Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics

In this paper we present computer-assisted proofs of a number of results in theoretical fluid dynamics and in quantum mechanics. An algorithm based on interval arithmetic yields provably correct eigenvalue enclosures and exclosures for non-self-adjoint boundary eigenvalue problems, the eigenvalues of which are highly sensitive to perturbations. We apply the algorithm to: the Orr-Sommerfeld equation with Poiseuille profile to prove the existence of an eigenvalue in the classically unstable region for Reynolds number R=5772.221818; the Orr-Sommerfeld equation with Couette profile to prove upper bounds for the imaginary parts of all eigenvalues for fixed R and wave number α; the problem of natural oscillations of an incompressible inviscid fluid in the neighbourhood of an elliptical flow to obtain information about the unstable part of the spectrum off the imaginary axis; Squire's problem from hydrodynamics; and resonances of one-dimensional Schrödinger operators

Modeling, Simulation and Application of Bacterial Transduction in Genetic Algorithms

At present, all methods in Evolutionary Computation are bioinspired in the fundamental principles of neo-Darwinism as well as on a vertical gene transfer. Thus, on a mechanism in which an organism receives genetic material from its ancestor. Horizontal, lateral or cross-population gene transfer is any process in which an organism transfers a genetic segment to another one that is not its offspring. Virus transduction is one of the key mechanisms of horizontal gene propagation in microorganism (e.g. bacteria). In the present paper, we model and simulate a transduction operator, exploring a possible role and usefulness of transduction in a genetic algorithm. The genetic algorithm including transduction has been named PETRI (abbreviation of Promoting Evolution Through Reiterated Infection). The efficiency and performance of this algorithm was evaluated using a benchmark function and the 0/1 knapsack problem. The utility was illustrated designing an AM radio receiver, optimizing the main features of the electronic components of the AM radio circuit as well as those of the radio enclosure. Our results shown how PETRI approaches to higher fitness values as transduction probability comes near to 100%. The conclusion is that transduction improves the performance of a genetic algorithm, assuming a population divided among several sub-populations or ‘bacterial colonies’

Finding Pairwise Intersections Inside a Query Range

We study the following problem: preprocess a set O of objects into a data structure that allows us to efficiently report all pairs of objects from O that intersect inside an axis-aligned query range Q. We present data structures of size $O(n({\rm polylog} n))$ and with query time $O((k+1)({\rm polylog} n))$ time, where k is the number of reported pairs, for two classes of objects in the plane: axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects and the query range are axis-aligned boxes in R^3, we present a data structures of size $O(n\sqrt{n}({\rm polylog} n))$ and query time $O((\sqrt{n}+k)({\rm polylog} n))$. When the objects and query are fat, we obtain $O((k+1)({\rm polylog} n))$ query time using $O(n({\rm polylog} n))$ storage

Orthogonal Range Reporting and Rectangle Stabbing for Fat Rectangles

In this paper we study two geometric data structure problems in the special case when input objects or queries are fat rectangles. We show that in this case a significant improvement compared to the general case can be achieved. We describe data structures that answer two- and three-dimensional orthogonal range reporting queries in the case when the query range is a \emph{fat} rectangle. Our two-dimensional data structure uses $O(n)$ words and supports queries in $O(\log\log U +k)$ time, where $n$ is the number of points in the data structure, $U$ is the size of the universe and $k$ is the number of points in the query range. Our three-dimensional data structure needs $O(n\log^{\varepsilon}U)$ words of space and answers queries in $O(\log \log U + k)$ time. We also consider the rectangle stabbing problem on a set of three-dimensional fat rectangles. Our data structure uses $O(n)$ space and answers stabbing queries in $O(\log U\log\log U +k)$ time.Comment: extended version of a WADS'19 pape

Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation

We present an algorithm for the rigorous integration of Delay Differential Equations (DDEs) of the form $x'(t)=f(x(t-\tau),x(t))$. As an application, we give a computer assisted proof of the existence of two attracting periodic orbits (before and after the first period-doubling bifurcation) in the Mackey-Glass equation