7,230 research outputs found
RANDOM GEOMETRIC GRAPHS AND ISOMETRIES OF NORMED SPACES
Given a countable dense subset S of a finite-dimensional normed space X, and 0 \u3c p \u3c 1, we form a random graph on S by joining, independently and with probability p, each pair of points at distance less than 1. We say that S is Rado if any two such random graphs are (almost surely) isomorphic. Bonato and Janssen showed that in ℓd∞ almost all S are Rado. Our main aim in this paper is to show that ℓd∞ is the unique normed space with this property: indeed, in every other space almost all sets S are non-Rado. We also determine which spaces admit some Rado set: this turns out to be the spaces that have an ℓ∞ direct summand. These results answer questions of Bonato and Janssen. A key role is played by the determination of which finite-dimensional normed spaces have the property that every bijective step-isometry (meaning that the integer part of distances is preserved) is in fact an isometry. This result may be of independent interest
Non-uniqueness of ergodic measures with full Hausdorff dimension on a Gatzouras-Lalley carpet
In this note, we show that on certain Gatzouras-Lalley carpet, there exist
more than one ergodic measures with full Hausdorff dimension. This gives a
negative answer to a conjecture of Gatzouras and Peres
Topological entanglement entropy in the second Landau level
The entanglement entropy of the incompressible states of a realistic quantum
Hall system in the second Landau level are studied by direct diagonalization.
The subdominant term to the area law, the topological entanglement entropy,
which is believed to carry information about topologic order in the ground
state, was extracted for filling factors nu = 12/5 and nu = 7/3. While it is
difficult to make strong conclusions about nu = 12/5, the nu = 7/3 state
appears to be very consistent with the topological entanglement entropy for the
k=4 Read-Rezayi state. The effect of finite thickness corrections to the
Coulomb potential used in the direct diagonalization are also systematically
studied.Comment: 5 pages, 4 figure
Influence of Co and Zr Content on Creation of Crystalline Phases in Rapidly-Cooled, Injection-Cast Alloys Fe70Zr8-xCoxNb2B20 (where x = 0, 2, 4, 6 or 8)
K-theory of noncommutative Bieberbach manifolds
We compute $K-theory of noncommutative Bieberbach manifolds, which quotients
of a three-dimensional noncommutative torus by a free action of a cyclic group
Z_N, N=2,3,4,6.Comment: 19 page
Finite type approximations of Gibbs measures on sofic subshifts
Consider a H\"older continuous potential defined on the full shift
A^\nn, where is a finite alphabet. Let X\subset A^\nn be a specified
sofic subshift. It is well-known that there is a unique Gibbs measure
on associated to . Besides, there is a natural nested
sequence of subshifts of finite type converging to the sofic subshift
. To this sequence we can associate a sequence of Gibbs measures
. In this paper, we prove that these measures weakly converge
at exponential speed to (in the classical distance metrizing weak
topology). We also establish a strong mixing property (ensuring weak
Bernoullicity) of . Finally, we prove that the measure-theoretic
entropy of converges to the one of exponentially fast.
We indicate how to extend our results to more general subshifts and potentials.
We stress that we use basic algebraic tools (contractive properties of iterated
matrices) and symbolic dynamics.Comment: 18 pages, no figure
Classical dynamics on graphs
We consider the classical evolution of a particle on a graph by using a
time-continuous Frobenius-Perron operator which generalizes previous
propositions. In this way, the relaxation rates as well as the chaotic
properties can be defined for the time-continuous classical dynamics on graphs.
These properties are given as the zeros of some periodic-orbit zeta functions.
We consider in detail the case of infinite periodic graphs where the particle
undergoes a diffusion process. The infinite spatial extension is taken into
account by Fourier transforms which decompose the observables and probability
densities into sectors corresponding to different values of the wave number.
The hydrodynamic modes of diffusion are studied by an eigenvalue problem of a
Frobenius-Perron operator corresponding to a given sector. The diffusion
coefficient is obtained from the hydrodynamic modes of diffusion and has the
Green-Kubo form. Moreover, we study finite but large open graphs which converge
to the infinite periodic graph when their size goes to infinity. The lifetime
of the particle on the open graph is shown to correspond to the lifetime of a
system which undergoes a diffusion process before it escapes.Comment: 42 pages and 8 figure
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