An efficient shooting algorithm for Evans function calculations in large systems

In Evans function computations of the spectra of asymptotically constant-coefficient linear operators, a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ODE. For small systems, a fast, shooting algorithm may be obtained by representing subspaces as single exterior products \cite{AS,Br.1,Br.2,BrZ,BDG}. For large systems, however, the dimension of the exterior-product space quickly becomes prohibitive, growing as $\binom{n}{k}$, where $n$ is the dimension of the system written as a first-order ODE and $k$ (typically $\sim n/2$) is the dimension of the subspace. We resolve this difficulty by the introduction of a simple polar coordinate algorithm representing ``pure'' (monomial) products as scalar multiples of orthonormal bases, for which the angular equation is a numerically optimized version of the continuous orthogonalization method of Drury--Davey \cite{Da,Dr} and the radial equation is evaluable by quadrature. Notably, the polar-coordinate method preserves the important property of analyticity with respect to parameters.Comment: 21 pp., two figure

Clustering for Different Scales of Measurement - the Gap-Ratio Weighted K-means Algorithm

This paper describes a method for clustering data that are spread out over large regions and which dimensions are on different scales of measurement. Such an algorithm was developed to implement a robotics application consisting in sorting and storing objects in an unsupervised way. The toy dataset used to validate such application consists of Lego bricks of different shapes and colors. The uncontrolled lighting conditions together with the use of RGB color features, respectively involve data with a large spread and different levels of measurement between data dimensions. To overcome the combination of these two characteristics in the data, we have developed a new weighted K-means algorithm, called gap-ratio K-means, which consists in weighting each dimension of the feature space before running the K-means algorithm. The weight associated with a feature is proportional to the ratio of the biggest gap between two consecutive data points, and the average of all the other gaps. This method is compared with two other variants of K-means on the Lego bricks clustering problem as well as two other common classification datasets.Comment: 13 pages, 6 figures, 2 tables. This paper is under the review process for AIAP 201

Напружено-деформований стан гофрованих циліндричних оболонок в уточненій постановці

Дано розв’язання задач про напружений стан гофрованих у поперечному перерiзi цилiндричних оболонок змiнної товщини в уточненiй постановцi з використанням методiв сплайн-колокацiї та дискретної ортогоналiзацiї. Дослiджено вплив частоти гофрування на розподiл полiв перемiщень та напружень.The solution of the problem on a stressed state of corrugated cross-section variable thickness cylindrical shells is given using the spline-approximation and discrete orthogonalization methods. The influence of the corrugation frequency on the fields of displacements and stresses is investigated

Numerical Analysis of Stress-Strain State of Orthotropic Plates in the Form of Arbitrary Convex Quadrangle

A numerical and analytical approach to solving problems of the stress-strain state of quadrangular orthotropic plates of complex shape has been proposed. Two-dimensional boundary value problem was solved using spline collocation and discrete orthogonalization methods after applying the appropriate domain transform. The influence of geometric shape of plate in different cases of boundary conditions on the displacement and stress fields is considered according to the refined theory. The results were compared with available data from other authors

Чисельне розв'язання задач про напружено-деформований стан сферичних оболонок змінної товщини в уточненій постановці

Дослiджено задачу про напружено-деформований стан сферичної ортотропної оболонки зi змiнною в одному координатному напрямку товщиною при рiзних граничних умовах в уточненiй постановцi. Розвинуто чисельно-аналiтичний пiдхiд, який базується на застосуваннi сплайн-апроксимацiї та методу дискретної ортогоналiзацiї. Напружено-деформований стан пологих ортотропних оболонок дослiджено у випадку змiнної товщини i збереженнi ваги.The problem of a stress-strain state of the orthotropic spherical shell of variable thickness in one dimension is carried out in a refined statement for different boundary conditions. The numericalanalytical method is developed based on using the spline-approximation and the discrete-orthogonalization methods. The stress-strain state of orthotropic shallow shells is investigated for the case of the variable thickness and preserving the weight