125 research outputs found
A chain morphism for Adams operations on rational algebraic K-theory
For any regular noetherian scheme X and every k>0, we define a chain morphism
between two chain complexes whose homology with rational coefficients is
isomorphic to the algebraic K-groups of X tensored by the field of rational
numbers. It is shown that these morphisms induce in homology the Adams
operations defined by Gillet and Soule or the ones defined by Grayson.Comment: 40 page
Adams operations on higher arithmetic K-theory
We construct Adams operations on the rational higher arithmetic K-groups of a
proper arithmetic variety. The definition applies to the higher arithmetic
K-groups given by Takeda as well as to the groups suggested by Deligne and
Soule, by means of the homotopy groups of the homotopy fiber of the regulator
map. They are compatible with the Adams operations on algebraic K-theory. The
definition relies on the chain morphism representing Adams operations in higher
algebraic K-theory given previously by the author. In this paper it is shown
that a slight modification of this chain morphism commutes strictly with the
representative of the Beilinson regulator given by Burgos and Wang
Power-law Kinetics and Determinant Criteria for the Preclusion of Multistationarity in Networks of Interacting Species
We present determinant criteria for the preclusion of non-degenerate multiple
steady states in networks of interacting species. A network is modeled as a
system of ordinary differential equations in which the form of the species
formation rate function is restricted by the reactions of the network and how
the species influence each reaction. We characterize families of so-called
power-law kinetics for which the associated species formation rate function is
injective within each stoichiometric class and thus the network cannot exhibit
multistationarity. The criterion for power-law kinetics is derived from the
determinant of the Jacobian of the species formation rate function. Using this
characterization we further derive similar determinant criteria applicable to
general sets of kinetics. The criteria are conceptually simple, computationally
tractable and easily implemented. Our approach embraces and extends previous
work on multistationarity, such as work in relation to chemical reaction
networks with dynamics defined by mass-action or non-catalytic kinetics, and
also work based on graphical analysis of the interaction graph associated to
the system. Further, we interpret the criteria in terms of circuits in the
so-called DSR-graphComment: To appear in SIAM Journal on Applied Dynamical System
Improving learning in large classes with online quizzes
In this project I address the problem of teaching mathematics to a large class of first-year students in life sciences. I focus on the introduction of quizzes during the lectures (clickers) and online quizzes for self study. The motivation to introduce quizzes is discussed, the practical aspects of the implementation is presented, and their effect on the students’ learning is assessed
Sign-sensitivities for reaction networks:an algebraic approach
This paper presents an algebraic framework to study sign-sensitivities for
reaction networks modeled by means of systems of ordinary differential
equations. Specifically, we study the sign of the derivative of the
concentrations of the species in the network at steady state with respect to a
small perturbation on the parameter vector. We provide a closed formula for the
derivatives that accommodates common perturbations, and illustrate its form
with numerous examples. We argue that, mathematically, the study of the
response to the system with respect to changes in total amounts is not well
posed, and that one should rather consider perturbations with respect to the
initial conditions. We find a sign-based criterion to determine, without
computing the sensitivities, whether the sign depends on the steady state and
parameters of the system. This is based on earlier results of so-called
injective networks. Finally, we address systems with multiple steady states and
the restriction to stable steady states.Comment: To appear in Mathematical Biosciences and Engineerin
Injectivity, multiple zeros, and multistationarity in reaction networks
Polynomial dynamical systems are widely used to model and study real
phenomena. In biochemistry, they are the preferred choice for modelling the
concentration of chemical species in reaction networks with mass-action
kinetics. These systems are typically parameterised by many (unknown)
parameters. A goal is to understand how properties of the dynamical systems
depend on the parameters. Qualitative properties relating to the behaviour of a
dynamical system are locally inferred from the system at steady state. Here we
focus on steady states that are the positive solutions to a parameterised
system of generalised polynomial equations. In recent years, methods from
computational algebra have been developed to understand these solutions, but
our knowledge is limited: for example, we cannot efficiently decide how many
positive solutions the system has as a function of the parameters. Even
deciding whether there is one or more solutions is non-trivial. We present a
new method, based on so-called injectivity, to preclude or assert that multiple
positive solutions exist. The results apply to generalised polynomials and
variables can be restricted to the linear, parameter-independent first
integrals of the dynamical system. The method has been tested in a wide range
of systems.Comment: Final version, Proceedings of the Royal Society
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