On homotopies with triple points of classical knots

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point $p$ of the cylinder is called {\em coherent} if all three branches intersect at $p$ pairwise with the same index. A {\em triple unknotting} of a classical knot $K$ is a homotopy which connects $K$ with the trivial knot and which has as singularities only coherent triple points. We give a new formula for the first Vassiliev invariant $v_2(K)$ by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that there are triple unknottings which are not homotopic as triple unknottings even if we allow more complicated singularities to appear in the homotopy of the homotopy.Comment: 10 pages, 13 figures, bugs in figures correcte

The Influence of Canalization on the Robustness of Boolean Networks

Time- and state-discrete dynamical systems are frequently used to model molecular networks. This paper provides a collection of mathematical and computational tools for the study of robustness in Boolean network models. The focus is on networks governed by $k$-canalizing functions, a recently introduced class of Boolean functions that contains the well-studied class of nested canalizing functions. The activities and sensitivity of a function quantify the impact of input changes on the function output. This paper generalizes the latter concept to $c$-sensitivity and provides formulas for the activities and $c$-sensitivity of general $k$-canalizing functions as well as canalizing functions with more precisely defined structure. A popular measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the $c$-sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.Comment: 16 pages, 2 figures, 3 table

Boolean networks synchronism sensitivity and XOR circulant networks convergence time

In this paper are presented first results of a theoretical study on the role of non-monotone interactions in Boolean automata networks. We propose to analyse the contribution of non-monotony to the diversity and complexity in their dynamical behaviours according to two axes. The first one consists in supporting the idea that non-monotony has a peculiar influence on the sensitivity to synchronism of such networks. It leads us to the second axis that presents preliminary results and builds an understanding of the dynamical behaviours, in particular concerning convergence times, of specific non-monotone Boolean automata networks called XOR circulant networks.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

Parameter estimation for Boolean models of biological networks

Boolean networks have long been used as models of molecular networks and play an increasingly important role in systems biology. This paper describes a software package, Polynome, offered as a web service, that helps users construct Boolean network models based on experimental data and biological input. The key feature is a discrete analog of parameter estimation for continuous models. With only experimental data as input, the software can be used as a tool for reverse-engineering of Boolean network models from experimental time course data.Comment: Web interface of the software is available at http://polymath.vbi.vt.edu/polynome

Higher order topological actions

In classical mechanics, an action is defined only modulo additive terms which do not modify the equations of motion; in certain cases, these terms are topological quantities. We construct an infinite sequence of higher order topological actions and argue that they play a role in quantum mechanics, and hence can be accessed experimentally.Comment: 7 page

Introduction to Strategy, Information, Technology, Economics, and Society (SITES) Minitrack

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The effect of negative feedback loops on the dynamics of Boolean networks

Feedback loops in a dynamic network play an important role in determining the dynamics of that network. Through a computational study, in this paper we show that networks with fewer independent negative feedback loops tend to exhibit more regular behavior than those with more negative loops. To be precise, we study the relationship between the number of independent feedback loops and the number and length of the limit cycles in the phase space of dynamic Boolean networks. We show that, as the number of independent negative feedback loops increases, the number (length) of limit cycles tends to decrease (increase). These conclusions are consistent with the fact, for certain natural biological networks, that they on the one hand exhibit generally regular behavior and on the other hand show less negative feedback loops than randomized networks with the same numbers of nodes and connectivity