Helly numbers of Algebraic Subsets of Rd\mathbb R^d

We study $S$-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in $\mathbb R^d$ with a proper subset $S\subset \mathbb R^d$. We contribute new results about their $S$-Helly numbers. We extend prior work for $S=\mathbb R^d$, $\mathbb Z^d$, and $\mathbb Z^{d-k}\times\mathbb R^k$; we give sharp bounds on the $S$-Helly numbers in several new cases. We considered the situation for low-dimensional $S$ and for sets $S$ that have some algebraic structure, in particular when $S$ is an arbitrary subgroup of $\mathbb R^d$ or when $S$ is the difference between a lattice and some of its sublattices. By abstracting the ingredients of Lov\'asz method we obtain colorful versions of many monochromatic Helly-type results, including several colorful versions of our own results.Comment: 13 pages, 3 figures. This paper is a revised version of what was originally the first half of arXiv:1504.00076v

Beyond Chance-Constrained Convex Mixed-Integer Optimization: A Generalized Calafiore-Campi Algorithm and the notion of SS-optimization

The scenario approach developed by Calafiore and Campi to attack chance-constrained convex programs utilizes random sampling on the uncertainty parameter to substitute the original problem with a representative continuous convex optimization with $N$ convex constraints which is a relaxation of the original. Calafiore and Campi provided an explicit estimate on the size $N$ of the sampling relaxation to yield high-likelihood feasible solutions of the chance-constrained problem. They measured the probability of the original constraints to be violated by the random optimal solution from the relaxation of size $N$. This paper has two main contributions. First, we present a generalization of the Calafiore-Campi results to both integer and mixed-integer variables. In fact, we demonstrate that their sampling estimates work naturally for variables restricted to some subset $S$ of $\mathbb R^d$. The key elements are generalizations of Helly's theorem where the convex sets are required to intersect $S \subset \mathbb R^d$. The size of samples in both algorithms will be directly determined by the $S$-Helly numbers. Motivated by the first half of the paper, for any subset $S \subset \mathbb R^d$, we introduce the notion of an $S$-optimization problem, where the variables take on values over $S$. It generalizes continuous, integer, and mixed-integer optimization. We illustrate with examples the expressive power of $S$-optimization to capture sophisticated combinatorial optimization problems with difficult modular constraints. We reinforce the evidence that $S$-optimization is "the right concept" by showing that the well-known randomized sampling algorithm of K. Clarkson for low-dimensional convex optimization problems can be extended to work with variables taking values over $S$.Comment: 16 pages, 0 figures. This paper has been revised and split into two parts. This version is the second part of the original paper. The first part of the original paper is arXiv:1508.02380 (the original article contained 24 pages, 3 figures

Rigidity analysis of protein structures and rapid simulations of protein motion

It is a common goal in biophysics to understand protein structural properties and their relationship to protein function. I investigated protein structural properties using three coarse graining methods: a rigidity analysis method First, a geometric simulation method Froda and normal mode analysis as implemented in Elnemo to identify the protein directions of motion. Furthermore, I also compared the results between the coarse graining methods with the results from molecular dynamics and from experiments that I carried out. The results from the rigidity analysis across a set of protein families presented in chapter 3 highlighted two different patterns of protein rigidity loss, i.e. "sudden" and "gradual". It was found that theses characteristic patterns were in line with the rigidity distribution of glassy networks. The simulations of protein motion by merging flexibility, rigidity and normal mode analyses presented in chapter 4 were able to identify large conformational changes of proteins using minimal computational resources. I investigated the use of RMSD as a measure to characterise protein motion and showed that, despite it is a good measure to identify structural differences when comparing the same protein, the use of extensive RMSD better captures the extend of motion of a protein structure. The in-depth investigation of yeast PDI mobility presented in chapter 5 confirmed former experimental results that predicted a large conformational change for this enzyme. Furthermore, the results predicted: a characteristic rigidity distribution for yeast PDI, a minimum and a maximum active site distance and a relationship between the energy cutoff, i.e. the number of hydrogen bonds part of the network of bonds, and protein mobility. The results obtained were tested against molecular dynamics simulations in chapter 6. The MD simulation also showed a large conformational change for yeast PDI but with a slightly different minimum and maximum inter-cysteine distance. Furthermore, MD was able to reveal new data, i.e. the most likely inter-cysteine distance. In order to test the accuracy of the coarse graining and MD simulations I carried out cross-linking experiments to test the minimum inter-cysteine distance predictions. The results presented in chapter 7 show that human PDI minimum distance is below 12Å whereas the yeast PDI minimum distance must be above 12Å as no cross-linking structures where found with the available (12Å long) cross-linkers

Non-critically squeezed light via spontaneous rotational symmetry breaking

We theoretically address squeezed light generation through the spontaneous breaking of the rotational invariance occuring in a type I degenerate optical parametric oscillator (DOPO) pumped above threshold. We show that a DOPO with spherical mirrors, in which the signal and idler fields correspond to first order Laguerre-Gauss modes, produces a perfectly squeezed vacuum with the shape of a Hermite-Gauss mode, within the linearized theory. This occurs at any pumping level above threshold, hence the phenomenon is non-critical. Imperfections of the rotational symmetry, due e.g. to cavity anisotropy, are shown to have a small impact, hence the result is not singular.Comment: 4 pages, 1 figure, replaced with resubmitted versio

Implicaciones genéticas de nuevos datos de Sr y Nd de rocas intrusivas del Arco Laramide en el Norte de Sonora, México

The Laramide Intrusive Arc constitutes a wide intrusive belt broadly parallel to the actual Sonora coastline. It was formed by the subduction of the Farallon Plate beneath the North-American Plate during the Late Cretaceous-Early Tertiary period. New isotopic data on rocks of this arc show initial 87Sr/86Sr and eNd isotopic values of 0.7066 to 0.7070 and -5 to -6, respectively, for two samples from Bacanora area; as well as 0.7074 to 0.7081 and -3 to -5.5 respectively, for five samples from Cananea, Mariquita and La Caridad areas. Isotopic ages were determined by U/Pb in zircons (95 Ma) and Ar/Ar in potassic feldspar (56 to 71 Ma) from a quartz monzonite porphyry, and by Ar/Ar in potassic feldspar (56 Ma) from another plutonic granodiorite, both from Bacanora. Initial 87Sr/86Sr and eNd values for samples reported in this study suggest that the Laramidic magmas had a large influence from the Proterozoic basement in northeastern Sonora. Four different isotopic zones are proposed for Sonora, according with Sr-Nd data of laramidic rocks and the substratum intruded.El Arco Intrusivo Laramide constituye un amplio cinturón de rocas intrusivas orientado burdamente paralelo a la costa actual de Sonora. Este cinturón se originó por la subducción de la Placa Farallón bajo la Placa de Norteamérica durante el período Cretácico Tardío – Terciario Temprano. Nuevos datos isotópicos en rocas de este arco indican valores iniciales de 87Sr/86Sr y de eNd de 0.7066 a 0.7070 y -5 a -6, respectivamente, para dos muestras del área de Bacanora; así como de 0.7074 a 0.7081 y -3 a -5.5, respectivamente, para cinco muestras de las áreas de Cananea, Mariquita y La Caridad. Se determinaron también edades isotópicas por U/Pb en zircones (95 Ma) y Ar/Ar en feldespato potásico (56 a 71 Ma) para un pórfido de monzonita de cuarzo, y por Ar/Ar en feldespato potásico (56 Ma) para otra granodiorita plutónica, ambas de Bacanora. Los valores iniciales de 87Sr/86Sr y eNd para las muestras reportadas en este estudio, sugieren que los magmas laramídicos tuvieron gran influencia del basamento proterozoico en el noreste de Sonora. Cuatro diferentes zonas isotópicas son propuestas para Sonora, de acuerdo con los datos de Sr-Nd de las rocas laramídicas y el sustrato intrusionado

Osteoma osteoide de localización escapular: descripción de un caso

Presentamos el caso de un paciente, con dolor en el hombro de predominio nocturno y que diagnosticamos como Osteoma Osteoide localizado en la escápula izquierda. El tratamiento con salicilatos no fue efectivo para mejorar el dolor y se procedió a la resección abierta y cauterización de la lesión. El resultado fue satisfactorio y actualmente el paciente permanece asintomático.We report a case of a patient with pain worse at night, located at the elbow. Osteoid Osteoma of the scapula was diagnosed. Pain was not relieved by Salycilates. Open surgical treatment and cauterization were the treatment elected in our case. Pain regressed inmediatly after surgery and the patient was asimptomatic since the surgery

Spontaneous symmetry breaking as a resource for noncritically squeezed light

In the last years we have proposed the use of the mechanism of spontaneous symmetry breaking with the purpose of generating perfect quadrature squeezing. Here we review previous work dealing with spatial (translational and rotational) symmetries, both on optical parametric oscillators and four-wave mixing cavities, as well as present new results. We then extend the phenomenon to the polarization state of the signal field, hence introducing spontaneous polarization symmetry breaking. Finally we propose a Jaynes-Cummings model in which the phenomenon can be investigated at the single-photon-pair level in a non-dissipative case, with the purpose of understanding it from a most fundamental point of view.Comment: Review for the proceedings of SPIE Photonics Europe. 11 pages, 5 figures

Deformation of anisotropic Fermi surfaces due to electron-electron interactions

We analyze the deformations of the Fermi surface induced by electron-electron interactions in anisotropic two dimensional systems. We use perturbation theory to treat, on the same footing, the regular and singular regions of the Fermi surface. It is shown that, even for weak local coupling, the self-energy presents a nontrivial behavior showing momentum dependence and interplay with the Fermi surface shape. Our scheme gives simple analytical expressions based on local features of the Fermi surface.Comment: 7 pages, 3 figure