Random trees between two walls: Exact partition function

We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that adjacent vertices have labels differing by +1 or -1. A non-trivial partition function is obtained whenever the target space is bounded by walls. We concentrate on the two cases where the target space is (i) the half-line bounded by a wall at the origin or (ii) a segment bounded by two walls at a finite distance. The general solution has a soliton-like structure involving elliptic functions. We derive the corresponding continuum scaling limit which takes the remarkable form of the Weierstrass p-function with constrained periods. These results are used to analyze the probability for an evolving population spreading in one dimension to attain the boundary of a given domain with the geometry of the target (i) or (ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main modifications in Sect. 5-6 and conclusio

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

An L1 Penalty Method for General Obstacle Problems

We construct an efficient numerical scheme for solving obstacle problems in divergence form. The numerical method is based on a reformulation of the obstacle in terms of an L1-like penalty on the variational problem. The reformulation is an exact regularizer in the sense that for large (but finite) penalty parameter, we recover the exact solution. Our formulation is applied to classical elliptic obstacle problems as well as some related free boundary problems, for example the two-phase membrane problem and the Hele-Shaw model. One advantage of the proposed method is that the free boundary inherent in the obstacle problem arises naturally in our energy minimization without any need for problem specific or complicated discretization. In addition, our scheme also works for nonlinear variational inequalities arising from convex minimization problems.Comment: 20 pages, 18 figure

Small-Signal Amplification of Period-Doubling Bifurcations in Smooth Iterated Maps

Various authors have shown that, near the onset of a period-doubling bifurcation, small perturbations in the control parameter may result in much larger disturbances in the response of the dynamical system. Such amplification of small signals can be measured by a gain defined as the magnitude of the disturbance in the response divided by the perturbation amplitude. In this paper, the perturbed response is studied using normal forms based on the most general assumptions of iterated maps. Such an analysis provides a theoretical footing for previous experimental and numerical observations, such as the failure of linear analysis and the saturation of the gain. Qualitative as well as quantitative features of the gain are exhibited using selected models of cardiac dynamics.Comment: 12 pages, 7 figure

Thermodynamic Analysis of Interacting Nucleic Acid Strands

Motivated by the analysis of natural and engineered DNA and RNA systems, we present the first algorithm for calculating the partition function of an unpseudoknotted complex of multiple interacting nucleic acid strands. This dynamic program is based on a rigorous extension of secondary structure models to the multistranded case, addressing representation and distinguishability issues that do not arise for single-stranded structures. We then derive the form of the partition function for a fixed volume containing a dilute solution of nucleic acid complexes. This expression can be evaluated explicitly for small numbers of strands, allowing the calculation of the equilibrium population distribution for each species of complex. Alternatively, for large systems (e.g., a test tube), we show that the unique complex concentrations corresponding to thermodynamic equilibrium can be obtained by solving a convex programming problem. Partition function and concentration information can then be used to calculate equilibrium base-pairing observables. The underlying physics and mathematical formulation of these problems lead to an interesting blend of approaches, including ideas from graph theory, group theory, dynamic programming, combinatorics, convex optimization, and Lagrange duality

Design and fabrication of a radiative actively cooled honeycomb sandwich structural panel for a hypersonic aircraft

The panel assembly consisted of an external thermal protection system (metallic heat shields and insulation blankets) and an aluminum honeycomb structure. The structure was cooled to temperature 442K (300 F) by circulating a 60/40 mass solution of ethylene glycol and water through dee shaped coolant tubes nested in the honeycomb and adhesively bonded to the outer skin. Rene'41 heat shields were designed to sustain 5000 cycles of a uniform pressure of + or - 6.89kPa (+ or - 1.0 psi) and aerodynamic heating conditions equivalent to 136 kW sq m (12 Btu sq ft sec) to a 422K (300 F) surface temperature. High temperature flexible insulation blankets were encased in stainless steel foil to protect them from moisture and other potential contaminates. The aluminum actively cooled honeycomb sandwich structural panel was designed to sustain 5000 cycles of cyclic in-plane loading of + or - 210 kN/m (+ or - 1200 lbf/in.) combined with a uniform panel pressure of + or - 6.89 kPa (?1.0 psi)

The effect of combined torsional and bending loads on a channel beam with one end restrained from warping

Channel beams have been designed with the main purpose of giving large resistance to bending while the torsional strengths are known to be relatively small. In this experiment, a cantilevered channel beam was loaded with a concentrated load at the free end, first through the experimentally determined shear center and then through the centroid of the cross section. For each loading condition, the strains were measured with the aid of SR-4 strain gages placed at intervals along the length of the beam. With the aid of a computer, the strains were converted into longitudinal stresses and these stresses were compared to the theoretically predicted values of longitudinal stresses. The derivation of the torsional stress equation as shown in the text Advanced Mechanics of Materials by Seely and Smith is briefly compared to the derivation of the torsional stress equation as shown in the text Strength of Materials by Timoshenko. The two derivations are discussed and, even though the derivations are completely different, the values of longitudinal stress obtained by the use of each equation agree very well with each other. The longitudinal stresses calculated from the experimentally obtained values or strain agree very closely with the theoretically predicted values of longitudinal stress --Abstract, page iii