Self-avoiding walks on a bilayer Bethe lattice

We propose and study a model of polymer chains in a bilayer. Each chain is confined in one of the layers and polymer bonds on first neighbor edges in different layers interact. We also define and comment results for a model with interactions between monomers on first neighbor sites of different layers. The thermodynamic properties of the model are studied in the grand-canonical formalism and both layers are considered to be Cayley trees. In the core region of the trees, which we may call a bilayer Bethe lattice, we find a very rich phase diagram in the parameter space defined by the two activities of monomers and the Boltzmann factor associated to the interlayer interaction between bonds or monomers. Beside critical and coexistence surfaces, there are tricritical, bicritical and critical endpoint lines, as well as higher order multicritical points.Comment: 21 pages, 10 figures. Journal of Statistical Mechanics: Theory and Experiment (in press

Grand canonical and canonical solution of self-avoiding walks with up to three monomers per site on the Bethe lattice

We solve a model of polymers represented by self-avoiding walks on a lattice which may visit the same site up to three times in the grand-canonical formalism on the Bethe lattice. This may be a model for the collapse transition of polymers where only interactions between monomers at the same site are considered. The phase diagram of the model is very rich, displaying coexistence and critical surfaces, critical, critical endpoint and tricritical lines, as well as a multicritical point. From the grand-canonical results, we present an argument to obtain the properties of the model in the canonical ensemble, and compare our results with simulations in the literature. We do actually find extended and collapsed phases, but the transition between them, composed by a line of critical endpoints and a line of tricritical points, separated by the multicritical point, is always continuous. This result is at variance with the simulations for the model, which suggest that part of the line should be a discontinuous transition. Finally, we discuss the connection of the present model with the standard model for the collapse of polymers (self-avoiding self-attracting walks), where the transition between the extended and collapsed phases is a tricritical point.Comment: 34 pages, including 10 figure

Inverse Additive Problems for Minkowski Sumsets II

The Brunn-Minkowski Theorem asserts that $\mu_d(A+B)^{1/d}\geq \mu_d(A)^{1/d}+\mu_d(B)^{1/d}$ for convex bodies $A,\,B\subseteq \R^d$, where $\mu_d$ denotes the $d$-dimensional Lebesgue measure. It is well-known that equality holds if and only if $A$ and $B$ are homothetic, but few characterizations of equality in other related bounds are known. Let $H$ be a hyperplane. Bonnesen later strengthened this bound by showing $$\mu_d(A+B)\geq (M^{1/(d-1)}+N^{1/(d-1)})^{d-1}(\frac{\mu_d(A)}{M}+\frac{\mu_d(B)}{N}),$$ where $M=\sup\{\mu_{d-1}((\mathbf x+H)\cap A)\mid \mathbf x\in \R^d\}$ and $N=\sup\{\mu_{d-1}((\mathbf y+H)\cap B)\mid \mathbf y\in \R^d\}$. Standard compression arguments show that the above bound also holds when $M=\mu_{d-1}(\pi(A))$ and $N=\mu_{d-1}(\pi(B))$, where $\pi$ denotes a projection of $\mathbb R^d$ onto $H$, which gives an alternative generalization of the Brunn-Minkowski bound. In this paper, we characterize the cases of equality in this later bound, showing that equality holds if and only if $A$ and $B$ are obtained from a pair of homothetic convex bodies by `stretching' along the direction of the projection, which is made formal in the paper. When $d=2$, we characterize the case of equality in the former bound as well

Logarithmic mathematical morphology: a new framework adaptive to illumination changes

A new set of mathematical morphology (MM) operators adaptive to illumination changes caused by variation of exposure time or light intensity is defined thanks to the Logarithmic Image Processing (LIP) model. This model based on the physics of acquisition is consistent with human vision. The fundamental operators, the logarithmic-dilation and the logarithmic-erosion, are defined with the LIP-addition of a structuring function. The combination of these two adjunct operators gives morphological filters, namely the logarithmic-opening and closing, useful for pattern recognition. The mathematical relation existing between ``classical'' dilation and erosion and their logarithmic-versions is established facilitating their implementation. Results on simulated and real images show that logarithmic-MM is more efficient on low-contrasted information than ``classical'' MM

Symmetry breaking and the random-phase approximation in small quantum dots

The random-phase approximation has been used to compute the properties of parabolic two-dimensional quantum dots beyond the mean-field approximation. Special emphasis is put on the ground state correlation energy, the symmetry restoration and the role of the spurious modes within the random-phase approximation. A systematics with the Coulombic interaction strength is presented for the 2-electron dot, while for the 6- and 12-electron dots selected cases are discussed. The validity of the random-phase approximation is assessed by comparison with available exact results.Comment: 9 pages, 4 embedded + 6 gif Figs. Published versio

Density of States for a Specified Correlation Function and the Energy Landscape

The degeneracy of two-phase disordered microstructures consistent with a specified correlation function is analyzed by mapping it to a ground-state degeneracy. We determine for the first time the associated density of states via a Monte Carlo algorithm. Our results are described in terms of the roughness of the energy landscape, defined on a hypercubic configuration space. The use of a Hamming distance in this space enables us to define a roughness metric, which is calculated from the correlation function alone and related quantitatively to the structural degeneracy. This relation is validated for a wide variety of disordered systems.Comment: Accepted for publication in Physical Review Letter

Mathematical morphology on tensor data using the Loewner ordering

The notions of maximum and minimum are the key to the powerful tools of greyscale morphology. Unfortunately these notions do not carry over directly to tensor-valued data. Based upon the Loewner ordering for symmetric matrices this paper extends the maximum and minimum operation to the tensor-valued setting. This provides the ground to establish matrix-valued analogues of the basic morphological operations ranging from erosion/dilation to top hats. In contrast to former attempts to develop a morphological machinery for matrices, the novel definitions of maximal/minimal matrices depend continuously on the input data, a property crucial for the construction of morphological derivatives such as the Beucher gradient or a morphological Laplacian. These definitions are rotationally invariant and preserve positive semidefiniteness of matrix fields as they are encountered in DT-MRI data. The morphological operations resulting from a component-wise maximum/minimum of the matrix channels disregarding their strong correlation fail to be rotational invariant. Experiments on DT-MRI images as well as on indefinite matrix data illustrate the properties and performance of our morphological operators

An integrated software system for geometric correction of LANDSAT MSS imagery

A system for geometrically correcting LANDSAT MSS imagery includes all phases of processing, from receiving a raw computer compatible tape (CCT) to the generation of a corrected CCT (or UTM mosaic). The system comprises modules for: (1) control of the processing flow; (2) calculation of satellite ephemeris and attitude parameters, (3) generation of uncorrected files from raw CCT data; (4) creation, management and maintenance of a ground control point library; (5) determination of the image correction equations, using attitude and ephemeris parameters and existing ground control points; (6) generation of corrected LANDSAT file, using the equations determined beforehand; (7) union of LANDSAT scenes to produce and UTM mosaic; and (8) generation of output tape, in super-structure format