Dynamic programming for graphs on surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2O(k·log k). Our approach combines tools from topological graph theory and analytic combinatorics.Postprint (updated version

Using web-based peer assessment in fostering deep learning in computer programming

Active learning is considered by many academics as an important and effective learning strategy. Students can improve the quality of their work by developing their higher cognitive skills through reflection on their own ideas and practice of analytic and evaluative skills. Peer assessment is one of the successful approaches which can be used to enhance this deep learning. In this paper we discuss a novel web-based peer assessment system to support computer programming courses. We discuss the educational rational for the system, and the deep learning theory, report on its deployment on large programming modules. The preliminary results indicate that the system has successfully helped students to develop their higher cognitive skills in learning computer programming

A real-time implementation of an advanced sensor failure detection, isolation, and accommodation algorithm

A sensor failure detection, isolation, and accommodation algorithm was developed which incorporates analytic sensor redundancy through software. This algorithm was implemented in a high level language on a microprocessor based controls computer. Parallel processing and state-of-the-art 16-bit microprocessors are used along with efficient programming practices to achieve real-time operation

The Nesterov-Todd Direction and its Relation to Weighted Analytic Centers

The subject of this report concerns differential-geometric properties of the Nesterov-Todd search direction for linear optimization over symmetric cones. In particular, we investigate the rescaled asymptotics of the associated flow near the central path. Our results imply that the Nesterov-Todd direction arises as the solution of a Newton system defined in terms of a certain transformation of the primal-dual feasible domain. This transformation has especially appealing properties which generalize the notion of weighted analytic centers for linear programming

An analytic approximation of the feasible space of metabolic networks

Assuming a steady-state condition within a cell, metabolic fluxes satisfy an under-determined linear system of stoichiometric equations. Characterizing the space of fluxes that satisfy such equations along with given bounds (and possibly additional relevant constraints) is considered of utmost importance for the understanding of cellular metabolism. Extreme values for each individual flux can be computed with Linear Programming (as Flux Balance Analysis), and their marginal distributions can be approximately computed with Monte-Carlo sampling. Here we present an approximate analytic method for the latter task based on Expectation Propagation equations that does not involve sampling and can achieve much better predictions than other existing analytic methods. The method is iterative, and its computation time is dominated by one matrix inversion per iteration. With respect to sampling, we show through extensive simulation that it has some advantages including computation time, and the ability to efficiently fix empirically estimated distributions of fluxes

Interactive visualization of a thin disc around a Schwarzschild black hole

In the first course of general relativity, the Schwarzschild spacetime is the most discussed analytic solution to Einstein's field equations. Unfortunately, there is rarely enough time to study the optical consequences of the bending of light for some advanced examples. In this paper, we present how the visual appearance of a thin disc around a Schwarzschild black hole can be determined interactively by means of an analytic solution to the geodesic equation processed on current high performance graphical processing units. This approach can, in principle, be customized for any other thin disc in a spacetime with geodesics given in closed form. The interactive visualization discussed here can be used either in a first course of general relativity for demonstration purposes only or as a thesis for an enthusiastic student in an advanced course with some basic knowledge of OpenGL and a programming language.Comment: 9 pages, 4 figure

Constructing Parsimonious Analytic Models for Dynamic Systems via Symbolic Regression

Developing mathematical models of dynamic systems is central to many disciplines of engineering and science. Models facilitate simulations, analysis of the system's behavior, decision making and design of automatic control algorithms. Even inherently model-free control techniques such as reinforcement learning (RL) have been shown to benefit from the use of models, typically learned online. Any model construction method must address the tradeoff between the accuracy of the model and its complexity, which is difficult to strike. In this paper, we propose to employ symbolic regression (SR) to construct parsimonious process models described by analytic equations. We have equipped our method with two different state-of-the-art SR algorithms which automatically search for equations that fit the measured data: Single Node Genetic Programming (SNGP) and Multi-Gene Genetic Programming (MGGP). In addition to the standard problem formulation in the state-space domain, we show how the method can also be applied to input-output models of the NARX (nonlinear autoregressive with exogenous input) type. We present the approach on three simulated examples with up to 14-dimensional state space: an inverted pendulum, a mobile robot, and a bipedal walking robot. A comparison with deep neural networks and local linear regression shows that SR in most cases outperforms these commonly used alternative methods. We demonstrate on a real pendulum system that the analytic model found enables a RL controller to successfully perform the swing-up task, based on a model constructed from only 100 data samples

Optimal Embeddings of Distance Regular Graphs into Euclidean Spaces

In this paper we give a lower bound for the least distortion embedding of a distance regular graph into Euclidean space. We use the lower bound for finding the least distortion for Hamming graphs, Johnson graphs, and all strongly regular graphs. Our technique involves semidefinite programming and exploiting the algebra structure of the optimization problem so that the question of finding a lower bound of the least distortion is reduced to an analytic question about orthogonal polynomials.Comment: 10 pages, (v3) some corrections, accepted in Journal of Combinatorial Theory, Series