8,359 research outputs found
The genotype-phenotype relationship in multicellular pattern-generating models - the neglected role of pattern descriptors
Background: A deep understanding of what causes the phenotypic variation arising from biological patterning
processes, cannot be claimed before we are able to recreate this variation by mathematical models capable of
generating genotype-phenotype maps in a causally cohesive way. However, the concept of pattern in a
multicellular context implies that what matters is not the state of every single cell, but certain emergent qualities
of the total cell aggregate. Thus, in order to set up a genotype-phenotype map in such a spatiotemporal pattern
setting one is actually forced to establish new pattern descriptors and derive their relations to parameters of the
original model. A pattern descriptor is a variable that describes and quantifies a certain qualitative feature of the
pattern, for example the degree to which certain macroscopic structures are present. There is today no general
procedure for how to relate a set of patterns and their characteristic features to the functional relationships,
parameter values and initial values of an original pattern-generating model. Here we present a new, generic
approach for explorative analysis of complex patterning models which focuses on the essential pattern features
and their relations to the model parameters. The approach is illustrated on an existing model for Delta-Notch
lateral inhibition over a two-dimensional lattice.
Results: By combining computer simulations according to a succession of statistical experimental designs,
computer graphics, automatic image analysis, human sensory descriptive analysis and multivariate data modelling,
we derive a pattern descriptor model of those macroscopic, emergent aspects of the patterns that we consider
of interest. The pattern descriptor model relates the values of the new, dedicated pattern descriptors to the
parameter values of the original model, for example by predicting the parameter values leading to particular
patterns, and provides insights that would have been hard to obtain by traditional methods.
Conclusion: The results suggest that our approach may qualify as a general procedure for how to discover and
relate relevant features and characteristics of emergent patterns to the functional relationships, parameter values
and initial values of an underlying pattern-generating mathematical model
Three-dimensional simplicial gravity and combinatorics of group presentations
We demonstrate how some problems arising in simplicial quantum gravity can be
successfully addressed within the framework of combinatorial group theory. In
particular, we argue that the number of simplicial 3-manifolds having a fixed
homology type grows exponentially with the number of tetrahedra they are made
of. We propose a model of 3D gravity interacting with scalar fermions, some
restriction of which gives the 2-dimensional self-avoiding-loop-gas matrix
model. We propose a qualitative picture of the phase structure of 3D simplicial
gravity compatible with the numerical experiments and available analytical
results.Comment: 24 page
Gibbs and Quantum Discrete Spaces
Gibbs measure is one of the central objects of the modern probability,
mathematical statistical physics and euclidean quantum field theory. Here we
define and study its natural generalization for the case when the space, where
the random field is defined is itself random. Moreover, this randomness is not
given apriori and independently of the configuration, but rather they depend on
each other, and both are given by Gibbs procedure; We call the resulting object
a Gibbs family because it parametrizes Gibbs fields on different graphs in the
support of the distribution. We study also quantum (KMS) analog of Gibbs
families. Various applications to discrete quantum gravity are given.Comment: 37 pages, 2 figure
Clustering comparison of point processes with applications to random geometric models
In this chapter we review some examples, methods, and recent results
involving comparison of clustering properties of point processes. Our approach
is founded on some basic observations allowing us to consider void
probabilities and moment measures as two complementary tools for capturing
clustering phenomena in point processes. As might be expected, smaller values
of these characteristics indicate less clustering. Also, various global and
local functionals of random geometric models driven by point processes admit
more or less explicit bounds involving void probabilities and moment measures,
thus aiding the study of impact of clustering of the underlying point process.
When stronger tools are needed, directional convex ordering of point processes
happens to be an appropriate choice, as well as the notion of (positive or
negative) association, when comparison to the Poisson point process is
considered. We explain the relations between these tools and provide examples
of point processes admitting them. Furthermore, we sketch some recent results
obtained using the aforementioned comparison tools, regarding percolation and
coverage properties of the Boolean model, the SINR model, subgraph counts in
random geometric graphs, and more generally, U-statistics of point processes.
We also mention some results on Betti numbers for \v{C}ech and Vietoris-Rips
random complexes generated by stationary point processes. A general observation
is that many of the results derived previously for the Poisson point process
generalise to some "sub-Poisson" processes, defined as those clustering less
than the Poisson process in the sense of void probabilities and moment
measures, negative association or dcx-ordering.Comment: 44 pages, 4 figure
Using the Hopf Algebra Structure of QFT in Calculations
We employ the recently discovered Hopf algebra structure underlying
perturbative Quantum Field Theory to derive iterated integral representations
for Feynman diagrams. We give two applications: to massless Yukawa theory and
quantum electrodynamics in four dimensions.Comment: 28 p, Revtex, epsf for figures, minor changes, to appear in
Phys.Rev.
Updown categories: Generating functions and universal covers
A poset can be regarded as a category in which there is at most one morphism
between objects, and such that at most one of Hom(c,c') and Hom(c',c) is
nonempty for distinct objects c,c'. If we keep in place the latter axiom but
allow for more than one morphism between objects, we have a sort of generalized
poset in which there are multiplicities attached to covering relations, and
possibly nontrivial automorphism groups. We call such a category an "updown
category". In this paper we give a precise definition of such categories and
develop a theory for them. We also give a detailed account of ten examples,
including updown categories of integer partitions, integer compositions, planar
rooted trees, and rooted trees.Comment: arXiv admin note: substantial text overlap with arXiv:math/040245
Some Exact Results on Bond Percolation
We present some exact results on bond percolation. We derive a relation that
specifies the consequences for bond percolation quantities of replacing each
bond of a lattice by bonds connecting the same adjacent
vertices, thereby yielding the lattice . This relation is used to
calculate the bond percolation threshold on . We show that this
bond inflation leaves the universality class of the percolation transition
invariant on a lattice of dimensionality but changes it on a
one-dimensional lattice and quasi-one-dimensional infinite-length strips. We
also present analytic expressions for the average cluster number per vertex and
correlation length for the bond percolation problem on the
limits of several families of -vertex graphs. Finally, we explore the effect
of bond vacancies on families of graphs with the property of bounded diameter
as .Comment: 33 pages latex 3 figure
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