On Cayley graphs of virtually free groups

In 1985, Dunwoody showed that finitely presentable groups are accessible. Dunwoody's result was used to show that context-free groups, groups quasi-isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually free. Using another theorem of Dunwoody of 1979, we study when a group is virtually free in terms of its Cayley graph and we obtain new proofs of the mentioned results and other previously depending on them

Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length

We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\operatorname{PTIME}$ of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous (time and space) elegant and simple characterization of $\operatorname{PTIME}$. This is the first time such classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of computable analysis. This extends to deterministic complexity classes above polynomial time. This may provide a new perspective on classical complexity, by giving a way to define complexity classes, like $\operatorname{PTIME}$, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations, i.e.~by using the framework of analysis

Reconsidering the Necessary Beings of Aquinas’s Third Way

Surprisingly few articles have focused on Aquinas’s particular conception of necessary beings in the Third Way, and many scholars have espoused inaccurate or incomplete views of that conception. My aim in this paper is both to offer a corrective to some of those views and, more importantly, to provide compelling answers to the following two questions about the necessary beings of the Third Way. First, how exactly does Aquinas conceive of these necessary beings? Second, what does Aquinas seek to accomplish in the third stage of the Third Way? In answering these questions, I challenge prominent contemporary understandings of the necessary beings of the Third Way

On the local-indicability cohen–lyndon theorem

For a group H and a subset X of H, we let HX denote the set {hxh?1 | h ? H, x ? X}, and when X is a free-generating set of H, we say that the set HX is a Whitehead subset of H. For a group F and an element r of F, we say that r is Cohen–Lyndon aspherical in F if F{r} is a Whitehead subset of the subgroup of F that is generated by F{r}. In 1963, Cohen and Lyndon (D. E. Cohen and R. C. Lyndon, Free bases for normal subgroups of free groups, Trans. Amer. Math. Soc. 108 (1963), 526–537) independently showed that in each free group each non-trivial element is Cohen–Lyndon aspherical. Their proof used the celebrated induction method devised by Magnus in 1930 to study one-relator groups. In 1987, Edjvet and Howie (M. Edjvet and J. Howie, A Cohen–Lyndon theorem for free products of locally indicable groups, J. Pure Appl. Algebra 45 (1987), 41–44) showed that if A and B are locally indicable groups, then each cyclically reduced element of A*B that does not lie in A ? B is Cohen–Lyndon aspherical in A*B. Their proof used the original Cohen–Lyndon theorem. Using Bass–Serre theory, the original Cohen–Lyndon theorem and the Edjvet–Howie theorem, one can deduce the local-indicability Cohen–Lyndon theorem: if F is a locally indicable group and T is an F-tree with trivial edge stabilisers, then each element of F that fixes no vertex of T is Cohen–Lyndon aspherical in F. Conversely, by Bass–Serre theory, the original Cohen–Lyndon theorem and the Edjvet–Howie theorem are immediate consequences of the local-indicability Cohen–Lyndon theorem. In this paper we give a detailed review of a Bass–Serre theoretical form of Howie induction and arrange the arguments of Edjvet and Howie into a Howie-inductive proof of the local-indicability Cohen–Lyndon theorem that uses neither Magnus induction nor the original Cohen–Lyndon theorem. We conclude with a review of some standard applications of Cohen–Lyndon asphericit

Power domains and iterated function systems

We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domain-theoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniqueness of the attractor of a weakly hyperbolic IFS and the invariant measure of a weakly hyperbolic IFS with probabilities, extending the classic results of Hutchinson for hyperbolic IFSs in this more general setting. We also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure, extending the corresponding algorithms for hyperbolic IFSs. We then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent IFS and obtain an algorithm to generate the invariant distribution on the digitised screen. The generalised Riemann integral is used to provide a formula for the expected value of almost everywh..

Characterizing time computational complexity classes with polynomial differential equations

In this paper we show that several classes of languages from computational complexity theory, such as EXPTIME, can be characterized in a continuous manner by using only polynomial differential equations. This characterization applies not only to languages, but also to classes of functions, such as the classes defining the Grzegorczyk hierarchy, which implies an analog characterization of the class of elementary computable functions and the class of primitive recursive functions.info:eu-repo/semantics/acceptedVersio