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 $V$ and $V'$ be vector spaces of dimension $n$ and $n'$, respectively. Let $k\in\{2,...,n-2\}$ and $k'\in\{2,...,n'-2\}$. We describe all isometric and $l$-rigid isometric embeddings of the Grassmann graph $\Gamma_{k}(V)$ in the Grassmann graph $\Gamma_{k'}(V')$

Gap solitons in almost periodic one-dimensional structures

We consider almost periodic stationary nonlinear Schr\"odinger equations in dimension $1$. 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

Geometrical characterization of semilinear isomorphisms of vector spaces and semilinear homeomorphisms of normed spaces

Let $V$ and $V'$ be vector spaces over division rings (possible infinite-dimensional) and let ${\mathcal P}(V)$ and ${\mathcal P}(V')$ be the associated projective spaces. We say that $f:{\mathcal P}(V)\to {\mathcal P}(V')$ is a PGL-{\it mapping} if for every $h\in {\rm PGL}(V)$ there exists $h'\in {\rm PGL}(V')$ such that $fh=h'f$. 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