Finite type invariants of integral homology 3-spheres: A survey

This is a survey on the current status of the study of finite type invariants of integral homology 3-spheres based on lectures given in the workshop on knot theory at Banach International Center of Mathematics, Warsaw, July 1995. As a new result, we show that the space of finite type invariants of integral homology 3-spheres is a graded polynomial algebra generated by invariants additive under the connected sum. We also discuss some open questions on this subject.Comment: 27 pages, amslatex. A new section was added surveying recent developments of the subject. To appear in the proceedings of Warsaw knot theory workshop, July-August 199

Shadowing matching errors for wave-front-like solutions

Consider a singularly perturbed system $$\epsilon u_t=\epsilon^2 u_{xx} + f(u,x,\epsilon),\quad u\in {\Bbb R}^n,x\in{\Bbb R},t\geq 0.$$ Assume that the system has a sequence of regular and internal layers occurring alternatively along the $x$-direction. These ``multiple wave'' solutions can formally be constructed by matched asymptotic expansions. To obtain a genuine solution, we derive a {\em Spatial Shadowing Lemma} which assures the existence of an exact solution that is near the formal asymptotic series provided (1) the residual errors are small in all the layers, and (2) the matching errors are small along the lateral boundaries of the adjacent layers. The method should work on some other systems like $\epsilon u_t=-(-\epsilon^2 D_{xx})^m u+ \dots.$Comment: 52 pages in a dvi fil

An Accelerated Proximal Coordinate Gradient Method and its Application to Regularized Empirical Risk Minimization

We consider the problem of minimizing the sum of two convex functions: one is smooth and given by a gradient oracle, and the other is separable over blocks of coordinates and has a simple known structure over each block. We develop an accelerated randomized proximal coordinate gradient (APCG) method for minimizing such convex composite functions. For strongly convex functions, our method achieves faster linear convergence rates than existing randomized proximal coordinate gradient methods. Without strong convexity, our method enjoys accelerated sublinear convergence rates. We show how to apply the APCG method to solve the regularized empirical risk minimization (ERM) problem, and devise efficient implementations that avoid full-dimensional vector operations. For ill-conditioned ERM problems, our method obtains improved convergence rates than the state-of-the-art stochastic dual coordinate ascent (SDCA) method

Dirac spin gapless semiconductors: Ideal platforms for massless and dissipationless spintronics and new (quantum) anomalous spin Hall effects

It is proposed that the new generation of spintronics should be ideally massless and dissipationless for the realization of ultra-fast and ultra-low-power spintronic devices. We demonstrate that the spin-gapless materials with linear energy dispersion are unique materials that can realize these massless and dissipationless states. Furthermore, we propose four new types of spin Hall effects which consist of spin accumulation of equal numbers of electrons and holes having the same or opposite spin polarization at the sample edge in Hall effect measurements, but with vanishing Hall voltage. These new Hall effects can be classified as (quantum) anomalous spin Hall effects. The physics for massless and dissipationless spintronics and the new spin Hall effects are presented for spin-gapless semiconductors with either linear or parabolic dispersion. New possible candidates for Dirac-type or parabolic type spin-gapless semiconductors are demonstrated in ferromagnetic monolayers of simple oxides with either honeycomb or square lattices.Comment: 5 pages, 7 figue