We introduce $\varepsilon $ -approximate versions of the notion of a Euclidean vector bundle for $\varepsilon \geq 0$ , which recover the classical notion of a Euclidean vector bundle when $\varepsilon = 0$ . In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $\varepsilon $ -approximate vector bundles can be used to represent classical vector bundles when $\varepsilon> 0$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.

We show that the maximum number of convex polygons in a triangulation of n points in the plane is O(1.5029n). This improves an earlier bound of O(1.6181n) established by van Kreveld, Löffler and Pach (2012), and almost matches the current best lower bound of Ω(1.5028n) due to the same authors. Given a planar straight-line graph G with n vertices, we also show how to compute efficiently the number of convex polygons in G.

A gravitational search algorithm (GSA) is a meta-heuristic development that is modelled on the Newtonian law of gravity and mass interaction. Here we propose a new hybrid algorithm called the Direct Gravitational Search Algorithm (DGSA), which combines a GSA that can perform a wide exploration and deep exploitation with the Nelder-Mead method, as a promising direct method capable of an intensification search. The main drawback of a meta-heuristic algorithm is slow convergence, but in our DGSA the standard GSA is run for a number of iterations before the best solution obtained is passed to the Nelder-Mead method to refine it and avoid running iterations that provide negligible further improvement. We test the DGSA on 7 benchmark integer functions and 10 benchmark minimax functions to compare the performance against 9 other algorithms, and the numerical results show the optimal or near optimal solution is obtained faster.

Neighbour search (NS) is the core of any implementations of smoothed particle hydrodynamics (SPH). In this paper,we present an efficient neighbour search method based on the plane sweep (PW) algorithm with N being the number of SPH particles. The resulting method, dubbed the PWNS method, is totally independent of grids (i.e., purely meshfree) and capable of treating variable smoothing length, arbitrary particle distribution and heterogenous kernels. Several state-of-the-art data structures and algorithms, e.g., the segment tree and the Morton code, are optimized and implemented. By simply allowingmultiple lines to sweep the SPH particles simultaneously from different initial positions, a parallelization of the PWNS method with satisfactory speedup and load-balancing can be easily achieved. That is, the PWNS SPH solver has a great potential for large scale fluid dynamics simulations.

Cubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in ${{\mathbb{R}}^{d}}$ in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris–Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology.