59,214 research outputs found
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 Assume that the
system has a sequence of regular and internal layers occurring alternatively
along the -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 Comment: 52 pages in a dvi fil
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
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
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