2,007 research outputs found
Covering and Separation for Permutations and Graphs
This is a thesis of two parts, focusing on covering and separation topics of extremal combinatorics and graph theory, two major themes in this area. They entail the existence and properties of collections of combinatorial objects which together either represent all objects (covering) or can be used to distinguish all objects from each other (separation). We will consider a range of problems which come under these areas. The first part will focus on shattering k-sets with permutations. A family of permutations is said to shatter a given k-set if the permutations cover all possible orderings of the k elements. In particular, we investigate the size of permutation families which cover t orders for every possible k-set as well as study the problem of determining the largest number of k-sets that can be shattered by a family with given size. We provide a construction for a small permutation family which shatters every k-set. We also consider constructions of large families which do not shatter any triple. The second part will be concerned with the problem of separating path systems. A separating path system for a graph is a family of paths where, for any two edges, there is a path containing one edge but not the other. The aim is to find the size of the smallest such family. We will study the size of the smallest separating path system for a range of graphs, including complete graphs, complete bipartite graphs, and lattice-type graphs. A key technique we introduce is the use of generator paths - constructed to utilise the symmetric nature of Kn. We continue this symmetric approach for bipartite graphs and study the limitations of the method. We consider lattice-type graphs as an example of the most efficient possible separating systems for any graph
Probabilistic Programming Interfaces for Random Graphs::Markov Categories, Graphons, and Nominal Sets
We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way.We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons
Logarithmic Quasimaps
We construct a proper moduli space which is a Deligne–Mumford stack parametrising quasimaps relative to a simple normal crossings divisor in any genus using logarithmic geometry. We show this moduli space admits a virtual fundamental class of the expected dimension leading to numerical invariants which agree with the theory of Battistella–Nabijou where the latter is defined
On the complexity of isomorphism problems for tensors, groups, and polynomials IV: linear-length reductions and their applications
Many isomorphism problems for tensors, groups, algebras, and polynomials were
recently shown to be equivalent to one another under polynomial-time
reductions, prompting the introduction of the complexity class TI (Grochow &
Qiao, ITCS '21; SIAM J. Comp., '23). Using the tensorial viewpoint, Grochow &
Qiao (CCC '21) then gave moderately exponential-time search- and
counting-to-decision reductions for a class of -groups. A significant issue
was that the reductions usually incurred a quadratic increase in the length of
the tensors involved. When the tensors represent -groups, this corresponds
to an increase in the order of the group of the form ,
negating any asymptotic gains in the Cayley table model.
In this paper, we present a new kind of tensor gadget that allows us to
replace those quadratic-length reductions with linear-length ones, yielding the
following consequences:
1. Combined with the recent breakthrough -time
isomorphism-test for -groups of class 2 and exponent (Sun, STOC '23),
our reductions extend this runtime to -groups of class and exponent
where .
2. Our reductions show that Sun's algorithm solves several TI-complete
problems over , such as isomorphism problems for cubic forms, algebras,
and tensors, in time .
3. Polynomial-time search- and counting-to-decision reduction for testing
isomorphism of -groups of class and exponent in the Cayley table
model. This answers questions of Arvind and T\'oran (Bull. EATCS, 2005) for
this group class, thought to be one of the hardest cases of Group Isomorphism.
4. If Graph Isomorphism is in P, then testing equivalence of cubic forms and
testing isomorphism of algebra over a finite field can both be solved in
time , improving from the brute-force upper bound
Minimal PD-sets for codes associated with the graphs Qm2, m even
Please read abstract in the article.The National Research Foundation of South Africahttp://link.springer.com/journal/2002021-12-08hj2021Mathematics and Applied Mathematic
Infinite Permutation Groups and the Origin of Quantum Mechanics
We propose an interpretation for the meets and joins in the lattice of
experimental propositions of a physical theory, answering a question of
Birkhoff and von Neumann in [1]. When the lattice is atomistic, it is
isomorphic to the lattice of definably closed sets of a finitary relational
structure in First Order Logic. In terms of mapping experimental propositions
to subsets of the atomic phase space, the meet corresponds to set intersection,
while the join is the definable closure of set union. The relational structure
is defined by the action of the lattice automorphism group on the atomic layer.
Examining this correspondence between physical theories and infinite group
actions, we show that the automorphism group must belong to a family of
permutation groups known as geometric Jordan groups. We then use the
classification theorem for Jordan groups to argue that the combined
requirements of probability and atomicism leave uncountably infinite Steiner
2-systems (of which projective spaces are standard examples) as the sole class
of options for generating the lattice of particle Quantum Mechanics.Comment: 23 page
Complexity of dynamical systems arising from random substitutions in one dimension
This thesis is based on three papers the author wrote while a PhD student, which concern different notions of complexity for dynamical systems arising from random substitutions.
Before presenting our main results, we first provide an introduction to random substitutions. In Chapter 2, we give the main definitions that we work with throughout, and prove several basic properties of random substitutions and their associated subshifts. We define the frequency measure corresponding to a random substitution, and prove a key result concerning such measures which will be of fundamental importance in our work.
Chapter 3 is based on a solo-author paper and concerns word complexity and topological entropy of random substitution subshifts. In contrast to previous work, we do not assume that the underlying random substitution is compatible. In our main results, we show that the subshift of a primitive random substitution has zero topological entropy if and only if it can be obtained as the subshift of a deterministic substitution -- answering in the affirmative an open question of Rust and Spindeler -- and provide a systematic approach to calculating the topological entropy for subshifts of constant length random substitutions. We also consider word complexity for constant length random substitutions and show that, without primitivity, the complexity function can exhibit features not possible in the deterministic or primitive random settings.
Chapters 4 and 5 are based on joint work with P. Gohlke, D. Rust and T. Samuel. These chapters focus on measure theoretic entropy and its relationship to topological entropy. In Chapter 4, we introduce a new measure of complexity for primitive random substitutions called measure theoretic inflation word entropy and show that this coincides with the measure theoretic entropy of the subshift with respect to the corresponding frequency measure. This allows the measure theoretic entropy to be explicitly calculated in many cases. In Chapter 5, we provide sufficient conditions under which a random substitution subshift supports a frequency measure of maximal entropy and, under more restrictive conditions, show that this measure is the unique measure of maximal entropy. Notably, we show that random substitutions can give rise to intrinsically ergodic subshifts that do not satisfy Bowen's specification property or the weaker specification property of Climenhaga and Thompson, thus providing an interesting new class of intrinsically ergodic subshifts. We conclude this chapter by showing that the random period doubling substitution is intrinsically ergodic.
Finally, Chapter 6 is based on joint work with A. Rutar. Here, we consider multifractal properties of frequency measures. Specifically, we study the multifractal spectrum and -spectrum of frequency measures corresponding to primitive and compatible random substitutions. We introduce a new notion called the inflation word -spectrum of a random substitution and show that this coincides with the -spectrum of the corresponding frequency measure for all . Under an additional assumption (recognisability) we show that the two notions coincide for all real q. Further, under these assumptions, we show that the multifractal formalism holds. The techniques we develop allow the -spectrum and multifractal spectrum to be obtained for many frequency measures
Projective Rectangles
A projective rectangle is like a projective plane that has different lengths
in two directions. We develop the basic theory of projective rectangles
including incidence properties, projective subplanes, configuration counts, a
partial Desargues's theorem, a construction from projective planes, and
alternative formulations. In sequels we study harmonic conjugation and the
graphs of lines and subplanes.Comment: 33 pp., 8 figure
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