Homogeneous Recurrence Research Articles

ON HOMOGENEOUS LINEAR RECURRENCE RELATIONS

We use the Z-transform to motivate the Baldoni et al algorithm to solve homogeneous linear recurrence relations, with applications to Fibonacci numbers and Chebyshev polynomials.

Linear Localization of Zero Modes in Weakly Coupled Non‐Hermitian Reservoirs

AbstractTopological and symmetry‐protected non‐Hermitian zero modes have attracted considerable interest in the past few years. Here, it is revealed that they can exhibit an unusual behavior when transitioning between the extended and localized regimes: When weakly coupled to a non‐Hermitian reservoir, such a zero mode displays a linearly decreasing amplitude as a function of space, which is not caused by an EP of a Hamiltonian, either of the entire system or the reservoir itself. Instead, this phenomenon is due to the non‐Bloch solution of a linear homogeneous recurrence relation, together with the underlying non‐Hermitian particle‐hole symmetry and the zeroness of its energy.

New Approaches for Solving Interpolation Problems and Homogeneous Linear Recurrence Relations

This article presents a new approach to address the resolution of homogeneous linear recurrences of higher order and interpolation problems. By establishing an explicit formula for the entries of the inverse of generalized Vandermonde matrices, a fresh perspective on these mathematical challenges is introduced. The study primarily focuses on linear recurrence relations and thoroughly investigates cases involving characteristic polynomials with both simple roots and roots of multiplicity. To illustrate the effectiveness and practicality of the proposed method, a comprehensive set of illustrative examples is provided, highlighting its applicability in solving a wide range of instances of linear recurrence relations. Additionally, the limitations of the formula are discussed, particularly in scenarios where its applicability may be restricted. The findings of this study contribute significantly to the existing literature, providing an alternative and promising approach for solving problems that rely on the inverse Vandermonde matrix. In conclusion, this article emphasizes the need for further research to explore the computational advantages of the proposed method and to extend its applicability to cases featuring characteristic polynomials with a single root of multiplicity greater than one. By expanding the knowledge in the field, this study offers valuable insights into the resolution of linear recurrences and interpolation problems, presenting a new perspective and expanding the existing knowledge in the field.

Triangular numbers and generalized fibonacci polynomial

Abstract In the present paper, we study triangular numbers. We focus on the linear homogeneous recurrence relation of degree 3 with constant coefficients for triangular numbers. Then we deal with the relationship between generalized Fibonacci polynomials and triangular numbers. We show that different properties of triangular numbers can be obtained by using this relationship. Finally, we examine the properties of the sequence A052529 that has strong relationships with triangular numbers.

Numerical evaluation and analysis of highly oscillatory singular Bessel transforms with a particular oscillator

This paper proposes and analyzes two affordable and efficient quadrature rules for the numerical approximation of the oscillatory Bessel transform ∫0bxα(b−x)βf(x)Jν(ωxγ)dx with algebraic singularities, where b,α,β,ν,ω,γ denote the given constants. Firstly, we derive the explicit formula and asymptotic estimation of the generalized moments ∫0bxα′(b−x)β′Jν(ωxγ)dx with α′,β′>−1 by means of the Meijer G function. Furthermore, we design a modified Filon-type method based on a two-endpoint Taylor interpolation polynomial. In particular, we also give a more efficient Clenshaw–Curtis–Filon-type method in view of a special Hermite interpolation polynomial at the Clenshaw–Curtis points. Moreover, this method is easily implemented by the fast Fourier transform and fast computation of the modified moments. The useful homogeneous recurrence relation of the required modified moments is derived by the Bessel equation and the properties of the Chebyshev polynomial. Importantly, the rigorous error analyses in inverse powers of ω for the proposed numerical methods are carried out in details. Some primary numerical experiments can confirm our theoretical analysis, and verify the accuracy and efficiency of the proposed numerical methods.

NOVEL FINDINGS ON CARDIORESPIRATORY ROLE SWAPPING BY ASSESSING NONLINEAR DYNAMICAL INTERACTIONS

This paper reports the innovative results of applying nonlinear dynamical theory to heart-rate (HR) and respiratory (RP) sequences to explore Yin–Yang (passive–active) interacting phenomenon of cardiorespiratory function. Organ systems inside human body interact mutually in the manner of swapping the active–passive role, according to Yin–Yang theory in traditional Chinese medicine. Yang as an active source rules over Yin as a passive load. Nonlinear inter-driven (NLID) factor, [Formula: see text], based on recurrence analysis is proposed in this study. NLID factor characterizes the nonlinear interactions between two nonlinear dynamical systems, cardiac and respiratory systems. NLID factor is computed by quantifying the HR and RP recurrent plots constructed from phase trajectories based on embedding theorem and phase-space reconstruction. Heterogeneous patterns of HR and RP recurrence plots in continuous-attention task reflect highly unpredictable system dynamics. Heterogeneous HR recurrence patterns are surprisingly associated with the nearly homogeneous RP recurrence plot at resting state, indicating rather weak cardiorespiratory interaction. HR recurrence in Zen exhibits notable regularity in spite of the irregular RP recurrence behaviors. NLID factor reveals prominent Yang energy delivered by RP system that dominates over the cardiac system in Zen-meditation group. Such findings draw our attention to the possibly extraordinary self-regularity power of heart (and autonomic nervous system) in the Zen-meditation state.

Symbolic Conversion of Holonomic Functions to Hypergeometric Type Power Series

A term $a_n$ is $m$-fold hypergeometric, for a given positive integer $m$, if the ratio $a_{n+m}/a_n$ is a rational function over a field $K$ of characteristic zero. We establish the structure of holonomic recurrence equation, i.e. linear and homogeneous recurrence equations having polynomial coefficients, that have $m$-fold hypergeometric term solutions over $K$, for any positive integer $m$. Consequently, we describe an algorithm, say $mfoldHyper$, that extends van Hoeij's algorithm (1998) which computes a basis of the subspace of hypergeometric $(m=1)$ term solutions of holonomic recurrence equations to the more general case of $m$-fold hypergeometric terms. We generalize the concept of hypergeometric type power series introduced by Koepf (1992), by considering linear combinations of Laurent-Puiseux series whose coefficients are $m$-fold hypergeometric terms. Thus thanks to $mfoldHyper$, we deduce a complete procedure to compute these power series; indeed, it turns out that every linear combination of power series with $m$-fold hypergeometric term coefficients, for finitely many values of $m$, is detected. On the other hand, we investigate an algorithm to represent power series of non-holonomic functions. The algorithm follows the same steps of Koepf's algorithm, but instead of seeking holonomic differential equations, quadratic differential equations are computed and the Cauchy product rule is used to deduce recurrence equations for the power series coefficients. This algorithm defines a normal function that yields together with enough initial values normal forms for many power series of non-holonomic functions. Therefore, non-trivial identities are automatically proved using this approach. This paper is accompanied by implementations in the Computer Algebra Systems (CAS) Maxima 5.44.0 and Maple 2019.

Hierarchical secret sharing scheme for WSN based on linear homogeneous recurrence relations

Wireless sensor network is a key technology in the sensing layer of the Internet of Things. Data security in wireless sensor network is directly related to the authenticity and validity of data transmitted in the Internet of Things. Due to the large number and different types of nodes in wireless sensor networks, layered secret key sharing technology is increasingly used in wireless sensor networks. In a hierarchical secret sharing scheme, participants are divided into sections with different permissions for each team, but the same permissions for participants in the same team. In this article, we follow the approach of the hierarchical secret sharing scheme derived from the linear homogeneous recurrence relations. We design a hierarchical multi-secret sharing scheme for wireless sensor networks on the basis of the elliptic curve public key cryptosystem combined with the linear homogeneous recurrence relations. In the proposed scheme, we do not make sure that the participants are half-truthful. In addition, the participants’ shadows can be reused. Our scheme is computational security. Only one share from each member is required in our hierarchical multi-secret sharing scheme. It is more suitable for wireless sensor networks compared to the up-to-date schemes.

A new efficient hierarchical multi-secret sharing scheme based on linear homogeneous recurrence relations

Hierarchical secret sharing is an important key management technique since it is specially customized for hierarchical organizations with different departments allocated with different privileges, such as the government agencies or companies. Hierarchical access structures have been widely adopted in secret sharing schemes, where efficiency is the primary consideration for various applications. How to design an efficient hierarchical secret sharing scheme is an important issue. A famous hierarchical secret sharing (HSS) scheme was proposed by Tassa based on Birkhoff interpolation. Later, based on the same method, many other HSS schemes were proposed. However, these schemes all depend on Polya’s condition, which is a necessary condition, not a sufficient condition. It cannot guarantee that Tassa’s HSS scheme always exists. Furthermore, this condition needs to check the non-singularity of many matrices.We propose a hierarchical multi-secret sharing scheme based on the linear homogeneous recurrence (LHR) relations and the one-way function. In our scheme, we select m linearly independent homogeneous recurrence relations. The participants in the highly-ranked subsets γ1,γ2,⋯,γj-1 join in the jth subset to construct the jth LHR relation. In addition, the proposed hierarchical multi-secret sharing scheme just requires one share for each participant. Besides, our scheme is both perfect and ideal. Furthermore, our scheme avoids many checks of the non-singularity of many matrices in the presented hierarchical secret sharing schemes. Although we need to publish more public values, our scheme reduces the computational complexity of the hierarchical secret sharing schemes from exponential time to polynomial time, i.e., O(nkm-1logn), which is relatively more efficient than schemes in the literature.

Connections Between Partitions and Divisors Related to the Parity of the Partition Function

Summary In this paper, the author gives two linear homogeneous recurrence relations to compute the values of Euler’s partition function p(n) when n is odd. These recurrences do not involve the generalized pentagonal numbers and provide new connections between the partition function p(n) and the positive divisor function for x = 0 and x = 1. Some relations involving the divisor functions . and the parity of the partition function p(n) are presented in this context.

Number of Subsets of the Set [n] Including No Three Consecutive Odd Integers.

For every natural number, let the sequence enumerate the number of subsets S of the set {1,2, . . . ,n } including no three consecutive odd integers. We give the generating function and the closed form formula of the sequence obtaining sixth order linear homogeneous recurrence relation with constant coefficients of the integer sequence. The sequence is associated with the Tribonacci sequence. The combinatorial representation of the sequence is obtained and limit of the ratios of consecutive terms of the sequence is found.

Relationships between spherical and bispherical harmonics, and an electrostatic T-matrix for dimers

Bispherical harmonics are the solutions to Laplace’s equation in bispherical coordinates. We investigate the relationships between spherical harmonics and bispherical harmonics in terms of radial inversion and derive new series expansions between the harmonics. The series coefficients are the Delannoy numbers encountered in combinatorics, and we exploit the series to derive for them a new second order homogeneous recurrence relation. We use the T-matrix/null field method to solve for the potential of two different spheres in an arbitrary static external electric field, as a series of bispherical harmonics where the coefficients are related via matrix transformations. The matrix elements are expressed as surface integrals of bispherical harmonics, for which analytic expressions are derived in terms of Legendre functions. The resonant values of the dielectric function are computed via an eigenvalue problem which is found to be more stable than the solution using difference equations. The rate of convergence of the matrix formulation is compared to the re-expansion method using spherical harmonics for a uniform field an a dipole in the gap; each series converges faster in a different region of space, which we discuss in terms of the boundaries of convergence of each solution and the image singularities of the scattered field.

An Efficient Compartmented Secret Sharing Scheme Based on Linear Homogeneous Recurrence Relations

Multipartite secret sharing schemes are those that have multipartite access structures. The set of the participants in those schemes is divided into several parts, and all the participants in the same part play the equivalent role. One type of such access structure is the compartmented access structure, and the other is the hierarchical access structure. We propose an efficient compartmented multisecret sharing scheme based on the linear homogeneous recurrence (LHR) relations. In the construction phase, the shared secrets are hidden in some terms of the linear homogeneous recurrence sequence. In the recovery phase, the shared secrets are obtained by solving those terms in which the shared secrets are hidden. When the global threshold is t , our scheme can reduce the computational complexity of the compartmented secret sharing schemes from the exponential time to polynomial time. The security of the proposed scheme is based on Shamir’s threshold scheme, i.e., our scheme is perfect and ideal. Moreover, it is efficient to share the multisecret and to change the shared secrets in the proposed scheme.

On a nonlinear relation for computing the overpartition function

Abstract In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function p (n). Computing p (n) by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the values of p (n) that requires only the values of p (k) with k ≤ n/2. This formula is combined with a known linear homogeneous recurrence relation for the overpartition function p (n) to obtain a simple and fast computation of the value of p (n). This new method uses only (large) integer arithmetic and it is simpler to program.

The impact of administrative prejudice upon the content of penal prohibition

The object of this research is the legal relations arising in the context of implementation of the norms of criminal law of the Russian Federation that establish liability for repeated administrative offenses. The subject of this research is the practice of application the criminal law norms of the Soviet and post-Soviet periods that regulate the institution of administrative prejudice, and acts of interpretation of the Russian Constitution, in which the Supreme Constitutional Court of the Russian Federation deals with the problems of the use of separate articles of the Criminal Code of the Russian Federation that contain the norms with administrative prejudice, and parts of interrelation between the institutions of administrative prejudice and recurrence of offenses. The subject of this research is also justification of existence the institution of administrative prejudice in the national criminal law, main flaws in interpretation of the articles that describe the norms of the institution of administrative prejudice in the text of the Criminal Code of the Russian Federation. Taking into consideration all shortcomings in interpretation of the articles of the Criminal Code of the Russian Federation, the author offers unified definition of the composition with administrative prejudice. It is suggested to reintroduce the concept of recurrent offense in the General Part of the Criminal Code of the Russian Federation, taking into account the fact of administrative liability, outstanding criminal record, or criminal record that has not been expunged. The case if the legislator deems it necessary to take into account not identical, but homogeneous recurrence should be stipulated in the note to the article of the Special Part. The author also offers to include the Article 16.1 into the General Part of the Criminal Code of the Russian Federation in the following wording: “The repeated offense is considered an act committed by a person who has previously been subjected to administrative penalty for similar type of offense, unless stipulated otherwise in the corresponding articles of the Special Part of the effective Code”.

New Curiosity Bivariate Quadratic Quaternionic Polynomials and Their Roots

Abstract We consider the second-order linear homogeneous quaternion recurrence solutions for some new curiosity bivariate quadratic quaternionic equations.

Analytical assessment of the frequency of natural vibrations of a truss with an arbitrary number of panels

The aim of the work is to derive a formula for the dependence of the first frequency of the natural oscillations of a planar statically determinate beam truss with parallel belts on the number of panels, sizes and masses concentrated in the nodes of the lower truss belt. Truss has a triangular lattice with vertical racks. The solution uses Maple computer math system operators. Methods. The basis for the upper estimate of the desired oscillation frequency of a regular truss is the energy method. As a form of deflection of the truss taken deflection from the action of a uniformly distributed load. Only vertical mass movements are assumed. The amplitude values of the deflection of the truss is calculated by the Maxwell - Mohrs formula. The forces in the rods are determined in symbolic form by the method of cutting nodes. The dependence of the solution on the number of panels is obtained by an inductive generalization of a series of solutions for trusses with a successively increasing number of panels. For sequences of coefficients of the desired formula, fourth-order homogeneous linear recurrence equations are compiled and solved. Results. The solution is compared with the numerical one, obtained from the analysis of the entire spectrum of natural frequencies of oscillations of the mass system located at the nodes of the truss. The frequency equation is compiled and solved using Eigenvalue search operators in the Maple system. It is shown that the obtained analytical estimate differs from the numerical solution by a fraction of a percent. Moreover, with an increase in the number of panels, the error of the energy method decreases monotonically. A simpler lower bound for the oscillation frequency according to the Dunkerley method is presented. The accuracy of the lower estimate is much lower than the upper estimate, depending on the size and number of panels.

ANALYTICAL CALCULATION OF DEFLECTION OF A MULTI-LATTICE TRUSS WITH AN ARBITRARY NUMBER OF PANELS

The scheme of a planar externally statically indeterminate truss with four supports is proposed. In analytical form, for several types of loads, the problem of forces in the rods and deflectionof the structure is solved, depending on the number of panels, the size and intensity of the load. The solution uses the Maple computer mathematics system. The deflectionat Midspan is determined using Maxwell – Mohr's formula, the forces in the rods – the method of cutting out nodes from the system of equilibrium equations for all nodes, which includes four reactions of the supports. By induction, a series of solutions for trusses with a consistently increasing number of panels is generalized to an arbitrary number of panels. For the elements of the sequences of coefficientare developed and are solved by homogeneous linear recurrence equations. The resulting formulas for the deflectio of the structure under various loads have the form of polynomials in the number of panels. A linear asymptotic solution for the number of panels is found. The kinematic degeneration of the structure and the distribution of node speeds corresponding to this case were found. The dependences of the reaction of supports and forces in the most compressed and stretched rods on the number of panels are determined.

Clenshaw–Curtis algorithms for an efficient numerical approximation of singular and highly oscillatory Fourier transform integrals

This paper investigates the implementation of Clenshaw–Curtis algorithms on singular and highly oscillatory integrals for efficient evaluation of the finite Fourier-type transform of integrands with endpoint singularities. In these methods, integrands are truncated by orthogonal polynomials and special function series term by term. Then their singularity types are computed using third and fourth-order homogeneous recurrence relations. The first approach reveals its efficiency for low, moderate and very high frequencies, whereas the second one, is more efficient for small values of frequencies. Moreover, all the results were found quite satisfactory. Algorithms and programming code in MATHEMATICA® 9.0 are provided for the implementation of methods for automatic computation on a computer. Lastly, illustrative numerical experiments and comparison of the proposed Clenshaw–Curtis algorithms to the steepest descent method are mentioned in support of our theoretical analysis in the examples section.

Linear recurrence sequences and the duality defect conjecture

It is conjectured that the dual variety of every smooth nonlinear subvariety of dimension $> \frac{2N}{3}$ in projective $N$-space is a hypersurface, an expectation known as the duality defect conjecture. This would follow from the truth of Hartshorne's complete intersection conjecture but nevertheless remains open for the case of subvarieties of codimension $> 2$. A combinatorial approach to proving the conjecture in the codimension $2$ case was developed by Holme, and following this approach Oaland devised an algorithm for proving the conjecture in the codimension $3$ case for particular $N$. This combinatorial approach gives a potential method of proving the duality defect conjecture in many of the cases by studying the positivity of certain homogeneous integer linear recurrence sequences. We give a generalization of the algorithm of Oaland to the higher codimension cases, obtaining with this bounds the degrees of counterexamples would have to satisfy, and using the relationship with recurrence sequences we prove that the conjecture holds in the codimension $3$ case when $N$ is odd.