Lagrange Inversion Research Articles

Lagrange expansion formula and its application in combinatorial mathematics

There is a uniqueness theorem for an implicit function. When a function satisfies certain conditions, the equation can determine a unique implicit function. However, we cannot determine the display expression of the implicit function according to this theorem. The Lagrange expansion formula just compensates for this disadvantage, which explicitly gives the display expression of the implicit function. This paper will give a simpler proof of the Lagrange expansion formula and provide examples to illustrate its application in combinatorial mathematics.

Vibration analysis of rotating variable-angle-tow composite cylindrical structures via high-fidelity shell models

The paper focuses on rotordynamics analyses of various rotating variable-angle-tow (VAT) composite cylindrical structures using both low- and high-fidelity structural models. To model the spatially varying composite shell structures, the Carrera Unified Formulation (CUF) is employed. The CUF provides a hierarchical and automated approach for developing shell finite element models. Various shell models are created using different kinematics models based on Taylor or Lagrange expansion functions. The linearized equations of motion include the Coriolis and initial stress contributions. Several VAT composite curved panels and thin and thick cylinders have been analyzed to validate the presented approach and provide benchmark solutions. The results are discussed and compared with commercial software solutions. The results demonstrate the good accuracy and reliability of the proposed numerical methodology.

Hexadecapole at the heart of nonlinear electromagnetic fields

In classical Maxwell’s electromagnetism, the monopole term of the electric field is proportional to r −2, while higher order multipole terms, sourced by anisotropic sources, fall-off faster. However, in nonlinear electromagnetism even a spherically symmetric field has multipole-like contributions. We prove that the leading subdominant term of the electric field, defined by nonlinear electromagnetic Lagrangian obeying Maxwellian weak field limit, in a static, spherically symmetric, asymptotically flat spacetime, is of the order O(r−6) as r→∞ . Moreover, using Lagrange inversion theorem and Faà di Bruno’s formula, we derive the series expansion of the electric field from the Taylor series of an analytic electromagnetic Lagrangian.

Geometrically nonlinear transient analyses of rotating structures through high-fidelity models

This work presents geometrically nonlinear transient analyses of various rotating blades. The structures are discretized through refined beams or multi-dimensional finite element models, which are generated using the Carrera Unified Formulation (CUF). The CUF offers a procedure to develop low- and high-fidelity one-dimensional, two-dimensional, and three-dimensional finite element models hierarchically and automatically. Various beam models were developed using different kinematics models based on Taylor or Lagrange expansion functions. Multi-dimensional models were obtained by merging beam and solid elements, exploiting the unique feature of Lagrange polynomials to have only pure displacements as unknowns. This property allows beam and solid elements to be coupled at the node level without requiring complicated mathematical formulations. By utilizing the Finite Element Method in conjunction with the CUF, the governing equations are written by including all rotation effects, namely the Coriolis term, spin-softening, and geometrical stiffening. In a total Lagrangian scenario, the Hilbert-Hughes-Taylor-α method and the iterative Newton-Raphson scheme are employed to solve the equations of motion. The proposed methodology has been applied to evaluate different blade configurations, comparing the solution obtained using linear, linearized, and nonlinear approaches. The results have been verified and validated by comparing them with existing solutions present in the literature.

Free vibration analysis of composite beams and laminated reinforced panels by refined dynamic stiffness method and CUF-based component-wise theory

This paper presents a comprehensive analysis of the free vibration behavior of composite beams and laminated reinforced panels. Employing high-order theories with displacement variables only, the investigation combines dynamic stiffness method (DSM) and the Carrera unified formulation (CUF). The 3D displacement field can be expanded in the framework of CUF as any order of generic unknown variables over the cross-section, in the case of beam theories. Specifically, Lagrange expansions (LE) of cross-sectional displacement variables are considered, enabling the refinement modeling of complex cross-sections with different layers (layer-wise) and components (component-wise). The governing differential equations and natural boundary conditions are derived using the principle of virtual displacements. Subsequently, an exact dynamic stiffness matrix is developed by establishing a relationship between the amplitudes of harmonically varying loads and the corresponding responses. The Wittrick–Williams algorithm is employed to solve the transcendental eigenvalue problem resulting from this approach. The Lagrange-based CUF(LE)-DSM outperforms other polynomials (such as Taylor)-based CUF in analyzing composite structures, enabling detailed analysis with various geometries, lamination schemes and boundary conditions. It delivers accurate results in reinforced panel analysis, requiring only 13% of the DOFs compared to 3D FE models, thereby reducing computational costs significantly, as confirmed by comparative studies and validations.

An Improved NLCS Algorithm Based on Series Reversion and Elliptical Model Using Geosynchronous Spaceborne–Airborne UHF UWB Bistatic SAR for Oceanic Scene Imaging

Geosynchronous spaceborne–airborne (GEO-SA) ultra-high-frequency ultra-wideband bistatic synthetic aperture radar (UHF UWB BiSAR) provides high-precision images for marine and polar environments, which are pivotal in glacier monitoring and sea ice thickness measurement for polar ocean mapping and navigation. Contrasting with traditional high-frequency BiSAR, it faces unique challenges, such as the considerable spatial variability, significant range–azimuth coupling, and vast volumes of echo data, which impede high-resolution image reconstruction. This paper presents an improved bistatic nonlinear chirp scaling (NLCS) algorithm for imaging oceanic scenes with GEO-SA UHF UWB BiSAR. This methodology extends the two-dimensional (2-D) spectrum up to the sixth order via the method of series reversion (MSR) to meet accuracy demands and then employs an elliptical model to elucidate the alterations in the azimuth frequency modulation (FM) rate mismatch. Initially, the imaging geometry and signal model are introduced, and then a separation of bistatic slant ranges based on the configuration is proposed. In addition, during range processing, after eliminating linear range cell migration (RCM), the derivation process for the sixth-order 2-D spectrum is detailed and an improved filter is applied to correct the high-order RCM. Finally, during azimuth processing, the causes of the FM rate mismatch are analyzed, a cubic perturbation function derived from the elliptical model is used for FM rate equalization, and a unified sixth-order filter is applied to complete the azimuth compression. Experimental results with point targets and natural oceanic scenes validate the outstanding efficacy of the proposed NLCS algorithm, particularly in imaging quality enhancements for GEO-SA UHF UWB BiSAR.

Dynamic stiffness method and CUF-based component-wise theories applied to free vibration analysis of solid beams, thin-walled structures and reinforced panels

In this paper, an exact dynamic stiffness formulation using higher-order theories with displacement variables only is presented and subsequently used to investigate the free vibration characteristics of solid beams, thin-walled structures and reinforced panel structures. In essence, higher-order displacement fields are developed by using the Carrera unified formulation (CUF), and by discretizing the cross-section kinematics with bilinear, cubic and fourth-order Lagrange polynomials. In particular, the component-wise (CW) approach based on Lagrange expansion is applied in which the solid part and thin-walled part are considered as two independent components that can be assembled. The principle of virtual displacements is used to derive the governing differential equations and the associated natural boundary conditions. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The explicit terms of the dynamic stiffness matrices are also presented. The Wittrick–Williams algorithm of the dynamic stiffness method (DSM) is applied with the explicit expressions of the J0 count for beam elements under all above support conditions. In return, there is no need to refine the element in the DSM, and thus, it becomes immensely efficient. The accuracy and efficiency of the proposed methodology are extensively assessed for different solid beams, thin-walled structures and reinforced panels and the results are compared with those appearing in published literature and also checked by 3D finite element (FE) solutions.

Application of Lagrange inversion to wall-crossing for Quot schemes on surfaces

Motivated by my work on enumerative invariants for Quot schemes, I related two power series obtained by two different means. One of them was computed using geometric arguments via virtual localization methods and the other one came from working with representation theoretic objects called vertex algebras. In this note, I give proof of the equality of the two power series by relying only on techniques related to Lagrange inversion. This makes my work on Quot schemes independent of the previous results in the literature and proves a new combinatorial identity.

A wide-beam NCS algorithm for multi-receiver SAS based on azimuth spectrum superposition

The existing multi-receiver synthetic aperture sonar (SAS) imaging algorithms are suitable for narrow-beam width, which will lead to a decrease in imaging quality under wide-beam condition and are not in line with the development needs of SAS. We propose a non-linear chirp scaling algorithm (NCSA) for wide beam multi-receiver SAS. Firstly, the point target reference spectrum (PTRS) of each receiver is obtained by the Lagrange inversion theorem (LIT), and then the under-sampled signal in the azimuth frequency domain is obtained through azimuth spectrum extension; Then, considering the cubic term of range frequency in the PTRS and the linear variation of equivalent frequency modulation slope with range, each receiver is imaged using the NCSA, and coherent superposition is performed in the azimuth frequency domain to eliminate spectrum aliasing caused by azimuth spectrum extension; Finally, the azimuth inverse transform is performed on the superimposed signal to obtain the focusing imaging. Computer simulation experiments and field data verify that this method is superior to the existing SAS imaging algorithm, improving the quality of wide-beam imaging, avoiding the interpolation operation of the traditional range-Doppler algorithm, and saving computation cost.

Some γ-positive polynomials arising from enumerations of the pseudo Schröder paths

We introduce a notion of pseudo Schröder paths, which are counted by the Schröder numbers. Using the symbolic method and the Lagrange inversion formula, we provide an alternative proof that the descent polynomials of separable permutations are γ-positive. We also present several classes of γ-positive polynomials via enumerating the pseudo Schröder paths according to several combinatorial statistics by applying Riordan arrays.

Sideband Harmonic Model Considering Sampling Delay Effect in the PMSM Drives With Regular Sampled SVPWM Technique under the Low Carrier Frequency Ratio

Regular sampled space vector pulse-width modulation (SVPWM) is a widely implemented technique in permanent magnet synchronous machine (PMSM) drives with the voltage-source inverter (VSI). The modulation technique would generate the sideband harmonics and increase electromagnetic loss, vibration, and acoustic noise. Therefore, a fast and accurate evaluation of sideband harmonics is particularly important. However, the carrier frequency ratio could be relatively low in some applications, which means it would cause sampling delay and influence the sideband harmonics. This paper analyzes the sampling delay effect in PMSM drives, and proposes a modified sideband harmonic model using Lagrange expansion and complex coordinate transformation for both symmetric and asymmetric regular sampling methods. Furthermore, the amplitudes and phases of whole sideband harmonic components are derived. Experiment results are carried out to validate the accuracy improvement of the modified model, which reveals that it could give a superior estimation of the sideband harmonics in the PMSM drives under the low carrier frequency ratio.

Run-and-tumble oscillator: Moment analysis of stationary distributions

When it comes to active particles, even an ideal gas model in a harmonic potential poses a mathematical challenge. An exception is a run-and-tumble particles (RTP) model in one dimension for which a stationary distribution is known exactly. The case of two dimensions is more complex, but the solution is possible. Incidentally, in both dimensions the stationary distributions correspond to a beta function. In three dimensions, a stationary distribution is not known but simulations indicate that it does not have a beta function form. The current work focuses on the three-dimensional RTP model in a harmonic trap. The main result of this study is the derivation of the recurrence relation for generating moments of a stationary distribution. These moments are then used to recover a stationary distribution using the Fourier–Lagrange expansion.

A new generalization of the zero-truncated negative binomial distribution by a Lagrange expansion with associated regression model and applications

A new generalization of the zero-truncated negative binomial distribution by a Lagrange expansion with associated regression model and applications

Lagrange Inversion Formula by Induction

We present a simple inductive proof of the Lagrange Inversion Formula.

Series reversion for electrical impedance tomography with modeling errors *

This work extends the results of Garde and Hyvönen (2022 Math. Comput. 91 1925–1953) on series reversion for Calderón’s problem to the case of realistic electrode measurements, with both the internal admittivity of the investigated body and the contact admittivity at the electrode-object interfaces treated as unknowns. The forward operator, sending the internal and contact admittivities to the linear electrode current-to-potential map, is first proven to be analytic. A reversion of the corresponding Taylor series yields a family of numerical methods of different orders for solving the inverse problem of electrical impedance tomography, with the possibility to employ different parametrizations for the unknown internal and boundary admittivities. The functionality and convergence of the methods is established only if the employed finite-dimensional parametrization of the unknowns allows the Fréchet derivative of the forward map to be injective, but we also heuristically extend the methods to more general settings by resorting to regularization motivated by Bayesian inversion. The performance of this regularized approach is tested via three-dimensional numerical examples based on simulated data. The effect of modeling errors related to electrode shapes and contact admittances is a focal point of the numerical studies.

A Novel Zero-Truncated Katz Distribution by the Lagrange Expansion of the Second Kind with Associated Inferences

In this article, the Lagrange expansion of the second kind is used to generate a novel zero-truncated Katz distribution; we refer to it as the Lagrangian zero-truncated Katz distribution (LZTKD). Notably, the zero-truncated Katz distribution is a special case of this distribution. Along with the closed form expression of all its statistical characteristics, the LZTKD is proven to provide an adequate model for both underdispersed and overdispersed zero-truncated count datasets. Specifically, we show that the associated hazard rate function has increasing, decreasing, bathtub, or upside-down bathtub shapes. Moreover, we demonstrate that the LZTKD belongs to the Lagrangian distribution of the first kind. Then, applications of the LZTKD in statistical scenarios are explored. The unknown parameters are estimated using the well-reputed method of the maximum likelihood. In addition, the generalized likelihood ratio test procedure is applied to test the significance of the additional parameter. In order to evaluate the performance of the maximum likelihood estimates, simulation studies are also conducted. The use of real-life datasets further highlights the relevance and applicability of the proposed model.

Normalized nodal distance-based J2-perturbed ground-track adjustment with apsidal-altitude constraints

This paper presents a novel analytical approach based on normalized nodal distance to obtain the entire set of solutions for the J2-perturbed ground-track adjustment problem under apsidal-altitude constraints. First, by introducing general nodes and considering apsidal-altitude and maximum fuel constraints, a conversion method is proposed to convert these constraints into equivalent linear inequality constraints on normalized nodal distance. A relation curve between post-maneuver apsidal altitudes is also provided. Next, two explicit analytical expressions are derived separately based on Lagrange's expansion for two crucial transcendental equations, one for the time-of-flight concerning the argument of latitude and the other for the impulse concerning the normalized nodal distance. Then, by obtaining a starting node, this constrained ground-track adjustment problem can be transformed into a simple distance assignment problem. Combining with the inequality constraints, the entire solution set concerning the normalized nodal distance can be obtained immediately. Additionally, for the case of an empty solution set, a method for determining the relative longitude coordinates of no-solution intervals is also proposed. The case study demonstrates the effectiveness and accuracy of the proposed method and finds that for a fixed departing point, the most fuel-efficient time-of-flight window is not always the maximum allowable window within the specified time range.

Significance of skewness and kurtosis on the solute dispersion in pulsatile Carreau–Yasuda fluid flow in a tube with wall absorption

Solute dispersion in Carreau–Yasuda fluid flow in a tube presented in Rana & Murthy (Proc. R. Soc. Lond.A, vol. 472, 2016, p. 20160294) was limited to a steady-state velocity profile due to the nonlinearity associated with the Yasuda parameter$a$with power-law exponent$n$. This limitation is overcome and the velocity profile is obtained for all values of the Yasuda parameter by using the Lagrange inversion theorem, which admits power series solution for the flow field. An analytical solution for the concentration distribution in the circular tube is obtained for the unsteady and pulsatile flow with$n\leq 1$and$\alpha <<1$and the numerical solution is presented for all values of$\alpha$and$n$. The solute dispersion is analysed analytically using the Sankarasubramanian–Gill generalized dispersion method and also using the Aris–Barton method of moments considering up to fourth-order moments. The solute dispersion is also simulated numerically by using a new class of computationally explicit Runge–Kutta method. The axial mean concentration of the solute is estimated by the exchange, convective and dispersion coefficients. The third- and fourth-order moments give rise to skewness and kurtosis revealing the deviation from the Gaussianity and reduction in the peak of the mean concentration profile at a small time of the solute injection. All time variations of these five moments against flow governing parameters are thoroughly investigated. The flow and dispersion regimes that are derived here for moments provide a good understanding of the solute dispersion in the tube. The increase in the Womersley frequency parameter led to a phase lag at each period. This work is the initiation of estimating the skewness and kurtosis in a non-yield stress fluid flow in a tube.

An analysis of the propagation of surface acoustic waves in a substrate covered by a metal T-plate with the Carrera unified formulation

In this work, the wave propagation of Rayleigh type through a periodic elastic element covered with a T-plate is analyzed. Viscous-spring artificial boundaries are used to satisfy boundary conditions of a periodic structure. By using the Carrera Unified Formulation (CUF), the various kinematics of the three-dimensional structure are consistently expressed and the propagation of waves within the model is obtained. Lagrange expansion is employed to generate the mathematical model of the structure. The numerical evaluation and the comparison with the results obtained by the analytical method and COMSOL involving different structures reveal that the results of this study are reliable and show that the model developed provides accurate results for the analysis of the periodic elastic structure of T-plates.

The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination

We revisit the secular 3D planetary three-body problem, aiming to provide a unified formalism representing all basic phenomena in the phase space as the mutual inclination between the planetary orbits increases. We propose a ‘book-keeping’ technique allowing to decompose the Hamiltonian as mathcal {H}_textrm{sec}=mathcal {H}_textrm{planar}+mathcal {H}_textrm{space}, with mathcal {H}_textrm{space} collecting all terms depending on the planets mutual inclination i_textrm{mut}. We numerically compare several models obtained by multipole (Legendre) or Laplace–Lagrange expansions of mathcal {H}_textrm{sec}, aiming to define suitable truncation orders for these models. We explore the transition, as i_textrm{mut} increases, from a ‘planar-like’ to a ‘Lidov–Kozai’ regime. Using a numerical example far from hierarchical limits, we find that the structure of the phase portraits of the (integrable) planar case is reproduced to a large extent also in the 3D case. A semi-analytical criterion allows to estimate the level of i_textrm{mut} up to which the dynamics remains nearly integrable. We propose a normal form method to compute the basic periodic orbits (apsidal corotation orbits A and B) in this regime. We explore the sequence of saddle-node and pitchfork bifurcations by which the A and B families are connected to the highly inclined periodic orbits of the Lidov–Kozai regime. Finally, we perform a numerical study of phase portraits for different planetary mass and distance ratios and qualitatively describe the approach to the corresponding hierarchical limits.