a random dynamical system

In this paper, we mainly investigate the geometric based relationship between the Katugampola fractional calculus and a Weierstrass-type function whose graph can be characterized as a fractal basin boundary for a random dynamical system. Using the potential-theoretic approach with some classical analytical tools, we have derived some kinds of fractal dimensions of the graph of the Katugampola fractional integral of this fractal function. It has been shown that there is a linear relationship between the order of the Katugampola fractional integral and the fractal dimension of the graph of this generalized Weierstrass function. Numerical results have also been provided to corroborate such linear connection.

Modeling Alternative Conformational States of Pseudo-Symmetric Solute Carrier Transporters using Methods from Machine Learning.

The Solute Carrier (SLC) superfamily of integral membrane proteins function to transport a wide array of solutes across the plasma and organelle membranes. SLC proteins also function as important drug transporters and as viral receptors. Despite being classified as a single superfamily, SLC proteins do not share a single common fold classification; however, most belong to multi-pass transmembrane helical protein fold families. SLC proteins populate different conformational states during the solute transport process, including outward open, intermediate (occluded), and inward open conformational states. For some SLC fold families this structural "flipping" corresponds to swapping between conformations of their N-terminal and C-terminal symmetry-related sub-structures. Conventional AlphaFold2 or Evolutionary Scale Modeling methods typically generate models for only one of these multiple conformational states of SLC proteins. Here we describe a fast and simple approach for modeling multiple conformational states of SLC proteins using a combined ESM - AF2 process. The resulting multi-state models are validated by comparison with sequence-based evolutionary co-variance data (ECs) that encode information about contacts present in the various conformational states adopted by the protein. We also explored the impact of mutations on conformational distributions of SLC proteins modeled by AlphaFold2 using both conventional and enhanced sampling methods. This approach for modeling conformational landscapes of pseudo-symmetric SLC proteins is demonstrated for several integral membrane protein transporters, including SLC35F2 the receptor of a feline leukemia virus envelope protein required for viral entry into eukaryotic cells.

Efficient Scheme for the Economic Heston–Hull–White Problem Using Novel RBF-FD Coefficients Derived from Multiquadric Function Integrals

This study presents an efficient method using the local radial basis function finite difference scheme (RBF-FD). The innovative coefficients are derived from the integrals of the multiquadric (MQ) function. Theoretical convergence rates for the coefficients used in function derivative approximation are provided. The proposed scheme utilizes RBF-FD estimations on three-point non-uniform stencils to construct the final approximation on a tensor grid for the 3D Heston–Hull–White (HHW) PDE, which is relevant in economics and mathematical finance. Numerical evidence and comparative analyses validate the results and the proposed scheme.

The Impact of Authenticity and Credibility Factors on Consumer Behavior in the Sustainable Fashion Industry on Weibo--Taking Micro-Influencers and Mega-Influencers as Examples

This study investigates the comparative influence of micro-influencers against mega-influencers on consumer behavior within the sustainable clothing industry on Weibo, and tries to reveal how authenticity and credibility factors shape purchasing decisions. This study utilizes qualitative research through semi-structured interviews with 8 participants (influencers and consumers) to highlight the integral key functions of influencers authenticity and marketing strategies in the sustainable fashion sector. It is found that micro-influencers who are believed to be genuine and more trustworthy, affect consumer buying decisions more than macro-influencers who have a wide reach but may not be seen as genuine enough. The study emphasizes the significance of strategic relationships with influencers, highlighting the imperative process of aligning with influencers whose values resonate with the brand and the audience. Nonetheless, the research realizes the limitations, namely, the small sample size and focus on a specific social media platform and proposes other studies to consider a wide population and different social media platforms.