Decomposing filtered chain complexes: Geometry behind barcoding algorithms

Adaptive algorithms with filtered regressor and filtered error

This paper presents a unified framework for the analysis of several discrete time adaptive parameter estimation algorithms, including RML with nonvanishing stepsize, several ARMAX identifiers, the Landau-style output error algorithms, and certain others for which no stability proof has yet appeared. A general algorithmic form is defined, incorporating a linear time-varying regressor '. filter a...

Nonabelian Algebraic Topology : filtered spaces , crossed complexes ,

Uncovering the hidden geometry behind metabolic networks.

Metabolism is a fascinating cell machinery underlying life and disease and genome-scale reconstructions provide us with a captivating view of its complexity. However, deciphering the relationship between metabolic structure and function remains a major challenge. In particular, turning observed structural regularities into organizing principles underlying systemic functions is a crucial task th...

Decomposing Correctness Proofs of Fault-Tolerant Algorithms

Proving the correctness of fault-tolerant algorithms is a tedious endeavor. Not only are the algorithms themselves complex, but nondeterministic fault transitions due to the environment compound the complexity. We demonstrate how to systematically decompose proofs of correctness. We first define a relation between the execution of an algorithm in a fault-generating environment and a fault-free ...

Second symmetric powers of chain complexes

We investigate Buchbaum and Eisenbud's construction of the second symmetric power $s_R(X)$ of a chain complex $X$ of modules over a commutative ring $R$. We state and prove a number of results from the folklore of the subject for which we know of no good direct references. We also provide several explicit computations and examples. We use this construction to prove the following vers...