Eisenstein Series and String Thresholds

We investigate the relevance of Eisenstein series for representing certain $G(Z)$-invariant string theory amplitudes which receive corrections from BPS states only. $G(Z)$ may stand for any of the mapping class, T-duality and U-duality groups $Sl(d,Z)$, $SO(d,d,Z)$ or $E_{d+1(d+1)}(Z)$ respectively. Using $G(Z)$-invariant mass formulae, we construct invariant modular functions on the symmetric space $K\backslash G(R)$ of non-compact type, with $K$ the maximal compact subgroup of $G(R)$, that generalize the standard non-holomorphic Eisenstein series arising in harmonic analysis on the fundamental domain of the Poincar\'e upper half-plane. Comparing the asymptotics and eigenvalues of the Eisenstein series under second order differential operators with quantities arising in one- and $g$-loop string amplitudes, we obtain a manifestly T-duality invariant representation of the latter, conjecture their non-perturbative U-duality invariant extension, and analyze the resulting non-perturbative effects. This includes the $R^4$ and $R^4 H^{4g-4}$ couplings in toroidal compactifications of M-theory to any dimension $D\geq 4$ and $D\geq 6$ respectively.Comment: Latex2e, 60 pages; v2: Appendix A.4 extended, 2 refs added, thms renumbered, plus minor corrections; v3: relation (1.7) to math Eis series clarified, eq (3.3) and minor typos corrected, final version to appear in Comm. Math. Phys; v4: misprints and Eq C.13,C.24 corrected, see note adde

A New Estimate of the Cutoff Value in the Bak-Sneppen Model

We present evidence that the Bak-Sneppen model of evolution on $N$ vertices requires $N^3$ iterates to reach equilibrium. This is substantially more than previous authors suggested (on the order of $N^2$). Based on that estimate, we present a novel algorithm inspired by previous rank-driven analyses of the model allowing for direct simulation of the model with populations of up to $N = 25600$ for $2\cdot N^3$ iterations. These extensive simulations suggest a cutoff value of $x^* = 0.66692 \pm 0.00003$, a value slightly lower than previously estimated yet still distinctly above $2/3$. We also study how the cutoff values $x^*_N$ at finite $N$ approximate the conjectured value $x^*$ at $N=\infty$. Assuming $x^*_N-x^*_\infty \sim N^{-\nu}$, we find that $\nu=0.978\pm 0.025$, which is significantly lower than previous estimates ($\nu\approx 1.4$).Comment: 18 figures, 12 page

Dynamics of Locally Coupled Oscillators with Next-Nearest-Neighbor Interaction

A theoretical description of decentralized dynamics within linearly coupled, one-dimensional oscillators (agents) with up to next-nearest-neighbor interaction is given. Conditions for stability of such system are presented. Our results indicate that the stable systems have response that grow at least linearly in the system size. We give criteria when this is the case. The dynamics of these systems can be described with traveling waves with strong damping in the high frequencies. Depending on the system parameters, two types of solutions have been found: damped oscillations and reflectionless waves. The latter is a novel result and a feature of systems with at least next-nearest-neighbor interactions. Analytical predictions are tested in numerical simulations

Scaffolding School Pupils’ Scientific Argumentation with Evidence-Based Dialogue Maps

This chapter reports pilot work investigating the potential of Evidence-based Dialogue Mapping to scaffold young teenagers’ scientific argumentation. Our research objective is to better understand pupils’ usage of dialogue maps created in Compendium to write scientific ex-planations. The participants were 20 pupils, 12-13 years old, in a summer science course for “gifted and talented” children in the UK. Through qualitative analysis of three case studies, we investigate the value of dialogue mapping as a mediating tool in the scientific reasoning process during a set of learning activities. These activities were published in an online learning envi-ronment to foster collaborative learning. Pupils mapped their discussions in pairs, shared maps via the online forum and in plenary discussions, and wrote essays based on their dialogue maps. This study draws on these multiple data sources: pupils’ maps in Compendium, writings in science and reflective comments about the uses of mapping for writing. Our analysis highlights the diversity of ways, both successful and unsuccessful, in which dialogue mapping was used by these young teenagers

Piecewise Linear Models for the Quasiperiodic Transition to Chaos

We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode-locking and the quasi-periodic transition to chaos. For instance, for these families, we obtain complete solutions to several questions still largely unanswered for families of smooth circle maps. Our main results describe (1) the sets of maps in these families having some prescribed rotation interval; (2) the boundaries between zero and positive topological entropy and between zero length and non-zero length rotation interval; and (3) the structure and bifurcations of the attractors in one of these families. We discuss the interpretation of these maps as low-order spline approximations to the classic ``sine-circle'' map and examine more generally the implications of our results for the case of smooth circle maps. We also mention a possible connection to recent experiments on models of a driven Josephson junction.Comment: 75 pages, plain TeX, 47 figures (available on request

Equation of State for Parallel Rigid Spherocylinders

The pair distribution function of monodisperse rigid spherocylinders is calculated by Shinomoto's method, which was originally proposed for hard spheres. The equation of state is derived by two different routes: Shinomoto's original route, in which a hard wall is introduced to estimate the pressure exerted on it, and the virial route. The pressure from Shinomoto's original route is valid only when the length-to-width ratio is less than or equal to 0.25 (i.e., when the spherocylinders are nearly spherical). The virial equation of state is shown to agree very well with the results of numerical simulations of spherocylinders with length-to-width ratio greater than or equal to 2

Polydispersity and ordered phases in solutions of rodlike macromolecules

We apply density functional theory to study the influence of polydispersity on the stability of columnar, smectic and solid ordering in the solutions of rodlike macromolecules. For sufficiently large length polydispersity (standard deviation $\sigma>0.25$) a direct first-order nematic-columnar transition is found, while for smaller $\sigma$ there is a continuous nematic-smectic and first-order smectic-columnar transition. For increasing polydispersity the columnar structure is stabilized with respect to solid perturbations. The length distribution of macromolecules changes neither at the nematic-smectic nor at the nematic-columnar transition, but it does change at the smectic-columnar phase transition. We also study the phase behaviour of binary mixtures, in which the nematic-smectic transition is again found to be continuous. Demixing according to rod length in the smectic phase is always preempted by transitions to solid or columnar ordering.Comment: 13 pages (TeX), 2 Postscript figures uuencode