Hyperbolic Polynomials Research Articles

Level crossings and turning points of random hyperbolic polynomials

In this paper, we show that the asymptotic estimate for the expected number of K‐level crossings of a random hyperbolic polynomial a1sinhx + a2sinh2x + ⋯+ansinhnx, where aj(j = 1, 2, …, n) are independent normally distributed random variables with mean zero and variance one, is (1/π)logn. This result is true for all K independent of x, provided . It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomial a1coshx + a2cosh2x + ⋯+ancoshnx, with aj(j = 1, 2, …, n) as before, is also (1/π)logn.

Combinatorics & Topology of Stratifications of The Space of Monic Polynomials With Real Coefficients

We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex calculating (co)homology of its one-point compactification and describe the homotopy type by order complexes of a class of posets of compositions. In the second part, we determine the homotopy type of the one-point compactification of the space of monic polynomials of fixed degree which have only real roots (i.e., hyperbolic polynomials) and at least one root is of multiplicity k. More generally, we describe the homotopy type of the one-point compactification of strata in the boundary of the set of hyperbolic polynomials, that are defined via certain restrictions on root multiplicities, by order complexes of posets of compositions. In general, the methods are combinatorial and the topological problems are mostly reduced to the study of partially ordered sets.

Recursively enumerable subsets of Rq in two computing models Blum-Shub-Smale machine and Turing machine

In this paper we compare recursively enumerable subsets of R q in two computing models over real numbers: the Blum-Shub-Smale machine and the oracle Turing machine. We prove that any Turing RE open subset of R q is a BSS RE set, while a Turing RE closed set may not be a BSS RE set. As an application we show that the Julia set of any computable hyperbolic polynomial is decidable in the Turing computing model.

Analog of the frobenius method for solving homogeneous integral equations of the quasi-potential approach in a relativistic configurational representation

A method is suggested for constructing a solution of the homogeneous integral equation of a three-dimensional, covariant, simultaneous theory of bound states formulated in a relativistic configurational representation. The method is based on representing the unknown solutions in the form of a combination of functions that includes hyperbolic functions and polynomials (degenerate power series). Using this method and the REDUCE system of analytical calculations, the exact conditions determining the mass spectrum and the exact wave functions for a two-particle relativistic system with a quasi-potential [tanh(πr/2)]/r are found.

Hyperbolic Polynomials and Interior Point Methods for Convex Programming

Hyperbolic polynomials have their origins in partial differential equations. We show in this paper that they have applications in interior point methods for convex programming. Each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone. We give an explicit representation of this cone in terms of polynomial inequalities. The function F(x) = −log p(x) is a logarithmically homogeneous self-concordant barrier function for the hyperbolicity cone with barrier parameter equal to the degree of p. The function F(x) possesses striking additional properties that are useful in designing long-step interior point methods. For example, we show that the long-step primal potential reduction methods of Nesterov and Todd and the surface-following methods of Nesterov and Nemirovskii extend to hyperbolic barrier functions. We also show that there exists a hyperbolic barrier function on every homogeneous cone.

An algorithm for solving the eigenvalue problem for a class of operator pencils

We study the generalized eigenvalue problem for a uniformly hyperbolic or totally hyperbolic polynomial operator pencil of arbitrary order. In the context of the variational approach to the problem of determining the eigenvalues of such pencils we propose and justify a recursive algorithm for finding an eigenpair. To accelerate the convergence we propose using the Rayleigh-Ritz procedure. We give data from a numerical implementation of the algorithm.

Definitizable Operators and Quasihyperbolic Operator Polynomials

The main concern of this paper is bounded operators A on a Hilbert space (with inner product (.,.)) which are selfadjoint in an indefinite scalar product (say [ x, y] = ( Gx, y)) and have entirely real spectrum. In addition, all points of spectrum are required to have "determinate type"; a notion refining earlier ideas of Krein, Langer, et al., which implies a strong stability property under perturbations. The central result states that the spectrum is of this type if and only if the operator in question is uniformly definitizable (i.e. Gp( A) ⪢ for some polynomial p). As a first application, characterizations of compact uniformly definitizable operators on Pontrjagin spaces are obtained. Then the basic ideas are extended to selfadjoint monic operator polynomials via their linearizations. In particular, a new class of "quasihyperbolic polynomials" (QHP) with real and determinate spectrum is introduced. It is shown that QHP have nontrivial monic factors. The special cases of strongly hyperbolic and quadratic polynomials are also discussed. In particular, a factorization theorem is proved for a class of "gyroscopically stabilized" quadratic polynomials, which originate in recent investigations of problems in mechanics.

Some properties of weighted hyperbolic polynomials

A new notion of weighted hyperbolic polynomials is introduced and their properties are discussed in this paper.

Level crossings of a random polynomial with hyperbolic elements

This paper provides an asymptotic estimate for the expected number of K-level crossings of a random hyperbolic polynomial g1 coshx + g2cosh2x + ---+ gcoshnx, where gj (j = 1, 2,...,n) are independent normally distributed random variables with mean zero, variance one and K is any constant independent of x . It is shown that the result for K = 0 remains valid as long as K _ K, = O(n) .

A theorem on the escape from the space of hyperbolic polynomials

A theorem on the escape from the space of hyperbolic polynomials

Real zero isolation for trigonometric polynomials

An exact and practical method for determining the number, location, and multiplicity of all real zeros of the trigonometric polynomials is described. All computations can be performed without loss of accuracy. lThe method is based on zero isolation techniques for algebraic polynomials. An efficient method for the calculation of the coefficients of a corresponding algebraic polynomial is stated. The complexity of trigonometric zero isolation depending on the degree and the coefficient size of the given trigonometric polynomial is analyzed. In an experimental evaluation, the performance of the method is compared to the performance of recently developed numeric techniques for the approximate determination of all roots of trigonometric polynomials. The case of exponential or hyperbolic polynomials is treated in an appendix.

Obreschkoff's theorem revisited: what convex sets are contained in the set of hyperbolic polynomials?

A real polynomial is hyperbolic if all its roots are real. We give a necessary and sufficient condition for a convex and compact set to be contained in the set of hyperbolic polynomials.

Gyroscopically Stabilized Systems: A Class Of Quadratic Eigenvalue Problems With Real Spectrum

AbstractEigenvalue problems for selfadjoint quadratic operator polynomials L(λ) = Iλ2 + Bλ+ C on a Hilbert space H are considered where B, C∈ℒ(H), C >0, and |B| ≥ kI + k-l C for some k >0. It is shown that the spectrum of L(λ) is real. The distribution of eigenvalues on the real line and other spectral properties are also discussed. The arguments rely on the well-known theory of (weakly) hyperbolic operator polynomials.

Singularities on the boundary of the hyperbolicity region

The singularities of hyperbolic polynomials (hypersurfaces) and the singularities of the boundary of the hyperbolicity region are investigated. Theorems on stabilization of these singularities in families with a fixed number of parameters and on their relationship with elliptic singularities are proved. The problems considered in this study are part of a research program focusing on singularities of boundaries of spaces of differential equations, proposed by V. I. Arnol'd.

Evaluating nonlinear models for microwave GaAs FETs

It is pointed out that although quite a number of nonlinear descriptions are available for modeling microwave GaAs MESFETs (GaAs FETs), their accuracy is much debated. Like their silicon bipolar counterparts, the GaAs FET models are based on experimental curve tracer measurements and a limited set of scattering parameters (S-parameters) measured at convenient bias points. Hence, it is impossible to be confident of their validity under conditions for which they have not been tested. This situation is complicated by a trend toward proprietary models, developed by software companies who are unwilling to share the details of their work with customers for fear of giving secrets away to potential competitors. Most of the GaAs FET models available today as part of a commercial design packages are characterized by a hyperbolic tangent or polynomial fit to the device current-voltage (I-V) curve, which is determined by actual measurement, not by studying the underlying physics. Thus, the different mathematical manipulations employed by the various developers yield significantly different results when applied to a nonlinear circuit analysis.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Spectral Methods for Initial Boundary Value Problems—An Alternative Approach

This paper presents a new approach to spectral methods for initial boundary value problems. A filtered version of the partial differential equation and the initial and boundary conditions at an overdetermined set of points are collocated. As an approximate solution, the function is chosen that belongs to an appropriate finite-dimensional space and minimizes a weighted average of the residuals at these points. It is proved that the approximate solution converges to the actual solution at a spectral rate of accuracy in both space and time. The proof is based on a priori energy estimates that have been proved for such systems. Although this method is restricted here to hyperbolic initial boundary value problems and Chebyshev polynomials, it generalizes to general initial boundary value problems, boundary value problems, and Gegenbauer polynomials.

Remarks on hyperbolic polynomials

Remarks on hyperbolic polynomials

Hyperbolic polynomials and Vandermonde mappings

Hyperbolic polynomials and Vandermonde mappings

An extension of the theory of orthogonal polynomials and Gaussian quadrature to trigonometric and hyperbolic polynomials

A class of functions was introduced by P. E. Koch (Research Report No. 72, Institute of Informatics, University of Oslo, 1982) that includes the algebraic, trigonometric, and hyperbolic polynomials, but retains sufficient similarities to the algebraic polynomials so that it was possible to generalize the Jackson theorems. This class need not satisfy the Haar condition. Some of the properties of orthogonal polynomials are extended here to this class. In particular, the recurrence relation, the Erdös-Turan theorem, the Dini condition for pointwise convergence of orthogonal expansions, and some of the basic properties of Gaussian quadrature are generalized.

Propagation of Singularities of Fundamental Solutions of Hyperbolic Mixed Problems

In this paper we shall deal with hyperbolic mixed problems with constant coefficients in a quarter-space and study the wave front sets of the fundamental solutions under the only assumption that the hyperbolic mixed problems are S-well posed. Recently Garnir has studied the wave front sets of fundamental solutions for hyperbolic systems [2]. The author was stimulated by his work. For the detailed literatures we refer the reader to [7], [8]. Now let us state our problems, assumptions and main results. Let R denote the ^-dimensional euclidean space and write x = (xl9 •••,o:n_1) for the coordinate x= (xl9 • • • , xn) in R n and £' — ($19 • • • , fn_a) , j = (f, £n+1) for the dual coordinate $ = (gl9 • • • , f n) . We shall also denote by JS+ the half-space {x = (x' , x^) e R; xn^>0}. For differentiation we will use the symbol D = i~ (d/dxl9 •-, d/dxn) . Let P-P(f) be a hyperbolic polynomial of order m of n variables f with respect to t?= (1,0, • • • , 0) in the sense of Carding, i.e.