The Number of Different Binary Functions Generated by NK-Kauffman Networks and the Emergence of Genetic Robustness

We determine the average number $\vartheta (N, K)$, of \textit{NK}-Kauffman networks that give rise to the same binary function. We show that, for $N \gg 1$, there exists a connectivity critical value $K_c$ such that $\vartheta(N,K) \approx e^{\phi N}$ ($\phi > 0$) for $K < K_c$ and $\vartheta(N,K) \approx 1$ for $K > K_c$. We find that $K_c$ is not a constant, but scales very slowly with $N$, as $K_c \approx \log_2 \log_2 (2N / \ln 2)$. The problem of genetic robustness emerges as a statistical property of the ensemble of \textit{NK}-Kauffman networks and impose tight constraints in the average number of epistatic interactions that the genotype-phenotype map can have.Comment: 4 figures 18 page

The computational complexity of Kauffman nets and the P versus NP problem

Complexity theory as practiced by physicists and computational complexity theory as practiced by computer scientists both characterize how difficult it is to solve complex problems. Here it is shown that the parameters of a specific model can be adjusted so that the problem of finding its global energy minimum is extremely sensitive to small changes in the problem statement. This result has implications not only for studies of the physics of random systems but may also lead to new strategies for resolving the well-known P versus NP question in computational complexity theory.Comment: 4 pages, no figure

Experimental approximation of the Jones polynomial with DQC1

We present experimental results approximating the Jones polynomial using 4 qubits in a liquid state nuclear magnetic resonance quantum information processor. This is the first experimental implementation of a complete problem for the deterministic quantum computation with one quantum bit model of quantum computation, which uses a single qubit accompanied by a register of completely random states. The Jones polynomial is a knot invariant that is important not only to knot theory, but also to statistical mechanics and quantum field theory. The implemented algorithm is a modification of the algorithm developed by Shor and Jordan suitable for implementation in NMR. These experimental results show that for the restricted case of knots whose braid representations have four strands and exactly three crossings, identifying distinct knots is possible 91% of the time.Comment: 5 figures. Version 2 changes: published version, minor errors corrected, slight changes to improve readabilit

Teleportation, Braid Group and Temperley--Lieb Algebra

We explore algebraic and topological structures underlying the quantum teleportation phenomena by applying the braid group and Temperley--Lieb algebra. We realize the braid teleportation configuration, teleportation swapping and virtual braid representation in the standard description of the teleportation. We devise diagrammatic rules for quantum circuits involving maximally entangled states and apply them to three sorts of descriptions of the teleportation: the transfer operator, quantum measurements and characteristic equations, and further propose the Temperley--Lieb algebra under local unitary transformations to be a mathematical structure underlying the teleportation. We compare our diagrammatical approach with two known recipes to the quantum information flow: the teleportation topology and strongly compact closed category, in order to explain our diagrammatic rules to be a natural diagrammatic language for the teleportation.Comment: 33 pages, 19 figures, latex. The present article is a short version of the preprint, quant-ph/0601050, which includes details of calculation, more topics such as topological diagrammatical operations and entanglement swapping, and calls the Temperley--Lieb category for the collection of all the Temperley--Lieb algebra with physical operations like local unitary transformation

Entwined Paths, Difference Equations and the Dirac Equation

Entwined space-time paths are bound pairs of trajectories which are traversed in opposite directions with respect to macroscopic time. In this paper we show that ensembles of entwined paths on a discrete space-time lattice are simply described by coupled difference equations which are discrete versions of the Dirac equation. There is no analytic continuation, explicit or forced, involved in this description. The entwined paths are `self-quantizing'. We also show that simple classical stochastic processes that generate the difference equations as ensemble averages are stable numerically and converge at a rate governed by the details of the stochastic process. This result establishes the Dirac equation in one dimension as a phenomenological equation describing an underlying classical stochastic process in the same sense that the Diffusion and Telegraph equations are phenomenological descriptions of stochastic processes.Comment: 15 pages, 5 figures Replacement 11/02 contains minor editorial change

Distinguishing scalar from pseudoscalar Higgs production at the LHC

In this letter we examine the production channels for the scalar or pseudoscalar Higgs plus two jets at the CERN Large Hadron Collider (LHC). We identify possible signals for distinguishing between a scalar and a pseudoscalar Higgs boson.Comment: 7 pages, REVTeX4, 4 eps figures. Figure 1 and 4 replaced. Typos corrected, additional reference adde

Lens Spaces and Handlebodies in 3D Quantum Gravity

We calculate partition functions for lens spaces L_{p,q} up to p=8 and for genus 1 and 2 handlebodies H_1, H_2 in the Turaev-Viro framework. These can be interpreted as transition amplitudes in 3D quantum gravity. In the case of lens spaces L_{p,q} these are vacuum-to-vacuum amplitudes \O -> \O, whereas for the 1- and 2-handlebodies H_1, H_2 they represent genuinely topological transition amplitudes \O -> T^2 and \O -> T^2 # T^2, respectively.Comment: 14 pages, LaTeX, 5 figures, uses eps

The Asymptotic Number of Attractors in the Random Map Model

The random map model is a deterministic dynamical system in a finite phase space with n points. The map that establishes the dynamics of the system is constructed by randomly choosing, for every point, another one as being its image. We derive here explicit formulas for the statistical distribution of the number of attractors in the system. As in related results, the number of operations involved by our formulas increases exponentially with n; therefore, they are not directly applicable to study the behavior of systems where n is large. However, our formulas lend themselves to derive useful asymptotic expressions, as we show.Comment: 16 pages, 1 figure. Minor changes. To be published in Journal of Physics A: Mathematical and Genera

Attractor-Specific and Common Expression Values in Random Boolean Network Models (with a Preliminary Look at Single-Cell Data)

Random Boolean Networks (RBNs for short) are strongly simplified models of gene regulatory networks (GRNs), which have also been widely studied as abstract models of complex systems and have been used to simulate different phenomena. We define the “common sea” (CS) as the set of nodes that take the same value in all the attractors of a given network realization, and the “specific part” (SP) as the set of all the other nodes, and we study their properties in different ensembles, generated with different parameter values. Both the CS and of the SP can be composed of one or more weakly connected components, which are emergent intermediate-level structures. We show that the study of these sets provides very important information about the behavior of the model. The distribution of distances between attractors is also examined. Moreover, we show how the notion of a “common sea” of genes can be used to analyze data from single-cell experiments

Abundances of Baade's Window Giants from Keck/HIRES Spectra: II. The Alpha- and Light Odd Elements

We report detailed chemical abundance analysis of 27 RGB stars towards the Galactic bulge in Baade's Window for elements produced by massive stars: O, Na, Mg, Al, Si, Ca and Ti. All of these elements are overabundant in the bulge relative to the disk, especially Mg, indicating that the bulge is enhanced in Type~II supernova ejecta and most likely formed more rapidly than the disk. We attribute a rapid decline of [O/Fe] to metallicity-dependent yields of oxygen in massive stars, perhaps connected to the Wolf-Reyet phenomenon. he explosive nucleosynthesis alphas, Si, Ca and Ti, possess identical trends with [Fe/H], consistent with their putative common origin. We note that different behaviors of hydrostatic and explosive alpha elements can be seen in the stellar abundances of stars in Local Group dwarf galaxies. We also attribute the decline of Si,Ca and Ti relative to Mg, to metallicity- dependent yields for the explosive alpha elements from Type~II supernovae. The starkly smaller scatter of [/Fe] with [Fe/H] in the bulge, as compared to the halo, is consistent with expected efficient mixing for the bulge. The metal-poor bulge [/Fe] ratios are higher than ~80% of the halo. If the bulge formed from halo gas, the event occured before ~80% of the present-day halo was formed. The lack of overlap between the thick and thin disk composition with the bulge does not support the idea that the bulge was built by a thickening of the disk driven by the bar. The trend of [Al/Fe] is very sensitive to the chemical evolution environment. A comparison of the bulge, disk and Sgr dSph galaxy shows a range of ~0.7 dex in [Al/Fe] at a given [Fe/H], presumably due to a range of Type~II/Type~Ia supernova ratios in these systems.Comment: 51 pages, 6 tables, 27 figures, submitte