How large are the level sets of the Takagi function?

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands.Comment: Added a new Section 5 with generalization of the main results; some new and corrected proofs of the old material; 29 pages, 3 figure

The Takagi function: a survey

This paper sketches the history of the Takagi function T and surveys known properties of T, including its nowhere-differentiability, modulus of continuity, graphical properties and level sets. Several generalizations of the Takagi function, in as far as they are based on the "tent map", are also discussed. The final section reviews a number of applications of the Takagi function to various areas of mathematics, including number theory, combinatorics and classical real analysis.Comment: 52 pages, 6 figure

Image-to-geometry registration using mutual correspondences

La tesi presenta un metodo innovativo per l'allineamento di immagini fotografiche su modelli tridimensionali basato su una tecnica semi automatica. L'approccio proposto sfrutta sia corrispondenze scelte da un utente che la similarità tra la vista del modello renderizzato e la foto originale misurata mediante tecniche di mutua informazione

Quality-aware performance analysis for multimedia MPSoC platforms

Ph.DDOCTOR OF PHILOSOPH

Signatures of quantumness: identification, quantification and dynamical preservation

2014 - 2015The quanti cation of quantumness is necessary to assess how much a physical system departs from a classical behaviour and thus gauge the quantum enhancement in opera- tional tasks such as information processing and computation. For arbitrary multiparti- cle systems, the quanti cation of quantumness typically involves nontrivial optimisation problems, and may require demanding tomographical techniques. We have developed an experimentally feasible approach to the evaluation of geometric measures of quantumness, according to which the distance from the state of the system to a suitable set of classi- cal states is considered. Our approach provides analytical results for particular classes of mixed states of N qubits, and computable lower bounds to global, partial, and gen- uine multiparticle entanglement, as well as to quantum coherence, for any general state. For global and partial entanglement, as well as quantum coherence, useful bounds have been obtained with minimum e ort, requiring local measurements in just three settings for any N. For genuine entanglement, a number of measurements scaling linearly with N is required. We have demonstrated the power of our approach to estimate and quantify di erent types of multiparticle entanglement in a variety of N-qubit states useful for quan- tum information processing and recently engineered in laboratories with quantum optics and trapped ion setups... [edited by author]XIV n.s