682 research outputs found
Eisenstein Series and String Thresholds
We investigate the relevance of Eisenstein series for representing certain
-invariant string theory amplitudes which receive corrections from BPS
states only. may stand for any of the mapping class, T-duality and
U-duality groups , or respectively.
Using -invariant mass formulae, we construct invariant modular functions
on the symmetric space of non-compact type, with the
maximal compact subgroup of , 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 -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 and
couplings in toroidal compactifications of M-theory to any
dimension and 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 vertices
requires iterates to reach equilibrium. This is substantially more than
previous authors suggested (on the order of ). 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 for iterations. These extensive simulations suggest a
cutoff value of , a value slightly lower than
previously estimated yet still distinctly above . We also study how the
cutoff values at finite approximate the conjectured value at
. Assuming , we find that
, which is significantly lower than previous estimates
().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 Pupilsâ Scientific Argumentation with Evidence-Based Dialogue Maps
This chapter reports pilot work investigating the potential of Evidence-based Dialogue Mapping to scaffold young teenagersâ scientific argumentation. Our research objective is to better understand pupilsâ 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 âgifted and talentedâ 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: pupilsâ 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 ) a direct first-order nematic-columnar transition is
found, while for smaller 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
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