Homological Algebra for Persistence Modules

Foundations of Computational Mathematics - We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either...     

Products of the addition of 1,4-naphthoquinone to an acecyclone

The major products from the exposure of tetracyclic keto-alcohol 1 to excess 1,4-naphthoquinone in the presence of acid were a hexacyclic quinone (6) and a heptacyclic dihydrobenzofuran derivative (8).     

Free and semi-inert cell attachments

Let be the space obtained by attaching a finite-type wedge of cells to a simply-connected, finite-type CW-complex.

We introduce the free and semi-inert conditions on the attaching map which broadly generalize the previously-studied inert condition. Under these conditions we determine as an -module and as an -algebra, respectively. Under a further condition we show that is generated by Hurewicz images.

As an example we study an infinite family of spaces constructed using only semi-inert cell attachments.

     

Separated Lie models and the homotopy Lie algebra

A simply connected topological space X has homotopy Lie algebra π_∗(ΩX)⊗Q. Following Quillen, there is a connected differential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X, and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property that we call being separated. The homology of a separated dgL has a particular form which lends itself to calculations.     

Categorification of Persistent Homology

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are $\mathbf {(\mathbb {R},\leq)}$ -indexed diagrams in some target category. A set of such diagrams has an interleaving distance, which we show generalizes the previously studied bottleneck distance. To illustrate the utility of this approach, we generalize previous stability results for persistence, extended persistence, and kernel, image, and cokernel persistence. We give a natural construction of a category of ε-interleavings of $\mathbf {(\mathbb {R},\leq)}$ -indexed diagrams in some target category and show that if the target category is abelian, so is this category of interleavings.     

Environmental effect on essential oil composition of Aloysia citriodora from Corrientes (Argentina)

Lemon verbena (Aloysia citriodora Palau) is indigenous to South America and was introduced into Europe. It is cultivated mainly due to the lemon-like aroma emitted from its leaves, which are utilized for the preparation of herbal tea reputed to have antispasmodic, antipyretic, sedative and digestive properties. In this work we introduce the enantiomeric distribution of sabinene and limonene by bidimensional gas chromatography (chiral GC-GC) as a genuine quantitative parameter in order to improve the knowledge so far available on A. citriodora oil. Multivariate analysis afforded information on the similarities and differences of wild and cultivated A. citriodora populations during different seasons in the same environmental conditions. The results indicated that it was possible to discard the environmental and seasonal effect on the chemical composition of A. citriodora for wild and cultivated materials belonging to the same genetic origin.     

Graph products of spheres, associative graded algebras and Hilbert series

Given a finite, simple, vertex-weighted graph, we construct a graded associative (noncommutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Noncommutative Gr?bner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment?Cangle complex. We apply our result to the loop-space homology of this space.     

Densities for random balanced sampling

A random balanced sample (RBS) is a multivariate distribution with n components X_k, each uniformly distributed on [-1,1], such that the sum of these components is precisely 0. The corresponding vectors lie in an (n-1)-dimensional polytope M(n). We present new methods for the construction of such RBS via densities over M(n) and these apply for arbitrary n. While simple densities had been known previously for small values of n (namely 2,3, and 4), for larger n the known distributions with large support were fractal distributions (with fractal dimension asymptotic to n as n→∞). Applications of RBS distributions include sampling with antithetic coupling to reduce variance, and the isolation of nonlinearities. We also show that the previously known densities (for n?4) are in fact the only solutions in a natural and very large class of potential RBS densities. This finding clarifies the need for new methods, such as those presented here.