77 research outputs found
Generic Regular Decompositions for Parametric Polynomial Systems
This paper presents a generalization of our earlier work in [19]. In this
paper, the two concepts, generic regular decomposition (GRD) and
regular-decomposition-unstable (RDU) variety introduced in [19] for generic
zero-dimensional systems, are extended to the case where the parametric systems
are not necessarily zero-dimensional. An algorithm is provided to compute GRDs
and the associated RDU varieties of parametric systems simultaneously on the
basis of the algorithm for generic zero-dimensional systems proposed in [19].
Then the solutions of any parametric system can be represented by the solutions
of finitely many regular systems and the decomposition is stable at any
parameter value in the complement of the associated RDU variety of the
parameter space. The related definitions and the results presented in [19] are
also generalized and a further discussion on RDU varieties is given from an
experimental point of view. The new algorithm has been implemented on the basis
of DISCOVERER with Maple 16 and experimented with a number of benchmarks from
the literature.Comment: It is the latest version. arXiv admin note: text overlap with
arXiv:1208.611
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
Tropical Support Vector Machine and its Applications to Phylogenomics
Most data in genome-wide phylogenetic analysis (phylogenomics) is essentially
multidimensional, posing a major challenge to human comprehension and
computational analysis. Also, we can not directly apply statistical learning
models in data science to a set of phylogenetic trees since the space of
phylogenetic trees is not Euclidean. In fact, the space of phylogenetic trees
is a tropical Grassmannian in terms of max-plus algebra. Therefore, to classify
multi-locus data sets for phylogenetic analysis, we propose tropical support
vector machines (SVMs). Like classical SVMs, a tropical SVM is a discriminative
classifier defined by the tropical hyperplane which maximizes the minimum
tropical distance from data points to itself in order to separate these data
points into sectors (half-spaces) in the tropical projective torus. Both hard
margin tropical SVMs and soft margin tropical SVMs can be formulated as linear
programming problems. We focus on classifying two categories of data, and we
study a simpler case by assuming the data points from the same category ideally
stay in the same sector of a tropical separating hyperplane. For hard margin
tropical SVMs, we prove the necessary and sufficient conditions for two
categories of data points to be separated, and we show an explicit formula for
the optimal value of the feasible linear programming problem. For soft margin
tropical SVMs, we develop novel methods to compute an optimal tropical
separating hyperplane. Computational experiments show our methods work well. We
end this paper with open problems.Comment: 27 pages, 6 figures, 2 table
Oscillations and bistability in a model of ERK regulation
This work concerns the question of how two important dynamical properties,
oscillations and bistability, emerge in an important biological signaling
network. Specifically, we consider a model for dual-site phosphorylation and
dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that
oscillations persist even as the model is greatly simplified (reactions are
made irreversible and intermediates are removed). Bistability, however, is much
less robust -- this property is lost when intermediates are removed or even
when all reactions are made irreversible. Moreover, bistability is
characterized by the presence of two reversible, catalytic reactions: as other
reactions are made irreversible, bistability persists as long as one or both of
the specified reactions is preserved. Finally, we investigate the maximum
number of steady states, aided by a network's "mixed volume" (a concept from
convex geometry). Taken together, our results shed light on the question of how
oscillations and bistability emerge from a limiting network of the ERK network
-- namely, the fully processive dual-site network -- which is known to be
globally stable and therefore lack both oscillations and bistability. Our
proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of
Newton polytopes arising from Hurwitz determinants, and recent
characterizations of multistationarity for networks having a steady-state
parametrization.Comment: 33 pages, 4 figures, 4 tables, 3 appendice
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