2,243 research outputs found
Low temperature solution of the Sherrington-Kirkpatrick model
We propose a simple scaling ansatz for the full replica symmetry breaking
solution of the Sherrington-Kirkpatrick model in the low energy sector. This
solution is shown to become exact in the limit x->0, x>>T of the Parisi replica
symmetry breaking scheme parameter x. The distribution function P(x,y) of the
frozen fields y has been known to develop a linear gap at zero temperature. We
integrate the scaling equations to find an exact numerical value for the slope
of the gap to be 0.3014046... We also use the scaling solution to devise an
inexpensive numerical procedure for computing finite timescale (x=1)
quantities. The entropy, the zero field cooled susceptibility and the local
field distribution function are computed in the low temperature limit with high
precision, barely achievable by currently available methods.Comment: 4 pages, 4 figure
Embeddings of Grassmann graphs
Let and be vector spaces of dimension and , respectively.
Let and . We describe all isometric
and -rigid isometric embeddings of the Grassmann graph in
the Grassmann graph
Geometrical characterization of semilinear isomorphisms of vector spaces and semilinear homeomorphisms of normed spaces
Let and be vector spaces over division rings (possible
infinite-dimensional) and let and be the
associated projective spaces. We say that is a PGL-{\it mapping} if for every there exists
such that . We show that for every PGL-bijection
the inverse mapping is a semicollineation. Also, we obtain an analogue of this
result for the projective spaces associated to normed spaces
Gap solitons in almost periodic one-dimensional structures
We consider almost periodic stationary nonlinear Schr\"odinger equations in
dimension . Under certain assumptions we prove the existence of nontrivial
finite energy solutions in the strongly indefinite case. The proof is based on
a carefull analysis of the energy functional restricted to the so-called
generalized Nehari manifold, and the existence and fine properties of special
Palais-Smale sequences. As an application, we show that certain one dimensional
almost periodic photonic crystals possess gap solitons for all prohibited
frequencies
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