A mathematical excursion: From the three-door problem to a Cantor-type perfect set

We start with a generalization of the well-known three-door problem:the n-door problem. The solution of this new problem leads us toa beautiful representation system for real numbers in (0,1] as alternated series, known in the literature as Pierce expansions. A closer look to Pierce expansions will take us to some metrical properties of sets defined through the Pierce expansions of its elements. Finally, these metrical properties will enable us to present 'strange' sets, similar to the classical Cantor set.

Formation of the long range Dpp morphogen gradient.

The TGF-β homolog Decapentaplegic (Dpp) acts as a secreted morphogen in the Drosophila wing disc, and spreads through the target tissue in order to form a long range concentration gradient. Despite extensive studies, the mechanism by which the Dpp gradient is formed remains controversial. Two opposing mechanisms have been proposed: receptor-mediated transcytosis (RMT) and restricted extracellular diffusion (RED). In these scenarios the receptor for Dpp plays different roles. In the RMT model it is essential for endocytosis, re-secretion, and thus transport of Dpp, whereas in the RED model it merely modulates Dpp distribution by binding it at the cell surface for internalization and subsequent degradation. Here we analyzed the effect of receptor mutant clones on the Dpp profile in quantitative mathematical models representing transport by either RMT or RED. We then, using novel genetic tools, experimentally monitored the actual Dpp gradient in wing discs containing receptor gain-of-function and loss-of-function clones. Gain-of-function clones reveal that Dpp binds in vivo strongly to the type I receptor Thick veins, but not to the type II receptor Punt. Importantly, results with the loss-of-function clones then refute the RMT model for Dpp gradient formation, while supporting the RED model in which the majority of Dpp is not bound to Thick veins. Together our results show that receptor-mediated transcytosis cannot account for Dpp gradient formation, and support restricted extracellular diffusion as the main mechanism for Dpp dispersal. The properties of this mechanism, in which only a minority of Dpp is receptor-bound, may facilitate long-range distribution.

The daily market for funds in Europe: Mathematical appendix

This paper includes the derivations of the main expressions in the paper ``The Daily Market for Funds in Europe: Has Something Changed With the EMU?'' by G. Pérez Quirós and H. Rodríguez Mendizábal.

Resurgence of HIV infection among men who have sex with men in Switzerland: mathematical modelling study.

BACKGROUND: New HIV infections in men who have sex with men (MSM) have increased in Switzerland since 2000 despite combination antiretroviral therapy (cART). The objectives of this mathematical modelling study were: to describe the dynamics of the HIV epidemic in MSM in Switzerland using national data; to explore the effects of hypothetical prevention scenarios; and to conduct a multivariate sensitivity analysis. METHODOLOGY/PRINCIPAL FINDINGS: The model describes HIV transmission, progression and the effects of cART using differential equations. The model was fitted to Swiss HIV and AIDS surveillance data and twelve unknown parameters were estimated. Predicted numbers of diagnosed HIV infections and AIDS cases fitted the observed data well. By the end of 2010, an estimated 13.5% (95% CI 12.5, 14.6%) of all HIV-infected MSM were undiagnosed and accounted for 81.8% (95% CI 81.1, 82.4%) of new HIV infections. The transmission rate was at its lowest from 1995-1999, with a nadir of 46 incident HIV infections in 1999, but increased from 2000. The estimated number of new infections continued to increase to more than 250 in 2010, although the reproduction number was still below the epidemic threshold. Prevention scenarios included temporary reductions in risk behaviour, annual test and treat, and reduction in risk behaviour to levels observed earlier in the epidemic. These led to predicted reductions in new infections from 2 to 26% by 2020. Parameters related to disease progression and relative infectiousness at different HIV stages had the greatest influence on estimates of the net transmission rate. CONCLUSIONS/SIGNIFICANCE: The model outputs suggest that the increase in HIV transmission amongst MSM in Switzerland is the result of continuing risky sexual behaviour, particularly by those unaware of their infection status. Long term reductions in the incidence of HIV infection in MSM in Switzerland will require increased and sustained uptake of effective interventions.

Expanding molecular modeling and design tools to non-natural sidechains.

Protein-protein interactions encode the wiring diagram of cellular signaling pathways and their deregulations underlie a variety of diseases, such as cancer. Inhibiting protein-protein interactions with peptide derivatives is a promising way to develop new biological and therapeutic tools. Here, we develop a general framework to computationally handle hundreds of non-natural amino acid sidechains and predict the effect of inserting them into peptides or proteins. We first generate all structural files (pdb and mol2), as well as parameters and topologies for standard molecular mechanics software (CHARMM and Gromacs). Accurate predictions of rotamer probabilities are provided using a novel combined knowledge and physics based strategy. Non-natural sidechains are useful to increase peptide ligand binding affinity. Our results obtained on non-natural mutants of a BCL9 peptide targeting beta-catenin show very good correlation between predicted and experimental binding free-energies, indicating that such predictions can be used to design new inhibitors. Data generated in this work, as well as PyMOL and UCSF Chimera plug-ins for user-friendly visualization of non-natural sidechains, are all available at http://www.swisssidechain.ch. Our results enable researchers to rapidly and efficiently work with hundreds of non-natural sidechains.

Comparing and validating hypothesis test procedures: Graphical and numerical tools

When the behaviour of a specific hypothesis test statistic is studied by aMonte Carlo experiment, the usual way to describe its quality is by givingthe empirical level of the test. As an alternative to this procedure, we usethe empirical distribution of the obtained \emph{p-}values and exploit itsinformation both graphically and numerically.

Informing the judgments of fingerprint analysts using quality metric and statistical assessment tools

The aim of this research was to evaluate how fingerprint analysts would incorporate information from newly developed tools into their decision making processes. Specifically, we assessed effects using the following: (1) a quality tool to aid in the assessment of the clarity of the friction ridge details, (2) a statistical tool to provide likelihood ratios representing the strength of the corresponding features between compared fingerprints, and (3) consensus information from a group of trained fingerprint experts. The measured variables for the effect on examiner performance were the accuracy and reproducibility of the conclusions against the ground truth (including the impact on error rates) and the analyst accuracy and variation for feature selection and comparison.¦The results showed that participants using the consensus information from other fingerprint experts demonstrated more consistency and accuracy in minutiae selection. They also demonstrated higher accuracy, sensitivity, and specificity in the decisions reported. The quality tool also affected minutiae selection (which, in turn, had limited influence on the reported decisions); the statistical tool did not appear to influence the reported decisions.

Nuclear well logging in soils. Calibration of slim hole nuclear tools gamma-gamma and neutron-thermal neutron.

The calculation of elasticity parameters by sonic and ultra sonic wave propagation in saturated soils using Biot's theory needs the following variables : forpiation density and porosity (p, ø), compressional and shear wave velocities (Vp, Vs), fluid density, viscosity and compressibility (Pfi Ilfi Ki), matrix density and compressibility (p" K), The first four parameters can be determined in situ using logging probes. Because fluid and matrix characteristics are not modified during core extraction, they can be obtained through laboratory measurements. All parameters necessitate precise calibrations in various environments and for specific range of values encountered in soils. The slim diameter of boreholes in shallow geophysics and the high cost of petroleum equipment demand the use of specific probes, which usually only give qualitative results. The measurement 'of density is done with a gamma-gamma probe and the measurement of hydrogen index, in relation to porosity, by a neutron probe. The first step of this work has been carried out in synthetic formations in the laboratory using homogeneous media of known density and porosity. To establish borehole corrections different casings have been used. Finally a comparison between laboratory and in situ data in cored holes of known geometry and casing has been performed.

Programming and analysing measurement experiments using the PicoCricket system

The aim of this project is to get used to another kind of programming. Since now, I used very complex programming languages to develop applications or even to program microcontrollers, but PicoCricket system is the evidence that we don’t need so complex development tools to get functional devices. PicoCricket system is the clear example of simple programming to make devices work the way we programmed it. There’s an easy but effective way to program small, devices just saying what we want them to do. We cannot do complex algorithms and mathematical operations but we can program them in a short time. Nowadays, the easier and faster we produce, the more we earn. So the tendency is to develop fast, cheap and easy, and PicoCricket system can do it.

Programa d'anàlisi d'imatges de provetes metal·logràfiques

L’objectiu del present TFM és explorar les possibilitats del programa matemàtic MATLAB i la seva eina Entorn de Disseny d’Interfícies Gràfiques d’Usuari (GUIDE), desenvolupant un programa d’anàlisi d’imatges de provetes metal·logràfiques que es pugui utilitzar per a realitzar pràctiques de laboratori de l’assignatura Tecnologia de Materials de la titulació de Grau en Enginyeria Mecatrònica que s’imparteix a la Universitat de Vic. Les àrees d’interès del treball són la Instrumentació Virtual, la programació MATLAB i les tècniques d’anàlisi d’imatges metal·logràfiques. En la memòria es posa un èmfasi especial en el disseny de la interfície i dels procediments per a efectuar les mesures. El resultat final és un programa que satisfà tots els requeriments que s’havien imposat en la proposta inicial. La interfície del programa és clara i neta, destinant molt espai a la imatge que s’analitza. L’estructura i disposició dels menús i dels comandaments ajuda a que la utilització del programa sigui fàcil i intuïtiva. El programa s’ha estructurat de manera que sigui fàcilment ampliable amb altres rutines de mesura, o amb l’automatització de les rutines existents. Al tractar-se d’un programa que funciona com un instrument de mesura, es dedica un capítol sencer de la memòria a mostrar el procediment de càlcul dels errors que s’ocasionen durant la seva utilització, amb la finalitat de conèixer el seu ordre de magnitud, i de saber-los calcular de nou en cas que variïn les condicions d’utilització. Pel que fa referència a la programació, malgrat que MATLAB no sigui un entorn de programació clàssic, sí que incorpora eines que permeten fer aplicacions no massa complexes, i orientades bàsicament a gràfics o a imatges. L’eina GUIDE simplifica la realització de la interfície d’usuari, malgrat que presenta problemes per tractar dissenys una mica complexos. Per altra banda, el codi generat per GUIDE no és accessible, cosa que no permet modificar manualment la interfície en aquells casos en els que GUIDE té problemes. Malgrat aquests petits problemes, la potència de càlcul de MATLAB compensa sobradament aquestes deficiències.

Evaluating solutions to process, view and listen mathematical formula within an accessible context

Proves de conversió de fòrmules matemàtiques des d'editors de text ofimàtics i des de Làtex. Visionat en HTML i MathML. El millor resultat s'aconsegueix amb MSWord+MathType i IE+MathPlayer.