Smooth Subspaces Research Articles

On totally smooth subspaces of Banach spaces: the Vlasov theorem revisited

On totally smooth subspaces of Banach spaces: the Vlasov theorem revisited

Time series k-means: A new k-means type smooth subspace clustering for time series data

Existing clustering algorithms are weak in extracting smooth subspaces for clustering time series data. In this paper, we propose a new k-means type smooth subspace clustering algorithm named Time Series k-means (TSkmeans) for clustering time series data. The proposed TSkmeans algorithm can effectively exploit inherent subspace information of a time series data set to enhance clustering performance. More specifically, the smooth subspaces are represented by weighted time stamps which indicate the relative discriminative power of these time stamps for clustering objects. The main contributions of our work include the design of a new objective function to guide the clustering of time series data and the development of novel updating rules for iterative cluster searching with respect to smooth subspaces. Based on a synthetic data set and five real-life data sets, our experimental results confirm that the proposed TSkmeans algorithm outperforms other state-of-the-art time series clustering algorithms in terms of common performance metrics such as Accuracy, Fscore, RandIndex, and Normal Mutual Information.

Characteristic properties of the Gurariy space

The Gurariy space G is defined by the property that for every pair of finite dimensional Banach spaces L ⊂ M, every isometry T: L → G admits an extension to an isomorphism \(\mathop T\limits^ \sim :M \to G\) with ‖T‖‖T−1‖ ≤ 1 + ∈. We investigate the question when we can take \(\mathop T\limits^ \sim \) to be also an isometry (i.e., ∈ = 0). We identify a natural class of pairs L ⊂ M such that the above property for this class with ∈ = 0 characterises the Gurariy space among all separable Banach spaces. We also show that the Gurariy space G is the only Lindenstrauss space such that its finite-dimensional smooth subspaces are dense in all subspaces.

Solvable groups, free divisors and nonisolated matrix singularities II: Vanishing topology

In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the "exceptional orbit varieties" for repesentations of solvable groups. Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors $E_n$, and we give an inductive procedure for computing the vanishing topology of the matrix singularities. The inductive procedure we use is an extension of that introduced by L\^{e}-Greuel for computing the Milnor number of an ICIS. Instead of linear subspaces, we use free divisors arising from the geometric configuration and which correspond to subgroups of the solvable groups. Here the vanishing topology involves a singular version of the Milnor fiber; however, it still has the good connectivity properties and is homotopy equivalent to a bouquet of spheres, whose number is called the singular Milnor number. We give formulas for this singular Milnor number in terms of singular Milnor numbers of various free divisors on smooth subspaces, which can be computed as lengths of determinantal modules. In addition to being applied to symmetric, general and skew-symmetric matrix singularities, the results are also applied to Cohen--Macaulay singularities defined as 2 x 3 matrix singularities. We compute the Milnor number of isolated Cohen--Macaulay surface singularities of this type in $\mathbb{C}^4$ and the difference of Betti numbers of Milnor fibers for isolated Cohen--Macaulay 3--fold singularities of this type in $\mathbb{C}^5$.

Smooth local subspace projection for nonlinear noise reduction

Many nonlinear or chaotic time series exhibit an innate broad spectrum, which makes noise reduction difficult. Local projective noise reduction is one of the most effective tools. It is based on proper orthogonal decomposition (POD) and works for both map-like and continuously sampled time series. However, POD only looks at geometrical or topological properties of data and does not take into account the temporal characteristics of time series. Here, we present a new smooth projective noise reduction method. It uses smooth orthogonal decomposition (SOD) of bundles of reconstructed short-time trajectory strands to identify smooth local subspaces. Restricting trajectories to these subspaces imposes temporal smoothness on the filtered time series. It is shown that SOD-based noise reduction significantly outperforms the POD-based method for continuously sampled noisy time series.

Bifurcation analysis of an impact oscillator with a one-sided elastic constraint near grazing

In this paper a linear oscillator undergoing impact with a secondary elastic support is studied experimentally and semi-analytically for near-grazing conditions. The experimentally observed bifurcations are explained with help from simulations based on mapping solutions between locally smooth subspaces. Smooth as well as nonsmooth bifurcations are observed, and the resulting atypical bifurcations are explained, often as an interplay between them. In order to understand the observed bifurcation scenarios, a global analysis is required, due to the influence of stable and unstable orbits which are born in distant bifurcations but become important at near-grazing conditions. The good degree of correspondence between experiment and theory fully justifies the modelling approach.

Dynamics of the archetypal piecewise linear oscillator close to grazing

A linear oscillator undergoing impact with a secondary support is studied numerically. Smooth as well as nonsmooth bifurcations are observed. The bifurcations are explained with help from simulations based on mapping solutions between locally smooth subspaces. The obtained results are verified by comparison with previous experimental studies. Experimental stability analysis is also presented, justifying the normal form maps used to show the response close to grazing for systems with and without prestress of the secondary spring. The high degree of correspondence lends support to the modelling approach, as does the close match between experiments and simulations.

An experimental study into the bilinear oscillator close to grazing

A linear oscillator undergoing impact with a secondary support is studied experimentally. Smooth as well as nonsmooth bifurcations are observed. The bifurcations are explained with help from simulations based on mapping solutions between locally smooth subspaces. Experimental stability studies are also presented, justifying the normal form maps used to show the response close to grazing for systems with and without prestress of the secondary spring. The high degree of correspondence lends support to the modelling approach, and the highly complicated response justifies continued study of this system.

Low-dimensional maps for piecewise smooth oscillators

Dynamics of the piecewise smooth nonlinear oscillators is considered, for which, general methodology of reducing multidimensional flows to low-dimensional maps is proposed. This includes a definition of piecewise smooth oscillator and creation of a global iterative map providing an exact solution. The global map is comprised of local maps, which are constructed in the smooth sub-regions of phase space. To construct this low-dimensional map, it is proposed to monitor the points of intersections of a chosen boundary between smooth subspaces by a trajectory. The dimension reduction is directly related to the dimension of the chosen boundary, and the lower its dimension is, the larger dimension reduction can be achieved. Full details are given for a drifting impact oscillator, where the five-dimensional flow is reduced to one-dimensional (1D) approximate analytical map. First an exact two-dimensional map has been formulated and analysed. A further reduction to 1D approximate map is introduced and discussed. Standard nonlinear dynamic analysis reveals a complex behaviour ranging from periodic oscillations to chaos, and co-existence of multiple attractors. Accuracy of the constructed maps is examined by comparing with the exact solutions for a wide range of the system parameters.

On smooth fuzzy subspaces

We introduce a new concept of smooth topological subspaces, which coincides with the usual definition in the case where μ = χY, Y ⊂ X. Also, we introduce some concepts such as q‐nbd systems, continuity, separation axioms, compactness, and connectedness in this sense. Also, various characterization for some fuzzy topological concepts in this sense are given.

An algebraic two‐level preconditioner for asymmetric, positive‐definite systems

AbstractA two‐level, linear algebraic solver for asymmetric, positive‐definite systems is developed using matrices arising from stabilized finite element formulations to motivate the approach. Supported by an analysis of a representative smoother, the parent space is divided into oscillatory and smooth subspaces according to the eigenvectors of the associated normal system. Using a mesh‐based aggregation technique, which relies only on information contained in the matrix, a restriction/prolongation operator is constructed. Various numerical examples, on both structured and unstructured meshes, are performed using the two‐level cycle as the basis for a preconditioner. Results demonstrate the complementarity between the smoother and the coarse‐level correction as well as convergence rates that are nearly independent of the problem size. Copyright © 2001 John Wiley & Sons, Ltd.

Variable-complexity genetic algorithm for topological design

Genetic algorithms are known to be appropriate for optimization in nonsmooth domains. A variable-complexity algorithm has been developed for use in topological design problems. The system starts with simple descriptions and adds degrees of freedom to allow development of more complex designs as optimization proceeds. The algorithm is applied in two different domains : design of low-speed nonplanar wings to minimize induced drag or total drag and design of structural trusses to minimize weight. In each domain, special constraint-handling schemes are introduced to improve performance. Calculus-based optimization in smooth subspaces is also employed to improve efficiency. These successful applications demonstrate that the variable-complexity algorithm is an important new tool for topological design.