Roots Of Polynomial Research Articles

Non-Oil Sector and Economic Growth in Nigeria: The National Accounts Perspective

This study examines the impact of expansion in non-oil sector on sustainable economic growth of Nigeria economy. The study sourced data from the Central bank of Nigeria (CBN) statistical bulletin covering the periods of 2000 – 2019. An economic growth model was formulated using the study variables and the model was estimated using vector auto-regression (VAR) techniques, other diagnostic tests such as Roots of Characteristic Polynomial for VAR model stability, Augmented Dickey-Fuller test for time series stationarity, and granger causality tests were conducted to ensure the reliability of the model estimates. The analysis revealed that the estimated model is stable while the VAR and variance decomposition results shows that real gross domestic product is strongly endogenous in the short run but weakly endogenous in the long run. Further findings suggest that in the long run non-oil sector is strongly endogenous to real gross domestic product (92% contribution). The study, therefore, recommends diversification of the Nigerian economy by focusing more attention on agriculture, solid minerals, and service sectors as they tend to influence economic growth in the long run. More so, improved frameworks of accounting in areas of non-oil revenues are desirable for the accountancy profession.

A unified prediction solution for vibro-acoustic analysis of composite laminated elliptical shells immersed in air

A semi-analytical method to conduct vibro-acoustic analysis of a composite laminated elliptical shell immersed in air is proposed. A variational method and multi-segment technique are used to formulate the dynamic model. The sound radiation of the exterior fluid field is calculated by a spectral Kirchhoff-Helmholtz integral formulation. The variables containing displacements and sound pressure are expanded by the combination of Fourier series and Chebyshev orthogonal polynomials. The collocation points are introduced to construct an algebraic system of acoustic integral equations, where these points are distributed on the roots of Chebyshev polynomials, and the non-uniqueness solution of system is eliminated by a combined Helmholtz integral. Numerical examples for sound radiation problems of composite laminated elliptical shells are presented and individual contributions of the circumferential modes to the acoustical results of composite laminated elliptical shells are also given. The effects of geometric and material parameters on sound radiation of composite laminated elliptical shells are also investigated.

A Comprehensive Analysis of Quantum Clustering : Finding All the Potential Minima

Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a σ value, a hyper-parameter which can be manually defined and manipulated to suit the application. Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an outstanding task because normally such expressions are impossible to solve analytically. However, we prove that if the points are all included in a square region of size σ, there is only one minimum. This bound is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new numerical approach “per block”. This technique decreases the number of particles by approximating some groups of particles to weighted particles. These findings are not only useful to the quantum clustering problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics and other applications.

A high accurate scheme for numerical simulation of two-dimensional mass transfer processes in food engineering

This paper contributes to develop a highly accurate numerical method for solving two-dimensional mass transfer equations during convective air drying of apple slices. The numerical scheme is based on the interpolating the solution of the mentioned equations over the roots of the orthogonal Jacobi polynomials (i.e., the Jacobi-Gauss-Lobatto points) in a nodal form. Moreover, for speeding up the procedure of numerical technique, operational matrices of differentiation are applied to discretize the derivatives of both spatial and temporal variables. After implementing the proposed technique, two-dimensional mass transfer equations would be transferred into the associated systems of linear algebraic equations which can be solved by appropriate iterative solvers such as robust Krylov subspace iterative methods. Some constructed artificial examples are provided to show the efficiency and applicability of the Jacobi pseudo-spectral method for solving two-dimensional mass transfer equations. Finally, a real example is considered and numerical results are validated by the experimental data which confirm the accuracy of the presented numerical approach.

Proposing a New Theorem to Determine If an Algebraic Polynomial Is Nonnegative in an Interval

We face the problem to determine whether an algebraic polynomial is nonnegative in an interval the Yau Number Theoretic Conjecture and Yau Geometric Conjecture is proved. In this paper, we propose a new theorem to determine if an algebraic polynomial is nonnegative in an interval. It improves Wang-Yau Lemma for wider applications in light of Sturm’s Theorem. Many polynomials can use the new theorem but cannot use Sturm’s Theorem and Wang-Yau Lemma to judge whether they are nonnegative in an interval. New Theorem also performs better than Sturm’s Theorem when the number of terms and degree of polynomials increase. Main Theorem can be used for polynomials whose coefficients are parameters and to any interval we use. It helps us to find the roots of complicated polynomials. The problem of constructing nonnegative trigonometric polynomials in an interval is a classical, important problem and crucial to many research areas. We can convert a given trigonometric polynomial to an algebraic polynomial. Hence, our proposed new theorem affords a new way to solve this classical, important problem.

On the Change of Distance Energy of Complete Bipartite Graph due to Edge Deletion

The distance energy of a graph is defined as the sum of absolute values of distance eigenvalues of the graph. The distance energy of a graph plays an important role in many fields. By constructing a new polynomial, we transform a problem on the sum of the absolute values of the roots of a quadratic polynomial into a problem on the largest root of a cubic polynomial. Hence, we give a new and shorter proof on the change of distance energy of a complete bipartite graph due to edge deletion, which was given by Varghese et al.

On the characteristic polynomial of (k,p)-Fibonacci sequence

Recently, Bednarz introduced a new two-parameter generalization of the Fibonacci sequence, which is called the (k,p)-Fibonacci sequence and denoted by (F_{k,p}(n))_{ngeq0}. In this paper, we study the geometry of roots of the characteristic polynomial of this sequence.

Optimal V-Plane Robust Stabilization Method for Interval Uncertain Fractional Order PID Control Systems

Robust stability is a major concern for real-world control applications. Realization of optimal robust stability requires a stabilization scheme, which ensures that the control system is stable and presents robust performance for a predefined range of system perturbations. This study presented an optimal robust stabilization approach for closed-loop fractional order proportional integral derivative (FOPID) control systems with interval parametric uncertainty and uncertain time delay. This stabilization approach, which is carried out in a v-plane, relies on the placement of the minimum angle system pole to a predefined target angle within the stability region of the first Riemann sheet. For this purpose, tuning of FOPID controller coefficients was performed to minimize a root angle error that is defined as the squared difference of minimum angle root of interval characteristic polynomials and the desired target angle within the stability region of the v-plane. To solve this optimization problem, a particle swarm optimization (PSO) algorithm was implemented. Findings of the study reveal that tuning of the target angle can also be used to improve the robust control performance of interval uncertain FOPID control systems. Illustrative examples demonstrated the effectiveness of the proposed v-domain, optimal, robust stabilization of FOPID control systems.

Maximal subgroup growth of some metabelian groups

Let G be a finitely generated group, and let denote the number of maximal subgroups of G of index n. An upper bound is given for the degree of maximal subgroup growth of all polycyclic metabelian groups G (i.e., for the degree of polynomial growth of ). A condition is given for when this upper bound is attained. Let where a generator of the on the right acts (by conjugation) on by multiplication by It is shown that grows like a polynomial of degree equal to the number of blocks in the rational canonical form of A. The leading term of this polynomial is the number of distinct roots (in ) of the characteristic polynomial of the smallest block.

Real forms of the complex Neumann system: A method for finding real roots of polynomial [formula omitted

The topology of Arnold–Liouville level sets of the real forms of the complex generic Neumann system depends indirectly on the positions of the roots of the special polynomial US(λ). For certain polynomials, the existence and positions of the real roots, according to the suitable parameters of the system, is not obvious.In the paper, a method for checking the existence and positions of the real roots of the polynomials US(λ) is given. The method and algorithm are based on searching of a positive solution of a system of linear equations. In case n=2, we provide a complete solution to the problem of existence of real roots for all special polynomials and determine the topology of the Arnold–Liouville level sets.

Generalized Relation between the Roots of Polynomial and Term of Recurrence Relation Sequence

Many researchers have been working on recurrence relation which is an important topic not only in mathematics but also in physics, economics and various applications in computer science. There are many useful results on recurrence relation sequence but there main problem to find any term of recurrence relation sequence we need to find all previous terms of recurrence relation sequence. There were many important theorems obtained on recurrence relations. In this paper we have given special identity for generalized kth order recurrence relation. These identities are very useful for finding any term of any order of recurrence relation sequence. Authors define a special formula in this paper by this we can find direct any term of a recurrence relation sequence. In this recurrence relation sequence to find any terms we need to find all previous terms so this result is very important. There is important property of a relation between coefficients of recurrence relation terms and roots of a polynomial for second order relation but in this paper, we gave this same property of recurrence relation of all higher order recurrence relation. So finally, we can say that this theorem is valid all order of recurrence relation only condition that roots are distinct. So, we can say that this paper is generalization of property of a relation between coefficients of recurrence relation terms and roots of a polynomial. Theorem: - Let C₁ and C₂ are arbitrary real numbers and suppose the equation <img src=image/13422456_01.gif> (1) Has X₁ and X₂ are distinct roots. Then the sequence <img src=image/13422456_25.gif> is a solution of the recurrence relation <img src=image/13422456_02.gif> (2) <img src=image/13422456_03.gif>. For n= 0, 1, 2 …where β₁ and β₂ are arbitrary constants. Proof: - First suppose that <img src=image/13422456_25.gif> of type <img src=image/13422456_04.gif> we shall prove <img src=image/13422456_25.gif> is a solution of recurrence relation (2). Since X₁, X₂ and X₃ are roots of equation (1) so all are satisfied equation (1) so we have<img src=image/13422456_05.gif>, <img src=image/13422456_06.gif>. Consider <img src=image/13422456_07.gif><img src=image/13422456_08.gif>. This implies <img src=image/13422456_09.gif>. So the sequence <img src=image/13422456_25.gif> is a solution of the recurrence relation. Now we will prove the second part of theorem. Let <img src=image/13422456_10.gif> is a sequence with three <img src=image/13422456_11.gif>. Let <img src=image/13422456_12.gif>. So <img src=image/13422456_13.gif> (3). <img src=image/13422456_14.gif> (4). Multiply by X₁ to (3) and subtracts from (4). We have <img src=image/13422456_15.gif> similarly we can find <img src=image/13422456_16.gif>. So we can say that values of β₁ and β₂ are defined as roots are distinct. So non- trivial values ofβ₁ and β₂ can find and we can say that result is valid. Example: Let <img src=image/13422456_25.gif> be any sequence such that <img src=image/13422456_17.gif> n≥3 and a₀=0, a₁=1, a₂=2. Then find a₁₀ for above sequence. Solution: The polynomial of above sequence is <img src=image/13422456_18.gif>. Solving this equation we have roots are 1, 2, and 3 using above theorem we have <img src=image/13422456_19.gif> (7). Using a₀=0, a₁=1, a₂=2 in (7) we have β₁+β₂+β₃=0 (8). β₁+2β₂+3β₂=1 (9).β₁+4β₂+9β₃=2 (10) Solving (8), (9) and (10) we have <img src=image/13422456_20.gif>, <img src=image/13422456_21.gif>, <img src=image/13422456_22.gif>. This implies <img src=image/13422456_23.gif>. Now put n=10 we have a₁₀=-27478. Recurrence relation is a very useful topic of mathematics, many problems of real life may be solved by recurrence relations, but in recurrence relation there is a major difficulty in the recurrence relation. If we want to find 100th term of sequence, then we need to find all previous 99 terms of given sequence, then we can get 100th term of sequence but above theorem is very useful if coefficients of recurrence relation of given sequence satisfies the condition of the above theorem, then we can apply above theorem and we can find direct any term of sequence without finding all previous terms.

A closed-form solution to multi-level image thresholding

This paper proposes an analytical formulation relying on the least square error. Similar results were also found for the cross correlation, within-class, and between-class variance. At first, a continuous distribution is hypothesized (for derivation purposes only) to produced a modified form of the well-known OTSU method. This hypothesis is “identical” to Otsu in terms of output performance and the need for an exhaustive search. However, apart from being derived from the continuous form, the proposed scheme requires less computational power. It turns out that the optimum threshold equals the average of the adjacent regions’ means. For some images, the scheme can result in multi-level thresholdeds. A direct form was then suggested to obtain a non-exhaustive solution. The idea is simply to approximate the non-continuous error function (used by the least square formulation and OTSU) with a forth order polynomial defined in the normalized gray intensity range [0,1]. The optimum threshold can then be found as a function of the roots of a second order polynomial whose coefficients are the solution of a 2x2 linear system. The performance of the proposed non-exhaustive solution is slightly inferior to OTSU in general, however; some images produced improved performance. Nevertheless, The proposed scheme can be easily generalized to the multi-level case without the need for an exhaustive search. For n+1 levels (i.e. n thresholds), the output is obtained by solving an nxn linear system followed by finding the roots of a n-order polynomial. The computational cost is clearly superior to the exhaustive search. In addition, as validated with some images, the performance is encouraging. Extension to the general clustering case is highly envolved with the exception of the two-level case (for any dimension) that has been successfully derived in this work.

Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applications

This paper seeks to further explore the distribution of the real roots of random polynomials with non-centered coefficients. We focus on polynomials where the typical values of the coefficients have power growth and count the average number of real zeros. Almost all previous results require coefficients with zero mean, and it is non-trivial to extend these results to the general case. Our approach is based on a novel comparison principle that reduces the general situation to the mean-zero setting. As applications, we obtain new results for the Kac polynomials, hyperbolic random polynomials, their derivatives, and generalizations of these polynomials. The proof features new logarithmic integrability estimates for random polynomials (both local and global) and fairly sharp estimates for the local number of real zeros.

Structured backward errors in linearizations

A standard approach to calculate the roots of a univariate polynomial is to compute the eigenvalues of an associated confederate matrix instead, such as, for instance, the companion or comrade matrix. The eigenvalues of the confederate matrix can be computed by Francis's QR algorithm. Unfortunately, even though the QR algorithm is provably backward stable, mapping the errors back to the original polynomial coefficients can still lead to huge errors. However, the latter statement assumes the use of a non-structure-exploiting QR algorithm. In [J. L. Aurentz et al., Fast and backward stable computation of roots of polynomials, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 942–973] it was shown that a structure-exploiting QR algorithm for companion matrices leads to a structured backward error in the companion matrix. The proof relied on decomposing the error into two parts: a part related to the recurrence coefficients of the basis (a monomial basis in that case) and a part linked to the coefficients of the original polynomial. In this article we prove that the analysis can be extended to other classes of comrade matrices. We first provide an alternative backward stability proof in the monomial basis using structured QR algorithms; our new point of view shows more explicitly how a structured, decoupled error in the confederate matrix gets mapped to the associated polynomial coefficients. This insight reveals which properties have to be preserved by a structure-exploiting QR algorithm to end up with a backward stable algorithm. We will show that the previously formulated companion analysis fits into this framework, and we analyze in more detail Jacobi polynomials (comrade matrices) and Chebyshev polynomials (colleague matrices).

Efficient Algorithm for Finding Roots of Error-Locator Polynomials

A novel method for finding roots of polynomials over finite fields has been proposed. This method is based on the cyclotomic discrete Fourier transform algorithm. The improvement is achieved by using the normalized cyclic convolutions, which have a small complexity and allow matrix decomposition, as well as methods of adapting the truncated normalized cyclic convolutions calculation. For small values of degree of the error-locator polynomial the novel method has not only the smallest multiplicative complexity, but the full computational complexity of this method is also less than with other methods. Thus, the multiplicative complexity of the novel method in comparison to the method of affine decomposition (the Fedorenko-Trifonov method) is up to ten times less, although the additive complexity is approximately 10-15% more. The novel method has matrix representation convenient for implementation.

Approximately half of the roots of a random Littlewood polynomial are inside the disk

We prove that for large $n$, all but $o(2^{n})$ polynomials of the form $P(z) = \sum_{k=0}^{n-1}\pm z^k$ have $n/2 + o(n)$ roots inside the unit disk. This solves a problem from Hayman's book 'Research Problems in Function Theory' (1967).

Periodic continued fractions over $S$-integers in number fields and Skolem’s $p$-adic method

We generalize the classical theory of periodic continued fractions (PCFs) over ${\mathbf Z}$ to rings ${\mathcal O}$ of $S$-integers in a number field. Let ${\mathcal B}=\{\beta, {\beta^*}\}$ be the multi-set of roots of a quadratic polynomial in ${\mathcal O}[x]$. We show that PCFs $P=[b_1,\ldots,b_N,\bar{a_1\ldots ,a_k}]$ of type $(N,k)$ potentially converging to a limit in ${\mathcal B}$ are given by ${\mathcal O}$-points on an affine variety $V:=V({\mathcal B})_{N,k}$ generically of dimension $N+k-2$. We give the equations of $V$ in terms of the continuant polynomials of Wallis and Euler. The integral points $V({\mathcal O})$ are related to writing matrices in $\textrm{SL}_2({\mathcal O})$ as products of elementary matrices. We give an algorithm to determine if a PCF converges and, if so, to compute its limit. Our standard example generalizes the PCF $\sqrt{2}=[1,\bar{2}]$ to the ${\mathbf Z}_2$-extension of ${\mathbf Q}$: $F_n={\mathbf Q}(\alpha_n)$, $\alpha_{n}:=2\cos(2\pi/2^{n+2})$, with integers ${\mathcal O}_n={\mathbf Z}[\alpha_n]$. We want to find the PCFs of $\alpha_{n+1}$ over ${\mathcal O}_{n}$ of type $(N,k)$ by finding the ${\mathcal O}_{n}$-points on $V({\mathcal B}_{n+1})_{N,k}$ for ${\mathcal B}_{n+1}:=\{\alpha_{n+1}, -\alpha_{n+1}\}$. There are three types $(N,k)=(0,3), (1,2), (2,1)$ such that the associated PCF variety $V({\mathcal B})_{N,k}$ is a curve; we analyze these curves. For generic ${\mathcal B}$, Siegel's theorem implies that each of these three $V({\mathcal B})_{N,k}({\mathcal O})$ is finite. We find all the ${\mathcal O}_n$-points on these PCF curves $V({\mathcal B}_{n+1})_{N,k}$ for $n=0,1$. When $n=1$ we make extensive use of Skolem's $p$-adic method for $p=2$, including its application to Ljunggren's equation $x^2 + 1 =2y^4$.

Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial systems

:The roots of random polynomials and of random polynomial systems have been extensively studied in the past. While the picture is quite complete in the case of polynomials, the case of systems is much harder. Kostlan, Shub, and Smale random polynomial systems were introduced in the nineties and the expectation of the number of their real roots is known since then. Only recently the asymptotic variance, as the degree goes to infinity, was obtained. In the present paper we obtain a Central Limit Theorem for the number of real roots of a square Kostlan-Shub-Smale system of any size as the degree goes to infinity.

HOMOTOPY CONTINUATION FOR THE SPECTRA OF PERSISTENT LAPLACIANS.

The p-persistent q-combinatorial Laplacian defined for a pair of simplicial complexes is a generalization of the q-combinatorial Laplacian. Given a filtration, the spectra of persistent combinatorial Laplacians not only recover the persistent Betti numbers of persistent homology but also provide extra multiscale geometrical information of the data. Paired with machine learning algorithms, the persistent Laplacian has many potential applications in data science. Seeking different ways to find the spectrum of an operator is an active research topic, becoming interesting when ideas are originated from multiple fields. In this work, we explore an alternative approach for the spectrum of persistent Laplacians. As the eigenvalues of a persistent Laplacian matrix are the roots of its characteristic polynomial, one may attempt to find the roots of the characteristic polynomial by homotopy continuation, and thus resolving the spectrum of the corresponding persistent Laplacian. We consider a set of simple polytopes and small molecules to prove the principle that algebraic topology, combinatorial graph, and algebraic geometry can be integrated to understand the shape of data.

A Note on Lienard-Chipart Criteria and its Application to Epidemic Models

An important part of the study of epidemic models is the local stability analysis of the equilibrium points. The linear algebra method which is commonly employed is the well-known Routh-Hurwitz criteria. The criteria give necessary and sufficient conditions for all of the roots of the characteristic polynomial to be negative or have negative real parts. To date, there are no epidemic models in the literature which employ Lienard-Chipart criteria. This note recommends an alternative linear algebra method namely Lienard-Chipart criteria, to significantly simplify the local stability analysis of epidemic models. Although Routh-Hurwitz criteria are a correct method for local stability analysis, Lienard-Chipart criteria have advantages over Routh-Hurwitz criteria. Using Lienard-Chipart criteria, only about half of the Hurwitz determinants inequalities are required, with the remaining conditions of each set concern with only the sign of the alternate coefficients of the characteristic polynomial. The Lienard-Chipart criteria are especially useful for polynomials with symbolic coefficients, as the determinants are usually significantly more complicated than original coefficients as degree of the polynomial increases. Lienard-Chipart criteria and Routh-Hurwitz criteria have similar performance for systems of dimension five or less. Theoretically, for systems of dimension higher than five, verifying Lienard-Chipart criteria should be much easier than verifying Routh-Hurwitz criteria and the advantage of Lienard-Chipart criteria may become clear. Examples of local stability analysis using Lienard-Chipart criteria for two recently proposed models are demonstrated to show the advantages of simplified Lienard-Chipart criteria over Routh-Hurwitz criteria.