A new recursive formula arising from a determinantal expression for weighted Delannoy numbers

2021 ◽  
Keyword(s):  
Recursive Formula ◽  
Delannoy Numbers ◽  
Determinantal Expression

scholarly journals A determinantal expression and a recursive relation of the Delannoy numbers

2021 ◽  
Vol 13 (2) ◽  
pp. 442-449 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Recursive Relation ◽  
Differentiable Functions ◽  
Delannoy Numbers ◽  
Derivatives Of ◽  
Determinantal Expression

Abstract In the paper, by a general and fundamental, but non-extensively circulated, formula for derivatives of a ratio of two differentiable functions and by a recursive relation of the Hessenberg determinant, the author finds a new determinantal expression and a new recursive relation of the Delannoy numbers. Consequently, the author derives a recursive relation for computing central Delannoy numbers in terms of related Delannoy numbers.

A New Generalization of Delannoy Numbers

2020 ◽  
Vol 51 (4) ◽  
pp. 1729-1735
Author(s):  
Muhammet Cihat Dağli
Keyword(s):  
Delannoy Numbers

scholarly journals A Recursive Formula for the Reliability of a r-Uniform Complete Hypergraph and Its Applications

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Ke Zhang ◽  
Haixing Zhao ◽  
Zhonglin Ye ◽  
Lixin Dong
Keyword(s):  
Recurrence Relations ◽  
Recursive Formula ◽  
Finite Graph ◽  
Hard Problem ◽  
Np Hard ◽  
Spanning Subgraph ◽  
Reliability Polynomial ◽  
Np Hard Problem

The reliability polynomial R(S,p) of a finite graph or hypergraph S=(V,E) gives the probability that the operational edges or hyperedges of S induce a connected spanning subgraph or subhypergraph, respectively, assuming that all (hyper)edges of S fail independently with an identical probability q=1-p. In this paper, we investigate the probability that the hyperedges of a hypergraph with randomly failing hyperedges induce a connected spanning subhypergraph. The computation of the reliability for (hyper)graphs is an NP-hard problem. We provide recurrence relations for the reliability of r-uniform complete hypergraphs with hyperedge failure. Consequently, we determine and calculate the number of connected spanning subhypergraphs with given size in the r-uniform complete hypergraphs.

scholarly journals Multi Body Dynamic Equations of Belt Conveyor and the Reasonable Starting Mode

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1489
Author(s):  
Yongbo Guo ◽  
Fansheng Wang
Keyword(s):  
Finite Element ◽  
Recursive Formula ◽  
Conservative System ◽  
Conveyor Belt ◽  
Dynamic Equations ◽  
Lagrange’S Equation ◽  
Rigid Finite Element Method ◽  
Potential Capacity ◽  
Multi Body ◽  
Body Dynamic

Based on the rigid finite element method and multibody dynamics, a discrete model of a flexible conveyor belt considering the material viscoelasticity is established. RFE (rigid finite element) and SDE (spring damping element) are used to describe the rigidity and flexibility of a conveyor belt. The dynamic differential equations of the RFE are derived by using Lagrange’s equation of the second kind of the non-conservative system. The generalized elastic potential capacity and generalized dissipation force of the SDE are considered. The forward recursive formula is used to construct the conveyor belt model. The validity of dynamic equations of conveyor belt is verified by field test. The starting mode of the conveyor is simulated by the model.

ESTIMATES FOR THE ERROR TERM IN A UNIFORM ASYMPTOTIC EXPANSION OF THE JACOBI POLYNOMIALS

2003 ◽  
Vol 01 (02) ◽  
pp. 213-241 ◽  
Author(s):  
R. WONG ◽  
Y.-Q. ZHAO
Keyword(s):  
Asymptotic Expansion ◽  
Explicit Expression ◽  
Canonical Form ◽  
Error Term ◽  
Jacobi Polynomials ◽  
Recursive Formula ◽  
Contour Integral ◽  
Integration By Parts ◽  
Uniform Asymptotic Expansion ◽  
Repeated Use

There are now several ways to derive an asymptotic expansion for [Formula: see text], as n → ∞, which holds uniformly for [Formula: see text]. One of these starts with a contour integral, involves a transformation which takes this integral into a canonical form, and makes repeated use of an integration-by-parts technique. There are two advantages to this approach: (i) it provides a recursive formula for calculating the coefficients in the expansion, and (ii) it leads to an explicit expression for the error term. In this paper, we point out that the estimate for the error term given previously is not sufficient for the expansion to be regarded as genuinely uniform for θ near the origin, when one takes into account the behavior of the coefficients near θ = 0. Our purpose here is to use an alternative method to estimate the remainder. First, we show that the coefficients in the expansion are bounded for [Formula: see text]. Next, we give an estimate for the error term which is of the same order as the first neglected term.

scholarly journals A new dynamic model to optimize the reliability of the series-parallel systems under warm standby components

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Amir Mohammad Fakoor Saghih ◽  
Azam Modares
Keyword(s):  
Dynamic Models ◽  
Failure Time ◽  
Recursive Formula ◽  
Reliability Function ◽  
Mode Analysis ◽  
Redundancy Allocation ◽  
Failure Time Distribution ◽  
Before And After ◽  
Common Technique ◽  
Warm Standby

<p style='text-indent:20px;'>Redundancy allocation problem (RAP) is a common technique for increasing the reliability of systems. In this paper, a new model for the RAP is introduced that takes into account the warm standby and mixed strategy, the model dynamics, and the type of the strategy in redundancy allocation problems. A recursive formula is first obtained for the reliability function in the dynamic warm standby and mixed redundancy strategies that leverages the success mode analysis and works for any arbitrary failure-time distribution. Failure rates for warm standby units change before and after their replacement with a damaged unit, and, therefore, the reliability function in warm standby varies with time (i.e., the model is dynamic). Although dynamic models are commonplace in practice, they are more challenging to assess than static models, which have been mainly considered in the literature. An optimization problem is then formulated to select the best redundancy strategy and redundancy levels. Genetic algorithm and particle swarm optimization are leveraged to solve the problem. Finally, the efficiency of the presented method is verified through a numerical example. The experimental results verify that the proposed model for RAP significantly improves the system reliability, which can be of vital importance for system designers.</p>

scholarly journals Some Properties of the $k$-bonacci Words on Infinite Alphabet

10.37236/9406 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Narges Ghareghani ◽  
Morteza Mohammad-Noori ◽  
Pouyeh Sharifani
Keyword(s):  
Fixed Point ◽  
Recursive Formula

The Fibonacci word $W$ on an infinite alphabet was introduced in [Zhang et al., Electronic J. Combinatorics 2017 24(2), 2-52] as a fixed point of the morphism $2i\rightarrow (2i)(2i+1)$, $(2i+1) \rightarrow (2i+2)$, $i\geq 0$. Here, for any integer $k>2$, we define the infinite $k$-bonacci word $W^{(k)}$ on the infinite alphabet as $\varphi_k^{\omega}(0)$, where the morphism $\varphi_k$ on the alphabet $\mathbb{N}$ is defined for any $i\geq 0$ and any $0\leq j\leq k-1$, by \begin{equation*} \varphi_k(ki+j) = \left\{ \begin{array}{ll} (ki)(ki+j+1) & \text{if } j = 0,\cdots ,k-2,\\ (ki+j+1)& \text{otherwise}. \end{array} \right. \end{equation*} We consider the sequence of finite words $(W^{(k)}_n)_{n\geq 0}$, where  $W^{(k)}_n$ is the prefix of $W^{(k)}$ whose length is the $(n+k)$-th $k$-bonacci number. We then provide a recursive formula for the number of palindromes that occur in different positions of $W^{(k)}_n$. Finally, we obtain the structure of all palindromes occurring in $W^{(k)}$ and based on this, we compute the palindrome complexity of $W^{(k)}$, for any $k>2$.

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