Tameness in generalized metric structures

We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and Zambrano on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections to classes of fuzzy structures, and structures on sheaves

The space of metric structures on hyperbolic groups

We study the metric and topological properties of the space $\mathscr{D}(G)$ of left-invariant hyperbolic pseudometrics on the non-elementary hyperbolic group $G$ that are quasi-isometric to a word metric, up to rough similarity. This space naturally contains the Teichm\"uller space in case $G$ is a surface group and the Culler-Vogtmann outer space when $G$ is a free group. Endowed with a natural metric reminiscent of the (symmetrized) Thurston's metric on Teichm\"uller space, we prove that $\mathscr{D}(G)$ is an unbounded contractible metric space and that $\mathrm{Out}(G)$ acts metrically properly by isometries on it. If we restrict ourselves to the subspace $\mathscr{D}_{\delta}(G)$ of the points represented by $\delta$-hyperbolic metrics with critical exponent 1, we prove that it is either empty or proper. We also prove continuity of the Bowen-Margulis map from $\mathscr{D}_{\delta}(G)$ into the space $\mathbb{P}\mathcal{C}urr(G)$ of projective geodesic currents on $G$, extending similar results for surface and free groups, and the continuity of the (normalized) mean distortion as a function on $\mathscr{D}(G)\times \mathscr{D}(G)$.Comment: Final version, to appear in the Journal of the London Mathematical Societ

Algebraic, Topological, and Mereological Foundations of Existential Granules

In this research, new concepts of existential granules that determine themselves are invented, and are characterized from algebraic, topological, and mereological perspectives. Existential granules are those that determine themselves initially, and interact with their environment subsequently. Examples of the concept, such as those of granular balls, though inadequately defined, algorithmically established, and insufficiently theorized in earlier works by others, are already used in applications of rough sets and soft computing. It is shown that they fit into multiple theoretical frameworks (axiomatic, adaptive, and others) of granular computing. The characterization is intended for algorithm development, application to classification problems and possible mathematical foundations of generalizations of the approach. Additionally, many open problems are posed and directions provided.Comment: 15 Pages. Accepted IJCRS 202

Statistical Analysis and Parameter Selection for Mapper

In this article, we study the question of the statistical convergence of the 1-dimensional Mapper to its continuous analogue, the Reeb graph. We show that the Mapper is an optimal estimator of the Reeb graph, which gives, as a byproduct, a method to automatically tune its parameters and compute confidence regions on its topological features, such as its loops and flares. This allows to circumvent the issue of testing a large grid of parameters and keeping the most stable ones in the brute-force setting, which is widely used in visualization, clustering and feature selection with the Mapper.Comment: Minor modification

Continuity In Enriched Categories And Metric Model Theory

We explore aspects of continuity as they manifest in two separate settings - metric model theory (continuous logic) and enriched categories - and interpret the former into the latter. One application of continuous logic is in proving that certain convergence results in analysis are in fact uniform across the choices of parameters: Avigad and Iovino outline a general method to deduce from a given convergence theorem that the convergence is uniform in a ``metastable\u27\u27 sense. While convenient, this method imposes strict requirements on the kinds of theorems allowed: in particular, any functions occurring in the theorem must be uniformly continuous. In aiming to apply to a broader class of examples the Avigad-Iovino approach, we construct a variant of continuous logic that is able to handle discontinuous functions in its domain of discourse. This logic weakens the usual continuity requirements for functions, but compensates by introducing a notion of ``linear structure\u27\u27 that mimics e.g. the vector space structure of Banach spaces. We use this logic to apply the Avigad-Iovino method to specific convergence results from functional analysis involving discontinuous functions, and obtain uniform metastable convergence in those examples. This is the project of the first part of this thesis. The second part of the thesis continues this study of continuity from a different angle, starting from where Lawvere shows that enriching a category over R with the appropriate monoidal structure turns that category into a metric space. He even muses on the notion of an ``R-valued logic\u27\u27, but does not make the connection to continuous logic (primarily because continuous logic did not yet exist). We introduce necessary structure that enables us to have a notion of ``uniform continuity\u27\u27 and ``continuous subobjects\u27\u27 in an enriched categorical setting, and use this to give an interpretation of continuous logic into a certain category of R-enriched categories

Nonpositive curvature and complex analysis

We discuss a few of the metrics that are used in complex analysis and potential theory, including the Poincaré, Carathéodory, Kobayashi, Hilbert, and quasihyperbolic metrics. An important feature of these metrics is that they are quite often negatively curved. We discuss what this means and when it occurs, and proceed to investigate some notions of nonpositive curvature, beginning with constant negative curvature (e.g. the unit disk with the Poincaré metric), and moving on to CAT(k) and Gromov hyperbolic spaces. We pay special attention to notions of the boundary at infinity

An Introduction to Set Theory and Topology

These notes are an introduction to set theory and topology. They are the result of teaching a two-semester course sequence on these topics for many years at Washington University in St. Louis. Typically the students were advanced undergraduate mathematics majors, a few beginning graduate students in mathematics, and some graduate students from other areas that included economics and engineering. The usual background for the material is an introductory undergraduate analysis course, mostly because it provides a solid introduction to Euclidean space Rn and practice with rigorous arguments — in particular, about continuity. Strictly speaking, however, the material is mostly self-contained. Examples are taken now and then from analysis, but they are not logically necessary for the development of the material. The only real prerequisite is the level of mathematical interest, maturity and patience needed to handle abstract ideas and to read and write careful proofs. A few very capable students have taken this course before introductory analysis (even, rarely, outstanding university freshmen) and invariably they have commented later on how material eased their way into analysis.https://openscholarship.wustl.edu/books/1020/thumbnail.jp

Amplitude spectrum distance: measuring the global shape divergence of protein fragments

International audienceBackground: In structural bioinformatics, there is an increasing interest in identifying and understanding the evolution of local protein structures regarded as key structural or functional protein building blocks. A central need is then to compare these, possibly short, fragments by measuring efficiently and accurately their (dis)similarity. Progress towards this goal has given rise to scores enabling to assess the strong similarity of fragments. Yet, there is still a lack of more progressive scores, with meaningful intermediate values, for the comparison, retrieval or clustering of distantly related fragments. Results: We introduce here the Amplitude Spectrum Distance (ASD), a novel way of comparing protein fragments based on the discrete Fourier transform of their C α distance matrix. Defined as the distance between their amplitude spectra, ASD can be computed efficiently and provides a parameter-free measure of the global shape dissimilarity of two fragments. ASD inherits from nice theoretical properties, making it tolerant to shifts, insertions, deletions, circular permutations or sequence reversals while satisfying the triangle inequality. The practical interest of ASD with respect to RMSD, RMSDd , BC and TM scores is illustrated through zinc finger retrieval experiments and concrete structure examples. The benefits of ASD are also illustrated by two additional clustering experiments: domain linkers fragments and complementarity-determining regions of antibodies.Conclusions: Taking advantage of the Fourier transform to compare fragments at a global shape level, ASD is an objective and progressive measure taking into account the whole fragments. Its practical computation time and its properties make ASD particularly relevant for applications requiring meaningful measures on distantly related protein fragments, such as similar fragments retrieval asking for high recalls as shown in the experiments, or for any application taking also advantage of triangle inequality, such as fragments clustering. ASD program and source code are freely available at: http://www.irisa.fr/dyliss/public/ASD/