Search

This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows one to transform the considered differential systems to the normal form of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms and determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method.

Based on the control strategies, transmission mechanism and clinical progression of COVID-19, we propose a compartmental model with time delays. Two time delays are introduced into the model to describe the incubation period of the disease and the quarantine period of uninfected individuals who have contacts with infected people. In order to reveal the spread rule for COVID-19, we study the threshold dynamics for the model. The basic reproduction number \begin{document}$ \mathcal R_0 $\end{document} is obtained. When \begin{document}$ \mathcal R_0 , the disease-free equilibrium is locally asymptotically stable, when \begin{document}$ \mathcal R_0>1 $\end{document} , the disease-free equilibrium is unstable and the disease is uniformly persistent. As the model's applications, we study COVID-19 transmission in the United States. The parameters are chosen to fit public data in the US. The numerical results indicate that an outbreak peak time in the US will appear in the middle of March. Sensitive analysis results show that enhancing the control measures, such as keeping social distance, wearing masks, isolation etc., can significantly contribute to the prevention and control of COVID-19 infection.

For an odd prime \begin{document}$ p $\end{document} and positive integers \begin{document}$ m $\end{document} and \begin{document}$ \ell $\end{document} , let \begin{document}$ \mathbb{F}_{p^m} $\end{document} be the finite field with \begin{document}$ p^{m} $\end{document} elements and \begin{document}$ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $\end{document} . Thus \begin{document}$ R_{\ell,m} $\end{document} is a finite commutative non-chain ring of order \begin{document}$ p^{2^{\ell} m} $\end{document} with characteristic \begin{document}$ p $\end{document} . In this paper, we aim to construct quantum codes from skew constacyclic codes over \begin{document}$ R_{\ell,m} $\end{document} . First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.

With view to applications to harmonic and stochastic analysis of infinite network/graph models, we introduce new tools for realizations and transforms of positive definite kernels (p.d.) \begin{document}$ K $\end{document} and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of factorizations: (i) Probabilistic: Starting with a positive definite kernel \begin{document}$ K $\end{document} we analyze associated Gaussian processes \begin{document}$ V $\end{document} . Properties of the Gaussian processes will be derived from certain factorizations of \begin{document}$ K $\end{document} , arising as a covariance kernel of \begin{document}$ V $\end{document} . (ii) Geometric analysis: We discuss families of measure spaces arising as boundaries for \begin{document}$ K $\end{document} . Our results entail an analysis of a partial order on families of p.d. kernels, a duality for operators and frames, optimization, Karhunen–Loeve expansions, and factorizations. Applications include a new boundary analysis for the Drury-Arveson kernel, and for certain fractals arising as iterated function systems; and an identification of optimal feature spaces in machine learning models.

Let \begin{document}$ p $\end{document} be a prime and let \begin{document}$ n $\end{document} be an integer with \begin{document}$ (n, p) = 1 $\end{document} . The Fermat quotient \begin{document}$ q_p(n) $\end{document} is defined as \begin{document}$ q_p(n)\equiv \frac{n^{p-1}-1}{p} \ (\bmod\ p), \quad 0\leq q_p(n)\leq p-1. $\end{document} We also define \begin{document}$ q_p(kp) = 0 $\end{document} for \begin{document}$ k\in \mathbb{Z} $\end{document} . Chen, Ostafe and Winterhof constructed the binary sequence \begin{document}$ E_{p^2} = \left(e_0, e_1, \cdots, e_{p^2-1}\right)\in \{0, 1\}^{p^2} $\end{document} as \begin{document}$ \begin{equation*} \begin{split} e_{n} = \left\{\begin{array}{ll} 0, & \hbox{if }\ 0\leq \frac{q_p(n)}{p} and studied the well-distribution measure and correlation measure of order \begin{document}$ 2 $\end{document} by using estimates for exponential sums of Fermat quotients. In this paper we further study the correlation measures of the sequence. Our results show that the correlation measure of order \begin{document}$ 3 $\end{document} is quite good, but the \begin{document}$ 4 $\end{document} -order correlation measure of the sequence is very large.

In this paper, by calculating the lower bounds on the nonlinearity of the derivatives of the following three classes of Boolean functions, we provide the tight lower bounds on the second-order nonlinearity of these Boolean functions: (1) \begin{document}$ f_1(x) = Tr_1^n(x^{2^{r+1}+2^r+1}) $\end{document} , where \begin{document}$ n = 2r+2 $\end{document} with even \begin{document}$ r $\end{document} ; (2) \begin{document}$ f_2(x) = Tr_1^n(\lambda x^{2^{2r}+2^{r+1}+1}) $\end{document} , where \begin{document}$ \lambda \in \mathbb{F}_{2^r}^* $\end{document} and \begin{document}$ n = 4r $\end{document} with even \begin{document}$ r $\end{document} ; (3) \begin{document}$ f_3(x,y) = yTr_1^n(x^{2^r+1})+Tr_1^n(x^{2^r+3}) $\end{document} , where \begin{document}$ (x, y)\in \mathbb{F}_{2^n}\times \mathbb{F}_2 $\end{document} , \begin{document}$ n = 2r $\end{document} with odd \begin{document}$ r $\end{document} . The results show that our bounds are better than previously known lower bounds in some cases.

This article focuses on the composite ${H}_{∞ }$ synchronization problem for jumping reaction-diffusion neural networks (NNs) with multiple kinds of disturbances. Due to the existence of disturbance effects, the performance of the aforementioned system would be degraded; therefore, improving the control performance of closed-loop NNs is the main goal of this article. Notably, for these disturbances, one of them can be described as a norm-bounded, and the other is generated by an exogenous model. In order to reject the above one kind of disturbance, a disturbance observer is developed. Furthermore, combining the disturbance observer approach and conventional state-feedback control scheme, a composite disturbance rejection controller is specifically designed to compensate for the influences of the disturbances. Then, some criteria are established based on the general Lyapunov stability theory, which can ensure that the synchronization error system is stochastically stable and satisfies a fixed $ {H}_{∞ } $ performance level. A simulation example is finally presented to verify the availability of our developed method.

This note proposes an adaptive second-order sliding mode (ASOSM) controller design by means of the Lyapunov method. The notable feature of the proposed algorithm is that it only needs boundedness of the uncertainties, while boundedness of the derivatives of uncertainties is not demanded. Under the proposed ASOSM control scheme, the gain can be dynamically tuned, which avoids gain overestimation. The finite-time stability of the closed-loop ASOSM dynamics is proved via the Lyapunov theory. Finally, the simulation results are shown to validate the theoretical analysis.

The unified criteria are analyzed on the global dissipativity and stability for the delayed fractional-order systems of multidimension-valued memristive neural networks (FSMVMNNs) in this article. First, based on the comprehensive knowledge about multidimensional algebra, fractional derivatives, and nonsmooth analysis, we establish the unified model for the studied FSMVMNNs in order to propose a more uniform method to analyze the dynamic behaviors of multidimensional neural networks. Then, by mainly applying the Lyapunov method, employing several new lemmas, and solving some mathematical difficulties, without any separation, we acquire the unified and concise criteria. The derived criteria have many advantages in a smaller calculation, lower conservatism, more diversity, and higher flexibility. Finally, we provide two numerical examples to express the availability and improvements of the theoretical results.

This article investigates the adaptive fuzzy control algorithm for a class of large-scale switched fractional-order nonlinear nonstrict feedback systems. In this algorithm, we utilize fuzzy-logic systems (FLSs) to approximate the complicated unknown nonlinear functions. Based on the fractional Lyapunov stability rules, a virtual control law is presented. A fuzzy adaptive decentralized control method is developed under the technique of the Lyapunov function. Under the operation of the proposed algorithm, the stability of the proposed systems and the control performance can be guaranteed. Finally, simulation results are presented to illustrate the feasibility and effectiveness of the proposed method.

Pole skipping is a recently discovered subtle effect in the thermal energy density retarded two point function at a special point in the complex $(\omega,p)$ planes. We propose that pole skipping is determined by the stress tensor contribution to many-body chaos, and the special point is at $(\omega,p)_\text{p.s.}= i \lambda^{(T)}(1,1/u_B^{(T)})$, where $\lambda^{(T)}=2\pi/\beta$ and $u_B^{(T)}$ are the stress tensor contributions to the Lyapunov exponent and the butterfly velocity respectively. While this proposal is consistent with previous studies conducted for maximally chaotic theories, where the stress tensor dominates chaos, it clarifies that one cannot use pole skipping to extract the Lyapunov exponent of a theory, which obeys $\lambda\leq \lambda^{(T)}$. On the other hand, in a large class of strongly coupled but non-maximally chaotic theories $u_B^{(T)}$ is the true butterfly velocity and we conjecture that $u_B\leq u_B^{(T)}$ is a universal bound. While it remains a challenge to explain pole skipping in a general framework, we provide a stringent test of our proposal in the large-$q$ limit of the SYK chain, where we determine $\lambda,\, u_B,$ and the energy density two point function in closed form for all values of the coupling, interpolating between the free and maximally chaotic limits. Since such an explicit expression for a thermal correlator is one of a kind, we take the opportunity to analyze many of its properties: the coupling dependence of the diffusion constant, the dispersion relations of poles, and the convergence properties of all order hydrodynamics., Comment: 39 pages, 13 figures

We consider a boundary value problem in a bounded domain involving a degenerate operator of the form $$L(u)=-\textrm{div} (a(x)\nabla u)$$ and a suitable nonlinearity $f$. The function $a$ vanishes on smooth 1-codimensional submanifolds of $\Omega$ where it is not allowed to be $C^{2}$. By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where $a$ vanishes.

In this paper, we stabilize a discrete-time Markov process evolving on a compact subset of $\mathbb{R}^d$ to an arbitrary target distribution that has an $L^\infty$ density and does not necessarily have a connected support on the state space. We address this problem by stabilizing the corresponding Kolmogorov forward equation, the \textit{mean-field model} of the system, using a density-dependent transition kernel as the control parameter. Our main application of interest is controlling the distribution of a multi-agent system in which each agent evolves according to this discrete-time Markov process. To prevent agent state transitions at the equilibrium distribution, which would potentially waste energy, we show that the Markov process can be constructed in such a way that the operator that pushes forward measures is the identity at the target distribution. In order to achieve this, the transition kernel is defined as a function of the current agent distribution, resulting in a nonlinear Markov process. Moreover, we design the transition kernel to be \textit{decentralized} in the sense that it depends only on the local density measured by each agent. We prove the existence of such

In this technical note, free-will arbitrary time terminal sliding mode control design is proposed. With such a design, it is possible to control the time duration of sliding in the sliding phase. Under the assumption that the chosen arbitrary time is greater than the reaching phase time (tr), it is shown that the nth order chain of integrators converges to zero within the selected arbitrary time. An algorithm is also proposed for the case when tr is not known to the designer. Stability analysis based on Lyapunov theory has been provided.

Multi-term variable-order fractional differential equations (VO-FDEs) are considered to be one of the tools to illustrate the behavior of transient-regime real-life phenomena precisely. The analytical solutions of VO-FDEs are generally very hard to obtain. Therefore, in this study, an approximation method for the solution of multi-term VO-FDEs is proposed using shifted Chebyshev polynomials of the third kind (SCP3). The method employs the operational matrices of SCP3 to proximate the VO-FDEs with a system of algebraic equations. The solution of which also yields the approximate result for the multi-term VO-FDE. An error analysis is also included. The SCP3 method is examined on several examples demonstrating the precision and prowess of the method.

The addition-min fuzzy relation inequalities could be applied to characterize the Peer-to-Peer (P2P) network system. This paper focuses on the problem that how many arbitrary (or specific) line faults a P2P network system is able to bear at most. In order to find the maximum number of tolerant line faults, we first study the consistency checking of the P2P network system with a given number of arbitrary line faults. Corresponding properties and illustrative example are also presented. Based on the consistency checking, we further propose the so-called Reordered-Vector-Based (RVB) algorithm for searching the maximum number of tolerant line faults in a given P2P network system. The validness of our proposed RVB algorithm is verified by formally proving and numerical example.

With the historical review, it can determine that plasma scientists devolved many numerical schemes in literature to solve the kinetic plasma models such as Vlasov- Maxwell system, but the fatal problem is that most of the designed algorithms violate the conservation laws. This issue is significant and cannot disregard because the Vlasov Maxwell system is connected with conservations properties and can evaluate after performing some effective mathematical operations. In this prospective, we formulated the Atangana-Baleanu fractional derivative with Caputo concepts based fractional-order plasma model to study the fractional performance of the plasma particles and further suggested a higher-order conservative numerical algorithm that follows the conservation laws, based on operational matrices theory which is evaluated by using the concepts of Shifted Gegenbauer estimations. The numerical convergence of the designed algorithm is also inspected to validate the competency and compatibility. The comprehensive analysis of the fractional performance of the plasma particles is also part of the study. Moreover, this concept can extend to the variable order and multi-dimensional problems for Vlasov and Boltzmann system.

This paper focuses on the modeling and stability analysis of the typical single-phase phase-locked loop (PLL) based on second-order generalized integrator (SOGI). Traditionally, SOGI-based PLL (SOGI-PLL) is linearized as the linear time-invariant (LTI) model for the stability analysis and parameter design. Yet, the PLL has periodical nature due to the periodical variation of the grid voltage. The traditional LTI model for PLL thereby reduces the precision of stability analysis. For addressing this problem, an accurate linear time-periodic (LTP) model of the single-phase SOGI-PLL is proposed in this paper. The proposed model can precisely characterize the coupling nature of phase, frequency, and amplitude estimated by SOGI-PLL. A precise stability region of SOGI-PLL thereby can be derived from the proposed LTP model. The effectiveness of the proposed LTP model for SOGI-PLL is verified by simulation and experimental tests.

For systems of linear differential equations whose dimension can be lowered, we estimate the growth of meromorphic vector solutions. As an essentially new feature, we can mention the fact that no additional restrictions are imposed on the order of growth of the coefficients of the system.

Many valued topologies on L-sets, providing a common framework for fuzzy topologies on fuzzy sets and the fixed-basis lattice-valued topologies on ordinary sets, are the main subject of the present study. In this paper, we have introduced many valued topological L-set-spaces, referring to L-sets equipped with many valued topologies, in an axiomatic way and focused on the vindication of their axiomatization in a categorical framework. In order to attain this objective, we apply the theory of Hohle of topological space objects formulated in terms of a partially ordered monad on an abstract category, and show that the category of many valued topological L-set-spaces is isomorphic to the category of topological space objects with respect to a partially ordered monad on a category of L-sets.

The aim of this paper is to deal with the boundedness of the Hardy-Littlewood maximal operator on variable exponent Morrey spaces of an integral form. As an application of the boundedness of the maximal operator, we establish Sobolev-type inequalities for Riesz potentials $$I_{\alpha (\cdot )}f$$ of variable order $$\alpha (\cdot )$$ of functions f in variable exponent Morrey spaces of an integral form.

Unsteady, 3D particle tracking velocimetry (PTV) data are applied as an inlet boundary condition in a direct numerical simulation (DNS). The considered flow case is a zero pressure gradient (ZPG) turbulent boundary layer (TBL) flow over a flat plate. The study investigates the agreement between the experimentally measured flow field and its simulated counterpart with a hybrid 3D inlet region. The DNS field inherits a diminishing contribution from the experimental field within the 3D inlet region, after which it is free to spatially evolve. Since the measurement does not necessarily provide a spectrally complete description of the turbulent field, the spectral recovery of the flow field is analyzed as the TBL evolves. The study summarizes the pre-processing methodology used to bring the experimental data into a form usable by the DNS as well as the numerical method used for simulation. Spectral and mean flow analysis of the DNS results show that turbulent structures with a characteristic length on the order of one average tracer particle nearest neighbor radius r¯NN or greater are well reproduced and stay correlated to the experimental field downstream of the hybrid inlet. For turbulent scales smaller than r¯NN, where experimental data are sparse, a relatively quick redevelopment of previously unresolved turbulent energy is seen. The results of the study indicate applicability of the approach to future DNS studies in which specific upstream or far field boundary conditions (BCs) are required and may provide the utility of decreasing high initialization costs associated with conventional inlet BCs., Deutsche Forschungsgemeinschaft, Projekt DEAL

Generalizations of attenuated ray transform (ART) and back-projection operators of tomography are suggested. The operators of ART of order m contain a complex-valued absorption, the weights of more general form, and the internal sources depending on time. Connections between ART of various orders are established, and corresponding differential equations and uniqueness theorems are obtained. We define the angular moments of ART of order m, representing generalization of back-projection operator, and establish connections between the moments of different orders. The generalized ART and angular moment operators have applications in integral geometry and tomography.

In this article we derive the expression of \textit{renormalized energies} for unit-valued harmonic maps defined on a smooth bounded domain in \(\mathbb{R}^2\) whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel-Brezis-H\'elein in order to describe the position of limiting Ginzburg-Landau vortices in simply connected domains. We show here, how a non-trivial topology of the domain modifies the expression of the renormalized energies. We treat the case of Dirichlet boundary conditions and Neumann boundary conditions as well.

In this paper, without dividing the fractional-order complex-valued coupled neural networks (FCVCNNs) into two real-valued systems, the problem of synchronization is investigated for FCVCNNs with time-varying coupling strength. To achieve synchronization, by updating coupling strength, two feasible adaptive protocols are designed, 1) a fractional-order adaptive strategy depending on global information; 2) a fractional-order adaptive law relying on the local information based on the connected dominating set theory. Additionally, instead of making the weighted average technique or the solution of the isolated node equation as synchronization reference value, direct error method is adopted to realize synchronization, which offers a flexible way in analysis of FCVCNNs. At last, a numerical simulation is provided to illustrate the validity of theoretical results.

A high-performance electro-absorption optical modulator based on the epsilon-near-zero (ENZ) effect is proposed. The structure consists of a waveguide with a silicon (Si) core over which a stack of graphene/HfO $_2$ /graphene/ITO/HfO $_2$ /graphene is grown, covered by a Si cladding. An external voltage is applied across the graphene layers to change the carrier concentration in the indium tin oxide (ITO) layers. Using a self-consistent theory, the required voltage to achieve the ENZ points in the ITO layers is calculated up to 3.42 V for an ITO thickness of 5 nm. The operation of the modulator is investigated using a three-dimensional finite-difference time-domain (FDTD) method, resulting in a modulation depth as high as 5.23 dB/ $\boldsymbol{\mu}$ m (5.36 dB/ $\boldsymbol{\mu }$ m) at a wavelength of 1.55 $\boldsymbol{\mu }$ m for the TE (TM) polarization, which ensures the polarization-insensitivity of our proposed modulator. It is also calculated that the insertion loss of the modulator is in the order of $\boldsymbol{ 2.5 \times 10^{-3}}$ dB/ $\boldsymbol{\mu }$ m that yields the figure of merit (FOM) of more than 1800. The outstanding features of our proposed modulator are mainly attributed to using the Si cladding layer instead of metal cladding. Furthermore, in contrast to the previously studied structures with metal electrodes, graphene layers significantly reduce the insertion loss.

In recent years, Rogers and Zudilin developed a method to write $L$-values attached to elliptic curves as periods. In order to apply this method to a broader collection of $L$-values, we study Eisenstein series and determine their Fourier series at cusps. Subsequently, we write the $L$-values of an elliptic curve of conductor 32 as an integral of Eisenstein series and evaluate the value at $k>1$ explicitly as a period. As a side result, we give simple integral expressions for the generating functions of $L(E,k)$ when even (or odd) $k$ runs over positive integers., 25 pages

Through the combination of contraction mapping and pseudo-spectral method, we propose a successive approximation technique to approximate the solution of a class of regulator equations with periodic exosystems and hyperbolic zero dynamics. In this scheme, the initial points of flows on the zero-error constrained manifolds are approximated successively as the fixed point of a contractive integral mapping. Accordingly, flows are obtained by utilizing the scaled Fourier-Gauss-Radau collocation method. Appropriate error analysis, in association with both regulation error and the error resulting from the approximate solution of the center manifold equation, is provided. Our analysis shows that the regulation error becomes negligible as we start the approximation process at an adequately large order and maintain it for a proper number of iterations.

Our main aim in this work is to find the operational matrix of fractional derivative and the operational matrix of distributed order fractional derivative for the Muntz-Legendre polynomials (MLPs). The operational matrix approach with the tau method or collocation method is applied to reduce the solution of the linear/nonlinear distributed order fractional differential equations (DFDEs) to a system of linear/nonlinear algebraic equations. Moreover, seven numerical examples are included to show the validity and applicability of the suggested methods.

As a special case of the Volterra system, the second-order Volterra (SOV) filter is very efficient for nonlinear system identification. The improved correntorpy based on the generalized Gaussian density function has been proven robust against impulsive noise. In this brief, we propose several SOV filters based on a recursive maximum correntropy (RMC) algorithm for nonlinear system identification. We first introduce a basic RMC algorithm, which faces a trade-off between filtering accuracy and tracking capability due to the use of a fixed forgetting factor (FFF). Two RMCs with variable FF (VFF) are further proposed to enhance the tracking ability. Simulation results demonstrate that our proposed algorithms outperform existing ones in impulsive noise environments and/or in time-varying systems.

Some research progress has been made on fractional-order continuous orthogonal moments (FrCOMs) in the past two years. Compared with integer-order continuous orthogonal moments (InCOMs), FrCOMs increase the number of affine invariants and effectively improve numerical stability. However, the existing types of FrCOMs are still very limited, of which all are planar image oriented. No report on stereoscopic images is available yet. To this end, in this paper, FrCOMs corresponding to various types of InCOMs are first deduced, and then, they are combined with trinion theory to construct trinion FrCOMs (TFrCOMs) applicable to stereoscopic images. Furthermore, the reconstruction performance and geometric invariance of TFrCOMs are analyzed theoretically and experimentally. Finally, an application in the stereoscopic image zero-watermarking algorithm is investigated to verify the superior performance of TFrCOMs.

The aim of this paper is to study higher-order Caginalp phase-field systems based on the Maxwell–Cattaneo law, instead of the classical Fourier law. More precisely, one obtains well-posedness results, as well as the existence of finite-dimensional attractors.

This article discusses the fault estimation (FE) and fault accommodation (FA) methods for fractional-order systems. First, two Lyapunov theorems of input-to-state practical stability are presented for fractional-order systems, based on which an adaptive FE/FA scheme is provided. Such a scheme ensures the faulty system is input-to-state practically stable (ISpS) with respect to estimation errors. Furthermore, new results are extended to fractional-order switched and interconnected systems: for the former, an FE/FA method is proposed which fully reveals the tradeoff between the value of the fractional order, each mode's dynamics, and switching law; for the latter, a cyclic-small-gain theorem of the fractional-order system is given under which both decentralized and distributed FE/FA methods are proposed.

This paper introduces the concept of measure chain theory in order to design event–triggered controllers for systems evolving on arbitrary time domains. The event–triggering mechanism is derived from input–to–state (ISS) stable Lyapunov functions defined on measure chains. These results not only unify existing event–triggered controller design procedures for systems on continuous and discrete time domains, but also extend them to non–homogeneous time domains. Numerical simulations for systems on different measure chains illustrate the effectiveness of the designed event–triggered controllers.

Let $G$ be a subgroup of ${\rm PGL}_2({\mathbb F}_q)$, where $q$ is any prime power, and let $Q \in {\mathbb F}_q[x]$ such that ${\mathbb F}_q(x)/{\mathbb F}_q(Q(x))$ is a Galois extension with group $G$. By explicitly computing the Artin map on unramified degree-1 primes in ${\mathbb F}_q(Q)$ for various groups $G$, interesting new results emerge about finite fields, additive polynomials, and conjugacy classes of ${\rm PGL}_2({\mathbb F}_q)$. For example, by taking $G$ to be a unipotent group, one obtains a new characterization for when an additive polynomial splits completely over ${\mathbb F}_q$. When $G = {\rm PGL}_2({\mathbb F}_q)$, one obtains information about conjugacy classes of ${\rm PGL}_2({\mathbb F}_q)$. When $G$ is the group of order 3 generated by $x \mapsto 1 - 1/x$, one obtains a natural tripartite symbol on ${\mathbb F}_q$ with values in ${\mathbb Z}/3{\mathbb Z}$. Some of these results generalize to ${\rm PGL}_2(K)$ for arbitrary fields $K$. Apart from the introduction, this article is written from first principles, with the aim to be accessible to graduate students or advanced undergraduates. An earlier draft of this article was published on the Math arXiv in June 2019 under the title {\it More structure theorems for finite fields}., Comment: Version 4 contains minor corrections and updates to the bibliograpy. Version 3 is a major revision, including a change in the title from "More structure theorems for finite fields" to "Explicit Artin maps into PGL2". The author thanks Xander Faber for insightful comments that led to the change in the title

The dynamics of neural networks has been widely concerned by scholars. However, most of the previous results on dynamical bifurcations are limited to few nodes coupling neural networks which modeled by differential equations with integer-order derivative, and few efforts have been contributed to studying the bifurcation behaviors of large-scale fractional-order neural networks. Furthermore, the structural characteristics of networks are also of great research value. Among them, the ring structure is a common phenomenon in neural networks. Although there are few papers on the bifurcation analysis of ring-structured neural networks recently, they consider only the case of a single ring. In this article, the dynamical analysis and design for a class of large-scale-delayed fractional-order neural networks with multiple rings and hub structure are investigated. First, the time delay is considered to be the bifurcation parameter and the formula of Coates' flow graph is adopted to obtain the characteristic equation of large-scale networks. Second, by analyzing the complex radial and circular connections of neurons, the delay-induced Hopf bifurcation sufficient conditions for the neural network are established. Finally, the theoretical results are substantiated by a number of numerical simulation experiments and the relationships between the onset of bifurcation and the fractional order, the number of neurons, and the number of rings are revealed.

This article presents a stability analysis of fractional-order tumor system that correlates tumor-immune with chemotherapy under Lyapunov's theory. More particularly, a finite-time stabilization of prescribed system with the consequence of quantized input and actuator fault is investigated. Further, the nonlinear tumor system is converted to Takagi-Sugeno(T-S) fuzzy model. Then based on the utilization of Gronwall's inequality, the necessary and sufficient conditions are derived with the assist of Lyapunov's candidates. To overcome the effect of external disturbances such as unwanted smoke, intake habits and tower radiations, the optimal level of performance which is reduced from the generalized performance extended dissipative is included. Finally, a numerical simulation is presented with graphical results to expound the superiority and validity of the prescribed controller.

Differentiation is an important task in control, observation and fault detection. Levant's differentiator is unique, since it is able to estimate exactly and robustly the derivatives of a signal with a bounded high-order derivative. However, the convergence time, although finite, grows unboundedly with the norm of the initial differentiation error, making it uncertain when the estimated derivative is exact. In this paper we propose an extension of Levant's differentiator so that the worst case convergence time can be arbitrarily assigned independently of the initial condition, i.e. the estimation converges in \emph{Fixed-Time}. We propose also a family of continuous differentiators and provide a unified Lyapunov framework for analysis and design.

This paper studies the delay bounds problem of second-order multi-agent systems with input and communication delays under the directed graph. Firstly, the systems with input delay only are studied and the delay margin of input delay is given. Then the critical value of communication delay is considered for the fixed input delay within the proposed scope. Then, the input delay and communication delay are analyzed simultaneously from geometric perspective. The explicit geometric description of the consensus crossing curves is obtained. Then the consensus regions can be completely determined.

In this paper, we consider a class of nonlocal equations where the convolution kernel is given by a Bessel potential symbol of order $\alpha$ for $\alpha > 1$. Based on the properties of the convolution operator, we apply a global bifurcation technique to show the existence of a highest, even, $2\pi$-periodic traveling-wave solution. The regularity of this wave is proved to be exactly Lipschitz., Comment: 22 pages. arXiv admin note: text overlap with arXiv:1810.00248 by other authors

This paper considers the steady-state performance of load balancing algorithms in a many-server system with distributed queues. The system has N servers, and each server maintains a local queue with buffer size b-1, i.e. a server can hold at most one job in service and b-1 jobs in the queue. Jobs in the same queue are served according to the first-in-first-out (FIFO) order. The system is operated in a heavy-traffic regime such that the workload per server is $λ = 1 - N^{-α}$ for 0.5≤ α

The absence of satisfactory results that address the stability analysis of nonlinear incommensurate Fractional-order Difference Systems (FoDSs) as well as the impossibility of establishing a proper Lyapunov function that helps us to reveal any stability outcome for those systems, are considered some significant nonlinear problems that should be taken into account. To overcome such dilemmas, the present work provides some novel convenient conditions that can be utilized to ensure the asymptotic stability of those systems. In particular, through utilizing some features of the Z-transform scheme, we intend to propose certain results that aim to reveal the stability of the nonlinear incommensurate fractional-order difference system formulated in the sense of the Caputo difference operator. These novel results are completely confirmed by demonstrating the stability of the numerical solutions for two nonlinear difference systems with incommensurate fractional-orders.

Improved Fractional Order Darwinian Particle Swarm Optimization (FODPSO) has been proposed to identify the parameters of solar PV cells and modules. For this purpose, the traditional PSO algorithm has been adapted and an improved one using the fractional order calculus has been especially designed as well. The proposed FODPSO together with the PSO has been applied in both the single and double diode models to describe the specified PV units for purpose of comparison. To examine the advantages of the proposed FODPSO over conventional PSO, two approaches have been performed; the first approach is implementing the FODPSO and PSO algorithms using input data obtained from previous international publications for different types of PV cells/modules, then comparing the FODPSO algorithm's results to the results of PSO and the other previous optimization techniques that were using the same input data and identical PV cells/modules. The second validation approach is achieved by performing real measurements in outdoor conditions for another two types of PV modules (mono and poly crystalline Silicon) and inserting these measurements' readings as input data to the PSO and FODPSO algorithms. A comparative study has also been done for both algorithms' results. Enhanced results have been achieved using the FODPSO technique. It is worth mentioning that during the development of this work, other efforts have been made with different techniques that may result in similar or even better results, yet this paper is still introducing for the first time the FODPSO technique (as a modified technique of the PSO) as applied on the PV modeling problem, in addition the paper presents the benefits of the proposed FODPSO over the conventional PSO and presents a comparison that shows the superiority of the FODPSO over the conventional PSO.

A highly accurate collocation technique for diffusion equations of Riesz distributed-order space-fractional is consider. A mixed Gauss-Lobatto and Gauss-Radau Legendre collocation technique is introduced to solve the diffusion equations of Riesz distributed-order space-fractional. A full theoretical discussion is given and numerical tests are listed to light the fulfillment and eligibility of the technique. The reliability of our technique to treat the diffusion equations of Riesz distributed-order space-fractional is revealed.

Mathematical modeling of infectious diseases with non-integer order getting attentions from scientists and researchers day by day. It is obvious that classical models in epidemiology can only be described through a fixed order while models in fractional order derivative are of not fixed order. Having non fixed order the fractional derivative becomes more powerful in modeling real life problems. In the recent era, different novel concepts regarding fractional operators such as the exponential decay and the Mittag–Leffler kernel have been introduced which overcome the limitations of the previous fractional order derivatives. These new operators have been found effective in modeling problems arising in science and engineering. A more recent operator in fractional calculus was introduced that is known as fractal-fractional operator. In this study, we consider this novel approach and apply it to an epidemic model of dengue fever and explore their dynamics. We show some important analysis for the dengue epidemic model in the presence of this new operator. The uniqueness and existence results will be shown. We show the simulation results for the considered model with a novel numerical approach which is not yet considered by anyone for such epidemic model. We obtain results for fractal model when fractional order is one and will have fractional solution when fractal order is one and have when both are present. We show that the fractal-fractional approach is much suitable for an epidemic model rather than fractional operator.

This paper investigates global dynamics in fractional-order dual inertial neural networks with time lags. Firstly, according to some crucial features of Mittag-Leffler functions and Banach contracting mapping principle, the existence and uniqueness of $ S $-asymptotically $ \omega $-periodic oscillation of the model are gained. Secondly, by using the comparison principle and the stability criteria of delayed Caputo fractional-order differential equations, global asymptotical stability of the model is studied. In the end, the feasibility and effectiveness of the obtained conclusions are supported by two numerical examples. There are few papers focus on $ S $-asymptotically $ \omega $-periodic dynamics in fractional-order dual inertial neural networks with time-varying lags, apparently, the works in this paper fill some of the gaps.

The atom-bond connectivity (ABC) index is a degree-based molecular descriptor that found diverse chemical applications. Characterizing trees with minimum ABC-index remained an elusive open problem even after serious attempts and is considered by some as one of the most intriguing open problems in mathematical chemistry. In this paper, we describe the exact structure of the extremal trees with sufficiently many vertices and we show how their structure evolves when the number of vertices grows. An interesting fact is that their radius is at most 5 and that all vertices except for one have degree at most 54. In fact, all but at most O ( 1 ) vertices have degree 1, 2, 4, or 53. Let γ n = min ⁡ { ABC ( T ) : T is a tree of order n } . It is shown that γ n = 1 365 1 53 ( 1 + 26 55 + 156 106 ) n + O ( 1 ) ≈ 0.67737178 n + O ( 1 ) .

A leak is a vertex that is not allowed to perform a force during the zero forcing process. Leaky forcing was recently introduced as a new variation of zero forcing in order to analyze how leaks in a network disrupt the zero forcing process. The l -leaky forcing number of a graph is the size of the smallest zero forcing set that can force a graph despite l leaks. A graph G is l -resilient if its zero forcing number is the same as its l -leaky forcing number. In this paper, we analyze l -leaky forcing and show that if an ( l − 1 ) -leaky forcing set B is robust enough, then B is an l -leaky forcing set. This provides the framework for characterizing l -leaky forcing sets. Furthermore, we consider structural implications of l -resilient graphs. We apply these results to bound the l -leaky forcing number of several graph families including trees, supertriangles, and grid graphs. In particular, we resolve a question posed by Dillman and Kenter concerning the upper bound on the 1-leaky forcing number of grid graphs.

The environment around us naturally represents number of its components in fractal structures. Some fractal patterns are also artificially simulated using real life mathematical systems. In this paper, we use the fractal operator combined to the fractional operator with both exponential and Mittag-leffler laws to analyze and solve generalized three-dimensional systems related to real life phenomena. Numerical solutions are provided in each case and applications to some related systems are given. Numerical simulations show the existence of the models' initial three-dimensional structure followed by its self- replication in fractal structure mathematically produced. The whole dynamics are also impacted by the fractional part of the operator as the derivative order changes.

In this paper, we study a projective multi-synchronization problem for fractional-order complex-valued coupled multi-stable neural networks (FCVCMNNs) with time-delays. Using a complex decomposition approach, FCVCMNNs are divided into their real and imaginary components. Our method uses certain conditions for each subnetwork to achieve the multiple local equilibrium points or stable periodic orbits that are exponentially stable, which, when combined with the Lyapunov functions method, result in FCVCMNNs that are projectively multi-synchronized. The FCVCMNNs, on the other hand, are examined directly through the use of the Lyapunov functional method and linear matrix inequality (LMI). Various new sufficient conditions in the form of complex-valued LMIs are presented for the projective multi-synchronization of the considered FCVCMNNs. As a final step, we provide two numerical simulations to verify the effectiveness of the main results derived in this paper.