In this research paper, the concept of hyperspherical/hyper-ellipsoidal separability is introduced. Method of arriving at the optimal hypersphere (maximizing margin) separating two classes is discussed. By projecting the quantized patterns into higher dimensional space (as in encoders of error correcting code), the patterns are made hyper-spherically separable. Single/multiple layers of spheric...

(Here we really mean the integral over Σ of the form obtained by pulling back ω under the inclusion map). Now suppose we have two such submanifolds, Σ0 and Σ1, which are (smoothly) homotopic. That is, we have a smooth map F : Σ × [0, 1] → M with F |Σ×{i} an immersion describing Σi for i = 0, 1. Then d(F∗ω) is a (k + 1)-form on the (k + 1)-dimensional oriented manifold with boundary Σ × [0, 1], ...

A class of local (in the space-time) conditions on the vorticity directions implying local regularity of weak solutions to the 3D Navier-Stokes equations is established. In all the preceding results, the relevant geometric conditions, although being local in nature, have been assumed uniformly throughout the spatial regions of high vorticity magnitude, and uniformly in time. In addition, simila...

Following Part I, this article continues to describe the calculation of the reach sets and tubes for linear control systems with time-varying coefficients and ellipsoidal hard bounds on the controls and initial states. It deals with parametrized families of internal ellipsoidal approximations constructed such that they touch the reach sets at every point of their boundary at any instant of time...

Six new species of coccidia are described from endangered Phelsuma spp. geckoes (Sauria: Gekkonidae) endemic to Mauritius, Indian Ocean. Five new species (3 Eimeria and 2 Isospora species) are described from Phelsuma rosagularis Vinson et Vinson; all lack a micropyle and an oocyst residuum, and all have a sporocyst residuum. Oocysts of Eimeria swinnertonae sp. n. are ellipsoidal, 22.2 x 17.8 (2...

We consider the 3D-Navier-Stokes system (NSS) on R3 without external forcing. After Fourier transform it becomes the system of non-linear integral equations. For one-parameter families of initial conditions A·c (0)(k) |k|2 it is known that if |A| is sufficiently small then NSS has global solution. We show that if c(0) satisfies some natural conditions at infinity then for sufficiently large A N...

The shock wave structure in a moderately dense gas is studied by numerically solving the one-dimensional Enskog kinetic equation. The numerical method is based on a combination of a nite difference scheme for the streaming term and a Monte Carlo quadrature technique to evaluate the collision integral. The obtained shock prooles are compared to the results of Navier-Stokes theory and to molecula...

In addition to conserving energy and enstrophy, the nonlinear terms of the twodimensional incompressible Navier–Stokes equation are well known to conserve the global integral of any continuously differentiable function of the scalar vorticity field. However, the phenomenological role of these additional inviscid invariants, including the issue as to whether they cascade to large or small scales...

A direct approach to calculating the hypersingular integral for a threedimensional Galerkin approximation is presented. The method does not employ either Stokes' Theorem or a regularization process to transform the integrand before the evaluation is carried out. Integrating two of the four dimensions analytically, the potentially divergent terms arising from the coincident and adjacent edge int...

Two issues that arise when solving Stokes flow problems with the hydrodynamical single-layer potential are addressed. First, the resulting boundary integral equation is singular, and second, discretizations lead to dense matrices. We discuss a well-posed modified equation which is equivalent for zero net-flux. Furthermore, we describe a multiscale basis that lead to sparse stiffness matrices. T...