Unique Positive Root Research Articles

On the balanced pantograph equation of mixed type

UDC 517.9 We consider the balanced pantograph equation (BPE) y ′ ( x ) + y ( x ) = ∑ k = 1 m p k y ( a k x ) , where a k , p k > 0 and ∑ k = 1 m p k = 1. It is known that if K = ∑ k = 1 m p k ln a k ≤ 0 then, under mild technical conditions, the BPE does not have bounded solutions that are not constant, whereas for K > 0 these solutions exist. In the present paper, we deal with a BPE of <em>mixed type</em>, i.e., a 1 < 1 < a m , and prove that, in this case, the BPE has a nonconstant solution y and that y ( x ) ∼ c x σ as x → ∞ , where c > 0 and σ is the unique positive root of the characteristic equation P ( s ) = 1 - ∑ k = 1 m p k a k - s = 0. We also show that y is unique (up to a multiplicative constant) among the solutions of the BPE that decay to zero as x → ∞ .

On the date of the epidemic peak.

Epidemiologists have used the timing of the peak of an epidemic to guide public health interventions. By determining the expected peak time, they can allocate resources effectively and implement measures such as quarantine, vaccination, and treatment at the right time to mitigate the spread of the disease. The peak time also provides valuable information for those modeling the spread of the epidemic and making predictions about its future trajectory. In this study, we analyze the time needed for an epidemic to reach its peak by presenting a straightforward analytical expression. Utilizing two epidemiological models, the first is a generalized $ SEIR $ model with two classes of latent individuals, while the second incorporates a continuous age structure for latent infections. We confirm the conjecture that the peak occurs at approximately $ T\sim(\ln N)/\lambda $, where $ N $ is the population size and $ \lambda $ is the largest eigenvalue of the linearized system in the first model or the unique positive root of the characteristic equation in the second model. Our analytical results are compared to numerical solutions and shown to be in good agreement.

Fullerenes via their counting polynomials

For a graph G with orbits O1,…,Ok, the orbit polynomial is defined as OG(x)=∑i=1kx|Oi|. In the current work, we will characterize all fullerenes concerning their pentagonal orbit polynomials (namely, the orbit polynomials for orbits of pentagons). Besides, the orbit polynomial, the orbit-entropy, the symmetry index and the unique positive root of orbit polynomial are computed for several infinite families of fullerenes. Finally, correlations between these graph measures are established to quantify their relationship.

Novel results on partial hosoya polynomials: An application in chemistry

This article deal with an investigation of certain properties of partial Hosoya polynomial and then computing this new polynomial for some family of well-known graphs. Also, we verify some results concerning the zeros location of this polynomial in a ring shaped region. In this way, we include not only new results as special cases, but also improve the results due to Dehmer et al. [12] as a particular case. In continuing, the unique positive root of a new version of partial Hosoya polynomial has been evaluated. Finally, we explore some functional measures based on the complex zeros of partial Hosoya polynomials and then we compute the Pearson correlation between these measures and well-known topological indices. Our results show that these new measures have a linear correlation with the Wiener index.

ON N-VERTEX CHEMICAL GRAPHS WITH A FIXED CYCLOMATIC NUMBER AND MINIMUM GENERAL RANDI´C INDEX

"The general Randi´c index of a graph G is defined as Rα(G) = P uv∈E(G)(dudv)α, where du and dv denote the degrees of the vertices u and v, respectively, α is a real number, and E(G) is the edge set of G. The minimum number of edges of a graph G whose removal makes G as acyclic is known as the cyclomatic number and it is usually denoted by ν. A graph with the maximum degree at most 4 is known as a chemical graph. For ν = 0, 1, 2 and α > 1, the problem of finding graph(s) with the minimum general Randi´c index Rα among all n-vertex chemical graphs with the cyclomatic number ν has already been solved. In this paper, this problem is solved for the case when ν ≥ 3, n ≥ 5(ν − 1), and 1 < α < α0, where α0 ≈ 11.4496 is the unique positive root of the equation 4(8α − 6α) + 4α − 9α = 0."

On the Roots of the Modified Orbit Polynomial of a Graph

The definition of orbit polynomial is based on the size of orbits of a graph which is OG(x)=∑ix|Oi|, where O1,…,Ok are all orbits of graph G. It is a well-known fact that according to Descartes’ rule of signs, the new polynomial 1−OG(x) has a positive root in (0,1), which is unique and it is a relevant measure of the symmetry of a graph. In the current work, several bounds for the unique and positive zero of modified orbit polynomial 1−OG(x) are investigated. Besides, the relation between the unique positive root of OG in terms of the structure of G is presented.

Network Analyzing by the Aid of Orbit Polynomial

This article aims to be a further contribution to the research on structural complexity networks. Here, we emphasize measures to determine symmetry. The so-called “orbit polynomial” is defined by OG(x)=∑iaixi, where ai is the number of orbits of size i. Furthermore, the graph polynomial 1−OG(x) has a unique positive root in the interval (0,1), which can be considered as a relevant measure of the symmetry of a graph. In the present paper, we studied some properties of the orbit polynomial with respect to the stabilizer elements of each vertex. Furthermore, we constructed graphs with a small number of orbits and characterized some classes of graphs in terms of calculating their orbit polynomials. We studied the symmetry structure of well-known real-world networks in terms of the orbit polynomial.

Implicit and explicit discrete‐time realizations of homogeneous differentiators

AbstractThis article presents explicit and implicit discrete‐time realizations for a class of homogeneous sliding mode‐based differentiators. The proposed approach relies on the method of exact discretization for linear systems with a zero‐order holder. The two discrete‐time schemes present the homogeneity property and, after a finite time, the accuracy of its continuous‐time counterpart, even in the presence of bounded noise. In comparison to the explicit realization, which makes it possible to determine the state of the system later from the state at present, for this case, the implicit method requires finding a solution by solving generalized equations that involve the current state of the system and two support variables. Therefore, this document proves that the polynomial's unique positive root defines the required solution as part of the main results. Hence, it is possible to introduce a non‐anticipative method to implement the implicit discrete‐time realization, including an appropriate root‐finding method for the polynomial. Finally, the simulation results include comparisons between the proposed implicit and explicit discrete methods with other existing schemes. Numerical studies clearly show that the implicit method supersedes the explicit one, consistent with the implicit and explicit time discretization of other continuous‐time algorithms.

On the zeros of the partial Hosoya polynomial of graphs

The partial Hosoya polynomial (or briefly the partial H-polynomial) can be used to construct the well-known Hosoya polynomial. The ith coefficient of this polynomial, defined for an arbitrary vertex u of a graph G, is the number of vertices at distance i from u. The aim of this paper is to determine the partial H-polynomial of several well-known graphs and, then, to investigate the location of their zeros. To pursue, we characterize the structure of graphs with the minimum and the maximum modulus of the zeros of partial H-polynomial. Finally, we define another graph polynomial of the partial H-polynomial, see [9]. Also, we determine the unique positive root of this polynomial for particular graphs.

The Orbit-Polynomial: A Novel Measure of Symmetry in Networks

Research on the structural complexity of networks has produced many useful results in graph theory and applied disciplines such as engineering and data analysis. This paper is intended as a further contribution to this area of research. Here we focus on measures designed to compare graphs with respect to symmetry. We do this by means of a novel characteristic of a graph G, namely an “orbit polynomial.”A typical term of this univariate polynomial is of the form czn, where c is the number of orbits of size n of the automorphism group of G. Subtracting the orbit polynomial from 1 results in another polynomial that has a unique positive root, which can serve as a relative measure of the symmetry of a graph. The magnitude of this root is indicative of symmetry and can thus be used to compare graphs with respect to that property. In what follows, we will prove several inequalities on the unique positive roots of orbit polynomials corresponding to different graphs, thus showing differences in symmetry. In addition, we present numerical results relating to several classes of graphs for the purpose of comparing the new symmetry measure with existing ones. Finally, it is applied to a set of isomers of the chemical compound adamantane C10H16. We believe that the measure can be quite useful for tackling applications in chemistry, bioinformatics, and structure-oriented drug design.

The Price of Stability of Weighted Congestion Games

We give exponential lower bounds on the Price of Stability (PoS) of weighted congestion games with polynomial cost functions. In particular, for any positive integer $d$ we construct rather simple games with cost functions of degree at most $d$ which have a PoS of at least $\varOmega(\Phi_d)^{d+1}$, where $\Phi_d\sim d/\ln d$ is the unique positive root of the equation $x^{d+1}=(x+1)^d$. This almost closes the huge gap between $\varTheta(d)$ and $\Phi_d^{d+1}$. Our bound extends also to network congestion games. We further show that the PoS remains exponential even for singleton games. More generally, we provide a lower bound of $\varOmega((1+1/\alpha)^d/d)$ on the PoS of $\alpha$-approximate Nash equilibria for singleton games. All our lower bounds hold for mixed and correlated equilibria as well. On the positive side, we give a general upper bound on the PoS of $\alpha$-approximate Nash equilibria, which is sensitive to the range $W$ of the player weights and the approximation parameter $\alpha$. We do this by explicitly constructing a novel approximate potential function, based on Faulhaber's formula, that generalizes Rosenthal's potential in a continuous, analytic way. From the general theorem, we deduce two interesting corollaries. First, we derive the existence of an approximate pure Nash equilibrium with PoS at most $(d+3)/2$; the equilibrium's approximation parameter ranges from $\varTheta(1)$ to $d+1$ in a smooth way with respect to $W$. Second, we show that for unweighted congestion games, the PoS of $\alpha$-approximate Nash equilibria is at most $(d+1)/\alpha$. Read More: https://epubs.siam.org/doi/10.1137/18M1207880

Towards detecting structural branching and cyclicity in graphs: A polynomial-based approach

Structural properties of graphs and networks have been investigated across scientific disciplines ranging from mathematics to structural chemistry. Structural branching, cyclicity and, more generally, connectedness are well-known examples of such properties. In particular, various graph measures for detecting structural branching and cyclicity have been investigated. These measures are of limited applicability since their interpretation relies heavily on a certain definition of structural branching. In this paper we define a related measure, taking an approach to measurement similar to that of Lovász and Pelikán (On the eigenvalues of trees, Periodica Mathematica Hungarica, Vol. 3 (1–2), 1973, 175–182). We define a complex valued polynomial which also has a unique positive root. Analytical and numerical results demonstrate that this measure can be interpreted as a structural branching and cyclicity measure for graphs. Our results generalize the work of Lovász and Pelikán since the measure we introduce is not restricted to trees.

Exponential growth rates of free and amalgamated products

We prove that there is a gap between \(\sqrt 2 and\left( {1 + \sqrt 5 } \right)/2\) for the exponential growth rate of nontrivial free products. For amalgamated products G = A*CB with ([A: C] − 1)([B: C] − 1) ≥ 2, we show that an exponential growth rate lower than \(\sqrt 2 \) can be achieved. Indeed, there are infinitely many amalgamated products for which the exponential growth rate is equal to ψ ≈ 1.325, where ψ is the unique positive root of the polynomial z3−z−1. One of these groups is \(PGL\left( {2,\mathbb{Z}} \right) \cong \left( {{C_2} \times {C_2}} \right){*_{{C_2}}}{D_6}\). However, under some natural conditions the lower bound can be put up to \(\sqrt 2 \). This answers two questions by Avinoam Mann [The growth of free products, Journal of Algebra 326, no. 1 (2011), 208–217]. We also prove that ψ is a lower bound for the minimal growth rates of a large class of Coxeter groups, including cofinite non-cocompact planar hyperbolic groups, which strengthens a result obtained earlier by William Floyd, who considered only standard Coxeter generators.

Bounds for eigenvalues of matrix polynomials with applications to scalar polynomials

We first generalize to complex matrix polynomials an improvement of an upper bound by Cauchy on the zeros of complex scalar polynomials. The bound requires the unique positive root of a real scalar polynomial of the same degree as the given complex scalar or matrix polynomial. We then create a recursive procedure to represent a matrix polynomial by another matrix polynomial of larger size, but of lower degree. We apply this procedure to scalar polynomials, and then apply the generalized improved Cauchy bound to their matrix polynomial representation, often obtaining a better bound by solving a real scalar polynomial equation of significantly reduced degree.

Asymptotic distribution of the numbers of vertices and arcs of the giant strong component in sparse random digraphs

Two models of a random digraph on n vertices, D(n; Prob(arc) = p) and D(n; number of arcs = m) are studied. In 1990, Karp for D(n;p) and independently T. Luczak for D(n;m = cn) proved that for c > 1, with probability tending to 1, there is an unique strong component of size of order n. Karp showed, in fact, that the giant component has likely size asymptotic to n 2 , where = (c) is the unique positive root of 1 = e c . In this paper we prove that, for both random digraphs, the joint distribution of the number of vertices and number of arcs in the giant strong component is asymptotically Gaussian with the same mean vector n (c), (c) := ( 2 ;c 2 ) and two distinct 2 2 covariance matrices, nB(c) and n B(c) +c( 0 (c)) T ( 0 (c))) . To this end, we introduce and analyze a randomized deletion process which determines the directed (1; 1)-core, the maximal digraph with minimum in-degree and out-degree at least 1. This (1; 1)-core contains all nontrivial strong components. However, we show that the likely numbers of peripheral vertices and arcs in the (1; 1)-core, those outside the largest strong component, are of polylog order, thus dwarfed by anticipated uctuations, on the scale of n 1=2 , of the giant component parameters. By approximating the likely realization of the deletion algorithm with a deterministic trajectory, we obtain our main result via exponential supermartingales and Fourier-based techniques.

The plastic number and its generalized polynomial

The polynomial $X^{3}-X-1$ has a unique positive root known as plastic number, which is denoted by $\rho$ and is approximately equal to $1.32471795$. In this note we study the zeroes of the generalized polynomial $X^{k}-\sum_{j=0}^{k-2}X^{j}$ for $k\geq 3$ and prove that its unique positive root $\lambda_{k}$ tends to the golden ratio $\phi=\frac{1+\sqrt{5}}{2}$ as $k \to \infty$. We also derive bounds on $\lambda_{k}$ in terms of Fibonacci numbers.

Optimal control of natural resources in mining industry

The paper focuses on the optimal control of natural resources in mining industry. The purpose is to propose an optimal extraction series of these resources during the lifetime of the Mine’s maintenance. Following the proposed optimal control model, a sensitivity analysis has been performed that includes the interest rate impact on the optimal solution. This study shows that the increasing of the interest rate stimulates faster extraction of the resources. The discounting factor induces that the resource has to be extracted faster but this effect is counterbalanced by the diminishing returns of the annual cash flow. At higher parameters of “alpha” close to one of the power function about 80% from the whole resource will be extracted during the first 4years of object/mine maintenance. An existence of unique positive root with respect to return of investment has been proposed and proved by two ways: by the “method of chords” and by using specialized software.

Zero-Error Capacity of a Class of Timing Channels

We analyze the problem of zero-error communication through timing channels that can be interpreted as discrete-time queues with bounded waiting times. The channel model includes the following assumptions: 1) Time is slotted, 2) at most $ N $ "particles" are sent in each time slot, 3) every particle is delayed in the channel for a number of slots chosen randomly from the set $ \{0, 1, \ldots, K\} $, and 4) the particles are identical. It is shown that the zero-error capacity of this channel is $ \log r $, where $ r $ is the unique positive real root of the polynomial $ x^{K+1} - x^{K} - N $. Capacity-achieving codes are explicitly constructed, and a linear-time decoding algorithm for these codes devised. In the particular case $ N = 1 $, $ K = 1 $, the capacity is equal to $ \log \phi $, where $ \phi = (1 + \sqrt{5}) / 2 $ is the golden ratio, and the constructed codes give another interpretation of the Fibonacci sequence.

An Efficient Continued Fraction Algorithm for Nonlinear Optimization and Its Computer Implementation

Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics and other sciences.There has been much attention to develop iterative methods for solving nonlinear equations in these years.New algorithms and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace. One of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field.Among wide various of papers have been published in the recent years, there are some progress about multi-step methods.These multi-step methods have been suggested by combining the well-known Newton's method with other methods. In this work, we develop a simple yet practical algorithm for solving nonlinear optimization problems by solving nonlinear equations with a good local convergence.The algorithm uses a continued fraction interpolation that can be easily implemented in software packages for achieving desired convergence orders.For the general $n$-point formula,the order of convergence rate of the presented algorithm is $\tau_n$, the unique positive root of the equation $x^n-x^{n-1}-\cdots-x-1=0$. Computational results ascertain that the developed algorithm is efficient and demonstrate equal or better performance as compared with other well known methods.

On an efficient method for the simultaneous approximation of polynomial multiple roots

An iterative method in parallel mode for the simultaneous determination of multiple roots of algebraic polynomials is stated together with its single-step variant. These methods are more efficient compared to all simultaneous methods based on fixed point relations. To attain very high computational efficiency, a suitable correction resulting from Li-Liao-Cheng?s two-point fourth order method of low computational complexity and Gauss-Seidel?s approach are applied. Considerable increase of the convergence rate is obtained applying only n additional polynomial evaluations per iteration, where n is the number of distinct roots. A special emphasis is given to the convergence analysis and computational efficiency of the proposed methods. The presented convergence analysis shows that the R-order of convergence of the proposed single-step method is at least 2 + ?v; where ?v?2 (4,6) is the unique positive root of the polynomial gv(t) = tn-4n-1 t-22n-1: The convergence order of the corresponding total-step method is six. Computational aspects and some numerical examples are given to demonstrate high computational efficiency and very fast convergence of the proposed methods.