2,924 research outputs found
Sequence variations of the 1-2-3 Conjecture and irregularity strength
Karonski, Luczak, and Thomason (2004) conjectured that, for any connected
graph G on at least three vertices, there exists an edge weighting from {1,2,3}
such that adjacent vertices receive different sums of incident edge weights.
Bartnicki, Grytczuk, and Niwcyk (2009) made a stronger conjecture, that each
edge's weight may be chosen from an arbitrary list of size 3 rather than
{1,2,3}. We examine a variation of these conjectures, where each vertex is
coloured with a sequence of edge weights. Such a colouring relies on an
ordering of the graph's edges, and so two variations arise -- one where we may
choose any ordering of the edges and one where the ordering is fixed. In the
former case, we bound the list size required for any graph. In the latter, we
obtain a bound on list sizes for graphs with sufficiently large minimum degree.
We also extend our methods to a list variation of irregularity strength, where
each vertex receives a distinct sequence of edge weights.Comment: Accepted to Discrete Mathematics and Theoretical Computer Scienc
Boolean Delay Equations: A simple way of looking at complex systems
Boolean Delay Equations (BDEs) are semi-discrete dynamical models with
Boolean-valued variables that evolve in continuous time. Systems of BDEs can be
classified into conservative or dissipative, in a manner that parallels the
classification of ordinary or partial differential equations. Solutions to
certain conservative BDEs exhibit growth of complexity in time. They represent
therewith metaphors for biological evolution or human history. Dissipative BDEs
are structurally stable and exhibit multiple equilibria and limit cycles, as
well as more complex, fractal solution sets, such as Devil's staircases and
``fractal sunbursts``. All known solutions of dissipative BDEs have stationary
variance. BDE systems of this type, both free and forced, have been used as
highly idealized models of climate change on interannual, interdecadal and
paleoclimatic time scales. BDEs are also being used as flexible, highly
efficient models of colliding cascades in earthquake modeling and prediction,
as well as in genetics. In this paper we review the theory of systems of BDEs
and illustrate their applications to climatic and solid earth problems. The
former have used small systems of BDEs, while the latter have used large
networks of BDEs. We moreover introduce BDEs with an infinite number of
variables distributed in space (``partial BDEs``) and discuss connections with
other types of dynamical systems, including cellular automata and Boolean
networks. This research-and-review paper concludes with a set of open
questions.Comment: Latex, 67 pages with 15 eps figures. Revised version, in particular
the discussion on partial BDEs is updated and enlarge
Micromechanics of fatigue in woven and stitched composites
The goals of this research program were to: (1) determine how microstructural factors, especially the architecture of reinforcing fibers, control stiffness, strength, and fatigue life in 3D woven composites; (2) identify mechanisms of failure; (3) model composite stiffness; (4) model notched and unnotched strength; and (5) model fatigue life. We have examined a total of eleven different angle and orthogonal interlock woven composites. Extensive testing has revealed that these 3D woven composites possess an extraordinary combination of strength, damage tolerance, and notch insensitivity in compression and tension and in monotonic and cyclic loading. In many important regards, 3D woven composites far outstrip conventional 2D laminates or stitched laminates. Detailed microscopic analysis of damage has led to a comprehensive picture of the essential mechanisms of failure and how they are related to the reinforcement geometry. The critical characteristics of the weave architecture that promote favorable properties have been identified. Key parameters are tow size and the distributions in space and strength of geometrical flaws. The geometrical flaws should be regarded as controllable characteristics of the weave in design and manufacture. In addressing our goals, the simplest possible models of properties were always sought, in a blend of old and new modeling concepts. Nevertheless, certain properties, especially regarding damage tolerance, ultimate failure, and the detailed effects of weave architecture, require computationally intensive stochastic modeling. We have developed a new model, the 'binary model,' to carry out such tasks in the most efficient manner and with faithful representation of crucial mechanisms. This is the final report for contract NAS1-18840. It covers all work from April 1989 up to the conclusion of the program in January 1993
Extreme value laws in dynamical systems under physical observables
Extreme value theory for chaotic dynamical systems is a rapidly expanding
area of research. Given a system and a real function (observable) defined on
its phase space, extreme value theory studies the limit probabilistic laws
obeyed by large values attained by the observable along orbits of the system.
Based on this theory, the so-called block maximum method is often used in
applications for statistical prediction of large value occurrences. In this
method, one performs inference for the parameters of the Generalised Extreme
Value (GEV) distribution, using maxima over blocks of regularly sampled
observations along an orbit of the system. The observables studied so far in
the theory are expressed as functions of the distance with respect to a point,
which is assumed to be a density point of the system's invariant measure.
However, this is not the structure of the observables typically encountered in
physical applications, such as windspeed or vorticity in atmospheric models. In
this paper we consider extreme value limit laws for observables which are not
functions of the distance from a density point of the dynamical system. In such
cases, the limit laws are no longer determined by the functional form of the
observable and the dimension of the invariant measure: they also depend on the
specific geometry of the underlying attractor and of the observable's level
sets. We present a collection of analytical and numerical results, starting
with a toral hyperbolic automorphism as a simple template to illustrate the
main ideas. We then formulate our main results for a uniformly hyperbolic
system, the solenoid map. We also discuss non-uniformly hyperbolic examples of
maps (H\'enon and Lozi maps) and of flows (the Lorenz63 and Lorenz84 models).
Our purpose is to outline the main ideas and to highlight several serious
problems found in the numerical estimation of the limit laws
The Impact Of Spike-Frequency Adaptation On Balanced Network Dynamics
A dynamic balance between strong excitatory and inhibitory neuronal inputs is hypothesized to play a pivotal role in information processing in the brain. While there is evidence of the existence of a balanced operating regime in several cortical areas and idealized neuronal network models, it is important for the theory of balanced networks to be reconciled with more physiological neuronal modeling assumptions. In this work, we examine the impact of spike-frequency adaptation, observed widely across neurons in the brain, on balanced dynamics. We incorporate adaptation into binary and integrate-and-fire neuronal network models, analyzing the theoretical effect of adaptation in the large network limit and performing an extensive numerical investigation of the model adaptation parameter space. Our analysis demonstrates that balance is well preserved for moderate adaptation strength even if the entire network exhibits adaptation. In the common physiological case in which only excitatory neurons undergo adaptation, we show that the balanced operating regime in fact widens relative to the non-adaptive case. We hypothesize that spike-frequency adaptation may have been selected through evolution to robustly facilitate balanced dynamics across diverse cognitive operating states
Collective dynamics in the presence of finite-width pulses
ACKNOWLEDGMENTS Afifurrahman was supported by the Ministry of Finance of the Republic of Indonesia through the Indonesia Endowment Fund for Education (LPDP) (Grant No. PRJ-2823/LPDP/2015).Peer reviewedPostprintPublisher PD
- …