Entire Functions Research Articles

On the Toeplitz Algebra in the Case of All Entire Functions and All Functions Holomorphic in the Unit Disc

On the Toeplitz Algebra in the Case of All Entire Functions and All Functions Holomorphic in the Unit Disc

TLS-handshake for Plug and Charge in vehicular communications

Electric vehicle technology has brought some important issues to order. Securing the communication between electric vehicle and charging station is an important one. The International Organization for Standardization released ISO 15118 standard to specify the communication between electric vehicles and electric vehicle supply equipment. The ISO 15118 standard requires Transport Layer Security function for secure charging communication between electric vehicle and charging station as mandatory. The Transport Layer Security requirements have been completely changed with the latest version of the standard (ISO 15118-20) and have not been clearly described and implemented yet. The implementation of the function causes an inevitable vulnerability due to non-functioning authentication resulting with the entire vehicle function and security system can suddenly stop working. For this reason, new Transport Layer Security protocol must urgently be analyzed, developed, integrated, and implemented.This study proposes solutions for the development, implementation, realization of Transport Layer Security handshake requirements and functions between electric vehicle and charging station with ISO 15118 protocol. The new and existing Transport Layer Security requirements, analyzed, evaluated and the new Transport Layer Security functions derived and fully described. Based on these requirements and functions, a Transport Layer Security communication sequence and processes are developed and implemented and validated on the test bench. The validation proved sufficient implementation of the Transport Layer Security functions (successful authentication, exchange of cipher suite parameters and generation of the TLS master key). The messages were prosperous encrypted and decrypted with the generated master key. Thus, Transport Layer Security handshake is developed, implemented, and validated with this study.

Comparative growth of an entire function and the integrated counting function of its zeros

Let $(\zeta_n)$ be a sequence of complex numbers such that $\zeta_n\to\infty$ as $n\to\infty$, $N(r)$ be the integrated counting function of this sequence, and let $\alpha$ be a positive continuous and increasing to $+\infty$ function on $\mathbb{R}$ for which $\alpha(r)=o(\log (N(r)/\log r))$ as $r\to+\infty$. It is proved that for any set $E\subset(1,+\infty)$ satisfying $\int_{E}r^{\alpha(r)}dr=+\infty$, there exists an entire function $f$ whose zeros are precisely the $\zeta_n$, with multiplicities taken into account, such that the relation $$ \liminf_{r\in E,\ r\to+\infty}\frac{\log\log M(r)}{\log r\log (N(r)/\log r)}=0 $$ holds, where $M(r)$ is the maximum modulus of the function $f$. It is also shown that this relation is best possible in a certain sense.

ON THE ITERATIONS AND THE ARGUMENT DISTRIBUTION OF MEROMORPHIC FUNCTIONS

AbstractThis paper consists of two parts. The first is to study the existence of a point a at the intersection of the Julia set and the escaping set such that a goes to infinity under iterates along Julia directions or Borel directions. Additionally, we find such points that approximate all Borel directions to escape if the meromorphic functions have positive lower order. We confirm the existence of such slowly escaping points under a weaker growth condition. The second is to study the connection between the Fatou set and argument distribution. In view of the filling disks, we show nonexistence of multiply connected Fatou components if an entire function satisfies a weaker growth condition. We prove that the absence of singular directions implies the nonexistence of large annuli in the Fatou set.

Nondefinability results with entire functions of finite order in polynomially bounded o-minimal structures

Nondefinability results with entire functions of finite order in polynomially bounded o-minimal structures

A new method of smoothness‐constrained magnetotelluric modelling with the utility of Pareto‐optimal multi‐objective particle swarm optimization

AbstractParticle swarm optimization, one of the modern global optimization methods, is attracting widespread interest because it overcomes the difficulties of conventional inversion techniques, such as trapping at a local minimum and/or initial model dependence. The main characteristic of particle swarm optimization is the large search space of parameters, which in a sense allows the exploration of the entire objective function space if the input parameters are properly chosen. However, in the case of a high‐dimensional model space, the numerical instability of the solution may increase and lead to unrealistic models and misinterpretations due to the sampling problem of particle swarm optimization. Therefore, smoothness‐constrained regularization techniques used for the objective function or model reduction techniques are required to stabilize the solution. However, weighting and combining objective function terms is partly a subjective process, as the regularization parameter is generally chosen based on some kind of criteria of how the smoothing constraints affect the data misfits. This means that it cannot be completely predefined but needs to be adjusted during the inversion process, which begins with the response of an initial model. In this paper, a new modelling approach is proposed to obtain a smoothness‐constrained model from magnetotelluric data utilizing multi‐objective particle swarm optimization based on the Pareto optimality approach without using a regularization parameter and combining several objective function terms. The presented approach was verified on synthetic models and an application with field data set from the Çanakkale–Tuzla geothermal field in Turkey. Findings from these analyses confirm the usefulness of the method as a new approach for all constrained inversions of geophysical data without the need to combine the objective function terms weighted by a regularization parameter.

Lower order and limiting directions of Julia sets of solutions to second order differential equations

In this paper, we consider the properties of entire solutions to second order differential equation(⁎)f″+Af′+Bf=0, where A(z) and B(z)≢0 are entire functions. Under certain assumptions on A(z) and B(z), we prove that every non-trivial solution f of equation (*) is of infinite lower order, and then obtain the measure estimation of the limiting directions of Julia sets for those infinite lower order entire solutions. The existence of Baker domain for f(n) is also discussed.

On hyperbolic dimension gap for entire functions

Polynomials and entire functions whose hyperbolic dimension is strictly smaller than the Hausdorff dimension of their Julia set are known to exist but in all these examples the latter dimension is maximal, i.e. equal to two. In this paper we show that there exist hyperbolic entire functions f f having Hausdorff dimension of the Julia set HD ⁡ ( J f ) > 2 \operatorname {HD} (\mathcal {J}_f)>2 and hyperbolic dimension H y p D i m ( f ) > H D ( J f ) \mathrm {HypDim}(f)>\mathrm {HD}(\mathcal {J}_f) .

CNN-GRU-FF: a double-layer feature fusion-based network intrusion detection system using convolutional neural network and gated recurrent units

Identifying and preventing malicious network behavior is a challenge for establishing a secure network communication environment or system. Malicious activities in a network system can seriously threaten users’ privacy and potentially jeopardize the entire network infrastructure and functions. Furthermore, cyber-attacks have grown in complexity and number due to the ever-evolving digital landscape of computer and network devices in recent years. Analyzing network traffic using network intrusion detection systems (NIDSs) has become an integral security measure in modern networks to identify malicious and suspicious activities. However, most intrusion detection datasets contain imbalance classes, making it difficult for most existing classifiers to achieve good performance. In this paper, we propose a double-layer feature extraction and feature fusion technique (CNN-GRU-FF), which uses a modified focal loss function instead of the traditional cross-entropy to handle the class imbalance problem in the IDS datasets. We use the NSL-KDD and UNSW-NB15 datasets to evaluate the effectiveness of the proposed model. From the research findings, it is evident our CNN-GRU-FF method obtains a detection rate of 98.22% and 99.68% using the UNSW-NB15 and NSL-KDD datasets, respectively while maintaining low false alarm rates on both datasets. We compared the proposed model’s performance with seven baseline algorithms and other published methods in literature. It is evident from the performance results that our proposed method outperforms the state-of-the-art network intrusion detection methods.

Entire Monogenic Functions of Given Proximate Order and Continuous Homomorphisms

Infinite order differential operators appear in different fields of mathematics and physics. In the past decade they turned out to play a crucial role in the theory of superoscillations and provided new insight in the study of the evolution as initial data for the Schrödinger equation. Inspired by the infinite order differential operators arising in quantum mechanics, in this paper we investigate the continuity of a class of infinite order differential operators acting on spaces of entire hyperholomorphic functions. Precisely, we consider homomorphisms acting on functions in the kernel of the Dirac operator. For this class of functions, often called monogenic functions, we introduce the proximate order and prove some fundamental properties. As an important application, we are able to characterize infinite order differential operators that act continuously on spaces of monogenic entire functions.

Bernstein Inequality for the Riesz Derivative of Order $$0<\alpha<1$$ of Entire Functions of Exponential Type in the Uniform Norm

Bernstein Inequality for the Riesz Derivative of Order $$0<\alpha<1$$ of Entire Functions of Exponential Type in the Uniform Norm

The high-order estimate of the entire function associated with inverse Sturm–Liouville problems

Abstract The inverse Sturm–Liouville problem with smooth potentials is considered. The high-order estimate of the entire function associated with two Sturm–Liouville problems is established. Applying this estimate expression to inverse Sturm–Liouville problems, we proved that the conclusion in [L. Amour, J. Faupin and T. Raoux, Inverse spectral results for Schrödinger operators on the unit interval with partial information given on the potentials, J. Math. Phys. 50 2009, 3, Article ID 033505] remains true for more general case.

Hutchinson’s intervals and entire functions from the Laguerre–Pólya class

We find intervals [α,β(α)]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[\\alpha , \\beta (\\alpha )]$$\\end{document} such that, if a univariate real polynomial or entire function f(z)=a0+a1z+a2z2+⋯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(z) = a_0 + a_1 z + a_2 z^2 + \\cdots $$\\end{document} with positive coefficients satisfies the conditions qk(f)=ak-12/(ak-2ak)∈[α,β(α)]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q_k(f) = {a_{k-1}^2}/({a_{k-2}a_{k}}) \\in [\\alpha , \\beta (\\alpha )]$$\\end{document} for all k⩾2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \\geqslant 2$$\\end{document}, then f belongs to the Laguerre–Pólya class. For instance, from Hutchinson’s theorem, one can observe that f belongs to the Laguerre–Pólya class (has only real zeros) when qk(f)∈[4,+∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q_k(f) \\in [4, + \\infty )$$\\end{document}. We are interested in finding those intervals which are not subsets of [4,+∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[4, + \\infty )$$\\end{document}.

The Theory of the Entire Algebraic Functions

Abstract Let $A$ be the integral closure of the ring of polynomials ${{\mathbb {C}}}[t]$, within the field of algebraic functions in one variable. We show that $A$ interprets the ring of integers. This contrasts with the analogue for finite fields, proved to have a decidable theory in [12] and [4].

Uniqueness of meromorphic functions and their powers in set sharing

Abstract The objective of this study is to determine whether the existence of shared sets S containing both meromorphic (entire) functions and their higher derivatives, as well as powers of meromorphic functions and their differential polynomials, could result in uniqueness. The main focus is on determining the precise solutions to various differential equations. This problem is being studied in a broader context, specifically in set sharing.

Entire function with its linear differential polynomial normally or weakly shares a set and some related topics

Entire function with its linear differential polynomial normally or weakly shares a set and some related topics

The link on extraneous non-repelling cycles of Schröder's methods of the first and second kind

Let Sf,n and Kf,n be the functions defined in Schröder's method of the first and second kind for an entire function f with given order n(n≥2), respectively. Based on unrefined algebra characterizations of Sf,n and Kf,n, we obtain some sufficient conditions on f such that both Sf,n and Kf,n possess given finite pairs of extraneous non-repelling cycles. Here, these conditions are a pair of equations, which have infinitely many polynomials or transcendental entire functions as its solutions. For obtaining some solutions f of such equations, we provide a step-by-step method. We start from any solution g in corresponding equations so that the function Sg,2 possesses the above finite pairs of extraneous cycles but all are super-attracting, and then f can be obtained by a series of formulas concerning the function g, points and multipliers of those cycles. Note that Sf,2=Kf,2 is identical with Newton's method for f. In a sense, this fact reveals that some extraneous super-attracting cycles of Newton's method imply certain extraneous non-repelling cycles of any method from the two families of methods. More generally, for any given orders n and m, some extraneous non-repelling cycles of Sf,n or Kf,n imply that of SF,m or KF,m for some entire functions f and F. These give a partial answer for the problem of finding possible link between the two families of methods, which was posed by Steven Smale in 1994.

Subharmonic additions to the Beurling–Malliavin Theorems. II. On the completeness radius

The Beurling–Malliavin Theorem on the multiplier, considered in a subharmonic framework in the first part of our work, already in its original classical version within the framework of entire functions of exponential type, allowed in the 1960s to completely solve the problem of the radius of completeness of exponential system in the form of a remarkable criterion and exclusively in geometric terms for the exponents of this exponential system without any additional restrictions on the relative position of these exponents. The exact formulations of the Beurling– Malliavin Theorem on the radius of completeness in the introduction are somewhat adapted as a problem of the possible minimum growth along the real axis R of subharmonic functions with given constraints on their Riesz distributions of masses. In this largely overview part of the paper, we discuss the Beurling–Malliavin Theorem on the radius of completeness, along with its somewhat more general subharmonic manifestations. Thus, our results from 2014-16 allow us to obtain significantly more subtle results with respect to the Beurling – Malliavin Theorem itself on the radius of completeness with a defect excess of no more than 1 or 2 for exponents in classical rigid Banach spaces of continuous functions on a segment of fixed length or functions with integrable module in the p-th degree on such segments.

A NEW PROGRAM FOR THE ENTIRE FUNCTIONS IN NUMBER THEORY

In this paper, we propose a new program for introducing the sign of the functional equation to present the entire functions of order one in number theory. We suggest some open problems for the zeros of these entire functions related to the completed Dedekind zeta function, completed quadratic Dirichlet L-functions, completed Ramanujan zeta function and completed automorphic L-function. These lead to the contribution to giving the deep understanding for diffusion equations associated with the entire functions of order one in number theory.

Distributions of zeros and masses of entire and subharmonic functions with restrictions on their growth along the strip

Let $\mathrm Z$ and $\mathrm W$ be distributions of points on the complex plane $\mathbb C$. The following problem dates back to F. Carlson, T. Carleman, L. Schwartz, A. F. Leont'ev, B. Ya. Levin, J.-P. Kahane, and others. For which $\mathrm Z$ and $\mathrm W$, for an entire function $g\neq 0$ of exponential type which vanishes on $\mathrm W$, there exists an entire function $f\neq 0$ of exponential type that vanishes on $\mathrm Z$ and is such that $|f|\leqslant |g|$ on the imaginary axis? The classical Malliavin-Rubel theorem of the early 1960s completely solves this problem for "positive" $\mathrm Z$ and $\mathrm W$ (which lie only on the positive semiaxis). Several generalizations of this criterion were established by the author of the present paper in the late 1980s for "complex" $\mathrm Z \subset \mathbb C$ and $\mathrm W\subset \mathbb C$ separated by angles from the imaginary axis, with some advances in the 2020s. In this paper, we solve more involved problems in a more general subharmonic framework for distributions of masses on $\mathbb C$. All the previously mentioned results can be obtained from the main results of this paper in a much stronger form (even for the initial formulation for distributions of points $\mathrm Z$ and $\mathrm W$ and entire functions $f$ and $g$ of exponential type). Some results of the present paper are closely related to the famous Beurling-Malliavin theorems on the radius of completeness and a multiplier.