Recursive Terms Research Articles

Augmented Lagrangian index-3 semi-recursive formulations with projections

Sensitivity analysis represents a powerful tool for the optimization of multibody system dynamics. The performance of a gradient-based optimization algorithm is strongly tied to the dynamic and the sensitivity formulations considered. The accuracy and efficiency are critical to any optimization problem, thus they are key factors in the selection of the dynamic and sensitivity analysis approaches used to compute an objective function gradient. Semi-recursive methods usually outperform global methods in terms of computational time, even though they involve sometimes demanding recursive procedures. Semi-recursive methods are well suited to be combined with different constraints enforcement schemes as the augmented Lagrangian index-3 formulation with velocity and acceleration projections (ALI3-P), taking advantage of the robustness, accurate fulfillment of constraint equations and the low computational burden. The sensitivity analysis of the semi-recursive ALI3-P formulation is studied in this document by means of the direct differentiation method. As a result, a semi-recursive ALI3-P sensitivity formulation is developed for an arbitrary reference point selection, and then two particular versions are unfolded and implemented in the general purpose multibody library MBSLIM, using as reference point the center of mass (RTdyn0) or the global origin of coordinates (RTdyn1). Besides, the detailed derivatives of the recursive terms are provided, which will be useful not only for the direct sensitivity formulation presented herein, but also for other sensitivity formulations relying on the same recursive expressions. The implementation has been tested in two numerical experiments, a five-bar benchmark problem and a buggy vehicle.

Functional equations with multiple recursive terms

In this paper, we study a functional equation for generating functions of the form f(z) = g(z) sum _{i=1}^M p_i f(alpha _i(z)) + K(z), viz. a recursion with multiple recursive terms. We derive and analyze the solution of this equation for the case that the alpha _i(z) are commutative contraction mappings. The results are applied to a wide range of queueing, autoregressive and branching processes.

A Krein space-based approach to event-triggered [formula omitted] filtering for linear discrete time-varying systems

This paper is concerned with the problem of event-triggered H∞ filtering for linear discrete time-varying (LDTV) systems. Using the lifting technique, we firstly establish an equivalent relationship with a certain equivalent minimum problem of indefinite quadratic form subject to LDTV systems with non-uniform sampling periods. Then, based on Krein space projection and innovation analysis, sufficient and necessary conditions for the existence of desired filter are derived and a feasible solution is obtained in terms of Riccati recursions. Thus, an algorithm based on the time-update and event-update recursions is given for the implementation of event-triggered H∞ filtering. Different from some existing results, a new event-triggered H∞ filtering scheme is provided so that the estimation error can be completely decoupled from the event-triggered transmission error. Moreover, the new proposed Krein space approach is less conservative and more computational attractive than the existing methods based on recursive linear inequality matrix. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.

Optimal linear filtering for networked control systems under time correlated fading channel and noise

In this paper, the optimal filtering problem for discrete-time networked control systems affected by fading channels is investigated where both the fading channel and channel noise are time-correlated. Each fading channel is modeled as a scalar uncertain stochastic discrete-time system, and the channel noise is described by a discrete-time linear system perturbed by white noise. By using multiple recursive terms and the correlations of the signals in the systems, an optimal linear filter is proposed such that it is implemented in a recursive manner. The optimal filter designed has time independent computation structure, hence is easier to be implemented in practice. A vehicle tracking example is provided to illustrate the effectiveness and performance of the theoretic results obtained.

Characterising Brouwer’s continuity by bar recursion on moduli of continuity

We identify bar recursion on moduli of continuity as a fundamental notion of constructive mathematics. We show that continuous functions from the Baire space $${{\mathbb {N}}}^{{\mathbb {N}}}$$ to the natural numbers $${\mathbb {N}}$$ which have moduli of continuity with bar recursors are exactly those functions induced by Brouwer operations. The connection between Brouwer operations and bar induction allows us to formulate several continuity principles on the Baire space stated in terms of bar recursion on continuous moduli which naturally characterise some variants of bar induction. These principles state that a certain kind of continuous function from $${{\mathbb {N}}}^{{\mathbb {N}}}$$ to $${\mathbb {N}}$$ admits a modulus of continuity with a bar recursor. The results for the Baire space are recast in the setting of the Cantor space $${{\{ 0,1 \}}}^{{\mathbb {N}}}$$ using the notion of fan recursor, a bar recursor for binary trees. This yields characterisations of uniformly continuous functions on the Cantor space and fan theorem in terms of fan recursors. The results for the Cantor space hold over the extensional version of intuitionistic arithmetic in all finite types ( $${\mathsf {E}}\text {-}\mathsf {HA}^\omega $$ ), and those for the Baire space hold over $${\mathsf {E}}\text {-}\mathsf {HA}^\omega $$ extended with the type for Brouwer operations. Our work places Spector’s bar recursion in a proper context of Brouwer’s intuitionistic mathematics and clarifies the connection between bar recursion and bar induction.

A novel semi-analytical solution to Jeffery-Hamel equation

A new approach based on the Adomian decomposition and the Fourier transform is introduced. The method suggests a solution for the well-known magneto-hydrodynamic (MHD) Jeffery-Hamel equation. Results of Adomian decomposition method combined with Fourier transform are compared with exact and numerical methods. The FTADM as an exclusive and new method satisfies all boundary and initial conditions over the entire spatial and temporal domains. Moreover, using the FTADM leads to rapid approach of approximate results toward the exact solutions is demonstrated. The second derivative of Jeffery-Hamel solution related to the similar number of items of recursive terms under a vast spatial domain shows the maximum error in the order of comparing to exact and numerical solutions. The results also imply that the FTADM can be considered as a precise approximation for solving the third-order nonlinear Jeffery-Hamel equations.

The expressiveness of looping terms in the semantic programming

We consider the language of $\Delta_0$-formulas with list terms interpreted over hereditarily finite list superstructures. We study the complexity of reasoning in extensions of the language of $\Delta_0$-formulas with non-standard list terms, which represent bounded list search, bounded iteration, and bounded recursion. We prove a number of results on the complexity of model checking and satisfiability for these formulas. In particular, we show that the set of $\Delta_0$-formulas with bounded recursive terms true in a given list superstructure $HW(\mathcal{M})$ is non-elementary (it contains the class kEXPTIME, for all $k\geqslant 1$). For $\Delta_0$-formulas with restrictions on the usage of iterative and recursive terms, we show lower complexity.

Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences

<p style='text-indent:20px;'>We consider three matrix models of order 2 with one random entry <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> and the other three entries being deterministic. In the first model, we let <inline-formula><tex-math id="M2">\begin{document}$ \epsilon\sim \rm{Bernoulli}\left(\frac{1}{2}\right) $\end{document}</tex-math></inline-formula>. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when <inline-formula><tex-math id="M3">\begin{document}$ \epsilon\sim \rm{Bernoulli}\left(p\right) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ p\in [0, 1] $\end{document}</tex-math></inline-formula> is a parameter. Both of these models are related to random Fibonacci sequences. In the last model, we compute the Lyapunov exponent exactly when the random entry is replaced with <inline-formula><tex-math id="M5">\begin{document}$\xi\epsilon$\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M6">\begin{document}$\epsilon$\end{document}</tex-math></inline-formula> is a standard Cauchy random variable and <inline-formula><tex-math id="M7">\begin{document}$\xi$\end{document}</tex-math></inline-formula> is a real parameter. We then use Monte Carlo simulations to approximate the variance in the CLT for both parameter models.

Recursive state estimation for linear systems with lossy measurements under time-correlated multiplicative noises

This paper is concerned with the recursive state estimation problem for a class of linear discrete-time systems with lossy measurements and time-correlated multiplicative noises (TCMNs). The lossy measurements result from one-step transmission delays and packet dropouts. Different from the traditional white multiplicative noises, TCMNs are included in the measurement model in order to reflect engineering practice. Utilizing the state augmentation approach, the system under investigation is first converted into a stochastic parameter system, and some new recursive terms (including the estimation for the product of state and multiplicative noises) are introduced to handle the difficulties caused by the TCMNs. Then, by the well-known projection theorem, recursive state estimation algorithms are developed in the sense of minimum mean-square error, which facilitate the design of the filter, the multi-step predictor and the smoothers. The proposed algorithms are explicitly dependant on the key system parameters including the covariances of the TCMNs, the occurrence probabilities of the transmission delays and the packet losses. Finally, simulation results illustrate the effectiveness of the presented estimation algorithms.

Data-sampling controllability of multi-agent systems

Abstract In this paper, we consider data-sampling controllability of multi-agent systems (MASs), where the interconnection topology is directed and weighted and the nodes have generic linear kinetic dynamics. First, the asynchronous data sampling protocols and synchronous data sampling protocols are proposed, respectively. Then the discussions focus on deriving the necessary and sufficient conditions for data-sampling controllability of MASs with a single leader and multiple leaders. The criteria for data-sampling controllability are derived in terms of recursions of matrix multiplication. For the single-leader situation, the asynchronous data-sampling controllability is equivalently transformed to be the output controllability of a discrete linear system. For the multiple-leader situation, the asynchronous data-sampling controllability is equivalently transformed to be the output controllability of a discrete time-delayed linear system. Finally, the synchronous data-sampling controllability is discussed as a special case of asynchronous data-sampling controllability. Two numerical examples are given to illustrate the effectiveness of the theoretical results.

Planning, execution, and revision in mathematics problem solving: Does the order of the phases matter?

The present study analysed the mathematical problem-solving processes, in terms of linearity and recursion, and the relationship with actual and self-perceived performances of a sample of 524 students of upper elementary students. The results showed a more linear than recursive process while performing the tasks, mainly characterized by continuity. The use of planning strategies before execution and the use of revision strategies after this phase were both significantly related to good performance, even if rates of success were low. The presence of a linear and hierarchical resolution process was related to students’ judgments of success, while recursion, or going back in the process, was associated with judgments of failure. Results are discussed in the light of current research on mathematics problem-solving.

Special Kähler geometry of the Hitchin system and topological recursion

We investigate the special K\"ahler geometry of the base of the Hitchin integrable system in terms of spectral curves and topological recursion. The Taylor expansion of the special K\"ahler metric about any point in the base may be computed by integrating the $g = 0$ Eynard-Orantin invariants of the corresponding spectral curve over cycles. In particular, we show that the Donagi-Markman cubic is computed by the invariant $W^{(0)}_3$. We use topological recursion to go one step beyond this and compute the symmetric quartic of second derivatives of the period matrix.

Recursive Terms in Semantic Programming

For constructing an enrichment of a language with restricted quantifiers, we extend the notion of term by the construction of the terms conditional and recursive with respect to lists. We show that the so-obtained extension of the language of formulas with restricted quantifiers over structures with hereditary finite lists is a conservative enrichment and the new terms are Δ-definable in the basic language.

Natural Cubic Spline Model for Estimating Volatility

Volatility measures the dispersion of returns for a market variable since a reasonable estimation of the volatility is an appropriate starting point for assessing investment risks and monetary policymaking. These risks are usually assessed by using the GARCH (1,1) model. However, the recursive term in this model makes finding the derivatives of the likelihood function mathematically intractable. In this study, the natural cubic spline model is used to estimate the volatility by fitting it to the absolute returns of the data. In estimating the parameters, the Maximum Likelihood method was applied while a simple algebra was used to find its derivatives. The damped Newton-Raphson method was then used to maximize the likelihood function with the R programming software. The proposed method was illustrated using the absolute returns of the crude oil prices data from West Texas Intermediate, and it showed similar results with the popular GARCH (1,1) model. The natural cubic spline can be an alternative for estimating the volatility of any financial time series data.

Mathematics in Caging of Robotics

It is a crucial problem in robotics field to cage an object using robots like multifingered hand. However the problem what is the caging for general geometrical objects and robots has not been well-described in mathematics though there were many rigorous studies on the methods how to cage an object by certain robots. In this article, we investigate the caging problem more mathematically and describe the problem in terms of recursion of the simple euclidean moves. Using the description, we show that the caging has the degree of difficulty which is closely related to a combinatorial problem and a wire puzzle. It implies that in order to capture an object by caging, from a practical viewpoint the difficulty plays an important role.

An Extended &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$H_{i}/H_\infty $ &lt;/tex-math&gt; &lt;/inline-formula&gt; Optimization Approach to Fault Detection of INS/GPS-Integrated System

This paper deals with the problem of robust fault detection (FD) for an integrated system of inertial navigation system and global position system (INS/GPS). Under the assumption of process disturbance and measurement noise being $l_{2}$ -norm bounded, a state-space model with potential additive fault is first introduced to describe the error dynamics of the INS/GPS integrated system. For the purpose of the FD, an observer-based FD filter (FDF) is used as a residual generator and the design of the FDF is formulated so as to find an observer gain matrix and a postfilter in the framework of $H_{i}/H_\infty $ optimization. By utilizing the techniques of innovation analysis and orthogonal projection, a solution to the extended $H_{i}/H_\infty $ -FDF is given in terms of Riccati recursions. Furthermore, a strategy of residual evaluation is also considered, which concerns the selection of an evaluation function and a threshold. Finally, a flight experiment is implemented to show the effectiveness of the proposed method.

Optimal Filtering for Discrete-Time Linear Systems With Time-Correlated Multiplicative Measurement Noises

In this note, the filtering problem for discrete-time linear systems with time-correlated multiplicative measurement noises is considered where the vector consisting of all the multiplicative measurement noises can be described by a linear system model with white noise. By introducing several new recursive terms, a novel algorithm for optimal filtering of the system under consideration is proposed in the sense of linear minimum mean-square error. The proposed algorithm is recursive and has time-independent complexity. Computer simulations are provided to illustrate the performance of the proposed algorithm.

Responding to Emergency in Democratic Athens: The Case of Anti-Tyranny Legislation

This article attempts to understand Athenian anti-tyranny laws as offering a democratic response to emergency. These laws sought to promote the killing of “tyrants” who overthrew the democracy and to punish those who collaborated with any non-democratic regime. I argue that the anti-tyranny laws reflect a broader series of measures enacted in the aftermath of the oligarchic coup of 411 BCE that sought to facilitate the collective action of the dēmos (people) as a means of maintaining and preserving Athenian democracy. Such legislation, I suggest, is best understood when situated within the context of a distinctly Athenian conception of the rule of law that viewed the relationship between the rule of law and popular sovereignty in recursive terms. The article places this Athenian response to emergency in dialogue with recent work on emergency politics in contemporary political theory.

On the greatest solution of equations in [formula omitted

It is shown that, a recursive equation X=RStX in the LLTS-oriented process calculus CLLR may have more than one consistent solution when X occurs in the scope of a conjunction in tX, and the recursive term 〈X|X=tX〉 is the greatest solution of this equation w.r.t. Lüttgen and Vogler's ready simulation whenever X is strongly guarded in tX.

HiH∞ optimization based fault detection for linear discrete time-varying systems with delayed state

This paper investigates the problem of fault detection for linear discrete time-varying systems with delayed state by using Hi/H∞ and/or Hi/H∞ optimization. A residual generator, which is the so-called Hi/H∞ fault detection filter (FDF), is designed by finding filter gain matrices and a post-filter. By utilizing projection and innovation re-organization technique, an optimal Hi/H∞ -FDF can be given in terms of recursions. Moreover, the obtained evaluation function is a unified solution and, in order to detect the presence of a fault, the selection of a threshold is dependent on the characteristics of unknown input and a prescribed false alarm rate. A numerical example is used to demonstrate the effectiveness of the proposed method.