Real Roots Research Articles

On connection between zeros and d-orthogonality

Connection between the interlacing of the zeros and the orthogonality of a given sequence of polynomials is done by K. Driver. In this paper, we attempt to extend this result to some particular cases of d-orthogonal polynomials. In fact, first, we characterize the 2-orthogonality of a given sequence , with the existence of a certain ratio expressed by means of the zeros of . Then, for the -fold symmetric polynomials, , such that has distinct positive real zeros, , we study the connection between the interlacing of these zeros, the d-orthogonality and the positivity of the ratio . Finally, we give necessary and sufficient conditions on the zeros of a given sequence , that will assure that this sequence satisfies a particular -order recurrence relation. Many examples to illustrate the obtained results are given.

Parameterization of minimal order strongly stabilizing controllers for two-link underactuated planar robot with single sensor

This paper investigates the minimal order of strongly stabilizing controllers for a two-link planar robot moving in a vertical plane with a single actuator and sensor, focusing on stabilizing the robot around its upright equilibrium point where both links are upright. By designing an output as the horizontal displacement of a point along the second link’s straight line, existing literature demonstrates that the robot’s strong stabilization problem is equivalent to that of a fourth-order linear plant, featuring two pairs of symmetric real poles and adjustable zeros. In contrast to an existing second-order strongly stabilizing controller tailored to a pair of symmetric real zeros around the origin within a given range, this paper achieves four main contributions. First, this paper obtains a parameterization of second-order strongly stabilizing controllers allowing the positive real zero in an expanded range. Second, this paper considers the plant with a pair of symmetric real zeros around the origin, a double origin zero, and a pair of conjugate imaginary zeros in a unified manner, and presents a more general parameterization of second-order strongly stabilizing controllers accommodating these three types of zeros in a more expanded range. Third, this paper provides a necessary and sufficient condition on these zeros (equivalently, all ranges of the observation point) for the existence of a second-order strongly stabilizing controller. Fourth, this paper demonstrates that a first-order stabilizing controller does not exist for the plant, regardless of its types of zeros, confirming that the minimal order of the strongly stabilizing controllers is two. This paper obtains all feasible regions for the parameters in each parameterization by explicitly expressing the range of each parameter using inequalities in a cascade structure, which makes the inequalities easy to check and verify. The results are applicable to both the Acrobot and Pendubot. The paper validates the theoretical findings through numerical investigations and simulations.

Amalgamation of real zero polynomials

With this article, we hope to launch the investigation of what we call the Real Zero Amalgamation Problem. Whenever a polynomial arises from another polynomial by substituting zero for some of its variables, we call the second polynomial an extension of the first one. The Real Zero Amalgamation Problem asks when two (multivariate real) polynomials have a common extension (called amalgam) that is a real zero polynomial. We show that the obvious necessary conditions are not sufficient. Our counterexample is derived in several steps from a counterexample to amalgamation of matroids by Poljak and Turzík. On the positive side, we show that even a degree-preserving amalgamation is possible in three very special cases with three completely different techniques. Finally, we conjecture that amalgamation is always possible in the case of two shared variables. The analogue in matroid theory is true by another work of Poljak and Turzík. This would imply a very weak form of the Generalized Lax Conjecture.

Qualitative properties, bifurcations and chaos of a discrete predator–prey system with weak Allee effect on the predator

In this work, the qualitative properties of the fixed points in the non-hyperbolic and degenerate cases, codimension-one bifurcations and Marotto’s chaos of a discrete predator–prey system with weak Allee effect on the predator are investigated. Utilizing the center manifold theorem and the reduction principle, the qualitative property of each fixed point in the non-hyperbolic case is explored. Based on the approximate flow theory, the qualitative property of the boundary fixed point in the degenerate case is investigated. Making use of the center manifold theorem and the bifurcation theory, all the potential codimension-one bifurcation types of the co-existence fixed point are explored, including flip bifurcation and Neimark–Sacker bifurcation. For each type of these bifurcations, not only the concise non-degenerate condition expressed by the system parameters is given, but also the analytical expression of the system orbit caused by the bifurcation is derived. Based on the non-negativity of the bifurcation critical values, the non-resonant and non-degenerate conditions for the occurrence of these bifurcations, the system parameter plane is divided into several regions using tools such as the polynomial complete discriminant system theory, real root separation theory, dichotomy method, implicit function theorem and Cardan’s formula. In each region, the direction and stability of these bifurcations are studied, which helps to theoretically explain the impact of Allee effect on the system. By proving that the co-existence fixed point is a snap-back repeller under appropriate conditions, it is obtained that the system is chaotic. Numerical simulations are completely consistent with all the theoretical analyses. The research results show that the impact of Allee effect on the population system depends not only on the assumption and construction of the system, but also on the system parameter and initial value of the system.

Quickly Computing Isotopy Type for Exponential Sums over Circuits (Extended Abstract)

Fewnomial Theory [Kho91] has established bounds on the number of connected components (a.k.a. pieces ) of a broad class of real analytic sets as a function of a particular kind of input complexity, e.g., the number of distinct exponent vectors among a generating set for the underlying ideal. Here, we pursue the algorithmic side: We show how to efficiently compute the exact isotopy type of certain (possibly singular) real zero sets, instead of just estimating their number of pieces. While we focus on the circuit case, our results form the foundation for an approach to the general case that we will pursue later.

Second order linear differential equations with a basis of solutions having only real zeros

Let A be a transcendental entire function of finite order. We show that, if the differential equation w″ + Aw = 0 has two linearly independent solutions with only real zeros, then the order of A must be an odd integer or one half of an odd integer. Moreover, A has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.

Limit theorems for random Dirichlet series

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series D(α;z)=∑n≥2(logn)α(ηn+iθn)/nz, properly scaled and normalized, where (ηn,θn)n∈N is a sequence of independent copies of a centered R2-valued random vector (η,θ) with a finite second moment and α>−1/2 is a fixed real parameter. As a consequence, we show that the point processes of complex and real zeros of D(α;z) converge vaguely, thereby obtaining a universality result. In the real case, that is, when P{θ=0}=1, we also prove a law of the iterated logarithm for D(α;z), properly normalized, as z→(1/2)+.

Application of Müntz Orthogonal Functions on the Solution of the Fractional Bagley–Torvik Equation Using Collocation Method with Error Stimate

This paper uses Müntz orthogonal functions to numerically solve the fractional Bagley–Torvik equation with initial and boundary conditions. Müntz orthogonal functions are defined on the interval 0 , 1 and have simple and distinct real roots on this interval. For the function f ∈ L 2 0 , 1 , we obtain the best unique approximation using Müntz orthogonal functions. We obtain the Riemann–Liouville fractional integral operator for Müntz orthogonal functions so that we can reduce the complexity of calculations and increase the speed of solving the problem, which can be seen in the process of running the Maple program. To solve the fractional Bagley–Torvik equation with initial and boundary conditions, we use Müntz orthogonal functions and consider simple and distinct real roots of Müntz orthogonal functions as collocation points. By using the Riemann–Liouville fractional integral operator that we define for the Müntz orthogonal functions, the process of numerically solving the fractional Bagley–Torvik equation that is solved using Müntz orthogonal functions is reduced, and finally, we reach a system of algebraic equations. By solving algebraic equations and obtaining the vector of unknowns, the fractional Bagley–Torvik equation is solved using Müntz orthogonal functions, and the error value of the method can be calculated. The low error value of this numerical solution method shows the high accuracy of this method. With the help of the Müntz functions, we obtain the error bound for the approximation of the function. We have obtained the error bounds for the numerical method using which we solved the fractional Bagley–Torvik equation with initial and boundary conditions. Finally, we have given a numerical example to show the accuracy of the solution of the method presented in this paper. The results of solving this example using Müntz orthogonal functions and comparing the results with other methods that have been used the solve this example show the higher accuracy of the method proposed in this paper.

On the cubic equation with its Siebeck–Marden–Northshield triangle and the quartic equation with its tetrahedron

The real roots of the cubic and quartic polynomials are studied geometrically with the help of their respective Siebeck–Marden–Northshield equilateral triangle and regular tetrahedron. The Viète trigonometric formulæfor the roots of the cubic are established through the rotation of the triangle by variation of the free term of the cubic. A very detailed complete root classification for the quartic x4+ax3+bx2+cx+d is proposed for which the conditions are imposed on the individual coefficients a, b, c, and d. The maximum and minimum lengths of the interval containing the four real roots of the quartic are determined in terms of a and b. The upper and lower root bounds for a quartic with four real roots are also found: no root can lie farther than (3/4)3a2−8b from −a/4. The real roots of the quartic are localized by finding intervals containing at most two roots. The end-points of these intervals depend on a and b and are roots of quadratic equations — which makes this localization helpful for quartic equations with complicated parametric coefficients.

Maxwell–Bloch equations without spectral broadening: the long-time asymptotics of an input pulse in a long two-level laser amplifier

We study the problem of propagation of an input electromagnetic pulse through a long two-level laser amplifier under trivial initial conditions. In this paper, we consider an unstable model described by the Maxwell–Bloch equations without spectral broadening. Previously, this model was studied by Manakov in (1982 Zh. Eksp. Teor. Fiz. 83 68–75) and together with Novokshenov in (1986 Teor. Mat. Fiz. 69 40–54). We consider this model in a more natural formulation as an initial-boundary (mixed) problem using a modern version of the inverse scattering transform method in the form of a suitable Riemann–Hilbert (RH) problem. The RH problem arises as a result of applying the Fokas–Its method of simultaneous analysis of the corresponding spectral problems for the Ablowitz–Kaup–Newell–Segur equations. This approach makes it possible to obtain rigorous asymptotic results at large times, which differ significantly from the previous ones. Differences take place both near the light cone and in the tail region, where a new type of solitons is found against an oscillating background. These solitons are physically relevant, their velocities are smaller than the speed of light. The number of such solitons can be either finite or infinite (in the latter case, the set of zeros has a condensation point at infinity). Such solitons cannot be reflectionless, they are generated by zeros of the reflection coefficient of the input pulse (and not by poles of the transmission coefficient). Thus our approach shows the presence of a new phenomenon in soliton theory, namely, the boundary condition (input pulse) of a mixed problem under trivial initial conditions can generate solitons due to the zeros of the reflection coefficient, while the poles of the transmission coefficient do not contribute to the asymptotics of the solution.

Exergy-Optimum coupling of radiant panels with heat pumps for minimum CO2 emission responsibility

This paper introduces an exergy-based optimization model for radiant heating or cooling panels coupled with heat pumps for minimum carbon dioxide emissions responsibility. The general idea that there exists an optimum tube spacing in radiant panels for minimum carbon dioxide emission responsibility in terms of simple life-cycle analysis and the first law of thermodynamics is challenged. Using the Rational Exergy Management Model, it is shown that a real optimum is unavailable according to the second law with or without using heat pumps or boilers for temperature peaking or chillers for cooling. It is hypothesized that the use of temperature-adjusting heat pumps is not an exergy-optimum choice. Through twelve sets of optimization constraints, identified in this paper, it is shown that potential optimality conditions concerning the increase of emission embodiments of the radiant panel material and labor with respect to narrover tube spacing versus an increase in the coefficient of performance of the heat pump, thus a decrease in operational emissions, lies outside the feasible design range. This paper also reminds that heat pumps destroy part of the high unit exergy of the electrical power demand by generating only low-exergy thermal energy. It is further shown that delegating solar photo-voltaic panels for thermal and electrical power supply as well as temperature peaking provides optimal solutions both for radiant panel heating and cooling, enabling wide use of low-enthalpy energy sources. Three case studies are presented. Case 1: Hydronic floor heating with temperature-peaking heat pump. This case has the highest emissions responsibility among other cases. It is + 0.095 kg CO2/kW-hexergy of heat delivered per unit panel area. Case 2: Oversized radiant floor heating with closer tube spacing without a heat pump. This case is responsible for + 0.035 kg CO2/kW-hexergy of heat delivered per unit panel area. 3-a: Heat-piped solar photo-voltaic-thermal panels with floor heating with heat pipes. This case has a negative emissions responsibility of −0.010 kg CO2/kW-hexergy of heat delivered. 3-b: This case is the radiant ceiling cooling option of Case 3-a. It is responsible for + 0.0010 kg CO2/kW-hexergy of sensible cooling. Results show that negative-carbon solar heating with radiant panels may be achieved with real roots available for optimum tube spacing. Optimal tube spacings for ceiling cooling panels also exist.

On the Zeros of Exponential Polynomials

We consider the problem of deciding the existence of real roots of real-valued exponential polynomials with algebraic coefficients. Such functions arise as solutions of linear differential equations with real algebraic coefficients. We focus on two problems: theZero Problem, which asks whether an exponential polynomial has a real root, and theInfinite Zeros Problem, which asks whether such a function has infinitely many real roots. Our main result is that for differential equations of order at most 8 the Zero Problem is decidable, subject to Schanuel’s Conjecture, while the Infinite Zeros Problem is decidable unconditionally. We show moreover that a decision procedure for the Infinite Zeros Problem at order 9 would yield an algorithm for computing the Lagrange constant of any given real algebraic number to arbitrary precision, indicating that it will be very difficult to extend our decidability results to higher orders.

A New Method to Calibrate Cardinal Temperatures for Eucalyptus Plantation

Developing a good understanding of the interactions between forest plantation growth and climate is essential for predicting the impact of climate change on terrestrial ecosystems and for assessing the adaptation and vulnerability of tree species. One such interaction, the response in growth rate of a forest stand to changes in temperature, may be described mathematically. Some models that run on monthly time steps assume a yearly optimum, minimum, and maximum temperature for simplicity, which may not represent well to actual forest growth. Here, we developed a finer-resolution methodology that encompasses monthly growth rates and temperature limits to calibrate the parameters for an envelope curve in Eucalyptus plantations in South America. Several polynomial curves were tested to determine temperature patterns, and their yearly tree growth patterns demonstrated that responses to temperature differed by as much as 10 °C among seasons. The best curve was a second-degree polynomial curve, whose extreme values indicated the optimum temperature and whose real roots limited the minimum and maximum temperatures for growth. This polynomial was fitted every month to describe yearly changes in optimum, maximum, and minimum temperatures. When fitted to annual data, it determined 7 °C, 19 °C, and 31 °C as the minimum, optimum, and maximum temperatures for tree growth, respectively. The monthly model predictions indicated that the minimum, optimum, and maximum temperatures lay between 8 °C and 16 °C, 18 °C and 22 °C, and 27 °C and 30 °C, respectively. These monthly temperature ranges can improve the estimation of productivity in process-based models. Our results contribute to the understanding of tree growth dynamics and its relationship to changes in temperature. Accurate ranges of temperature can be used to improve productivity predictions in new expanding planting regions with no previous information or to suggest a regionalization for potential species.

On the zeros of certain polynomials and entire functions

Entire functions whose coefficients are polynomials having real and negative roots are built from Touchard polynomials. The particular set of polynomials is shown to have purely complex roots, where we show a connection of these polynomials with certain approximations of the Riemann’s zeta function. Also a certain class of Fourier transforms is shown to have only real roots.

Rates of convergence for the number of zeros of random trigonometric polynomials

In this paper, we quantify the rate of convergence between the distribution of number of zeros of random trigonometric polynomials (RTP) with i.i.d. centered random coefficients and the number of zeros of a stationary centered Gaussian process G, whose covariance function is given by the sinc function. First, we find the convergence of the RTP towards G in the Wasserstein-1 distance, which in turn is a consequence of Donsker Theorem. Then, we use this result to derive the rate of convergence between their respective number of zeros. Since the number of real zeros of the RTP is not a continuous function, we use the Kac-Rice formula to express it as the limit of an integral and, in this way, we approximate it by locally Lipschitz continuous functions.

L-Index in Joint Variables: Sum and Composition of an Entire Function with a Function With a Vanished Gradient

The composition H(z)=f(Φ(z)) is studied, where f is an entire function of a single complex variable and Φ is an entire function of n complex variables with a vanished gradient. Conditions are presented by the function Φ providing boundedness of the L-index in joint variables for the function H, if the function f has bounded l-index for some positive continuous function l and L(z)=l(Φ(z))(max{1,|Φz1′(z)|},…,max{1,|Φzn′(z)|}),z∈Cn. Such a constrained function L allows us to consider a function Φ with a nonempty zero set. The obtained results complement earlier published results with Φ(z)≠0. Also, we study a more general composition H(w)=G(Φ(w)), where G:Cn→C is an entire function of the bounded L-index in joint variables, Φ:Cm→Cn is a vector-valued entire function, and L:Cn→R+n is a continuous function. If the L-index of the function G equals zero, then we construct a function L˜:Cm→R+m such that the function H has bounded L˜-index in the joint variables z1,…,zn. The other group of our results concern a sum of entire functions in several variables. As a general case, a sum of functions with bounded index is not of bounded index. The same is also valid for the multidimensional case. We found simple conditions proving that f1(z1)+f2(z2) belongs to the class of functions having bounded index in joint variables z1,z2. We formulate some open problems based on the deduced results and on the usage of fractional differentiation operators in the theory of functions with bounded index.

Newton’s method in practice, II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees

We present an algorithm, based on Newton’s method, for finding all roots of univariate complex polynomials so that the observed complexity is linear in the degree, up to logarithmic factors. Unlike the usual Newton method, which finds at most one root at a time, it is global in the sense that it attempts to find all roots of polynomials simultaneously.We demonstrate the feasibility of this algorithm by employing it to find all roots of several families of polynomials of degrees up to more than one billion (109). In all cases, the observed (empirical) complexity for finding all roots of a polynomial of degree d – measured either as the number of Newton iterations or computing time – was between O(dlnd) and O(dln3d), with small constants.

Isolating all the real roots of a mixed trigonometric-polynomial

Mixed trigonometric-polynomials (MTPs) are functions of the form f(x,sin⁡x,cos⁡x) where f is a trivariate polynomial with rational coefficients, and the argument x ranges over the reals. In this paper, an algorithm “isolating” all the real roots of an MTP is provided and implemented. It automatically divides the real roots into two parts: one consists of finitely many roots in an interval [μ−,μ+] while the other consists of countably many roots in R﹨[μ−,μ+]. For the roots in [μ−,μ+], the algorithm returns isolating intervals and corresponding multiplicities while for those greater than μ+, it returns finitely many mutually disjoint small intervals Ii⊂[−π,π], integers ci>0 and multisets of root multiplicity {mj,i}j=1ci such that any root >μ+ is in the set (∪i∪k∈N(Ii+2kπ)) and any interval Ii+2kπ⊂(μ+,∞) contains exactly ci distinct roots with multiplicities m1,i,...,mci,i, respectively. The efficiency of the algorithm is shown by experiments. The method used to isolate the roots in [μ−,μ+] is applicable to any other bounded interval as well. The algorithm takes advantages of the weak Fourier sequence technique and deals with the intervals period-by-period without scaling the coordinate so to keep the length of the sequence short. The new approaches can easily be modified to decide whether there is any root, or whether there are infinitely many roots in unbounded intervals of the form (−∞,a) or (a,∞) with a∈Q.

An Efficient Hardware Implementation for Complex Square Root Calculation Using a PWL Method

In this paper, we propose a methodology for computing the square root of a complex number based on a piecewise linear (PWL) approximation method. The proposed method relies on a software-based segmentor that automatically divides the three real square root functions used in complex square root computation into the fewest segments with a predefined fractional bit width and maximum absolute error (MAE). The coefficients, including the start point, end point, slope and y-intercept of each segment, are stored for use in the implementation of the hardware design. The proposed fully pipelined circuit is coded in the Verilog hardware description language (HDL). The results of synthesis in TSMC (Taiwan Semiconductor Manufacturing Company) 65-nm CMOS technology show that our design achieves savings of 64.21% in area, 16.67% in delay and 65.08% in power compared to the existing methods. Moreover, implementation results on an FPGA (Field-Programmable Gate Array) platform (XC7Z020-CLG400) show that the proposed design reduces the number of LUTs by 29.38%, delay by 28.57% and power consumption by 53.47%.

The Constant K and the Gaussian temporal evolution for COVID-19 & new wave of COVID -X

This paper is an invitation from the South Florida Journal of Development[1] for the republication of an original version published in reference (A - Balloni & Winter, 2020). This paper presents a slightly updated version of the original publication and includes "Recommendations2" concerns about a possible new wave of COVID-X in the section. This republication has been allowed by the IJONEST Journal and iLSET conference (A, B, C: Balloni, A. J. & Winter, R. (2020). The paper presents the Gaussian temporal evolution of Corona Virus, the temporal average constant Ktemporal average constant = Kt. The Kt and its standard deviation come from the 52 experimental Gaussian distribution analysis -histogram-. We have analyzed all histograms from 185 countries presented in the reference (Johns Hopkins, 2020), and we found 52 countries have a definite trend toward an experimental Gaussian profile. As a result, we found Kt = K52 countries = (35 ± 5) days - average & standard deviation-. Using an experimental Gaussian got in reference (Johns Hopkins, 2020), we also calculate the temporal evolution of the World, the constant Kworld. We found Kworld = Kw = (47 ± ½) days. Finally, up to April 20/2020, only 52 (C: Balloni, A. J. & Winter, R. (2020)) of 185 countries present trends toward an experimental Gaussian profile (Johns Hopkins, 2020). The main conclusion from this short communication is that the standard deviation found -Kt = (35 ± 5) days-is very low, which is very good. Therefore, we may conclude that the maximum spread of the Corona contamination should occur in "a maximum" of up to 40 days from the first registered contamination and, in the worst scenario, up to 30 days. Regarding the Brazil peak of contamination, on March 10/2020, we carried out A PREVISION, and in that time, we affirmed by reasoning that the peak in Brazil would be around 10-15 April/2020. We confirm this prevision, as presented in the ILSET/2020 Congress (C: Balloni, A. J. & Winter, R. (2020)). For our next prevision, the decrease from contamination must trend to zero within 30-40 days after the peak contamination, and we also confirm this prevision as presented in the ILSET/2020 Congress (C: Balloni, A. J. & Winter, R. (2020)). These are the most critical situations because the real zero takes a while to get null, and the contamination starts all over if no personal safety, such as social reclusion & mask protection use, are adopted. Finally, we have observed that the Kw has a shifting in the function of the time, and our finding explains this. The last section, "Recommendations, item 2," presents concerns about a possible new wave of COVID-X.