Exponential Formulas for the Jacobians and Jacobian Matrices of Analytic Maps

Let $F=(F_1, F_2, ... F_n)$ be an $n$-tuple of formal power series in $n$ variables of the form $F(z)=z+ O(|z|^2)$. It is known that there exists a unique formal differential operator A=\sum_{i=1}^n a_i(z)\frac {\p}{\p z_i} such that $F(z)=exp (A)z$ as formal series. In this article, we show the Jacobian ${\cal J}(F)$ and the Jacobian matrix $J(F)$ of $F$ can also be given by some exponential formulas. Namely, ${\cal J}(F)=\exp (A+\triangledown A)\cdot 1$, where \triangledown A(z)= \sum_{i=1}^n \frac {\p a_i}{\p z_i}(z), and $J(F)=\exp(A+R_{Ja})\cdot I_{n\times n}$, where $I_{n\times n}$ is the identity matrix and $R_{Ja}$ is the multiplication operator by $Ja$ for the right. As an immediate consequence, we get an elementary proof for the known result that ${\cal J}(F)\equiv 1$ if and only if $\triangledown A=0$. Some consequences and applications of the exponential formulas as well as their relations with the well known Jacobian Conjecture are also discussed.Comment: Latex, 17 page

Formulating genome-scale kinetic models in the post-genome era.

The biological community is now awash in high-throughput data sets and is grappling with the challenge of integrating disparate data sets. Such integration has taken the form of statistical analysis of large data sets, or through the bottom-up reconstruction of reaction networks. While progress has been made with statistical and structural methods, large-scale systems have remained refractory to dynamic model building by traditional approaches. The availability of annotated genomes enabled the reconstruction of genome-scale networks, and now the availability of high-throughput metabolomic and fluxomic data along with thermodynamic information opens the possibility to build genome-scale kinetic models. We describe here a framework for building and analyzing such models. The mathematical analysis challenges are reflected in four foundational properties, (i) the decomposition of the Jacobian matrix into chemical, kinetic and thermodynamic information, (ii) the structural similarity between the stoichiometric matrix and the transpose of the gradient matrix, (iii) the duality transformations enabling either fluxes or concentrations to serve as the independent variables and (iv) the timescale hierarchy in biological networks. Recognition and appreciation of these properties highlight notable and challenging new in silico analysis issues

Summation-By-Parts Operators and High-Order Quadrature

Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. This property can be useful in constructing energy-stable discretizations of partial differential vequations. SBP operators are defined by a weight matrix and a difference operator, with the latter designed to approximate $d/dx$ to a specified order of accuracy. The accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP definition. We show that SBP weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes. The accuracy of SBP quadrature extends to curvilinear domains provided the Jacobian is approximated with the same SBP operator used for the quadrature. This quadrature has significant implications for SBP-based discretizations; for example, the discrete norm accurately approximates the $L^{2}$ norm for functions, and multi-dimensional SBP discretizations accurately mimic the divergence theorem.Comment: 18 pages, 3 figure

Synchronization and partial synchronization of linear maps

We study synchronization of low-dimensional ($d=2,3,4$) chaotic piecewise linear maps. For Bernoulli maps we find Lyapunov exponents and locate the synchronization transition, that numerically is found to be discontinuous (despite continuously vanishing Lyapunov exponent(s)). For tent maps, a limit of stability of the synchronized state is used to locate the synchronization transition that numerically is found to be continuous. For nonidentical tent maps at the partial synchronization transition, the probability distribution of the synchronization error is shown to develop highly singular behavior. We suggest that for nonidentical Bernoulli maps (and perhaps some other discontinuous maps) partial synchronization is merely a smooth crossover rather than a well defined transition. More subtle analysis in the $d=4$ case locates the point where the synchronized state becomes stable. In some cases, however, a riddled basin attractor appears, and synchronized and chaotic behaviors coexist. We also suggest that similar riddling of a basin of attractor might take place in some extended systems where it is known as stable chaos.Comment: 7 pages, eps figures include