Some results on orbifold quotients and related objects

The quotient of a finite-dimensional Euclidean space by a finite linear group inherits different structures from the initial space, e.g. a topology, a metric and a piecewise linear structure. The question when such a quotient is a manifold leads to the study of finite groups generated by reflections and rotations, i.e. by orthogonal transformations whose fixed point subspace has codimension one or two. We classify such groups and thereby complete earlier results by M. A. Mikhaîlova from the 70s and 80s. Moreover, we show that a finite group is generated by reflections and) rotations if and only if the corresponding quotient is a Lipschitz-, or equivalently, a piecewise linear manifold (with boundary). For the proof of this statement we show in addition that each piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly. This solves a challenge by Thurston and confirms a conjecture by Kwasik and Lee. In the topological category a counterexample to the above mentioned characterization is given by the binary icosahedral group. We show that this is the only counterexample up to products. In particular, we answer the question by Davis of when the underlying space of an orbifold is a topological manifold. As a corollary of our results we generalize a fixed point theorem by Steinberg on unitary reflection groups to finite groups generated by reflections and rotations. As an application thereof we answer a question by Petrunin on quotients of spheres.

Tensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometry

This thesis deals with tensor completion for the solution of multidimensional inverse problems. We study the problem of reconstructing an approximately low rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, non-uniform sampling strategies, and parameter selection algorithms are developed. We derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property (RIP) based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by non-uniform sampling from a Parseval tight frame. We show how tensor completion can be used to solve multidimensional inverse problems arising in NMR relaxometry. Algorithms are developed for regularization parameter selection, including accelerated k-fold cross-validation and generalized cross-validation. These methods are validated on experimental and simulated data. We also derive condition number estimates for nonnegative least squares problems. Tensor recovery promises to significantly accelerate N-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments. Our methods could also be applied to other inverse problems arising in machine learning, image processing, signal processing, computer vision, and other fields.

Análise da estrutura e transporte do zooplâncton no estuário do Rio Caravelas (Bahia-Brasil) através do uso do zooscan

Spatial-temporal dynamics of zooplankton in the Caravelas river estuary (Bahia, Brazil). The survey was conducted in order to describe the zooplankton community of the estuary Caravelas (Bahia, Brazil), to quantify and relate the patterns of horizontal and vertical transport with the type of tide (neap and spring) and tidal phase (flood and ebb). Zooplankton samples were collected with the aid of a suction pump (300L), filtered in plankton nets (300μm) and fixed in saline formalin 4%. Samples were collected at a fixed point (A1), near the mouth of the estuary, with samples taken at neap tides and spring tides during the dry and rainy seasons. Samples were collected for 13 hours, at intervals of 1 hour in 3 depths: surface, middle and bottom. Simultaneous collection of biological, we measured the current velocity, temperature and salinity of the water through CTD. In the laboratory, samples were selected for analysis in estereomicroscope, with 25 groups identified, with Copepoda getting the highest number of species. The 168 samples obtained from temporal samples were subsampled and processed on equipment ZooScan, with the aid of software ZooProcess at the end were generated 458.997 vingnettes. 8 taxa were identified automatically, with 16 classified as a semi-automatic. The group Copepoda, despite the limited taxonomic refinement ZooScan, obtained 2 genera and 1 species identified automatically. Among the seasons dry and wet groups Brachyura (zoea), Chaetognatha, and the Calanoid copepods (others), Temora spp., Oithona spp. and Euterpina acutifrons were those who had higher frequency of occurrence, appearing in more than 70% of the samples. Copepoda group showed the largest percentage of relative abundance in both seasons. There was no seasonal variation of total zooplankton, with an average density of 7826±4219 org.m-3 in the dry season, and 7959±3675 org.m-3 in the rainy season, neither between the types and phases of the tides, but seasonal differences were significant recorded for the main zooplankton groups. Vertical stratification was seen for the major zooplankton groups (Brachyura, Chaetognatha, Calanoida (other), Oithona spp, Temora spp. e Euterpina acutifrons). The scale of this stratification varied with the type (square or tide) and tidal phase (flood or ebb). The instantaneous transport was more influenced by current velocity, with higher values observed in spring tides to the total zooplankton, however, there was a variation of this pattern depending on the zooplankton group. According to the data import and export of total zooplankton, the outflow of organisms of the estuary was higher than the input. The results suggest that the estuary of Caravelas may influence the dynamics of organic matter to the adjacent coast, with possible consequences in National Marine Park of Abrolhos

Selbstgesteuertes Lernen mit Kompetenzrastern. Ein theoretischer Blick auf das Potenzial eines pädagogischen Instruments zum individualisierten Lernen

Kompetenzraster sind pädagogische Instrumente, die zum kompetenzorientierten, individualisierten und selbstgesteuerten Lernen in beruflichen Schulen eingesetzt werden. Sie werden üblicherweise im Rahmen eines pädagogischen Gesamtkonzeptes genutzt, indem die Raster oft ein zentrales Instrument in einem komplexen Gefüge schulischer Lern- und Lehrprozesse sind. Kompetenzraster sind häufig der Fixpunkt, an dem sich andere Instrumente (wie Checklisten und Lernjobs) orientieren und sie definieren die Ausgangs- und Zielpunkte der Lernprozesse. Dabei werden den Schülern üblicherweise Freiheitsgrade eingeräumt, so dass sie (mit-) entscheiden ob, was, wann, wie und woraufhin sie lernen. Die schulische Arbeit mit den Rastern kann als ein Versuch angesehen werden, die Lernenden in den Mittelpunkt pädagogischen Denkens und Handelns zu stellen. Dieser Beitrag hat das Ziel, selbstgesteuertes Lernen aus einer distanzierten, vom einzelnen pragmatischen Modell abstrahierenden und eher theoretischen Perspektive auf das individualisierte Lernen mit Kompetenzrastern zu beziehen. Im Kern wird ein Systematisierungsansatz entwickelt, in dem die komplexen Zusammenhänge des Lernens mit Kompetenzrastern im Kontext von selbstgesteuertem Lernen dargestellt werden. Damit soll ein Beitrag zur Elaboration des Lernens mit Kompetenzrastern in beruflichen Schulen geleistet werden. Konkret wird die folgende Frage fokussiert: Was können Kompetenzraster im Rahmen selbstgesteuerten Lernens leisten? (DIPF/Orig.)

Delays in activity based neural networks

In this paper we study the effect of two distinct discrete delays on the dynamics of a Wilson-Cowan neural network. This activity based model describes the dynamics of synaptically interacting excitatory and inhibitory neuronal populations. We discuss the interpretation of the delays in the language of neurobiology and show how they can contribute to the generation of network rhythms. First we focus on the use of linear stability theory to show how to destabilise a fixed point, leading to the onset of oscillatory behaviour. Next we show for the choice of a Heaviside nonlinearity for the firing rate that such emergent oscillations can be either synchronous or anti-synchronous depending on whether inhibition or excitation dominates the network architecture. To probe the behaviour of smooth (sigmoidal) nonlinear firing rates we use a mixture of numerical bifurcation analysis and direct simulations, and uncover parameter windows that support chaotic behaviour. Finally we comment on the role of delays in the generation of bursting oscillations, and discuss natural extensions of the work in this paper.

Análise da estrutura e transporte do zooplâncton no estuário do Rio Caravelas (Bahia-Brasil) através do uso do zooscan

Spatial-temporal dynamics of zooplankton in the Caravelas river estuary (Bahia, Brazil). The survey was conducted in order to describe the zooplankton community of the estuary Caravelas (Bahia, Brazil), to quantify and relate the patterns of horizontal and vertical transport with the type of tide (neap and spring) and tidal phase (flood and ebb). Zooplankton samples were collected with the aid of a suction pump (300L), filtered in plankton nets (300μm) and fixed in saline formalin 4%. Samples were collected at a fixed point (A1), near the mouth of the estuary, with samples taken at neap tides and spring tides during the dry and rainy seasons. Samples were collected for 13 hours, at intervals of 1 hour in 3 depths: surface, middle and bottom. Simultaneous collection of biological, we measured the current velocity, temperature and salinity of the water through CTD. In the laboratory, samples were selected for analysis in estereomicroscope, with 25 groups identified, with Copepoda getting the highest number of species. The 168 samples obtained from temporal samples were subsampled and processed on equipment ZooScan, with the aid of software ZooProcess at the end were generated 458.997 vingnettes. 8 taxa were identified automatically, with 16 classified as a semi-automatic. The group Copepoda, despite the limited taxonomic refinement ZooScan, obtained 2 genera and 1 species identified automatically. Among the seasons dry and wet groups Brachyura (zoea), Chaetognatha, and the Calanoid copepods (others), Temora spp., Oithona spp. and Euterpina acutifrons were those who had higher frequency of occurrence, appearing in more than 70% of the samples. Copepoda group showed the largest percentage of relative abundance in both seasons. There was no seasonal variation of total zooplankton, with an average density of 7826±4219 org.m-3 in the dry season, and 7959±3675 org.m-3 in the rainy season, neither between the types and phases of the tides, but seasonal differences were significant recorded for the main zooplankton groups. Vertical stratification was seen for the major zooplankton groups (Brachyura, Chaetognatha, Calanoida (other), Oithona spp, Temora spp. e Euterpina acutifrons). The scale of this stratification varied with the type (square or tide) and tidal phase (flood or ebb). The instantaneous transport was more influenced by current velocity, with higher values observed in spring tides to the total zooplankton, however, there was a variation of this pattern depending on the zooplankton group. According to the data import and export of total zooplankton, the outflow of organisms of the estuary was higher than the input. The results suggest that the estuary of Caravelas may influence the dynamics of organic matter to the adjacent coast, with possible consequences in National Marine Park of Abrolhos

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, their second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants as well as Schur complements of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasi-determinants, lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm, those that relate the Baker and adjoint Baker functions to ratios of Miwa shifted tau-functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R^D, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

Efficient algorithms for beacon routing in polygons

In a paper by Biro et al. [7], a novel twist on guarding in art galleries is introduced. A beacon is a fixed point with an attraction pull that can move points within the polygon. Points move greedily to monotonically decrease their Euclidean distance to the beacon by moving straight towards the beacon or sliding on the edges of the polygon. The beacon attracts a point if the point eventually reaches the beacon. Unlike most variations of the art gallery problem, the beacon attraction has the intriguing property of being asymmetric, leading to separate definitions of attraction region and inverse attraction region. The attraction region of a beacon is the set of points that it attracts. For a given point in the polygon, the inverse attraction region is the set of beacon locations that can attract the point. We first study the characteristics of beacon attraction. We consider the quality of a "successful" beacon attraction and provide an upper bound of $\sqrt{2}$ on the ratio between the length of the beacon trajectory and the length of the geodesic distance in a simple polygon. In addition, we provide an example of a polygon with holes in which this ratio is unbounded. Next we consider the problem of computing the shortest beacon watchtower in a polygonal terrain and present an $O(n \log n)$ time algorithm to solve this problem. In doing this, we introduce $O(n \log n)$ time algorithms to compute the beacon kernel and the inverse beacon kernel in a monotone polygon. We also prove that $\Omega(n \log n)$ time is a lower bound for computing the beacon kernel of a monotone polygon. Finally, we study the inverse attraction region of a point in a simple polygon. We present algorithms to efficiently compute the inverse attraction region of a point for simple, monotone, and terrain polygons with respective time complexities $O(n^2)$, $O(n \log n)$ and $O(n)$. We show that the inverse attraction region of a point in a simple polygon has linear complexity and the problem of computing the inverse attraction region has a lower bound of $\Omega(n \log n)$ in monotone polygons and consequently in simple polygons.

Unbounded solutions for functional problems on the half-line

This paper presents an existence and localization result of unbounded solutions for a second-order differential equation on the half-line with functional boundary conditions. By applying unbounded upper and lower solutions, Green's functions and Schauder fixed point theorem, the existence of at least one solution is shown for the above problem. One example and one application to an Emden-Fowler equation are shown to illustrate our results.

Unbounded solutions for functional problems on the half-line

This paper presents an existence and localization result of unbounded solutions for a second-order differential equation on the half-line with functional boundary conditions. By applying unbounded upper and lower solutions, Green's functions and Schauder fixed point theorem, the existence of at least one solution is shown for the above problem. One example and one application to an Emden-Fowler equation are shown to illustrate our results.

Multi-scalar critical theories: first steps for a non perturbative functional renormalization group analysis

In this work the fundamental ideas to study properties of QFTs with the functional Renormalization Group are presented and some examples illustrated. First the Wetterich equation for the effective average action and its flow in the local potential approximation (LPA) for a single scalar field is derived. This case is considered to illustrate some techniques used to solve the RG fixed point equation and study the properties of the critical theories in D dimensions. In particular the shooting methods for the ODE equation for the fixed point potential as well as the approach which studies a polynomial truncation with a finite number of couplings, which is convenient to study the critical exponents. We then study novel cases related to multi field scalar theories, deriving the flow equations for the LPA truncation, both without assuming any global symmetry and also specialising to cases with a given symmetry, using truncations based on polynomials of the symmetry invariants. This is used to study possible non perturbative solutions of critical theories which are extensions of known perturbative results, obtained in the epsilon expansion below the upper critical dimension.

Gibbs random graphs on point processes

Consider a discrete locally finite subset Gamma of R(d) and the cornplete graph (Gamma, E), with vertices Gamma and edges E. We consider Gibbs measures on the set of sub-graphs with vertices Gamma and edges E` subset of E. The Gibbs interaction acts between open edges having a vertex in common. We study percolation properties of the Gibbs distribution of the graph ensemble. The main results concern percolation properties of the open edges in two cases: (a) when Gamma is sampled from a homogeneous Poisson process; and (b) for a fixed Gamma with sufficiently sparse points. (c) 2010 American Institute of Physics. [doi:10.1063/1.3514605]

Impairment of tangible fixed assets in the Portuguese listed companies: Disclosures

A Work Project, presented as part of the requirements for the Award of a Masters Degree in Management from the NOVA – School of Business and Economics

Skeletal Muscle Mitochondrial and Lipid Droplet Volume Density: Validity of Electron Microscopy Point-counting Measurements

Mitochondrial (M) and lipid droplet (L) volume density (vd) are often used in exercise research. Vd is the volume of muscle occupied by M and L. The means of calculating these percents are accomplished by applying a grid to a 2D image taken with transmission electron microscopy; however, it is not known which grid best predicts these values. PURPOSE: To determine the grid with the least variability of Mvd and Lvd in human skeletal muscle. METHODS: Muscle biopsies were taken from vastus lateralis of 10 healthy adults, trained (N=6) and untrained (N=4). Samples of 5-10mg were fixed in 2.5% glutaraldehyde and embedded in EPON. Longitudinal sections of 60 nm were cut and 20 images were taken at random at 33,000x magnification. Vd was calculated as the number of times M or L touched two intersecting grid lines (called a point) divided by the total number of points using 3 different sizes of grids with squares of 1000x1000nm sides (corresponding to 1µm2), 500x500nm (0.25µm2) and 250x250nm (0.0625µm2). Statistics included coefficient of variation (CV), 1 way-BS ANOVA and spearman correlations. RESULTS: Mean age was 67 ± 4 yo, mean VO2peak 2.29 ± 0.70 L/min and mean BMI 25.1 ± 3.7 kg/m2. Mean Mvd was 6.39% ± 0.71 for the 1000nm squares, 6.01% ± 0.70 for the 500nm and 6.37% ± 0.80 for the 250nm. Lvd was 1.28% ± 0.03 for the 1000nm, 1.41% ± 0.02 for the 500nm and 1.38% ± 0.02 for the 250nm. The mean CV of the three grids was 6.65% ±1.15 for Mvd with no significant differences between grids (P>0.05). Mean CV for Lvd was 13.83% ± 3.51, with a significant difference between the 1000nm squares and the two other grids (P<0.05). The 500nm squares grid showed the least variability between subjects. Mvd showed a positive correlation with VO2peak (r = 0.89, p < 0.05) but not with weight, height, or age. No correlations were found with Lvd. CONCLUSION: Different size grids have different variability in assessing skeletal muscle Mvd and Lvd. The grid size of 500x500nm (240 points) was more reliable than 1000x1000nm (56 points). 250x250nm (1023 points) did not show better reliability compared with the 500x500nm, but was more time consuming. Thus, choosing a grid with square size of 500x500nm seems the best option. This is particularly relevant as most grids used in the literature are either 100 points or 400 points without clear information on their square size.

Singular Bott-Chern classes and the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions

We study the singular Bott-Chern classes introduced by Bismut, Gillet and Soulé. Singular Bott-Chern classes are the main ingredient to define direct images for closed immersions in arithmetic K-theory. In this paper we give an axiomatic definition of a theory of singular Bott-Chern classes, study their properties, and classify all possible theories of this kind. We identify the theory defined by Bismut, Gillet and Soulé as the only one that satisfies the additional condition of being homogeneous. We include a proof of the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions that generalizes a result of Bismut, Gillet and Soulé and was already proved by Zha. This result can be combined with the arithmetic Grothendieck-Riemann-Roch theorem for submersions to extend this theorem to arbitrary projective morphisms. As a byproduct of this study we obtain two results of independent interest. First, we prove a Poincaré lemma for the complex of currents with fixed wave front set, and second we prove that certain direct images of Bott-Chern classes are closed.