Reply to the comment by Jacobs and Thorpe

Reply to a comment on "Infinite-Cluster geometry in central-force networks", PRL 78 (1997), 1480. A discussion about the order of the rigidity percolation transition.Comment: 1 page revTe

Combinatorial models of rigidity and renormalization

We first introduce the percolation problems associated with the graph theoretical concepts of $(k,l)$-sparsity, and make contact with the physical concepts of ordinary and rigidity percolation. We then devise a renormalization transformation for $(k,l)$-percolation problems, and investigate its domain of validity. In particular, we show that it allows an exact solution of $(k,l)$-percolation problems on hierarchical graphs, for $k\leq l<2k$. We introduce and solve by renormalization such a model, which has the interesting feature of showing both ordinary percolation and rigidity percolation phase transitions, depending on the values of the parameters.Comment: 22 pages, 6 figure

Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe Lattices

We show that negative of the number of floppy modes behaves as a free energy for both connectivity and rigidity percolation, and we illustrate this result using Bethe lattices. The rigidity transition on Bethe lattices is found to be first order at a bond concentration close to that predicted by Maxwell constraint counting. We calculate the probability of a bond being on the infinite cluster and also on the overconstrained part of the infinite cluster, and show how a specific heat can be defined as the second derivative of the free energy. We demonstrate that the Bethe lattice solution is equivalent to that of the random bond model, where points are joined randomly (with equal probability at all length scales) to have a given coordination, and then subsequently bonds are randomly removed.Comment: RevTeX 11 pages + epsfig embedded figures. Submitted to Phys. Rev.

Toward the next generation of research into small area effects on health : a synthesis of multilevel investigations published since July 1998.

To map out area effects on health research, this study had the following aims: (1) to inventory multilevel investigations of area effects on self rated health, cardiovascular diseases and risk factors, and mortality among adults; (2) to describe and critically discuss methodological approaches employed and results observed; and (3) to formulate selected recommendations for advancing the study of area effects on health. Overall, 86 studies were inventoried. Although several innovative methodological approaches and analytical designs were found, small areas are most often operationalised using administrative and statistical spatial units. Most studies used indicators of area socioeconomic status derived from censuses, and few provided information on the validity and reliability of measures of exposures. A consistent finding was that a significant portion of the variation in health is associated with area context independently of individual characteristics. Area effects on health, although significant in most studies, often depend on the health outcome studied, the measure of area exposure used, and the spatial scale at which associations are examined

Indexed induction and coinduction, fibrationally.

This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a sound coinduction rule for any data type arising as the final coalgebra of a functor, thus relaxing Hermida and Jacobs’ restriction to polynomial data types. For this we introduce the notion of a quotient category with equality (QCE), which both abstracts the standard notion of a fibration of relations constructed from a given fibration, and plays a role in the theory of coinduction dual to that of a comprehension category with unit (CCU) in the theory of induction. Second, we show that indexed inductive and coinductive types also admit sound induction and coinduction rules. Indexed data types often arise as initial algebras and final coalgebras of functors on slice categories, so our key technical results give sufficent conditions under which we can construct, from a CCU (QCE) U : E -> B, a fibration with base B/I that models indexing by I and is also a CCU (QCE)

Experimental Demonstration of a Quantum Circuit using Linear Optics Gates

One of the main advantages of an optical approach to quantum computing is the fact that optical fibers can be used to connect the logic and memory devices to form useful circuits, in analogy with the wires of a conventional computer. Here we describe an experimental demonstration of a simple quantum circuit of that kind in which two probabilistic exclusive-OR (XOR) logic gates were combined to calculate the parity of three input qubits.Comment: v2 is final PRA versio