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

Computationally efficient solutions (less computation time) for the Colebrook equation are important for the simulation of pipeline networks. However, the friction law resistance formula has an implicit form with respect to the friction factor. In the present study, computationally efficient accurate explicit 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 computationally efficient as existing analytic solutions but provides a better accuracy in estimating the friction head loss.

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.

Lagrange Inversion and Combinatorial Species with Uncountable Color Palette

We prove a multivariate Lagrange-Good formula for functionals of uncountably many variables and investigate its relation with inversion formulas using trees. We clarify the cancellations that take place between the two aforementioned formulas and draw connections with similar approaches in a range of applications.

Explicit results for the distribution of the number of customers served during a busy period for $M^X/PH/1$ queue

<p style='text-indent:20px;'>We give analytically explicit solutions for the distribution of the number of customers served during a busy period for the <inline-formula><tex-math id="M1">\begin{document}$ M^X/PH/1 $\end{document}</tex-math></inline-formula> queues when initiated with <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> customers. When customers arrive in batches, we present the functional equation for the Laplace transform of the number of customers served during a busy period. Applying the Lagrange inversion theorem, we provide a refined result to this functional equation. From a phase-type service distribution, we obtain the distribution of the number of customers served during a busy period for various special cases such as exponential, Erlang-k, generalized Erlang, hyperexponential, Coxian, and interrupted Poisson process. The results are exact, rapid and vigorous, owing to the clarity of the expressions. Moreover, we also consider computational results for several service-time distributions using our method. Phase-type distributions can approximate any non-negative valued distribution arbitrarily close, making them a useful practical stochastic modelling tool. These distributions have eloquent properties which make them beneficial in the computation of performance models.</p>

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.