150 research outputs found
Doubly transitive lines II: Almost simple symmetries
We study lines through the origin of finite-dimensional complex vector spaces
that enjoy a doubly transitive automorphism group. This paper, the second in a
series, classifies those lines that exhibit almost simple symmetries. To
perform this classification, we introduce a general recipe involving Schur
covers to recover doubly transitive lines from their automorphism group
Automated Semantic Classification of French Verbs
The aim of this work is to explore (semi-)automatic means to create a Levin-type classification of French verbs, suitable for Natural Language Processing. For English, a classification based on Levin's method is VerbNet (Kipper 2005). VerbNet is an extensive digital verb lexicon which systematically extends Levin's classes while ensuring that class members have a common semantics and share a common set of syntactic frames and thematic roles. In this work we reorganise the verbs from three French syntax lexicons, namely Volem, the Grammar-Lexicon (Ladl) and Dicovalence, into VerbNet-like verb classes using the technique of Formal Concept Analysis. We automatically acquire syntactic-semantic verb class and diathesis alternation information. We create large scale verb classes and compare their verb and frame distributions to those of VerbNet. We discuss possible evaluation schemes and finally focus on an evaluation methodology with respect to VerbNet, of which we present the theoretical motivation and analyse the feasibility on a small hand-built example
Spectral characterizations of complex unit gain graphs
While eigenvalues of graphs are well studied, spectral analysis of complex unit gain graphs is still in its infancy. This thesis considers gain graphs whose gain groups are gradually less and less restricted, with the ultimate goal of classifying gain graphs that are characterized by their spectra. In such cases, the eigenvalues of a gain graph contain sufficient structural information that it might be uniquely (up to certain equivalence relations) constructed when only given its spectrum. First, the first infinite family of directed graphs that is – up to isomorphism – determined by its Hermitian spectrum is obtained. Since the entries of the Hermitian adjacency matrix are complex units, these objects may be thought of as gain graphs with a restricted gain group. It is shown that directed graphs with the desired property are extremely rare. Thereafter, the perspective is generalized to include signs on the edges. By encoding the various edge-vertex incidence relations with sixth roots of unity, the above perspective can again be taken. With an interesting mix of algebraic and combinatorial techniques, all signed directed graphs with degree at most 4 or least multiplicity at most 3 are determined. Subsequently, these characterizations are used to obtain signed directed graphs that are determined by their spectra. Finally, an extensive discussion of complex unit gain graphs in their most general form is offered. After exploring their various notions of symmetry and many interesting ties to complex geometries, gain graphs with exactly two distinct eigenvalues are classified
Reconstruction of graph colourings
A -deck of a (coloured) graph is a multiset of its induced -vertex
subgraphs. Given a graph , when is it possible to reconstruct with high
probability a uniformly random colouring of its vertices in colours from
its -deck? In this paper, we study this question for grids and random
graphs. Reconstruction of random colourings of -dimensional -grids from
the deck of their -subgrids is one of the most studied colour reconstruction
questions. The 1-dimensional case is motivated by the problem of reconstructing
DNA sequences from their `shotgunned' stretches. It was comprehensively studied
and the above reconstruction question was completely answered in the '90s. In
this paper, we get a very precise answer for higher . For every
and every , we present an almost linear algorithm that reconstructs
with high probability a random -colouring of vertices of a -dimensional
-grid from the deck of all its -subgrids for every and prove that the random -colouring is not
reconstructible with high probability if .
This answers the question of Narayanan and Yap (that was asked for )
on "two-point concentration" of the minimum so that -subgrids determine
the entire colouring. Next, we prove that with high probability a uniformly
random -colouring of vertices of a uniformly random graph is
reconstructible from its full -deck if and is not
reconstructible with high probability if . We further
show that the colour reconstruction algorithm for random graphs can be modified
and used for graph reconstruction: we prove that with high probability
is reconstructible from its full -deck if
while it is not reconstructible with high probability if
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