Cluster Synchronization in Variable-Order Fractional Community Network via Intermittent Control

In this paper, the cluster synchronization of a variable-order fractional community network with nonidentical dynamics is investigated. For achieving the cluster synchronization, intermittent controllers are designed, and the sufficient conditions with respect to system parameters, intermittent control instants and control gains are derived based on stability theory of fractional-order system and linear matrix inequalities (LMIs). To avoid verifying the LMIs, a corresponding simple corollary is provided. Finally, a numerical example is performed to verify the derived result.

Parametrized bar recursion: a unifying framework for realizability interpretations of classical dependent choice

During the last 20 years or so, a wide range of realizability interpretations of classical analysis have been developed. In many cases, these are achieved by extending the base interpreting system of primitive recursive functionals with some form of bar recursion, which realizes the negative translation of either countable or countable dependent choice. In this work, we present the many variants of bar recursion used in this context as instantiations of a parametrized form of backward recursion, and give a uniform proof that under certain conditions this recursor realizes a corresponding family of parametrized dependent choice principles. From this proof, the soundness of most of the existing bar recursive realizability interpretations of choice, including those based on the Berardi–Bezem–Coquand functional, modified realizability and the more recent products of selection functions of Escardó and Oliva, follows as a simple corollary. We achieve not only a uniform framework in which familiar realizability interpretations of choice can be compared, but show that these represent just simple instances of a large family of potential interpretations of dependent choice principles.

Exact Distance Colouring in Trees

For an integer q ⩾ 2 and an even integer d, consider the graph obtained from a large complete q-ary tree by connecting with an edge any two vertices at distance exactly d in the tree. This graph has clique number q + 1, and the purpose of this short note is to prove that its chromatic number is Θ((d log q)/log d). It was not known that the chromatic number of this graph grows with d. As a simple corollary of our result, we give a negative answer to a problem of van den Heuvel and Naserasr, asking whether there is a constant C such that for any odd integer d, any planar graph can be coloured with at most C colours such that any pair of vertices at distance exactly d have distinct colours. Finally, we study interval colouring of trees (where vertices at distance at least d and at most cd, for some real c > 1, must be assigned distinct colours), giving a sharp upper bound in the case of bounded degree trees.

Big free groups are almost free

It is shown that the big free group (the set of countably-long words over a countable alphabet) is almost free, in the sense that any function from the alphabet to a compact topological group factors through a homomorphism. This statement is in fact a simple corollary of the more general result proven below on the extendability of homomorphisms from subgroups (of a certain kind) of the big free group to a compact topological group.

Esencia, definición y la tesis de identidad en Aristóteles, Metaphysica Z 4-6

To clarify Aristotle’s “Identity Thesis” in Metaphysics Z 6, according to which each strictly definable item must be identical to its own essence (1031a15-16, 1032a4- 6), I show that it is a simple corollary of some “logical” theorems drawn from the Organon, which have a direct impact on Z 4’s explanation that identity is based on the possession of essence. As I claim, Z 6’s innovation consists of a clarification about substance as that which - because of having essence - can fulfill the identity- condition. Therefore, Z 6’s argument targets principally “non-Identity” as a consequence of Platonic separation. Further, I argue against the commonly held view that the Identity Thesis stands on the hylomorphism of Z’s later chapters.

On the Inequality for Volume and Minkowskian Thickness

Given a centrally symmetric convex body B in , we denote by ℳd(B) the Minkowski space (i.e., finite dimensional Banach space) with unit ball B. Let K be an arbitrary convex body in ℳd(B). The relationship between volume V(K) and the Minkowskian thickness (= minimal width) ΔB(K) of K can naturally be given by the sharp geometric inequality V(K) ≥ α(B) · ΔB(K)d, where α(B) > 0. As a simple corollary of the Rogers-Shephard inequality we obtain that with equality on the left attained if and only if B is the difference body of a simplex and on the right if B is a cross-polytope. The main result of this paper is that for d = 2 the equality on the right implies that B is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach–Mazur distance to the regular hexagon.

Pragmatic Aspects of Type-Directed Partial Evaluation

Type-directed partial evaluation stems from the residualization of static values in dynamic contexts, given their type and the type of their free variables. Its algorithm coincides with the algorithm for coercing<br />a subtype value into a supertype value, which itself coincides with<br />Berger and Schwichtenberg's normalization algorithm for the simply typed lambda-calculus. Type-directed partial evaluation thus can be used to specialize a compiled, closed program, given its type.<br />Since Similix, let-insertion is a cornerstone of partial evaluators for call-by-value procedural languages with computational effects (such as divergence). It prevents the duplication of residual computations, and more generally maintains the order of dynamic side effects in the residual program. This article describes the extension of type-directed partial evaluation<br />to insert residual let expressions. This extension requires the<br />user to annotate arrow types with effect information. It is achieved by delimiting and abstracting control, comparably to continuation-based specialization in direct style. It enables type-directed partial evaluation of programs with effects (e.g., a definitional lambda-interpreter for an imperative language) that are in direct style. The residual programs<br />are in A-normal form. A simple corollary yields CPS (continuation-passing style) terms instead. We illustrate both transformations with two interpreters for Paulson's Tiny language, a classical example in partial evaluation.

An alternative proof of Diestel's theorem

We show that Diestel's theorem on weak compactness of subsets of L1,(μ, X) can be derived as a simple corollary of James's theorem. It is a pleasure to acknowledge several stimulating conserversations with Dave Emmons and the remarks of an anonymous referee. Errors are, of course, solely mine.

Liberty and Harm to Others1

This paper is part of an extended program of research with two aims: to understand Mill's moral and political philosophy, and to develop a utilitarian doctrine in its most plausible or least vulnerable form, in order to identify what, if anything, may reasonably be said for or against utilitarianism as a general type of doctrine. These aims fit together because I have found when trying to interpret Mill that he suggests lines of development somewhat different and in some ways more promising than the usual versions of utilitarianism.The Principle of liberty is not a simple corollary of utilitarianism, and Mill argues for it. I am not concerned with his arguments for it, however, so much as I am with his applications of it, since they tell us how it is to be understood.

Causation, Transparency, and Emphasis

It is often said that singular causal statements express a relationship between one event and another or between a fact and an event. This is a very strong view, which has the following simple corollary: singular causal statements whose cause-term purports to refer to an event (or a fact) and whose effect-term purports to refer to an event express a relationship between an event (or a fact) and an event.Thus, both Davidson and Kim would claim that the singular causal Statement(1) Socrates’ drinking hemlock at dusk caused his death expresses a relationship between two events, referred to, respectively, by the expressions “Socrates’ drinking hemlock at dusk” and “his death.” For Kim, but not for Davidson, an event is analyzable as a thing's having a property at or during a time. The event of Socrates’ drinking hemlock at dusk consists of Socrates’ at a certain time having the property of drinking hemlock at dusk. I shall not here try to choose between their respective theories of events but will only note that both theorists would say that (1) expresses a relationship between events however the latter are to be construed.