Search Results (7)

We consider two specific families of binomial trees and forests: simply generated binomial d-ary trees and forests versus their increasing phylogenetic version, with tree nodes in increasing order from the root to any of its leaves. The analysis (both pre-asymptotic and asymptotic) consists of some of the main statistical features of their total progenies. We take advantage of the fact that the random distribution of those trees are obtained while weighting the counts of the underlying combinatorial trees. We finally briefly stress a rich alternative randomization of combinatorial trees and forests, based on the ratio of favorable count outcomes to the total number of possible ones. Full article

To compute the maximum power point (MPP) from physical parameters of the single-diode model (SDM), it is necessary to solve a transcendental equation using numerical methods. This is computationally expensive and can lead to divergence problems. An alternative is to develop analytical approximations which can be accurate enough for engineering problems and simpler to use. Therefore, this paper presents approximations for computing the MPP of single-junction solar cells. Two special cases are considered: (i) SDM with only series resistance, and (ii) SDM with only shunt resistance. Power series closed-form expressions for the MPP are obtained using perturbation theory and the Lagrange inversion theorem. Validation of the formulas is performed using experimental data from six different technologies obtained from the NREL database and comparing the results with the numerical solution of the SDM and three approximations from the literature. The results show an absolute percentage error (APE) of less than 0.035% with respect to the real MPP measurements. In cases with limited computational resources, this value could be further improved by using a higher- or lower-order power-series approximation. Full article

A new variance formula is developed using the generalized inverse of an increasing function. Based on the variance formula, a new entropy formula for any uncertain variable is provided. Most of the entropy formulas in the literature are special cases of the new entropy formula. Using the new entropy formula, the maximum entropy distribution for unimodel entropy of uncertain variables is provided without using the Euler–Lagrange equation. Full article

By establishing new multiple inverse series relations (with their connection coefficients being given by higher derivatives of fixed multivariate analytic functions), we illustrate a general framework to provide new proofs for MacMahon’s master theorem and the multivariate expansion formula due to Good (1960). Further multivariate extensions of the derivative identities due to Pfaff (1795) and Cauchy (1826) will be derived and the generalized multifold convolution identities due to Carlitz (1977) will be reviewed. Full article

In this invited survey-cum-expository review article, we present a brief and comprehensive account of some general families of linear and bilinear generating functions which are associated with orthogonal polynomials and such other higher transcendental functions as (for example) hypergeometric functions and hypergeometric polynomials in one, two and more variables. Many of the results as well as the methods and techniques used for their derivations, which are presented here, are intended to provide incentive and motivation for further research on the subject investigated in this article. Full article

Special polynomials play an important role in several subjects of mathematics, engineering, and theoretical physics. Many problems arising in mathematics, engineering, and mathematical physics are framed in terms of differential equations. In this paper, we introduce the family of the Lagrange-based hypergeometric Bernoulli polynomials via the generating function method. We state some algebraic and differential properties for this family of extensions of the Lagrange-based Bernoulli polynomials, as well as a matrix-inversion formula involving these polynomials. Moreover, a generating relation involving the Stirling numbers of the second kind was derived. In fact, future investigations in this subject could be addressed for the potential applications of these polynomials in the aforementioned disciplines. Full article

Ultra−high spatial resolution, which can bring more detail to ground observation, is a constant pursuit of the modern space−borne synthetic aperture radar. However, the exact imaging in this case has always been a complex technical problem due to its complicated imaging geometry and signal structure. To achieve those applications’ strict requirements, a novel ultra−high resolution imaging algorithm based on an accurate high−order 2−D spectrum is presented in this paper. The only first two Doppler parameters needed as range models in the defective spectrum are replaced by a polynomial range model, which can derive coefficients from the relative motion between the radar and the targets. Then, the new spectrum is calculated through the Lagrange inversion formula. Based on this, the novel imaging algorithm is elaborated in detail as follows: The range high−order term of the spectrum is compensated completely, and the range chirp rate space variance is eliminated by the cubic phase term. Two steps of range cell migration correct are applied in this algorithm before and after the range compression; one is the traditional linear chirp scaling method, and another is the interpolation to correct the quadratic range cell migration introduced by the range chirp rate equalization. The simulation results illustrate that the proposed algorithm can handle the exact imaging processing with a 0.25 m resolution around the azimuth and range in 2 km $\times$ 6 km, which validates the feasibility of the proposed algorithm. Full article