Growth rates of permutation grid classes, tours on graphs, and the spectral radius

Monotone grid classes of permutations have proven very effective in helping to determine structural and enumerative properties of classical permutation pattern classes. Associated with grid class Grid(M) is a graph, G(M), known as its "row-column" graph. We prove that the exponential growth rate of Grid(M) is equal to the square of the spectral radius of G(M). Consequently, we utilize spectral graph theoretic results to characterise all slowly growing grid classes and to show that for every γ ≥ 2 + √5 there is a grid class with growth rate arbitrarily close to γ. To prove our main result, we establish bounds on the size of certain families of tours on graphs. In the process, we prove that the family of tours of even length on a connected graph grows at the same rate as the family of "balanced" tours on the graph (in which the number of times an edge is traversed in one direction is the same as the number of times it is traversed in the other direction)

Finding and Combining Indicable Subgroups of Big Mapping Class Groups

We explicitly construct new subgroups of the mapping class groups of an uncountable collection of infinite-type surfaces, including, but not limited to, right-angled Artin groups, free groups, Baumslag-Solitar groups, mapping class groups of other surfaces, and a large collection of wreath products. For each such subgroup $H$ and surface $S$, we show that there are countably many non-conjugate embeddings of $H$ into $\text{Map}(S)$; in certain cases, there are uncountably many such embeddings. The images of each of these embeddings cannot lie in the isometry group of $S$ for any hyperbolic metric and are not contained in the closure of the compactly supported subgroup of $\text{Map}(S)$. In this sense, our construction is new and does not rely on previously known techniques for constructing subgroups of mapping class groups. Notably, our embeddings of $\text{Map}(S')$ into $\text{Map}(S)$ are not induced by embeddings of $S'$ into $S$. Our main tool for all of these constructions is the utilization of special homeomorphisms of $S$ called shift maps, and more generally, multipush maps.Comment: 31 pages, 19 figures. Results have been improved to show countably many non-conjugate embeddings of each subgroup we construc

A Decomposition Theorem for Unitary Group Representations on Kaplansky-Hilbert Modules and the Furstenberg-Zimmer Structure Theorem

In this paper, a decomposition theorem for (covariant) unitary group representations on Kaplansky-Hilbert modules over Stone algebras is established, which generalizes the well-known Hilbert space case (where it coincides with the decomposition of Jacobs, de Leeuw and Glicksberg). The proof rests heavily on the operator theory on Kaplansky-Hilbert modules, in particular the spectral theorem for Hilbert-Schmidt homomorphisms on such modules. As an application, a generalization of the celebrated Furstenberg-Zimmer structure theorem to the case of measure-preserving actions of arbitrary groups on arbitrary probability spaces is established.Comment: Comments welcom

Coalgebras of topological types

In This work, we focus on developing the basic theory of coalgebras over the category Top (the category of topological spaces with continuous maps). We argue that, besides Set, the category Top is an interesting base category for coalgebras. We study some endofunctors on Top, in particular, Vietoris functor and P-Vietoris Functor (where P is a set of propositional letters) that due to Hofmann et. al. [42] can be considered as the topological versions of the powerset functor and P-Kripke functor, respectively. We define the notion of compact Kripke structures and we prove that Kripke homomorphisms preserve compactness. Our definition of "compact Kripke structure" coincides with the notion of "modally saturated structures" introduced in Fine [27]. We prove that the class of compact Kripke structures has Hennessy-Milner property. As a consequence we show that in this class of Kripke structures, bihavioral equivalence, modal equivalence and Kripke bisimilarity all coincide.Furthermore, we generalize the notion of descriptive structures defined in Venema et. al. [11] by introducing a notion Vietoris models. We identify Vietoris models as coalgebras for the P-Vietoris functor on the category Top. One can see that each compact Kripke model can be modified to a Vietoris model. This yields an adjunction between the category of Vietoris structures (VS) and the category of compact Kripke structurs (CKS). Moreover, we will prove that the category of Vietoris models has a terminal object. We study the concept of a Vietoris bisimulation between Vietoris models, and we will prove that the closure of a Kripke bisimulation between underlying Kripke models of two Vietoris models is a Vietoris bisimulation. In the end, it will be shown that in the class of Vietoris models, Vietoris bisimilarity, bihavioral equivalence, modal equivalence, all coincide

Mathematical Foundations for a Compositional Account of the Bayesian Brain

This dissertation reports some first steps towards a compositional account of active inference and the Bayesian brain. Specifically, we use the tools of contemporary applied category theory to supply functorial semantics for approximate inference. To do so, we define on the `syntactic' side the new notion of Bayesian lens and show that Bayesian updating composes according to the compositional lens pattern. Using Bayesian lenses, and inspired by compositional game theory, we define fibrations of statistical games and classify various problems of statistical inference as corresponding sections: the chain rule of the relative entropy is formalized as a strict section, while maximum likelihood estimation and the free energy give lax sections. In the process, we introduce a new notion of `copy-composition'. On the `semantic' side, we present a new formalization of general open dynamical systems (particularly: deterministic, stochastic, and random; and discrete- and continuous-time) as certain coalgebras of polynomial functors, which we show collect into monoidal opindexed categories (or, alternatively, into algebras for multicategories of generalized polynomial functors). We use these opindexed categories to define monoidal bicategories of cilia: dynamical systems which control lenses, and which supply the target for our functorial semantics. Accordingly, we construct functors which explain the bidirectional compositional structure of predictive coding neural circuits under the free energy principle, thereby giving a formal mathematical underpinning to the bidirectionality observed in the cortex. Along the way, we explain how to compose rate-coded neural circuits using an algebra for a multicategory of linear circuit diagrams, showing subsequently that this is subsumed by lenses and polynomial functors.Comment: DPhil thesis; as submitted. Main change from v1: improved treatment of statistical games. A number of errors also fixed, and some presentation improved. Comments most welcom

Tight polynomial worst-case bounds for loop programs

In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple programming language - representing non-deterministic imperative programs with bounded loops, and arithmetics limited to addition and multiplication - it is possible to decide precisely whether a program has certain growth-rate properties, in particular whether a computed value, or the program's running time, has a polynomial growth rate. A natural and intriguing problem was to move from answering the decision problem to giving a quantitative result, namely, a tight polynomial upper bound. This paper shows how to obtain asymptotically-tight, multivariate, disjunctive polynomial bounds for this class of programs. This is a complete solution: whenever a polynomial bound exists it will be found. A pleasant surprise is that the algorithm is quite simple; but it relies on some subtle reasoning. An important ingredient in the proof is the forest factorization theorem, a strong structural result on homomorphisms into a finite monoid

Metabolic Network Alignments and their Applications

The accumulation of high-throughput genomic and proteomic data allows for the reconstruction of the increasingly large and complex metabolic networks. In order to analyze the accumulated data and reconstructed networks, it is critical to identify network patterns and evolutionary relations between metabolic networks. But even finding similar networks becomes computationally challenging. The dissertation addresses these challenges with discrete optimization and the corresponding algorithmic techniques. Based on the property of the gene duplication and function sharing in biological network,we have formulated the network alignment problem which asks the optimal vertex-to-vertex mapping allowing path contraction, vertex deletion, and vertex insertions. We have proposed the first polynomial time algorithm for aligning an acyclic metabolic pattern pathway with an arbitrary metabolic network. We also have proposed a polynomial-time algorithm for patterns with small treewidth and implemented it for series-parallel patterns which are commonly found among metabolic networks. We have developed the metabolic network alignment tool for free public use. We have performed pairwise mapping of all pathways among five organisms and found a set of statistically significant pathway similarities. We also have applied the network alignment to identifying inconsistency, inferring missing enzymes, and finding potential candidates