Recurrence Research Articles

Autonomous longitudinal separation control of aircraft in random wind fields based on MPC

Aiming at the problem of autonomous longitudinal separation under random disturbance of flight path, the random factor of high-altitude wind often leads to poor robustness of longitudinal separation between two aircraft. This paper proposes an autonomous longitudinal separation control method for aircraft based on model predictive control. Firstly, the nonlinear kinematic differential equation of the wind field difference between the two aircraft and the longitudinal separation is established, and the linear time-varying prediction model is derived. The longitudinal separation and path deviation distance of the two aircraft are selected as the optimization objectives, the high-altitude wind is the random disturbance, and the true airspeed and yaw angle of the leading aircraft are measurable. Terminal equality constraints are added to the air safety and aircraft performance constraints to maintain the stability of the system. To verify the effectiveness of the algorithm, within the prescribed simulation time of 120 seconds, this paper sets three groups of different expected intervals 12, 13, 14km. Through the designed MPC controller, by controlling the true airspeed and yaw angle of the trailing aircraft within the rolling time domain period, the interval curve between the two aircraft is relatively smooth and is always not less than the minimum safety interval of 10 km. It stabilizes at the expected target interval at the 74th, 90th and 118th seconds respectively, and begins to return to the route at the 58th, 74th and 95th seconds, and finally returns to the center line of the route; two groups of wind field control groups are set up, one group is the forecast wind intensity increased by 2 times, and the other group is the turbulent wind disturbance intensity increased by 8 times. Both can smoothly and stably establish the expected interval of 12 km at the 61st and 72nd seconds respectively.

Enhancing COVID-19 Forecasting in Dagestan with Quasi-linear Recurrence Equations by using GLDM Algorithm

Enhancing COVID-19 Forecasting in Dagestan with Quasi-linear Recurrence Equations by using GLDM Algorithm

Common values of linear recurrences related to Shank's simplest cubics

Let A,B,C∈Z not all zeroes and let F(u,n)=F(A,B,C,u,n) be the linear recursive sequence, which is defined by the initial terms F(u,0)=A,F(u,1)=B,F(u,2)=C and whose characteristic polynomial is Daniel Shanks simplest cubic Su(X)=X3−(u−1)X2−(u+2)X−1,u∈Z. We prove that there exists an effectively computable constant c depending only on L=max⁡{|A|,|B|,|C|} such that if |F(A,B,C,u,n)|=|F(A,B,C,u,m)| holds for some integers u,n,m with n≠m then |n|,|m|<c. For the choices (A,B,C)∈{(0,0,1),(1,−1,1)} we solve the above equations completely. At the end we give an outlook to the equation F(0,0,1,u,n)=F(0,0,1,v,m) for some fixed integers n,m.

Fractional order forestry resource conservation model featuring chaos control and simulations for toxin activity and human-caused fire through modified ABC operator

In this work, we proposed a nonlinear mathematical model with fractional-order differential equations employed to illustrate the impacts of depleted forestry resources with the effect of toxin activity and human-caused fire. The numerical and theoretical outcomes are based on the consideration of using a modified ABC-fractional-order depleted forestry resources dynamical system. In the theoretical aspect, we examination of solution positivity, existence, and uniqueness it makes use of Banach’s fixed point and the Leray Schauder nonlinear alternative theorem. The consecutive recursive sequences are purposefully designed to verify the existence of a solution to the depletion of forestry resources as delineated. To showcase the specificity and stability of the solution within the Hyers–Ulam framework, we employ the concepts and findings of functional analysis. Chaos control will stabilize the system following its equilibrium points by applying the regulate for linear responses technique. Using Lagrange polynomials insight of modified ABC-fractional-order, we conduct simulations and present a comparative analysis in graphical form with classical and integer derivatives. Results also demonstrate the impact of different parameters used in a model that is designed on the system, they provide more understanding and a better approach for real-life problems. Our results demonstrate the significant effects of toxic and fire activities produced by humans on forest ecosystems. More accurate management techniques are made possible by the modified ABC operator’s effectiveness in capturing the long-term effects of these disturbances. The findings highlight how crucial it is to use fractional calculus in ecological modeling to comprehend and manage the intricacies of forest preservation in the face of human pressures. To ensure the sustainable management of forest resources in the face of escalating environmental difficulties, this research offers policymakers and environmental managers a fresh paradigm for creating more robust and adaptive conservation policies.

Soft Computing-Based Generalized Least Deviation Method Algorithm for Modeling and Forecasting COVID-19 using Quasilinear Recurrence Equations

Soft Computing-Based Generalized Least Deviation Method Algorithm for Modeling and Forecasting COVID-19 using Quasilinear Recurrence Equations

A hybrid model of recursive cellular automata, DNA sequences, and chaotic system for image encryption

A hybrid model of recursive cellular automata, DNA sequences, and chaotic system for image encryption

Exact solution for mode shapes of cracked nonuniform axially functionally graded beams

This paper established the exact formulas for mode shapes of cracked nonuniform axially functionally graded (AFG) beams. The formula of the mode shape of cracked nonuniform AFG beam is derived in the form of a power presented as a recurrent relation. The power series solution of cracked beam is the same with the intact beam where the effect of cracks contributes to only the first four coefficients of the power series. The derivation of the formula of mode shapes is presented in details. Numerical simulations show that these effects on the natural frequency and mode shape are quite significant. The natural frequency and mode shape are more sensitive to cracks located at positions with low elastic modulus, small cross section and high mass density. For the case of tapered AFG cantilever beam, it is important to consider not only cracks located close to the fixed end but also cracks located close to the free end, especially when the free end has lower elastic modulus and higher mass density.

Branching processes with immigration in a random environment—The Grincevičius–Grey setup

We determine the tail asymptotics of the stationary distribution of a branching process with immigration in a random environment, when the immigration distribution dominates the offspring distribution. The assumptions are the same as in the Grincevičius–Grey theorem for the stochastic recurrence equation.

Application and Reversibility of Three Dimensional Cellular Automata

In this study, we obtain the characteristic matrices of three-dimensional cellular automata under the null boundary condition. We examine the inverse of characteristic matrices. We obtain a recurrence equation to determine under what conditions the matrix is invertible. Thanks to this equation, we can calculate the inverse of large-dimensional matrices. Finally, we give some applications of cellular automata. We find the minimal polynomial of the characteristic matrix. We find the cycle length and transition length of the characteristic matrix with the help of minimal polynomials. We also find the attractive points of the characteristic matrix. Finally, we draw the State Transition diagram with the results we obtained.

(Un)Solvable loop analysis

AbstractAutomatically generating invariants, key to computer-aided analysis of probabilistic and deterministic programs and compiler optimisation, is a challenging open problem. Whilst the problem is in general undecidable, the goal is settled for restricted classes of loops. For the class of solvable loops, introduced by Rodríguez-Carbonell and Kapur (in: Proceedings of the ISSAC, pp 266–273, 2004), one can automatically compute invariants from closed-form solutions of recurrence equations that model the loop behaviour. In this paper we establish a technique for invariant synthesis for loops that are not solvable, termed unsolvable loops. Our approach automatically partitions the program variables and identifies the so-called defective variables that characterise unsolvability. Herein we consider the following two applications. First, we present a novel technique that automatically synthesises polynomials from defective monomials, that admit closed-form solutions and thus lead to polynomial loop invariants. Second, given an unsolvable loop, we synthesise solvable loops with the following property: the invariant polynomials of the solvable loops are all invariants of the given unsolvable loop. Our implementation and experiments demonstrate both the feasibility and applicability of our approach to both deterministic and probabilistic programs.

(Almost) complete characterization of the stability of a discrete-time Hawkes process with inhibition and memory of length two

Abstract We consider a Poisson autoregressive process whose parameters depend on the past of the trajectory. We allow these parameters to take negative values, modelling inhibition. More precisely, the model is the stochastic process $(X_n)_{n\ge0}$ with parameters $a_1,\ldots,a_p \in \mathbb{R}$ , $p\in\mathbb{N}$ , and $\lambda \ge 0$ , such that, for all $n\ge p$ , conditioned on $X_0,\ldots,X_{n-1}$ , $X_n$ is Poisson distributed with parameter $(a_1 X_{n-1} + \cdots + a_p X_{n-p} + \lambda)_+$ . This process can be regarded as a discrete-time Hawkes process with inhibition and a memory of length p. In this paper we initiate the study of necessary and sufficient conditions of stability for these processes, which seems to be a hard problem in general. We consider specifically the case $p = 2$ , for which we are able to classify the asymptotic behavior of the process for the whole range of parameters, except for boundary cases. In particular, we show that the process remains stochastically bounded whenever the solution to the linear recurrence equation $x_n = a_1x_{n-1} + a_2x_{n-2} + \lambda$ remains bounded, but the converse is not true. Furthermore, the criterion for stochastic boundedness is not symmetric in $a_1$ and $a_2$ , in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.

Some infinite series summations involving linear recurrence relations of order 2 and 3

This paper extends known results of second and third order recursive sequences through extensive formulations of properties of the roots of their characteristic equations, some are old but most are new. They are applied to novel studies of $\sum_{n=0}^{\infty}{\frac{a_{mn}}{10^{n+1}}, \ m=1,2,3}$, including their convergence criteria, and applied to many standard sequences, as particular cases of a generic $\left\{a_n\right\}$. The detailed development of the algebra of the pertinent theorems, and their associated lemmas and corollaries, should open up new vistas for interested number theorists with the concluding results on series values.

New Proof of the Property of Stirling Number Based on Fubini Polynomials

The main purpose of this article is using the elementary methods and the properties of the Fubini polynomials to study the congruence properties of a signless Stirling number of the first kind and solve a conjecture proposed by J. H. Zhao and Z. Y. Chen. Without a doubt, the novel approach employed in this work provides a useful reference for researching the congruence properties of other nonlinear binary recursive sequences.

Using Mathematical Models and Extensions to Solve a Robot’s Equidistant Returning Path

The main purpose of this project is to study the Rusty the Robot problem where a robot travels equidistantly from the starting point back to the starting point. We mainly use mathematical models such as trigonometric functions, recursive sequences, and mathematical induction to explore, deduce and demonstrate findings, while using mathematical computer software for calculation and verification. In the beginning, Geogebra was used to explore the original question and allow us to obtain some properties and results. Then, the topic was extended and the relationship between the number of steps and angles was explored, as well as the recursive relationship between the landing points during the travel process. Next, we altered the starting positions of Rusty to see how this affects our findings. Finally, the two intersecting straight lines are extended into three straight lines intersecting at the same point. Based on the angles between them, we discuss the number of returning paths from the starting point to the starting point in an equidistant traveling method. Based on the mathematical model constructed, this study constructs a return path diagram for the robot to return to the starting point in an equidistant manner according to different starting point positions and obtains the recursive relationship of the relative positions on this return path. During the research process of this study, some interesting mathematical theories were obtained. It is expected that these results can be applied to related fields of AI robot movement patterns in the future.

Compiling Recurrences over Dense and Sparse Arrays

We present a framework for compiling recurrence equations into native code. In our framework, users specify a system of recurrences, the types of data structures that store inputs and outputs, and scheduling commands for optimization. Our compiler then lowers these specifications into native code that respects the dependencies in the recurrence equations. Our compiler can generate code over both sparse and dense data structures, and determines if the recurrence system is solvable with the provided scheduling primitives. We evaluate the performance and correctness of the generated code on several recurrences, from domains as diverse as dense and sparse matrix solvers, dynamic programming, graph problems, and sparse tensor algebra. We demonstrate that the generated code has competitive performance to hand-optimized implementations in libraries. However, these handwritten libraries target specific recurrences, specific data structures, and specific optimizations. Our system, on the other hand, automatically generates implementations from recurrences, data formats, and schedules, giving our system more generality than library approaches.

Dynamic partial correlation models

We introduce a new scalable model for dynamic conditional correlation matrices based on a recursion of dynamic bivariate partial correlation models. By exploiting the model’s recursive structure and the theory of perturbed stochastic recurrence equations, we establish stationarity, ergodicity, and filter invertibility in the multivariate setting using conditions for bivariate slices of the data only. From this, we establish consistency and asymptotic normality of the maximum likelihood estimator for the model’s static parameters. The new model outperforms benchmarks like the t-cDCC and the multivariate t-GAS, both in simulations and in an in-sample and out-of-sample asset pricing application to US stock returns.

A recurrence algorithm of time-dependent analysis for concrete bridges

This paper proposes an algorithm for time-dependent effects in the finite element analysis of concrete bridges. Based on the primary influence of concrete shrinkage and creep on bridge structures, the calculation formulas for the equivalent nodal load increments of beam elements caused by the shrinkage and creep effects were deduced according to the initial strain method. To simplify the calculation process and improve calculation speed, the formulas for shrinkage and creep in the bridge design code were fitted with an exponential function, and the recurrence equations for shrinkage and creep calculation were obtained. Thus, a finite element method formulation for concrete shrinkage and creep calculated according to the divided time steps was established. The algorithm was programmed into a self-developed program, and the correctness of the program's calculation was verified through 20 years of on-site monitoring results of a concrete girder cable-stayed bridge with a main span of 500 m. The method proposed in this paper can substantially improve the speed of the program to calculate the time-dependent effects during the construction and operation of large-span concrete bridges, while ensuring the reliability of the calculation results, and is easy to be applied in various types of finite element method programs.

Subspace Identification of Bridge Frequencies Based on the Dimensionless Response of a Two-Axle Vehicle.

As an essential reference to bridge dynamic characteristics, the identification of bridge frequencies has far-reaching consequences for the health monitoring and damage evaluation of bridges. This study proposes a uniform scheme to identify bridge frequencies with two different subspace-based methodologies, i.e., an improved Short-Time Stochastic Subspace Identification (ST-SSI) method and an improved Multivariable Output Error State Space (MOESP) method, by simply adjusting the signal inputs. One of the key features of the proposed scheme is the dimensionless description of the vehicle-bridge interaction system and the employment of the dimensionless response of a two-axle vehicle as the state input, which enhances the robustness of the vehicle properties and speed. Additionally, it establishes the equation of the vehicle biaxial response difference considering the time shift between the front and the rear wheels, theoretically eliminating the road roughness information in the state equation and output signal effectively. The numerical examples discuss the effects of vehicle speeds, road roughness conditions, and ongoing traffic on the bridge identification. According to the dimensionless speed parameter Sv1 of the vehicle, the ST-SSI (Sv1 < 0.1) or MOESP (Sv1 ≥ 0.1) algorithm is applied to extract the frequencies of a simply supported bridge from the dimensionless response of a two-axle vehicle on a single passage. In addition, the proposed methodology is applied to two types of long-span complex bridges. The results show that the proposed approaches exhibit good performance in identifying multi-order frequencies of the bridges, even considering high vehicle speeds, high levels of road surface roughness, and random traffic flows.

English Teachers Teaching Models In English Class

The quality of learning depends on the instructor, serving as a basic calculate in deciding its adequacy. This think about points to create a educator competency show through Lesson Consider (LS) exercises. Show competence alludes to the direction of Indonesian Service of Instruction and Culture No 6565 in 2020. This consider connected the subjective graphic inquire about plan. It utilized the lesson consider approach in creating EFL teacher's proficient information competence, proficient learning hone competence, and career proficient advancement. The members included four EFL Instructors at Senior Tall School 10 Ternate City Indonesia. They comprised of one male and three females. Four EFL instructors were chosen through purposive examining with four criteria: (a) sex, (b) age, (c) instructing encounter (d) instructive foundation, and, (e) work status. Moreover, the think about information were gotten through three LS stages (arrange, teach/do, and reflect/see) for four cycles. A 4-point Likert scale ((1=Thriving; 2= Competent; 3 = Worth; and 4 = progressed) was utilized to degree lesson consider profile perspectives on the improvement of EFL Teachers' Demonstrate competencies and it is an impact on students' communication and basic considering abilities. Information was picked up by a 4-point Likert scale calculated with recurrence and introduction equation through Microsoft Exceed expectations Windows 10. The result appeared that LS exercises affected the improvement of information competencies, proficient learning hones, and proficient advancement of EFL instructors in learning English. It too moves forward communication, basic considering, and understudy excitement for English learning in lesson. This consider gives openings for instructors to share information, great learning hones, and encounter, as well as reflect and learn to collaborate professionally. This think about speaks to a oneof a kind commitment by joining three models of educator competence as an exertion to alter the Autonomous Educational programs.

On Protected Quasi-Metrics

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, the Michael line, and the Khalimsky line, among others. Our motivation is due, in part, to the fact that a successful improvement of the classical Banach fixed-point theorem obtained by Suzuki does not admit a natural and full quasi-metric extension, as we have noted in a recent article. Thus, and with the help of this new structure, we obtained a fixed-point theorem in the framework of Smyth-complete quasi-metric spaces that generalizes Suzuki’s theorem. Combining right completeness with partial ordering properties, we also obtained a variant of Suzuki’s theorem, which was applied to discuss types of difference equations and recurrence equations.