Computable Analysis of Differential Equations (Invited Talk)

In this talk, we discuss some algorithmic aspects of the local and global existence theory for various ordinary and partial differential equations. We will present a sample of results and give some idea of the motivation and general philosophy underlying these results

From Interval Computations to Constraint-Related Set Computations: Towards Faster Estimation of Statistics and ODEs under Interval and p-Box Uncertainty (Invited Talk)

Interval computations estimate the uncertainty of the result of data processing in situations in which we only know the upper bounds $Delta$ on the measurement errors. In this case, based on the measurement result $widetilde x$, we can only conclude that the actual (unknown) value $x$ of the desired quantity is in the interval $[widetilde x-Delta,widetilde x+Delta]$. In interval computations, at each intermediate stage of the computation, we have intervals of possible values of the corresponding quantities. As a result, we often have bounds with excess width. To remedy this problem, in our previous papers, we proposed an extension of interval technique to {it set computations}, where on each stage, in addition to intervals of possible values of the quantities, we also keep sets of possible values of pairs (triples, etc.). In this paper, we show that in several practical problems, such as estimating statistics (variance, correlation, etc.) and solutions to ordinary differential equations (ODEs) with given accuracy, this new formalism enables us to find estimates in feasible (polynomial) time

Semilattices, Domains, and Computability (Invited Talk)

As everyone knows, one popular notion of Scott domain is defined as a bounded complete algebraic cpo. These are closely related to algebraic lattices: (i) A Scott domain becomes an algebraic lattice with the adjunction of an (isolated) top element. (ii) Every non-empty Scott-closed subset of an algebraic lattice is a Scott domain. Moreover, the isolated ($=$ compact) elements of an algebraic lattice form a semilattice (under join). This semilattice has a zero element, and, provided the top element is isolated, it also has a unit element. The algebraic lattice itself may be regarded as the ideal completion of the semilattice of isolated elements. This is all well known. What is not so clear that is that there is an easy-to-construct domain of countable semilattices giving isomorphic copies of all countably based domains. This approach seems to have advantages over both ``information systems\u27\u27 or more abstract lattice formulations, and it makes definitions of solutions to domain equations very elementary to justify. The ``domain of domains\u27\u27 also has an immediate computable structure

Computability and Complexity of Julia Sets (Invited Talk)

Studying dynamical systems is key to understanding a wide range of phenomena ranging from planetary movement to climate patterns to market dynamics. Various numerical tools have been developed to address specific questions about dynamical systems, such as predicting the weather or planning the trajectory of a satellite. However, the theory of computation behind these problems appears to be very difficult to develop. In fact, little is known about computability of even the most natural problems arising from dynamical systems. In this talk I will survey the recent study of the computational properties of dynamical systems that arise from iterating quadratic polynomials on the complex plane. These give rise to the amazing variety of fractals known as Julia sets, and are closely connected to the Mandelbrot set. Julia sets are perhaps the most drawn objects in Mathematics due to their fascinating fractal structure. The theory behind them is even more fascinating, and the dynamical systems generating them are in many ways archetypal. I will present both positive and negative results on the computability and complexity of Julia sets. In conclusion of the talk I will discuss possible future directions and challenges in the study of the computability and complexity of dynamical systems

Canonical General Relativity on a Null Surface with Coordinate and Gauge Fixing

We use the canonical formalism developed together with David Robinson to st= udy the Einstein equations on a null surface. Coordinate and gauge conditions = are introduced to fix the triad and the coordinates on the null surface. Toget= her with the previously found constraints, these form a sufficient number of second class constraints so that the phase space is reduced to one pair of canonically conjugate variables: \Ac_2\and\Sc^2. The formalism is related to both the Bondi-Sachs and the Newman-Penrose methods of studying the gravitational field at null infinity. Asymptotic solutions in the vicinity of null infinity which exclude logarithmic behavior require the connection to fall off like $1/r^3$ after the Minkowski limit. This, of course, gives the previous results of Bondi-Sachs and Newman-Penrose. Introducing terms which fall off more slowly leads to logarithmic behavior which leaves null infinity intact, allows for meaningful gravitational radiation, but the peeling theorem does not extend to $\Psi_1$ in the terminology of Newman-Penrose. The conclusions are in agreement with those of Chrusciel, MacCallum, and Singleton. This work was begun as a preliminary study of a reduced phase space for quantization of general relativity.Comment: magnification set; pagination improved; 20 pages, plain te

Bridging the GUI gap with reactive values and relations

There are at present two ways to write GUIs for functional code. One is to use standard GUI toolkits, with all the benefits they bring in terms of feature completeness, choice of platform, conformance to platform-specific look-and-feel, long-term viability, etc. However, such GUI APIs mandate an imperative programming style for the GUI and related parts of the application. Alternatively, we can use a functional GUI toolkit. The GUI can then be written in a functional style, but at the cost of foregoing many advantages of standard toolkits that often will be of critical importance. This paper introduces a light-weight framework structured around the notions of reactive values and reactive relations . It allows standard toolkits to be used from functional code written in a functional style. We thus bridge the gap between the two worlds, bringing the advantages of both to the developer. Our framework is available on Hackage and has been been validated through the development of non-trivial applications in a commercial context, and with different standard GUI toolkits

Rationale and methods of the European Study on Cardiovascular Risk Prevention and Management in Daily Practice (EURIKA)

<p>Abstract</p> <p>Background</p> <p>The EURIKA study aims to assess the status of primary prevention of cardiovascular disease (CVD) across Europe. Specifically, it will determine the degree of control of cardiovascular risk factors in current clinical practice in relation to the European guidelines on cardiovascular prevention. It will also assess physicians' knowledge and attitudes about CVD prevention as well as the barriers impeding effective risk factor management in clinical practice.</p> <p>Methods/Design</p> <p>Cross-sectional study conducted simultaneously in 12 countries across Europe. The study has two components: firstly at the physician level, assessing eight hundred and nine primary care and specialist physicians with a daily practice in CVD prevention. A physician specific questionnaire captures information regarding physician demographics, practice settings, cardiovascular prevention beliefs and management. Secondly at the patient level, including 7641 patients aged 50 years or older, free of clinical CVD and with at least one classical risk factor, enrolled by the participating physicians. A patient-specific questionnaire captures information from clinical records and patient interview regarding sociodemographic data, CVD risk factors, and current medications. Finally, each patient provides a fasting blood sample, which is sent to a central laboratory for measuring serum lipids, apolipoproteins, hemoglobin-A1c, and inflammatory biomarkers.</p> <p>Discussion</p> <p>Primary prevention of CVD is an extremely important clinical issue, with preventable circulatory diseases remaining the leading cause of major disease burden. The EURIKA study will provide key information to assess effectiveness of and attitudes toward primary prevention of CVD in Europe. A transnational study creates opportunities for benchmarking good clinical practice across countries and improving outcomes. (ClinicalTrials.gov number, NCT00882336.)</p