43,393 research outputs found
Coalgebraic Geometric Logic: Basic Theory
Using the theory of coalgebra, we introduce a uniform framework for adding
modalities to the language of propositional geometric logic. Models for this
logic are based on coalgebras for an endofunctor on some full subcategory of
the category of topological spaces and continuous functions. We investigate
derivation systems, soundness and completeness for such geometric modal logics,
and we we specify a method of lifting an endofunctor on Set, accompanied by a
collection of predicate liftings, to an endofunctor on the category of
topological spaces, again accompanied by a collection of (open) predicate
liftings. Furthermore, we compare the notions of modal equivalence, behavioural
equivalence and bisimulation on the resulting class of models, and we provide a
final object for the corresponding category
Integrals and Valuations
We construct a homeomorphism between the compact regular locale of integrals
on a Riesz space and the locale of (valuations) on its spectrum. In fact, we
construct two geometric theories and show that they are biinterpretable. The
constructions are elementary and tightly connected to the Riesz space
structure.Comment: Submitted for publication 15/05/0
A localic theory of lower and upper integrals
An account of lower and upper integration is given. It is constructive in the sense of geometric logic. If the integrand takes its values in the non-negative lower reals, then its lower integral with respect to a valuation is a lower real. If the integrand takes its values in the non-negative upper reals,then its upper integral with respect to a covaluation and with domain of
integration bounded by a compact subspace is an upper real. Spaces of valuations and of covaluations are defined.
Riemann and Choquet integrals can be calculated in terms of these lower and upper integrals
Ten Misconceptions from the History of Analysis and Their Debunking
The widespread idea that infinitesimals were "eliminated" by the "great
triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an
uninterrupted chain of work on infinitesimal-enriched number systems. The
elimination claim is an oversimplification created by triumvirate followers,
who tend to view the history of analysis as a pre-ordained march toward the
radiant future of Weierstrassian epsilontics. In the present text, we document
distortions of the history of analysis stemming from the triumvirate ideology
of ontological minimalism, which identified the continuum with a single number
system. Such anachronistic distortions characterize the received interpretation
of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note:
text overlap with arXiv:1108.2885 and arXiv:1110.545
Cities as emergent models: the morphological logic of Manhattan and Barcelona
This paper is set to unveil several particulars about the logic embedded in the diachronic model of city
growth and the rules which govern the emergence of urban spaces. The paper outlines an attempt to
detect and define the generative rules of a growing urban structure by means of evaluation techniques.
The initial approach in this regards will be to study the evolution of existing urban regions or cities which
in our case are Manhattan and Barcelona and investigate the rules and causes of their emergence and
growth. The paper will concentrate on the spatial aspect of the generative rules and investigate their
behaviour and dimensionality. Several Space Syntax evaluation methods will be implemented to capture
the change of spatial configurations within the growing urban structures. In addition, certain spatial
elements will be isolated and tested aiming to illustrate their influence on the main spatial structures.
Both urban regions were found to be emergent products of a bottom up organic growth mostly
distinguished in the vicinities of the first settlements. Despite the imposition of a uniform grid on both
cities in later stages of their development these cities managed to deform the regularity in the preplanned
grid in an emergent manner to end up with an efficient model embodied in their current spatial
arrangement. The paper reveals several consistencies in the spatial morphology of both urban regions
and provides explanation of these regularities in an approach to extract the underlying rules which
contributed to the growth optimization process
Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow
Fermat, Leibniz, Euler, and Cauchy all used one or another form of
approximate equality, or the idea of discarding "negligible" terms, so as to
obtain a correct analytic answer. Their inferential moves find suitable proxies
in the context of modern theories of infinitesimals, and specifically the
concept of shadow. We give an application to decreasing rearrangements of real
functions.Comment: 35 pages, 2 figures, to appear in Notices of the American
Mathematical Society 61 (2014), no.
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