100,744 research outputs found
On the Problem of Computing the Probability of Regular Sets of Trees
We consider the problem of computing the probability of regular languages of
infinite trees with respect to the natural coin-flipping measure. We propose an
algorithm which computes the probability of languages recognizable by
\emph{game automata}. In particular this algorithm is applicable to all
deterministic automata. We then use the algorithm to prove through examples
three properties of measure: (1) there exist regular sets having irrational
probability, (2) there exist comeager regular sets having probability and
(3) the probability of \emph{game languages} , from automata theory,
is if is odd and is otherwise
Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model
We present a computer-assisted proof of heteroclinic connections in the
one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a
fourth-order parabolic partial differential equation subject to homogeneous
Neumann boundary conditions, which contains as a special case the celebrated
Cahn-Hilliard equation. While the attractor structure of the latter model is
completely understood for one-dimensional domains, the diblock copolymer
extension exhibits considerably richer long-term dynamical behavior, which
includes a high level of multistability. In this paper, we establish the
existence of certain heteroclinic connections between the homogeneous
equilibrium state, which represents a perfect copolymer mixture, and all local
and global energy minimizers. In this way, we show that not every solution
originating near the homogeneous state will converge to the global energy
minimizer, but rather is trapped by a stable state with higher energy. This
phenomenon can not be observed in the one-dimensional Cahn-Hillard equation,
where generic solutions are attracted by a global minimizer
Recovery thresholds in the sparse planted matching problem
We consider the statistical inference problem of recovering an unknown
perfect matching, hidden in a weighted random graph, by exploiting the
information arising from the use of two different distributions for the weights
on the edges inside and outside the planted matching. A recent work has
demonstrated the existence of a phase transition, in the large size limit,
between a full and a partial recovery phase for a specific form of the weights
distribution on fully connected graphs. We generalize and extend this result in
two directions: we obtain a criterion for the location of the phase transition
for generic weights distributions and possibly sparse graphs, exploiting a
technical connection with branching random walk processes, as well as a
quantitatively more precise description of the critical regime around the phase
transition.Comment: 19 pages, 8 figure
Characterizing the quantum field theory vacuum using temporal Matrix Product states
In this paper we construct the continuous Matrix Product State (MPS)
representation of the vacuum of the field theory corresponding to the
continuous limit of an Ising model. We do this by exploiting the observation
made by Hastings and Mahajan in [Phys. Rev. A \textbf{91}, 032306 (2015)] that
the Euclidean time evolution generates a continuous MPS along the time
direction. We exploit this fact, together with the emerging Lorentz invariance
at the critical point in order to identify the matrix product representation of
the quantum field theory (QFT) vacuum with the continuous MPS in the time
direction (tMPS). We explicitly construct the tMPS and check these statements
by comparing the physical properties of the tMPS with those of the standard
ground MPS. We furthermore identify the QFT that the tMPS encodes with the
field theory emerging from taking the continuous limit of a weakly perturbed
Ising model by a parallel field first analyzed by Zamolodchikov.Comment: The results presented in this paper are a significant expansion of
arXiv:1608.0654
A generalization of the integer linear infeasibility problem
Does a given system of linear equations with nonnegative constraints have an
integer solution? This is a fundamental question in many areas. In statistics
this problem arises in data security problems for contingency table data and
also is closely related to non-squarefree elements of Markov bases for sampling
contingency tables with given marginals. To study a family of systems with no
integer solution, we focus on a commutative semigroup generated by a finite
subset of and its saturation. An element in the difference of the
semigroup and its saturation is called a ``hole''. We show the necessary and
sufficient conditions for the finiteness of the set of holes. Also we define
fundamental holes and saturation points of a commutative semigroup. Then, we
show the simultaneous finiteness of the set of holes, the set of non-saturation
points, and the set of generators for saturation points. We apply our results
to some three- and four-way contingency tables. Then we will discuss the time
complexities of our algorithms.Comment: This paper has been published in Discrete Optimization, Volume 5,
Issue 1 (2008) p36-5
On the freezing of variables in random constraint satisfaction problems
The set of solutions of random constraint satisfaction problems (zero energy
groundstates of mean-field diluted spin glasses) undergoes several structural
phase transitions as the amount of constraints is increased. This set first
breaks down into a large number of well separated clusters. At the freezing
transition, which is in general distinct from the clustering one, some
variables (spins) take the same value in all solutions of a given cluster. In
this paper we study the critical behavior around the freezing transition, which
appears in the unfrozen phase as the divergence of the sizes of the
rearrangements induced in response to the modification of a variable. The
formalism is developed on generic constraint satisfaction problems and applied
in particular to the random satisfiability of boolean formulas and to the
coloring of random graphs. The computation is first performed in random tree
ensembles, for which we underline a connection with percolation models and with
the reconstruction problem of information theory. The validity of these results
for the original random ensembles is then discussed in the framework of the
cavity method.Comment: 32 pages, 7 figure
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