103,677 research outputs found
Minimal Permutation Representations of Classes of Semidirect Products of Groups
Given a finite group , the smallest such that embeds into the symmetric group is referred to as the minimal degree. Much of the accumulated literature focuses on the interplay between minimal degrees and direct products. This thesis extends this to cover large classes of semidirect products. Chapter 1 provides a background for minimal degrees - stating and proving a number of essential theorems and outlining relevant previous work, along with some small original results. Chapter 2 calculates the minimal degrees for an infinite class of semidirect products - specifically the semidirect products of elementary abelian groups by groups of prime order not dividing the order of the base group. This is established using vector space theory, including a number of novel techniques. The utility of this research is then demonstrated by answering an existing problem in the field of minimal degrees in a new and potentially generalisable way
Minimal Permutation Representations of Classes of Semidirect Products of Groups
Given a finite group , the smallest such that embeds into the symmetric group is referred to as the minimal degree. Much of the accumulated literature focuses on the interplay between minimal degrees and direct products. This thesis extends this to cover large classes of semidirect products. Chapter 1 provides a background for minimal degrees - stating and proving a number of essential theorems and outlining relevant previous work, along with some small original results. Chapter 2 calculates the minimal degrees for an infinite class of semidirect products - specifically the semidirect products of elementary abelian groups by groups of prime order not dividing the order of the base group. This is established using vector space theory, including a number of novel techniques. The utility of this research is then demonstrated by answering an existing problem in the field of minimal degrees in a new and potentially generalisable way
Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces
Let be a proper, geodesically complete CAT(0) space under a proper,
non-elementary, isometric action by a group with a rank one element.
We construct a generalized Bowen-Margulis measure on the space of unit-speed
parametrized geodesics of modulo the -action.
Although the construction of Bowen-Margulis measures for rank one
nonpositively curved manifolds and for CAT(-1) spaces is well-known, the
construction for CAT(0) spaces hinges on establishing a new structural result
of independent interest: Almost no geodesic, under the Bowen-Margulis measure,
bounds a flat strip of any positive width. We also show that almost every point
in , under the Patterson-Sullivan measure, is isolated in
the Tits metric. (For these results we assume the Bowen-Margulis measure is
finite, as it is in the cocompact case).
Finally, we precisely characterize mixing when has full limit set: A
finite Bowen-Margulis measure is not mixing under the geodesic flow precisely
when is a tree with all edge lengths in for some .
This characterization is new, even in the setting of CAT(-1) spaces.
More general (technical) versions of these results are also stated in the
paper.Comment: v2: 26 pages, 1 figure. Theorems stated in much more generality (in
particular, the cocompactness hypothesis was removed almost everywhere), also
a number of proofs dropped. This is the July 2015 version that was accepted
for publication in Ergodic Theory and Dynamical Systems. v1: 39 pages, 1
figur
On the alleged simplicity of impure proof
Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim
On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers
The systems of arithmetic discussed in this work are non-elementary theories. In this
paper, natural numbers are characterized axiomatically in two dierent ways.
We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and
mathematician, in 1932. The axioms W are those of ordered sets without largest
element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier
A formally verified proof of the prime number theorem
The prime number theorem, established by Hadamard and de la Vall'ee Poussin
independently in 1896, asserts that the density of primes in the positive
integers is asymptotic to 1 / ln x. Whereas their proofs made serious use of
the methods of complex analysis, elementary proofs were provided by Selberg and
Erd"os in 1948. We describe a formally verified version of Selberg's proof,
obtained using the Isabelle proof assistant.Comment: 23 page
Model Checking Lower Bounds for Simple Graphs
A well-known result by Frick and Grohe shows that deciding FO logic on trees
involves a parameter dependence that is a tower of exponentials. Though this
lower bound is tight for Courcelle's theorem, it has been evaded by a series of
recent meta-theorems for other graph classes. Here we provide some additional
non-elementary lower bound results, which are in some senses stronger. Our goal
is to explain common traits in these recent meta-theorems and identify barriers
to further progress. More specifically, first, we show that on the class of
threshold graphs, and therefore also on any union and complement-closed class,
there is no model-checking algorithm with elementary parameter dependence even
for FO logic. Second, we show that there is no model-checking algorithm with
elementary parameter dependence for MSO logic even restricted to paths (or
equivalently to unary strings), unless E=NE. As a corollary, we resolve an open
problem on the complexity of MSO model-checking on graphs of bounded max-leaf
number. Finally, we look at MSO on the class of colored trees of depth d. We
show that, assuming the ETH, for every fixed d>=1 at least d+1 levels of
exponentiation are necessary for this problem, thus showing that the (d+1)-fold
exponential algorithm recently given by Gajarsk\`{y} and Hlin\u{e}n\`{y} is
essentially optimal
On Constructive Axiomatic Method
In this last version of the paper one may find a critical overview of some
recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure
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