Spurious Zeros Research Articles

${\mathbf{U}} = \mathbf{C}^{1/2}$ and Its Invariants in Terms of $\mathbf{C}$ and Its Invariants

We consider $N\times N$ tensors for $N= 3,4,5,6$. In the case $N=3$, it is desired to find the three principal invariants $i_1, i_2, i_3$ of $\bf U$ in terms of the three principal invariants $I_1, I_2, I_3$ of ${\bf C}={\bf U}^2$. Equations connecting the $i_\alpha$ and $I_\alpha$ are obtained by taking determinants of the factorisation \[\lambda^2{\bf I}- {\bf C} = (\lambda{\bf I}- {\bf U}) (\lambda{\bf I}+ {\bf U})\] and comparing coefficients. On eliminating $i_2$ we obtain a quartic equation with coefficients depending solely on the $I_\alpha$ whose largest root is $i_1$. Similarly, we may obtain a quartic equation whose largest root is $i_2$. For $N=4$ we find that $i_2$ is once again the largest root of a quartic equation and so all the $i_\alpha$ are expressed in terms of the $I_\alpha$. Then $\bf U$ and ${\bf U}^{-1}$ are expressed solely in terms of $\bf C$, as for $N=3$. For $N= 5$ we find, but do not exhibit, a twentieth degree polynomial of which $i_1$ is the largest root and which has four spurious zeros. We are unable to express the $i_\alpha$ in terms of the $I_\alpha$ for $N=5$. Nevertheless, $\bf U$ and ${\bf U}^{-1}$ are expressed in terms of powers of $\bf C$ with coefficients now depending on the $i_\alpha$. For $N=6$ we find, but do not exhibit, a 32 degree polynomial which has largest root $i_1^2$. Sixteen of these roots are relevant but the other 16, which we exhibit, are spurious. $\bf U$ and ${\bf U}^{-1}$ are expressed in terms of powers of $\bf C$. The cases $N>6$ are discussed. Keywords: Continuum mechanics, polar decomposition, tensor square roots, principal invariants, cubic equations, quartic equations, equations of degree 16

Behavior of the Roots of the Taylor Polynomials of Riemann’s $$\xi $$ ξ Function with Growing Degree

We establish a uniform approximation result for the Taylor polynomials of Riemann’s \(\xi \) function valid in the entire complex plane as the degree grows. In particular, we identify a domain growing with the degree of the polynomials on which they converge to Riemann’s \(\xi \) function. Using this approximation, we obtain an estimate of the number of “spurious zeros” of the Taylor polynomial that lie outside of the critical strip, which leads to a Riemann–von Mangoldt type formula for the number of zeros of the Taylor polynomials within the critical strip. Super-exponential convergence of Hurwitz zeros of the Taylor polynomials to bounded zeros of the \(\xi \) function are also established. Finally, we explain how our approximation techniques can be extended to a collection of analytic L-functions.

The supercritical regime in the normal matrix model with cubic potential

The normal matrix model with a cubic potential is ill-defined and it develops a critical behavior in finite time. We follow the approach of Bleher and Kuijlaars to reformulate the model in terms of orthogonal polynomials with respect to a Hermitian form. This reformulation was shown to capture the essential features of the normal matrix model in the subcritical regime, namely that the zeros of the polynomials tend to a number of segments (the motherbody) inside a domain (the droplet) that attracts the eigenvalues in the normal matrix model.In the present paper we analyze the supercritical regime and we find that the large n behavior is described by the evolution of a spectral curve satisfying the Boutroux condition. The Boutroux condition determines a system of contours Σ1, consisting of the motherbody and whiskers sticking out of the domain. We find a second critical behavior at which the original motherbody shrinks to a point at the origin and only the whiskers remain.In the regime before the second criticality we also give strong asymptotics of the orthogonal polynomials by means of a steepest descent analysis of a 3×3 matrix valued Riemann–Hilbert problem. It follows that the zeros of the orthogonal polynomials tend to Σ1, with the exception of at most three spurious zeros.

Zeros of sections of exponential sums

We derive the large n asymptotics of zeros of sections of a generic exponential sum. We divide all the zeros of the n-th section of the exponential sum into which approach, as n → ∞, the zeros of the exponential sum, and which go to infinity as n → ∞. We show that the spurious zeros, after scaling down by the factor of n, approach a rosette, a finite collection of curves on the complex plane, resembling the rosette. We derive also the large n asymptotics of the transitional the intermediate zeros between genuine and spurious ones. Our results give an extension to the classical results of Szego about the large n asymptotics of zeros of sections of the exponential, sine, and cosine functions.

On truncated Taylor series and the position of their spurious zeros

A truncated Taylor series, or a Taylor polynomial, which may appear when treating the motion of gravity water waves, is obtained by truncating an infinite Taylor series for a complex, analytical function. For such a polynomial the position of the complex zeros is considered in case the Taylor series has a finite radius of convergence. It is of interest to find whether the moduli of the zeros are close to the radius of convergence. We therefore discuss various upper and lower bounds for the moduli given in the literature and present a new procedure for their estimation. Finally the results obtained are related to an old German paper. It investigates how zeros of partial sums of power series will condensate near the circle of convergence.

Computation of system zeros with balancing

The numerical computation of the transmission zeros of a linear, time-invariant system requires the solution of a generalized eigenvalue problem ( GEP) Mx= λNx with N=block diag ( I,0). The Emami-Naeini Van Dooren (ENVD) algorithm for the computation of system zeros avoids the computation of spurious zeros by isolating the finite zeros of a system prior to numerical solution the corresponding GEP. Unfortunately, the ENVD algorithm can be sensitive to the dynamic range of system coefficients. We present a numerical procedure for reduction of the dynamic range of coefficients in a zero-computation problem that preserves the underlying structure N=block diag ( I,0) of the zero-computation GEP.

Resummation of classical and semiclassical periodic-orbit formulas.

The convergence properties of cycle expanded periodic orbit expressions for the spectra of classical and semiclassical time evolution operators have been studied for the open three disk billiard. We present evidence that both the classical and the semiclassical Selberg zeta function have poles. Applying a Pad\'{e} approximation on the expansions of the full Euler products, as well as on the individual dynamical zeta functions in the products, we calculate the leading poles and the zeros of the improved expansions with the first few poles removed. The removal of poles tends to change the simple linear exponential convergence of the Selberg zeta functions to an $\exp\{-n^{3/2}\}$ decay in the classical case and to an $\exp\{-n^2\}$ decay in the semiclassical case. The leading poles of the $j$th dynamical zeta function are found to equal the leading zeros of the $j+1$th one: However, in contrast to the zeros, which are all simple, the poles seem without exception to be {\em double}\/. The poles are therefore in general {\em not}\/ completely cancelled by zeros, which has earlier been suggested. The only complete cancellations occur in the classical Selberg zeta function between the poles (double) of the first and the zeros (squared) of the second dynamical zeta function. Furthermore, we find strong indications that poles are responsible for the presence of spurious zeros in periodic orbit quantized spectra and that these spectra can be greatly improved by removing the leading poles, e.g. by using the Pad\'{e} technique.

Noninvasive detection of coronary stenoses before and after angioplasty using eigenvector methods

Previous studies done by our group suggest that partially occluded coronary arteries may generate sounds due to turbulent blood flow. To support these previous findings the frequency spectra of diastolic heart sounds are compared before and after angioplastic surgery. Since the low-level sounds associated with partially occluded coronary arteries are contaminated with considerable background noise, traditional FFT analysis may not produce accurate frequency spectra. Indeed, in a previous study using the same data, no significant differences were found in the diastolic heart sounds before and after angioplastic surgery. In this study, three eigenvector methods (Pisarenko, MUSIC, and Minimum-Norm) have been selected to generate the frequency spectra because of their higher resolution, particularly in the presence of noise. Although the Pisarenko method produced spurious zeros and could not be used, the other two methods produced spectra showing, in most cases, a marked decrease in high-frequency spectral components following angioplasty.