Noncooperative algorithms in self-assembly

We show the first non-trivial positive algorithmic results (i.e. programs whose output is larger than their size), in a model of self-assembly that has so far resisted many attempts of formal analysis or programming: the planar non-cooperative variant of Winfree's abstract Tile Assembly Model. This model has been the center of several open problems and conjectures in the last fifteen years, and the first fully general results on its computational power were only proven recently (SODA 2014). These results, as well as ours, exemplify the intricate connections between computation and geometry that can occur in self-assembly. In this model, tiles can stick to an existing assembly as soon as one of their sides matches the existing assembly. This feature contrasts with the general cooperative model, where it can be required that tiles match on \emph{several} of their sides in order to bind. In order to describe our algorithms, we also introduce a generalization of regular expressions called Baggins expressions. Finally, we compare this model to other automata-theoretic models.Comment: A few bug fixes and typo correction

Three osculating walkers

We consider three directed walkers on the square lattice, which move simultaneously at each tick of a clock and never cross. Their trajectories form a non-crossing configuration of walks. This configuration is said to be osculating if the walkers never share an edge, and vicious (or: non-intersecting) if they never meet. We give a closed form expression for the generating function of osculating configurations starting from prescribed points. This generating function turns out to be algebraic. We also relate the enumeration of osculating configurations with prescribed starting and ending points to the (better understood) enumeration of non-intersecting configurations. Our method is based on a step by step decomposition of osculating configurations, and on the solution of the functional equation provided by this decomposition

On the functions counting walks with small steps in the quarter plane

Models of spatially homogeneous walks in the quarter plane ${\bf Z}_+^{2}$ with steps taken from a subset $\mathcal{S}$ of the set of jumps to the eight nearest neighbors are considered. The generating function $(x,y,z)\mapsto Q(x,y;z)$ of the numbers $q(i,j;n)$ of such walks starting at the origin and ending at $(i,j) \in {\bf Z}_+^{2}$ after $n$ steps is studied. For all non-singular models of walks, the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are continued as multi-valued functions on ${\bf C}$ having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the 51 walks which admit a certain infinite group of birational transformations of ${\bf C}^2$, the interval $]0,1/|\mathcal{S}|[$ of variation of $z$ splits into two dense subsets such that the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are shown to be holonomic for any $z$ from the one of them and non-holonomic for any $z$ from the other. This entails the non-holonomy of $(x,y,z)\mapsto Q(x,y;z)$, and therefore proves a conjecture of Bousquet-M\'elou and Mishna.Comment: 40 pages, 17 figure

Exact enumeration of Hamiltonian circuits, walks, and chains in two and three dimensions

We present an algorithm for enumerating exactly the number of Hamiltonian chains on regular lattices in low dimensions. By definition, these are sets of k disjoint paths whose union visits each lattice vertex exactly once. The well-known Hamiltonian circuits and walks appear as the special cases k=0 and k=1 respectively. In two dimensions, we enumerate chains on L x L square lattices up to L=12, walks up to L=17, and circuits up to L=20. Some results for three dimensions are also given. Using our data we extract several quantities of physical interest

Comparison of in vitro static and dynamic assays to evaluate the efficacy of an antimicrobial drug combination against Staphylococcus aureus

An easily implementable strategy to reduce treatment failures in severe bacterial infections is to combine already available antibiotics. However, most in vitro combination assays are performed by exposing standard bacterial inocula to constant concentrations of antibiotics over less than 24h, which can be poorly representative of clinical situations. The aim of this study was to assess the ability of static and dynamic in vitro Time-Kill Studies (TKS) to identify the potential benefits of an antibiotic combination (here, amikacin and vancomycin) on two different inoculum sizes of two S. aureus strains. In the static TKS (sTKS), performed by exposing both strains over 24h to constant antibiotic concentrations, the activity of the two drugs combined was not significantly different the better drug used alone. However, the dynamic TKS (dTKS) performed over 5 days by exposing one strain to fluctuating concentrations representative of those observed in patients showed that, with the large inoculum, the activities of the drugs, used alone or in combination, significantly differed over time. Vancomycin did not kill bacteria, amikacin led to bacterial regrowth whereas the combination progressively decreased the bacterial load. Thus, dTKS revealed an enhanced effect of the combination on a large inoculum not observed in sTKS. The discrepancy between the sTKS and dTKS results highlights that the assessment of the efficacy of a combination for severe infections associated with a high bacterial load could be demanding. These situations probably require the implementation of dynamic assays over the entire expected treatment duration rather than the sole static assays performed with steady drug concentrations over 24h

Scaling of the atmosphere of self-avoiding walks

The number of free sites next to the end of a self-avoiding walk is known as the atmosphere. The average atmosphere can be related to the number of configurations. Here we study the distribution of atmospheres as a function of length and how the number of walks of fixed atmosphere scale. Certain bounds on these numbers can be proved. We use Monte Carlo estimates to verify our conjectures. Of particular interest are walks that have zero atmosphere, which are known as trapped. We demonstrate that these walks scale in the same way as the full set of self-avoiding walks, barring an overall constant factor

Pharmacokinetics and pharmacodynamics of a therapeutic dose of unfractionated heparin (200 U/kg) administered subcutaneously or intravenously to healthy dogs

Objectives: To evaluate the effects of 200 U/kg of sodium unfractionated heparin (UFH) on coagulation times in dogs after IV and SC administration and to compare these effects with plasma heparin concentrations assessed by its anti Xa activity. Methods: 200 U/kg of UFH were administered Intravenously (IV) and Subcutaneously (SC) to five healthy adult Beagle dogs with a wash out period of at least 3 days. Activated Partial Thromboplastin Time (APTT), Prothrombin Time (PT) and plasma anti-factor Xa (aXa) activity were determined in serial blood samples. Results: After IV injection, PT remained unchanged except for a slight increase in one dog; APTT was not measurable (> 60 s) for 45 to 90 min, then decreased regularly and returned to baseline values between 150 and 240 min. High plasma heparin concentrations were observed (C max = 4.64±1.4 aXa U/mL) and decreased according to a slightly concave-convex pattern on a semi-logarithmic curve but returned to baseline slightly more slowly (t240 to t300 min). After SC administration, APTT was moderately prolonged (mean±SD prolongation: 1.55±0.28 x APTT t0, range [1.35-2.01]) between 1 and 4 hours after administration. Plasma anti-factor Xa activity reached a maximum of 0.56±0.20 aXa U/mL, range: [0.42 - 0.9] after 132±26.8 min and this lasted for 102±26.8 min. Heparin concentrations were grossly correlated to APTT; prolongation of APTT of 120 to 160% corresponded to plasma heparin concentrations range of 0.3-0.7 aXa U/mL, considered as the therapeutic range in human medicine. Conclusions: As in human, pharmacokinetic of UFH in dogs is non linear. Administration of 200 U/kg of UFH SC in healthy dogs results in sustained plasma heparin concentrations in accordance with human recommendations for thrombosis treatment or prevention, without excessively increased bleeding risks. In these conditions, APTT can be used as a surrogate to assess plasma heparin concentrations. This has to be confirmed in diseased animals

Series expansions of the percolation probability on the directed triangular lattice

We have derived long series expansions of the percolation probability for site, bond and site-bond percolation on the directed triangular lattice. For the bond problem we have extended the series from order 12 to 51 and for the site problem from order 12 to 35. For the site-bond problem, which has not been studied before, we have derived the series to order 32. Our estimates of the critical exponent $\beta$ are in full agreement with results for similar problems on the square lattice, confirming expectations of universality. For the critical probability and exponent we find in the site case: $q_c = 0.4043528 \pm 0.0000010$ and $\beta = 0.27645 \pm 0.00010$; in the bond case: $q_c = 0.52198\pm 0.00001$ and $\beta = 0.2769\pm 0.0010$; and in the site-bond case: $q_c = 0.264173 \pm 0.000003$ and $\beta = 0.2766 \pm 0.0003$. In addition we have obtained accurate estimates for the critical amplitudes. In all cases we find that the leading correction to scaling term is analytic, i.e., the confluent exponent $\Delta = 1$.Comment: 26 pages, LaTeX. To appear in J. Phys.

Lattice Point Generating Functions and Symmetric Cones

We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically contrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out the specific cases of a symmetry group of type A (previously known) and types B and D (new). We obtain several applications of the special cases in type B, including identities involving permutation statistics and lecture hall partitions.Comment: 19 page

Integrability of graph combinatorics via random walks and heaps of dimers

We investigate the integrability of the discrete non-linear equation governing the dependence on geodesic distance of planar graphs with inner vertices of even valences. This equation follows from a bijection between graphs and blossom trees and is expressed in terms of generating functions for random walks. We construct explicitly an infinite set of conserved quantities for this equation, also involving suitable combinations of random walk generating functions. The proof of their conservation, i.e. their eventual independence on the geodesic distance, relies on the connection between random walks and heaps of dimers. The values of the conserved quantities are identified with generating functions for graphs with fixed numbers of external legs. Alternative equivalent choices for the set of conserved quantities are also discussed and some applications are presented.Comment: 38 pages, 15 figures, uses epsf, lanlmac and hyperbasic