Simple Zeros Research Articles

Simple zeros of $\operatorname{GL}(2)$ $L$-functions

Let f \in S_{k}(\Gamma_{1}(N)) be a primitive holomorphic form of arbitrary weight k and level N . We show that the completed L -function of f has \Omega(T^{\delta}) simple zeros with imaginary part in [-T, T] , for any \delta <{2/27} . This is the first power bound in this problem for f of non-trivial level, where previously the best results were \Omega(\log\log\log{T}) for N odd, due to Booker, Milinovich, and Ng, and infinitely many simple zeros for N even, due to Booker. In addition, for f of trivial level ( N=1 ), we also improve an old result of Conrey and Ghosh on the number of simple zeros.

Chebyshev subdivision and reduction methods for solving multivariable systems of equations

We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in Rn. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. We prove the method has quadratic convergence locally near simple zeros of the system. It also finds all nonsimple zeros, but convergence to those zeros is not guaranteed to be quadratic. We also analyze the arithmetic complexity and the numerical stability of the algorithm and provide numerical evidence in dimensions up to five that the method is both fast and accurate on a wide range of problems. Our tests show that the algorithm outperforms other standard methods on the problem of finding all real zeros in a bounded domain. Our Python implementation of the algorithm is publicly available at https://github.com/tylerjarvis/RootFinding.

Chebyshev’s Method for Multiple Zeros of Analytic Functions: Convergence, Dynamics and Real-World Applications

This paper deals with the convergence and dynamics of Chebyshev’s method for simple and multiple zeros of analytic functions. We establish a local convergence theorem that provides error estimates and exact domains of initial approximations to guarantee the Q-cubic convergence of the method right from the first iteration. Applications to some real-world problems as well as theoretical and numerical comparison with the famous Halley’s method are also provided.

The uniform distribution modulo one of certain subsequences of ordinates of zeros of the zeta function

AbstractOn the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $1/2+i\gamma$ of the Riemann zeta function, we show that the sequence \begin{equation*}\Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad \frac{ \log\big(| \zeta^{(m_{\gamma })} (\frac12+ i{\gamma }) | / (\!\log{{\gamma }} )^{m_{\gamma }}\big)}{\sqrt{\frac12\log\log {\gamma }}} \in [a, b] \Bigg\},\end{equation*} where the ${\gamma }$ are arranged in increasing order, is uniformly distributed modulo one. Here a and b are real numbers with $a<b$ , and $m_\gamma$ denotes the multiplicity of the zero $1/2+i{\gamma }$ . The same result holds when the ${\gamma }$ ’s are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers $\gamma (\!\log T)/2\pi$ with ${\gamma }\in \Gamma_{[a, b]}$ and $0<{\gamma }\leq T$ .

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.

A New Ehrlich-Type Sixth-Order Simultaneous Method for Polynomial Complex Zeros

This paper presents a new iterative method for the simultaneous determination of simple polynomial zeros. The proposed method is obtained from the combination of the third-order Ehrlich iteration with an iterative correction derived from Li's fourth-order method for solving nonlinear equations. The combined method developed has order of convergence six. Some examples are presented to illustrate the convergence and efficiency of the proposed Ehrlich-type method with Li correction for the simultaneous approximation of polynomial zeros.

The magic renormalisability of affine Gaudin models

We study the renormalisation of a large class of integrable σ-models obtained in the framework of affine Gaudin models. They are characterised by a simple Lie algebra g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{g} $$\\end{document} and a rational twist function φ(z) with simple zeros, a double pole at infinity but otherwise no further restrictions on the pole structure. The crucial tool used in our analysis is the interpretation of these integrable theories as E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{E} $$\\end{document}-models, which are σ-models studied in the context of Poisson-Lie T-duality and which are known to be at least one- and two-loop renormalisable. The moduli space of E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{E} $$\\end{document}-models still contains many non-integrable theories. We identify the submanifold formed by affine Gaudin models and relate its tangent space to curious matrices and semi-magic squares. In particular, these results provide a criteria for the stability of these integrable models under the RG-flow. At one loop, we show that this criteria is satisfied and derive a very simple expression for the RG-flow of the twist function, proving a conjecture made earlier in the literature.

A Special Interval Newton Step for Solving Nonlinear Equations

The problem of finding the zeros of a nonlinear equation has been extensively and thoroughly studied. Point methods constitute a significant and extensive category of techniques, allowing for the efficient finding of zeros with arbitrary precision under specific conditions. However, the limitation of these methods is that they typically yield a single zero. An alternative approach employs Interval Analysis, leveraging its properties to provide reliable and with certainty inclusions of all zeros within a given search interval. Interval methods, such as the Interval Newton method, exhibit quadratic convergence to the corresponding inclusions when monotonicity and simple zeros exist. Nonetheless, there exist pathological cases, like the existence of multiple zeros, where the obtained inclusions cannot be bounded with arbitrary precision, necessitating the adoption of bisection schemes to refine the search interval. These schemes not only increase computational time and cost but also result in a higher number of enclosures, enclosing sometimes the same zero more than once. The main objective of this work is to enhance the applicability of Interval Newton method in cases where no efficient alternative are available. Thus, in this paper, the Interval Newton method is studied and an adjusted perturbation technique is proposed to address the cases where multiple zeros exist. In particular, the given function is vertically shifted. Then, the Interval Newton operator is applied once to this shifted function. The resulting enclosures are then used to efficiently partition the search interval. The successful application of the Interval Newton method is expected to improve overall performance and reduce reliance on bisection schemes. Experimental results on a set of problems demonstrate the effectiveness of the proposed technique.

A new family of Sakurai–Torii–Sugiura type iterative methods with high order of convergence

In this paper, we introduce a new family of iterative methods for finding simultaneously all zeros (multiple or simple) of a polynomial. The proposed family is constructed by combining the known Sakurai–Torii–Sugiura iteration function with an arbitrary iteration function. We provide a detailed convergence analysis in the following two directions: local convergence if the polynomial has multiple zeros with known multiplicity and semilocal convergence if the polynomial has only simple zeros. As an application, we study the convergence of several particular iterative methods with high order of convergence.

Masur–Veech volumes and intersection theory: The principal strata of quadratic differentials

We describe a conjectural formula via intersection numbers for the Masur–Veech volumes of strata of quadratic differentials with prescribed zero orders, and we prove the formula for the case when the zero orders are odd. For the principal strata of quadratic differentials with simple zeros, the formula reduces to computing the top Segre class of the quadratic Hodge bundle, which can be further simplified to certain linear Hodge integrals. An appendix proves that the intersection of this class with ψ-classes can be computed by Eynard–Orantin topological recursion. As applications, we analyze numerical properties of Masur–Veech volumes, area Siegel–Veech constants, and sums of Lyapunov exponents of the principal strata for fixed genus and varying number of zeros, which settles the corresponding conjectures due to Grivaux and Hubert, Fougeron, and elaborated in Andersen et al. We also describe conjectural formulas for area Siegel–Veech constants and sums of Lyapunov exponents for arbitrary affine invariant submanifolds, and verify them for the principal strata.

On $$SL(2,\mathbb{R})$$-Cocycles over Irrational Rotations with Secondary Collisions

We consider a skew product $F_{A} = (\sigma_{\omega}, A)$ over irrational rotation $\sigma_{\omega}(x) = x + \omega$ of a circle $\mathbb{T}^{1}$. It is supposed that the transformation $A: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ being a $C^{1}$-map has the form $A(x) = R(\varphi(x)) Z(\lambda(x))$, where $R(\varphi)$ is a rotation in $\mathbb{R}^{2}$ over the angle $\varphi$ and $Z(\lambda)= diag\{\lambda, \lambda^{-1}\}$ is a diagonal matrix. Assuming that $\lambda(x) \ge \lambda_{0} > 1$ with a sufficiently large constant $\lambda_{0}$ and the function $\varphi$ be such that $\cos \varphi(x)$ possesses only simple zeroes, we study hyperbolic properties of the cocycle generated by $F_{A}$. We apply the critical set method to show that, under some additional requirements on the derivative of the function $\varphi$, the secondary collisions compensate weakening of the hyperbolicity due to primary collisions and the cocycle generated by $F_{A}$ becomes hyperbolic in contrary to the case when secondary collisions can be partially eliminated.

High-order soliton solutions and their dynamics in the inhomogeneous variable coefficients Hirota equation

A series of new soliton solutions is presented for the inhomogeneous variable coefficient Hirota equation by using the Riemann–Hilbert method and transformation relationship. Firstly, through a standard dressing procedure, the N-soliton matrix associated with the simple zeros in the Riemann–Hilbert problem for the Hirota equation is constructed. Then the N-soliton matrix of the inhomogeneous variable coefficient Hirota equation can be obtained by a special relationship transformation from the N-soliton matrix of the Hirota equation. Next, using the generalized Darboux transformation, the high-order soliton solutions corresponding to the elementary high-order zeros in the Riemann–Hilbert problem for the Hirota equation can be derived. Similarly, employing the relationship transformation mentioned above can lead to the high-order soliton solutions of the inhomogeneous variable coefficient Hirota equation. In addition, the collision dynamics of Hirota and inhomogeneous variable coefficient Hirota equations are analyzed; the asymptotic behaviors for multi-solitons and long-term asymptotic estimates for the high-order one-soliton of the Hirota equation are concretely calculated. Most notably, by analyzing the dynamics of the multi-solitons and high-order solitons of the inhomogeneous variable coefficient Hirota equation, we discover numerous new waveforms such as heart-shaped periodic wave solutions, O-shaped periodic wave solutions etc. that have never been reported before, which are crucial in theory and practice.

A sufficient condition for a complex polynomial to have only simple zeros and an analog of Hutchinson's theorem for real polynomials

A sufficient condition for a complex polynomial to have only simple zeros and an analog of Hutchinson's theorem for real polynomials

Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

For a given polynomial P with simple zeros, and a given semiclassical weight w, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of P. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of P. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of P with respect to w allows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasi-orthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite–Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems. We also discuss the discrete-to-continuous transition of these models in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.

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.

General multi-soliton and higher-order soliton solutions for a novel nonlocal Lakshmanan–Porsezian–Daniel equation

The inverse scattering transformation for a novel nonlocal Lakshmanan–Porsezian–Daniel (LPD) equation with rapidly decaying initial data is studied in the framework of Riemann–Hilbert problem. Firstly, a novel integrable nonlocal LPD equation corresponding to a \(3\times 3\) Lax pair is proposed. Secondly, the inverse scattering process with a novel left-right \(3\times 3\) matrix Riemann–Hilbert(RH) problem is constructed. The analytical properties and symmetry relations for the Jost functions and scattering data are considerably different from the local ones. Due to the special symmetry properties for the nonlocal LPD equation, the zeros of the RHP problem are purely imaginary or occur in pairs. With different types and configuration of zeros, the soliton formula is provided and the rich dynamical behaviors for the three kinds of multi-solitons for the novel nonlocal LPD equation are demonstrated. Third, by a technique of adding perturbed parameters and limiting process, the formula of higher-order solitons for the nonlocal LPD equation is exhibited. Lastly, the plots of diverse higher-order solitons and various solutions corresponding to different combinations of the following zeros: purely imaginary higher-order zeros, purely imaginary simple zeros, pairs of non-purely imaginary simple zeros and pairs of non-purely imaginary higher-order zeros are displayed.

Tight relative t-designs on two shells in hypercubes, and Hahn and Hermite polynomials

Relative $t$-designs in the $n$-dimensional hypercube $\mathcal{Q}_n$ are equivalent to weighted regular $t$-wise balanced designs, which generalize combinatorial $t$-$(n,k,\lambda)$ designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean $t$-designs on two concentric spheres, in this paper we discuss tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells. We show under a mild condition that such a relative $t$-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells are rare for large $t$.

A general approach to the study of the convergence of Picard iteration with an application to Halley's method for multiple zeros of analytic functions

In this paper, we define a new wide class of iteration functions and then we use it to prove a general convergence theorem that provides exact domain of initial approximations to guarantee the high Q-order of convergence of Picard iteration generated by this class of functions. As an application of this theorem, we prove some local convergence theorems about the famous Halley's method for simple and multiple zeros of analytic functions. All obtained results are supplied with a priori and a posteriori error estimates as well as with assessments of the asymptotic error constants.

A Family of Derivative Free Algorithms for Multiple-Roots of Van Der Waals Problem

There are a good number of higher-order iterative methods for computing multiple zeros of nonlinear equations in the available literature. Most of them required first or higher-order derivatives of the involved function. No doubt, high-order derivative-free methods for multiple zeros are more difficult to obtain in comparison with simple zeros and with first order derivatives. This study presents an optimal family of fourth order derivative-free techniques for multiple zeros that requires just three evaluations of function ϕ, per iteration. The approximations of the derivative/s are based on symmetric divided differences. We also demonstrate the application of new algorithms on Van der Waals, Planck law radiation, Manning for isentropic supersonic flow and complex root problems. Numerical results reveal that the proposed derivative-free techniques are more efficient in comparison terms of CPU, residual error, computational order of convergence, number of iterations and the difference between two consecutive iterations with other existing methods.

Improved two-step Newton’s method for computing simple multiple zeros of polynomial systems

Improved two-step Newton’s method for computing simple multiple zeros of polynomial systems