Recursive Properties Research Articles

Virtual cyclic cellular automata, finite group actions and recursive properties

The aim of these notes is three fold. First we introduce virtual cyclic cellular automata and show that the inverse of a reversible (2R+1)-cyclic cellular automaton with periodic boundary conditions is a virtual cyclic cellular automaton. These virtual automata have two special characteristics: they have active and non active cells at specific steps of times and they reflect certain periodicity. We will relate these particularities with a finite cyclic group action on the cellular automaton, and prove that the inverse transition dipolynomial is an invariant dipolynomial under this action. Secondly, we use a recursive estimation of neighbours (REN) algorithm to produce direct examples of virtual cyclic cellular automata, which moreover generalize some of the cellular automata used in applications like collective control or traffic patterns. We also propose a new REN algorithm which allows us to reinterprete a (2R+1)-cyclic cellular automata as a recursive sequence originated from the elementary cellular automaton with base rule 150, and which motivate us to introduce a new notion of a recursive Wolfram number for a (2R+1)-cyclic cellular automaton. Finally we show that this recursive Wolfram number can be computed by the new REN algorithm applied to the base rule 150 and its complementary 105 rule.

An MILP approach for persistent coverage tasks with multiple robots and performance guarantees

Multiple robots are increasingly being considered in a variety of tasks requiring continuous surveillance of a dynamic area, examples of which are environmental monitoring, and search and rescue missions. Motivated by these applications, in this paper we consider the multi-robot persistent coverage control problem over a grid environment. The goal is to ensure a desired lower bound on the coverage level of each cell in the grid, that is decreasing at a given rate for unoccupied cells. We consider a finite set of candidate poses for the agents and introduce a directed graph with nodes representing their admissible poses. We formulate a persistent coverage control problem as a MILP problem that aims to maximize the coverage level of the cells over a finite horizon. To solve the problem, we design a receding horizon scheme (RHS) and prove its recursive feasibility property by introducing a set of time-varying terminal constraints to the problem. These terminal constraints ensure that the agents are always able to terminate their plans in pre-determined closed trajectories. A two-step method is proposed for the construction of the closed trajectories, guaranteeing the satisfaction of the coverage level lower bound constraint, when the resulting closed trajectories are followed repeatedly. Due to the special structure of the problem, agents are able to visit every cell in the grid repeatedly within a worst-case visitation period. Finally, we provide a computational time analysis of the problem for different simulated scenarios and demonstrate the performance of the RHS problem by an illustrative example.

Partial (In)Completeness in abstract interpretation: limiting the imprecision in program analysis

Imprecision is inherent in any decidable (sound) approximation of undecidable program properties. In abstract interpretation this corresponds to the release of false alarms, e.g., when it is used for program analysis and program verification. As all alarming systems, a program analysis tool is credible when few false alarms are reported. As a consequence, we have to live together with false alarms, but also we need methods to control them. As for all approximation methods, also for abstract interpretation we need to estimate the accumulated imprecision during program analysis. In this paper we introduce a theory for estimating the error propagation in abstract interpretation, and hence in program analysis. We enrich abstract domains with a weakening of a metric distance. This enriched structure keeps coherence between the standard partial order relating approximated objects by their relative precision and the effective error made in this approximation. An abstract interpretation is precise when it is complete. We introduce the notion of partial completeness as a weakening of precision. In partial completeness the abstract interpreter may produce a bounded number of false alarms. We prove the key recursive properties of the class of programs for which an abstract interpreter is partially complete with a given bound of imprecision. Then, we introduce a proof system for estimating an upper bound of the error accumulated by the abstract interpreter during program analysis. Our framework is general enough to be instantiated to most known metrics for abstract domains.

The Gauss sums involving 24-order character and their recursive properties

<abstract><p>The main purpose of this paper is to use elementary and analytic methods to study the calculating problem of one kind of Gauss sums and obtain an exact computational formula for it.</p></abstract>

A Young diagram expansion of the hexagonal Wilson loop (amplitude) in mathcal{N} = 4 SYM

We shall interpret the null hexagonal Wilson loop (or, equivalently, six gluon scattering amplitude) in 4D mathcal{N} = 4 Super Yang-Mills, or, precisely, an integral representation of its matrix part, via an ADHM-like instanton construction. In this way, we can apply localisation techniques to obtain combinatorial expressions in terms of Young diagrams. Then, we use our general formula to obtain explicit expressions in several explicit cases. In particular, we discuss those already available in the literature and find exact agreement. Moreover, we are capable to determine explicitly the denominator (poles) of the matrix part, and find some interesting recursion properties for the residues, as well.

A New Gauss Sum and Its Recursion Properties

In this paper, we introduce a new Gauss sum, and then we use the elementary and analytic methods to study its various properties and prove several interesting three-order linear recursion formulae for it.

Recursive properties of generalized ordinary differential equations and applications

The aim of this paper is to obtain results on topological properties of flows for nonnegative time in the framework of generalized ODEs. We define the concept of generalized semiflow and we present some recursive properties as minimality and recurrence. Using correspondence theorems, we translate our results to measure differential equations which are known to be special cases of generalized ODEs. Some examples are presented to illustrate the theory.

Decentralized closed-loop optimization for 6-DOF self-assembly satellites

A decentralized closed-loop optimization based on the difference of convex programming is proposed to assemble large telescopes autonomously in-orbit. This study focuses on assembly problems of mirror satellites with six degrees of freedom. Firstly, the hp-Radau pseudospectral collocation method is adopted to increase computational efficiency. Subsequently, the nonconvex constraints are convexified by applying undominated decomposition and sequential convex programming. Additionally, the fixed trajectory assumption enables the translational optimization problem to be a decentralized problem with convergence-guaranteed proof. The six degrees of freedom optimization problem is decoupled by utilizing the proposed algorithm in translational and rotational planning, and coupled by thruster allocation method. To improve the robustness of the system, the algorithm is wrapped in the model predictive control framework, and the recursive feasible property is ensured using the outer-bounding tube. Therefore, the closed-loop optimal control is robust under additive uncertainties. In the numerical experiments, mirror satellites are released from a high-accuracy fitting carrier satellite to accomplish the assembly, and these satellites are utilized to verify the high-speed algorithm and the effectiveness of the closed-loop optimization.

Continuous phase transitions on Galton–Watson trees

AbstractDistinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton–Watson trees. For example, let $\mathcal{T}_1$ be the event that a Galton–Watson tree is infinite and let $\mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $\mathcal{T}_1$ holds if and only if $\mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $\mathcal{T}_2$ holds if and only if $\mathcal{T}_2$ holds for at least two of these trees. The probability of $\mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $\mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a non-zero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterise the event undergoing the phase transition.

Complete formalization of taxa with their names, contents and descriptions improves taxonomic databases and access to the taxonomic knowledge they support

As concepts, taxa should be addressed through their name, their description (definition by intension) or their content (definition by extension). Each part of this taxonomic triplet is necessary but none alone sufficient to fully formalize a taxon without ambiguity. However, in practice, taxa are most often only addressed through their name and their extension, circumscribing them by their components (child-taxa) that are added or excluded following the progress of knowledge and discoveries. Here we show that without modifying its content, moving a taxon in the classification affects fundamentally its definition by intension, while the taxon remains considered the same. During taxonomic revisions, a taxonomic concept is not only extended or restricted according to its components, but it can be definitively modified when its contextualization in a classification changes. Because of the recursive property of the classification, defining a taxon by intension is equivalent to addressing its classification, and the taxon is necessarily contextualized. Accordingly, a taxon is therefore only fully defined as a triplet, documenting at the same time its name, its content and its classification, each issue varying independently from each other. If one wants to correctly report or aggregate taxonomic knowledge in taxonomic databases, managing these three issues independently is therefore necessary to assure the relational integrity in the database. Particularly, all aspects managing the taxon contextualization (e.g. its classification) and its eventual changes should be included in taxonomic databases and some recommendations are provided. As discussed, consequences in the taxonomic treatment of the related data that follows the taxa (not the name!) remain underestimated and still need to be better understood and formalized. As complex human concepts, taxa modelization for the digital world remains a real challenge and shows that taxonomy is a much more complex science than usually expected.

The Recursive Properties of the Error Term of the Fourth Power Mean of the Generalized Cubic Gauss Sums

In this paper, we use the analytic methods and the properties of the classical Gauss sums to study the properties of the error term of the fourth power mean of the generalized cubic Gauss sums and give two recurrence formulae for it.

Influence of time-dependent seismic hazard on structural design

Recent works on seismic hazard have introduced the concept of time-dependent seismic hazard and different models have been proposed to predict the inter-arrival time between consecutive events. Currently, the reliability assessment of structures and relevant design rules proposed by the codes are based on the Poisson recursive model, for which the frequency of the occurrence of seismic events does not change over time. This paper presents results on the impact of Time-Dependent Seismic Hazard on structural design, by evaluating the strength required by the structure (seismic capacity) for different time intervals elapsing from the last event. “Seismic capacity” is understood here as the capacity required to provide a fixed reliability level, measured by the failure rate. Two different seismic scenarios have been investigated and results concerning different site-to-source distance, capacity dispersion of the structure and different recurrence properties of the time-dependent source have been discussed. Finally, the impact of recursive properties of time-dependent model is analysed and discussed. The results obtained from the analyses highlight a significant influence of time-dependent hazard properties on the structural capacity required to attain a target reliability, and give evidence to the different roles played by the parameters considered in the analysis. Within the set of the considered case studies, the ratios between seismic capacities evaluated by the time-dependent and non-time-dependent model span the range [0,1.32] in the first scenario, where inter-arrival time varies from 0 to two times the mean return period. The second scenario involves multiple sources and observed ratios were in the range [0.84,1.23], extreme values are relevant to inter-arrival times equal to 139 y and 371 y, respectively.

Approximating the [formula omitted]th root by composite rational functions

A landmark result from rational approximation theory states that x1∕p on [0,1] can be approximated by a type-(n,n) rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev functions (for the square root and sign functions), we investigate approximating x1∕p by composite rational functions of the form rk(x,rk−1(x,rk−2(⋯(x,r1(x,1))))). While this class of rational functions ceases to contain the minimax (best) approximant for p≥3, we show that it achieves approximately pth-root exponential convergence with respect to the degree. Moreover, crucially, the convergence is doubly exponential with respect to the number of degrees of freedom, suggesting that composite rational functions are able to approximate x1∕p and related functions (such as |x| and the sector function) with exceptional efficiency.

Analysis of Longitudinal Guided Wave Propagation in the Functionally Graded Hollow Cylinder Using State-Vector Formalism and Legendre Polynomial Hybrid Approach

A numerical approach for analyzing the guided wave propagation behavior in a functionally graded material (FGM) hollow cylinder is presented. Based on the three-dimensional linear elasticity theory, the state matrix form of displacement and stress component is introduced into the differential governing equation. Combining with boundary conditions, the dispersion characteristic equations can be constructed. Meanwhile, taking the orthogonal completeness of Legendre series into consideration, the exhausting root findings of transcendental equation can be avoided, and it transforms the redundant integral operators into analytical expressions by means of its recursive properties. Moreover, it is not only unnecessary to stratify FGMs, and introducing the univariate nonlinear regression to modelling the arbitrary gradient distribution, and transforming the redundant integral operator into an analytical expression by means of recursive property. It should be pointed out that the coefficient matrix consists only zero and constant terms, which are related to the elastic constants and normalized wave number. And it provides a fast and effectively approach to extracting the dispersion curves, displacement distribution and stress profile, simultaneously. Meanwhile, the proposed method can get rid of the exhausted root-finding algorithm, and accordingly distinguish the different modes and the dispersion curves simultaneously. Finally, the typical non-stratified computation of dispersion curves of longitudinal guided wave for FGM hollow cylinder is realized. The accuracy of the proposed method is further demonstrated through comparison with the available data from Legendre polynomial method. Several numerical cases about iron based alumina FGM cylinder are studied, and the convergence of the present polynomial approach is discussed. Then the effect of the circumferential wave number, gradient variation, diameter-thickness ratio on the dispersion characteristics are illustrated. Moreover, the distribution characteristics of displacement and stress profiles of F(1,1) mode at a given frequency are discussed in detail. It is shown that the state-vector formalism and Legendre polynomial hybrid approach is a powerful tool in solving wave motions in FGM elastic cylinder, which lays a theoretical foundation for the non-destructive evaluation and quantitative characterization of the gradient structure characteristics of functionally graded hollow cylinder.

Recurrence Relations for Knot Polynomials of Twist Knots

This paper gives HOMFLY polynomials and Kauffman polynomials L and F of twist knots as recurrence relations, respectively, and also provides some recursive properties of them.

An Incremental and Backward-Conflict Guided Method for Unfolding Petri Nets

The unfolding technique of Petri net can characterize the real concurrency and alleviate the state space explosion problem. Thus, it is greatly suitable to analyze/check some potential errors in concurrent systems. During the unfolding process of a Petri net, the calculations of configurations, cuts, and cut-off events are the key factors for the unfolding efficiency. However, most of the unfolding methods do not specify a highly efficient calculations on them. In this paper, we reveal some recursive relations and structural properties of these factors. Subsequently, we propose an improved method for computing configurations and cuts. Meanwhile, backward conflicts are used to guide the calculations of cut-off events. Moreover, a case study and a series of experiments are done to illustrate the effectiveness and application scenarios of our methods.

A functional equation with polynomial solutions and application to Neural Networks

We construct and discuss a functional equation with contraction property. The solutions are real univariate polynomials. The series solving the natural fixed point iterations have immediate interpretation in terms of Neural Networks with recursive properties and controlled accuracy.

Robust Nonlinear Predictive Control through qLPV embedding and Zonotope Uncertainty Propagation

This paper presents a novel robust Model Predictive Control (MPC) algorithm for nonlinear systems represented through quasi-Linear Parameter Varying (qLPV) models. The nominal MPC predictions are made considering a frozen scheduling parameter guess, which is computationally cheaper than nonlinear predictions, while zonotopes bound the disturbance propagation along the prediction. These sets are computed with respect to the bounds of the variation of scheduling parameters, offering reduced conservatism of the closed-loop dynamics and ensuring input-to-state stability and recursive feasibility properties. A DC-DC converter benchmark example is used to illustrate the advantages of the proposed method.

Determinantal Expressions and Recursive Relations for the Bessel Zeta Function and for a Sequence Originating from a Series Expansion of the Power of Modified Bessel Function of the First Kind

In the paper, by virtue of a general formula for any derivative of the ratio of two differentiable functions, with the aid of a recursive property of the Hessenberg determinants, the authors establish determinantal expressions and recursive relations for the Bessel zeta function and for a sequence originating from a series expansion of the power of modified Bessel function of the first kind.

Circumferential thermoelastic Lamb wave in fractional order cylindrical plates

AbstractAn improved Legendre polynomial series approach (AILPSA) is presented to investigate the circumferential thermoelastic Lamb wave in a fractional order orthotropic cylindrical plate. In the AILPSA, the analytical integration is developed based on the orthogonality and recursive properties of the Legendre polynomial to simplify the integral computation involved in the solving progress. As a consequence, the computational efficiency is improved significantly. Results are compared with those from the previous article to confirm the validity of the introduced method. Using the AILPSA, guided wave characteristics in various fractional order orthotropic cylindrical plates are investigated by solving the eigenvalues and eigenvectors of the system of algebraic equations. The influences of fractional order and radius‐thickness ratio on dispersion curve and displacement, temperature amplitudes are illustrated. The results show that the fractional order almost has no effect on the phase velocities of quasi‐elastic wave modes, but has notable effect on their attenuations. A smaller fractional order means a smaller temperature amplitude for the quasi‐elastic wave mode, but means a larger temperature amplitude for the thermal wave mode. A smaller radius thickness ratio indicates a smaller attenuation and a larger temperature amplitude.