Roots Of Polynomial Research Articles

CONDITIONS OF MONOTONE APPROXIMATION OF RAMSEY CURVES AND THEIR MODIFICATIONS

The algorithm for approximating the experimental data of the Ramsey curve and its modifications has been developed, which provides a monotonic increase of the approximating function in the interval [0;\infty) and an existence of a given number of inflection points. The Ramsey curve belongs to the family of logistic curves that are widely used in modeling of limited increasing processes in various subject fields. The classical Ramsey curve has two parameters and has a left constant asymmetry. It is also known that its three-parameter modification provides the possibility of displacement along the axes of ordinate. The extensive practical use of the Ramsey curve with both two and more parameters for approximating experimental dependences is restrained by the frequent loss by this curve of the logistic shape when approximating without additional restrictions on the relationships between its parameters. The article discusses modifications of the Ramsey curve with three and five parameters. The first and second derivatives of the studied modifications of the Ramsey function have a special structure. They are products of polynomial and exponential functions. This allows using Sturm's theorem on the number of polynomial roots in a given interval to control the shape of the approximating curve. It has been shown that with an increase in the number of parameters for the modified curve, the number of possible combinations of restrictions on the values of the parameters ensuring the preservation of its like shape increases significantly. The solution to the approximation problem in this case consists of solving a sequence of conditional global optimization problems with different constraints and choosing a solution that provides the smallest approximation error. Also, the studies of the accuracy of estimating the parameters of the Ramsey curve in accordance with the accuracy of the experimental data have been carried out. In order to simulate the presence of measurement errors, the values of a normally distributed random variable with a mathematical expectation equal to zero and different values of the standard deviation for different series of computational experiments were added to the values of the deterministic sequence. Computational experiments have shown a significant sensitivity of the values of the Ramsey function parameters to the measurement accuracy of experimental data.

Two-dimensional grid-free compressive beamforming with spherical microphone arrays

Compressive beamforming based on microphone array measurements and compressive sensing theory is an emerging and practical acoustic source identification technology. The grid-free compressive beamforming that treats the target source region as a continuum has attracted tremendous attention due to the advantage of circumventing the basis mismatch defect caused by the gridding. The grid-free methods have been developed for the linear and the planar microphone array measurements to identify sources in a local region. In contrast, for the spherical microphone array measurements suitable for 360° panoramic identification, the current research mainly focuses on grid-based methods. In this paper, we propose a two-dimensional grid-free compressive beamforming method with spherical microphone arrays. First of all, the relationship between the spherical harmonic function vector and the two-dimensional Fourier basis function vector is determined. Then, the mathematical model relating the measured sound pressures to the source distribution is built under a continuous setting. Next, a source sparsity constraint is imposed based on the total variation norm to solve the mathematical model, and its dual problem is deduced. Subsequently, the dual problem is solved by positive semidefinite programming. Finally, the DOAs of sources are estimated through polynomial rooting and utilizing the dual optimal variables, and the source strengths are quantified via least-square fitting. An experimental case is conducted to examine the practicability and superiority of the proposed method to the existing grid-based compressive beamforming method. Plenty of Monte Carlo simulations are carried out to disclose the influence of the source coherence, the angular distance between sources, the noise, the number of snapshots and the number of microphones. This study offers a new approach for the accurate and 360° panoramic identification of acoustic sources.

Powerful method to evaluate the mass gaps of free-particle quantum critical systems

We present a numerical method for the evaluation of the mass gap, and the low-lying energy gaps, of a large family of free-fermionic and free-parafermionic quantum chains. The method is suitable for some generalizations of the quantum Ising and XY models with multispin interactions. We illustrate the method by considering the Ising quantum chains with uniform and random coupling constants. The mass gaps of these quantum chains are obtained from the largest root of a characteristic polynomial. We also show that the Laguerre bound, for the largest root of a polynomial, used as an initial guess for the largest root in the method, gives us estimates for the mass gaps sharing the same leading finite-size behavior as the exact results. This opens an interesting possibility of obtaining precise critical properties very efficiently which we explore by studying the critical point and the paramagnetic Griffiths phase of the quantum Ising chain with random couplings. In this last phase, we obtain the effective dynamical critical exponent as a function of the distance-to-criticality. Finally, we compare the mass gap estimates derived from the Laguerre bound and the strong-disorder renormalization-group method. Both estimates require comparable computational efforts, with the former having the advantage of being more accurate and also being applicable away from infinite-randomness fixed points. We believe this method is a relevant tool for tackling critical quantum chains with and without quenched disorder.

The Expected Number of Roots over The Field of p-adic Numbers

Abstract We study the roots of a random polynomial over the field of $p$-adic numbers. For a random monic polynomial with i.i.d. coefficients in ${\mathbb{Z}}_p$, we obtain an estimate for the expected number of roots of this polynomial. In particular, if the coefficients take the values $\pm 1$ with equal probability, the expected number of $p$-adic roots converges to $(p-1)/(p+1)$ as the degree of the polynomial tends to $\infty $.

Random polynomials: Central limit theorems for the real roots

The number of real roots has been a central subject in the theory of random polynomials and random functions since the fundamental papers of Littlewood, Offord, and Kac in the 1940s. The main task here is to determine the limiting distribution of this random variable. In 1974, Maslova famously proved a central limit theorem (CLT) for the number of real roots of Kac polynomials. It has remained the only limiting theorem available for the number of real roots for more than four decades. In this paper, using a new approach, we derive a general CLT for the number of real roots of a large class of random polynomials with coefficients growing polynomially. Our result both generalizes and strengthens Maslova’s theorem.

Supersingular j-invariants and the class number of ℚ(−p)

Let [Formula: see text] be a prime. Let [Formula: see text] be the discriminant of an imaginary quadratic order. Assume that [Formula: see text] and [Formula: see text]. We compute the number of [Formula: see text]-roots of the class polynomials [Formula: see text]. Suppose [Formula: see text], we prove that two class polynomials [Formula: see text] and [Formula: see text] have a common root in [Formula: see text] if and only if [Formula: see text] is a perfect square. Furthermore, any three class polynomials do not have a common root in [Formula: see text]. As an application, we propose a deterministic algorithm for computing the class number of [Formula: see text].

Generalization of Multiplication M-Sequences over Fp and Its Reciprocal Sequences

Mp-Sequences or M-Sequence over Fp not used so much in current time as binary M-Sequences and it is pending with the difficult to construct there coders and decoders of Mp-Sequences further these reasons there is expensive values to construct them but the progress in the technical methods will be lead to fast using these sequences in different life’s ways, and these sequences give more collection of information and distribution them on the input and output links of the communication channels, building new systems with more complexity, larger period, and security. In current article we will study the construction of the multiplication Mp-Sequence {zn}and its linear equivalent, this sequences are as multiple two sequences, the first sequence{Sn}is an arbitrary Mp-Sequence and the second sequence {ζn} reciprocal sequence of the first sequence {Sn}, length of the sequence {zn}, period, orthogonal and the relations between the coefficients and roots of the characteristic polynomial of f(x) and it’s reciprocal polynomial g(x) and compare these properties with corresponding properties in M-Sequences.

Classes of Dynamic Systems with Various Combinations of Multipliers in Their Reciprocal Polynomial Right Parts

A family of differential dynamic systems is considered on a real plane of their phase variables x, y. The main common feature of systems under consideration is: every particular system includes equations with polynomial right parts of the third order in one equation and of the second order in another one. These polynomials are mutually reciprocal, i.e., their decompositions into forms of lower orders do not contain common multipliers. The whole family of dynamic systems has been split into subfamilies according to the numbers of different reciprocal multipliers in the decompositions and depending on an order of sequence of different roots of polynomials. Every subfamily has been studied in a Poincare circle using Poincare mappings. A plan of the investigation for each selected subfamily of dynamic systems includes the following steps. We determine a list of singular points of systems of the fixed subfamily in a Poincare circle. For every singular point in the list, we use the notions of a saddle (S) and node (N) bundles of adjacent to this point semi trajectories, of a separatrix of the singular point, and of a topo dynamical type of the singular point (its TD – type). Further we split the family under consideration to subfamilies of different hierarchical levels with proper numbers. For every chosen subfamily we reveal topo dynamical types of singular points and separatrices of them. We investigate the behavior of separatrices for all singular points of systems belonging to the chosen subfamily. Very important are: a question of a uniqueness of a continuation of every given separatrix from a small neighborhood of a singular point to all the lengths of this separatrix, as well as a question of a mutual arrangement of all separatrices in a Poincare circle Ω. We answer these questions for all subfamilies of studied systems. The presented work is devoted to the original study. The main task of the work is to depict and describe all different in the topological meaning phase portraits in a Poincare circle, possible for the dynamical differential systems belonging to a broad family under consideration, and to its numerical subfamilies of different hierarchical levels. This is a theoretical work, but due to special research methods it may be useful for applied studies of dynamic systems with polynomial right parts. Author hopes that this work may be interesting and useful for researchers as well as for students and postgraduates. As a result, we describe and depict phase portraits of dynamic systems of a taken family and outline the criteria of every portrait appearance.

Direction-of-Arrival Estimation of Electromagnetic Wave Impinging on Spherical Antenna Array in the Presence of Mutual Coupling Using a Multiple Signal Classification Method

A spherical antenna array (SAA) is the configuration of choice in obtaining an antenna array with isotropic characteristics. An SAA has the capacity to receive an electromagnetic wave (EM) with equal intensity irrespective of the direction-of-arrival (DoA) and polarization. Therefore, the DoA estimation of electromagnetic (EM) waves impinging on an SAA with unknown mutual coupling needs to be considered. In the spherical domain, the traditional multiple signal classification algorithm (SH-MUSIC) is faced with a computational complexity problem. This paper presents a one-dimensional MUSIC method (1D-MUSIC) for the estimation of the azimuth and elevation angles. An intermediate mapping matrix that exists between Fourier series and the spherical harmonic function is designed, and the Fourier series Vandermonde structure is used for the realization of the polynomial rooting technique. This mapping matrix can be computed prior to the DoA estimation, and it is only a function of the array configuration. Based on the mapping matrix, the 2-D angle search is transformed into two 1-D angle findings. Employing the features of the Fourier series, two root polynomials are designed for the estimation of the elevation and azimuth angles, spontaneously. The developed method avoids the 2-D spectral search, and angles are paired in automation. Both numerical simulation results, and results from experimental measured data (i.e., with mutual coupling effect incorporated), show the validity, potency, and potential practical application of the developed algorithm.

Free Fermions Behind the Disguise

An invaluable method for probing the physics of a quantum many-body spin system is a mapping to noninteracting effective fermions. We find such mappings using only the frustration graph G of a Hamiltonian H, i.e., the network of anticommutation relations between the Pauli terms in H in a given basis. Specifically, when G is (even-hole, claw)-free, we construct an explicit free-fermion solution for H using only this structure of G, even when no Jordan–Wigner transformation exists. The solution method is generic in that it applies for any values of the couplings. This mapping generalizes both the classic Lieb–Schultz–Mattis solution of the XY model and an exact solution of a spin chain recently given by Fendley, dubbed “free fermions in disguise.” Like Fendley’s original example, the free-fermion operators that solve the model are generally highly nonlinear and nonlocal, but can nonetheless be found explicitly using a transfer operator defined in terms of the independent sets of G. The associated single-particle energies are calculated using the roots of the independence polynomial of G, which are guaranteed to be real by a result of Chudnovsky and Seymour. Furthermore, recognizing (even-hole, claw)-free graphs can be done in polynomial time, so recognizing when a spin model is solvable in this way is efficient. In a crucial step to proving our result, we additionally prove that there exists a hierarchy of commuting conserved charges for models whose frustration graphs are claw-free only, and hence these models are integrable. Finally, we give several example families of solvable and integrable models for which no Jordan–Wigner solution exists, and we give a detailed analysis of such a spin chain having 4-body couplings using this method.

On the distribution of the roots of polynomial $z^k-z^{k-1}-\dots - z-1$

We consider the polynomial $f_k(z) = z^k-z^{k-1}-\cdots -z-1$ for $k\ge 2$ which arises as the characteristic polynomial of the $k$-generalized Fibonacci sequence. In this short paper, we give estimates for the absolute values of the roots of $f_k(z)$ which lie inside the unit disk.

Some Applications of Wagner's Weighted Subgraph Counting Polynomial

We use Wagner's weighted subgraph counting polynomial to prove that the partition function of the anti-ferromagnetic Ising model on line graphs is real rooted and to prove that roots of the edge cover polynomial have absolute value at most $4$. We more generally show that roots of the edge cover polynomial of a $k$-uniform hypergraph have absolute value at most $2^k$, and discuss applications of this to the roots of domination polynomials of graphs. We moreover discuss how our results relate to efficient algorithms for approximately computing evaluations of these polynomials.

Discovering Geometric Inequalities: The Concourse of GeoGebra Discovery, Dynamic Coloring and Maple Tools

Recently developed GeoGebra tools for the automated deduction and discovery of geometric statements combine in a unique way computational (real and complex) algebraic geometry algorithms and graphic features for the introduction and visualization of geometric statements. In our paper we will explore the capabilities and limitations of these new tools, through the case study of a classic geometric inequality, showing how to overcome, by means of a double approach, the difficulties that might arise attempting to ‘discover’ it automatically. On the one hand, through the introduction of the dynamic color scanning method, which allows to visualize on GeoGebra the set of real solutions of a given equation and to shed light on its geometry. On the other hand, via a symbolic computation approach which currently requires the (tricky) use of a variety of real geometry concepts (determining the real roots of a bivariate polynomial p(x,y) by reducing it to a univariate case through discriminants and Sturm sequences, etc.), which leads to a complete resolution of the initial problem. As the algorithmic basis for both instruments (scanning, real solving) are already internally available in GeoGebra (e.g., via the Tarski package), we conclude proposing the development and merging of such features in the future progress of GeoGebra automated reasoning tools.

An evaluation of weighted polynomial interpolation with certain conditions on the roots of Hermite polynomial

The purpose of this paper is to construct a polynomialRnof degree atmost 3n−2 satisfying weighted (0,2; 0) interpolation under certain condi-tions at the zeroes ofHnandH′n, whereHnstands for Hermite polynomial.Furthermore, we prove a convergence theorem forRn

Roots of Second Order Polynomials with Real Coefficients in Elliptic Scator Algebra

The roots of second order polynomials with real coefficients are obtained in the S^{1+2} scator set. Explicit formulae are computed in terms of the polynomial coefficients. Although the scator product does not distribute over addition, the lack of distributivity is surmountable in order to find the zeros of the polynomial. The structure of the solutions and their distribution in 1+2 dimensional scator space are illustrated and discussed. There exist six, two, or eight solutions, depending on the value of polynomial coefficients. Four of these roots only exist in the hypercomplex S^{1+2}\S^{1+1} set.

DOA estimation in beamspace for monostatic MIMO radar utilizing real polynomial rooting

A novel angle estimation method in beamspace for monostatic multiple-input multiple-output (MIMO) radar based on a real polynomial rooting technique is presented. In order to eliminate the redundancy of the received data, the reduced-dimensional matrix is used to reduce the dimension of the received data. To further reduce computational complexity, the reduced-dimensional data is transformed into the beamspace. In order to avoid computationally expensive spectrum search, we will introduce conformal transformation to determine the extreme value of the MUSIC spectrum by utilizing the real polynomial rooting technique. Finally, the DOA of the target can be obtained from these extreme values. Numerical results demonstrate the effectiveness of the proposed algorithm.

On μ-symmetric polynomials

We study functions of the roots of an integer polynomial [Formula: see text] with [Formula: see text] distinct roots [Formula: see text] of multiplicity [Formula: see text], [Formula: see text]. Traditionally, root functions are studied via the theory of symmetric polynomials; we generalize this theory to [Formula: see text]-symmetric polynomials. We initiate the study of the vector space of [Formula: see text]-symmetric polynomials of a given degree [Formula: see text] via the concepts of [Formula: see text]-gist and [Formula: see text]-ideal. In particular, we are interested in the root [Formula: see text]. The D-plus discriminant of [Formula: see text] is [Formula: see text]. This quantity appears in the complexity analysis of the root clustering algorithm of Becker et al. (ISSAC 2016). We conjecture that [Formula: see text] is [Formula: see text]-symmetric, which implies [Formula: see text] is rational. To explore this conjecture experimentally, we introduce algorithms for checking if a given root function is [Formula: see text]-symmetric. We design three such algorithms: one based on Gröbner bases, another based on canonical bases and reduction, and the third based on solving linear equations. Each of these algorithms has variants that depend on the choice of a basis for the [Formula: see text]-symmetric functions. We implement these algorithms (and their variants) in Maple and experiments show that the latter two algorithms are significantly faster than the first.

On the Degree of the GCD of Random Polynomials over a Finite Field

In this paper, we focus on the degree of the greatest common divisor ( gcd ) of random polynomials over F q . Here, F q is the finite field with q elements. Firstly, we compute the probability distribution of the degree of the gcd of random and monic polynomials with fixed degree over F q . Then, we consider the waiting time of the sequence of the degree of gcd functions. We compute its probability distribution, expectation, and variance. Finally, by considering the degree of a certain type gcd , we investigate the probability distribution of the number of rational (i.e., in F q ) roots (counted with multiplicity) of random and monic polynomials with fixed degree over F q .

Algorithms and Programs for Calculating the Roots of Polynomial of One or Two Variables

Algorithms and Programs for Calculating the Roots of Polynomial of One or Two Variables

Lengths of roots of polynomials in a Hahn field

Lengths of roots of polynomials in a Hahn field