Double Zeros Research Articles

Inverse scattering and soliton dynamics for the mixed Chen–Lee–Liu derivative nonlinear Schrödinger equation

Under investigation in this letter is a mixed Chen–Lee–Liu derivative nonlinear Schrödinger equation which can be considered as the simplest model to approximate the dynamics of weakly nonlinear and dispersive waves, taking into account the self-steepening effect. The inverse scattering transform under the zero boundary conditions and analytical scattering coefficients with an arbitrary number of simple and double zeros is detailedly discussed. Particularly, the inverse problem is solved by the study of a matrix Riemann–Hilbert problem. As a consequence, we present the general solution for the potential, and explicit expression for the reflectionless potential.

Goodness-of-fit testing for meta-analysis of rare binary events

Random-effects (RE) meta-analysis is a crucial approach for combining results from multiple independent studies that exhibit heterogeneity. Recently, two frequentist goodness-of-fit (GOF) tests were proposed to assess the fit of RE model. However, they tend to perform poorly when assessing rare binary events. Under a general binomial-normal framework, we propose a novel GOF test for the meta-analysis of rare events. Our method is based on pivotal quantities that play an important role in Bayesian model assessment. It further adopts the Cauchy combination idea proposed in a 2019 JASA paper, to combine dependent p-values computed using posterior samples from Markov Chain Monte Carlo. The advantages of our method include clear conception and interpretation, incorporation of all data including double zeros without the need for artificial correction, well-controlled Type I error, and generally improved ability in detecting model misfits compared to previous GOF methods. We illustrate the proposed method via simulation and three real data applications.

Double and triple-pole solutions for the third-order flow equation of the Kaup-Newell system with zero/nonzero boundary conditions

In this work, the double and triple-pole solutions for the third-order flow equation of Kaup-Newell system (TOFKN) with zero boundary conditions (ZBCs) and non-zero boundary conditions (NZBCs) are investigated by means of the Riemann-Hilbert (RH) approach stemming from the inverse scattering transformation. Starting from spectral problem of the TOFKN, the analyticity, symmetries, asymptotic behavior of the Jost function and scattering matrix, the matrix RH problem with ZBCs and NZBCs are constructed. Then the obtained RH problem with ZBCs and NZBCs can be solved in the case of scattering coefficients with double or triple zeros, and the reconstruction formula of potential, trace formula as well as theta condition are also derived correspondingly. Specifically, the general formulas of N-double and N-triple poles solutions with ZBCs and NZBCs are derived systematically by means of determinants. The vivid plots and dynamics analyses for double and triple-pole soliton solutions with the ZBCs as well as double and triple-pole interaction solutions with the NZBCs are exhibited in details. Compared with the most classical second-order flow Kaup-Newell system, we find the third-order dispersion and quintic nonlinear term of the Kaup-Newell system change the trajectory and velocity of solutions. Furthermore, the asymptotic states of the 1-double poles soliton solution and the 1-triple poles soliton solution are analyzed when t tends to infinity.

Design of Modified Jacobi Microstrip Lowpass Filter for L-Band Application

In this brief, a design of microstrip lowpass filter with the 1 GHz cutoff frequency is revisited, where a class of modified Jacobi polynomials is used for the filter design. The optimization procedure for these filters includes the passband return loss minimization and the roll-off factor enhancement using the magnitude correction of the transmission coefficient. Mentioned correction is performed by introducing the multiple transmission zeros such that LC ladder prototype filter can be realized symmetrically. Three seventh degree filters for L-band applications are realized and simulated: allpole, with a single and double transmission zeros. It is shown that, among these three stepped impedance lowpass filters realizations, the filter with simple transmission zeros offers the best performance.

Riemann-Hilbert approach for the Kundu equation with non-vanishing boundary conditions: Simple and double poles

The inverse scattering transform for the Kundu equation with non-vanishing boundary conditions at infinity is studied via the Riemann-Hilbert approach. A detailed spectral analysis for analyticity, symmetry properties, and asymptotic behavior of the modified Jost eigenfunctions and scattering matrix is discussed. And an appropriate matrix Riemann-Hilbert problem is formulated. A closed system is solved for the explicit N-soliton solutions to the Kundu equation in the case of the reflectionless potential. Besides, the trace formula for the analytic scattering coefficients and the so-called theta condition for the phase difference between the boundary values are derived. A consistent framework is constructed to obtain solutions associated with double zeros to the analytic scattering coefficients. The solitons in the simple and double poles case of the Chen-Lee-Liu equation as a reduced example are studied, including any combination of dark, bright and breather solitons, and the dynamical properties of these solutions are displayed and analyzed.

Riemann–Hilbert problem and dynamics of soliton solutions of the fifth-order nonlinear Schrödinger equation

The theory of inverse scattering is developed to investigate the initial-value problem for the fifth-order nonlinear Schrödinger (foNLS) equation under the zero boundary conditions at infinity. The spectral analysis is performed in the direct scattering process, including the establishment of the analytical, asymptotic and symmetric properties of the scattering matrix and the Jost functions. In the inverse scattering process, a suitable Riemann–Hilbert (RH) problem is successfully established by using the modified eigenfunctions and scattering data, and the relationship between the potential function and the solution of the RH problem is successfully established. In order to further analyze the propagation behavior of the solutions of the foNLS equation, we present some new phenomena of studying three-soliton solutions corresponding to simple zeros and two-soliton solutions corresponding to double zeros in scattered data.

A new form of general soliton solutions and multiple zeros solutions for a higher-order Kaup–Newell equation

Due to the fact that the higher-order Kaup–Newell (KN) system has more complex and diverse solutions than the classical second-order flow KN system, the research on it has attracted much attention. In this paper, we consider a higher-order KN equation with third-order dispersion and fifth-order nonlinearity. Based on the theory of the inverse scattering, the matrix Riemann–Hilbert problem is established. Through the dressing method, the solution matrix with simple zeros without reflection is constructed. In particular, a new form of solution is given which is more direct and simpler than previous methods. In addition, through the determinant solution matrix, the vivid diagrams and dynamic analysis of single-soliton solution and two-soliton solution are given in detail. Finally, by using the technique of limit, we construct the general solution matrix in the case of multiple zeros, and the examples of solutions for the cases of double zeros, triple zeros, single–double zeros, and double–double zeros are especially shown.

Timing-Optimized Hardware Implementation to Accelerate Polynomial Multiplication in the NTRU Algorithm

Post-quantum cryptographic algorithms have emerged to secure communication channels between electronic devices faced with the advent of quantum computers. The performance of post-quantum cryptographic algorithms on embedded systems has to be evaluated to achieve a good trade-off between required resources (area) and timing. This work presents two optimized implementations to speed up the NTRUEncrypt algorithm on a system-on-chip. The strategy is based on accelerating the most time-consuming operation that is the truncated polynomial multiplication. Hardware dedicated modules for multiplication are designed by exploiting the presence of consecutive zeros in the coefficients of the blinding polynomial. The results are validated on a PYNQ-Z2 platform that includes a Zynq-7000 SoC from Xilinx and supports a Python-based programming environment. The optimized version that exploits the presence of double, triple, and quadruple consecutive zeros offers the best performance in timing, in addition to considerably reducing the possibility of an information leakage against an eventual attack on the device, making it practically negligible.

Dispersive CFT sum rules

We give a unified treatment of dispersive sum rules for four-point correlators in conformal field theory. We call a sum rule “dispersive” if it has double zeros at all double-twist operators above a fixed twist gap. Dispersive sum rules have their conceptual origin in Lorentzian kinematics and absorptive physics (the notion of double discontinuity). They have been discussed using three seemingly different methods: analytic functionals dual to double-twist operators, dispersion relations in position space, and dispersion relations in Mellin space. We show that these three approaches can be mapped into one another and lead to completely equivalent sum rules. A central idea of our discussion is a fully nonperturbative expansion of the correlator as a sum over Polyakov-Regge blocks. Unlike the usual OPE sum, the Polyakov-Regge expansion utilizes the data of two separate channels, while having (term by term) good Regge behavior in the third channel. We construct sum rules which are non-negative above the double-twist gap; they have the physical interpretation of a subtracted version of “superconvergence” sum rules. We expect dispersive sum rules to be a very useful tool to study expansions around mean-field theory, and to constrain the low-energy description of holographic CFTs with a large gap. We give examples of the first kind of applications, notably we exhibit a candidate extremal functional for the spin-two gap problem.

The Riemann-Hilbert approach for the focusing Hirota equation with single and double poles

We study the Cauchy problem of the focusing Hirota equation with fastly decay initial value at infinity via Riemann-Hilbert (RH) approach. The Jost function and scattering matrix related to the initial value are obtained by spectral analysis, and their analytical, asymptotic and symmetric properties are analyzed. The solution of the Hirota equation is derived by solving the corresponding RH problem, which contains two cases of scattering data with single zeros and double zeros. The general form of solutions with no reflection potential is obtained. The two types of the discrete spectrum are selected for the purpose of comparing the influence of different eigenvalues on solution propagation behavior.

Application of Floquet theory to formation flying in elliptic orbits via a Hamiltonian structure-preserving control

In this study, an improved Hamiltonian structure-preserving controller is applied for formation flying in elliptic orbits. The Lyapunov-Floquet theory is applied to the Tschauner-Hempel (TH) equation in order to generate its transformed form with two zero eigenvalues and two pairs of imaginary eigenvalues. Then, an improved Hamiltonian structure-preserving (HSP) control is introduced to the transformed system to eliminate the instability of double zeros. All bounded trajectories can be generated by basic solutions, which effectively reveal the following performances of the controller, such as the relationship between the frequencies of basic motions and control gains are detected and the wide selection of gains for stability. Local and global optimizations on gains are discussed based on different performances of the controller. One focuses on the sensitivity of output with respect to input of controller and the other aims at a minimal fuel consumption for the maximum likelihood. The ergodic relationship between the fuel consumption and initial condition is established for the minimal consumption. To avoid in-orbit collision, a collision-free strategy based on the artificial potential function is proposed to enhance the proposed controller, which make sure a minimum safety distance for the formation.

On the Zeros of Euler Product Dirichlet Functions

We study a class of Dirichlet functions obtained as analytic continuation across the line of convergence of Dirichlet series which can be written as Euler products. This class includes that of Dirichlet L-functions. The problem of the existence of multiple zeros for this last class is outstanding. It is tacitly accepted, yet not proved that the Riemann Zeta function, which belongs to this class, does not possess multiple zeros. In a previous study we provided an example of Dirichlet function having double zeros, but that function is not an Euler product function. In this paper we deal with Euler product functions and by using the geometric properties of the mapping realized by these functions, we tackle the problem of the multiplicity of their zeros.

Passive Controller Realization of a Biquadratic Impedance With Double Poles and Zeros as a Seven-Element Series–Parallel Network for Effective Mechanical Control

This work is concerned with the passive realizability problem for a biquadratic impedance with double poles and zeros as a seven-element series–parallel $RLC$ (damper-spring-inerter) network, which is a series or a parallel connection of two networks: a three-element series–parallel network and a four-element series–parallel network. First, a necessary and sufficient condition is obtained for the realizability of a biquadratic impedance with double poles and zeros as a series–parallel network containing no more than six elements (see Theorem 1 ). Then, the main theorem is presented (see Theorem 2 ), which is derived for three-reactive-element, four-reactive-element, and five-reactive-element settings, respectively (see Lemmas 2 – 4 ). The results can be directly utilized to physically construct a passive mechanism containing dampers, springs, and inerters, which can be regarded as a passive controller for some mechanical control systems.

A Separation in Modulus Property of the Zeros of a Partial Theta Function

We consider the partial theta function $$\theta (q,z): = \sum\nolimits_{j = 0} \infty {{q {j(j + 1)/2}}{z j}} $$ , where z ∈ ℂ is a variable and q ∈ ℂ, 0 < |q| < 1, is a parameter. Set $$\alpha 0: = \sqrt 3 /2\pi = 0.2756644477....$$ We show that, for n ≥ 5, for |q| ≤ 1 − 1/(α0n) and for k ≥ n there exists a unique zero ξk of θ(q,.) satisfying the inequalities |q|−k+1/2 < |ξk| < |q|−k−1/2; all these zeros are simple ones. The moduli of the remaining n−1 zeros are ≤ |q|−n+1/2. A spectral value of q is a value for which θ(q,.) has a multiple zero. We prove the existence of the spectral values 0.4353184958... ± i 0.1230440086... for which θ has double zeros −5.963... ± i 6.104....

The Geometry of the Mappings by General Dirichlet Series

We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other information which has been waiting for a rigorous proof. Such a task is partially fulfilled in this paper, where we succeeded among other things, to prove a theorem about general Dirichlet series having as corollary the Speiser’s theorem. We have also proved that those functions do not possess multiple zeros of order higher than 2 and the double zeros have very particular locations. Moreover, their derivatives have only simple zeros. With these results at hand, we revisited GRH for a simplified proof.

Borg's Periodicity Theorems for First-Order Self-Adjoint Systems with Complex Potentials

AbstractA self-adjoint first-order system with Hermitian π-periodic potential Q(z), integrable on compact sets, is considered. It is shown that all zeros of are double zeros if and only if this self-adjoint system is unitarily equivalent to one in which Q(z) is π/2-periodic. Furthermore, the zeros of are all double zeros if and only if the associated self-adjoint system is unitarily equivalent to one in which Q(z) = σ2Q(z)σ2. Here, Δ denotes the discriminant of the system and σ0, σ2 are Pauli matrices. Finally, it is shown that all instability intervals vanish if and only if Q = rσ0 + qσ2, for some real-valued π-periodic functions r and q integrable on compact sets.

Sampling and Reconstruction in Shift Invariant Spaces of B $B$ -Spline Functions

A Kadec-type theorem is proved for functions belonging to the shift invariant space V(Qr)$V(Q_{r})$, where Qr$Q_{r}$ denotes the B$B$-spline function of even order r$r$. It is also shown that if a non-zero function fźV(Qm)$f\in V(Q_{m})$ has infinitely many simple or double zeros on the real axis which are separated, then there exists at least one pair of consecutive zeros whose distance apart is greater than or equal to a certain number which depends on Krein---Favard constants. Further, iterative reconstruction algorithms are provided for functions in V(Qm)$V(Q_{m})$. Finally, a sampling density theorem for irregular Gabor-type frames {MyjTxi,jg(l):l=0,1,ź,kź1,i,jźZ}$\{M_{y_{j}}T_{x_{i,j}}g^{(l)}:l=0, 1, \dots , {k-1}, i, j \in \mathbb{Z} \}$ is proved for gźV(Qm)źL1(R)$g\in V(Q_{m})\cap L^{1}( \mathbb{R})$ under certain sufficient conditions on g$g$.

Inverse Scattering Transform for the Defocusing Manakov System with Nonzero Boundary Conditions

The inverse scattering transform for the defocusing Manakov system with nonzero boundary conditions at infinity is rigorously studied. Several new results are obtained: (i) The analyticity of the Jost eigenfunctions is investigated, and precise conditions on the potential that guarantee such analyticity are provided. (ii) The analyticity of the scattering coefficients is established. (iii) The behavior of the eigenfunctions and scattering coefficients at the branch points is discussed. (iv) New symmetries are derived for the analytic eigenfunctions (which differ from those in the scalar case). (v) These symmetries are used to obtain a rigorous characterization of the discrete spectrum and to rigorously derive the symmetries of the associated norming constants. (vi) The asymptotic behavior of the Jost eigenfunctions is derived systematically. (vii) A general formulation of the inverse scattering problem as a Riemann--Hilbert problem is presented. (viii) Precise results guaranteeing the existence and uniqueness of solutions of the Riemann--Hilbert problem are provided. (ix) Explicit relations among all reflection coefficients are given, and all entries of the scattering matrix are determined in the case of reflectionless solutions. (x) A compact, closed-form expression is presented for general soliton solutions, including any combination of dark-dark and dark-bright solitons. (xi) A consistent framework is formulated for obtaining solutions corresponding to double zeros of the analytic scattering coefficients, leading to double poles in the Riemann--Hilbert problem, and such solutions are constructed explicitly.

A Double Zeros Compensated Direct Fast Feedback Current Driver for Medium to Large AMOLED Displays

A current feedback driver with double zeros compensation is proposed for medium to large AMOLED displays. The zeros for lead compensation are implemented with switched-capacitor circuits to reduce chip area. The selection rules for compensation capacitors and zero capacitors are described to obtain wide bandwidth and high speed. The proposed driver has a settling time of 7 μs for the panel load of 10 kΩ/100 pF. De-multiplexing operation is realized to drive two channels using one proposed driver. The chip area is 120 × 183 μm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> per two channels in a 0.35 μm CMOS process.

Algebras Generated by Two Quadratic Elements

Let K be a field of any characteristic and let R be an algebra generated by two elements satisfying quadratic equations. Then R is a homomorphic image of F = K ⟨x, y | x 2 + ax + b = 0, y 2 + cy + d = 0⟩ for suitable a, b, c, d ∈ K. We establish that F can be embedded into the 2 × 2 matrix algebra with entries from the polynomial algebra over the algebraic closure of K and that F and satisfy the same polynomial identities as K-algebras. When the quadratic equations have double zeros, our result is a partial case of more general results by Ufnarovskij, Borisenko, and Belov from the 1980s. When each of the equations has different zeros, we improve a result of Weiss, also from the 1980s.