7,136 research outputs found
On the nonexistence of Smith-Toda complexes
Let p be a prime. The Smith-Toda complex V(k) is a finite spectrum whose
BP-homology is isomorphic to BP_*/(p,v_1,...,v_k). For example, V(-1) is the
sphere spectrum and V(0) the mod p Moore spectrum. In this paper we show that
if p > 5, then V((p+3)/2) does not exist and V((p+1)/2), if it exists, is not a
ring spectrum. The proof uses the new homotopy fixed point spectral sequences
of Hopkins and Miller.Comment: 10 pages, AMSLate
Invariant Discretization Schemes Using Evolution-Projection Techniques
Finite difference discretization schemes preserving a subgroup of the maximal
Lie invariance group of the one-dimensional linear heat equation are
determined. These invariant schemes are constructed using the invariantization
procedure for non-invariant schemes of the heat equation in computational
coordinates. We propose a new methodology for handling moving discretization
grids which are generally indispensable for invariant numerical schemes. The
idea is to use the invariant grid equation, which determines the locations of
the grid point at the next time level only for a single integration step and
then to project the obtained solution to the regular grid using invariant
interpolation schemes. This guarantees that the scheme is invariant and allows
one to work on the simpler stationary grids. The discretization errors of the
invariant schemes are established and their convergence rates are estimated.
Numerical tests are carried out to shed some light on the numerical properties
of invariant discretization schemes using the proposed evolution-projection
strategy
On Static and Dynamic Heterogeneities in Water
We analyze differences in dynamics and in properties of the sampled potential
energy landscape between different equilibrium trajectories, for a system of
rigid water molecules interacting with a two body potential. On entering in the
supercooled region, differences between different realizations enhance and
survive even when particles have diffused several time their average distance.
We observe a strong correlation between the mean square displacement of the
individual trajectories and the average energy of the sampled landscape
Tensor renormalization group approach to 2D classical lattice models
We describe a simple real space renormalization group technique for two
dimensional classical lattice models. The approach is similar in spirit to
block spin methods, but at the same time it is fundamentally based on the
theory of quantum entanglement. In this sense, the technique can be thought of
as a classical analogue of DMRG. We demonstrate the method - which we call the
tensor renormalization group method - by computing the magnetization of the
triangular lattice Ising model.Comment: 4 pages, 7 figure
A low complexity algorithm for non-monotonically evolving fronts
A new algorithm is proposed to describe the propagation of fronts advected in
the normal direction with prescribed speed function F. The assumptions on F are
that it does not depend on the front itself, but can depend on space and time.
Moreover, it can vanish and change sign. To solve this problem the Level-Set
Method [Osher, Sethian; 1988] is widely used, and the Generalized Fast Marching
Method [Carlini et al.; 2008] has recently been introduced. The novelty of our
method is that its overall computational complexity is predicted to be
comparable to that of the Fast Marching Method [Sethian; 1996], [Vladimirsky;
2006] in most instances. This latter algorithm is O(N^n log N^n) if the
computational domain comprises N^n points. Our strategy is to use it in regions
where the speed is bounded away from zero -- and switch to a different
formalism when F is approximately 0. To this end, a collection of so-called
sideways partial differential equations is introduced. Their solutions locally
describe the evolving front and depend on both space and time. The
well-posedness of those equations, as well as their geometric properties are
addressed. We then propose a convergent and stable discretization of those
PDEs. Those alternative representations are used to augment the standard Fast
Marching Method. The resulting algorithm is presented together with a thorough
discussion of its features. The accuracy of the scheme is tested when F depends
on both space and time. Each example yields an O(1/N) global truncation error.
We conclude with a discussion of the advantages and limitations of our method.Comment: 30 pages, 12 figures, 1 tabl
- …