Lagrange Inversion Research Articles

Computationally Efficient and Accurate Solution for Colebrook Equation Based on Lagrange Theorem

Abstract Computationally efficient solutions (less computation time) for the Colebrook equation are essential for simulating pipeline networks. However, the friction law resistance formula has an implicit form for the friction factor. In this study, a computationally efficient and accurate solution for the friction head loss in pipeline networks is developed using the Lagrange inversion theorem. The results are in the form of fast converging power series. Truncated and regressed expressions are obtained using two and three terms of the expanded series that have maximum relative errors of 0.149% and 0.040%, respectively. The proposed solution is as accurate as existing analytic solutions but is computationally more efficient in estimating the friction head loss.

Coalescense with arbitrary-parameter kernels and monodisperse initial conditions: A study within combinatorial framework

For this work, we studied a finite system of discreet-size aggregating particles for two types of kernels with arbitrary parameters, a condensation (or branched-chain polymerization) kernel, $K(i,j)=(A+i)(A+j)$, and a linear combination of the constant and additive kernels, $K(i,j)=A+i+j$. They were solved under monodisperse initial conditions in the combinatorial approach where discreet time is counted as subsequent states of the system. A generating function method and Lagrange inversion were used for derivations. Expressions for an average number of clusters of a given size and its corresponding standard deviation were obtained and tested against numerical simulation. High precision of the theoretical predictions can be observed for a wide range of $A$ and coagulation stages, excepting post-gel phase in the case of the condensation kernel (a giant cluster presence is preserved). For appropriate $A$, these two kernels reproduced known results of the constant, additive and product kernels. Beside a previously solved linear-chain kernel, they extend the number of arbitrary-parameter kernels solved in the combinatorial approach.

Efficient strong Unified Formulation for stress analysis of non-prismatic beam structures

The Unified Formulation (UF) has gained attention as a powerful tool for efficient design of structural components. Due to the inherent flexibility of its kinematics representation, arbitrary shape functions can be selected in different dimensions to achieve a high-fidelity characterisation of structural response under load. Despite this merit, the classical isoparametric description of UF limits the application to prismatic structures. The weak-form anisoparametric approach adopted to overcome this limitation in a recent work by Patni et al. proves to be versatile yet computationally challenging owing to the expensive computation of its UF stiffness matrix by means of full volume integrals. We propose a strong-form anisoparametric UF (SUF) based on the Serendipity Lagrange Expansion (SLE) cross-sectional finite element and differential quadrature beam element. The main objective of the SUF is to achieve an efficient computation of the UF stiffness matrix by restricting Gauss operations to the variable cross-sections of non-prismatic structures in a discrete sense, thus eliminating the need for full volume integrals. When assessed against weak-form based UF, ABAQUS FE and analytical solutions, the static analysis of non-prismatic beam-like structures under different loads by the SUF is shown to be accurate, numerically stable, and computationally more efficient than state-of-the-art methods.

Focus Improvement of Airborne High-Squint Bistatic SAR Data Using Modified Azimuth NLCS Algorithm Based on Lagrange Inversion Theorem

In this paper, a modified azimuth nonlinear chirp scaling (NLCS) algorithm is derived for high-squint bistatic synthetic aperture radar (BiSAR) imaging to solve its inherent difficult issues, including the large range cell migration (RCM), azimuth-dependent Doppler parameters, and the sensibility of the higher order terms. First, using the Lagrange inversion theorem, an accurate spectrum suitable for processing airborne high-squint BiSAR data is introduced. Different from the spectrum that is based on the method of series reversion (MSR), it is allowed to derive the bistatic stationary phase point while retaining the double square root (DSR) of the slant range history. Based the spectrum, a linear RCM correction is used to remove the most of the linear RCM components and mitigate the range-azimuth coupling, and, then, bulk secondary range compression is implemented to compensate the residual RCM and cross-coupling terms. Following this, a modified azimuth NLCS operation is applied to eliminate the azimuth-dependence of Doppler parameters and equalize the azimuth frequency modulation for azimuth compression. The experimental results, with better focusing performance, prove the high accuracy and effectiveness of the proposed algorithm.

Buckling and post-buckling of anisotropic flat panels subjected to axial and shear in-plane loadings accounting for classical and refined structural and nonlinear theories

This article investigates the large deflection and post-buckling of composite plates by employing the Carrera Unified Formulation (CUF). As a consequence, the geometrically nonlinear governing equations and the relevant incremental equations are derived in terms of fundamental nuclei, which are invariant of the theory approximation order. By using the Lagrange expansion functions across the laminate thickness and the classical finite element (FE) approximation, layerwise (LW) refined plate models are implemented. The Newton–Raphson linearization scheme with the path-following method based on the arc-length constraint is employed to solve geometrically nonlinear composite plate problems. In this study, different composite plates subjected to large deflections/rotations and post-buckling are analysed, and the corresponding equilibrium curves are compared with the results in the available literature or the traditional FEM-based solutions. The effects of various parameters, such as stacking sequence, number of layers, loading conditions, and edge conditions are demonstrated. The accuracy and reliability of the proposed method for solving the composite plates’ geometrically nonlinear problems are verified.

Focusing Translational-Variant Bistatic Forward- Looking SAR Data Using the Modified Omega-K Algorithm

Accurate 2-D frequency spectrum (2-D FS) with acceleration, two-way range coupling terms, and spatial-variant Doppler parameters are the main problems to be solved in translational-variant bistatic forward-looking synthetic aperture radar (SAR) (TV BFSAR) with curved trajectory. For these issues, a modified omega-K imaging algorithm is derived in this article. The maximum usage of 2-D FS based on the method of series reversion (MSR) is achieved by linear range cell migration correction, and 2-D FS is linearized in bistatic range by using high-order polynomial fitting. Then, a method of azimuth resampling is introduced to implement compensation of spatial-variant Doppler parameters. Different from other bistatic omega-K methods, our newly proposed method focuses on the small-aperture data in the azimuth frequency domain to avoid azimuth aliasing without padding zeros and uses the frequency focusing position to study the model of spatial-variant phase. Simulation results and real data verify the effectiveness of the proposed method.

Moving Target Detection and Parameter Estimation via a Modified Imaging STAP with a Large Baseline in Multistatic GEO SAR

With the development trends of multistatic spaceborne synthetic aperture radar (SAR), geosynchronous SAR (GEO SAR) employing several formation-flying small satellites also has great potential for remote sensing. The small satellites can cooperate to acquire multi-channel data for moving target detection and parameter estimation in strong clutters. However, multistatic GEO SAR has large satellite spacing and a curved trajectory, which induce the near-field effects and channels out of alignment, respectively, bringing about challenges for the spatial adaptive processing. These problems produce a high-order term in the multi-channel slant range model, making the traditional model and adaptive processing method invalid. In this paper, to meet the requirement of SAR focusing, we firstly derive a fourth-order slant range model and a third-order path difference model for multistatic GEO SAR. Secondly, based on the derived model, the principle of stationary phase and series reversion method are utilized to derive the spatial steering vector for a moving target, which is a basis of spatial adaptive processing in the range-Doppler domain. Thirdly, the time-domain match filtering is constructed based on the fourth-order slant range model to image the moving target. Additionally, the moving targets are detected in the image domain. The motion parameter is estimated by iteratively maximizing the output signal to clutter and noise ratio (SCNR) through the range of possible target velocities. Finally, considering that the GEO SAR is still in development, the computer simulations are carried out to verify the effectiveness and evaluate the performance.

A novel AFNCS algorithm for super-resolution SAR in curve trajectory

With the improvement of SAR resolution, super-resolution SAR imaging is more and more widely used in the all-time and all-weather video surveillance and remote sensing imaging. One implementation of super-resolution SAR is that radar works in the spotlight mode. In the case with highly squinted angle and acceleration, the azimuth space variance and the coupling between range and azimuth will become serious in super-resolution imaging. Thus, an azimuth frequency nonlinear chirp scaling algorithm is proposed to solve this problem. Based on the acceleration model, the accurate 2-D spectrum is performed by adopting the method of series reversion. The space-variance of missile-borne SAR in curved flight path is analyzed and an azimuth polynomial phase filter is constructed to make the coefficients of the perturbation function has sufficient flexibility to eliminate the spatial-variant couple terms between range and azimuth. In addition, the gradient operation is used to expand the space-variant coefficients of azimuth modulation term, and the perturbation function is applied to eliminate the space-variant in azimuth direction. The proposed focusing method can process the missile-borne SAR data obtained in spotlight mode in highly squinted angle with high efficiency. The simulation results verify the effectiveness of the proposed imaging approach. The integration of the research in this paper and the deep learning will further pave the way of super-resolution SAR imaging applications in disaster monitoring, security and surveillance.

Approximate analytical analysis of longitudinal natural frequencies of a cracked beam

PurposeIn this paper, an approximate analytical approach is developed for the determination of natural longitudinal frequencies of a cantilever-cracked beam based on the Lagrange inversion theorem.Design/methodology/approachThe crack is modeled by an equivalent axial spring with stiffness according to Castigliano's theorem. Thus, an implicit frequency equation corresponding to cantilever-cracked bar is obtained. The resulting equation is solved using the Lagrange inversion theorem.FindingsEffect of different crack depths and crack positions on natural frequencies of the cracked beam is analyzed. It is shown that an increase in the crack depth ratio produces a decrease in the fundamental longitudinal natural frequency of a cracked bar. Furthermore, approximate analytical results are compared with those obtained numerically as well as from experimental tests.Originality/valueA new approximate analytical expression of a fundamental longitudinal frequency, as a function of crack depth and crack location, is obtained.

Lagrange Inversion Theorem for Dirichlet series

We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [5].

Bistatic-Range-Doppler-Aperture Wavenumber Algorithm for Forward-Looking Spotlight SAR With Stationary Transmitter and Maneuvering Receiver

Bistatic forward-looking spotlight synthetic aperture radar with stationary transmitter and maneuvering receiver (STMR-BFSSAR) is a promising sensor for various applications, such as the automatic navigation and landing of maneuvering vehicles. Because of the bistatic forward-looking configuration and the receiver’s maneuvers, conventional image formation algorithms suffer from high computational complexity or small size of a well-focused scene if applied to STMR-BFSSAR. In this article, we propose a wavenumber-domain algorithm for STMR-BFSSAR image formation, which is termed the bistatic-range-Doppler-aperture wavenumber algorithm (BDWA). First, a novel range model in bistatic-range and Doppler-aperture coordinate space instead of conventional Cartesian coordinate space is established by employing the elliptic polar coordinate system and the method of series reversion. The novel range model not only makes the echo’s samples to be regular along the direction of the bistatic-range wavenumber axis but also constructs a curved wavefront close to the true wavefront. Second, an operation termed wavenumber-domain gridding is conceived to regularize the echo’s samples along the Doppler-aperture wavenumber axis, which can be implemented by 1-D interpolation. The proposed algorithm significantly outperforms the conventional algorithms in terms of computational complexity and scene size limits. Both point and distributed targets are simulated for two STMR-BFSSAR systems with different parameters. The simulation results verify the validity and superiority of the proposed BDWA.

Properties of the \u0454-expansion, Lagrange inversion and associahedra and the O (1) model

We discuss properties of the є-expansion in d = 4 − є dimensions. Using Lagrange inversion we write down an exact expression for the value of the Wilson-Fisher fixed point coupling order by order in є in terms of the beta function coefficients. The є-expansion is combinatoric in the sense that the Wilson-Fisher fixed point coupling at each order depends on the beta function coefficients via Bell polynomials. Using certain properties of Lagrange inversion we then argue that the є-expansion of the Wilson-Fisher fixed point coupling equally well can be viewed as a geometric expansion which is controlled by the facial structure of associahedra. We then write down an exact expression for the value of anomalous dimensions at the Wilson-Fisher fixed point order by order in є in terms of the coefficients of the beta function and anomalous dimensions. We finally use our general results to compute the values for the Wilson-fisher fixed point coupling and critical exponents for the scalar O (1) symmetric model to O(є7).

Convergence of Cluster and Virial expansions for Repulsive Classical Gases

We study the convergence of cluster and virial expansions for systems of particles subject to positive two-body interactions. Our results strengthen and generalize existing lower bounds on the radii of convergence and on the value of the pressure. Our treatment of the cluster coefficients is based on expressing the truncated weights in terms of trees and partition schemes, and generalize to soft repulsions previous approaches for models with hard exclusions. Our main theorem holds in a very general framework that does not require translation invariance and is applicable to models in general measure spaces. Our virial results, stated only for homogeneous single-space systems, rely on an approach due to Ramawadh and Tate. The virial coefficients are computed using Lagrange inversion techniques but only at the level of formal power series, thereby yielding diagrammatic expressions in terms of trees, rather than the doubly connected diagrams traditionally used. We obtain a new criterion that strengthens, for repulsive interactions, the best criterion previously available (proposed by Groeneveld and proven by Ramawadh and Tate). We illustrate our results with a few applications showing noticeable improvements in the lower bound of convergence radii.

ANALYTICALLY CLOSED-FORM SOLUTIONS FOR THE DISTRIBUTION OF A NUMBER OF CUSTOMERS SERVED DURING A BUSY PERIOD FOR SPECIAL CASES OF THE GEO/G/1 QUEUE

AbstractThis paper presents the distribution of the number of customers served during a busy period for special cases of the Geo/G/1 queue when initiated with m customers. We analyze the system under the assumptions of a late arrival system with delayed access and early arrival system policies. It is not easy to invert the functional equation for the number of customers served during a busy period except for the simple case Geo/Geo/1 queue, as stated by several researchers. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We find the distribution of the number of customers served during a busy period for various service-time distributions such as geometric, deterministic, binomial, negative binomial, uniform, Delaporte, discrete phase-type and interrupted Bernoulli process. We compute the mean and variance of these distributions and also give numerical results. Due to the clarity of the expressions, the computations are very fast and robust. We also show that in the limiting case, the results tend to the analogous continuous-time counterparts.

High-Order AVO Inversion for Effective Pore-Fluid Bulk Modulus Based on Series Reversion and Bayesian Theory

Pore-fluid identification is one of the key technologies in seismic exploration. Fluid indicators play important roles in pore-fluid identification. For sandstone reservoirs, the effective pore-fluid bulk modulus is more susceptible to pore-fluid than other fluid indicators. AVO (amplitude variation with offset) inversion is an effective way to obtain fluid indicators from seismic data directly. Nevertheless, current methods lack a high-order AVO equation for a direct, effective pore-fluid bulk modulus inversion. Therefore, based on the Zoeppritz equations and Biot–Gassmann theory, we derived a high-order P-wave AVO approximation for an effective pore-fluid bulk modulus. Series reversion and Bayesian theory were introduced to establish a direct non-linear P-wave AVO inversion method. By adopting this method, the effective pore-fluid bulk modulus, porosity, and density can be inverted directly from seismic data. Numerical simulation results demonstrate the precision of our proposed method. Model and field data evaluations show that our method is stable and feasible.

A Range Doppler Imaging Algorithm Based on the MSR Spectrum for the Multi-Aperutre Syntactic Sonar

For ignoring the azimuth variation of the “non-hop-go-hop” time, the existing imaging algorithm for multi-aperture have a poor imaging in moderate squint, so this paper proposes a multi-aperture range Doppler imaging algorithm based on the spectrum of method of series reversion. The exact range history is approximated once, and the error can be adjusted by the order of Taylor series expansion. Then, the analytical expression of the two-dimensional spectrum is obtained by the method of series reversion. In signal processing, the signal of the single aperture is handle with the squint range Doppler imaging algorithm, and the azimuth spectrum is superposed to each aperture, to obtain the image. Finally, the effectiveness and correctness of the algorithm is provided by the simulation experiments.

Evaluation of the bending response of compact and thin-walled FG beams with CUF

The study deals with the stress evaluations of the functionally graded beams (FGBs) using refined beam models. The refined beam models are obtained within the framework of the Carrera unified formulation (CUF), where expansions are used across the cross-section to enrich the kinematics of the finite elements and obtain 3 D-like accuracy of the solution. Analyses of FGBs are performed by using different structural theories based on Taylor-like and Lagrange expansions. The results are compared with the literature and 3 D finite element solutions. It is observed that calculation costs are significantly decreased in finite element analyses performed with CUF.

Efficient Strip-Mode SAR Raw-Data Simulator of Extended Scenes Included Moving Targets Based on Reversion of Series

Efficient Strip-Mode SAR Raw-Data Simulator of Extended Scenes Included Moving Targets Based on Reversion of Series

A Generalized Wavefront-Curvature-Corrected Polar Format Algorithm to Focus Bistatic SAR Under Complicated Flight Paths

Bistatic synthetic aperture radar (BiSAR) imaging is faced with two major challenges: large scene imaging and adaptability to unideal platform motion in practice. In order to deal with these two problems, a generalized wavefront-curvature-corrected polar format algorithm (PFA) is proposed in this article. The traditional PFA is little restricted on geometry configuration and platform motion, but its application to large scene imaging is limited by the far-field planar wavefront assumption. To solve this limitation, this article derives the phase error caused by wavefront curvature and analyzes its influence on both geometric distortion and defocusing effect in detail. Based on the analysis, we present a wavefront curvature completely correcting method through space-variant phase compensation using the analytical wavefront curvature phase in wavenumber-domain, which is derived through method of series reversion. What's more, an efficient realization of the space-variant phase compensation based on two-stage image division is given to avoid high overlap rate in the traditional image division method. The proposed method can obtain well focused and geometric undistorted image for BiSAR under complicated flight paths, and it also keeps the logarithmic complexity of traditional PFA. The effectiveness of the proposed method is verified by numerical simulations.

Labeled binary trees, subarrangements of the Catalan arrangements, and Schur positivity

In 1995, the first author introduced a multivariate generating function G that tracks the distribution of ascents and descents in labeled binary trees. In addition to proving that G is symmetric, he conjectured that G is Schur positive. We prove this conjecture by expanding G positively in terms of ribbon Schur functions. We obtain this expansion using a weight-preserving bijection whose inverse is inspired by the Push-Glide algorithm of Préville-Ratelle and Viennot. In fact, this weight-preserving bijection allows us to establish a stronger version of the first author's conjecture showing that the generating function restricted to labeled binary trees with a fixed canopy is still Schur positive.We also discuss applications in the setting of hyperplane arrangements. We show that a certain specialization of G equals the Frobenius characteristic of the natural Sn-action on regions of the semiorder arrangement, which we then expand in terms of the Frobenius characteristics of Foulkes characters. We also construct an Sn-action on regions of the Linial arrangement using a set of trees studied by Bernardi, and subsequently compute the character of this action by employing Lagrange inversion. The resulting expression generalizes Postnikov's formula for the number of regions in the Linial arrangement. As a final application, we prove γ-nonnegativity for the distribution of the number of right edges over local binary search trees.