1,506 research outputs found
Duality and canonical extensions for stably compact spaces
We construct a canonical extension for strong proximity lattices in order to
give an algebraic, point-free description of a finitary duality for stably
compact spaces. In this setting not only morphisms, but also objects may have
distinct pi- and sigma-extensions.Comment: 29 pages, 1 figur
New distinguished classes of spectral spaces: a survey
In the present survey paper, we present several new classes of Hochster's
spectral spaces "occurring in nature", actually in multiplicative ideal theory,
and not linked to or realized in an explicit way by prime spectra of rings. The
general setting is the space of the semistar operations (of finite type),
endowed with a Zariski-like topology, which turns out to be a natural
topological extension of the space of the overrings of an integral domain,
endowed with a topology introduced by Zariski. One of the key tool is a recent
characterization of spectral spaces, based on the ultrafilter topology, given
in a paper by C. Finocchiaro in Comm. Algebra 2014. Several applications are
also discussed
Semantic spaces
Any natural language can be considered as a tool for producing large
databases (consisting of texts, written, or discursive). This tool for its
description in turn requires other large databases (dictionaries, grammars
etc.). Nowadays, the notion of database is associated with computer processing
and computer memory. However, a natural language resides also in human brains
and functions in human communication, from interpersonal to intergenerational
one. We discuss in this survey/research paper mathematical, in particular
geometric, constructions, which help to bridge these two worlds. In particular,
in this paper we consider the Vector Space Model of semantics based on
frequency matrices, as used in Natural Language Processing. We investigate
underlying geometries, formulated in terms of Grassmannians, projective spaces,
and flag varieties. We formulate the relation between vector space models and
semantic spaces based on semic axes in terms of projectability of subvarieties
in Grassmannians and projective spaces. We interpret Latent Semantics as a
geometric flow on Grassmannians. We also discuss how to formulate G\"ardenfors'
notion of "meeting of minds" in our geometric setting.Comment: 32 pages, TeX, 1 eps figur
Towards an infinitary logic of domains : Abramsky logic for transition systems
We give a new characterization of sober spaces in terms of their completely distributive lattice of saturated sets. This characterization is used to extend Abramsky's results about a domain logic for transition systems. The Lindenbaum algebra generated by the Abramsky finitary logic is a distributive lattice dual to an SFP-domain obtained as a solution of a recursive domain equation. We prove that the Lindenbaum algebra generated by the infinitary logic is a completely distributive lattice dual to the same SFP-domain. As a consequence soundness and completeness of the infinitary logic is obtained for a class of transition systems that is computational interesting
The notion of dimension in geometry and algebra
This talk reviews some mathematical and physical ideas related to the notion
of dimension. After a brief historical introduction, various modern
constructions from fractal geometry, noncommutative geometry, and theoretical
physics are invoked and compared.Comment: 29 pages, a revie
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