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This monograph is a review of Poisson geometry, Lie groupoids and Lie algebroids theory and they relations. We present a summary of the main known results in this areas and as a return of the striking connection of this topics, we present the geometric approach to Conns linearization theorem present in [Crainic and Fernandes 2011]. Este trabalho é uma revisão bibliografica nas areas de geometria de Poisson e teoria de grupoides e algebroides de Lie. Como resultado final desta interação apresentamos a abordagem geometrica para o teorema de linearização de Conn para estruturas de Poisson apresentada em [Crainic and Fernandes 2011].

We reconsider the rational Calogero–Moser system from the point of view of bi-Hamiltonian geometry. By using geometrical tools of the latter, we explicitly construct set(s) of spectral canonical coordinates, that is, complete sets of Darboux coordinates defined by the eigenvalues and the eigenvectors of the Lax matrix.

In the first part of this letter it will be shown that the post-Lie Magnus expansion can be interpreted as a crossed morphism between two (local) Lie group. The second part will be devoted to present two combinatorial methods, both based on special tubings on planar trees, to compute the coefficients of this remarkable formal series., Comment: 25 pages, many pictures

In this note we further explore the properties of universal enveloping algebras associated to a post-Lie algebra. Emphasizing the role of the Magnus expansion, we analyze the properties of group like-elements belonging to (suitable completions of) those Hopf algebras. Of particular interest is the case of post-Lie algebras defined in terms of solutions of modified classical Yang–Baxter equations. In this setting we will study factorization properties of the aforementioned group-like elements.

The performance of an Unsupervised Online feature Selection (UOS) algorithm was investigated for the selection of training features of multispectral images acquired from a dairy product (vanilla cream) stored under isothermal conditions. The selected features were further used as input in a support vector machine (SVM) model with linear kernel for the determination of the microbiological quality of vanilla cream. Model training (n = 65) was based on two batches of cream samples provided directly by the manufacturer and stored at different isothermal conditions (4, 8, 12, and 15 &deg, C), whereas model testing (n = 132) and validation (n = 48) were based on real life conditions by analyzing samples from different retail outlets as well as expired samples from the market. Qualitative analysis was performed for the discrimination of cream samples in two microbiological quality classes based on the values of total viable counts [TVC &le, 2.0 log CFU/g (fresh samples) and TVC &ge, 6.0 log CFU/g (spoiled samples)]. Results exhibited good performance with an overall accuracy of classification for the two classes of 91.7% for model validation. Further on, the model was extended to include the samples in the TVC range 2&ndash, 6 log CFU/g, using 1 log step to define the microbiological quality of classes in order to assess the potential of the model to estimate increasing microbial populations. Results demonstrated that high rates of correct classification could be obtained in the range of 2&ndash, 5 log CFU/g, whereas the percentage of erroneous classification increased in the TVC class (5,6) that was close to the spoilage level of the product. Overall, the results of this study demonstrated that the UOS algorithm in tandem with spectral data acquired from multispectral imaging could be a promising method for real-time assessment of the microbiological quality of vanilla cream samples.

In these notes we review and further explore the Lie enveloping algebra of a post-Lie algebra. From a Hopf algebra point of view, one of the central results, which will be recalled in detail, is the existence of second Hopf algebra structure. By comparing group-like elements in suitable completions of these two Hopf algebras, we derive a particular map which we dub post-Lie Magnus expansion. These results are then considered in the case of Semenov-Tian-Shansky’s double Lie algebra, where a post-Lie algebra is defined in terms of solutions of modified classical Yang–Baxter equation. In this context, we prove a factorization theorem for group-like elements. An explicit exponential solution of the corresponding Lie bracket flow is presented, which is based on the aforementioned post-Lie Magnus expansion.

In these notes we review and further explore the Lie enveloping algebra of a post-Lie algebra. From a Hopf algebra point of view, one of the central results, which will be recalled in detail, is the existence of a second Hopf algebra structure. By comparing group-like elements in suitable completions of these two Hopf algebras, we derive a particular map which we dub post-Lie Magnus expansion. These results are then considered in the case of Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined in terms of solutions of modified classical Yang-Baxter equation. In this context, we prove a factorization theorem for group-like elements. An explicit exponential solution of the corresponding Lie bracket flow is presented, which is based on the aforementioned post-Lie Magnus expansion.

We present a multistage approach to detection and classification of mammographic lesions that is independent of accurate extraction of their contours. The ultimate goal is to discriminate malignant tumors from benign lesions and normal parenchymal tissue in a realistic scenario of lesion candidates automatically detected in mammograms. Local analysis of the Gaussian curvature and of the phase response of multidirectional Gabor filters is performed for identification of suspicious focal areas. The detection of lesions and the classification of malignant tumors are performed in series, respectively, via a differential approach to analysis of the tissue surrounding the candidates and via quantification of nonstationarity and spatial dependence of pixel values within circular and annular regions of interest. A unified 3D free-response receiver operating characteristic framework is applied for global analysis of the two binary categorization problems in series. The system was tested on a total of 2105 full-field digital and screen-film mammograms from three different datasets, including abnormal mammograms with 560 malignant tumors and 639 benign lesions, masses, or architectural distortion, and 1010 normal mammograms. For sensitivity of detection of malignant tumors in the range of 0.70–0.81, the range of falsely detected malignant tumors was 0.82–3.47 per image, with a series of two stages of classification, including stepwise logistic regression for selection of features, Fisher linear discriminant analysis, and two-fold cross-validation.

Humans can communicate their emotions by modulating facial expressions or the tone of their voice. Albeit numerous applications exist that enable machines to read facial emotions and recognize the content of verbal messages, methods for speech emotion recognition are still in their infancy. Yet, fast and reliable applications for emotion recognition are the obvious advancement of present 'intelligent personal assistants', and may have countless applications in diagnostics, rehabilitation and research. Taking inspiration from the dynamics of human group decision-making, we devised a novel speech emotion recognition system that applies, for the first time, a semi-supervised prediction model based on consensus. Three tests were carried out to compare this algorithm with traditional approaches. Labeling performances relative to a public database of spontaneous speeches are reported. The novel system appears to be fast, robust and less computationally demanding than traditional methods, allowing for easier implementation in portable voice-analyzers (as used in rehabilitation, research, industry, etc.) and for applications in the research domain (such as real-time pairing of stimuli to participants' emotional state, selective/differential data collection based on emotional content, etc.).

PurposeThe purpose of this paper is to prove the validity of the front‐tracking variant of the lattice Boltzmann method (LBM) to simulate free surface hydraulic flows (i.e. dam break flows).Design/methodology/approachIn this paper, an algorithm for free surface simulations with the LBM method is presented. The method is chosen for its computational efficiency and ability to deal with complex geometries. The LBM is combined to a surface‐tracking technique applied to a fixed Eulerian mesh in order to simulate free surface flows.FindingsThe numerical method is then validated against two typical cases of environmental‐hydraulic interest (i.e. dam break) by comparing LBM results with experimental data available in literature. The results show that the model is able to reproduce the observed water levels and the wave fronts with reasonable accuracy in the whole period of the transient simulations, thus highlighting that the present method may be a promising tool for practical dam break analyses.Originality/valueEven if the main philosophy of the proposed method is equal to the volume of fluid technique usually coupled to Navier‐Stokes models, no additional differential equation is needed to determine the relative volume fraction of the two phases, or phase fraction, in each computational cell, as the free‐surface tracking is automatically performed. This results in a method very simple to be coded with high computational efficiency. The results presented in this paper are the first, to the best of the authors' knowledge, in the field of hydraulic engineering.

Mammographic images suffer from low contrast and signal dependent noise, and a very small size of tumoral signs is not easily detected, especially for an early diagnosis of breast cancer. In this context, many methods proposed in literature fail for lack of generality. In particular, too weak assumptions on the noise model, e.g., stationary normal additive noise, and an inaccurate choice of the wavelet family that is applied, can lead to an information loss, noise emphasizing, unacceptable enhancement results, or in turn an unwanted distortion of the original image aspect. In this paper, we consider an optimal wavelet thresholding, in the context of Discrete Dyadic Wavelet Transforms, by directly relating all the parameters involved in both denoising and contrast enhancement to signal dependent noise variance (estimated by a robust algorithm) and to the size of cancer signs. Moreover, by performing a reconstruction from a zero-approximation in conjunction with a Gaussian smoothing filter, we are able to extract the background and the foreground of the image separately, as to compute suitable contrast improvement indexes. The whole procedure will be tested on high resolution X-ray mammographic images and compared with other techniques. Anyway, the visual assessment of the results by an expert radiologist will be also considered as a subjective evaluation.

Mammography is the most effective method for the early detection of breast diseases. However, the typical diagnostic signs such as microcalcifications and masses are difficult to detect because mammograms are low-contrast and noisy images. In this paper, a novel algorithm for image denoising and enhancement based on dyadic wavelet processing is proposed. The denoising phase is based on a local iterative noise variance estimation. Moreover, in the case of microcalcifications, we propose an adaptive tuning of enhancement degree at different wavelet scales, whereas in the case of mass detection, we developed a new segmentation method combining dyadic wavelet information with mathematical morphology. The innovative approach consists of using the same algorithmic core for processing images to detect both microcalcifications and masses. The proposed algorithm has been tested on a large number of clinical images, comparing the results with those obtained by several other algorithms proposed in the literature through both analytical indexes and the opinions of radiologists. Through preliminary tests, the method seems to meaningfully improve the diagnosis in the early breast cancer detection with respect to other approaches.

In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical R-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation. publishedVersion

We hypothesize that quantification of structural similarity or dissimilarity between paired mammographic regions can be effective in detecting asymmetric signs of breast cancer. Bilateral masking procedures are applied for this purpose by using automatically detected anatomical landmarks. Changes in structural information of the extracted regions are investigated using spherical semivariogram descriptors and correlation-based structural similarity indices in the spatial and complex wavelet domains. The spatial distribution of grayscale values as well as of the magnitude and phase responses of multidirectional Gabor filters are used to represent the structure of mammographic density and of the directional components of breast tissue patterns, respectively. A total of 188 mammograms from the DDSM and mini-MIAS databases, consisting of 47 asymmetric cases and 47 normal cases, were analyzed. For the combined dataset of mammograms, areas under the receiver operating characteristic curves of 0.83, 0.77, and 0.87 were obtained, respectively, with linear discriminant analysis, the Bayesian classifier, and an artificial neural network with radial basis functions, using the features selected by stepwise logistic regression and leave-one-patient-out cross-validation. Two-view analysis provided accuracy up to 0.94, with sensitivity and specificity of 1 and 0.88, respectively.

The concept ‘sensitivity’ has multiple and sometimes incompatible usages and definitions, as they can be found in the scientific and technical literature. A strategy is proposed toward a conceptual framework in which sensitivity is qualitatively intended as a feature of a black box behavior and quantitatively is defined according to specific evaluation types (interval/ratio, ordinal, nominal) for both deterministic and stochastic behaviors. The proposed formal definitions characterize stochastic sensitivity as constituted of “effective” and “confounding” components, that can be simultaneously present and contribute to a desirable and unwanted increment of global sensitivity respectively. Two examples taken from the context of imaging systems and image-based measuring systems, in which sensitivity is computed in presence of non-negligible uncertainty sources, provide some hints on the usefulness of the proposed framework.

The processing of intensity data from terrestrial laser scanners has attracted considerable attention in recent years. Accurate calibrated intensity could give added value for laser scanning campaigns, for example, in producing faithful 3D colour models of real targets and classifying easier and more reliable automatic tools. In cultural heritage area, the purely geometric information provided by the vast majority of currently available scanners is not enough for most applications, where indeed accurate colorimetric data is needed. This paper presents a remote calibration method for self-registered RGB colour data provided by a 3D tristimulus laser scanner prototype. Such distinguishing colour information opens new scenarios and problems for remote colorimetry. Using piecewise cubic Hermite polynomials, a quadratic model with nonpolynomial terms for reducing inaccuracies occurring in remote colour measurement is implemented. Colorimetric data recorded by the prototype on certified diffusive targets is processed for generating a remote Lambertian model used for assessing the accuracy of the proposed algorithm. Results concerning laser scanner digitizations of artworks are reported to confirm the effectiveness of the method. © 2015 Alessandro Danielis et al.

A fuzzy logic system (FLS) can be completely described through its behavioral function. If the analytical expression of this function is known, then the system synthesis consists in the realization of a structure implementing it in the most convenient way. So, the problem of the FLS design can be considered as a function approximation problem. Many methods have been proposed in literature to minimize a certain error parameter. Nevertheless, sometimes the specifications of the problem need some further analytical properties of the approximating function. This paper analyzes multiple-input-single-output (MISO) FLSs with polynomial membership functions inspecting the case of two input variables. In particular, it will be shown what characteristics such systems should have in order to respect the constraints imposed on the derivatives of the implemented function, such as their continuity up to a desired order and even their value in the grid points of the input space. Furthermore, we prove that these systems can uniformly approximate, under certain conditions, any multivariate function with a desired approximation accuracy both on the target function and its partial derivatives. Proper illustrative examples will show the behavior of these FLSs comparing their performance with other methods proposed in literature.

We prove that insertion-elimination Lie algebra of Feynman graphs, in the ladder case, has a natural interpretation in terms of a certain algebra of infinite dimensional matrices. We study some aspects of its representation theory and we discuss some relations with the representation of the Heisenberg algebra, Comment: LaTex, 17 pages, typos corrected, to appear in LMP

We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n,R). The relation with the Lax formalism is also discussed., 19 pages, no figures, minor changes and remarks added

We present a novel method to detect asymmetry in mammograms based upon bilateral analysis of the spatial distribution of density within paired mammographic strips. Various differential measures of spatial correlation of gray-scale values were computed with reference to the position of the nipple for a set of 128 pairs of mammograms from the Digital Database for Screening Mammography (DDSM). Features were selected by stepwise logistic regression and the leave-one-patient-out method was used for cross-validation of results. An area under the receiver operating characteristic curve of 0.87 (SE = 0.08) was achieved by using an artificial neural network classifier with radial basis functions.

Given a (smooth) action of a Lie group G on Rd we construct a DGA whose Maurer-Cartan elements are in one to one correspondence with some class of defomations of the (induced) G-action on the ring of formal power series with coefficients in the ring of smooth functions on Rd. In the final part of this note we discuss the cohomological obstructions to the existence and to the uniqueness (in a sense to be clarified) of such deformations., 11 pages, accepted for publication in the Journal of Mathematical Physics. This submission supersedes arXiv:1202.0886

In this note we give an explicit formula for a family of deformation quantizations for the momentum map associated with the cotangent lift of a Lie group action on Rd. This family of deformations is parametrized by the formal G-systems introduced in B. Dherin and I. Mencattini [“G-systems and deformations of G-actions on Rd,” J. Math. Phys. 55, 011702 (2014)] (see also B. Dherin and I. Mencattini [“Quantization of (volume-preserving) actions on Rd,” e-print arXiv:1202.0886]) and allows us to obtain classical invariant Hamiltonians that quantize without anomalies with respect to the quantizations of the action prescribed by the formal G-systems.

Segmentation of the breast region is a fundamental step in any system for computerized analysis of mammograms. In this work, we propose a novel procedure for the estimation of the breast skin-line based upon multidirectional Gabor filtering. The method includes an adaptive values-of-interest (VOI) transformation, extraction of the skin-air ribbon by Otsu's thresholding method and the Euclidean distance transform, Gabor filtering with 18 real kernels, and a step for suppression of false edge points using the magnitude and phase responses of the filters. On a test set of 361 images from different acquisition modalities (screen-film and full-field digital mammograms), the average Hausdorff and polyline distances obtained were 2.85 mm and 0.84 mm, respectively, with reference to the ground-truth boundaries provided by an expert radiologist. When compared with the results obtained by other state-of-the-art methods on the same set of images and with respect to the same ground-truth boundaries, our method mostly outperformed the other approaches. The results demonstrate the effectiveness and robustness of the proposed algorithm.

We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the Hochschild-Serre spectral sequence for Lie algebras to the holomorphic Lie algebroid setting. As an application, we show that the hypercohomology of the Atiyah algebroid of a line bundle has a natural Hodge structure., 22 pages. v2: 26 pages. The material has been reorganized and the exposition improved. Slightly modified title. v3: Typos corrected, exposition streamlined, introduction rewritten

We show that there exists a morphism between a group $\Gamma^{\mathrm{alg}}$ introduced by G. Wilson and a quotient of the group of tame symplectic automorphisms of the path algebra of a quiver introduced by Bielawski and Pidstrygach. The latter is known to act transitively on the phase space $\mathcal{C}_{n,2}$ of the Gibbons-Hermsen integrable system of rank 2, and we prove that the subgroup generated by the image of $\Gamma^{\mathrm{alg}}$ together with a particular tame symplectic automorphism has the property that, for every pair of points of the regular and semisimple locus of $\mathcal{C}_{n,2}$, the subgroup contains an element sending the first point to the second.

We propose a method to analyze bilateral asymmetry in mammograms based upon systematic comparison of paired mammographic strips of the right and left breasts of a patient. Similarity between corresponding directional structures was quantified by applying measures of similarity to the masked strips. A novel application of Moran's index was designed to measure the angular covariance between rose diagrams related to the phase and magnitude responses of multidirectional Gabor filters. A set of 128 mammograms from the DDSM database, including 32 normal and 32 asymmetric pairs, was used to validate the procedure. The leave-one-patient-out method was used for cross-validation of the results. The best result, with an area under the receiver operating characteristic curve of 0.8435, was achieved using similarity measures on craniocaudal views and Fisher-linear discriminant analysis. The results indicate that the proposed techniques can be applied for computer-aided detection of bilateral asymmetry.

The paper emphasizes the importance that fundamental concepts in measurement science are defined according to a structured strategy, which provides both a general, qualitative characterization and a specific, type-related, quantitative definition. As a significant case, the concept 'sensitivity' is discussed and a definition for it proposed.

In this paper, a novel approach for classification of breast masses is presented that quantifies the texture of masses without relying on accurate extraction of their contours. Two novel feature descriptors based on 2D extensions of the reverse arrangement (RA) and Mantel's tests were designed for this purpose. Measures of radial correlation and radial trend were extracted from the original gray-scale values as well as from the Gabor magnitude response of 146 regions of interest, including 120 benign masses and 26 malignant tumors. Four classifiers, Fisher-linear discriminant analysis, Bayesian, support vector machine, and an artificial neural network based on radial basis functions (ANN-RBF), were employed to predict the diagnosis, using stepwise logistic regression for feature selection and the leave-one-patient-out method for cross-validation. The ANN-RBF resulted in an area under the receiver operating characteristic curve of 0.93. The experimental results show the effectiveness of the proposed approach.

Automatic tools for detection and identification of lung and lesion from high-resolution CT (HRCT) are becoming increasingly important both for diagnosis and for delivering high-precision radiation therapy. However, development of robust and interpretable classifiers still presents a challenge especially in case of non-small cell lung carcinoma (NSCLC) patients. In this paper, we have attempted to devise such a classifier by extracting fuzzy rules from texture segmented regions from HRCT images of NSCLC patients. A fuzzy inference system (FIS) has been constructed starting from a feature extraction procedure applied on overlapping regions from the same organs and deriving simple if–then rules so that more linguistically interpretable decisions can be implemented. The proposed method has been tested on 138 regions extracted from CT scan images acquired from patients with lung cancer. Assuming two classes of tissues C1 (healthy tissues) and C2 (lesion) as negative and positive, respectively; preliminary results report an AUC = 0.98 for lesions and AUC = 0.93 for healthy tissue, with an optimal operating condition related to sensitivity = 0.96, and specificity = 0.98 for lesions and sensitivity 0.99, and specificity = 0.94 for healthy tissue. Finally, the following results have been obtained: false-negative rate (FNR) = 6 % (C1), FNR = 2 % (C2), false-positive rate (FPR) = 4 % (C1), FPR = 3 % (C2), true-positive rate (TPR) = 94 %, (C1) and TPR = 98 % (C2).

In the recent years, fuzzy variables (FVs) and random-fuzzy variables (RFVs) have been proposed to represent the measurement results with their associated uncertainty. However, up to now, the different authors do not yet agree in the mathematical way FVs should be composed together, so different approaches have been proposed. This paper compares these approaches to find their advantages and disadvantages and shows a new proposal that is supposed to hopefully overcome the disadvantages of the original approaches.

Given a special Kahler manifold M , we give a new, direct proof of the relationship between the quaternionic structure on T ∗ M and the variation of Hodge structures on T C M .

In the recent years, fuzzy variables and random-fuzzy variables have been proposed to represent the measurement results with their associated uncertainty. However, up to now, the different authors do not yet agree in the mathematical way fuzzy variables should be composed together, so that different approaches have been proposed. This paper compares these approaches, in order to find their advantages and disadvantages and shows a new proposal, that is supposed to overcome, hopefully, the disadvantages of the original ones.

The paper deals with noise power variation that occurs when Discrete Dyadic Wavelet Transform (DDWT) is applied to signals affected by Wide Sense Stationary (WSS) additive white noise owing to the use of a non orthonormal expansion. An exact relationship between the noise variance in the original signal and the noise variance in the wavelet coefficients at a generic level is derived. This relationship is crucial in the application of wavelet thresholding for signal denoising to properly select the threshold in each subband.

The problem of uncertainty representation and propagation in the context of statistically correlated variables is commonly addressed by means of Monte Carlo simulation as recommended in IEC-ISO Guide. Moreover, in a recent literature, fuzzy sets have proved to be a valid alternative in the case of independent variables. Unfortunately, the problem of modelling statistically correlated variables, by means of fuzzy sets, is still an open problem. Since it is well known that T-norms are the natural way of combining fuzzy variables into a nonfuzzy function f, in this paper, we investigate how to generalize the class of T-norms, making it dependent from correlation coefficient in order to emulate different statistical correlation degree among variables. In order to validate the model a comparison with central limit theorem will be accomplished in the case of zero correlation while a practical example will be provided in order to compare the proposed method with Montecarlo simulation and with that obtained by uncertainty propagation described in IEC-ISO Guide.

Uncertainty modelling in measurement represents a crucial task since the final result of a measurement process cannot be expressed by a single value, but by a distribution of values over an interval within which the measurements lie with a given confidence level. A classical Type-1 fuzzy set could be the natural choice for uncertainty model, since a Membership Function (MF) intrinsically embeds the concepts of confidence interval and confidence level. Moreover, from computational aspects, working on fuzzy sets is much more easily than a Montecarlo simulation, that is actually the recommended approach. However, both the approaches need reliable and specific assumptions on the probability distribution of the input variables. So, in many practical cases, a Type-2 fuzzy set is needed in order to overcome this problem. In this paper, we will provide a full description of how a Type-2 MF can be built in order to model the uncertainty of a variable and how to evaluate uncertainty propagation through a generic function. A practical example of this representation will be also provided and compared with a Montecarlo simulation.

The scientific community and the standardization world are exploring the concepts of nominal evaluation and nominal property in the metrological context. In this paper we show that measurement can be framed in a structural context that is independent of the algebraic structure assumed for the measurand and the measured values. In such a framework the basic metrological concepts are consistently applied also to the nominal case.

We continue our investigation into the insertion-elimination Lie algebra of Feynman graphs in the ladder case, emphasizing the structure of this Lie algebra relevant for future applications in the study of Dyson-Schwinger equations. We work out the relation of this Lie algebra to some classical infinite dimensional Lie algebra and we determine its cohomology., 24 pages, LaTex, typos corrected

We prove that an integrable system over a symplectic manifold, whose symplectic form is covariantly constant w.r.t. the Gauss-Manin connection, carries a natural hyper-symplectic structure. Moreover, a special Kaehler structure is induced on the base manifold., LaTeX file, 7 pages; to be published in Journal of Geometry and Physics

Noise estimation is an important issue in image processing because it is a fundamental step in many algorithms for noise suppression and then for image restoration. In the literature, many approaches have been presented in order to obtain good results. This paper presents a novel method suitable for obtaining a good estimation if the type of noise distribution is known. In particular, the algorithm provides the variance of the noise distribution and the proof that the distribution itself matches the foreseen one. The algorithm has been tested on different images affected by Gaussian noise. The simulations show results better than those obtained with other approaches.

In this thesis we propose a description of the operad defining post-Lie algebras in terms of rooted trees and we discuss some applications of such a construction. In particular, we re-derive both the free post-Lie algebra defined in [22] and the main result of the paper [8]. Furthermore, a possible extension of the concept of symmetric brace algebra to the category of the post-Lie algebras is proposed. Nessa tese propomos a descrição da operad que define as álgebras pós-Lie em termos de árvores enraizadas e discutimos algumas aplicações dessa construção. Em particular, nós obtemos novamente a álgebra pós-Lie livre definida em [22] e o resultado principal do artigo [8]. Além disso, uma possível extensão do conceito de álgebra brace simétrica à categoria de álgebras pós-Lie é apresentada.

A quantização geométrica e um método desenvolvido para prover uma construção geométrica que relacione a mecânica clássica com a quântica. O primeiro passo consiste em apresentar uma forma simplética, \'omega\'!, sobre uma variedade simplética, M, como a forma curvatura da conexão abla de um brado linear, L, sobre M. As funções sobre M operam como seções de L. Mas o espaço de todas as seções é grande demais. Queremos considerar seções constantes em certa direção, com respeito a derivada covariante dada por abla, e para isso precisamos o conceito de polarizações, essas seções são chamadas de seções polarizadas. Para obter uma estrutura de espaco de Hilbert nestas seções, precisamos de certos objetos chamados de meias densidades. Além disso, também temos um empareamento sesquilinear entre seções de polarizações diferentes. Neste trabalho, primeiramente consideraremos o empareamento para seções polarizadas adaptadas a polarizações reais não transversais, como método para obter aplicações integrais entre estes espaços de Hilbert que em combinação com a convolução do par grupóide M x \' M BARRA\', pode definir um produto integral de funções definidas na variedade simplética. Este produto, no caso do plano euclidiano e do plano de Bieliavsky, coincide com produto de Weyl integral e o produto de Bieliavsky, respectivamente. Jáa no caso do plano hiperbólico, este tipo de polarizações reais não são transversais nem são não transversais, dessa forma, escolhemos o empareamento entre uma polarização real e uma polarização holomorfa do par grupóide, as quais são transversais, para obter um produto integral no plano hiperbólico, que no caso do plano euclidiano e o produto de Weyl The geometric quantization is a method developed to provide a geometrical construction relating classical to quantum mechanics. The first step consists of realizing the symplectic form, \'omega\', on a symplectic manifold, M, as the curvature form of a line bundle, L, over M. The functions on M then operate as sections of L. However, the space of all sections of L is too large. One wants to consider sections which are constant in certain directions (polarized sections) and for that one needs to introduce the concept of a polarization. To get a Hilbert space structure on the polarized sections, one needs to consider objects known as half densities. In this work, first we consider a sesquilinear pairing between objects associated to certain different polarizations, which are nontransverse real polarizations, to obtain integral applications between their associated Hilbert spaces, and to use the convolution of the pair groupoid M x \' M BARRA\' to obtain an integral product of functions on M. In the euclidian plane case, we recover the integral Weyl product and, in the Bieliavsky plane case, we obtain the Bieliavsky product. On the other hand, for the hyperbolic plane, such real polarizations are neither transverse nor nontransverse, so we use the pairing between a real polarization and a holomorphic polarization, which are transverse polarizations on the pair groupoid, to obtain an integral product of functions on the hyperbolic plane. This same procedure, in the euclidian plane case, also produces the integral Weyl product