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Nuove idee alle origini dell'algebra moderna
No full textIn questo lavoro ho cercato di esporre alcune tra le idee principali che
portarono alla nascita dell'algebra moderna. Ho fatto riferimento
principalmente alle idee di Galois, di Hamilton e di Grassmann e ho cercato
di evidenziare come i loro ruoli siano stati utili nell'elaborazione di
nuovi concetti
Il problema dei due cammini disgiunti: teoria e algoritmi risolutivi
No full textIl presente lavoro ha come oggetto principale il problema dei due cammini disgiunti. Una semplice descrizione del problema viene data a partire da un grafo nel quale vengono selezionate due coppie di vertici. Lo scopo della trattazione è conoscere in quali grafi sia possibile trovare due cammini disgiunti che colleghino le due coppie di vertici, e successivamente ricavare algoritmi che permettano di individuarli in caso di risposta affermativa
Syntomic regulators
Get PDFIn this work we introduce different p-adic cohomology theories with a particular ispiration to the Deligne-Beilinson one in the trascendental situation. We will introduce the concept of regulator map as an "higher cycle map" from the higher Chow groups to cohomology theories that we choose to deal with. Our goal is to develop a p-adic theory introduced by Gros and provide a regulator map for it. Some aspect are still open, but this theory allows us to have a connection between Besser and p-adic étale cohomolgy theories in terms of these regulator maps
On symplectic structures in information geometry
No full textInformation Geometry (Amari) gives us a framework to investigate
probability theory and statistics using notions of differential geometry.
The principal object of Information Geometry is the concept of divergence
function which acts as a pseudo-distance between two probability
distributions because it does not satisfy the symmetry property. Divergence functions are often used to define a Riemannian statistical structure on finite dimensional manifold M endowed with a dual coordinate system, as we will show for the prototypical Exponential Family which will arise from the solutions of the Maximum Entropy Principle (MEP). Since the final aim of this thesis is to study the role of the symplectic structures in the context of the Information Geometry, it will be useful the notion of yoke
(Barndorff-Nielsen), i.e. a generalization of the divergence function, that
given a manifold M permits to define via pull-back a symplectic structure
on M^2 from the canonical one over T^*M. Moreover thanks to the
Maslov-Hormander Theorem it will be possible to obtain a local description
of Lagrangian submanifolds, which are fundamental objects in Symplectic
Geometry, and to study the prototype of a statistical manifold generated by
a divergence function on which to construct a symplectic structure, i.e.
the Exponential family M(h,k) with the KL. Since the MEP is a powerful tool
that allows to single out a unique probability distribution for given
constraints, we will conclude our work studying the Lagrangian ubmanifolds
generated from the maximization problem given by the MEP with nonlinear
constraints by the application of the MaslovHormander Theorem to the
Lagrange Multipliers method (LMM). There will be also a study of the
Lagrangian submanifolds given from the solutions of the LMM in T^*M(h, k)
and the corresponding image of the submanifold to M(h, k)^2
A machine learning approach to football betting market
No full textIn the Soccer Betting Market it is very hard to find a strategy to get a
high return. However, by leveraging Machine Learning (ML) we are able to
find a good strategy. We firstly use simple ML algorithms to predict the
result of each match, and then we combine them into more complicated ones.
Betting on the result of each of these models gives us a profit and loss,
so that we have different assets and we can construct a portfolio. Using
Optimization and ML techniques we find how to give reasonable weights to
the assets in our portfolio in order to get a well performing asset, that
will be our final strategy. We will see that this is an even better proxy
of the Growth Optimal Portfolio (GOP) then the equally weighted portfolio,
whose performance is usually very hard to beat
