Periodic points of holomorphic maps via Lefschetz numbers

In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension n and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of CP(n) of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker's theorem.

Faire la ville juste : une analyse "in itinere" de la maîtrise publique d'ouvrage du projet urbain Carré de Soie (métropole lyonnaise)

Dans certaines portions des agglomérations (poches de pauvreté de centre-ville, couronnes suburbaines dégradées, espaces périurbains sans aménité), un cumul entre des inégalités sociales (pauvreté, chômage, etc.) et environnementales (exposition au bruit, aux risques industriels, etc.) peut être observé. La persistance de ces inégalités croisées dans le temps indique une tendance de fond : la capacité d'accéder à un cadre de vie de qualité n'est pas équitablement partagée parmi les individus. Ce constat interroge : comment se créent ces inégalités ? Comment infléchir cette tendance et faire la ville plus juste ?¦Apporter des réponses à cette problématique nécessite d'identifier les facteurs de causalités qui entrent en jeu dans le système de (re)production des inégalités urbaines. Le fonctionnement des marchés foncier et immobilier, la « tyrannie des petites décisions » et les politiques publiques à incidence spatiale sont principalement impliqués. Ces dernières, agissant sur tous les éléments du système, sont placées au coeur de ce travail. On va ainsi s'intéresser précisément à la manière dont les collectivités publiques pilotent la production de la ville contemporaine, en portant l'attention sur la maîtrise publique d'ouvrage (MPO) des grands projets urbains.¦Poser la question de la justice dans la fabrique de la ville implique également de questionner les référentiels normatifs de l'action publique : à quelle conception de la justice celle-ci doit- elle obéir? Quatre perspectives (radicale, substantialiste, procédurale et intégrative) sont caractérisées, chacune se traduisant par des principes d'action différenciés. Une méthodologie hybride - empruntant à la sociologie des organisations et à l'analyse des politiques publiques - vient clore le volet théorique, proposant par un détour métaphorique d'appréhender le projet urbain comme une pièce de théâtre dont le déroulement dépend du jeu d'acteurs.¦Cette méthodologie est utilisée dans le volet empirique de la recherche, qui consiste en une analyse de la MPO d'un projet urbain en cours dans la première couronne de l'agglomération lyonnaise : le Carré de Soie. Trois grands objectifs sont poursuivis : descriptif (reconstruire le scénario), analytique (évaluer la nature de la pièce : conte de fée, tragédie ou match d'improvisation ?) et prescriptif (tirer la morale de l'histoire). La description de la MPO montre le déploiement successif de quatre stratégies de pilotage, dont les implications sur les temporalités, le contenu du projet (programmes, morphologies) et les financements publics vont être déterminantes. Sur la base de l'analyse, plusieurs recommandations peuvent être formulées - importance de l'anticipation et de l'articulation entre planification et stratégie foncière notamment - pour permettre à la sphère publique de dominer le jeu et d'assurer la production de justice par le projet urbain (réalisation puis entretien des équipements et espaces publics, financement de logements de qualité à destination d'un large éventail de populations, etc.). Plus généralement, un décalage problématique peut être souligné entre les territoires stratégiques pour le développement de l'agglomération et les capacités de portage limitées des communes concernées. Ce déficit plaide pour le renforcement des capacités d'investissement de la structure intercommunale.¦La seule logique du marché (foncier, immobilier) mène à la polarisation sociale et à la production d'inégalités urbaines. Faire la ville juste nécessite une forte volonté des collectivités publiques, laquelle doit se traduire aussi bien dans l'ambition affichée - une juste hiérarchisation des priorités dans le développement urbain - que dans son opérationnalisation - une juste maîtrise publique d'ouvrage des projets urbains.¦Inner-city neighborhoods, poor outskirts, and peri-urban spaces with no amenities usually suffer from social and environmental inequalities, such as poverty, unemployment, and exposure to noise and industrial hazards. The observed persistence of these inequalities over time points to an underlying trend - namely, that access to proper living conditions is fundamentally unequal, thus eliciting the question of how such inequalities are effected and how this trend can be reversed so as to build a more equitable city.¦Providing answers to such questions requires that the causal factors at play within the system of (re)production of urban inequalities be identified. Real estate markets, "micromotives and macrobehavior", and public policies that bear on space are mostly involved. The latter are central in that they act on all the elements of the system. This thesis therefore focuses on the way public authorities shape the production of contemporary cities, by studying the public project ownership of major urban projects.¦The study of justice within the urban fabric also implies that the normative frames of reference of public action be questioned: what conception of justice should public action refer to? This thesis examines four perspectives (radical, substantialist, procedural, and integrative) each of which results in different principles of action. This theoretical part is concluded by a hybrid methodology that draws from sociology of organizations and public policy analysis and that suggests that the urban project may be understood as a play, whose outcome hinges on the actors' acting.¦This methodology is applied to the empirical analysis of the public project ownership of an ongoing urban project in the Lyon first-ring suburbs: the Carré de Soie. Three main objectives are pursued: descriptive (reconstructing the scenario), analytical (assessing the nature of the play - fairy tale, tragedy or improvisation match), and prescriptive (drawing the moral of the story). The description of the public project ownership shows the successive deployment of four control strategies, whose implications on deadlines, project content (programs, morphologies), and public funding are significant. Building on the analysis, several recommendations can be made to allow the public sphere to control the process and ensure the urban project produces equity (most notably, anticipation and articulation of planning and real- estate strategy, as well as provision and maintenance of equipment and public spaces, funding of quality housing for a wide range of populations, etc.). More generally, a gap can be highlighted between those territories that are strategic to the development of the agglomeration and the limited resources of the municipalities involved. This deficit calls for strengthening the investment abilities of the intermunicipal structure.¦By itself, the real-estate market logic brings about social polarization and urban inequalities. Building an equitable city requires a strong will on the part of public authorities, a will that must be reflected both in the stated ambition - setting priorities of urban development equitably - and in its implementation managing urban public projects fairly.

Monnaie, citoyenneté et globalisation financière. Les citoyens entre protection sociale et promesses de la capitalisation boursière

La monnaie a été étudiée par des économistes hétérodoxes, des sociologues et des historiens qui ont souligné ses rapports à l'ordre collectif, mais elle n'est que rarement analysée sous l'angle de la citoyenneté. Notre thèse propose une réflexion théorique sur quatre types de fonctions (politique, symbolique, socioéconomique et psychoaffective) qui permettent à la monnaie de jouer un rôle de médiation de la citoyenneté. A partir d'une perspective qui combine les apports de l'économie politique internationale et de l'école de la régulation, nous montrons que cette médiation ne mobilise pas seulement des mécanismes sociopolitiques nationaux, mais aussi des mécanismes internationaux qui rétroagissent sur la sphère domestique des États et affectent leur capacité à définir leur régime de citoyenneté. Cette relation est analysée dans le contexte de l'institutionnalisation du système monétaire international de Bretton Woods (1944) et du développement de la globalisation financière depuis les années 1970. Si la monnaie a été mise au service d'un principe de protection des droits sociaux des citoyens contre les pressions financières extérieures après la Seconde guerre mondiale, elle contribue aujourd'hui à l'ouverture de la sphère domestique des Etats aux flux de capitaux transnationaux et à la création d'un ordre politique et juridique favorable aux droits des investisseurs. Cette dynamique est impulsée par l'essor de nouveaux intermédiaires financiers (notamment les agences de notation et les investisseurs institutionnels) et l'émergence concomitante d'une nouvelle forme d'Etat légitimée à partir d'un discours politique néolibéral insistant sur la quête de compétitivité, la réduction de la protection sociale et la responsabilisation individuelle. Elle se traduit par la privatisation des régimes de retraite et le développement des politiques d'éducation financière qui incitent les citoyens à se comporter en « preneurs de risques » actifs et responsables, assurant eux-mêmes leur sécurité économique à travers le placement de leur épargne retraite sur les marchés financiers. Nous soulignons toutefois les difficultés institutionnelles, cognitives et socioéconomiques qui rendent cette transformation de la citoyenneté contradictoire et problématique. Money has been studied by heterodox economists, sociologists and historians who stressed its relationship to collective order. However, it has hardly been analysed from the viewpoint of its relationship to citizenship. We propose a theoretical account of four types of functions (political, symbolic, socioeconomic and psychoaffective) enabling money to operate as a mediation of citizenship. From a perspective that combines the contributions of international political economy and the regulation school, we show that this mediation mobilises not only national sociopolitical mechanisms, but also international mechanisms which feed back on the domestic sphere of states and affect their capacity to define their regime of citizenship. This relationship is analysed in the context of the institutionalisation of the international monetary system of Bretton Woods (1944) and the development of financial globalization since the 1970s. If money has served to protect the social rights of citizens against external financial pressures after the Second World War, today it contributes to the opening of the domestic sphere of states to transnational capital flows and to the creation of a political and legal order favorable to the rights of investors. This dynamic is driven by the rise of new financial intermediaries (in particular rating agencies and institutional investisors) and the simultaneous emergence of a new form of state legitimized from a neoliberal political discourse emphasizing the quest for competitiveness, reduced social protection and individual responsibilization. It results in the privatization of pension systems and the development of policies of financial education that encourage citizens to behave as active and responsible « risk takers », ensuring their own economic security through the investment of their savings retirement on financial markets. However, we emphasize the institutional, cognitive and socioeconomic difficulties that make this transformation of citizenship contradictory and problematic. - Money has been studied by heterodox economists, sociologists and historians who stressed its relationship to collective order. However, it has hardly been analysed from the viewpoint of its relationship to citizenship. We propose a theoretical account of four types of functions (political, symbolic, socioeconomic and psychoaffective) enabling money to operate as a mediation of citizenship. From a perspective that combines the contributions of international political economy and the regulation school, we show that this mediation mobilises not only national sociopolitical mechanisms, but also international mechanisms which feed back on the domestic sphere of states and affect their capacity to define their regime of citizenship. This relationship is analysed in the context of the institutionalisation of the international monetary system of Bretton Woods (1944) and the development of financial globalization since the 1970s. If money has served to protect the social rights of citizens against external financial pressures after the Second World War, today it contributes to the opening of the domestic sphere of states to transnational capital flows and to the creation of a political and legal order favorable to the rights of investors. This dynamic is driven by the rise of new financial intermediaries (in particular rating agencies and institutional investisors) and the simultaneous emergence of a new form of state legitimized from a neoliberal political discourse emphasizing the quest for competitiveness, reduced social protection and individual responsibilization. It results in the privatization of pension systems and the development of policies of financial education that encourage citizens to behave as active and responsible « risk takers », ensuring their own economic security through the investment of their savings retirement on financial markets. However, we emphasize the institutional, cognitive and socioeconomic difficulties that make this transformation of citizenship problematic.

A model of cellular dosimetry for macroscopic tumors in radiopharmaceutical therapy.

PURPOSE: In the radiopharmaceutical therapy approach to the fight against cancer, in particular when it comes to translating laboratory results to the clinical setting, modeling has served as an invaluable tool for guidance and for understanding the processes operating at the cellular level and how these relate to macroscopic observables. Tumor control probability (TCP) is the dosimetric end point quantity of choice which relates to experimental and clinical data: it requires knowledge of individual cellular absorbed doses since it depends on the assessment of the treatment's ability to kill each and every cell. Macroscopic tumors, seen in both clinical and experimental studies, contain too many cells to be modeled individually in Monte Carlo simulation; yet, in particular for low ratios of decays to cells, a cell-based model that does not smooth away statistical considerations associated with low activity is a necessity. The authors present here an adaptation of the simple sphere-based model from which cellular level dosimetry for macroscopic tumors and their end point quantities, such as TCP, may be extrapolated more reliably. METHODS: Ten homogenous spheres representing tumors of different sizes were constructed in GEANT4. The radionuclide 131I was randomly allowed to decay for each model size and for seven different ratios of number of decays to number of cells, N(r): 1000, 500, 200, 100, 50, 20, and 10 decays per cell. The deposited energy was collected in radial bins and divided by the bin mass to obtain the average bin absorbed dose. To simulate a cellular model, the number of cells present in each bin was calculated and an absorbed dose attributed to each cell equal to the bin average absorbed dose with a randomly determined adjustment based on a Gaussian probability distribution with a width equal to the statistical uncertainty consistent with the ratio of decays to cells, i.e., equal to Nr-1/2. From dose volume histograms the surviving fraction of cells, equivalent uniform dose (EUD), and TCP for the different scenarios were calculated. Comparably sized spherical models containing individual spherical cells (15 microm diameter) in hexagonal lattices were constructed, and Monte Carlo simulations were executed for all the same previous scenarios. The dosimetric quantities were calculated and compared to the adjusted simple sphere model results. The model was then applied to the Bortezomib-induced enzyme-targeted radiotherapy (BETR) strategy of targeting Epstein-Barr virus (EBV)-expressing cancers. RESULTS: The TCP values were comparable to within 2% between the adjusted simple sphere and full cellular models. Additionally, models were generated for a nonuniform distribution of activity, and results were compared between the adjusted spherical and cellular models with similar comparability. The TCP values from the experimental macroscopic tumor results were consistent with the experimental observations for BETR-treated 1 g EBV-expressing lymphoma tumors in mice. CONCLUSIONS: The adjusted spherical model presented here provides more accurate TCP values than simple spheres, on par with full cellular Monte Carlo simulations while maintaining the simplicity of the simple sphere model. This model provides a basis for complementing and understanding laboratory and clinical results pertaining to radiopharmaceutical therapy.

La financiación de la Renta Básica, un modelo de microsimulación

[spa] La financiación es uno de los aspectos de la Renta Básica en que más se ha avanzado en los últimos 10 o 12 años. Por Renta Básica aquí se entenderá en todo momento un ingreso pagado por el estado a cada miembro de pleno derecho de la sociedad o residente, incluso si no quiere trabajar de forma remunerada, sin tomar en consideración si es rico o pobre, o dicho de otra forma, independientemente de cuáles puedan ser las otras posibles fuentes de renta, y sin importar con quien conviva. Diversas e interesantes investigaciones para ámbitos geográficos distintos han hecho acto de aparición y han alimentado un vivo debate acerca de las formas de financiar la Renta Básica. La Renta Básica puede ser pagada por parte de distintas instituciones públicas. La propuesta de financiación de la Renta Básica a la que se refiere este artículo consiste en una reforma en profundidad del actual Impuesto de la Renta de las Personas Físicas (IRPF). El estudio opta por este camino porque se ha tenido acceso a datos individualizados del IRPF de Cataluña, pero también porque este impuesto es especialmente útil para apreciar, por ejemplo, la redistribución resultante de la renta. Está basado en un programa de microsimulación específicamente diseñado para este objetivo, aplicado sobre una base de datos provenientes de una muestra de 110.474 declaraciones del IRPF de Cataluña, para evaluar diferentes opciones de políticas de integración de impuestos y prestaciones que incluyan una Renta Básica. Este estudio muestra que la reforma propuesta es viable en términos económicos y que el impacto en la distribución de la renta sería muy progresivo, como después se especificará.

EADock: docking of small molecules into protein active sites with a multiobjective evolutionary optimization.

In recent years, protein-ligand docking has become a powerful tool for drug development. Although several approaches suitable for high throughput screening are available, there is a need for methods able to identify binding modes with high accuracy. This accuracy is essential to reliably compute the binding free energy of the ligand. Such methods are needed when the binding mode of lead compounds is not determined experimentally but is needed for structure-based lead optimization. We present here a new docking software, called EADock, that aims at this goal. It uses an hybrid evolutionary algorithm with two fitness functions, in combination with a sophisticated management of the diversity. EADock is interfaced with the CHARMM package for energy calculations and coordinate handling. A validation was carried out on 37 crystallized protein-ligand complexes featuring 11 different proteins. The search space was defined as a sphere of 15 A around the center of mass of the ligand position in the crystal structure, and on the contrary to other benchmarks, our algorithm was fed with optimized ligand positions up to 10 A root mean square deviation (RMSD) from the crystal structure, excluding the latter. This validation illustrates the efficiency of our sampling strategy, as correct binding modes, defined by a RMSD to the crystal structure lower than 2 A, were identified and ranked first for 68% of the complexes. The success rate increases to 78% when considering the five best ranked clusters, and 92% when all clusters present in the last generation are taken into account. Most failures could be explained by the presence of crystal contacts in the experimental structure. Finally, the ability of EADock to accurately predict binding modes on a real application was illustrated by the successful docking of the RGD cyclic pentapeptide on the alphaVbeta3 integrin, starting far away from the binding pocket.

Linear random knots and their scaling behavior

We present here a nonbiased probabilistic method that allows us to consistently analyze knottedness of linear random walks with up to several hundred noncorrelated steps. The method consists of analyzing the spectrum of knots formed by multiple closures of the same open walk through random points on a sphere enclosing the walk. Knottedness of individual "frozen" configurations of linear chains is therefore defined by a characteristic spectrum of realizable knots. We show that in the great majority of cases this method clearly defines the dominant knot type of a walk, i.e., the strongest component of the spectrum. In such cases, direct end-to-end closure creates a knot that usually coincides with the knot type that dominates the random closure spectrum. Interestingly, in a very small proportion of linear random walks, the knot type is not clearly defined. Such walks can be considered as residing in a border zone of the configuration space of two or more knot types. We also characterize the scaling behavior of linear random knots.

Standardizing the diagnosis of inhalation injury using a descriptive score based on mucosal injury criteria.

INTRODUCTION: Inhalation injury is an important determinant of outcome in patients with major burns. However the diagnostic criteria remain imprecise, preventing objective comparisons of published data. The aims were to evaluate the utility of an inhalation score based on mucosal injury, while assessing separately the oro-pharyngeal sphere (ENT) and tracheobronchial tree (TB) in patients admitted to the ICU with a suspicion of inhalation injury. METHODS: Prospective observational study in 100 patients admitted with suspicion of inhalation injury among 168 consecutive burn admissions to the ICU of a university hospital. Inclusion criteria, endoscopic airway assessment during the first hours. ENT/TB lesion grading was 1: oedema, hyperemia, hypersecretion, 2: bullous mucosal detachment, erosion, exudates, 3: profound ulcers, necrosis. RESULTS: Of the 100 patients (age 42±17 years, burns 23±19%BSA), 79 presented an ENT inhalation injury ≥ENT1 (soot present in 24%): 36 had a tracheobronchial extension, 33 having a grade ≥TB1. Burned vibrissae: 10 patients "without" suffered ENT injury, while 6 patients "with" had no further lesions. Length of mechanical ventilation was strongly associated with the first 24 hrs' fluid resuscitation volume (p<0.0001) and the presence of inhalation injury (p=0.03), while the ICU length of stay was correlated with the %BSA. Soot was associated with prolonged mechanical ventilation (p=0.0115). There was no extubation failure. CONCLUSIONS: The developed inhalation score was simple to use, providing a unified language, and drawing attention to upper airway involvement. Burned vibrissae and suspected history proved to be insufficient diagnostic criteria. Further studies are required to validate the score in a larger population.

Some common indexes of group diversity: upper boundaries

Workgroup diversity can be conceptualized as variety, separation, or disparity. Thus, the proper operationalization of diversity depends on how a diversity dimension has been defined. Analytically, the minimal diversity must be obtained when there are no differences on an attribute among the members of a group, however maximal diversity has a different shape for each conceptualization of diversity. Previous work on diversity indexes indicated maximum values for variety (e.g., Blau"s index and Teachman"s index), separation (e.g., standard deviation and mean Euclidean distance), and disparity (e.g., coefficient of variation and the Gini coefficient of concentration), although these maximum values are not valid for all group characteristics (i.e., group size and group size parity) and attribute scales (i.e., number of categories). We demonstrate analytically appropriate upper boundaries for conditional diversity determined by some specific group characteristics, avoiding the bias related to absolute diversity. This will allow applied researchers to make better interpretations regarding the relationship between group diversity and group outcomes.

Tire traces: discrimination and classification of pyrolysis-GC/MS profiles

Tire traces can be observed on several crime scenes as vehicles are often used by criminals. The tread abrasion on the road, while braking or skidding, leads to the production of small rubber particles which can be collected for comparison purposes. This research focused on the statistical comparison of Py-GC/MS profiles of tire traces and tire treads. The optimisation of the analytical method was carried out using experimental designs. The aim was to determine the best pyrolysis parameters regarding the repeatability of the results. Thus, the pyrolysis factor effect could also be calculated. The pyrolysis temperature was found to be five time more important than time. Finally, a pyrolysis at 650 °C during 15 s was selected. Ten tires of different manufacturers and models were used for this study. Several samples were collected on each tire, and several replicates were carried out to study the variability within each tire (intravariability). More than eighty compounds were integrated for each analysis and the variability study showed that more than 75% presented a relative standard deviation (RSD) below 5% for the ten tires, thus supporting a low intravariability. The variability between the ten tires (intervariability) presented higher values and the ten most variant compounds had a RSD value above 13%, supporting their high potential of discrimination between the tires tested. Principal Component Analysis (PCA) was able to fully discriminate the ten tires with the help of the first three principal components. The ten tires were finally used to perform braking tests on a racetrack with a vehicle equipped with an anti-lock braking system. The resulting tire traces were adequately collected using sheets of white gelatine. As for tires, the intravariability for the traces was found to be lower than the intervariability. Clustering methods were carried out and the Ward's method based on the squared Euclidean distance was able to correctly group all of the tire traces replicates in the same cluster than the replicates of their corresponding tire. Blind tests on traces were performed and were correctly assigned to their tire source. These results support the hypothesis that the tested tires, of different manufacturers and models, can be discriminated by a statistical comparison of their chemical profiles. The traces were found to be not differentiable from their source but differentiable from all the other tires present in the subset. The results are promising and will be extended on a larger sample set.

Different likelihood ratio approaches to evaluate the strength of evidence of MDMA tablet comparisons

Two likelihood ratio (LR) approaches are presented to evaluate the strength of evidence of MDMA tablet comparisons. The first one is based on a more 'traditional' comparison of MDMA tablets by using distance measures (e.g., Pearson correlation distance or a Euclidean distance). In this approach, LRs are calculated using the distribution of distances between tablets of the same-batch and that of different-batches. The second approach is based on methods used in some other fields of forensic comparison. Here LRs are calculated based on the distribution of values of MDMA tablet characteristics within a specific batch and from all batches. The data used in this paper must be seen as examples to illustrate both methods. In future research the methods can be applied to other and more complex data. In this paper, the methods and their results are discussed, considering their performance in evidence evaluation and several practical aspects. With respect to evidence in favor of the correct hypothesis, the second method proved to be better than the first one. It is shown that the LRs in same-batch comparisons are generally higher compared to the first method and the LRs in different-batch comparisons are generally lower. On the other hand, for operational purposes (where quick information is needed), the first method may be preferred, because it is less time consuming. With this method a model has to be estimated only once in a while, which means that only a few measurements have to be done, while with the second method more measurements are needed because each time a new model has to be estimated.

Generation of symmetric periodic orbits by a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2 in R3

In this paper we will find a continuous of periodic orbits passing near infinity for a class of polynomial vector fields in R3. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane and that possess a “generalized heteroclinic loop” formed by two singular points e+ and e− at infinity and their invariant manifolds � and . � is an invariant manifold of dimension 1 formed by an orbit going from e− to e+, � is contained in R3 and is transversal to . is an invariant manifold of dimension 2 at infinity. In fact, is the 2–dimensional sphere at infinity in the Poincar´e compactification minus the singular points e+ and e−. The main tool for proving the existence of such periodic orbits is the construction of a Poincar´e map along the generalized heteroclinic loop together with the symmetry with respect to .

Symmetric periodic orbits near a heteroclinic loop in R3 formed by two singular points, a semistable periodic orbit and their invariant manifolds

In this paper we consider C1 vector fields X in R3 having a “generalized heteroclinic loop” L which is topologically homeomorphic to the union of a 2–dimensional sphere S2 and a diameter connecting the north with the south pole. The north pole is an attractor on S2 and a repeller on . The equator of the sphere is a periodic orbit unstable in the north hemisphere and stable in the south one. The full space is topologically homeomorphic to the closed ball having as boundary the sphere S2. We also assume that the flow of X is invariant under a topological straight line symmetry on the equator plane of the ball. For each n ∈ N, by means of a convenient Poincar´e map, we prove the existence of infinitely many symmetric periodic orbits of X near L that gives n turns around L in a period. We also exhibit a class of polynomial vector fields of degree 4 in R3 satisfying this dynamics.