28,320 research outputs found
The Rotor-Router Model on Regular Trees
The rotor-router model is a deterministic analogue of random walk. It can be
used to define a deterministic growth model analogous to internal DLA. We show
that the set of occupied sites for this model on an infinite regular tree is a
perfect ball whenever it can be, provided the initial rotor configuration is
acyclic (that is, no two neighboring vertices have rotors pointing to one
another). This is proved by defining the rotor-router group of a graph, which
we show is isomorphic to the sandpile group. We also address the question of
recurrence and transience: We give two rotor configurations on the infinite
ternary tree, one for which chips exactly alternate escaping to infinity with
returning to the origin, and one for which every chip returns to the origin.
Further, we characterize the possible "escape sequences" for the ternary tree,
that is, binary words a_1 ... a_n for which there exists a rotor configuration
so that the k-th chip escapes to infinity if and only if a_k=1.Comment: v2 incorporates referee comments, clarifies that the results of
section 2 apply also to multigraph
Neutral Aggregation in Finite Length Genotype space
The advent of modern genome sequencing techniques allows for a more stringent
test of the neutrality hypothesis of Darwinian evolution, where all individuals
have the same fitness. Using the individual based model of Wright and Fisher,
we compute the amplitude of neutral aggregation in the genome space, i.e., the
probability of finding two individuals at genetic (hamming) distance k as a
function of genome size L, population size N and mutation probability per base
\nu. In well mixed populations, we show that for N\nu\textless{}1/L, neutral
aggregation is the dominant force and most individuals are found at short
genetic distances from each other. For N\nu\textgreater{}1 on the contrary,
individuals are randomly dispersed in genome space. The results are extended to
geographically dispersed population, where the controlling parameter is shown
to be a combination of mutation and migration probability. The theory we
develop can be used to test the neutrality hypothesis in various ecological and
evolutionary systems
X-ray studies on crystalline complexes involving amino acids and peptides. XVII. Chirality and molecular aggregation: the crystal structures of DL-arginine DL-glutamate monohydrate and DL-arginine DL-aspartate
DL-Arginine DL-glutamate monohydrate and DL-arginine DL-aspartate, the first DL-DL amino acid-amino acid complexes to be prepared and x-ray analyzed, crystallize in the space group P1 with a = 5.139(2), b = 10.620(1), c = 14.473(2) Å, α = 101.34(1)°, β = 94.08(2)°, γ = 91.38(2)° and a = 5.402(3), b = 9.933(3), c = 13.881(2) Å, α = 99.24(2)°, β = 99.73(3)°, γ = 97.28(3)° , respectively. The structures were solved using counter data and refined to R values of 0.050 and 0.077 for 1827 and 1739 observed reflections, respectively. The basic element of aggregation in both structures is an infinite chain made up of pairs of molecules. Each pair, consisting of a L- and a D-isomer, is stabilized by two centrosymmetrically or nearly centrosymmetrically related hydrogen bonds involving the α-amino and the α-carboxylate groups. Adjacent pairs in the chain are then connected by specific guanidyl-carboxylate interactions. The infinite chains are interconnected through hydrogen bonds to form molecular sheets. The sheets are then stacked along the shortest cell translation. The interactions between sheets involve two head-to-tail sequences in the glutamate complex and one such sequence in the aspartate complex. However, unlike in the corresponding LL and DL complexes, head-to-tail sequences are not the central feature of molecular aggregation in the DL-DL complexes. Indeed, fundamental differences exist among the aggregation patterns in the LL, the LD, and the DL-DL complexes
Rotor walks on general trees
The rotor walk on a graph is a deterministic analogue of random walk. Each
vertex is equipped with a rotor, which routes the walker to the neighbouring
vertices in a fixed cyclic order on successive visits. We consider rotor walk
on an infinite rooted tree, restarted from the root after each escape to
infinity. We prove that the limiting proportion of escapes to infinity equals
the escape probability for random walk, provided only finitely many rotors send
the walker initially towards the root. For i.i.d. random initial rotor
directions on a regular tree, the limiting proportion of escapes is either zero
or the random walk escape probability, and undergoes a discontinuous phase
transition between the two as the distribution is varied. In the critical case
there are no escapes, but the walker's maximum distance from the root grows
doubly exponentially with the number of visits to the root. We also prove that
there exist trees of bounded degree for which the proportion of escapes
eventually exceeds the escape probability by arbitrarily large o(1) functions.
No larger discrepancy is possible, while for regular trees the discrepancy is
at most logarithmic.Comment: 32 page
Divergent mathematical treatments in utility theory
In this paper I study how divergent mathematical treatments affect mathematical modelling, with a special focus on utility theory. In particular I examine recent work on the ranking of information states and the discounting of future utilities, in order to show how, by replacing the standard analytical treatment of the models involved with one based on the framework of Nonstandard Analysis, diametrically opposite results are obtained. In both cases, the choice between the standard and nonstandard treatment amounts to a selection of set-theoretical parameters that cannot be made on purely empirical grounds. The analysis of this phenomenon gives rise to a simple logical account of the relativity of impossibility theorems in economic theory, which concludes the paper
Games with Delays. A Frankenstein Approach
We investigate infinite games on finite graphs where the information flow is
perturbed by nondeterministic signalling delays. It is known that such
perturbations make synthesis problems virtually unsolvable, in the general
case. On the classical model where signals are attached to states, tractable
cases are rare and difficult to identify.
Here, we propose a model where signals are detached from control states, and
we identify a subclass on which equilibrium outcomes can be preserved, even if
signals are delivered with a delay that is finitely bounded. To offset the
perturbation, our solution procedure combines responses from a collection of
virtual plays following an equilibrium strategy in the instant- signalling game
to synthesise, in a Frankenstein manner, an equivalent equilibrium strategy for
the delayed-signalling game
Statistical Mechanics of DNA-Mediated Colloidal Aggregation
We present a statistical mechanical model of aggregation in colloidal systems
with DNA mediated interactions. We obtain a general result for the two-particle
binding energy in terms of the hybridization free energy of DNA and
two model dependent properties: the average number of available DNA bridges
\left and the effective DNA conccentration . We calculate
these parameters for a particular DNA bridging scheme. The fraction of all the
-mers, including the infinite aggregate, are shown to be universal functions
of a single parameter directly related to the two-particle binding energy. We
explicitly take into account the partial ergodicity of the problem resulting
from the slow DNA binding-unbinding dynamics, and introduce the concept of
angular localization of DNA linkers. In this way, we obtain a direct link
between DNA thermodynamics and the global aggregation and melting properties in
DNA-colloidal systems. The results of the theory are shown to be in
quantitative agreement with two recent experiments with particles of micron and
nanometer size.
PACS numbers: 81.16.Dn, 82.20.Db, 68.65.-k, 87.14.GgComment: 12 pages, 6 figures, v2: added reference, expanded conclusion, added
journal re
Complex patterns on the plane: different types of basin fractalization in a two-dimensional mapping
Basins generated by a noninvertible mapping formed by two symmetrically
coupled logistic maps are studied when the only parameter \lambda of the system
is modified. Complex patterns on the plane are visualised as a consequence of
basins' bifurcations. According to the already established nomenclature in the
literature, we present the relevant phenomenology organised in different
scenarios: fractal islands disaggregation, finite disaggregation, infinitely
disconnected basin, infinitely many converging sequences of lakes, countable
self-similar disaggregation and sharp fractal boundary. By use of critical
curves, we determine the influence of zones with different number of first rank
preimages in the mechanisms of basin fractalization.Comment: 19 pages, 11 figure
Broadcasting Automata and Patterns on Z^2
The Broadcasting Automata model draws inspiration from a variety of sources
such as Ad-Hoc radio networks, cellular automata, neighbourhood se- quences and
nature, employing many of the same pattern forming methods that can be seen in
the superposition of waves and resonance. Algorithms for broad- casting
automata model are in the same vain as those encountered in distributed
algorithms using a simple notion of waves, messages passed from automata to au-
tomata throughout the topology, to construct computations. The waves generated
by activating processes in a digital environment can be used for designing a
vari- ety of wave algorithms. In this chapter we aim to study the geometrical
shapes of informational waves on integer grid generated in broadcasting
automata model as well as their potential use for metric approximation in a
discrete space. An explo- ration of the ability to vary the broadcasting radius
of each node leads to results of categorisations of digital discs, their form,
composition, encodings and gener- ation. Results pertaining to the nodal
patterns generated by arbitrary transmission radii on the plane are explored
with a connection to broadcasting sequences and ap- proximation of discrete
metrics of which results are given for the approximation of astroids, a
previously unachievable concave metric, through a novel application of the
aggregation of waves via a number of explored functions
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