881 research outputs found
On the dimension of polynomial semirings
In our previous work, motivated by the study of tropical polynomials, a
definition for prime congruences was given for an arbitrary commutative
semiring. It was shown that for additively idempotent semirings this class
exhibits some analogous properties to prime ideals in ring theory. The current
paper focuses on the resulting notion of Krull dimension, which is defined as
the length of the longest chain of prime congruences. Our main result states
that for any additively idempotent semiring , the semiring of polynomials
and the semiring of Laurent polynomials , we have
Graph-theoretic analysis of multistationarity using degree theory
Biochemical mechanisms with mass action kinetics are often modeled by systems
of polynomial differential equations (DE). Determining directly if the DE
system has multiple equilibria (multistationarity) is difficult for realistic
systems, since they are large, nonlinear and contain many unknown parameters.
Mass action biochemical mechanisms can be represented by a directed bipartite
graph with species and reaction nodes. Graph-theoretic methods can then be used
to assess the potential of a given biochemical mechanism for multistationarity
by identifying structures in the bipartite graph referred to as critical
fragments. In this article we present a graph-theoretic method for conservative
biochemical mechanisms characterized by bounded species concentrations, which
makes the use of degree theory arguments possible. We illustrate the results
with an example of a MAPK network
The European Citizens' Initiative: A First Assessment
Designed in order to enable the effective use of this new instrument and to guarantee equal access
Large Covariance Estimation by Thresholding Principal Orthogonal Complements
This paper deals with the estimation of a high-dimensional covariance with a
conditional sparsity structure and fast-diverging eigenvalues. By assuming
sparse error covariance matrix in an approximate factor model, we allow for the
presence of some cross-sectional correlation even after taking out common but
unobservable factors. We introduce the Principal Orthogonal complEment
Thresholding (POET) method to explore such an approximate factor structure with
sparsity. The POET estimator includes the sample covariance matrix, the
factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding
estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator
(Cai and Liu, 2011) as specific examples. We provide mathematical insights when
the factor analysis is approximately the same as the principal component
analysis for high-dimensional data. The rates of convergence of the sparse
residual covariance matrix and the conditional sparse covariance matrix are
studied under various norms. It is shown that the impact of estimating the
unknown factors vanishes as the dimensionality increases. The uniform rates of
convergence for the unobserved factors and their factor loadings are derived.
The asymptotic results are also verified by extensive simulation studies.
Finally, a real data application on portfolio allocation is presented
On the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle
Protein phosphorylation cycles are important mechanisms of the post
translational modification of a protein and as such an integral part of
intracellular signaling and control. We consider the sequential phosphorylation
and dephosphorylation of a protein at two binding sites. While it is known that
proteins where phosphorylation is processive and dephosphorylation is
distributive admit oscillations (for some value of the rate constants and total
concentrations) it is not known whether or not this is the case if both
phosphorylation and dephosphorylation are distributive. We study four
simplified mass action models of sequential and distributive phosphorylation
and show that for each of those there do not exist rate constants and total
concentrations where a Hopf bifurcation occurs. To arrive at this result we use
convex parameters to parameterize the steady state and Hurwitz matrices
Identifying parameter regions for multistationarity
Mathematical modelling has become an established tool for studying the
dynamics of biological systems. Current applications range from building models
that reproduce quantitative data to identifying systems with predefined
qualitative features, such as switching behaviour, bistability or oscillations.
Mathematically, the latter question amounts to identifying parameter values
associated with a given qualitative feature.
We introduce a procedure to partition the parameter space of a parameterized
system of ordinary differential equations into regions for which the system has
a unique or multiple equilibria. The procedure is based on the computation of
the Brouwer degree, and it creates a multivariate polynomial with parameter
depending coefficients. The signs of the coefficients determine parameter
regions with and without multistationarity. A particular strength of the
procedure is the avoidance of numerical analysis and parameter sampling.
The procedure consists of a number of steps. Each of these steps might be
addressed algorithmically using various computer programs and available
software, or manually. We demonstrate our procedure on several models of gene
transcription and cell signalling, and show that in many cases we obtain a
complete partitioning of the parameter space with respect to multistationarity.Comment: In this version the paper has been substantially rewritten and
reorganised. Theorem 1 has been reformulated and Corollary 1 adde
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