Symmetric Zeros Research Articles

Quantitative Green’s function estimates for lattice quasi-periodic Schrödinger operators

In this paper, we establish quantitative Green's function estimates for some higher dimensional lattice quasi-periodic (QP) Schr\"odinger operators. The resonances in the estimates can be described via a pair of symmetric zeros of certain functions and the estimates apply to the sub-exponential type non-resonant conditions. As the application of quantitative Green's function estimates, we prove both the arithmetic version of Anderson localization and the $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for such QP Schr\"odinger operators. This gives an affirmative answer to Bourgain's problem in\cite{Bou00}.

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.

Five texture zeros in the lepton sector and neutrino oscillations at DUNE

In this work, we have assumed special structures for the charged and neutral mass matrices in the lepton sector, inspired by structures for the up and down quark mass matrices that result by assuming a certain number of symmetrical zeros in their entries named texture zeros. A prediction of the lepton mixing matrix results from the rotation matrices that diagonalize the mass matrices in the neutral and charged lepton sectors. The use of texture zeros reduces the number of spurious parameters to the minimal ones needed to explain observations i.e., charged lepton masses and neutrino oscillation parameters. Specifically, we have considered the case of five texture zeros and we have confronted the resulting lepton mixing matrices with current measurements in the neutrino sector. Finally, sensitivities to the independent parameters in the mixing predicted by the nonequivalent forms were studied using simulated events at the DUNE neutrino oscillation experiment. We have found that DUNE is sensitive to nonzero $CP$ violation allowed in the models.

A True Inline Coaxial-Cavity Filter With Two Symmetric Zeros

This work deals with the design of true inline filters with two symmetric transmission zeros. A new filters category named “path filters” is introduced (including the considered filters), its general features are discussed and analyzed and a general method for the synthesis of the low-pass prototype network is proposed. To validate the proposed design approach, a coaxial-cavity filter with a true inline topology exhibiting two symmetrical transmission zeros has been designed and fabricated. The realized prototype shows a response in good agreement with the theory.

On the Non-Trivial Zeros of Dirichlet Functions

The purpose of this research is to extend to the functions obtained by meromorphic continuation of general Dirichlet series some properties of the functions in the Selberg class, which are all generated by ordinary Dirichlet series. We wanted to put to work the powerful tool of the geometry of conformal mappings of these functions, which we built in previous research, in order to study the location of their non-trivial zeros. A new approach of the concept of multiplier in Riemann type of functional equation was necessary and we have shown that with this approach the non-trivial zeros of the Dirichlet function satisfying a Reimann type of functional equation are either on the critical line, or they are two by two symmetric with respect to the critical line. The Euler product general Dirichlet series are defined, a wide class of such series is presented, and finally by using geometric and analytic arguments it is proved that for Euler product functions the symmetric zeros with respect to the critical line must coincide.

How to count the number of zeros that a polynomial has on the unit circle?

The classical problem of counting the number of real zeros of a real polynomial was solved a long time ago by Sturm. The analogous problem of counting the number of zeros that a polynomial has on the unit circle is, however, still an open problem. In this paper, we show that the second problem can be reduced to the first one through the use of a suitable pair of Möbius transformations – often called Cayley transformations – that have the property of mapping the unit circle onto the real line and vice versa. Although the method applies to arbitrary complex polynomials, we discuss in detail several classes of polynomials with symmetric zeros as, for instance, the cases of self-conjugate, self-adjoint, self-inversive, self-reciprocal and skew-reciprocal polynomials. Finally, an application of this method to Salem polynomials and to polynomials with small Mahler measure is also discussed.

The Reentrant Four-Layer Quasi-Elliptic Bandstop Filter

The novel microwave quasi-elliptic bandstop filter, which uses the initially uncoupled strip transmission lines, is considered in this work. The proposed filter is based on the reentrant structure, where the metallic body with a floating potential is asymmetric. Generally speaking, the internal and external dielectric fillings must have their unequal relative permittivities. As a result, additional symmetrical reflection zeros are reached at the lower and upper pass bands of the stop band leading to the quasi-elliptic function response that improves the filter selectivity. The general transverse electromagnetic (TEM) circuit model for the proposed filter in terms of a series connection of the multi-ports is presented and then used to predict the initial electrical and geometrical parameters. An experimental printed circuit prototype has been manufactured and evaluated to validate the design concept. The measured filter parameters correlate well with the simulation-derived ones and that increases the degree of freedom in the fabrication of microwave frequency selective components.

Routh-type table test for zero distribution of polynomials with commensurate fractional and integer degrees

This paper mainly presents Routh-type table test methods for zero distribution of polynomials with commensurate fractional degrees on the left-half plane, right-half plane and imaginary axis in the complex plane. The proposed tabular methods are derived for extension and generalization of the Routh test, which is widely used in controls for zero distribution of polynomials with integer degrees. Singular cases are discussed and handled efficiently and simply. Necessary and sufficient conditions for the second singular case are completely analyzed in terms of symmetric zeros. A particular property is revealed that a polynomial with commensurate fractional degrees without pure imaginary zero may still be stable in the presence of the second singular case, which is impossible for a real polynomial with integer degrees. Furthermore, we present a test to solve the zero distribution problem with respect to general sector region for polynomials with commensurate fractional degrees and real/complex coefficients. Finally, numerical examples are given to illustrate the correctness and effectiveness of the results. The proposed methods have broad application areas, including various systems, circuits and control.

Robust transfer function identification via an enhanced magnitude vector fitting algorithm

The study introduces an enhanced version of the magnitude vector fitting (magVF) algorithm, a robust procedure for the identification of a transfer function from magnitude frequency domain data. The approach is based on the rational approximation of the magnitude square function with enforcement of symmetric poles and zeros, followed by the elimination of poles and zeros located in the right half-plane. The obtained transfer function is stable and of minimum-phase shift type. Robustness and accuracy of the basic magVF algorithm are enhanced by enforcing that the magnitude square rational function is non-negative definite and by introducing a new method to remove purely imaginary conjugate poles from the approximation. Practical application of the proposed approach is demonstrated for measured transformer responses and transmission line propagation functions.

Generalized parallel-cascaded microring networks for spectral engineering applications

The transfer characteristics of parallel-cascaded arrays of generalized two-port microring networks with interstage phase shifts are investigated for potential applications in spectral engineering. It is found that these structures can exactly realize optical transfer functions with quadrantal symmetric zeros if the microring networks are even-symmetric. The proposed filter architectures have advantages in that the microring networks have all-positive coupling coefficients, and the array can always be reduced to the canonical form that contains only one interstage π phase-shift element. A method for exactly synthesizing cascaded symmetric microring networks based on the transfer matrix factorization technique is presented, and an example of an eighth-order generalized Chebyshev filter is given to illustrate the application of these microring lattice architectures for realizing very high-order optical transfer functions.

On the regularity of growth of canonical products with real zeros

We obtain a criterion for totally regular growth of canonical products with real symmetric zeros, which involves a generalization of the notion of condensation index of a sequence of positive numbers of a finite upper density.

Quadratically Convergent Method for Simultaneously Approaching the Roots of Polynomial Solutions of a Class of Differential Equations: Application to Orthogonal Polynomials

This paper investigates the application of the method introduced by L. Pasquini (1989) for simultaneously approaching the zeros of polynomial solutions to a class of second-order linear homogeneous ordinary differential equations with polynomial coefficients to a particular case in which these polynomial solutions have zeros symmetrically arranged with respect to the origin. The method is based on a family of nonlinear equations which is associated with a given class of differential equations. The roots of the nonlinear equations are related to the roots of the polynomial solutions of differential equations considered. Newton's method is applied to find the roots of these nonlinear equations. In (Pasquini, 1994) the nonsingularity of the roots of these nonlinear equations is studied. In this paper, following the lines in (Pasquini, 1994), the nonsingularity of the roots of these nonlinear equations is studied. More favourable results than the ones in (Pasquini, 1994) are proven in the particular case of polynomial solutions with symmetrical zeros. The method is applied to approximate the roots of Hermite–Sobolev type polynomials and Freud polynomials. A lower bound for the smallest positive root of Hermite–Sobolev type polynomials is given via the nonlinear equation. The quadratic convergence of the method is proven. A comparison with a classical method that uses the Jacobi matrices is carried out. We show that the algorithm derived by the proposed method is sometimes preferable to the classical QR type algorithms for computing the eigenvalues of the Jacobi matrices even if these matrices are real and symmetric.

Quark mass textures within a finite non-Abelian dicyclic group.

Using as a flavor symmetry a finite nonabelian dicyclic $Q_{2N}$ group, we show how to derive quark mass matrices with two arrangements of symmetric texture zeros which are phenomenologically viable. Three other such acceptable textures in the recent literature are unattainable in this approach and hence disfavored. We assume massive vector-like fermions and Higgs singlets transforming as judiciously-chosen $Q_{2N}$ doublets and use the tree-level mass generation mechanism of Froggatt and Nielsen.

A current-mode biquadratic amplitude equalizer

A boost biquad transfer function with two symmetrical zeros on the real axis is derived; it provides a technique for amplitude equalization with a second-order continuous-timegm-C section using current injection. The synthesis procedure yields a differential boost biquad consisting of three double-input OTAs and a single-input high-gmboost OTA. SPICE simulations show that for boost gain up to 20 dB the phase of the biquad is essentially unaffected. The presented technique is found suitable for amplitude equalization of disk-drive read channels operating in the range of 100 MHz.