17 research outputs found
On One-Rule Grid Semi-Thue Systems
International audienceThe family of one-rule grid semi-Thue systems, introduced by Alfons Geser, is the family of one-rule semi-Thue systems such that there exists a letter c that occurs as often in the left-hand side as the right-hand side of the rewriting rule. We prove that for any one-rule grid semi-Thue system S, the set S(w) of all words obtainable from w using repeatedly the rewriting rule of S is a constructible context-free language. We also prove the regularity of the set Loop(S) of all words that start a loop in a one-rule grid semi-Thue systems S.La famille des systèmes de semi-Thue à une seule règle "en grille", introduite par Alfons Geser, est la famille des systèmes de réécriture de mots pour lesquels il existe une lettre apparaissant autant de fois dans la partie gauche et dans la partie droite de leur unique règle. Nous prouvons que, pour tout système S de cette famille, l'ensemble S(w) des mots obtenus à partir du mot w en appliquant itérativement la règle de réécriture de S est un langage algébrique constructible. Nous prouvons également que l'ensemble Loop(S) des mots qui sont à l'origine d'une boucle de réécriture pour un systèmes de semi-Thue à une seule règle "en grille" S est un langage régulier
Flip-sort and combinatorial aspects of pop-stack sorting
Flip-sort is a natural sorting procedure which raises fascinating
combinatorial questions. It finds its roots in the seminal work of Knuth on
stack-based sorting algorithms and leads to many links with permutation
patterns. We present several structural, enumerative, and algorithmic results
on permutations that need few (resp. many) iterations of this procedure to be
sorted. In particular, we give the shape of the permutations after one
iteration, and characterize several families of permutations related to the
best and worst cases of flip-sort. En passant, we also give some links between
pop-stack sorting, automata, and lattice paths, and introduce several tactics
of bijective proofs which have their own interest.Comment: This v3 just updates the journal reference, according to the
publisher wis
Irreversible stochastic processes on lattices
Models for irreversible random or cooperative filling of lattices are required to describe many processes in chemistry and physics. Since the filling is assumed to be irreversible, even the stationary, saturation state is not in equilibrium. The kinetics and statistics of these processes are described by recasting the master equations in infinite hierarchial form. Solutions can be obtained by implementing various techniques involving, e.g., truncation or formal density expansions. Refinements in these solution techniques are presented;Problems considered include random dimer, trimer, and tetramer filling of 2D lattices, random dimer filling of a cubic lattice, competi- tive filling of two or more species, and the effect of a random distribu- tion of inactive sites on the filling. We also consider monomer filling of a linear lattice with nearest neighbor cooperative effects and solve for the exact cluster-size distribution for cluster sizes up to the asymptotic regime. Additionally, we develop a technique to directly determine the asymptotic properties of the cluster-size distribution;Finally, we consider cluster growth via irreversible aggregation involving random walkers. In particular, we provide explicit results for the large-lattice-size asymptotic behavior of trapping probabilities and average walk lengths for a single walker on a lattice with multiple;traps. Procedures for exact calculation of these quantities on finite lattices are also developed; *DOE Report IS-T-1230. This work was performed under Contract W-7405-eng-82 with the Department of Energy
Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of Symmetry with Non-integer
When studying quantum field theories and lattice models, it is often useful
to analytically continue the number of field or spin components from an integer
to a real number. In spite of this, the precise meaning of such analytic
continuations has never been fully clarified, and in particular the symmetry of
these theories is obscure. We clarify these issues using Deligne categories and
their associated Brauer algebras, and show that these provide logically
satisfactory answers to these questions. Simple objects of the Deligne category
generalize the notion of an irreducible representations, avoiding the need for
such mathematically nonsensical notions as vector spaces of non-integer
dimension. We develop a systematic theory of categorical symmetries, applying
it in both perturbative and non-perturbative contexts. A partial list of our
results is: categorical symmetries are preserved under RG flows; continuous
categorical symmetries come equipped with conserved currents; CFTs with
categorical symmetries are necessarily non-unitary.Comment: 59 pages, many figures; v3 typos fixed, version submitted to journa
Neuronal computation on complex dendritic morphologies
When we think about neural cells, we immediately recall the wealth of electrical
behaviour which, eventually, brings about consciousness. Hidden deep in the
frequencies and timings of action potentials, in subthreshold oscillations, and in
the cooperation of tens of billions of neurons, are synchronicities and emergent behaviours
that result in high-level, system-wide properties such as thought and cognition.
However, neurons are even more remarkable for their elaborate morphologies,
unique among biological cells. The principal, and most striking, component of neuronal
morphologies is the dendritic tree.
Despite comprising the vast majority of the surface area and volume of a
neuron, dendrites are often neglected in many neuron models, due to their sheer
complexity. The vast array of dendritic geometries, combined with heterogeneous
properties of the cell membrane, continue to challenge scientists in predicting neuronal
input-output relationships, even in the case of subthreshold dendritic currents.
In this thesis, we will explore the properties of neuronal dendritic trees, and
how they alter and integrate the electrical signals that diffuse along them. After
an introduction to neural cell biology and membrane biophysics, we will review
Abbott's dendritic path integral in detail, and derive the theoretical convergence
of its infinite sum solution. On certain symmetric structures, closed-form solutions
will be found; for arbitrary geometries, we will propose algorithms using various
heuristics for constructing the solution, and assess their computational convergences
on real neuronal morphologies. We will demonstrate how generating terms for the
path integral solution in an order that optimises convergence is non-trivial, and how a computationally-significant number of terms is required for reasonable accuracy.
We will, however, derive a highly-efficient and accurate algorithm for application to
discretised dendritic trees. Finally, a modular method for constructing a solution in
the Laplace domain will be developed