/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

We define a new topological summary for data that we call the persistence landscape. Since this summary lies in a vector space, it is easy to combine with tools from statistics and machine learning, in contrast to the standard topological summaries. Viewed as a random variable with values in a Banach space, this summary obeys a strong law of large numbers and a central limit theorem. We show how a number of standard statistical tests can be used for statistical inference using this summary. We also prove that this summary is stable and that it can be used to provide lower bounds for the bottleneck and Wasserstein distances.

/pdf/statistical-topological-data-analysis-using-persistence-2hbwqc96bo.pdf

Statistical topological data analysis using persistence landscapes

Advances in mathematics

Plants and animals produce modular developmental units in a periodic fashion. In plants, lateral roots form as repeating units along the root primary axis; however, the developmental mechanism regulating this process is unknown. We found that cyclic expression pulses of a reporter gene mark the position of future lateral roots by establishing prebranch sites and that prebranch site production and root bending are periodic. Microarray and promoter-luciferase studies revealed two sets of genes oscillating in opposite phases at the root tip. Genetic studies show that some oscillating transcriptional regulators are required for periodicity in one or both developmental processes. This molecular mechanism has characteristics that resemble molecular clock–driven activities in animal species.

/pdf/oscillating-gene-expression-determines-competence-for-2rufwtkfxv.pdf

Oscillating Gene Expression Determines Competence for Periodic Arabidopsis Root Branching

Biochemical systems analysis: A study of function and design in molecular biology

Toric dynamical systems

Toric dynamical systems are known as complex balancing mass action systems in the mathematical chemistry literature, where many of their remarkable properties have been established. They include as special cases all deficiency zero systems and all detailed balancing systems. One feature is that the steady state locus of a toric dynamical system is a toric variety, which has a unique point within each invariant polyhedron. We develop the basic theory of toric dynamical systems in the context of computational algebraic geometry and show that the associated moduli space is also a toric variety. It is conjectured that the complex balancing state is a global attractor. We prove this for detailed balancing systems whose invariant polyhedron is two-dimensional and bounded.

https://www.math.tamu.edu/~annejls/multisite.pdf

Chemical Reaction Systems with Toric Steady States

We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity conditions expressed in terms of Jacobian matrices and determinants. In the context of chemical reaction networks with power-law kinetics, our results can be used to preclude as well as to guarantee multiple positive steady states. In the context of real algebraic geometry, our work recognizes a prior result of Craciun, Garcia-Puente, and Sottile, together with work of two of the authors, as the first partial multivariate generalization of the classical Descartes' rule, which bounds the number of positive real roots of a univariate real polynomial in terms of the number of sign variations of its coefficients.

/pdf/sign-conditions-for-injectivity-of-generalized-polynomial-3c7m5t6qxi.pdf

Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry

Which reaction networks, when taken with mass-action kinetics, have the capacity for multiple steady states? There is no complete answer to this question, but over the last 40 years various criteria have been developed that can answer this question in certain cases. This work surveys these developments, with an emphasis on recent results that connect the capacity for multistationarity of one network to that of another. In this latter setting, we consider a network N that is embedded in a larger network G , which means that N is obtained from G by removing some subsets of chemical species and reactions. This embedding relation is a significant generalization of the subnetwork relation. For arbitrary networks, it is not true that if N is embedded in G , then the steady states of N lift to G . Nonetheless, this does hold for certain classes of networks; one such class is that of fully open networks. This motivates the search for embedding-minimal multistationary networks : those networks which admit multiple steady states but no proper, embedded networks admit multiple steady states. We present results about such minimal networks, including several new constructions of infinite families of these networks.

/pdf/a-survey-of-methods-for-deciding-whether-a-reaction-network-4bbu9uo0mo.pdf

Anne Shiu

Papers

Toric dynamical systems

Toric dynamical systems

Chemical Reaction Systems with Toric Steady States

Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry

A Survey of Methods for Deciding Whether a Reaction Network is Multistationary