Series Expansion Research Articles

Towards a turnkey approach for unbiased Monte Carlo estimation of smooth functions of expectations

Summary Given a smooth function f, we develop a general approach to turn Monte Carlo samples with expectation m into an unbiased estimate of f (m). Specifically, we develop estimators that are based on randomly truncating the Taylor series expansion of f and estimating the coefficients of the truncated series. We derive their properties and propose a strategy to set their tuning parameters—which depend on m—automatically, with a view to make the whole approach simple to use. We develop our methods for the specific functions f (x) = log x and f (x) = 1/x, as they arise in several statistical applications such as maximum likelihood estimation of latent variable models and Bayesian inference for un-normalised models. Detailed numerical studies are performed for a range of applications to determine how competitive and reliable the proposed approach is.

Combining Physics and Mathematics Learning: A Taylor Series Analysis of an Oscillating Magnetic Field

In this work, we present a simple and low-cost experiment designed to study the oscillations of the magnetic field created by a cylindrical magnet under two different conditions: far and short distances from the magnetic sensor. A Taylor’s series expansion of the magnetic field function has been done to study the convergence of the polynomial series to the real field in both situations. To carry out the experiment, a small cylindrical magnet has been attached to an oscillating and well-known spring-mass system. The resulting oscillating magnetic field has been registered with the smartphone by using the magnetometer sensor. A very good agreement has been obtained between the theoretical model for the magnetic field and the experimental data collected with the sensor located near and far from a cylindrical magnet and along its longitudinal axis.

When the Future Perfectly Reflects the Past in Celestial Mechanics

Newton’s equations of celestial mechanics for the N-body problem possess a continuum of solutions in which the future trajectories are a perfect reflection of their past. These solutions evolve from zero initial velocities of the N bodies. Consequently, the future gravitational forces acting on the N bodies are also a perfect reflection of their past. The proof is carried out via Taylor series expansions.

Phenomenology of the semileptonic Σb*0→Σc+ℓν¯ℓ transition within QCD sum rules

We conduct an investigation on the spin 32→12 semileptonic weak transition of single heavy baryons for the exclusive decay Σb*0→Σc+ℓν¯ℓ in three possible lepton channels within the three point QCD sum rule method. We compute the responsible form factors of this semileptonic decay by incorporating both perturbative and nonperturbative contributions of the operator product expansion series up to a mass dimension six. Having acquired the form factors, the decay widths of the processes in all lepton channels are determined. Our findings as well as possible future experimental information can be employed in order to check the SM predictions and explore the possibility of new physics in heavy baryonic decay channels. Published by the American Physical Society 2025

Towards whole brain mapping of the haemodynamic response function.

Functional magnetic resonance imaging time-series are conventionally processed by linear modelling the evoked response as the convolution of the experimental conditions with a stereotyped haemodynamic response function (HRF). However, the neural signal in response to a stimulus can vary according to task, brain region, and subject-specific conditions. Moreover, HRF shape has been suggested to carry physiological information. The BOLD signal across a range of sensorial and cognitive tasks was fitted using a sine series expansion, and modelled signals were deconvolved, thus giving rise to a task-specific deconvolved HRF (dHRF), which was characterized in terms of amplitude, latency, time-to-peak and full-width at half maximum for each task. We found that the BOLD response shape changes not only across activated regions and tasks, but also across subjects despite the age homogeneity of the cohort. Largest variabilities were observed in mean amplitude and latency across tasks and regions, while time-to-peak and full width at half maximum were relatively more consistent. Additionally, the dHRF was found to deviate from canonicity in several brain regions. Our results suggest that the choice of a standard, uniform HRF may be not optimal for all fMRI analyses and may lead to model misspecifications and statistical bias.

A novel method for analyzing foot motion during circumduction using an electromagnetic tracking system.

Circumduction of the hindfoot does not occur primarily in one of the traditional anatomic planes and can be difficult to describe precisely. The purpose of this study was to measure foot bone motion quickly and objectively to subsequently characterize differences among feet of varying shapes. As such, we have developed a quantitative characterization of foot bone motion during circumduction using electromagnetic tracking sensors. Five of these sensors were attached to the foot on specific bony landmarks, and one was attached to a footplate. The lower leg was held by padded clamps in a custom non-ferrous jig, and the foot was moved through a full range of circumduction. To describe the motion of the bones of the foot during circumduction, the sensor positions were fitted to 2D ellipses and 3D curves. A repeatability study on multiple feet (n = 7) demonstrated that multiple raters (n = 3) introduced more error than a single rater; therefore, a single rater was used for all subsequent data collection. Results from five neutrally aligned subjects demonstrated that bone motion was quantifiable by fitted ellipse parameters. Additional modeling with a paraboloid surface described the motion with improved accuracy. A further reduction in error was obtained using a 3D eighth-order Fourier series expansion fit. This method holds promise as a means for characterizing differences in foot bone motion among foot types during a clinical exam.

Quark density in lattice QC2D at imaginary and real chemical potential

We study lattice two-color QCD with two flavors of staggered fermions at imaginary and real quark chemical potential μq and T>Tc. We employ various methods of extrapolation of the quark number density from imaginary to real quark chemical potentials μq, including series expansions as well as analytic continuation based on phenomenological models, and study their accuracy by comparing the results to the lattice data. Below the Roberge-Weiss temperature, T<TRW, we find that the cluster expansion model provides an accurate analytic continuation of the baryon number density in the studied range of chemical potentials. On the other hand, the behavior of the reconstructed canonical partition functions indicates that the available models may require corrections at high quark densities. At T>TRW we show that the analytic continuation to the real values of μq based on trigonometric functions works equally well with the conventional method based on the Taylor expansion in powers of μq. Published by the American Physical Society 2025

Weyl gauge invariant DBI action in conformal geometry

We construct the analog of the Dirac-Born-Infeld (DBI) action in Weyl conformal geometry in d dimensions and obtain a general theory of gravity with Weyl gauge symmetry of dilatations (Weyl-DBI). This is done in the formulation of conformal geometry in d dimensions, suitable for a gauge theory, in which this geometry is . The Weyl-DBI action is a special gauge theory in that it has the same gauge invariant expression with couplings in any dimension d, so it has no need for a UV regulator (be it an additional field, higher derivative operator or DR subtraction scale) for which reason we argue it is Weyl-anomaly free. For d=4, the leading order of a series expansion of the Weyl-DBI action recovers exactly the gauge invariant Weyl quadratic gravity action associated to this geometry, that is Weyl anomaly-free; this is broken spontaneously and Einstein-Hilbert gravity is recovered in the broken phase, with Λ>0. The rest of the terms in the series expansion of the Weyl-DBI action are of nonperturbative nature and can, in principle, be recovered by quantum corrections in Weyl quadratic gravity in d=4 in a gauge invariant (geometric) regularization automatically provided by the Weyl-DBI action. If the Weyl gauge boson is not dynamical the Weyl-DBI action recovers in the leading order the conformal gravity action. All fields and scales have , with no added matter, scalar field compensators or UV regulators. Published by the American Physical Society 2025

Mie scattering near the Fröhlich mode

We consider the scattering of electromagnetic radiation by a spherical particle, known as Mie scattering. Of particular interest is the case where the particle is small, compared to an optical wavelength. A series expansion in the particle radius then gives an approximation for the otherwise very complicated scattering coefficients. However, for certain values of the permittivity of the particle, the approximation for electric multipole radiation breaks down. This is known as the Fröhlich resonance. On one hand, the scattering coefficients diverge, and on the other hand, they violate conservation of energy. We derive a new expression for the small-radius limit, which does not have these problems. We find that there exists a resonant radius for certain parameters, and the resonance is shifted as compared to the common expression. We show that our approximation is excellent by comparing it to the exact solution.

Observer-based wing rock motion suppression using the Fourier series expansion as uncertainty approximator

The motion of wing rock varies unpredictably with changes in the angle of attack (AOA) and is highly susceptible to external disturbances. This paper presents a robust adaptive control strategy based on an observer, utilizing Fourier Series (FS) polynomials to estimate uncertainties and suppress wing rock oscillations. The unmodeled dynamics, external disturbances, and uncertainties are first modeled using the Fourier series. The adaptation laws, obtained through analyzing the stability, are subsequently used to adjust its coefficients. A distinctive feature of the suggested design is that it does not require a precise model of the plant or any data about roll rate signal measurements and uncertainties, which enhances cost-effectiveness and applicability in real-world systems. The paper employs the Lyapunov lemma to ensure that the error signals in the controlled system stay uniformly ultimately bounded (UUB). Simulation results showcase the effectiveness and versatility of the proposed approach. The outcomes are compared with two advanced approximation techniques to demonstrate the precision and effectiveness of the suggested controller design.

Determination of the Kinetic Parameters of Condensed Phase Reactions Using Chebyshev Series Expansion

AbstractIn the investigation of condensed phase reactions, obtaining kinetic parameters is vital for understanding reaction behavior and optimizing conditions. To achieve this, differential methods have been devised, yet due to the instability of calculating instantaneous reaction rates through numerical differentiation, they have been less commonly utilized. In this study, the extraction of smooth reaction rate curves from highly noisy experimental data via the Chebyshev series expansion (CSE) approach is explained. Furthermore, a novel combined kinetic analysis is developed to determine reaction kinetic parameters utilizing the Chebyshev series expansion. By employing the new method, kinetic parameters can be accurately deduced by performing multiple linear regression analysis on kinetic data generated from reactions. The CSE has consistently exhibited exceptional accuracy in approximating the conversion function. The primary advantage of the new method lies in its ability to accurately determine unique values for kinetic parameters, including activation energy, pre‐exponential factor, and conversion function, without prior knowledge of the reaction mechanism. The new method has been validated using kinetic data from a simulated reaction and poly(methyl methacrylate) thermal degradation. To facilitate readers in applying the new methods to various kinetic data, the GNU Octave/MATLAB codes have been made publicly available.

Solving the Parametric Eigenvalue Problem by Taylor Series and Chebyshev Expansion

Solving the Parametric Eigenvalue Problem by Taylor Series and Chebyshev Expansion

Nonlinear optimal control of heat pumps in electric vehicles

Heat pumps can be used for thermal management of electric vehicles (EVs) in a low-cost and environmentally friendly manner. Heat pumps can exploit the heat that is diffused by the vehicle’s batteries so as to perform temperature control of the passengers cabin. A heat-pump-based thermal management system of EVs has the dual objective of (a) stabilizing temperature in the passenger cabin at the desirable levels so as to ensure the passengers’ comfort and (b) stabilizing temperature at the batteries so as to avoid degradation of the batteries’ power storage capacity and of their regenerative charging capability. So far, nonlinear model predictive control (NMPC) methods have been mainly applied for the nonlinear optimal control problem of heat pumps in EVs. In this article a novel nonlinear optimal control method is proposed for solving the nonlinear optimal control problem of heat pumps. This method is computationally simple, has clear implementation stages, and is of proven global stability. The dynamic model of the heat pump undergoes approximate linearization with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. Linearization takes place at each sampling instance, around a temporary operating point that is defined by the present value of the heat pump’s state vector and by the last sampled value of the control inputs vector. For the approximately linearized model of the heat pump an H-infinity stabilizing controller is designed. The feedback gains of the controller are computed through the solution of an algebraic Riccati equation, at each time step of the control method. The global stability properties of the control scheme are proven through Lyapunov analysis. Moreover, to enable state estimation-based control, the H-infinity Kalman filter is used as a robust observer. The method achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs.

Convergence Analysis of Jarratt-like Methods for Solving Nonlinear Equations for Thrice-Differentiable Operators

The main goal of this paper is to study Jarratt-like iterative methods to obtain their order of convergence under weaker conditions. Generally, obtaining the pth-order convergence using the Taylor series expansion technique needed at least p+1 times differentiability of the involved operator. However, we obtain the fourth- and sixth-order for Jarratt-like methods using up to the third-order derivatives only. An upper bound for the asymptotic error constant (AEC) and a convergence ball are provided. The convergence analysis is developed in the more general setting of Banach spaces and relies on Lipschitz-type conditions, which are required to control the derivative. The results obtained are examined using numerical examples, and some dynamical system concepts are discussed for a better understanding of convergence ideas.

On Certain New Functional Generalizations of the Hardy and Levinson Integral Inequalities

Recent results have provided important functional generalizations, extensions and improvements of the Hardy and Levinson integral inequalities. However, they require some assumptions on the main functions, such as monotonicity or convexity assumptions, which remain somewhat restrictive. In this article, we propose two new ideas of functional generalizations, one based on a series expansion approach and the other on an integral approach. Both achieve the goal of offering adaptable generalizations and extensions of the Hardy and Levinson integral inequalities. They are formulated in two different general theorems, which are proved in detail. Several examples of new integral inequalities are derived.

Oscillation Flow of Viscous Electron Fluids in Conductors of Rectangular Cross-Section

The article presents results of an analytical and numerical modeling of electron fluid motion and heat generation in a rectangular conductor at an alternating electric potential. The analytical solution is based on the series expansion solution (Fourier method) and double series solution (method of eigenfunction decomposition). The numerical solution is based on the lattice Boltzmann method (LBM). An analytical solution for the electric current was obtained. This enables estimating the heat generation in the conductor and determining the influence of the parameters characterizing the conductor dimensions, the parameter M (phenomenological transport time describing momentum-nonconserving collisions), the Knudsen number (mean free path for momentum-nonconserving) and the Sh number (frequency) on the heat generation rate as an electron flow passes through a conductor.

Inhomogeneous and simultaneous Diophantine approximation in Cantor series expansions

Inhomogeneous and simultaneous Diophantine approximation in Cantor series expansions

Numerical Results for Gauss-Seidel Iterative Algorithm Based on Newton Methods for Unconstrained Optimization Problems

Optimization problems play a crucial role in various fields such as economics, engineering, and computer science. They involve finding the best value (maximum or minimum) of an objective function. In unconstrained optimization problems, the goal is to find a point where the function’s value reaches a maximum or minimum without being restricted by any conditions. Currently, there are many different methods to solve unconstrained optimization problems, one of which is the Newton method. This method is based on using a second-order Taylor series expansion to approximate the objective function. By calculating the first derivative (gradient) and second derivative (Hessian matrix) of the function, the Newton method determines the direction and step size to find the extrema. This method has a very fast convergence rate when near the solution and is particularly effective for problems with complex mathematical structures. In this paper, we introduce a Gauss-Seidel-type algorithm implemented for the Newton and Quasi-Newton methods, which is an efficient approach for finding solutions to optimization problems when the objective function is a convex functional. We also present some computational results for the algorithm to illustrate the convergence of the method.

Method of measuring precession details on a coordinate measuring machine

The article focuses on the development and implementation of an effective methodology for measuring high-precision parts on coordinate measuring machines (CMM). The proposed approach addresses the challenges associated with complex geometry measurements under variable environmental conditions by combining advanced mathematical modeling techniques with adaptive error compensation algorithms. The mathematical foundation is based on the application of tensor formalism in Riemannian space, which allows for more precise model-ing of geometric errors using Christoffel symbols and covariant derivatives of the error po-tential. This approach significantly improves the accuracy of spatial positioning. A refined stochastic error model, incorporating Stratonovich integrals and fractional Brownian motion, provides a more accurate description of random processes occurring in the measurement sys-tem. To enhance overall measurement accuracy, an adaptive correction algorithm is pro-posed, based on Itô stochastic differential equations with Fourier-Bessel series expansion, which ensures efficient compensation of systematic errors. In addition, the measurement tra-jectory optimization is formulated as a variational problem, taking into account holonomic and non-holonomic constraints, enabling the optimal positioning strategy. A thermoelastic deformation model based on sixth-rank tensors and Green’s functions is developed to account for temperature-induced deformations, ensuring the reliability of measurements under ther-mal instability. Experimental verification confirmed the effectiveness of the proposed method-ology, demonstrating a 15-20% reduction in systematic errors, a 10-15% decrease in random errors, and an approximately 20% improvement in the accuracy of measurement uncertainty estimation. The proposed methodology combines theoretical advancements with practical so-lutions, providing a robust tool for improving the accuracy and reliability of coordinate measurements in industrial metrology applications.

Longitudinal leading-twist distribution amplitude of the P11 -state b1(1235) meson and its implications on B→b1(1235)ℓ+νℓ decays

In the paper, we derive the ξ moments ⟨ξ2;b1n;∥⟩ of the longitudinal leading-twist distribution amplitude ϕ2;b1∥ for the P11-state b1(1235) meson by using the QCD sum rules under the background field theory. Considering the contributions from the vacuum condensates up to dimension six, its first two nonzero ξ moments at the scale 1 GeV are ⟨ξ2;b11;∥⟩=−0.647−0.113+0.118 and ⟨ξ2;b13;∥⟩=−0.328−0.052+0.055, respectively. Using those moments, we then fix the Gegenbauer expansion series of ϕ2;b1∥ and apply it to calculate B→b1(1235) transition form factors (TFFs) that are derived by using the QCD light-cone sum rules. Those TFFs are then extrapolated to the physically allowable q2 range via the simplified series expansion. As for the branching fractions, we obtain B(B¯0→b1(1235)+e−ν¯e)=2.179−0.422+0.553×10−4, B(B0→b1(1235)−μ+νμ)=2.166−0.415+0.544×10−4, B(B+→b1(1235)0e+νe)=2.353−0.456+0.597×10−4, and B(B+→b1(1235)0μ+νμ)=2.339−0.448+0.587×10−4, respectively. Published by the American Physical Society 2025