Ricardo Restrepo

Reconstruction and Clustering in Random Constraint Satisfaction Problems

Random instances of constraint satisfaction problems (CSPs) appear to be hard for all known algorithms when the number of constraints per variable lies in a certain interval. Contributing to the general understanding of the structure of the solution space of a CSP in the satisfiable regime, we formulate a set of technical conditions on a large family of random CSPs and prove bounds on three most interesting thresholds for the density of such an ensemble: namely, the satisfiability threshold, the threshold for clustering of the solution space, and the threshold for an appropriate reconstruction problem on the CSPs. The bounds become asymptoticlally tight as the number of degrees of freedom in each clause diverges. The families are general enough to include commonly studied problems such as random instances of Not-All-Equal SAT, k-XOR formulae, hypergraph 2-coloring, and graph k-coloring. An important new ingredient is a condition involving the Fourier expansion of clauses, which characterizes the class of ...

Memory allocation in distributed storage networks

We consider the problem of distributing a file in a network of storage nodes whose storage budget is limited but at least equals the size file. We first generate T encoded symbols (from the file) which are then distributed among the nodes. We investigate the optimal allocation of T encoded packets to the storage nodes such that the probability of reconstructing the file by using any r out of n nodes is maximized. Since the optimal allocation of encoded packets is difficult to find in general, we find another objective function which well approximates the original problem and yet is easier to optimize. We find the optimal symmetric allocation for all coding redundancy constraints using the equivalent approximate problem. We also investigate the optimal allocation in random graphs. Finally, we provide simulations to verify the theoretical results.

Donor impurity-related linear and nonlinear intraband optical absorption coefficients in quantum ring: effects of applied electric field and hydrostatic pressure

AbstractThe linear and nonlinear intraband optical absorption coefficients in GaAs three-dimensional single quantum rings are investigated. Taking into account the combined effects of hydrostatic pressure and electric field, applied along the growth direction of the heterostructure, the energies of the ground and first excited states of a donor impurity have been found using the effective mass approximation and a variational method. The energies of these states are examined as functions of the dimensions of the structure, electric field, and hydrostatic pressure. We have also investigated the dependencies of the linear, nonlinear, and total optical absorption coefficients as a function of incident photon energy for several configurations of the system. It is found that the variation of distinct sizes of the structure leads to either a redshift and/or a blueshift of the resonant peaks of the intraband optical spectrum. In addition, we have found that the application of an electric field leads to a redshift, whereas the influence of hydrostatic pressure leads to a blueshift (in the case of on-ring-center donor impurity position) of the resonant peaks of the intraband optical spectrum.

Photoionization cross section and binding energy of single dopant in hollow cylindrical core/shell quantum dot

In this study, we have investigated the confined donor impurity in a hollow cylindrical-shell quantum dot. The charges are assumed to be completely confined to the interior of the shell with rigid walls. Within the framework of the effective-mass approximation and by using a simple variational approach, we have computed the donor binding energy as a function of the shell sizes in order to study the behavior of the electron-impurity attraction for a very small thickness. Our results show that the binding energy of a donor impurity placed at the center of cylindrical core/shell dots depends strongly on the shell size. The binding energy increases when the shell-wideness becomes smaller and shows the same behavior as in a simple cylindrical quantum dot. A special case has been studied, which corresponds to the ratio between the inner and outer radii near to one (a/b → 1) for which our model gives a non-significant behavior of the impurity binding energy. This fact implies the existence of a critical value (a...

Improved Mixing Condition on the Grid for Counting and Sampling Independent Sets

The hard-core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics, a multicast model of calls in communication networks, and as a weighted independent set problem in combinatorics, probability and theoretical computer science. In this model, each independent set

Intraexcitonic Energy Transition in GaAs Quantum Wells

I

Linear and nonlinear optical absorption coefficients in GaAs/Ga1 xAlxAs concentric double quantum rings: Effects of hydrostatic pressure and aluminum concentration

in a graph

Donor impurity-related linear and nonlinear optical absorption coefficients in GaAs/Ga1−xAlxAs concentric double quantum rings: Effects of geometry, hydrostatic pressure, and aluminum concentration

G

Improved mixing condition on the grid for counting and sampling independent sets

is weighted proportionally to

Effects of hydrostatic pressure on the nonlinear optical properties of a donor impurity in a GaAs quantum ring

\lambda^{|I|}