Negative Zeros Research Articles

IMC based modified Smith predictor for second order delay dominated processes with RHP

A modified Smith predictor (MSP) with IMC based tuning is proposed for second order delay dominated processes. These processes often exhibit unstable behaviour due to right half poles (RHP) with positive or negative zeros. The control scheme incorporates forward path PI/PID controller with a lead–lag filter and a PD controller in the feedback path. IMC technique having single tuning parameter is utilized to design the forward path controller, while the feedback path controller is tuned by Routh stability criterion. The proposed scheme ascertains quick set point following without overshoot and smooth disturbance rejection, resulting superior closed-loop performance compared to recent MSP schemes. Validation is made based on performance indices and stability margins.

A domain free of the zeros of the partial theta function

The partial theta function is the sum of the series \medskip\centerline{$\displaystyle\theta (q,x):=\sum\nolimits _{j=0}^{\infty}q^{j(j+1)/2}x^j$,}\medskip\noi where $q$is a real or complex parameter ($|q|<1$). Its name is due to similaritieswith the formula for the Jacobi theta function$\Theta (q,x):=\sum _{j=-\infty}^{\infty}q^{j^2}x^j$. The function $\theta$ has been considered in Ramanujan's lost notebook. Itfinds applicationsin several domains, such as Ramanujan type$q$-series, the theory of (mock) modular forms, asymptotic analysis, statistical physics, combinatorics and most recently in the study of section-hyperbolic polynomials,i.~e. real polynomials with all coefficients positive,with all roots real negative and all whose sections (i.~e. truncations)are also real-rooted.For each $q$ fixed,$\theta$ is an entire function of order $0$ in the variable~$x$. When$q$ is real and $q\in (0,0.3092\ldots )$, $\theta (q,.)$ is a function of theLaguerre-P\'olyaclass $\mathcal{L-P}I$. More generally, for $q \in (0,1)$, the function $\theta (q,.)$ is the product of a realpolynomialwithout real zeros and a function of the class $\mathcal{L-P}I$. Thus it isan entire function withinfinitely-many negative, with no positive and with finitely-many complexconjugate zeros. The latter are known to belongto an explicitly defined compact domain containing $0$ andindependent of $q$ while the negative zeros tend to infinity as ageometric progression with ratio $1/q$. A similar result is true for$q\in (-1,0)$ when there are also infinitely-many positive zeros.We consider thequestion how close to the origin the zeros of the function $\theta$ can be.In the generalcase when $q$ is complex it is truethat their moduli are always larger than $1/2|q|$. We consider the case when $q$ is real and prove that for any $q\in (0,1)$,the function $\theta (q,.)$ has no zeros on the set $$\displaystyle \{x\in\mathbb{C}\colon |x|\leq 3\} \cap \{x\in\mathbb{C}\colon {\rm Re} x\leq 0\}\cap \{x\in\mathbb{C}\colon |{\rm Im} x|\leq 3/\sqrt{2}\}$$which containsthe closure left unit half-disk and is more than $7$ times larger than it.It is unlikely this result to hold true for the whole of the lefthalf-disk of radius~$3$. Similar domains do not exist for $q\in (0,1)$, Re$x\geq 0$, for$q\in (-1,0)$, Re$x\geq 0$ and for $q\in (-1,0)$, Re$x\leq 0$. We show alsothat for $q\in (0,1)$, the function $\theta (q,.)$ has no real zeros $\geq -5$ (but one can find zeros larger than $-7.51$).

On entire functions from the Laguerre-Polya I class with non-monotonic second quotients of Taylor coefficients

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we define its second quotients of Taylor coefficients as $q_k (f):= \frac{a_{k-1}^2}{a_{k-2}a_k}, k \geq 2.$ In the present paper, we study entire functions of order zerowith non-monotonic second quotients of Taylor coefficients. We consider those entire functions for which the even-indexed quotients are all equal and the odd-indexed ones are all equal:$q_{2k} = a>1$ and $q_{2k+1} = b>1$ for all $k \in \mathbb{N}.$We obtain necessary and sufficient conditions under which such functions belong to the Laguerre-P\'olya I class or, in our case, have only real negative zeros. In addition, we illustrate their relation to the partial theta function.

Ratios of Entire Functions and Generalized Stieltjes Functions

Monotonicity properties of the ratio $$\begin{aligned} \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, \end{aligned}$$where f is an entire function are investigated. Earlier results for Euler’s gamma function and other entire functions of genus 1 are generalised to entire functions of genus p with negative zeros. Derivatives of order comparable to p of the expression above are related to generalised Stieltjes functions of order \(p+1\). Our results are applied to the Barnes multiple gamma functions. We also show how recent results on the behaviour of Euler’s gamma function on vertical lines can be sharpened and generalised to functions of higher genus. Finally a connection to the so-called Prouhet-Tarry-Escott problem is described.

On Classifying At Risk Latent Zeros Using Zero Inflated Models

Count data often have excess zeros in many clinical studies. These zeros usually represent disease-free state. Although disease (event) free at the time, some of them might be at a of having the putative outcome while others may be at or no such risk. We postulate these zeros as a one of the two types, either as 'low risk' or as 'high risk' zeros for the disease process in question. Low zeros can arise due to the absence of factors for disease initiation/progression and/or due to very early stage of the disease. High zeros can arise due to the presence of significant factors for disease initiation/ progression or could be, in rare situations, due to misclassification, more specific diagnostic tests, or below the level of detection. We use zero inflated models which allows us to assume that zeros arise from one of the two separate latent processes-one giving low-risk zeros and the other high-risk zeros and subsequently propose a strategy to identify and classify them as such. To illustrate, we use data on the number of involved nodes in breast cancer patients. Of the 1152 patients studied, 38.8% were node- negative (zeros). The model predicted that about a third (11.4%) of negative nodes are high risk and the remaining (27.4%) are at low risk of nodal positivity. Posterior probability based classification was more appropriate compared to other methods. Our approach indicates that some node negative patients may be re-assessed for their diagnosis about nodal positivity and/or for future clinical management of their disease. The approach developed here is applicable to any scenario where the disease or outcome can be characterized by count-data.

On the determination of the number of positive and negative polynomial zeros and their isolation

Abstract A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree n can be restricted with significantly better determinacy than that provided by the Descartes rule of signs. The method also allows the isolation of the zeros of the polynomial quite successfully, and the determined root bounds are significantly narrower than the Cauchy and the Lagrange bounds. The method relies on solving equations of degree smaller than that of the given polynomial. One can determine analytically the exact number of positive and negative zeros of a polynomial of degree up to and including five and also fully isolate the zeros of the polynomial analytically, and with one of the variations of the method, one can analytically approach polynomials of degree up to and including nine by solving equations of degree not more than four. For polynomials of higher degree, either of the two variations of the method should be applied recursively. Numerous examples are given. Presented is the full classification of the roots of the cubic equation, together with their isolation intervals – especially important for various scientific models for which the coefficients of the equation might be functions of the model parameters. An application of the method to a quartic equation with variable coefficients (resulting from the study of wave–current interactions in the physically realistic model of azimuthal two-dimensional non-viscous flows with piecewise constant vorticity in a two-layer fluid with a flat bed and a free surface) has been demonstrated. A step-by-step algorithm for each version of the method has also been presented.

Quantized ΔS = 2 Excitation Spectra by Confinement in an S = 1 Spin Chain

We calculate the dynamical spin-structure factor of the $S=1$ Ising spin chain with negative single-ion anisotropy in magnetic fields using the infinite time-evolving-block-decimation algorithm. We show that when a transverse magnetic field is applied, both the $\Delta S=2$ excitation continuum and one-magnon mode appear in the low-lying excitation. When a longitudinal magnetic field is further applied, the excitation continuum changes into quantized excitation spectra. The quantized $\Delta S=2$ excitation spectra originate from the confinement of two domain walls, each of which carries $\Delta S=1$. The quantized excitation energies are explained by the negative zeros in the Airy function.

On Zeros of Some Entire Functions

Let $$\begin{aligned} A_{q}^{(\alpha )}(a;z)=\sum _{k=0}^{\infty }\frac{(a;q)_{k}q^{\alpha k^2} z^k}{(q;q)_{k}}, \end{aligned}$$ where $$\alpha >0,~0<q<1.$$ In a paper of Ruiming Zhang, he asked under what conditions the zeros of the entire function $$A_{q}^{(\alpha )}(a;z)$$ are all real and established some results on the zeros of $$A_{q}^{(\alpha )}(a;z)$$ which present a partial answer to that question. In the present paper, we will set up some results on certain entire functions which includes that $$A_{q}^{(\alpha )}(q^l;z),~l\ge 2$$ has only infinitely many negative zeros that gives a partial answer to Zhang’s question. In addition, we establish some results on zeros of certain entire functions involving the Rogers–Szegő polynomials and the Stieltjes–Wigert polynomials.

Quantized excitation spectra by magnon confinement in quasi-one-dimensional S=1 spin systems

We apply the infinite time-evolving-block-decimation algorithm to calculate the dynamical spin-structure factors of the quasi-one-dimensional (q1D) S=1 antiferromagnetic spin system with the single-ion anisotropy and the bond alternation. We find that excitation continuum originating from magnons is quantized, when the staggered field induced by the weak inter-chain interaction is taken into account. The excitation energies of the quantized excitation spectra are well explained by negative zeros of the Airy functions, when the easy-axis anisotropy is strong and the ground state is located deep in the N\'eel phase. This quantization of the magnon continuum is a counterpart of the spinon confinement, which has been recently discussed in q1D S=1/2 antiferromagnets. We further show that, when the staggered field exists, the quantized excitation spectra appear the phase boundary between the Haldane phase and the N\'eel phase of the phase diagram without the staggered field. However, the quantized excitation spectra disappear in the singlet dimer phase.

On the sum of the Lah numbers and zeros of the Kummer confluent hypergeometric function

Abstract In the paper, the authors find the sum of the Lah numbers and make sure that the Kummer confluent hypergeometric function 1F1(n+ 1; 2; z) has only n− 1real and negative zeros.

On complete monotonicity of certain special functions

Given an entire function f ( z ) f(z) that has only negative zeros, we shall prove that all the functions of type f ( m ) ( x ) / f ( n ) ( x ) , m &gt; n f^{(m)}(x)/f^{(n)}(x),\ m&gt;n are completely monotonic. Examples of this type are given for Laguerre polynomials, ultraspherical polynomials, Jacobi polynomials, Stieltjes-Wigert polynomials, q q -Laguerre polynomials, Askey-Wilson polynomials, Ramanujan function, q q -exponential functions, q q -Bessel functions, Euler’s gamma function, Airy function, modified Bessel functions of the first and the second kind, and the confluent basic hypergeometric series.

A Simple Method to Make the Quadruple Tank System Near Linear

Quadruple tank liquid level systems are popular in testing multivariable control systems for multivariable processes with positive or negative zeros. The liquid level system is nonlinear and it will help to illustrate the robustness of control systems. However, due to nonlinearity, it can be cumbersome to obtain process parameters for testing linear control systems. Perturbation sizes are limited for valid linearized process models, requiring level sensors with high pre-cision. A simple method where the outlet orifice is replaced to a long tube is proposed here. The effluent flow rate becomes proportional to the liquid level due to the friction loss of long tube and the liquid level system shows near lin-ear dynamics. It is applied to the quadruple tank system for easier experiments.

One helpful property of functions generating Pólya frequency sequences

In this work we study the solutions of the equation $z^pR(z^k)=\alpha$ with nonzero complex $\alpha$, integer $p,k$ and $R(z)$ generating a (possibly doubly infinite) totally positive sequence. It is shown that the zeros of $z^pR(z^k)-\alpha$ are simple (or at most double in the case of real $\alpha^k$) and split evenly among the sectors $\{\frac jk \pi\le\operatorname{Arg} z\le\frac {j+1}k \pi\}$, $j=0,\dots, 2k-1$. Our approach rests on the fact that $z(\ln z^{p/k}R(z) )'$ is an $\mathcal R$-function (i.e. maps the upper half of the complex plane into itself). This result guarantees the same localization to zeros of entire functions $f(z^k)+z^p g(z^k)$ and $g(z^k)+z^{p}f(z^k)$ provided that $f(z)$ and $g(-z)$ have genus $0$ and only negative zeros. As an application, we deduce that functions of the form $\sum_{n=0}^\infty (\pm i)^{n(n-1)/2}a_n z^{n}$ have simple zeros distinct in absolute value under a certain condition on the coefficients $a_n\ge 0$. This includes the "disturbed exponential" function corresponding to $a_n= q^{n(n-1)/2}/n!$ when $0<q\le 1$, as well as the partial theta function corresponding to $a_n= q^{n(n-1)/2}$ when $0<q\le q_*\approx 0.7457224107$.

On a partial theta function and its spectrum

The bivariate series defines a partial theta function. For fixed q (∣q∣ &lt; 1), θ(q, ·) is an entire function. For q ∈ (–1, 0) the function θ(q, ·) has infinitely many negative and infinitely many positive real zeros. There exists a sequence of values of q tending to –1+ such that has a double real zero (the rest of its real zeros being simple). For k odd (respectively, k even) has a local minimum (respectively, maximum) at , and is the rightmost of the real negative zeros of (respectively, for k sufficiently large is the second from the left of the real negative zeros of ). For k sufficiently large one has . One has and .

On the double zeros of a partial theta function

The series θ(q,x):=∑j=0∞qj(j+1)/2xj converges for q∈[0,1), x∈R, and defines a partial theta function. For any fixed q∈(0,1) it has infinitely many negative zeros. For q taking one of the spectral values q˜1, q˜2, … (where 0.3092493386…=q˜1<q˜2<⋯<1, limj→∞⁡q˜j=1) the function θ(q,.) has a double zero yj which is the rightmost of its real zeros (the rest of them being simple). For q≠q˜j the partial theta function has no multiple real zeros. We prove that q˜j=1−π/2j+(log⁡j)/8j2+O(1/j2) and yj=−eπe−(log⁡j)/4j+O(1/j).

Asymptotics of the spectrum of partial theta function

The series \(\theta (q,x):=\sum _{j=0}^{\infty }q^{j(j+1)/2}x^j\) converges for \(q\in [0,1)\), \(x\in \mathbb R \), and defines a partial theta function. For any fixed \(q\in (0,1)\) it has infinitely many negative zeros. For \(q\) taking one of the spectral values \(\tilde{q}_1\), \(\tilde{q}_2\), \(\ldots \) (where \(0.3092493386\ldots =\tilde{q}_1<\tilde{q}_2<\cdots <1\), \(\lim _{j\rightarrow \infty }\tilde{q}_j=1\)) the function \(\theta (q,.)\) has a double zero \(y_j\) which is the rightmost of its real zeros (the rest of them being simple). For \(q\ne \tilde{q}_j\) the partial theta function has no multiple real zeros. We prove that \(\tilde{q}_j=1-(\pi /2j)+o(1/j)\) and that \(\lim _{j\rightarrow \infty }y_j=-e^{\pi }=-23.1407\ldots \).

Extension of Inverses of Entire Functions of Genus 1 and 2 to the Upper Half Plane

Any entire function of genus 1 which is positive on the positive real axis and which has only negative zeros decreases on some unbounded interval of the positive axis. The inverse of its reciprocal is shown to have an extension from that interval to a Pick-function in the upper half plane. A similar result holds for a class of entire functions of genus 2 and in particular the inverse function of Barnes’ double gamma function on a certain interval of the positive axis can be extended to a Pick-function. These results are proved using positive and negative definite kernels.

On the zeros of a partial theta function

The series θ(q,x):=∑j=0∞qj(j+1)/2xj converges for q∈[0,1), x∈R and defines a partial theta function. For any q∈(0,1) fixed it has infinitely many negative zeros. For countably many values q˜j of q said to form the spectrum of θ (where 0.3092493386…=q˜1<q˜2<⋯<1, limj→∞q˜j=1) the function θ(q,.) has a double zero which is the rightmost of its real zeros (the rest of them being simple). For q≠q˜j it has no multiple real zeros. For q∈(q˜N,q˜N+1) the function θ(q,.) has exactly N complex conjugate pairs of zeros counted with multiplicity (we set q˜0=0). If ξkl denote the zeros of ∂θl/∂xl(q,.) in the order of decreasing, then limk→∞ξklqk=−1 and limk→∞ξk+1l/ξkl=q.

Positive stable realizations of fractional continuous-time linear systems

Positive stable realizations of fractional continuous-time linear systemsConditions for the existence of positive stable realizations with system Metzler matrices for fractional continuous-time linear systems are established. A procedure based on the Gilbert method for computation of positive stable realizations of proper transfer matrices is proposed. It is shown that linear minimum-phase systems with real negative poles and zeros always have positive stable realizations.

Conditions for a Class of Rational Approximants of Fractional Differentiators/Integrators to Enjoy the Interlacing Property

AbstractFractional-order controllers are based on fractional differentiators and integrators, which extend classical integral or derivative actions to improve performance and robustness of feedback loops. The fractional (irrational) operators are usually realized by continued fractions leading to rational approximations. Stable and minimum-phase transfer functions, with interlaced distribution of real zeros and poles, are also requested as suitable approximations. This paper proves that two classes of continued-fractions-based approximations have negative zeros and poles enjoying the interlacing property, for any fractional order v, with |v| < 1, and for any degree, m, of the approximating transfer function.