1,698 research outputs found
Identifying the parametric occurrence of multiple steady states for some biological networks
We consider a problem from biological network analysis of determining regions
in a parameter space over which there are multiple steady states for positive
real values of variables and parameters. We describe multiple approaches to
address the problem using tools from Symbolic Computation. We describe how
progress was made to achieve semi-algebraic descriptions of the
multistationarity regions of parameter space, and compare symbolic results to
numerical methods. The biological networks studied are models of the
mitogen-activated protein kinases (MAPK) network which has already consumed
considerable effort using special insights into its structure of corresponding
models. Our main example is a model with 11 equations in 11 variables and 19
parameters, 3 of which are of interest for symbolic treatment. The model also
imposes positivity conditions on all variables and parameters.
We apply combinations of symbolic computation methods designed for mixed
equality/inequality systems, specifically virtual substitution, lazy real
triangularization and cylindrical algebraic decomposition, as well as a
simplification technique adapted from Gaussian elimination and graph theory. We
are able to determine multistationarity of our main example over a
2-dimensional parameter space. We also study a second MAPK model and a symbolic
grid sampling technique which can locate such regions in 3-dimensional
parameter space.Comment: 60 pages - author preprint. Accepted in the Journal of Symbolic
Computatio
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
Splitting Behavior of -Polynomials
We analyze the probability that, for a fixed finite set of primes S, a
random, monic, degree n polynomial f(x) with integer coefficients in a box of
side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with
splitting field over the rationals having Galois group ; (ii) the
polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii)
f(x) has a prescribed splitting type at each prime p in S.
The limit probabilities as are described in terms of values of
a one-parameter family of measures on , called splitting measures, with
parameter evaluated at the primes p in S. We study properties of these
measures. We deduce that there exist degree n extensions of the rationals with
Galois closure having Galois group with a given finite set of primes S
having given Artin symbols, with some restrictions on allowed Artin symbols for
p<n. We compare the distributions of these measures with distributions
formulated by Bhargava for splitting probabilities for a fixed prime in
such degree extensions ordered by size of discriminant, conditioned to be
relatively prime to .Comment: 33 pages, v2 34 pages, introduction revise
The notion of dimension in geometry and algebra
This talk reviews some mathematical and physical ideas related to the notion
of dimension. After a brief historical introduction, various modern
constructions from fractal geometry, noncommutative geometry, and theoretical
physics are invoked and compared.Comment: 29 pages, a revie
COMPUTER TOOLS FOR SOLVING MATHEMATICAL PROBLEMS: A REVIEW
The rapid development of digital computer hardware and software has had a dramatic influence on mathematics, and contrary. The advanced hardware and modern sophistical software such as computer visualization, symbolic computation, computerassisted proofs, multi-precision arithmetic and powerful libraries, have provided resolving many open problems, a huge very difficult mathematical problems, and discovering new patterns and relationships, far beyond a human capability. In the first part of the paper we give a short review of some typical mathematical problems solved by computer tools. In the second part we present some new original contributions, such as intriguing consequence of the presence of roundoff errors, distribution of zeros of random polynomials, dynamic study of zero-finding methods, a new three-point family of methods for solving nonlinear equations and two algorithms for the inclusion of a simple complex zero of a polynomial
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