1,397 research outputs found
Special Algorithm for Stability Analysis of Multistable Biological Regulatory Systems
We consider the problem of counting (stable) equilibriums of an important
family of algebraic differential equations modeling multistable biological
regulatory systems. The problem can be solved, in principle, using real
quantifier elimination algorithms, in particular real root classification
algorithms. However, it is well known that they can handle only very small
cases due to the enormous computing time requirements. In this paper, we
present a special algorithm which is much more efficient than the general
methods. Its efficiency comes from the exploitation of certain interesting
structures of the family of differential equations.Comment: 24 pages, 5 algorithms, 10 figure
Quasi-isometries Between Groups with Two-Ended Splittings
We construct `structure invariants' of a one-ended, finitely presented group
that describe the way in which the factors of its JSJ decomposition over
two-ended subgroups fit together.
For groups satisfying two technical conditions, these invariants reduce the
problem of quasi-isometry classification of such groups to the problem of
relative quasi-isometry classification of the factors of their JSJ
decompositions. The first condition is that their JSJ decompositions have
two-ended cylinder stabilizers. The second is that every factor in their JSJ
decompositions is either `relatively rigid' or `hanging'. Hyperbolic groups
always satisfy the first condition, and it is an open question whether they
always satisfy the second.
The same methods also produce invariants that reduce the problem of
classification of one-ended hyperbolic groups up to homeomorphism of their
Gromov boundaries to the problem of classification of the factors of their JSJ
decompositions up to relative boundary homeomorphism type.Comment: 61pages, 6 figure
Invariance of immersed Floer cohomology under Lagrangian surgery
We show that cellular Floer cohomology of an immersed Lagrangian brane is
invariant under smoothing of a self-intersection point if the quantum valuation
of the weakly bounding cochain vanishes and the Lagrangian has dimension at
least two. The chain-level map replaces the two orderings of the
self-intersection point with meridianal and longitudinal cells on the handle
created by the surgery, and uses a bijection between holomorphic disks
developed by Fukaya-Oh-Ohta-Ono. Our result generalizes invariance of
potentials for certain Lagrangian surfaces in
Dimitroglou-Rizell--Ekholm--Tonkonog, and implies the invariance of Floer
cohomology under mean curvature flow with this type of surgery, as conjectured
by Joyce.Comment: 100 pages. This version has minor corrections (one which was in the
isomorphism of Floer cohomologies, but which did not affect the main result.
Deformations of asymptotically cylindrical G_2 manifolds
We prove that for a 7-dimensional manifold M with cylindrical ends the moduli
space of exponentially asymptotically cylindrical torsion-free G_2 structures
is a smooth manifold (if non-empty), and study some of its local properties. We
also show that the holonomy of the induced metric of an exponentially
asymptotically cylindrical G_2 manifold M is exactly G_2 if and only if its
fundamental group is finite and neither M nor any double cover of M is
homeomorphic to a cylinder.Comment: 31 pages, corrected proof of proposition 6.2
Deciding the consistency of non-linear real arithmetic constraints with a conflict driven search using cylindrical algebraic coverings
We present a new algorithm for determining the satisfiability of conjunctions
of non-linear polynomial constraints over the reals, which can be used as a
theory solver for satisfiability modulo theory (SMT) solving for non-linear
real arithmetic. The algorithm is a variant of Cylindrical Algebraic
Decomposition (CAD) adapted for satisfiability, where solution candidates
(sample points) are constructed incrementally, either until a satisfying sample
is found or sufficient samples have been sampled to conclude unsatisfiability.
The choice of samples is guided by the input constraints and previous
conflicts.
The key idea behind our new approach is to start with a partial sample;
demonstrate that it cannot be extended to a full sample; and from the reasons
for that rule out a larger space around the partial sample, which build up
incrementally into a cylindrical algebraic covering of the space. There are
similarities with the incremental variant of CAD, the NLSAT method of Jovanovic
and de Moura, and the NuCAD algorithm of Brown; but we present worked examples
and experimental results on a preliminary implementation to demonstrate the
differences to these, and the benefits of the new approach
Algorithmically generating new algebraic features of polynomial systems for machine learning
There are a variety of choices to be made in both computer algebra systems
(CASs) and satisfiability modulo theory (SMT) solvers which can impact
performance without affecting mathematical correctness. Such choices are
candidates for machine learning (ML) approaches, however, there are
difficulties in applying standard ML techniques, such as the efficient
identification of ML features from input data which is typically a polynomial
system. Our focus is selecting the variable ordering for cylindrical algebraic
decomposition (CAD), an important algorithm implemented in several CASs, and
now also SMT-solvers. We created a framework to describe all the previously
identified ML features for the problem and then enumerated all options in this
framework to automatically generation many more features. We validate the
usefulness of these with an experiment which shows that an ML choice for CAD
variable ordering is superior to those made by human created heuristics, and
further improved with these additional features. We expect that this technique
of feature generation could be useful for other choices related to CAD, or even
choices for other algorithms with polynomial systems for input.Comment: To appear in Proc SC-Square Workshop 2019. arXiv admin note:
substantial text overlap with arXiv:1904.1106
Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition
There has been recent interest in the use of machine learning (ML) approaches
within mathematical software to make choices that impact on the computing
performance without affecting the mathematical correctness of the result. We
address the problem of selecting the variable ordering for cylindrical
algebraic decomposition (CAD), an important algorithm in Symbolic Computation.
Prior work to apply ML on this problem implemented a Support Vector Machine
(SVM) to select between three existing human-made heuristics, which did better
than anyone heuristic alone. The present work extends to have ML select the
variable ordering directly, and to try a wider variety of ML techniques.
We experimented with the NLSAT dataset and the Regular Chains Library CAD
function for Maple 2018. For each problem, the variable ordering leading to the
shortest computing time was selected as the target class for ML. Features were
generated from the polynomial input and used to train the following ML models:
k-nearest neighbours (KNN) classifier, multi-layer perceptron (MLP), decision
tree (DT) and SVM, as implemented in the Python scikit-learn package. We also
compared these with the two leading human constructed heuristics for the
problem: Brown's heuristic and sotd. On this dataset all of the ML approaches
outperformed the human made heuristics, some by a large margin.Comment: Accepted into CICM 201
Relative elegance and cartesian cubes with one connection
We establish a Quillen equivalence between the Kan-Quillen model structure
and a model structure, derived from a model of a cubical type theory, on the
category of cartesian cubical sets with one connection. We thereby identify a
second model structure which both constructively models homotopy type theory
and presents infinity-groupoids, the first known example being the equivariant
cartesian model of Awodey-Cavallo-Coquand-Riehl-Sattler.Comment: 60 pages. Comments welcome
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