49 research outputs found
Integrating Singular Functions on the Sphere
We obtain rigorous results concerning the evaluation of integrals on the two
sphere using complex methods. It is shown that for regular as well as singular
functions which admit poles, the integral can be reduced to the calculation of
residues through a limiting procedure.Comment: 15 pages, revte
Complex-linear invariants of biochemical networks
The nonlinearities found in molecular networks usually prevent mathematical
analysis of network behaviour, which has largely been studied by numerical
simulation. This can lead to difficult problems of parameter determination.
However, molecular networks give rise, through mass-action kinetics, to
polynomial dynamical systems, whose steady states are zeros of a set of
polynomial equations. These equations may be analysed by algebraic methods, in
which parameters are treated as symbolic expressions whose numerical values do
not have to be known in advance. For instance, an "invariant" of a network is a
polynomial expression on selected state variables that vanishes in any steady
state. Invariants have been found that encode key network properties and that
discriminate between different network structures. Although invariants may be
calculated by computational algebraic methods, such as Gr\"obner bases, these
become computationally infeasible for biologically realistic networks. Here, we
exploit Chemical Reaction Network Theory (CRNT) to develop an efficient
procedure for calculating invariants that are linear combinations of
"complexes", or the monomials coming from mass action. We show how this
procedure can be used in proving earlier results of Horn and Jackson and of
Shinar and Feinberg for networks of deficiency at most one. We then apply our
method to enzyme bifunctionality, including the bacterial EnvZ/OmpR osmolarity
regulator and the mammalian
6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase glycolytic regulator,
whose networks have deficiencies up to four. We show that bifunctionality leads
to different forms of concentration control that are robust to changes in
initial conditions or total amounts. Finally, we outline a systematic procedure
for using complex-linear invariants to analyse molecular networks of any
deficiency.Comment: 36 pages, 6 figure
Linear Toric Fibrations
These notes are based on three lectures given at the 2013 CIME/CIRM summer
school. The purpose of this series of lectures is to introduce the notion of a
toric fibration and to give its geometrical and combinatorial
characterizations. Polarized toric varieties which are birationally equivalent
to projective toric bundles are associated to a class of polytopes called
Cayley polytopes. Their geometry and combinatorics have a fruitful interplay
leading to fundamental insight in both directions. These notes will illustrate
geometrical phenomena, in algebraic geometry and neighboring fields, which are
characterized by a Cayley structure. Examples are projective duality of toric
varieties and polyhedral adjunction theory
Tropical surface singularities
In this paper, we study tropicalisations of singular surfaces in toric
threefolds. We completely classify singular tropical surfaces of
maximal-dimensional type, show that they can generically have only finitely
many singular points, and describe all possible locations of singular points.
More precisely, we show that singular points must be either vertices, or
generalized midpoints and baricenters of certain faces of singular tropical
surfaces, and, in some cases, there may be additional metric restrictions to
faces of singular tropical surfaces.Comment: A gap in the classification was closed. 37 pages, 29 figure
Singular Tropical Hypersurfaces
We study the notion of singular tropical hypersurfaces of any dimension. We characterize the singular points in terms of tropical Euler derivatives and we give an algorithm to compute all singular points. We also describe non-transversal intersection points of planar tropical curves
Weighted integral formulas on manifolds
We present a method of finding weighted Koppelman formulas for -forms
on -dimensional complex manifolds which admit a vector bundle of rank
over , such that the diagonal of has a defining
section. We apply the method to \Pn and find weighted Koppelman formulas for
-forms with values in a line bundle over \Pn. As an application, we
look at the cohomology groups of -forms over \Pn with values in
various line bundles, and find explicit solutions to the \dbar-equation in
some of the trivial groups. We also look at cohomology groups of -forms
over \Pn \times \Pm with values in various line bundles. Finally, we apply
our method to developing weighted Koppelman formulas on Stein manifolds.Comment: 25 page
Residue currents associated with weakly holomorphic functions
We construct Coleff-Herrera products and Bochner-Martinelli type residue
currents associated with a tuple of weakly holomorphic functions, and show
that these currents satisfy basic properties from the (strongly) holomorphic
case, as the transformation law, the Poincar\'e-Lelong formula and the
equivalence of the Coleff-Herrera product and the Bochner-Martinelli type
residue current associated with when defines a complete intersection.Comment: 28 pages. Updated with some corrections from the revision process. In
particular, corrected and clarified some things in Section 5 and 6 regarding
products of weakly holomorphic functions and currents, and the definition of
the Bochner-Martinelli type current
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
A Linear Algebra Approach for Detecting Binomiality of Steady State Ideals of Reversible Chemical Reaction Networks
Motivated by problems from Chemical Reaction Network Theory, we investigate
whether steady state ideals of reversible reaction networks are generated by
binomials. We take an algebraic approach considering, besides concentrations of
species, also rate constants as indeterminates. This leads us to the concept of
unconditional binomiality, meaning binomiality for all values of the rate
constants. This concept is different from conditional binomiality that applies
when rate constant values or relations among rate constants are given. We start
by representing the generators of a steady state ideal as sums of binomials,
which yields a corresponding coefficient matrix. On these grounds we propose an
efficient algorithm for detecting unconditional binomiality. That algorithm
uses exclusively elementary column and row operations on the coefficient
matrix. We prove asymptotic worst case upper bounds on the time complexity of
our algorithm. Furthermore, we experimentally compare its performance with
other existing methods
Statistical-Thermodynamic Model for Light Scattering from Eye Lens Protein Mixtures
We model light-scattering cross sections of concentrated aqueous mixtures of the bovine eye lens proteins γB- and α-crystallin by adapting a statistical-thermodynamic model of mixtures of spheres with short-range attractions. The model reproduces measured static light scattering cross sections, or Rayleigh ratios, of γB-α mixtures from dilute concentrations where light scattering intensity depends on molecular weights and virial coefficients, to realistically high concentration protein mixtures like those of the lens. The model relates γB-γB and γB-α attraction strengths and the γB-α size ratio to the free energy curvatures that set light scattering efficiency in tandem with protein refractive index increments. The model includes (i) hard-sphere α-α interactions, which create short-range order and transparency at high protein concentrations, (ii) short-range attractive plus hard-core γ-γ interactions, which produce intense light scattering and liquid-liquid phase separation in aqueous γ-crystallin solutions, and (iii) short-range attractive plus hard-core γ-α interactions, which strongly influence highly non-additive light scattering and phase separation in concentrated γ-α mixtures. The model reveals a new lens transparency mechanism, that prominent equilibrium composition fluctuations can be perpendicular to the refractive index gradient. The model reproduces the concave-up dependence of the Rayleigh ratio on α/γ composition at high concentrations, its concave-down nature at intermediate concentrations, non-monotonic dependence of light scattering on γ-α attraction strength, and more intricate, temperature-dependent features. We analytically compute the mixed virial series for light scattering efficiency through third order for the sticky-sphere mixture, and find that the full model represents the available light scattering data at concentrations several times those where the second and third mixed virial contributions fail. The model indicates that increased γ-γ attraction can raise γ-α mixture light scattering far more than it does for solutions of γ-crystallin alone, and can produce marked turbidity tens of degrees celsius above liquid-liquid separation