Tesis (Álgebra)
https://hdl.handle.net/11441/10807
2022-05-25T20:17:50ZCategorías de descenso simplicial
https://hdl.handle.net/11441/99174
Categorías de descenso simplicial
2007-12-19T00:00:00ZEstudio de dos invariantes en álgebras de Lie filiformes complejas y clasificación a partir de estos
https://hdl.handle.net/11441/99168
Estudio de dos invariantes en álgebras de Lie filiformes complejas y clasificación a partir de estos
• En las álgebras de lie filiformes complejas se estudian de manera detallada los subíndices definidos por: i = , que es equivalente a i = inf es conmutativo), que es equivalente a j = inf , dos invariantes respecto de bases adaptadas, y se prueban algunas de sus propiedades, demostramos que toda algebra de lie filiforme compleja no modelo, tiene un producto principal, demostramos que: 4 , también demostramos que un algebra de lie filiforme compleja está definida si se conocen los productos ( ) ( ) e introducimos el concepto de algebras cortadas para probar que ciertas algebras no son isomorfas entre sí.
Estos invariantes van a permitirnos realizar la clasificación de las algebras de lie filiformes complejas atendiendo a la terna (i, j, n), donde i, j son los invariantes mencionados y n la dimensión del algebra.
1995-05-01T00:00:00ZAspectos diofánticos y computacionales de la torsión racional en curvas elípticas
https://hdl.handle.net/11441/98965
Aspectos diofánticos y computacionales de la torsión racional en curvas elípticas
2006-12-13T00:00:00ZQuantum álgorithms for the combinatorial invariants of numerical semigroups
https://hdl.handle.net/11441/87264
Quantum álgorithms for the combinatorial invariants of numerical semigroups
It was back in spring 2014 when the author of this doctoral dissertation was
finishing its master thesis, whose main objective was the understanding of
Peter W. Shor’s most praised result, a quantum algorithm capable of
factoring integers in polynomial time. During the development of this master
thesis, me and my yet-tobe doctoral advisor studied the main aspects of
quantum computing from a purely algebraic perspective. This research
eventually evolved into a sufficiently thorough canvas capable of explaining
the main aspects and features of the mentioned algorithm from within an
undergraduate context.
Just after its conclusion, we seated down and elaborated a research plan for
a future Ph.D. thesis, which would expectantly involve quantum computing but
also a branch of algebra whose apparently innocent definitions hide some
really hard problems from a computational perspective: the theory of
numerical semigroups. As will be seen later, the definition of numerical
semigroup does not involve sophisticated knowledge from any somewhat obscure
and distant branch of the tree of mathematics. Nonetheless, a number of
combinatorial problems associated with these numerical semigroups are
extremely hard to solve, even when the size of the input is relatively
small. Some examples of these problems are the calculations of the Frobenius
number, the Apéry set, and the Sylvester denumerant, all of them bearing the
name of legendary mathematicians.
This thesis is the result of our multiple attempts to tackle those
combinatorial problems with the help of a hypothetical quantum computer.
First, Chapter 2 is devoted to numerical semigroups and computational
complexity theory, and is divided into three sections. In Section 2.1, we
give the formal definition of a numerical semigroup, along with a
description of the main problems involved with them. In Section 2.2, we
sketch the fundamental concepts of complexity theory, in order to understand
the true significance within the inherent hardness concealed in the
resolution of those problems. Finally, in Section 2.3 we prove the
computational complexity of the problems we aim to solve.
Chapter 3 is the result of our outline of the theory of quantum computing.
We give the basic definitions and concepts needed for understanding the
particular place that quantum computers occupy in the world of Turing
machines, and also the main elements that compose this particular model of
computation: quantum bits and quantum entanglement. We also explain the two
most common models of quantum computation, namely quantum circuits and
adiabatic quantum computers. For all of them we give mathematical
definitions, but always having in mind the physical experiments from which
they stemmed.
Chapter 4 is also about quantum computing, but from an algorithmical
perspective. We present the most important quantum algorithms to date in a
standardized way, explaining their context, their impact and consequences,
while giving a mathematical proof of their correctness and worked-out
examples. We begin with the early algorithms of Deutsch, Deutsch-Jozsa, and
Simon, and then proceed to explain their importance in the dawn of quantum
computation. Then, we describe the major landmarks: Shor’s factoring,
Grover’s search, and quantum counting.
Chapter 5 is the culmination of all previously explained concepts, as it
includes the description of various quantum algorithms capable of solving
the main problems inside the branch of numerical semigrops. We present
quantum circuit algorithms for the Sylvester denumerant and the numerical
semigroup membership, and adiabatic quantum algorithms for the Ap´ery Set
and the Frobenius problem. We also describe a C++ library called numsem,
specially developed within the context of this doctoral thesis and which
helps us to study the computational hardness of all previously explained
problems from a classical perspective.
This thesis is intended to be autoconclusive at least in the main branches
of mathematics in which it is supported; that is to say numerical
semigroups, computational complexity theory, and quantum computation.
Nevertheless, for the majority of concepts explained here we give references
for the interested reader that wants to delve more into them.
2019-01-01T00:00:00Z