Permutation Classes of Polynomial Growth

A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterised as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and sufficient conditions on sets of forbidden permutations which ensure that the associated pattern class is of polynomial growth is determined. A catalogue of all such sets of forbidden permutations having three or fewer elements is provided together with bounds on the degrees of the associated enumerating polynomials.Comment: 17 pages, 4 figure

Validity of the Adiabatic Approximation

We analyze the validity of the adiabatic approximation, and in particular the reliability of what has been called the "standard criterion" for validity of this approximation. Recently, this criterion has been found to be insufficient. We will argue that the criterion is sufficient only when it agrees with the intuitive notion of slowness of evolution of the Hamiltonian. However, it can be insufficient in cases where the Hamiltonian varies rapidly but only by a small amount. We also emphasize the distinction between the adiabatic {\em theorem} and the adiabatic {\em approximation}, two quite different although closely related ideas.Comment: 4 pages, 1 figur

Percolation in Directed Scale-Free Networks

Many complex networks in nature have directed links, a property that affects the network's navigability and large-scale topology. Here we study the percolation properties of such directed scale-free networks with correlated in- and out-degree distributions. We derive a phase diagram that indicates the existence of three regimes, determined by the values of the degree exponents. In the first regime we regain the known directed percolation mean field exponents. In contrast, the second and third regimes are characterized by anomalous exponents, which we calculate analytically. In the third regime the network is resilient to random dilution, i.e., the percolation threshold is p_c->1.Comment: Latex, 5 pages, 2 fig

Developmental gene regulatory network architecture across 500 million years of echinoderm evolution

Evolutionary change in morphological features must depend on architectural reorganization of developmental gene regulatory networks (GRNs), just as true conservation of morphological features must imply retention of ancestral developmental GRN features. Key elements of the provisional GRN for embryonic endomesoderm development in the sea urchin are here compared with those operating in embryos of a distantly related echinoderm, a starfish. These animals diverged from their common ancestor 520-480 million years ago. Their endomesodermal fate maps are similar, except that sea urchins generate a skeletogenic cell lineage that produces a prominent skeleton lacking entirely in starfish larvae. A relevant set of regulatory genes was isolated from the starfish Asterina miniata, their expression patterns determined, and effects on the other genes of perturbing the expression of each were demonstrated. A three-gene feedback loop that is a fundamental feature of the sea urchin GRN for endoderm specification is found in almost identical form in the starfish: a detailed element of GRN architecture has been retained since the Cambrian Period in both echinoderm lineages. The significance of this retention is highlighted by the observation of numerous specific differences in the GRN connections as well. A regulatory gene used to drive skeletogenesis in the sea urchin is used entirely differently in the starfish, where it responds to endomesodermal inputs that do not affect it in the sea urchin embryo. Evolutionary changes in the GRNs since divergence are limited sharply to certain cis-regulatory elements, whereas others have persisted unaltered

Epidemic dynamics in finite size scale-free networks

Many real networks present a bounded scale-free behavior with a connectivity cut-off due to physical constraints or a finite network size. We study epidemic dynamics in bounded scale-free networks with soft and hard connectivity cut-offs. The finite size effects introduced by the cut-off induce an epidemic threshold that approaches zero at increasing sizes. The induced epidemic threshold is very small even at a relatively small cut-off, showing that the neglection of connectivity fluctuations in bounded scale-free networks leads to a strong over-estimation of the epidemic threshold. We provide the expression for the infection prevalence and discuss its finite size corrections. The present work shows that the highly heterogeneous nature of scale-free networks does not allow the use of homogeneous approximations even for systems of a relatively small number of nodes.Comment: 4 pages, 2 eps figure

Monetary policy and economic activity: a postwar review

Monetary policy ; Financial markets ; Economic conditions