43 research outputs found
Proofs of Urysohn's Lemma and the Tietze Extension Theorem via the Cantor function
Urysohn's Lemma is a crucial property of normal spaces that deals with
separation of closed sets by continuous functions. It is also a fundamental
ingredient in proving the Tietze Extension Theorem, another property of normal
spaces that deals with the existence of extensions of continuous functions.
Using the Cantor function, we give alternative proofs for Urysohn's Lemma and
the Tietze Extension Theorem.Comment: A slightly modified version has been accepted for publication in the
Bulletin of the Australian Mathematical Societ
Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing nonlinearity
Given the unit ball of (), we study smooth
positive singular solutions to . Here ,
is critical for Sobolev embeddings, and . When and , the profile at the singularity was fully
described by Caffarelli-Gidas-Spruck. We prove that when and ,
besides this profile, two new profiles might occur. We provide a full
description of all the singular profiles. Special attention is accorded to
solutions such that and
. The particular case
requires a separate analysis which we also perform
Simplified Coalgebraic Trace Equivalence
The analysis of concurrent and reactive systems is based to a large degree on
various notions of process equivalence, ranging, on the so-called
linear-time/branching-time spectrum, from fine-grained equivalences such as
strong bisimilarity to coarse-grained ones such as trace equivalence. The
theory of concurrent systems at large has benefited from developments in
coalgebra, which has enabled uniform definitions and results that provide a
common umbrella for seemingly disparate system types including
non-deterministic, weighted, probabilistic, and game-based systems. In
particular, there has been some success in identifying a generic coalgebraic
theory of bisimulation that matches known definitions in many concrete cases.
The situation is currently somewhat less settled regarding trace equivalence. A
number of coalgebraic approaches to trace equivalence have been proposed, none
of which however cover all cases of interest; notably, all these approaches
depend on explicit termination, which is not always imposed in standard
systems, e.g. LTS. Here, we discuss a joint generalization of these approaches
based on embedding functors modelling various aspects of the system, such as
transition and braching, into a global monad; this approach appears to cover
all cases considered previously and some additional ones, notably standard LTS
and probabilistic labelled transition systems
A population biological model with a singular nonlinearity
summary:We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form where is a bounded smooth domain of with , , , , and , , and are positive parameters. Here is a continuous function. This model arises in the studies of population biology of one species with representing the concentration of the species. We discuss the existence of a positive solution when satisfies certain additional conditions. We use the method of sub-supersolutions to establish our results
Asymmetric Combination of Logics is Functorial: A Survey
Asymmetric combination of logics is a formal process that develops the characteristic features of a specific logic on top of another one. Typical examples include the development of temporal, hybrid, and probabilistic dimensions over a given base logic. These examples are surveyed in the paper under a particular perspectiveâthat this sort of combination of logics possesses a functorial nature. Such a view gives rise to several interesting questions. They range from the problem of combining translations (between logics), to that of ensuring property preservation along the process, and the way different asymmetric combinations can be related through appropriate natural transformations
Logics for contravariant simulations
Covariant-contravariant simulation and conformance simulation are two generalizations of the simple notion of simulation which aim at capturing the fact that it is not always the case that âthe larger the number of behaviors, the betterâ. Therefore, they can be considered to be more adequate to express the fact that a system is a correct implementation of some specification. We have previously shown that these two more elaborated notions fit well within the categorical framework developed to study the notion of simulation in a generic way. Now we show that their behaviors have also simple and natural logical characterizations, though more elaborated than those for the plain simulation semantics