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For positive integers n ≥ m , let P n , m (x) : = ∑ j = 0 m n j x j = n 0 + n 1 x + ... + n m x m be the truncated binomial expansion of (1 + x) n consisting of all terms of degree ≤ m. It is conjectured that for n > m + 1 , the polynomial P n , m (x) is irreducible. We confirm this conjecture when 2 m ≤ n < (m + 1) 1 0. Also we show for any r ≥ 1 0 and 2 m ≤ n < (m + 1) r + 1 , the polynomial P n , m (x) is irreducible when m ≥ max { 1 0 6 , 2 r 3 }. Under the explicit abc-conjecture, for a fixed m , we give an explicit n 0 , n 1 depending only on m such that ∀ n ≥ n 0 , the polynomial P n , m (x) is irreducible. Further ∀ n ≥ n 1 , the Galois group associated to P n , m (x) is the symmetric group S m. [ABSTRACT FROM AUTHOR]

In this article, we propose E-Bayes estimators of the parameter of xgamma distribution under squared error loss function, general entropy loss function, and linear exponential loss function for progressive type II censored data with binomial removals. The proposed estimators, maximum likelihood estimator, and corresponding Bayes estimators are compared in terms of their risks based on simulated samples from xgamma distribution. The proposed methodology is illustrated on two real data sets of bile duct cancer data and the endurance of deep-groove ball bearings data. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new q-analogue of the binomial identity:... Our proof relies on a weight-preserving and sign-reversing involution due to Guo and Zhang. [ABSTRACT FROM AUTHOR]

In a recent paper the authors proved a nonuniform local limit theorem concerning normal approximation of the point probabilities P (S = k) when S = ∑ i = 1 n X i and X 1 , X 2 , ... , X n are independent Bernoulli random variables that may have different success probabilities. However, their main result contained an undetermined constant, somewhat limiting its applicability. In this paper we give a nonuniform bound in the same setting but with explicit constants. Our proof uses Stein's method and, in particular, the K-function and concentration inequality approaches. We also prove a new uniform local limit theorem for Poisson binomial random variables that is used to help simplify the proof in the nonuniform case. [ABSTRACT FROM AUTHOR]

In this paper, a new threshold INAR(1) model based on modified negative binomial operator with random coefficient is proposed. Basic probabilistic and statistical properties of this process are established. Then the conditional least squares (CLS) and the conditional maximum likelihood (CML) methods are applied to estimate the model parameters when the threshold value is known or not. The asymptotic properties of the CLS-estimator and the CML-estimator are also been discussed. A method to test the constancy of the autoregressive parameters is provided. As an illustration, a simulation study is conducted to illustrate the performances of these estimators and present an empirical analysis of monthly counts of break and enter non-dwelling in Bellingen. [ABSTRACT FROM AUTHOR]

Following the Kubo-Ando theory of operator means we consider the weighted geometric mean $ A \#_t B $ A # t B of $ n \times n $ n × n upper triangular matrices A and B whose main diagonals are all 1, named the upper unipotent matrices. We also present its binomial expansion \[ A \#_t B = \sum_{k=0}^{n-1} \begin{pmatrix} t\\ k\\ \end{pmatrix} A(A^{-1} B - I)^{k}, \quad t \in \mathbb{R}. \] A # t B = ∑ k = 0 n − 1 ( t k ) A (A − 1 B − I) k , t ∈ R. Showing that the weighted geometric mean is a geodesic of symmetry in the symmetric space equipped with point reflection, known as the Loos symmetric space, we derive several binomial identities on the Lie group of upper unipotent (resp. the Lie algebra of nilpotent) matrices. [ABSTRACT FROM AUTHOR]

Broadcasting is a fundamental problem in the information dissemination area. In classical broadcasting, a message must be sent from one network member to all other members as rapidly as feasible. Although it has been demonstrated that this problem is NP-Hard for arbitrary graphs, it has several applications in various fields. As a result, the universal lists model, replicating real-world restrictions like the memory limits of nodes in large networks, is introduced as a branch of this problem in the literature. In the universal lists model, each node is equipped with a fixed list and has to follow the list regardless of the originator. In this study, we focus on the non-adaptive branch of universal lists broadcasting. In this regard, we establish the optimal broadcast time of k -ary trees and binomial trees under the non-adaptive model and provide an upper bound for complete bipartite graphs. We also improved a general upper bound for trees under the same model and showed that our upper bound cannot be improved in general. [ABSTRACT FROM AUTHOR]

The gamma-Poisson and beta-binomial mixture distributions are used for analyzing count-valued data, and the estimation of the hyper-parameters including the shape and/or scale parameters is important in the empirical Bayes inference. The maximum likelihood method requires the nested loops for solving the non-linear equations at each step of iteration in the EM algorithm. To avoid the extra loops, we derive the closed-form updating procedures at each step of iteration by using the score-adjusted method. The performance is compared by simulation with the maximum likelihood estimators. [ABSTRACT FROM AUTHOR]

We derive Abel's generalization of the binomial theorem and use it to present a short proof of Cayley's theorem on the number of trees on n labeled vertices. [ABSTRACT FROM AUTHOR]

We prove a nonuniform local limit theorem concerning approximation of the point probabilities P (S = k) , where S = ∑ i = 1 n X i , and X 1 , ... , X n are independent Bernoulli random variables with possibly different success probabilities. Our proof uses Stein's method, in particular, the zero bias transformation and concentration inequality approaches. [ABSTRACT FROM AUTHOR]

A combinatorial property that characterizes Cohen-Macaulay binomial edge ideals has long been elusive. A recent conjecture ties the Cohen-Macaulayness of a binomial edge ideal J G to special disconnecting sets of vertices of its underlying graph G , called cut sets. More precisely, the conjecture states that J G is Cohen-Macaulay if and only if J G is unmixed and the collection of the cut sets of G is an accessible set system. In this paper we prove the conjecture theoretically for all graphs with up to 12 vertices and develop an algorithm that allows to computationally check the conjecture for all graphs with up to 15 vertices and all blocks with whiskers where the block has at most 11 vertices. This significantly extends previous computational results. [ABSTRACT FROM AUTHOR]

Using the Fourier–Gegenbauer series, we prove several identities that generalize known results. In particular, it is proved, that ∑ n = 0 ∞ 1 4 n ( 2 n n ) z − 2 n ( z − 1 / 2 n ) ( z n ) 3 = tan ⁡ (π z) π for all complex numbers z such that ℜ (z) > − 1 2 and z ∉ 1 2 + Z . In addition, for all nonnegative integers m, we obtain the Ramanujan-type series 2 π = ∑ n = m ∞ (− 1) n − m (1 + 4 n) 64 n ( 2 n − 2 m n − m ) ( 2 n n ) ( 2 m + 2 n m + n ) . 2 4 m + 5 π 2 = ∑ n = 0 ∞ (1 + 2 m + 4 n) (2 m + 1) (2 m + 3) (m + n + 1) 2 (2 n − 1) 2 256 n ( 2 n n ) 2 ( 2 m + 2 n m + n ) 2. Which are known in the case m=0. [ABSTRACT FROM AUTHOR]

In meta‐analyses of rare events, it can be challenging to obtain a reliable estimate of the pooled effect, in particular when the meta‐analysis is based on a small number of studies. Recent simulation studies have shown that the beta‐binomial model is a promising candidate in this situation, but have thus far only investigated its performance in a frequentist framework. In this study, we aim to make the beta‐binomial model for meta‐analysis of rare events amenable to Bayesian inference by proposing prior distributions for the effect parameter and investigating the models' robustness to different specifications of priors for the scale parameter. To evaluate the performance of Bayesian beta‐binomial models with different priors, we conducted a simulation study with two different data generating models in which we varied the size of the pooled effect, the degree of heterogeneity, the baseline probability, and the sample size. Our results show that while some caution must be exercised when using the Bayesian beta‐binomial in meta‐analyses with extremely sparse data, the use of a weakly informative prior for the effect parameter is beneficial in terms of mean bias, mean squared error, and coverage. For the scale parameter, half‐normal and exponential distributions are identified as candidate priors in meta‐analysis of rare events using the Bayesian beta‐binomial model. [ABSTRACT FROM AUTHOR]

We prove that level binomial edge ideals with regularity 2 and pseudo-Gorenstein binomial edge ideals with regularity 3 are cones, and we describe them completely. Also, we characterize level and pseudo-Gorenstein binomial edge ideals of bipartite graphs. [ABSTRACT FROM AUTHOR]

We prove a conjecture by Ene, Herzog, and Hibi (2011) that the Betti numbers of the binomial edge ideal J G of a closed graph G coincide with the Betti numbers of its lex initial ideal M G . We describe the Betti numbers of the ideal M G . [ABSTRACT FROM AUTHOR]

In this article, we introduce a new member to Poisson-X family namely, the Poisson-power Lindley distribution. The statistical as well as the distributional properties of the new distribution are studied. The flexibility of the distribution is illustrated by means of real data sets. We also introduce a reliability test plan for acceptance or rejection of a lot of products submitted for inspection when lifetimes follow the new distribution. The minimum sample size using binomial and Poisson approximations, operating characteristic values and minimum ratios of the true value and the required value of the parameter with a given producer's risk are also developed with respect to the newly introduced sampling plans. A real data example is also given to illustrate the sampling plan developed. [ABSTRACT FROM AUTHOR]

In this paper, we derive a Mittag-Leffler function for real index and establish solutions of special type of fractional order differential equations (FDEs). The same concept is extended to discrete case by replacing polynomials into factorial polynomials and differentiation into l-difference operator. Moreover, numerical examples of our results are stated to validate our findings. The acquired results here have the ability to generate a wide range of formulas in relation to newer results. [ABSTRACT FROM AUTHOR]

In this paper, we established an effective formula for Dnestrovskii functions based on the binomial expansion theorem which frequently appears in a weakly relativistic loss cone distribution, relativistic, magnetized, and thermal plasmas. The obtained formulas are expressed in terms of binomial coefficients and incomplete Gamma functions. Therefore, the proposed approach is a fully useful method to evaluate Dnestrovskii functions and related physical properties of plasmas. Calculation results of the Dnestrovskii functions are compared with other numerical and theoretical models and indicated acceptable agreement. The results of the calculations demonstrate that the derived formula is valid for a wide range of parameters. [ABSTRACT FROM AUTHOR]

The Chavy–Waddy–Kolokolnikov model for the description of bacterial colonies is considered. In order to establish if the mathematical model is integrable, the Painlevé test is conducted for the nonlinear ordinary differential equation which corresponds to the fourth-order partial differential equation. The restrictions on the mathematical model parameters for ordinary differential equations to pass the Painlevé test are obtained. It is determined that the method of the inverse scattering transform does not solve the Cauchy problem for the original mathematical model, since the corresponding nonlinear ordinary differential equation passes the Painlevé test only when its solution is stationary. In the case of the stationary solution, the first integral of the equation is obtained, which makes it possible to represent the general solution in the quadrature form. The stability of the stationary points of the investigated mathematical model is carried out and their classification is proposed. Periodic and solitary stationary solutions of the Chavy–Waddy–Kolokolnikov model are constructed for various parameter values. To build analytical solutions, the method of the simplest equations is also used. The solutions, obtained in the form of a truncated expansion in powers of the logistic function, are represented as a closed formula using the formula for the Newton binomial. [ABSTRACT FROM AUTHOR]

The passages of literacy criticism in Photius' Bibliotheca have been studied by segmenting their elements, which prevents discernment of the passage structure and the precise meaning of terms. The objective of this paper is to demonstrate, firstly, the existence of two- and three-element groups (binomials and trinomials) in Photius' passages and, secondly, the usefulness of these groups as interpretation tools. These goals will lead to a better understanding of not only the work's composition but the meaning of the text. [ABSTRACT FROM AUTHOR]

In this paper certain results on Ramanujan’s continued fractions have been discussed. [ABSTRACT FROM AUTHOR]

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results. [ABSTRACT FROM AUTHOR]

This article, titled "Proof Without Words: Inversion Number of Binary Words," explores the concept of inversion number in binary words. It defines the Hamming weight and reversed position sum of a binary vector and introduces the concept of inversion number as the count of certain pairs of elements in the vector. The article provides a visual proof of an identity related to inversion number, which is crucial for understanding Cauchy's binomial theorem. The theorem is briefly explained, and a proof using the concept of inversion number is presented. The text is a mathematical equation and proof, emphasizing the significance of considering the reversed position sum (Rev_PS) in simplifying the proof. The publisher's note states that Springer Nature remains neutral in terms of jurisdictional claims and institutional affiliations. The author of the text is N. Uday Kiran. [Extracted from the article]

A method of recovering the potential of the fourth-order binomial operator on a half-interval [1/2, 1] using a known potential on another half-interval [0, 1/2] and the eigenvalues of the self-adjoint boundary problem on the whole interval [0, 1] is proposed. [ABSTRACT FROM AUTHOR]

We study a special class of higher order generalized geometric polynomials. Based on our combinatorial interpretation of labeled barred preferential arrangements, we prove several recursions. We also study the polynomials from a probabilistic point of view, and show how our polynomials can be written in terms of the expectation of a random descending factorial involving the negative binomial process. Using techniques of probability theory, we derive identities, in particular we extend Nelsen's theorem. [ABSTRACT FROM AUTHOR]

A polygonal chain P defined by the sequence (A1, A2, ..., Ak) is said to be perpendicularly inscribed in a simple polygon Q if the line segment AiAi+1 is perpendicular to the side of Q in which Ai lies. If P closes after n steps, then P is a perpendicularly inscribed polygon in P. The goal of this paper is to determine the number of perpendicularly inscribed polygons that intersect a given side of a regular polygon with an odd number of sides. This is done using circular permutations with repetition and partitions of an integer, and some special cases are calculated via circulant matrices and the Binomial Theorem. A method for finding such polygons, based on Banach's Fixed Point Theorem, is also developed. [ABSTRACT FROM AUTHOR]

We investigate a generalization of Fubini numbers. We present the combinatorial interpre-tation as barred preferential arrangements with some additional conditions on the blocks. We provide a proof for a generalization of Nelsen's Theorem. We consider these numbers from a probabilistic view point and demonstrate how they can be written in terms of the expectation of random descending factorial involving the negative binomial process. [ABSTRACT FROM AUTHOR]

In this work, we study the effects of geometric confinement on the point statistics in a quasi-low-dimensional system. Specifically, we focus on the nearest-neighbor statistics. Accordingly, we have performed comprehensive numerical simulations of binomial point process on quasi-one-dimensional rectangle strips for different values of the confinement ratio defined as the ratio of the strip width to the mean nearest-neighbor distance. We found that the nearest-neighbor distance distributions (NNDDs) conform to an extreme value Weibull distribution with the shape parameter depending on the confinement ratio, while the process intensity remains constant. This finding reveals the reduction of effective spatial degrees of freedom in a quasi-low-dimensional system under the geometric confinement. The scale dependence of the number of effective spatial degrees of freedom is found to obey the crossover ansatz. We stress that the functional form of the crossover ansatz is determined by the nature of the studied point process. Accordingly, different physical processes in the quasi-low-dimensional system obey different crossover ansatzes. The relevance of these results for quasi-low-dimensional systems is briefly highlighted. [ABSTRACT FROM AUTHOR]

q-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, q-analogs of various probability distributions have been introduced over the years, including the binomial distribution. Here, I propose a new refinement of the binomial distribution by way of the quantum binomial theorem (also known as the noncommutative q-binomial theorem), where the q is a formal variable in which information related to the sequence of successes and failures in the underlying binomial experiment is encoded in its exponent. [ABSTRACT FROM AUTHOR]

This black box study assessed the performance of forensic firearms examiners in the United States. It involved three different types of firearms and 173 volunteers who performed a total of 8640 comparisons of both bullets and cartridge cases. The overall false‐positive error rate was estimated as 0.656% and 0.933% for bullets and cartridge cases, respectively, while the rate of false negatives was estimated as 2.87% and 1.87% for bullets and cartridge cases, respectively. The majority of errors were made by a limited number of examiners. Because chi‐square tests of independence strongly suggest that error probabilities are not the same for each examiner, these are maximum‐likelihood estimates based on the beta‐binomial probability model and do not depend on an assumption of equal examiner‐specific error rates. Corresponding 95% confidence intervals are (0.305%, 1.42%) and (0.548%, 1.57%) for false positives for bullets and cartridge cases, respectively, and (1.89%, 4.26%) and (1.16%, 2.99%) for false negatives for bullets and cartridge cases, respectively. The results of this study are consistent with prior studies, despite its comprehensive design and challenging specimens. [ABSTRACT FROM AUTHOR]

This paper introduces and studies a new class of analytic p-valent functions in the open symmetric unit disc involving the Sălăgean-type q-difference operator. Furthermore, we present several interesting subordination results, coefficient inequalities, fractional q-calculus applications, and distortion theorems. [ABSTRACT FROM AUTHOR]

We propose a rule-based statistical design for combination dose-finding trials with two agents. The Ci3 + 3 design is an extension of the i3 + 3 design with simple up-and-down decision rules comparing the observed toxicity rates and equivalence intervals that define the maximum tolerated dose combination. Ci3 + 3 consists of two stages to allow fast and efficient exploration of the dose-combination space. Statistical inference is restricted to a beta-binomial model for dose evaluation, and the entire design is built upon a set of fixed rules. We show via simulation studies that the Ci3 + 3 design exhibits similar and comparable operating characteristics to more complex designs utilizing model-based inferences. Implementation of Ci3 + 3 for practical trials is simple for the first stage, where the up-and-down decisions may be carried out using a decision table based on the preselected escalation path and i3 + 3. The second stage is not simpler than model-based designs, however, since it also requires computation of posterior probabilities based on a Bayesian model. We believe that the Ci3 + 3 design may provide an alternative choice to help simplify the design and conduct of combination dose-finding trials in practice. [ABSTRACT FROM AUTHOR]

We evaluate, in closed forms, twelve infinite series that contain a free variable x and binomial coefficients ( 3 n + ϵ n) in the denominators. Several summation formulae resembling Gosper's sum are derived as consequences. [ABSTRACT FROM AUTHOR]

New practically applicable estimates are constructed for the accuracy of approximation to the distributions of negative binomial random sums in the case where the shape parameter r > 0 of the negative binomial distribution satisfies the condition r ⩽ δ/2, where 2 + δ is the maximum order of the moment of random summands, δ ∈ (0, 1]. [ABSTRACT FROM AUTHOR]

For testing the null hypothesis of a marginal binomial distribution of bounded count data, we derive novel and flexible goodness-of-fit (GoF) tests. We propose two general approaches to construct moment-based test statistics. The first one relies on properties of higher-order factorial moments, while the second one uses a so-called Stein identity being satisfied under the null. For a broad class of stationary time series processes of bounded counts with joint bivariate binomial distributions of lagged time series values, we derive the limiting distributions of the proposed GoF-test statistics. Among others, our setup covers the binomial autoregressive model, but includes also other binomial time series obtained, e. g. by superpositioning independent binary time series. The test performance under the null and under different alternatives is investigated in simulations. Two data examples are used to illustrate the application of the novel GoF-tests in practice. [ABSTRACT FROM AUTHOR]

We introduce a new model for sums of exchangeable binary random variables. The proposed distribution is an approximation to the exact distributional form, and relies on the theory of completely monotone functions and the Laplace transform of a gamma distribution function. Using Monte Carlo methods, we show that this new model compares favorably to the beta-binomial model with respect to estimating the success probability of the Bernoulli trials and the correlation between any two variables in the exchangeable set. We apply the new methodology to two classic data sets and the results are summarized. [ABSTRACT FROM AUTHOR]

Zero-inflated models have become a popular tool for assessing relationships between explanatory variables and a zero-inflated count outcome. In these models, regression coefficients have latent class interpretations, where latent classes correspond to a susceptible subpopulation with observations generated from a count distribution and a non susceptible subpopulation that provides only zeros. However, it is often of interest to evaluate covariates effects in the overall mixture population, that is, on the marginal mean of the zero-inflated count. Marginal zero-inflated models, such as the marginal zero-inflated Poisson models, have been developed for that purpose. They specify independent submodels for the susceptibility probability and the marginal mean of the count response. When the count outcome is bounded, it is tempting to formulate a marginal zero-inflated binomial model in the same fashion. This, however, is not possible, due to inherent constraints that relate, in the zero-inflated binomial model, the susceptibility probability and the latent and marginal means of the count outcome. In this paper, we propose a new marginal zero-inflated binomial regression model that accommodates these constraints. We investigate the maximum likelihood estimator in this model, both theoretically and by simulations. An application to the analysis of health-care demand is provided for illustration. [ABSTRACT FROM AUTHOR]

We consider the binomial random set model [ n ] p where each element in { 1 , ... , n } is chosen independently with probability p : = p (n). We show that for essentially all regimes of p and very general conditions for a matrix A and a column vector b , the count of specific integer solutions to the system of linear equations A x = b with the entries of x in [ n ] p follows a (conveniently rescaled) normal limiting distribution. This applies among others to the number of solutions with every variable having a different value, as well as to a broader class of so-called non-trivial solutions in homogeneous strictly balanced systems. Our proof relies on the delicate linear algebraic study both of the subjacent matrices and the corresponding ranks of certain submatrices, together with the application of the method of moments in probability theory. [ABSTRACT FROM AUTHOR]

Purpose: The authors investigated the effects of the characteristics of reviews, reviewers and corporate factors on review helpfulness and assessed the role of culture in moderating these relationships. Design/methodology/approach: A research model was established based on the elaboration likelihood and information adoption models. To empirically analyze this research model, 10,611 TripAdvisor reviews from 9 countries were collected. In addition, a zero-inflated negative binomial model and multilevel analysis were employed in consideration of the data characteristics. Findings: The results revealed that review depth had a positive effect on review helpfulness, and review ratings and reviewer expertise had a negative effect. As a corporate characteristic, hotel size had a negative effect on review helpfulness. In addition, the effects of review rating, reviewer expertise and hotel rating exhibited significant differences based on the moderating effects of uncertainty avoidance and power distance level. Originality/value: The results of this study expand the review helpfulness literature by explaining the inconsistent findings of previous studies via cultural theory. In addition, past research in this field has mainly focused on analyzing only review and reviewer characteristics, while this study demonstrated that company size negatively affects review helpfulness based on the signaling theory. Finally, this study contributes to cultural comparison literature by discovering that the processing of review information by consumers differs according to their cultural background. [ABSTRACT FROM AUTHOR]

Although differential evolution (DE) algorithms perform well on a large variety of complicated optimization problems, only a few theoretical studies are focused on the working principle of DE algorithms. To make the first attempt to reveal the function of binomial crossover, this paper aims to answer whether it can reduce the approximation error of evolutionary algorithms. By investigating the expected approximation error and the probability of not finding the optimum, we conduct a case study comparing two evolutionary algorithms with and without binomial crossover on two classical benchmark problems: OneMax and Deceptive. It is proven that using binomial crossover leads to the dominance of transition matrices. As a result, the algorithm with binomial crossover asymptotically outperforms that without crossover on both OneMax and Deceptive, and outperforms on OneMax, however, not on Deceptive. Furthermore, an adaptive parameter strategy is proposed which can strengthen the superiority of binomial crossover on Deceptive. [ABSTRACT FROM AUTHOR]

The current loop is a fundamental building block of cylindrically symmetric magnetic calculations. However, the off-axis magnetic field involves the subtraction of elliptic integrals of the first and second kind, which is computationally expensive, hard to manipulate in equations, and difficult to visualize. By conducting a different binomial series expansion on the original loop integral, a series solution is created, which can be simplified to a set of approximation functions with useful characteristics: exactly correct along the axis and at distance, while in the current loop itself the relative error is limited, computationally simple, highly accurate, and easy to visualize for behavior or symmetry. For the radial field, the first and second orders fit where the parameters are optimized to minimize the relative peak error. Note the symmetry that occurs by expressing as function of the scaled expansion term $W$ , which ranges from 0 to 1, allowing maximum relative errors of 0.025 for first order and 2.9E-4 for second order. For the axial field first order, due to the subtraction of terms, the accuracy is only modest but the second order has a 9E-4 maximum relative error, with zero error at the loop, on the $z =0$ loop plane. The axial field relative error stays low within the loop, but the outside stays low for any h until it falls to 1% of the $z =0$ plane value, then it loses accuracy as $z$ is near where the field direction reverses sign due to slight differences in the zero crossing predicated coordinate, increasing again in accuracy as $z$ moves further from the reveal. Higher order approximations for both radial and axial add more $W$ terms with increasing accuracy—for radial, the third order is 1.8E-5, fourth order is 2.3E-6, and fifth order is 4.9E-7, while axial goes as third 4.6E-5, fourth 6E-6, and fifth 1.3E-6. Relative error plots in 3-D space are presented for all orders of approximations. The simplicity of these functions suggests new ways combining loops to optimize such things as field uniformity. [ABSTRACT FROM AUTHOR]

Motivated by a problem proposed by Seiffert a quarter of century ago, we explicitly evaluate binomial sums with Pell and Lucas polynomials as weight functions. Their special cases result in several interesting identities concerning Fibonacci and Lucas numbers. [ABSTRACT FROM AUTHOR]

We consider the binomial decomposition of ordered weighted averaging (OWA) functions proposed by Calvo and De Baets (1998) in the framework of Choquet integration. Our aim in the paper is to further investigate the equivalence between the two representations of OWA functions involved in the binomial decomposition: the usual canonical representation in terms of the order statistics, and the binomial representation in terms of the binomial OWA functions. We describe and discuss in detail the linear transformations that relate the coefficients of these two equivalent representations: the original expression of the weights in terms of the coefficients of the binomial representation, and its inverse, the expression of those coefficients in terms of the weights. In both cases simple and direct proofs are presented. Moreover, we consider the linear transformations between the two representations in the general linear algebra framework of unconstrained linear combinations of binomial OWA functions. In this perspective we obtain compact matrix expressions for the linear transformations, which also offer new insight on the geometry of the coefficient constraints in the binomial representation of OWA functions. [ABSTRACT FROM AUTHOR]

We present a calculus and a probabilistic proofs of the binomial theorem. We believe that the material presented in this article would be most suitable to students with some background in calculus and discrete mathematics. It can be used by instructors looking for interesting applications in a calculus or a discrete mathematics/probability course. [ABSTRACT FROM AUTHOR]

We propose two-stage and sequential procedures to estimate the unknown parameter N of a binomial distribution with unknown parameter p, when we reinforce data with an independent sample of a negative binomial experiment having the same p. [ABSTRACT FROM AUTHOR]