We present a novel feature selection algorithm for the k-means clustering problem. Our algorithm is randomized and, assuming an accuracy parameter ε ∈ (0, 1), selects and appropriately rescales in an unsupervised manner Θ(k log(k/ε)/ε) features from a dataset of arbitrary dimensions. We prove that, if we run any γ-approximate k-means algorithm (γ ≥ 1) on the features selected using our method, ...

We give a nonstandard variant of Jordan’s proof of the Jordan curve theorem which is free of the defects his contemporaries criticized and avoids the epsilontic burden of the classical proof. The proof is selfcontained, except for the Jordan theorem for polygons taken for granted.

in this study, a numerical computational fluid dynamics study is conducted in order to predict the aerodynamic forces on the np car. the turbulent air flow around the car is modeled using the realizable k-ε model. first, results are validated against those presented for the ahmed’s body. next, the fluid flow around the car is simulated for different car speeds ( to mph) and flow directions ( to...

The commonly used linear k-ε turbulence model is shown to be incapable of accurate prediction of turbulent flows, where non-isotropy is dominant. Two examples of non-isotropic flows, which have a wide range of applications in marine waters, are saline water flow and the stratified flows due to temperature gradients. These relate to stratification and consequently, variation of density through...

A natural generalization of graph Ramsey theory is the study of unavoidable subgraphs in large colored graphs. In this paper, we find a minimal family of unavoidable graphs in two-edgecolored graphs. Namely, for a positive even integer k, let Sk be the family of two-edge-colored graphs on k vertices such that one of the colors forms either two disjoint Kk/2s or simply one Kk/2. Bollobás conject...

We give algorithms for approximation by low-rank positive semidefinite (PSD) matrices. For symmetric input matrix A ∈ Rn×n, target rank k, and error parameter ε > 0, one algorithm finds with constant probability a PSD matrix Ỹ of rank k such that ‖A− Ỹ ‖2F ≤ (1+ε)‖A−Ak,+‖ 2 F , where Ak,+ denotes the best rank-k PSD approximation to A, and the norm is Frobenius. The algorithm takes time O(nnz(A...