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Closeness of Markov binomial distribution to the set of infinitely divisible distributions is measured in ℓr and Lr metrics. For a Markov chain with finite number of states, we construct a compound Poisson approximation of order o(n−1/2) in the total-variation metric. The characteristic function method is used for the proof. [ABSTRACT FROM AUTHOR]

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]

The Reliability is one of the key characteristics of software that operates flawlessly and in accordance with needs of users. The assessment of Reliability is very important but it is complicated. The one-parameter exponential class failure intensity function is used in this article to quantify the model and assess the software Reliability. The scale parameter and the number of existing total failures are the model's parameters. Using the Bayesian approach, the estimators of parameters are obtained under the assumption that gamma priors are suitable to provide prior information of the parameters. Using risk efficiencies computed under squared error loss, the performance of proposed estimators is studied with their corresponding maximum likelihood estimators. The suggested Bayes estimators are found to outperform over the equivalent maximum likelihood estimators. [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 this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different kinds of binomial coefficients. We also present some novel relations between sums with two and three binomial coefficients. In the course of exploration, we rediscover a few isolated results existing in the literature, commonly presented as problem proposals. [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]

Almost poised 2ϕ1-series extended with two integer parameters are investigated by means of the linearization method. Four analytical formulae are established and fourteen closed formulae are presented as examples. [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 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]

We present a tree-based approach to the Pull-to-Par model for call options on zero-coupon bonds presented in Tomas and Yu (2021). The binomial approach presented is a simple alternative to the original model solution. The model presented converges to the stochastic process given in Tomas and Yu. Some illustrative comparison values to the original model for calls and puts are given. A discussion of American option pricing and the addition of coupons is also presented and illustrated. [ABSTRACT FROM AUTHOR]

In the last seven decades, the method of moments (MoM) has become an invaluable tool in the field of polymerization reaction engineering, due to the simplicity of translating a complex set of population balance equations (PBE) into a system with a limited number of equations. In this work, we offer an overview of the MoM, describing the derivation of the moment equations in a basic kinetic mechanism. Some tools and strategies for the derivation of the moment equations are reviewed and explained in detail, such as the binomial theorem, the method of series expansion and pattern identification (SEPI), extensively used by the community, and a graphical approach of summation inversion. The treatment for multivariate distributions is also exposed, taking advantage of the complete and partial moment techniques. The derivation of the MoM contribution by kinetic mechanisms beyond the basic ones or involving special difficulties, such as depropagation, long chain branching (LCB), random chain scission, LCB and β‐scission, short chain branching (SCB) and scission, internal double bond (IDB) (polymerization), termination by combination, reversible deactivation radical polymerization (RDRP), and intermolecular transesterification reactions (ITRs), are explained in a tutorial way. Additionally, the fundamentals of the MoM in copolymerization and the application of the pseudo‐homopolymerization approach are briefly described. An introduction of the MoM to emulsion polymerization is also presented. Finally, some advanced applications of MoM in recent works are exposed: MoM models with chain‐length dependent or diffusion‐controlled termination, and the extension of the MoM for the prediction of the molecular weight distribution (MWD). [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]

We compute here the sums of infinite series of four types: P1 n=0 ... for positive integers k ≼ 9. We also look at their alternating forms and at some related series. Binomial expansions and log-sine integrals are used for the purpose. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the family of binomials xaι ∙ ∙ ∙ xmm -- y11 ∙ ∙ ∙ ynn with gcd(aι,..., am, 6χ,..., bn) = 1 is irreducible by identifying the connection between the irreducibility of a binomial in C[x1, . . ., xm, y1, . . ., yn] and C(x2, . . ., xm, y1, . . ., yn)[x1]. Then we show that the necessary and sufficient conditions for the irreducibility of this family of binomials is equivalent to the existence of a unimodular matrix Ui with integer entries such that (a1, . . ., am, b1, . . ., bn)T = Uiei for i ∈ {1, . . ., m + n}, where ei is the standard basis vector. [ABSTRACT FROM AUTHOR]

Let G = K m , n − m be a complete bipartite graph on [ n ] with n − m ≥ m and JG its binomial edge ideal in the polynomial ring S = K [ x 1 , ... , x n , y 1 , ... , y n ] over a field K. For any t ≥ 2 , it is proved that depth (S / J G t) is equal to n + 1 if m = 1 and 4 if m ≥ 2 . [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]

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]

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 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]

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 paper, we obtain some improvements and generalizations of Young's inequalities as the following:... for m ∈ N+ and 0 < v ≤ τ < 1. In addition, we obtain new result of Young's inequality by using the expansions of the functions (1-v)+vx-xv with 0 < x < 2. [ABSTRACT FROM AUTHOR]

We consider the problem of interval estimation of the odds ratio. An asymptotic confidence interval is widely applied in economics, medicine, sociology, etc. Unfortunately, this confidence interval has a poor coverage probability, significantly smaller than the nominal confidence level. In this paper, a new confidence interval is proposed. Its construction requires only information on the sizes of samples and the sample odds ratio. The coverage probability of the proposed confidence interval is at least the nominal confidence level. [ABSTRACT FROM AUTHOR]