19 research outputs found
On Systems of Equations over Free Partially Commutative Groups
Version 2: Corrected Section 3.3: instead of lexicographical normal forms we
now use a normal form due to V. Diekert and A. Muscholl. Consequent changes
made and some misprints corrected.
Using an analogue of Makanin-Razborov diagrams, we give an effective
description of the solution set of systems of equations over a partially
commutative group (right-angled Artin group) . Equivalently, we give a
parametrisation of , where is a finitely generated group.Comment: 117 pages, 22 figure
Topics in the arithmetic of hypersurfaces and K3 surfaces
This thesis is a collection of various results related to the arithmetic of K3 surfaces and hypersurfaces which were obtained by the author during the course of his PhD studies.
The first part is related to Artin's conjecture on hypersurfaces over p-adic fields and solves the following question using tools from logarithmic geometry: Let f:X->Y be a proper, dominant morphism of smooth varieties over a number field k. When is it true that for almost all places v of k, the fibre X_P over any point P in Y(k_v) contains a zero-cycle of degree 1?
The second part proves new cases of Mazur's conjecture on the topology of rational points. Let E be an elliptic curve over Q with j-invariant 1728. For a class of elliptic pencils which are quadratic twists of E by quartic polynomials, the rational points on the projective line with positive rank fibres are dense in the real topology. This extends results obtained by Rohrlich and Kuwata-Wang for quadratic and cubic polynomials. We also give a proof of Mazur's conjecture for the Kummer surface associated to the product of two elliptic curves without any restrictions on the j-invariants.
The third and largest part presents a cohomological framework for determining the full Brauer group of a variety over a number field with torsion-free geometric Picard group. It investigates the middle cohomology of weighted diagonal hypersurfaces and implements the framework in the case of degree 2 K3 surfaces over Q which are double covers of the projective plane ramified in a diagonal sextic curve.Open Acces
Postquantum Br\`{e}gman relative entropies and nonlinear resource theories
We introduce the family of postquantum Br\`{e}gman relative entropies, based
on nonlinear embeddings into reflexive Banach spaces (with examples given by
reflexive noncommutative Orlicz spaces over semi-finite W*-algebras,
nonassociative L spaces over semi-finite JBW-algebras, and noncommutative
L spaces over arbitrary W*-algebras). This allows us to define a class of
geometric categories for nonlinear postquantum inference theory (providing an
extension of Chencov's approach to foundations of statistical inference), with
constrained maximisations of Br\`{e}gman relative entropies as morphisms and
nonlinear images of closed convex sets as objects. Further generalisation to a
framework for nonlinear convex operational theories is developed using a larger
class of morphisms, determined by Br\`{e}gman nonexpansive operations (which
provide a well-behaved family of Mielnik's nonlinear transmitters). As an
application, we derive a range of nonlinear postquantum resource theories
determined in terms of this class of operations.Comment: v2: several corrections and improvements, including an extension to
the postquantum (generally) and JBW-algebraic (specifically) cases, a section
on nonlinear resource theories, and more informative paper's titl
New Directions for Contact Integrators
Contact integrators are a family of geometric numerical schemes which
guarantee the conservation of the contact structure. In this work we review the
construction of both the variational and Hamiltonian versions of these methods.
We illustrate some of the advantages of geometric integration in the
dissipative setting by focusing on models inspired by recent studies in
celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282
A Contribution to Metric Diophantine Approximation : the Lebesgue and Hausdorff Theories
This thesis is concerned with the theory of Diophantine approximation from the point of
view of measure theory. After the prolegomena which conclude with a number of conjectures set
to understand better the distribution of rational points on algebraic planar curves, Chapter 1
provides an extension of the celebrated Theorem of Duffin and Schaeffer. This enables one to
set a generalized version of the Duffin–Schaeffer conjecture. Chapter 2 deals with the topic of
simultaneous approximation on manifolds, more precisely on polynomial curves. The aim is
to develop a theory of approximation in the so far unstudied case when such curves are not
defined by integer polynomials. A new concept of so–called “liminf sets” is then introduced in
Chapters 3 and 4 in the framework of simultaneous approximation of independent quantities.
In short, in this type of problem, one prescribes the set of integers which the denominators of
all the possible rational approximants of a given vector have to belong to. Finally, a reasonably
complete theory of the approximation of an irrational by rational fractions whose numerators
and denominators lie in prescribed arithmetic progressions is developed in chapter 5. This
provides the first example of a Khintchine type result in the context of so–called uniform
approximation
The word problem and combinatorial methods for groups and semigroups
The subject matter of this thesis is combinatorial semigroup theory. It includes material, in no particular order, from combinatorial and geometric group theory, formal language theory, theoretical computer science, the history of mathematics, formal logic, model theory, graph theory, and decidability theory.
In Chapter 1, we will give an overview of the mathematical background required to state the results of the remaining chapters. The only originality therein lies in the exposition of special monoids presented in §1.3, which uni.es the approaches by several authors.
In Chapter 2, we introduce some general algebraic and language-theoretic constructions which will be useful in subsequent chapters. As a corollary of these general methods, we recover and generalise a recent result by Brough, Cain & Pfei.er that the class of monoids with context-free word problem is closed under taking free products.
In Chapter 3, we study language-theoretic and algebraic properties of special monoids, and completely classify this theory in terms of the group of units. As a result, we generalise the Muller-Schupp theorem to special monoids, and answer a question posed by Zhang in 1992.
In Chapter 4, we give a similar treatment to weakly compressible monoids, and characterise their language-theoretic properties. As a corollary, we deduce many new results for one-relation monoids, including solving the rational subset membership problem for many such monoids. We also prove, among many other results, that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem.
In Chapter 5, we study context-free graphs, and connect the algebraic theory of special monoids with the geometric behaviour of their Cayley graphs. This generalises the geometric aspects of the Muller-Schupp theorem for groups to special monoids. We study the growth rate of special monoids, and prove that a special monoid of intermediate growth is a group
Entwurf funktionaler Implementierungen von Graphalgorithmen
Classic graph algorithms are usually presented and analysed
in imperative programming languages.
Imperative programming languages are well-suited for the description of
a program flow,
in which the order in which the operations are performed is important.
One common example of such a description is the successive,
typically destructive modification of objects.
This kind of iteration often occurs in the context of graph algorithms
that deal with a certain kind of optimisation.
In functional programming,
the order of execution is abstracted
and problem solutions are described
as compositions of intermediate solutions.
Additionally,
functional programming languages are referentially transparent
and thus destructive updates of objects are discouraged.
The development of purely functional graph algorithms begins with the
decomposition of a given problem into simpler problems.
In many cases
the solutions of these partial problems can be used to solve
different problems as well.
What is more,
this compositionality allows exchanging functions
for more efficient or more comprehensible versions with little effort.
An algebraic approach with a focus on relation algebra as defined by Tarski
is used as an intermediate step in this dissertation.
One advantage of this approach is the formality of the resulting specifications.
Despite their formality,
the resulting expressions are still readable,
because the algebraic operations have intuitive interpretations.
Another advantage is that the specification is executable,
once the necessary operations are implemented.
This dissertation presents the basics of the algebraic approach in the
functional programming language Haskell.
Using this foundation,
some exemplary graph-theoretic problems are solved in the presented
framework.
Finally,
optimisations of the presented implementations are discussed
and pointers are provided to further problems
that can be solved using the above methods.Klassische Graphalgorithmen werden ĂĽblicherweise in imperativen
Programmiersprachen
beschrieben und analysiert.
Imperative Programmiersprachen eignen sich gut,
um Programmabläufe zu beschreiben,
in welchen die Reihenfolge der Operationen
wichtig ist.
Dies betrifft insbesondere die schrittweise,
in der Regel destruktive Veränderung von Objekten,
wie sie häufig im Falle von Optimierungsproblemen
auf Graphen vorkommt.
In der funktionalen Programmierung abstrahiert man von einer festen
Berechnungsreihenfolge und beschreibt Problemlösungen als
Kompositionen von Teillösungen.
Ferner sind funktionale Programmiersprachen referentiell transparent,
sodass destruktive Veränderungen nur bedingt möglich sind.
Die Entwicklung rein funktionaler Graphalgorithmen setzt bei der Zerlegung
der bestehenden Probleme in einfachere Probleme an.
Oftmals können Lösungen dieser Teilprobleme auch in anderen
Situationen eingesetzt werden.
Darüber hinaus erlaubt es diese Kompositionalität,
einzelne Funktionen mit wenig Aufwand durch effizientere
oder verständlichere Fassungen
auszutauschen.
Als Zwischenschritt in der Entwicklung wird in dieser Dissertation
ein algebraischer Ansatz basierend auf der Relationenalgebra im Sinne von Tarski
verwendet.
Ein Vorteil dieses Ansatzes ist die
Formalität der entstehenden Spezifikationen.
Trotz ihrer Formalität bleiben die entstehenden Ausdrücke oft
leserlich,
weil die algebraischen Operationen
anschauliche Interpretationen zulassen.
Ein weiterer Vorteil ist,
dass Spezifikationen ausfĂĽhrbar werden,
sobald bestimmte Basisoperationen implementiert sind.
In dieser Dissertation werden Grundlagen einer Implementierung
des algebraischen Ansatzes in der
funktionalen Programmiersprache Haskell behandelt.
Ausgehend hiervon werden exemplarisch einige
Probleme der Graphentheorie gelöst.
SchlieĂźlich werden Optimierungen der vorgestellten Implementierungen
und weitere Probleme,
welche mit den obigen Methoden lösbar sind, diskutiert