6 research outputs found
Reconstruction of functions from minors
The central notion of this thesis is the minor relation on functions of several arguments. A function f: A^n→B is called a minor of another function g: A^m→B if f can be obtained from g by permutation of arguments, identification of arguments, and introduction of inessential arguments. We first provide some general background and context to this work by presenting a brief survey of basic facts and results concerning different aspects of the minor relation, placing some emphasis on the author’s contributions to the field.
The notions of functions of several arguments and minors give immediately rise to the following reconstruction problem: Is a function f: A^n→B uniquely determined, up to permutation of arguments, by its identification minors, i.e., the minors obtained by identifying a pair of arguments? We review known results – both positive and negative – about the reconstructibility of functions from identification minors, and we outline the main ideas of the proofs, which often amount to formulating and solving reconstruction problems for other kinds of mathematical objects.
We then turn our attention to functions determined by the order of first occurrence, and we are interested in the reconstructibility of such functions. One of the main results of this thesis states that the class of functions determined by the order of first occurrence is weakly reconstructible. Some reconstructible subclasses are identified; in particular, pseudo-Boolean functions determined by the order of first occurrence are reconstructible.
As our main tool, we introduce the notion of minor of permutation. This is a quotient-like construction for permutations that parallels minors of functions and has some similarities to permutation patterns. We develop the theory of minors of permutations, focusing on Galois connections induced by the minor relation and on the interplay between permutation groups and minors of permutations. Our results will then find applications in the analysis of the reconstruction problem of functions determined by the order of first occurrence
Reconstructing multisets over commutative groupoids and affine functions over nonassociative semirings
A reconstruction problem is formulated for multisets over commutative
groupoids. The cards of a multiset are obtained by replacing a pair of its
elements by their sum. Necessary and sufficient conditions for the
reconstructibility of multisets are determined. These results find an
application in a different kind of reconstruction problem for functions of
several arguments and identification minors: classes of linear or affine
functions over nonassociative semirings are shown to be weakly reconstructible.
Moreover, affine functions of sufficiently large arity over finite fields are
reconstructible.Comment: 18 pages. Int. J. Algebra Comput. (2014
T-Branes and Monodromy
We introduce T-branes, or "triangular branes," which are novel non-abelian
bound states of branes characterized by the condition that on some loci, their
matrix of normal deformations, or Higgs field, is upper triangular. These
configurations refine the notion of monodromic branes which have recently
played a key role in F-theory phenomenology. We show how localized matter
living on complex codimension one subspaces emerge, and explain how to compute
their Yukawa couplings, which are localized in complex codimension two. Not
only do T-branes clarify what is meant by brane monodromy, they also open up a
vast array of new possibilities both for phenomenological constructions and for
purely theoretical applications. We show that for a general T-brane, the
eigenvalues of the Higgs field can fail to capture the spectrum of localized
modes. In particular, this provides a method for evading some constraints on
F-theory GUTs which have assumed that the spectral equation for the Higgs field
completely determines a local model.Comment: 110 pages, 5 figure