Evaluating parametric holonomic sequences using rectangular splitting

We adapt the rectangular splitting technique of Paterson and Stockmeyer to the problem of evaluating terms in holonomic sequences that depend on a parameter. This approach allows computing the $n$-th term in a recurrent sequence of suitable type using $O(n^{1/2})$ "expensive" operations at the cost of an increased number of "cheap" operations. Rectangular splitting has little overhead and can perform better than either naive evaluation or asymptotically faster algorithms for ranges of $n$ encountered in applications. As an example, fast numerical evaluation of the gamma function is investigated. Our work generalizes two previous algorithms of Smith.Comment: 8 pages, 2 figure

Computing hypergeometric functions rigorously

We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions ${}_0F_1$, ${}_1F_1$, ${}_2F_1$ and ${}_2F_0$ (or the Kummer $U$-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function ${}_pF_q$ and computation of high-order parameter derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case E-G in section 8.5 (table 6, figure 2); adjusted paper siz

Polynomial tuning of multiparametric combinatorial samplers

Boltzmann samplers and the recursive method are prominent algorithmic frameworks for the approximate-size and exact-size random generation of large combinatorial structures, such as maps, tilings, RNA sequences or various tree-like structures. In their multiparametric variants, these samplers allow to control the profile of expected values corresponding to multiple combinatorial parameters. One can control, for instance, the number of leaves, profile of node degrees in trees or the number of certain subpatterns in strings. However, such a flexible control requires an additional non-trivial tuning procedure. In this paper, we propose an efficient polynomial-time, with respect to the number of tuned parameters, tuning algorithm based on convex optimisation techniques. Finally, we illustrate the efficiency of our approach using several applications of rational, algebraic and P\'olya structures including polyomino tilings with prescribed tile frequencies, planar trees with a given specific node degree distribution, and weighted partitions.Comment: Extended abstract, accepted to ANALCO2018. 20 pages, 6 figures, colours. Implementation and examples are available at [1] https://github.com/maciej-bendkowski/boltzmann-brain [2] https://github.com/maciej-bendkowski/multiparametric-combinatorial-sampler

Arbitrary-precision computation of the gamma function

We discuss the best methods available for computing the gamma function $\Gamma(z)$ in arbitrary-precision arithmetic with rigorous error bounds. We address different cases: rational, algebraic, real or complex arguments; large or small arguments; low or high precision; with or without precomputation. The methods also cover the log-gamma function $\log \Gamma(z)$, the digamma function $\psi(z)$, and derivatives $\Gamma^{(n)}(z)$ and $\psi^{(n)}(z)$. Besides attempting to summarize the existing state of the art, we present some new formulas, estimates, bounds and algorithmic improvements and discuss implementation results

Autonomous robot systems and competitions: proceedings of the 12th International Conference

This is the 2012’s edition of the scientific meeting of the Portuguese Robotics Open (ROBOTICA’ 2012). It aims to disseminate scientific contributions and to promote discussion of theories, methods and experiences in areas of relevance to Autonomous Robotics and Robotic Competitions. All accepted contributions are included in this proceedings book. The conference program has also included an invited talk by Dr.ir. Raymond H. Cuijpers, from the Department of Human Technology Interaction of Eindhoven University of Technology, Netherlands.The conference is kindly sponsored by the IEEE Portugal Section / IEEE RAS ChapterSPR-Sociedade Portuguesa de Robótic

Learning to represent surroundings, anticipate motion and take informed actions in unstructured environments

Contemporary robots have become exceptionally skilled at achieving specific tasks in structured environments. However, they often fail when faced with the limitless permutations of real-world unstructured environments. This motivates robotics methods which learn from experience, rather than follow a pre-defined set of rules. In this thesis, we present a range of learning-based methods aimed at enabling robots, operating in dynamic and unstructured environments, to better understand their surroundings, anticipate the actions of others, and take informed actions accordingly

The handbook of zonoid calculus

In this work, we present a new method of computation that we call zonoid calculus. It is based on a particular class of convex bodies called zonoids and on a representation of zonoids using random vectors. Concretely, this is a recipe to build multilinear maps on spaces of zonoids from multilinear maps on the underlying vector spaces. We call this recipe the fundamental theorem of zonoid calculus (FTZC). Using this and the wedge product in the exterior algebra we build the zonoid algebra, that is a structure of algebra on the space of convex bodies of the exterior algebra of a vector space. We show how this relates to the notion of mixed volume on one side and to random determinants on the other. This produces new inequalities on expected absolute determinants. We also show how this applies in two detailed examples: fiber bodies and non centered Gaussian determinants. We then use FTZC to produce a new function on zonoids of a complex vector space that we call the mixed J-volume. We uncover a link between the zonoid algebra and the algebra of valuations on convex bodies. We prove that the wedge product of zonoids extends Alesker’s product of smooth valuations. Finally we apply the previous results to integral geometry in two different context. First we show how, in Riemannian homogeneous spaces, the expected volume of random intersections can be computed in the zonoid algebra. We use this to produce a new inequality modelled on the Alexandrov–Fenchel inequality, and to compute formulas for random intersection of real submanifolds in complex projective space. Secondly, we prove how a Kac-Rice type formula can relate to the zonoid algebra and a certain zonoid section. We use this to study the expected volume of random submanifolds given as the zero set of a random function. We again produce an inequality on the densities of expected volume modelled on the Alexandrov–Fenchel inequality, as well as a general Crofton formula in Finsler geometry

Quantum logic and entanglement by neutral Rydberg atoms: methods and fidelity

Quantum gates and entanglement based on dipole-dipole interactions of neutral Rydberg atoms are relevant to both fundamental physics and quantum information science. The precision and robustness of the Rydberg-mediated entanglement protocols are the key factors limiting their applicability in experiments and near-future industry. There are various methods for generating entangling gates by exploring the Rydberg interactions of neutral atoms, each equipped with its own strengths and weaknesses. The basics and tricks in these protocols are reviewed, with specific attention paid to the achievable fidelity and the robustness to the technical issues and detrimental innate factors.Comment: 57 pages, 10 figure