Recurrence Relations Research Articles

Intrinsic Solutions to the Biquadratic Diophantine Equations

Abstract: The Biquadratic Diophantine equation with five unknowns has non-zero unique intrinsic integral solutions obtained. The recurrence relation between the solutions is also obtained.

Computational aspects of simultaneous Gaussian quadrature

AbstractIn this paper, we derive a new method to compute the nodes and weights of simultaneous n-point Gaussian quadrature rules. The method is based on the eigendecomposition of the banded lower Hessenberg matrix that contains the coefficients of the recurrence relations for the corresponding multiple orthogonal polynomials. The novelty of the approach is that it uses the property of total nonnegativity of this matrix associated with the particular considered multiple orthogonal polynomials, in order to compute its eigenvalues and eigenvectors in a numerically stable manner. The overall complexity of the computation of all the nodes and weights is $$\mathcal {O}(n^2)$$ O ( n 2 ) .

Crushing characteristics and particle size distribution of steelmaking slag according to repetitive compaction loads

Steelmaking slag, which is used as a construction material, can be crushed during compaction or repetitive loading in construction work. This crushing has a considerable influence on its physical and mechanical properties. Therefore, this study aims to evaluate/estimate the crushing properties and particle size distribution of steelmaking slag under various compaction loads. Accordingly, compaction tests were conducted by subjecting steelmaking slag with 10 different particle sizes to an increasing number of compaction cycles (1–15 times) to simulate repetitive compaction loading. The compaction test results were analyzed by considering the particle size distribution, particle size distribution curve, breakage factors, and fractal dimension. The results revealed that these factors could be influenced by changing the voids in steelmaking slag by altering the repetitive compaction load and particle size. An equation for predicting the particle size distribution from the compaction energy and particle size based on the sequence limit with the rational and the recurrence formulas was proposed. The proposed equation demonstrated excellent accuracy, and it can be utilized in efficiently predicting the particle size distribution curve.

On q-Hermite Polynomials with Three Variables: Recurrence Relations, q-Differential Equations, Summation and Operational Formulas

In the present study, we use several identities from the q-calculus to define the concept of q-Hermite polynomials with three variables and present their associated formalism. Many properties and new results of q-Hermite polynomials of three variables are established, including their generation function, series description, summation equations, recurrence relationships, q-differential formula and operational rules.

Q-Difference Recurrence Relations of Aleph Function with Generalization to nth Derivative

q-Difference Recurrence Relations of Aleph Function with Generalization to nth Derivative

Interlacing Property of a Family of Generating Polynomials over Dyck Paths

In the study of a tantalizing symmetry on Catalan objects, Bóna et al. introduced a family of polynomials $\{W_{n,k}(x)\}_{n\geq k\geq 0}$ defined by$$W_{n,k}(x)=\sum_{m=0}^{k}w_{n,k,m}x^{m},$$where $w_{n,k,m}$ counts the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. They proposed two conjectures on the interlacing property of these polynomials, one of which states that $\{W_{n,k}(x)\}_{n\geq k}$ is a Sturm sequence for any fixed $k\geq 1$, and the other states that $\{W_{n,k}(x)\}_{1\leq k\leq n}$ is a Sturm-unimodal sequence for any fixed $n\geq 1$. In this paper, we obtain certain recurrence relations for $W_{n,k}(x)$, and further confirm their conjectures.

On the Two-Variable Analogue Matrix of Bessel Polynomials and Their Properties

In this paper, we explore a study focused on a two-variable extension of matrix Bessel polynomials. We initiate the discussion by introducing the matrix Bessel polynomials involving two variables and derive specific differential formulas and recurrence relations associated with them. Additionally, we present a segment detailing integral formulas for the extended matrix Bessel polynomials. Lastly, we introduce the Laplace–Carson transform for the two-variable matrix Bessel polynomial analogue.

BSAlign: A Library for Nucleotide Sequence Alignment.

Increasing the accuracy of the nucleotide sequence alignment is an essential issue in genomics research. Although classic dynamic programming (DP) algorithms (e.g., Smith-Waterman and Needleman-Wunsch) guarantee to produce the optimal result, their time complexity hinders the application of large-scale sequence alignment. Many optimization efforts that aim to accelerate the alignment process generally come from three perspectives: redesigning data structures [e.g., diagonal or striped Single Instruction Multiple Data (SIMD) implementations], increasing the number of parallelisms in SIMD operations (e.g., difference recurrence relation), or reducing search space (e.g., banded DP). However, no methods combine all these three aspects to build an ultra-fast algorithm. In this study, we developed a Banded Striped Aligner (BSAlign) library that delivers accurate alignment results at an ultra-fast speed by knitting a series of novel methods together to take advantage of all of the aforementioned three perspectives with highlights such as active F-loop in striped vectorization and striped move in banded DP. We applied our new acceleration design on both regular and edit distance pairwise alignment. BSAlign achieved 2-fold speed-up than other SIMD-based implementations for regular pairwise alignment, and 1.5-fold to 4-fold speed-up in edit distance-based implementations for long reads. BSAlign is implemented in C programing language and is available at https://github.com/ruanjue/bsalign.

Real stable polynomials and the alternatingly increasing property

In this paper, we first present a criterion for the real stability of recursive polynomials by applying the characterization of Borcea and Brändén concerning linear operators preserving stability. As applications, we derive some real stability results occurred in the literature in a unified manner. Moreover, we prove the real-rootedness of peak polynomials in bicolored dyck paths and ternary polynomials.Second, based on the h-polynomials from combinatorial geometry, we use real stability to get two criteria for interlacing property of zeros between a polynomial and its reciprocal polynomial, which in particular imply the alternatingly increasing property of the original polynomial or its reciprocal polynomial. These criteria extend a result of Brändén and Solus and unify such properties for many combinatorial polynomials, including ascent polynomials for k-ary words, descent polynomials on signed Stirling permutations and colored permutations, q-analog of descent polynomials on colored permutations and descent polynomials on color-signed permutations. Furthermore, we also obtain a recurrence relation and zero-interlacing property of q-analog of descent polynomials on colored permutations that extend some results of Brändén and Brenti. In addition, as an application of weak Hurwitz stability, we get the alternatingly increasing property and zero-interlacing property for two kinds of peak polynomials on the dual set of Stirling permutations.

On cover times of Markov chains

. We consider the cover time of a discrete-time homogenous Markov chain, which is the time needed by the Markov chain to visit all its states. We analyze both the distribution and the moments of the cover time, and we are interested in exact results instead of asymptotic values of the mean cover time, which are generally considered in the literature. We first obtain several general results on the hitting time and the cover time of a subset of the state space, both in terms of distribution and moments. These results are then applied to particular graphs, namely the generalized cycle graph, the complete graph, and the generalized path graph. They lead to recurrence or analytic relations for the distribution and the mean value of their cover times.

Study of multi-dimensional problems arising in wave propagation using a hybrid scheme

Many scientific phenomena are linked to wave problems. This paper presents an effective and suitable technique for generating approximation solutions to multi-dimensional problems associated with wave propagation. We adopt a new iterative strategy to reduce the numerical work with minimum time efficiency compared to existing techniques such as the variational iteration method (VIM) and homotopy analysis method (HAM) have some limitations and constraints within the development of recurrence relation. To overcome this drawback, we present a Sawi integral transform (S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}$$\\end{document}T) for constructing a suitable recurrence relation. This recurrence relation is solved to determine the coefficients of the homotopy perturbation strategy (HPS) that leads to the convergence series of the precise solution. This strategy derives the results in algebraic form that are independent of any discretization. To demonstrate the performance of this scheme, several mathematical frameworks and visual depictions are shown.

Some Properties of Generalized Apostol-Type Frobenius–Euler–Fibonacci Polynomials

In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. Subsequently, we present summation formulas, Stirling–Fibonacci numbers of the second kind, and relationships for these polynomials and numbers. Finally, we define the new family of the generalized Apostol-type Frobenius–Euler–Fibonacci matrix and obtain some factorizations of this newly established matrix. Using Mathematica, the computational formulae and graphical representation for the mentioned polynomials are obtained.

Bi-stability in femtosecond laser ablation by MHz bursts

In this work, a bi-stable behavior of laser ablation efficiency and quality was controlled by fluence and burst length. The plasma shielding of incoming laser radiation caused sudden jumps with a significant decrease in ablation efficiency for every even number of pulses in the burst. The attenuation of incoming laser radiation by plasma created by the previous pulse was incorporated into the toy model of burst ablation efficiency. The mathematical recurrence relation has been derived for the first time, binding ablation efficiency for the next pulse with the efficiency of the previous pulse, which predicts bi-stability, as well as sudden jumps occurring in ablation efficiency depending on the number of pulses in burst with the response to changes of the control parameter of peak laser fluence in the pulse. The modeling results using new recurrence relation showed stable and bi-stable ablation efficiency depending on burst fluence and the number of pulses, which agreed well with experimental data. The extremely efficient laser ablation has been achieved by optimizing the shielding effect using three pulses in the burst.

Integral values of generating functions of recursive sequences

Suppose that a0,a1,… is an integer sequence which satisfies a recurrence relation with constant coefficients, and let T(x)=f(x)/g(x) be its generating function, where f(x) and g(x) have no common factors in Z[x]. In this article, we study the problem of finding the rational values of x such that T(x) is an integer. We say that such a number is good for the sequence. Our first main result is that if g(x) has at least two different irreducible factors, or if g(x) has a single irreducible factor of degree at least 3, then the sequence has only finitely many good values. We also study sequences of the form 0,1,… for which the recurrence relation has order 2. Among other results, we show that under a mild condition on the recurrence relation, the sequence has infinitely many good values, and we give a constructive method to find all of them.

Cobalancing Numbers: Another Way of Demonstrating Their Properties

In this study, previously obtained cobalancing numbers are considered from a different perspective, and the properties of the numbers are re-examined. The main purpose is to change the recurrence relation of cobalancing numbers and calculate some relations and properties in a more diverse and easier way. The reason that led us to this method is that the recurrence relation of cobalancing numbers has a second-order but non-homogeneous difference equations. Thus, it will be much easier to find the Binet formula, generating function, sum formulas, and many other relations with a sequence that is homogeneous and has a third-degree recurrence relation. Also some identities that have not been found before in the sequence are also included in this study.

Model order reduction based on Laguerre orthogonal polynomials for parabolic equation constrained optimal control problems

In this paper, two model order reduction methods based on Laguerre orthogonal polynomials for parabolic equation constrained optimal control problems are studied. The spatial discrete scheme of the cost function subject to a parabolic equation is obtained by Galerkin approximation, and then the coupled ordinary differential equations of the optimal original state and adjoint state with initial value and final value conditions are obtained by Pontryagin's minimum principle. For this original system, we propose two kinds of model order reduction methods based on the differential recurrence formula and integral recurrence formula of Laguerre orthogonal polynomials, respectively, and they are totally different from these existing researches on model order reduction only for initial value problems. Furthermore, we prove the coefficient-matching properties of the outputs between the reduced system and the original system. Finally, two numerical examples are given to verify the feasibility of the proposed methods.

Generalized Padovan Sequences and Figurate Numbers

Generalized Padovan Sequences are more general class of sequences which are viewed as general form of the classical Padovan sequences named after English mathematician Richard Padovan. Figurate numbers are special class of numbers which occur in various problems in number theory. In this paper, we discuss the limiting ratio of the recurrence relation formed by generalizing Padovan sequence and considering Figurate numbers as coefficients in two possible cases. We find very interesting numbers turning out to be limiting ratios under our assumptions.

Fiboquadratic numbers and Rithmomachia

Abstract This paper shows our discovery of the fiboquadratic numbers, quadratic Fibonacci numbers in rithmomachia. This board game consists in performing mathematical operations to attack an opponent through geometric numbered pieces. We will show that the numbers on the board pieces are the first terms of six sequences that we will define through a recursive pattern of superpartients. We then prove that this extension is the solution of a recurrence relation with initial conditions, and therefore, is unique. The fiboquadratic numbers we introduce in this work are not limited to the context of this board. In general, we can generate any fiboquadratic sequence with arbitrary initial conditions.

Two-point resistances in symmetric bracelet resistor networks: accurate estimates and exact expressions

Resistor networks, used to model new types of natural or artificial matter, also provide generic examples for practising the methods of physics for obtaining estimates, revealing the main properties of a system and deriving exact expressions. Symmetric bracelet resistor networks are constructed by connecting n identical resistors in a circle, and then connecting two such circles by another set of n identical resistors. First, using van Steenwijk’s method, we establish that the equivalent resistance or two-point resistance (TPR) between any two nodes is derived when the layer-to-layer resistance R 0n is known. We then determine R 0n by an elementary recurrence relation which converges rapidly to its large n limit. Using this reference value of R 0n , accurate estimates of other TPRs follow for all values of n, characterised by a leading 1/n variation. In addition, exact explicit expressions of the TPRs can be calculated for any value of n. These networks, prototypes of three-dimensional networks considered in research, can be used to illustrate the diversity of the physical approach, the power of elementary methods, and to learn to be comfortable with approximations. Easy to make and use for experimental tests, they can support hands-on activities and conceptual changes.

Multiple Zeta Values and Bell Polynomials

In this short paper, we establish that certain multiple zeta values satisfy a recurrence relation and can be written in terms of Bell polynomials.