We study the coefficients of algebraic functions∑n≥0f nzn. First, we recall the too-little-known fact that these coefficientsf n always admit a closed form. Then we study their asymptotics, known to beof the type f n ~ CA nn α. When the function is a power seriesassociated to a context-free grammar, we solve a folklore conjecture: thecritical exponents α cannot be 1/3 or −5/2; they in factbelong to a proper subset of the dyadic numbers. We initiate the study of theset of possible values for A. We extend what Philippe Flajoletcalled the Drmota–Lalley–Woods theorem (which states thatα=−3/2 when the dependency graph associated to thealgebraic system defining the function is strongly connected). We fullycharacterize the possible singular behaviours in the non-strongly connectedcase. As a corollary, the generating functions of certain lattice paths andplanar maps are not determined by a context-free grammar (i.e.,their generating functions are not ℕ-algebraic). We give examples ofGaussian limit laws (beyond the case of theDrmota–Lalley–Woods theorem), and examples of non-Gaussianlimit laws. We then extend our work to systems involving non-polynomial entirefunctions (non-strongly connected systems, fixed points of entire functions withpositive coefficients). We give several closure properties forℕ-algebraic functions. We end by discussing a few extensions of ourresults (infinite systems of equations, algorithmic aspects).

The depth of a trie has been deeply studied when the source which produces the words is a simple source (a memoryless source or a Markov chain). When a source is simple but not an unbiased memoryless source, the expectation and the variance are both of logarithmic order and their dominant terms involve characteristic objects of the source, for instance the entropy. Moreover, there is an asymptotic Gaussian law, even though the speed of convergence towards the Gaussian law has not yet been precisely estimated. The present paper describes a ‘natural’ class of general sources, which does not contain any simple source, where the depth of a random trie, built on a set of words independently drawn from the source, has the same type of probabilistic behaviour as for simple sources: the expectation and the variance are both of logarithmic order and there is an asymptotic Gaussian law. There are precise asymptotic expansions for the expectation and the variance, and the speed of convergence toward the Gaussian law is optimal. The paper first provides analytical conditions on the Dirichlet series of probabilities of a general source under which this Gaussian law can be derived: a pole-free region where the series is of polynomial growth. In a second step, the paper focuses on sources associated with dynamical systems, called dynamical sources, where the Dirichlet series of probabilities is expressed with the transfer operator of the dynamical system. Then, the paper extends results due to Dolgopyat, already generalized by Baladi and Vallée, and shows that the previous analytical conditions are fulfilled for ‘most’ dynamical sources, provided that they ‘strongly differ’ from simple sources. Finally, the present paper describes a class of sources not containing any simple source, where the trie depth has the same type of probabilistic behaviour as for simple sources, even with more precise estimates.

We describe a general framework for realistic analysis of sorting algorithms, and we apply it to the average-case analysis of three basic sorting algorithms (QuickSort, InsertionSort, BubbleSort). Usually the analysis deals with the mean number of key comparisons, but here we view keys as words produced by the same source, which are compared via their symbols in lexicographic order. The ‘realistic’ cost of the algorithm is now the total number of symbol comparisons performed by the algorithm, and, in this context, the average-case analysis aims to provide estimates for the mean number of symbol comparisons used by the algorithm. For sorting algorithms, and with respect to key comparisons, the average-case complexity of QuickSort is asymptotic to 2n log n, InsertionSort to n 2/4 and BubbleSort to n 2/2. With respect to symbol comparisons, we prove that their average-case complexity becomes Θ (n log2 n), Θ(n 2), Θ (n 2 log n). In these three cases, we describe the dominant constants which exhibit the probabilistic behaviour of the source (namely entropy and coincidence) with respect to the algorithm.

In this paper we propose a dynamic image-based visual servoing (IBVS) control for a rotary wing unmanned aerial vehicle (UAV) which directly accounts for the vehicle's underactuated dynamic model. The motion control objective is to follow parallel lines and is motivated by power line inspection tasks where the UAV's relative position and orientation to the lines are controlled. The design is based on a virtual camera whose motion follows the onboard physical camera but which is constrained to point downwards independent of the vehicle's roll and pitch angles. A set of image features is proposed for the lines projected into the virtual camera frame. These features are chosen to simplify the interaction matrix which in turn leads to a simpler IBVS control design which is globally asymptotically stable. The proposed scheme is adaptive and therefore does not require depth estimation. Simulation results are presented to illustrate the performance of the proposed control and its robustness to calibration parameter error.

When compared to serial manipulators, parallel manipulators have small workspaces mainly due to their closed-loop structure. As opposed to type I singularities (or inverse kinematic singularities) that are generally encountered at the workspace boundaries, type II singularities characteristically arise within the workspace, and around them, the inverse dynamic solution becomes unbounded. Hence, a desired trajectory passing through a type II singular position cannot be achieved by the manipulator, and its useful workspace becomes further and substantially reduced. It has been previously shown in the literature that if the trajectory is replanned in such a way that the dynamic equations of motion of the manipulator are consistent at a type II singularity, i.e. if the trajectory is consistent, then the manipulator passes through this singular configuration in a controllable manner, while the inverse dynamic solution remains finite. An inconsistent trajectory, on the other hand, is stated in the literature to be unrealizable. However, although seems a promising technique, trajectory replanning itself is also a deviation from the originally desired trajectory, and there might be cases in applications where, due to some task-specific reasons, the desired trajectory, although inconsistent, is not allowed to be replanned to satisfy the consistency conditions. In this paper, a method of singularity robust balancing is proposed for parallel manipulators passing through type II singular configurations while following inconsistent trajectories. By this means, an originally unrealizable inconsistent trajectory passing through a type II singularity can be followed without any deviation, while the required actuator forces remain bounded after the manipulator is balanced according to the design methodology presented in this study. The effectiveness of the introduced method is shown through numerical simulations considering a planar 3-DOF 2-PRR parallel manipulator under different balancing scenarios.