Zeros Of Function Research Articles

HIGHER CORRELATIONS AND THE ALTERNATIVE HYPOTHESIS

AbstractThe Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functions of the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in $\tfrac{1}{2}\mathbb{Z}$, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known about zeros of the zeta function.

A sampling theorem by perturbing the zeros of a sine-type function

ABSTRACT In this paper, we present a generalisation of the classical Shannon sampling theorem that allows for sampling sets that are perturbations of the set of zeros of a sine-type function. Such sampling sets may be non-equidistant and non-periodic.

Optimal eighth-order iterative methods for approximating multiple zeros of nonlinear functions

It is well known that the optimal iterative methods are of more significance than the non-optimal ones in view of their efficiency and convergence speed. There are only a few number of optimal iterative methods for finding multiple zeros with eighth order of convergence. In this paper, we propose a new family of optimal eighth order convergent iterative methods for multiple roots with known multiplicity. We present an extensive convergence analysis which confirms theoretically eighth-order convergence of the presented scheme. Moreover, we consider several real life problems that contain simple as well as multiple zeros in order to compare our proposed methods with the existing eighth-order iterative schemes. Some dynamical aspects of the presented methods are also discussed. Finally, we conclude on the basis of obtained numerical results that the proposed family of iterative methods perform better than the existing methods in terms of residual error, computational order of convergence and difference between the two consecutive iterations.

Quantum mechanical analogue of optical microdisk resonators

Abstract The complex wave number plane furnishes us with a natural way to seek resonant modes of quantum potentials and optical cavities. In this work we present the solution of the quantum mechanical analogue of an optical microdisk resonator. The rich dynamical behavior of the states is shown for a circular potential barrier in response to continuous variations of the parameters of the potential. In particular, the flows of the resonances go to limit points which are zeros of Hankel functions in complex wave number coordinates and zeros of Bessel functions in an alternative complex potentiodynamic coordinate system.

Polynomial-based method for determining coast-terminating zero of fuel-optimal time-fixed trajectory

This paper presents a fail-safe polynomial-based method for determining the coast-terminating zero (CTZ) of fuel-optimal, time-fixed, low-thrust trajectories. The CTZ of the fuel-optimal bang-bang control is decided by the zeros of the switching function (SF). The state and costate differential equations along the coast arcs of the optimal trajectory have closed-form solutions; thus efficiently determining the CTZ is important. Existing methods suffer potential failures or are limited to time-free problems. The polynomial-based method consists of three steps. First, the SF is simplified by a coordinate transformation that eliminates the three orientation-related orbital elements. Then, the SF is rewritten as a combination of power and trigonometric functions of the argument eccentric anomaly, and the coefficients in the SF are simply evaluated using the states and costates at the periapsis. Finally, the SF is described as a polynomial of the eccentric anomaly by replacing the trigonometric functions with polynomials. The roots of the polynomial can provide an accurate estimation of the CTZ. Based on a Monte Carlo simulation, SF curves whose greatest possible numbers of zeros are $1,2,\ldots,7$ are listed. For each curve, the CTZ is solved by both the presented polynomial-based method and the existing sampling-searching method. The performance comparison demonstrates that the polynomial-based method provides superior reliability and efficiency.

THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE

For each$t\in \mathbb{R}$, we define the entire function$$\begin{eqnarray}H_{t}(z):=\int _{0}^{\infty }e^{tu^{2}}\unicode[STIX]{x1D6F7}(u)\cos (zu)\,du,\end{eqnarray}$$where$\unicode[STIX]{x1D6F7}$is the super-exponentially decaying function$$\begin{eqnarray}\unicode[STIX]{x1D6F7}(u):=\mathop{\sum }_{n=1}^{\infty }(2\unicode[STIX]{x1D70B}^{2}n^{4}e^{9u}-3\unicode[STIX]{x1D70B}n^{2}e^{5u})\exp (-\unicode[STIX]{x1D70B}n^{2}e^{4u}).\end{eqnarray}$$Newman showed that there exists a finite constant$\unicode[STIX]{x1D6EC}$(thede Bruijn–Newman constant) such that the zeros of$H_{t}$are all real precisely when$t\geqslant \unicode[STIX]{x1D6EC}$. The Riemann hypothesis is equivalent to the assertion$\unicode[STIX]{x1D6EC}\leqslant 0$, and Newman conjectured the complementary bound$\unicode[STIX]{x1D6EC}\geqslant 0$. In this paper, we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that$\unicode[STIX]{x1D6EC}<0$and then analyzing the dynamics of zeros of$H_{t}$(building on the work of Csordas, Smith and Varga) to obtain increasingly strong control on the zeros of$H_{t}$in the range$\unicode[STIX]{x1D6EC}<t\leqslant 0$, until one establishes that the zeros of$H_{0}$are in local equilibrium, in the sense that they locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeros of the Riemann zeta function, such as the pair correlation estimates of Montgomery.

Mean, mode or median? The score mean

A continuous random variable can be uniquely represented by the score function of distribution, i.e. a scalar-valued function describing the influence of a data item on the typical value of the distribution. There are at most two relevant scalar-valued scores based on two basic transformations: the “natural” one, useful for estimating parameters of parametric distributions by the generalized moment method; and the “universal” one, which is a basis for the score function of distribution. The score mean, the zero of the score function of distribution, represents the typical value of the distribution in the sense of the value which is most likely to be sampled. Both scores are often identical to each other. The derivative of the score function of distribution is the weight function of this distribution. The main result is that sums of score random variables obey the central limit theorem and can be used for directly estimating the score mean without use of any current estimator.

Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

Abstract We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural “asymptotic gcd” inequality of Pasten and the second author, and consider moving targets versions of our results.

On the curves of zeros of Lommel functions of two variables

ABSTRACTIn this paper, we study some properties of the locus of zeros of the Lommel function of two variables . Furthermore, we find the regions on which the curves of zeros of lie, when ν is large.

On lineability of additive surjective functions

We prove that the class of additive perfectly everywhere surjective functions contains (with the exception of the zero function) a vector space of maximal possible dimension ($2^\cont$). Additionally, we show under the assumption of regularity of $\cont$ that the family of additive everywhere surjective functions that are not strongly everywhere surjective contains (with the exception of the zero function) a vector space of dimension $\cont^+$.

Correction to: New asymptotic formulas for sums over zeros of functions from the Selberg class, Ann. Sci. Math. Québec 34 (2010), no 1, 85–108

In this note we make corrections to the above mentioned paper. First, we correct the statement of the main theorem (Theorem 1.1).

Existence and Uniqueness of Zeros for Vector-Valued Functions with K-Adjustability Convexity and Their Applications

In this paper, we introduce the new concepts of K-adjustability convexity and strictly K-adjustability convexity which respectively generalize and extend the concepts of K-convexity and strictly K-convexity. We establish some new existence and uniqueness theorems of zeros for vector-valued functions with K-adjustability convexity. As their applications, we obtain existence theorems for the minimization problem and fixed point problem which are original and quite different from the known results in the existing literature.

Twin prime correlations from the pair correlation of Riemann zeros

We establish, via a heuristic Fourier inversion calculation, that the Hardy–Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height E on the critical line. Previously it was known that the Hardy–Littlewood conjecture implies the pair correlation formula, and we show that the reverse implication also holds. An averaged form of the Hardy–Littlewood conjecture is obtained by inverting the limit of the two-point correlation function and the precise form of the conjecture is found by including asymptotically lower order terms in the two-point correlation function formula.

On Algebro-Geometric Simply-Periodic Solutions of the KdV Hierarchy

In this paper, we show that as $$\tau \rightarrow \sqrt{-1}\infty $$, any zero of the Lame function converges to either $$\infty $$ or a finite point p satisfying $${\text {Re}}p=\frac{1}{2}$$ and $$e^{2\pi i p}$$ being an algebraic number. Our proof is based on studying a special family of simply-periodic KdV potentials with period 1, i.e. algebro-geometric simply-periodic solutions of the KdV hierarchy. We show that except the pole 0, all other poles of such KdV potentials locate on the line $${\text {Re}}z=\frac{1}{2}$$. We also compute explicitly the eigenvalue set of the corresponding $$L^2[0,1]$$ eigenvalue problem for such KdV potentials, thus extends Takemura’s works (Commun Math Phys 235:467–494, 2003) and (Electron J Differ Equ 2004(15):1–30, 2004). Our main idea is to apply the classification result for simply-periodic KdV potentials by Gesztesy et al. (Trans Am Math Soc 358:603–656, 2006) and the Darboux transformation.

The bifurcation diagram of an elliptic Kirchhoff-type equation with respect to the stiffness of the material

We study a superlinear and subcritical Kirchhoff type equation which is variational and depends upon a real parameter $\lambda$. The nonlocal term forces some of the fiber maps associated with the energy functional to have two critical points. This suggest multiplicity of solutions and indeed we show the existence of a local minimum and a mountain pass type solution. We characterize the first parameter $\lambda_0^*$ for which the local minimum has non-negative energy when $\lambda\ge \lambda_0^*$. Moreover we characterize the extremal parameter $\lambda^*$ for which if $\lambda>\lambda^*$, then the only solution to the Kirchhoff equation is the zero function. In fact, $\lambda^*$ can be characterized in terms of the best constant of Sobolev embeddings. We also study the asymptotic behavior of the solutions when $\lambda\downarrow 0$.

A data method using the inner temperature difference to improve the stability of the inverse heat conduction problems

This article proposes a method to construct two series systems for improving the stability of the inverse heat conduction problems (IHCP) in a finite slab. The transfer function between the surface heat flux or temperature and the inner temperature difference is respectively obtained by Laplace transform technique firstly. Then the series systems which can solve IHCP based on the inner temperature difference are constructed by replacing the unsuitable zero and pole points of the transfer function approximated by è approximation. Finally the effects of the series systems are evaluated by a typical example. The results of the evaluation show that this method can obtain the surface heat flux and temperature by the inner temperature difference, and enhance the response speed of the measurement system at the same time. In addition this method can also improve the signal to noise ratio (SNR) of the inverse solutions by selectively amplifying the high SNR parts of the inner temperature difference. The present work provides an effective method to improve the stability of IHCP.

Method for Localizing the Zeros of Analytic Functions Based on the Krawczyk Operator

The problem of localizing the zeros of analytic functions using the Krawczyk operator is considered. Formulas are obtained for computing the image of the Krawczyk operator for analytic functions. Based on this, a new algorithm for localizing the zeros is proposed. The use of the method is demonstrated by numerical examples including the search for the zeros of the Riemann zeta function.

Phase Transitions and Zeros of the Partition Function: An introduction

In this revision, I provide a brief and pedagogical introduction to the relation of phase transitions and the zeros of the statistical mechanical partition function. DOI: http://dx.doi.org/10.17807/orbital.v11i2.1362

On the Nevanlinna characteristic of confluent hypergeometric functions

ABSTRACTA confluent hypergeometric function (Kummer's function) is a generalized hypergeometric series introduced by Kummer in 1837 [De integralibus quibusdam definitis et seriebus infinitis. J Reine Angew Math (in Latin). 1837;17:228–242], given by , which are of great applications in statistics, mathematical physics, engineering and so on. In this paper, we investigate some properties of Kummer's function from viewpoint of value distribution theory. Specifically, two different growth orders are obtained for and , which are corresponding to the degenerated and non-degenerated cases respectively. Moreover, we obtain an asymptotic estimate of characteristic function and calculate the logarithmic derivative , the distribution of zeros of Kummer's function is also discussed. Finally, we show Kummer's function and an entire function are uniquely determined by their c-values.

Torus partition function of the six-vertex model from algebraic geometry

We develop an efficient method to compute the torus partition function of the six-vertex model exactly for finite lattice size. The method is based on the algebro-geometric approach to the resolution of Bethe ansatz equations initiated in a previous work, and on further ingredients introduced in the present paper. The latter include rational Q-system, primary decomposition, algebraic extension and Galois theory. Using this approach, we probe new structures in the solution space of the Bethe ansatz equations which enable us to boost the efficiency of the computation. As an application, we study the zeros of the partition function in a partial thermodynamic limit of M × N tori with N ≫ M. We observe that for N → ∞ the zeros accumulate on some curves and give a numerical method to generate the curves of accumulation points.