The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2ε) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue λ(ε) and the eigenvector angular frequency k(ε) for shells with Dirichlet boundary conditions along the lateral boundary, and natural boundary conditions on th...

It is shown that for every K > 0 and ε ∈ (0, 1/2) there exist N = N(K) ∈ N and D = D(K, ε) ∈ (1,∞) with the following properties. For every metric space (X, d) with doubling constant at most K, the metric space (X, d1−ε) admits a bi-Lipschitz embedding into R with distortion at most D. The classical Assouad embedding theorem makes the same assertion, but with N →∞ as ε→ 0.

Developing boundary layer flow and the flow over a backward-facing step are studied here. The LRN PANS k − ε [1] and the WALE turbulence model [2] are used as SGS models. The PANS (Partially AveragedNavier-Stokes) modelling approach [3, 4] is a modified k − ε model that can operate in both RANS and LES mode. An extension of PANS based on a four-equation k− ε− ζ − f model was recently proposed [...

More than twenty years ago Erdős conjectured [4] that a triangle-free graph G of chromatic number k ≥ k0(ε) contains cycles of at least k2−ε different lengths as k →∞. In this paper, we prove the stronger fact that every triangle-free graph G of chromatic number k ≥ k0(ε) contains cycles of 1 64 (1− ε)k 2 log k4 consecutive lengths, and a cycle of length at least 14 (1− ε)k 2 log k. As there ex...

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...