409 research outputs found
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
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
Anyons in Geometric Models of Matter
We show that the "geometric models of matter" approach proposed by the first
author can be used to construct models of anyon quasiparticles with fractional
quantum numbers, using 4-dimensional edge-cone orbifold geometries with
orbifold singularities along embedded 2-dimensional surfaces. The anyon states
arise through the braid representation of surface braids wrapped around the
orbifold singularities, coming from multisections of the orbifold normal bundle
of the embedded surface. We show that the resulting braid representations can
give rise to a universal quantum computer.Comment: 22 pages LaTe
Bounds on data limits for all-to-all comparison from combinatorial designs
In situations where every item in a data set must be compared with every
other item in the set, it may be desirable to store the data across a number of
machines in such a way that any two data items are stored together on at least
one machine. One way to evaluate the efficiency of such a distribution is by
the largest fraction of the data it requires to be allocated to any one
machine. The all-to-all comparison (ATAC) data limit for machines is a
measure of the minimum of this value across all possible such distributions. In
this paper we further the study of ATAC data limits. We observe relationships
between them and the previously studied combinatorial parameters of fractional
matching numbers and covering numbers. We also prove a lower bound on the ATAC
data limit that improves on one of Hall, Kelly and Tian, and examine the
special cases where equality in this bound is possible. Finally, we investigate
the data limits achievable using various classes of combinatorial designs. In
particular, we examine the cases of transversal designs and projective
Hjelmslev planes.Comment: 16 pages, 1 figur
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Dissolving four-manifolds and positive scalar curvature
We prove that many simply connected symplectic four-manifolds dissolve after
connected sum with only one copy of .
For any finite group G that acts freely on the three-sphere we construct
closed smooth four-manifolds with fundamental group G which do not admit
metrics of positive scalar curvature, but whose universal covers do admit such
metrics.Comment: 13 pages; to appear in Mathematische Zeitschrif
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