Special compositions in affinely connected spaces without a torsion

2000 Mathematics Subject Classification: 53B05, 53B99.

Polynomial automorphisms over finite fields: Mimicking tame maps by the Derksen group

2010 Mathematics Subject Classification: 14L99, 14R10, 20B27.

Lousa digital no ensino de matemática

Using the digital whiteboard and its resources , classes become much more motivating and interesting than those that only require the chalk or whiteboard. In the face of still images , or moving, this technology has caused a broader interaction between the student, the content taught and the teacher bringing positive and significant changes to education. In this work, we discuss the technological evolution of education, new educational technologies, the digital blackboard and your pedagogical applications inmath classes and howthe teacher can mount it using resources available at the school and other relatively low cost. Finally, we will address the concept of homography and how it is applied in its operation.

Lousa digital no ensino de matemática

Using the digital whiteboard and its resources , classes become much more motivating and interesting than those that only require the chalk or whiteboard. In the face of still images , or moving, this technology has caused a broader interaction between the student, the content taught and the teacher bringing positive and significant changes to education. In this work, we discuss the technological evolution of education, new educational technologies, the digital blackboard and your pedagogical applications inmath classes and howthe teacher can mount it using resources available at the school and other relatively low cost. Finally, we will address the concept of homography and how it is applied in its operation.

Some near horizon studies onWeyl and Vaidya spacetimes

This thesis comprises some studies on the Weyl, Vaidya and Weyl distorted Schwarzschild (WDS) spacetimes. The main focal areas are : a) construction of near horizon metric(NHM) for WDS spacetime and subsequently a "stretched horizon" prescribed by the membrane formalism for black holes, b) application of membrane formalism and construction of stretched horizons for Vaidya spacetime and c) using the thin shell formalism to construct an asymptotically flat spacetime with a Weyl interior where the construction does not violate energy conditions. For a), a standard formalism developed in [1] has been used wherein the metric is expanded as a Taylor series in ingoing Gaussian null coordinates with the affine parameter as the expansion parameter. This expansion is used to construct a timelike "stretched horizon" just outside the true horizon to facilitate some membrane formalism studies, the theory for which was first introduced in [2]. b) applies the membrane formalism to Vaidya spacetime and also extends a part of the work done in [1] in which event horizon candidates were located perturbatively. Here, we locate stretched horizons in close proximity to every event horizon candidate located in [1]. c) is an attempt to induce Weyl distortions with a thin shell of matter in an asymptotically flat spacetime without violating energy conditions.

Learning from Geometry

Subspaces and manifolds are two powerful models for high dimensional signals. Subspaces model linear correlation and are a good fit to signals generated by physical systems, such as frontal images of human faces and multiple sources impinging at an antenna array. Manifolds model sources that are not linearly correlated, but where signals are determined by a small number of parameters. Examples are images of human faces under different poses or expressions, and handwritten digits with varying styles. However, there will always be some degree of model mismatch between the subspace or manifold model and the true statistics of the source. This dissertation exploits subspace and manifold models as prior information in various signal processing and machine learning tasks.

A near-low-rank Gaussian mixture model measures proximity to a union of linear or affine subspaces. This simple model can effectively capture the signal distribution when each class is near a subspace. This dissertation studies how the pairwise geometry between these subspaces affects classification performance. When model mismatch is vanishingly small, the probability of misclassification is determined by the product of the sines of the principal angles between subspaces. When the model mismatch is more significant, the probability of misclassification is determined by the sum of the squares of the sines of the principal angles. Reliability of classification is derived in terms of the distribution of signal energy across principal vectors. Larger principal angles lead to smaller classification error, motivating a linear transform that optimizes principal angles. This linear transformation, termed TRAIT, also preserves some specific features in each class, being complementary to a recently developed Low Rank Transform (LRT). Moreover, when the model mismatch is more significant, TRAIT shows superior performance compared to LRT.

The manifold model enforces a constraint on the freedom of data variation. Learning features that are robust to data variation is very important, especially when the size of the training set is small. A learning machine with large numbers of parameters, e.g., deep neural network, can well describe a very complicated data distribution. However, it is also more likely to be sensitive to small perturbations of the data, and to suffer from suffer from degraded performance when generalizing to unseen (test) data.

From the perspective of complexity of function classes, such a learning machine has a huge capacity (complexity), which tends to overfit. The manifold model provides us with a way of regularizing the learning machine, so as to reduce the generalization error, therefore mitigate overfiting. Two different overfiting-preventing approaches are proposed, one from the perspective of data variation, the other from capacity/complexity control. In the first approach, the learning machine is encouraged to make decisions that vary smoothly for data points in local neighborhoods on the manifold. In the second approach, a graph adjacency matrix is derived for the manifold, and the learned features are encouraged to be aligned with the principal components of this adjacency matrix. Experimental results on benchmark datasets are demonstrated, showing an obvious advantage of the proposed approaches when the training set is small.

Stochastic optimization makes it possible to track a slowly varying subspace underlying streaming data. By approximating local neighborhoods using affine subspaces, a slowly varying manifold can be efficiently tracked as well, even with corrupted and noisy data. The more the local neighborhoods, the better the approximation, but the higher the computational complexity. A multiscale approximation scheme is proposed, where the local approximating subspaces are organized in a tree structure. Splitting and merging of the tree nodes then allows efficient control of the number of neighbourhoods. Deviation (of each datum) from the learned model is estimated, yielding a series of statistics for anomaly detection. This framework extends the classical {\em changepoint detection} technique, which only works for one dimensional signals. Simulations and experiments highlight the robustness and efficacy of the proposed approach in detecting an abrupt change in an otherwise slowly varying low-dimensional manifold.

On surfaces with prescribed shape operator

The problem of immersing a simply connected surface with a prescribed shape operator is discussed. I show that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface with one family of principal curves being geodesic, the space of such realizations is a convex set in an affine space of dimension at most 3. The cases where this maximum dimension of realizability is achieved are analyzed and it is found that there are two such families of shape operators, one depending essentially on three arbitrary functions of one variable and another depending essentially on two arbitrary functions of one variable. The space of realizations is discussed in each case, along with some of their remarkable geometric properties. Several explicit examples are constructed.

Oligocene to early Miocene benthic foraminifera in the eastern Equatorial Pacific

We investigated Oligocene and early Miocene benthic foraminiferal faunas (> 105 µm in size) from Ocean Drilling Program (Leg 199) Site 1218 (4826 m water depth and ~3300 to ~4000 m paleo-water depth) and Site 1219 (5063 m water depth and ~4200 to ~4400 m paleo-water depth) to understand the response of abyssal benthic foraminifera to mid-Oligocene glacial events in the eastern Equatorial Pacific Ocean. Two principal factor assemblages were recognized. The Factor 1 assemblage (common Nuttallides umbonifer) is related to either an influx of the Southern Component Water (SCW), possibly carbonate undersaturated, or a decrease in seasonality of the food supply from the surface ocean. The Factor 2 assemblage is characterized by typical deep-sea taxa living under variable trophic conditions, possibly with a seasonal component in food supply. The occurrence of abyssal benthic foraminifera faunas during the mid-Oligocene depends on either the effect of SCW or the seasonality of food resources. The Factor 1 assemblage was most common near 76Ol-C11r, 73Ol-C10rn and 67Ol-C9n (ca. 30.2, 29.1 and 26.8 Ma respectively by Pälike et al. (2006, doi:10.1126/science.1133822)). This indicates that the effect of SCW increased or the seasonal input of food from the surface ocean to benthic environments was weakened close to these glacial events. In contrast, the huge export flux of small biogenic carbonate particles close to these glacial events might be responsible for carbonate-rich sediments buffering carbonate undersaturation. Changes in deep-water masses or the periodicity of food supply from the surface ocean and variation in surface carbonate production affected by orbital forcing had an impact on the mid-Oligocene faunas of abyssal benthic foraminifera around the intervals of glacial events in the eastern Equatorial Pacific Ocean. The Factor 1 assemblage decreased sharply at ? 30 Ma (29.8 Ma by Pälike et al. (2006), 30.0 Ma by CK95) and returned to dominance after ? 29 Ma (28.6 Ma by Pälike et al. (2006), 28.8 Ma by CK95). It is likely that the effect of SCW (possibly carbonate undersaturated) has intensified since the late Oligocene. The faunal transition of benthic foraminifera in the eastern Equatorial Pacific Ocean at ~29 Ma might be attributable to the influence of Northern Component Water (NCW) input to the Southern Ocean and the subsequent formation of SCW at about that time.

Relative abundance of euphausiids (Crustacea) in water samples of the Arabian Sea (Table 2)

In the present paper, the ecology and feeding habits of euphausiids are described. The samples were taken at the time of the NE-monsoon (1964/65) by R. V. "Meteor" in the Arabian Sea and adjacent waters. 24 species were determined. According to distribution of the species, the following marine areas can be distinguished: Arabian Sea: 24 species, dominant are Euphausia diomedeae, E. tenera, E. distinguenda, Stylocheiron carinatum. Gulf of Aden: 10 species, dominant are Euphausia diomedeae, E. distinguenda. Red Sea: 6 species, dominant are Euphausia diomedeae, E. distinguenda. Gulf of Oman : 5 Species, dominant are Euphausia distinguenda, Pseudeupbaufia latifrons. Persian Gulf: 1 species - Pseudeuphausia latifrons. The total number of euphausiids indicate the biomass of this group. High densities of euphausiids (200-299 and > 300 individuals/100 m**3) occur in the innermost part of the Gulf cf Aden, in the area south of the equator near the African east coast, near Karachi (Indian west coast) and in the Persian Gulf. Comparison with data relating to production biology confirms that these are eutrophic zones which coincide with areas in which upwelling occurs at the time of the NE-monsoon. The central part of the Arabian Sea differs from adjacent waters by virtue of less dense euphausiid populations (> 199 individuals/100 m**3). Measurements relating to production biology demonstrate a relatively low concentration of primary food sources. Food material was ascertained by analysis of stomach content. The following omnivorous species were examined: Euphausia diomedeae, E. distinguenda, E. tenera, Pseudeuphausia latifrons and Thysanopoda tricuspidata. Apart from crustacean remains large numbers of Foraminifera, Radiolaria, tintinnids, dinoflagellates were found in the stomachs. Quantitatively crustaceans form the most important item in the diet. Food selection on the basis of size and form appears to be restricted to certain genera of tintinnids. The genera Stylocheiron and Nematoscelis are predators. Only crustacean remains were found in the stomachs of Stylocheiron abbreviatum, whereas Radiolaria, Foraminifera and tintinnids occurred to some extent in Nematasceli sp. Different euphausiids in the food chain in the Arabian Sea. In omnivorous species the position is variable, since they not only feed by filtering autotrophic and heterotrophic Protista, but also by predation on zooplankton. Carnivorous species without filtering apparatus feed exclusively on zooplankton of the size of copepods. Only these species are well established as occupying a higher position in the food chain. The parasitic protozoan Tbalassomyces fagei was found on Euphausia diomedeae, E. fenera, E. distinguenda and E. sanzoi.

Classificazione delle curve ellittiche viste come curve algebriche piane

Nel presente lavoro è affrontato lo studio delle curve ellittiche viste come curve algebriche piane, più precisamente come cubiche lisce nel piano proiettivo complesso. Dopo aver introdotto nella prima parte le nozioni di Superfici compatte e orientabili e curve algebriche, tramite il teorema di classificazione delle Superfici compatte, se ne fornisce una preliminare classificazione basata sul genere della superficie e della curva, rispettivamente. Da qui, segue la definizione di curve ellittiche e uno studio più dettagliato delle loro pricipali proprietà, quali la possibilità di definirle tramite un'equazione affine nota come equazione di Weierstrass e la loro struttura intrinseca di gruppo abeliano. Si fornisce quindi un'ulteriore classificazione delle cubiche lisce, totalmente differente da quella precedente, che si basa invece sul modulo della cubica, invariante per trasformazioni proiettive. Infine, si considera un aspetto computazionale delle curve ellittiche, ovvero la loro applicazione nel campo della Crittografia. Grazie alla struttura che esse assumono sui campi finiti, sotto opportune ipotesi, i crittosistemi a chiave pubblica basati sul problema del logaritmo discreto definiti sulle curve ellittiche, a parità di sicurezza rispetto ai crittosistemi classici, permettono l'utilizzo di chiavi più corte, e quindi meno costose computazionalmente. Si forniscono quindi le definizioni di problema del logaritmo discreto classico e sulle curve ellittiche, ed alcuni esempi di algoritmi crittografici classici definiti su quest'ultime.

Situações-problema no ensino de matemática financeira

O presente trabalho descreve uma proposta de atividade educacional direcionada para professores de Matemática, envolvendo situações-problema no ensino de Matemática Financeira para ser aplicado com alunos do Ensino Médio. Tais atividades tem como objetivo fornecer um contexto real, no qual o estudante esteja inserido. O trabalho se divide em quatro partes: a introdução de uma situaçãoproblema envolvendo juros simples, o conhecimento matemático, a resolução da situação-problema e a proposta de atividade educacional. Diferenciando-se do que usualmente é encontrado nos livros didáticos, a proposta aqui apresentada propõe estudar conteúdos matemáticos de forma articulada, envolvendo o conceito de porcentagem vinculado com funções lineares e juros simples com função afim e progressão aritmética. Dessa forma, é apresentada uma sequência de aulas envolvendo situações-problema através de atividades, adequadas para os alunos.

EIR – sistema inteligente para detecção e classificação de calcificações em mamografias

Dissertação (mestrado)—Universidade de Brasília, Faculdade Gama, Programa de Pós-Graduação em Engenharia Biomédica, 2016.

Energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions

We develop the energy norm a-posteriori error estimation for hp-version discontinuous Galerkin (DG) discretizations of elliptic boundary-value problems on 1-irregularly, isotropically refined affine hexahedral meshes in three dimensions. We derive a reliable and efficient indicator for the errors measured in terms of the natural energy norm. The ratio of the efficiency and reliability constants is independent of the local mesh sizes and weakly depending on the polynomial degrees. In our analysis we make use of an hp-version averaging operator in three dimensions, which we explicitly construct and analyze. We use our error indicator in an hp-adaptive refinement algorithm and illustrate its practical performance in a series of numerical examples. Our numerical results indicate that exponential rates of convergence are achieved for problems with smooth solutions, as well as for problems with isotropic corner singularities.

Guided Migration and Collective Behavior: Cell Dynamics during Cancer Progression

This dissertation focuses on gaining understanding of cell migration and collective behavior through a combination of experiment, analysis, and modeling techniques. Cell migration is a ubiquitous process that plays an important role during embryonic development and wound healing as well as in diseases like cancer, which is a particular focus of this work. As cancer cells become increasingly malignant, they acquire the ability to migrate away from the primary tumor and spread throughout the body to form metastatic tumors. During this process, changes in gene expression and the surrounding tumor environment can lead to changes in cell migration characteristics. In this thesis, I analyze how cells are guided by the texture of their environment and how cells cooperate with their neighbors to move collectively. The emergent properties of collectively moving groups are a particular focus of this work as collective cell dynamics are known to change in diseases such as cancer. The internal machinery for cell migration involves polymerization of the actin cytoskeleton to create protrusions that---in coordination with retraction of the rear of the cell---lead to cell motion. This actin machinery has been previously shown to respond to the topography of the surrounding surface, leading to guided migration of amoeboid cells. Here we show that epithelial cells on nanoscale ridge structures also show changes in the morphology of their cytoskeletons; actin is found to align with the ridge structures. The migration of the cells is also guided preferentially along the ridge length. These ridge structures are on length scales similar to those found in tumor microenvironments and as such provide a system for studying the response of the cells' internal migration machinery to physiologically relevant topographical cues. In addition to sensing surface topography, individual cells can also be influenced by the pushing and pulling of neighboring cells. The emergent properties of collectively migrating cells show interesting dynamics and are relevant for cancer progression, but have been less studied than the motion of individual cells. We use Particle Image Velocimetry (PIV) to extract the motion of a collectively migrating cell sheet from time lapse images. The resulting flow fields allow us to analyze collective behavior over multiple length and time scales. To analyze the connection between individual cell properties and collective migration behavior, we compare experimental flow fields with the migration of simulated cell groups. Our collective migration metrics allow for a quantitative comparison between experimental and simulated results. This comparison shows that tissue-scale decreases in collective behavior can result from changes in individual cell activity without the need to postulate the existence of subpopulations of leader cells or global gradients. In addition to tissue-scale trends in collective behavior, the migration of cell groups includes localized dynamic features such as cell rearrangements. An individual cell may smoothly follow the motion of its neighbors (affine motion) or move in a more individualistic manner (non-affine motion). By decomposing individual motion into both affine and non-affine components, we measure cell rearrangements within a collective sheet. Finally, finite-time Lyapunov exponent (FTLE) values capture the stretching of the flow field and reflect its chaotic character. Applying collective migration analysis techniques to experimental data on both malignant and non-malignant human breast epithelial cells reveals differences in collective behavior that are not found from analyzing migration speeds alone. Non-malignant cells show increased cooperative motion on long time scales whereas malignant cells remain uncooperative as time progresses. Combining multiple analysis techniques also shows that these two cell types differ in their response to a perturbation of cell-cell adhesion through the molecule E-cadherin. Non-malignant MCF10A cells use E-cadherin for short time coordination of collective motion, yet even with decreased E-cadherin expression, the cells remain coordinated over long time scales. In contrast, the migration behavior of malignant and invasive MCF10CA1a cells, which already shows decreased collective dynamics on both time scales, is insensitive to the change in E-cadherin expression.