Terms Of Zeros Research Articles

Efficient spectral tests for multiple recursive generators

. Large-order maximum-period Multiple Recursive Generators (MRGs) have become popular in the area of computer simulation. They have the nice properties of high-dimensional equi-distribution, generating efficiency, long period, and portability. The spectral test is a commonly used criterion for ranking pseudo-random number generators. Procedures for computing the spectral test values of MRGs are available in the literature but may not be efficient when the order of the MRG is large. In this article, we propose a novel method for the spectral test computation of MRGs that is simple, intuitive, and particularly efficient for MRGs with few non zero terms. With the proposed method, we are able to provide a list of ready-to-use “better” generators with respect to the spectral test performance among the DX generators of order k for various values of k.

On uniqueness of \(L\)-functions in terms of zeros of strong uniqueness polynomial

In this article, we have mainly focused on the uniqueness problem of an \(L\)-function \(\mathcal{L}\) with an \(L\)-function or a meromorphic function \(f\) under the condition of sharing the sets, generated from the zero set of some strong uniqueness polynomials. We have introduced two new definitions, which extend two existing important definitions of URSM and UPM in the literature and the same have been used to prove one of our main results. As an application of the result, we have exhibited a much improved and extended version of a recent result of Khoai-An-Phuong [23]. Our remaining results are about the uniqueness of \(L\)-function under weighted sharing of sets generated from the zeros of a suitable strong uniqueness polynomial, which improve and extend some results in [12].

Strong Left Inversion of Linear Systems and Input Reconstruction

Left inversion of linear time-invariant systems aims at identifying the input acting on a system, for the zero initial state, from partial information on the input and the state. In the present contribution, we propose to generalize such left inversion for linear systems in various directions, that take arbitrary initial state into consideration to address both exact and asymptotic input reconstruction. Necessary and sufficient algebraic conditions are given to achieve such strong left inversion properties. Complete characterizations of introduced concepts in terms of system zeros are provided. Relationship between inversion and input reconstruction is therefore investigated, with emphasis on causal realization. Conditions for input observer existence are proposed, and a constructive causal design for an asymptotically convergent input observer is presented.

Inverse resonance scattering for massless Dirac operators on the real line

We consider massless Dirac operators on the real line with compactly supported potentials. We solve two inverse problems: in terms of zeros of reflection coefficient and in terms of poles of reflection coefficients (i.e. resonances). Moreover, we prove the following: 1) a zero of the reflection coefficient can be arbitrarily shifted, such that we obtain the sequence of zeros of the reflection coefficient for another compactly supported potential, 2) the set of “isoresonance potentials” is described, 3) the forbidden domain for resonances is estimated, 4) asymptotics of the resonances counting function is determined, 5) these results are applied to canonical systems.

The covering radius of permutation designs

A notion of t-designs in the symmetric group on n letters was introduced by Godsil in 1988. In particular, t-transitive sets of permutations form a t-design. We derive upper bounds on the covering radius of these designs, as a function of n and t and in terms of the largest zeros of Charlier polynomials.

Analytical Modeling of Three-Section Coaxial Stepped Impedance Resonators for the Design of Compact Tx Bandpass Filters

This article focuses on the synthesis equations of three-section coaxial stepped impedance resonators (SIRs) for the design of Tx bandpass filters and on the benefits of adding a third section compared with using a two-section SIR. In this topology, the three coaxial structures fit inside ones another. In terms of the electrical scheme, the model is a succession of coaxial sections in a cascade, where the ground conductor of one becomes the central core of the next and vice versa. An advantageous property of SIRs is that they allow the size and the distance from the next harmonic to be controlled. The SIR effect depends on the ratio of the characteristic impedances of two adjacent coaxial TEM sections. For a resonator based on three sections, two impedance ratios need to be considered to electrically characterize the resonator in terms of fundamental resonant frequency and transmission zeros. Thanks to the equations obtained by modeling its frequency behavior, we are able to show the added value of changing such a resonator from two to three sections. This makes it possible to enlarge the minimal gap between the cylinders, decreasing the sensitivity to technological variations in manufacture and improving immunity to multipactor phenomena under constrained conditions of height and bandwidth. It also makes the first transmission zero frequency and fundamental frequency independent, as the relative proximity of these two frequencies has an impact on the quality factor and the power handling of the structure. To demonstrate the advantages, we designed an example of a three-section fourth-order filter in the L-band. This was compared with equivalent two-section filters, considering electrical performances and power handling, particularly the multipactor effect. This filter was fabricated in aluminum and tested. The experimental results are presented and discussed.

An asymptotic expansion for a Lambert series associated to the symmetric square L-function

Hafner and Stopple proved a conjecture of Zagier that the inverse Mellin transform of the symmetric square [Formula: see text]-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function [Formula: see text]. Later, Chakraborty et al. extended this phenomenon for any Hecke eigenform over the full modular group. In this paper, we study an asymptotic expansion of the Lambert series [Formula: see text] where [Formula: see text] is the [Formula: see text]th Fourier coefficient of a Hecke eigenform [Formula: see text] of weight [Formula: see text] over the full modular group.

A graph-oriented approach to address generically flat outputs in structured LTI discrete-time systems

This paper addresses difference flatness for structured LTI discrete-time systems. Two forms of necessary and sufficient conditions for an output to be a structural flat output are given. First, a preliminary result algebraically defines a flat output in terms of invariant zeros regardless whether an LTI system is structured or not. Next, the conditions are expressed in terms of graphical conditions to define a structural flat output. Checking for the graphical conditions calls for algorithms that have polynomial-time complexity and that are commonly used for digraphs. The tractability of the conditions is illustrated on several examples.

Lambert series associated to Hilbert modular form

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion.

Simulation of the Charge Carriers Distribution in the Active Region of the P-I-N-Diodes by the Perturbation Theory Methods

A mathematical model of the electron-hole plasma stationary distribution in the active region (i-region) of p-i-n-diodes in the dif-fusion-drift approximation is proposed. The model is represented in the form of a nonlinear singularly perturbed boundary value prob-lem for the system of equations of the electron-hole currents conti-nuity, the Poisson equation and the corresponding boundary condi-tions. The decomposition of the nonlinear boundary value problem of modeling the stationary distribution of charge carriers in the plasma of p-i-n-diodes is carried out on the basis of the solutions asymptotic representation. The model problem is reduced to a se-quence of the linear boundary value problems with a characteristic separation of the main (regular) components of the asymptotics and a boundary corrections. It was found that the formulation of the problem for finding the zero term of the asymptotics regular part coincides with the classical formulation of the p-i-n-diodes charac-teristics modeling problem, which is carried out in the approxima-tion of the ambipolar diffusion (approximation of a self-consistent electrostatic field).The proposed mathematical model and the method of its linearization make it possible to determing the main components in the diffusion-drift process and to study their role. For example, it becomes possible to study (including by analytical methods) the behavior of plasma in the p-i-, n-i-contacts zones. The results of the study are aimed at developing methods for de-signing p-i-n-diode structures, used, in particular, as active ele-ments of the signals switches of a microwave data transmission systems and the corresponding protective devices.

An exact formula for a Lambert series associated to a cusp form and the Möbius function

In 1981, Zagier conjectured that the constant term of the automorphic form $$ y^{12}|\Delta (z)|^2$$ , that is, $$ a_0(y):=y^{12} \sum _{n=1}^{\infty } \tau ^2(n) \exp ({-4 \pi ny}), $$ where $$\tau (n)$$ is the nth Fourier coefficient of the Ramanujan cusp form $$\Delta (z)$$ , has an asymptotic expansion when $$y \rightarrow 0^{+}$$ and it can be expressed in terms of the non-trivial zeros of the Riemann zeta function $$\zeta (s)$$ . This conjecture was settled by Hafner and Stopple, and later Chakraborty, Kanemitsu, and the second author have extended this result for any Hecke eigenform over the full modular group. In this paper, we investigate a Lambert series associated to the Fourier coefficients of a cusp form and the Mobius function $$\mu (n)$$ . We present an exact formula for the Lambert series and interestingly the main term is in terms of the non-trivial zeros of $$\zeta (s)$$ , and the error term is expressed as an infinite series involving the generalized hypergeometric series $${}_2F_1(a,b;c;z)$$ . As an application of this exact form, we also establish an asymptotic expansion of the Lambert series.

Insights into the multiplicity-induced-dominancy for scalar delay-differential equations with two delays

It has been observed in recent works that, for several classes of linear time-invariant time-delay systems of retarded or neutral type with a single delay, if a root of its characteristic equation attains its maximal multiplicity, then this root is the rightmost spectral value, and hence it determines the exponential behavior of the system, a property usually referred to as multiplicity-induced-dominancy (MID). In this paper, we investigate the MID property for one of the simplest cases of systems with two delays, a scalar delay-differential equation of first order with two delayed terms of order zero. We discuss the standard approach based on the argument principle for establishing the MID property for single-delay systems and some of its limitations in the case of our simple system with two delays, before proposing a technique based on crossing imaginary roots that allows to conclude that the MID property holds in our setting.

Lee-Yang theory, high cumulants, and large-deviation statistics of the magnetization in the Ising model

We investigate the Ising model in one, two, and three dimensions using a cumulant method that allows us to determine the Lee-Yang zeros from the magnetization fluctuations in small lattices. By doing so with increasing system size, we are able to determine the convergence point of the Lee-Yang zeros in the thermodynamic limit and thereby predict the occurrence of a phase transition. The cumulant method is attractive from an experimental point of view since it uses fluctuations of measurable quantities, such as the magnetization in a spin lattice, and it can be applied to a variety of equilibrium and non-equilibrium problems. We show that the Lee-Yang zeros encode important information about the rare fluctuations of the magnetization. Specifically, by using a simple ansatz for the free energy, we express the large-deviation function of the magnetization in terms of Lee-Yang zeros. This result may hold for many systems that exhibit a first-order phase transition.

Design of tunable microstrip diplexer with reconfigurable filtering characteristics based on dual‐mode square loop resonators

A novel compact microstrip diplexer with tunable centre frequencies and reconfigurable filtering characteristics at both channels is presented. The designed diplexer is constructed by coupling two dual-mode square loop resonators (DMSLRs) having different electrical lengths. The DMSLRs are investigated by virtue of the coupled line analysis to evaluate the resonance frequencies, transmission zeros and out-band insertion loss level more conveniently. Centre frequencies of both channels can be independently tuned by the varactor diodes utilised instead of perturbation and reference capacitances. Depending on the specialised values of perturbation and reference capacitances, filtering characteristics of both channels can be reconfigured in terms of transmission zeros. The designed diplexer was fabricated and tested for the experimental studies and the measurements were obtained in a good agreement with the predicted results. Tuning ranges of the measured centre frequency in the first channel are between 1.23 and 1.43 and 1.35 and 1.64 GHz for the filtering characteristics with right- and left-side transmission zeros, respectively. In the second channel, they are 2.07–2.31 and 2.14–2.51 GHz for the filtering characteristics with real- and imaginary-axis transmission zeros, respectively. Isolation between two channels has been obtained better than 22 dB.

Charged reflecting shells supporting non-minimally coupled massless scalar field configurations

We study analytically the physical and mathematical properties of spatially regular massless scalar field configurations which are non-minimally coupled to the electromagnetic field of a spherically symmetric charged reflecting shell. In particular, the Klein–Gordon wave equation for the composed charged-reflecting-shell-nonminimally-coupled-linearized-massless-scalar-field system is solved analytically. Interestingly, we explicitly prove that the discrete resonance spectrum {R_{text {s}}(Q,alpha ,l;n)}^{n=infty }_{n=1} of charged shell radii that can support the non-minimally coupled massless scalar fields can be expressed in a remarkably compact form in terms of the characteristic zeros of the Bessel function (here Q, alpha , and l are respectively the electric charge of the central supporting shell, the dimensionless non-minimal coupling parameter of the Maxwell-scalar theory, and the angular harmonic index of the supported scalar configuration).

A Hybrid Low-Cost Bandpass Filter With SAW Resonators and External Lumped Inductors Using a Dual-Coupling Scheme

A class of hybrid low-cost bandpass filter (BPF) with only two kinds of circuit components, straightforward design procedure, and miniaturized circuit footprint is presented. The proposed BPF is based on a hybrid module composed of one type of surface-acoustic-wave (SAW) resonator and one external lumped inductor. Compared with traditional SAW-resonator-based BPF devices, the use of a single-SAW-resonator model for its design reduces the cost and complexity by avoiding the need for well-matched SAW-resonator pairs. The engineered filter topology consists of a dual-path coupling scheme, in which one coupling path is simply a series-type SAW resonator and another path is made up of two SAW resonators that are intercoupled by means of an external inductor in the middle. Such coupling architecture generates two additional transmission zeros (TZs) that allow to increase the out-of-band power-rejection levels and enlarge the stopband bandwidth. A circuit analysis based on the even-/odd-mode method is applied to derive its equivalent circuit that can be directly matched to a conventional transversal filter topology. The obtained equivalent circuit readily illustrates the operational mechanisms of the proposed filter principle in terms of reflection zeros and TZs. Subsequently, the quality factors of its even- and odd-mode-subnetwork resonators are analytically derived, and the obtained numerical results determine the tradeoff existing when choosing the external-inductor value among various performance metrics. For experimental validation purposes, two 418-MHz BPF prototypes formed by one and two in-series-cascade-connected second-order filtering units, respectively, are manufactured and tested. The circuit footprints of these filters are largely reduced by using a three-layer lamination technique for their physical implementations. The measured results show sharp-rejection filtering responses and are in close agreement with the theoretical predictions.

On the complex magnitude of Dirichlet beta function

In this article, we derive an expression for the complex magnitude of the Dirichlet beta function $\beta(s)$ represented as a Euler prime product and compare with similar results for the Riemann zeta function. We also obtain formulas for $\beta(s)$ valid for an even and odd $k$th positive integer argument and present a set of generated formulas for $\beta(k)$ up to $11$th order, including Catalan's constant and compute these formulas numerically. Additionally, we derive a second expression for the complex magnitude of $\beta(s)$ valid in the critical strip from which we obtain a formula for the Euler-Mascheroni constant expressed in terms of zeros of the Dirichlet beta function on the critical line. Finally, we investigate the asymptotic behavior of the Euler prime product on the critical line.

Temporal response of polymer waveguide beam scanner with thermo-optic phase-modulator array.

Solid-state light detection and ranging, capable of performing beam scanning without using any mechanical moving parts, requires a phase-modulator array. Polymers facilitate the fabrication of efficient phase modulators with low drive power, owing to their high thermo-optic (TO) effect and low thermal conductivity. We designed and fabricated a polymeric phase-modulator array and analyzed the temporal response of the TO phase modulator. The frequency response of the phase modulator was measured for a Mach-Zehnder interferometer (MZI), and the transfer function was modeled in terms of multiple poles and zeros. The frequency response of a fabricated beam-scanning device incorporating the TO phase modulator was also measured. The temporal response of the beam scanner was confirmed to coincide well with that of the MZI modulator. The device exhibited a fast rise time of 12 ms, accompanied by slight power variations appearing for a long period (over hundreds of seconds), which originated from the inherent viscoelastic effect of the polymer materials.

Computation of Sums of Natural Powers of the Inverses of Roots of an Equation Connected with a Spectral Problem

We consider equations arising in the oblique derivative spectral problem of the form$$ az{J}_v^{\prime }(z)+b{J}_v(z)=0,\kern1em z\in \mathbb{C}, $$ where ν, a, b ∈ ℂ are parameters such that |a|+|b| > 0 and Jν(z) is the Bessel function. For roots of the equation we prove summation relations. The results obtained agree with the theory of Rayleigh sums which are calculated in terms of zeros of the Bessel functions.

Euler's Formula-Based Stability Analysis for Wideband Harmonic Resonances

During propagation along medium- or high-voltage feeders, the currents generated by distributed generation power plants (DGPP) can exhibit harmonic tendencies to deteriorate over wide frequency ranges. These phenomena are construed as wideband harmonic resonances. Although the shapes of these harmonic resonances can be delineated by using the Π-type feeder model with hyperbolic functions, the involved sinh or cosh terms can hinder researchers from obtaining the relative stability of the systems by applying classic stability criteria. To this end, we proposed transforming the hyperbolic functions via Euler's formula for complex numbers. These transformations enable us to evaluate the relative stability of the system in terms of poles and zeros in the complex plane. We find that: first, the underlying causes of the wideband harmonic resonances correlate with the motion patterns of the eigenvalues on the complex plane; second, a recommended coordinated damper restricts the motion ranges of the eigenvalues as the system frequency varies, and this restriction guarantees the invariability of the time-domain performance of the systems. Simulation, experimental results, and a case study verify the applicability of the proposed stability-analysis method.