On moments-preserving cosine families and semigroups in C[0,1]C[0,1]

We use the newly developed Kelvin's method of images \cite{kosinusy,kelvin} to show existence of a unique cosine family generated by a restriction of the Laplace operator in $C[0,1]$, that preserves the first two moments. We characterize the domain of its generator by specifying its boundary conditions. Also, we show that it enjoys inherent symmetry properties, and in particular that it leaves the subspaces of odd and even functions invariant. Furthermore, we provide information on long-time behavior of the related semigroup.Comment: 20 pages, 2 figure

Probabilistic Convergence and Stability of Random Mapper Graphs

We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $\mathbb{X}$ equipped with a continuous function $f: \mathbb{X} \rightarrow \mathbb{R}$. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line $\mathbb{R}$. We then introduce a variant of the classic mapper graph of Singh et al.~(2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of $(\mathbb{X}, f)$ when it is applied to points randomly sampled from a probability density function concentrated on $(\mathbb{X}, f)$. Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (2018), we first show that the mapper graph of $(\mathbb{X}, f)$, a constructible $\mathbb{R}$-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of $(\mathbb{X},f)$ to the mapper of a super-level set of a probability density function concentrated on $(\mathbb{X}, f)$. Finally, building on the approach of Bobrowski et al.~(2017), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data

Long-term monitoring of SO2 quiescent degassing from Nyiragongo’s lava lake

The activity of open-vent volcanoes with an active lava-lake, such as Nyiragongo, is characterized by persistent degassing, thus continuous monitoring of the rate, volume and fate of their gas emissions is of great importance to understand their geophysical state and their potential impact. We report results of SO2 emission measurements from Nyiragongo conducted between 2004 and 2012 with a network of ground-based scanning-DOAS (Differential Optical Absorption Spectroscopy) remote sensors. The mean SO2 emission rate is found to be 13 ± 9 kg s−1, similar to that observed in 1959. Daily emission rate has a distribution close to log-normal and presents large inter-day variability, reflecting the dynamics of percolation of magma batches of heterogeneous size distribution and changes in the effective permeability of the lava lake. The degassed S content is found to be between 1000 and 2000 ppm from these measurements and the reported magma flow rates sustaining the lava lake. The inter-annual trend and plume height statistics indicate stability of a quiescently degassing lava lake during the period of study

Cosine families and semigroups really differ

We reveal three surprising properties of cosine families, distinguishing them from semigroups of operators: (1) A single trajectory of a cosine family is either strongly continuous or not measurable. (2) Pointwise convergence of a sequence of equibounded cosine families implies that the convergence is almost uniform for time in the entire real line; in particular, cosine families cannot be perturbed in a singular way. (3) A non-constant trajectory of a bounded cosine family does not have a limit at infinity; in particular, the rich theory of asymptotic behaviour of semigroups does not have a counterpart for cosine families. In addition, we show that equibounded cosine families that converge strongly and almost uniformly in time may fail to converge uniformly.Adam Bobrowski and Wojciech Chojnack