Toruses and meshes include graphs of many varieties of topologies, with lines, rings, and hypercubes being special cases. Given a d-dimensional torus or mesh G and a c-dimensional torus or mesh H of the same size, we study the problem of embedding G in H to minimize the dilation cost. For increasing dimension cases (d < c ) in which the shapes of G and H satisfy the condition of expansion, the ...

Reconstruction of a dynamical system from a time series requires the selection of two parameters, the embedding dimension de and the embedding lag τ . Many competing criteria to select these parameters exist, and all are heuristic. Within the context of modeling the evolution operator of the underlying dynamical system, we show that one only need be concerned with the product deτ . We introduce...

The embedding dimensions of normal and epileptic electroencephalogram (EEG) time series are analyzed by two different methods, Cao's method and differential entropy method. The results of the two methods indicate consistently that the embedding dimensions of EEG signals during seizure will change and become different from that of normal EEG signals, and the embedding dimensions will vary intens...

Consider a sum-product normed space, i.e. a space of the form Y = `1 ⊗ X , where X is another normed space. Each element in Y consists of a length-n vector of elements in X , and the norm of an element in Y is the sum of the norms of its coordinates. In this paper we show a constant-distortion embedding from the normed space `1 ⊗X into a lower-dimensional normed space ` ′ 1 ⊗ X , where n′ n is ...

The reconstruction of a dynamical system from a time series requires the selection of two parameters: the embedding dimension de and the embedding lag τ. Many competing criteria to select these parameters exist, and all are heuristic. Within the context of modelling the evolution operator of the underlying dynamical system, we show that one only need be concerned with the product deτ. We introd...

In this short article, we describe how the correlation of typical diffusion processes arising e.g. in financial modelling can be exploited – by means of asymptotic analysis of principal components – to make Feynman-Kac PDEs of high dimension computationally tractable. We explore the links to dimension adaptive sparse grids [GG03], anchored ANOVA decompositions and dimension-wise integration [GH...

The Fibonacci dimension fdim(G) of a graph G was introduced in [1] as the smallest integer d such that G admits an isometric embedding into Γd, the d-dimensional Fibonacci cube. The Fibonacci dimension of the resonance graphs of catacondensed benzenoid systems is studied. This study is inspired by the fact, that the Fibonacci cubes are precisely the resonance graphs of a subclass of the catacon...