STABLE PIECEWISE POLYNOMIAL VECTOR FIELDS

Let N = {y > 0} and S = {y < 0} be the semi-planes of R-2 having as common boundary the line D = {y = 0}. Let X and Y be polynomial vector fields defined in N and S, respectively, leading to a discontinuous piecewise polynomial vector field Z = (X, Y). This work pursues the stability and the transition analysis of solutions of Z between N and S, started by Filippov (1988) and Kozlova (1984) and reformulated by Sotomayor-Teixeira (1995) in terms of the regularization method. This method consists in analyzing a one parameter family of continuous vector fields Z(epsilon), defined by averaging X and Y. This family approaches Z when the parameter goes to zero. The results of Sotomayor-Teixeira and Sotomayor-Machado (2002) providing conditions on (X, Y) for the regularized vector fields to be structurally stable on planar compact connected regions are extended to discontinuous piecewise polynomial vector fields on R-2. Pertinent genericity results for vector fields satisfying the above stability conditions are also extended to the present case. A procedure for the study of discontinuous piecewise vector fields at infinity through a compactification is proposed here.

Rigidity for piecewise smooth homeomorphisms on the circle

We find conditions for two piecewise 'C POT.2+V' homeomorphisms f and g of the circle to be 'C POT.1' conjugate. Besides the restrictions on the combinatorics of the maps (we assume that the maps have bounded combinatorics), and necessary conditions on the one-side derivatives of points where f and g are not differentiable, we also assume zero mean-nonlinearity for f and g.

Globally solvable systems of complex vector fields

We consider a class of involutive systems of n smooth vector fields on the n + 1 dimensional torus. We obtain a complete characterization for the global solvability of this class in terms of Liouville forms and of the connectedness of all sublevel and superlevel sets of the primitive of a certain 1-form in the minimal covering space.

Flow Visualization with Quantified Spatial and Temporal Errors Using Edge Maps

Robust analysis of vector fields has been established as an important tool for deriving insights from the complex systems these fields model. Traditional analysis and visualization techniques rely primarily on computing streamlines through numerical integration. The inherent numerical errors of such approaches are usually ignored, leading to inconsistencies that cause unreliable visualizations and can ultimately prevent in-depth analysis. We propose a new representation for vector fields on surfaces that replaces numerical integration through triangles with maps from the triangle boundaries to themselves. This representation, called edge maps, permits a concise description of flow behaviors and is equivalent to computing all possible streamlines at a user defined error threshold. Independent of this error streamlines computed using edge maps are guaranteed to be consistent up to floating point precision, enabling the stable extraction of features such as the topological skeleton. Furthermore, our representation explicitly stores spatial and temporal errors which we use to produce more informative visualizations. This work describes the construction of edge maps, the error quantification, and a refinement procedure to adhere to a user defined error bound. Finally, we introduce new visualizations using the additional information provided by edge maps to indicate the uncertainty involved in computing streamlines and topological structures.

EQUIVALENCE OF REAL MILNOR FIBRATIONS FOR QUASI-HOMOGENEOUS SINGULARITIES

We show that for real quasi-homogeneous singularities f : (R-m, 0) -> (R-2, 0) with isolated singular point at the origin, the projection map of the Milnor fibration S-epsilon(m-1) \ K-epsilon -> S-1 is given by f/parallel to f parallel to. Moreover, for these singularities the two versions of the Milnor fibration, on the sphere and on a Milnor tube, are equivalent. In order to prove this, we show that the flow of the Euler vector field plays and important role. In addition, we present, in an easy way, a characterization of the critical points of the projection (f/parallel to f parallel to) : S-epsilon(m-1) \ K-epsilon -> S-1.

Petrogenesis of Cenozoic volcanic rocks in the NW sector of the Gharyan volcanic field, Libya

The north-western sector of the Gharyan volcanic field (northern Libya) consists of trachytic-phonolitic domes emplaced between similar to 41 and 38 Ma, and small-volume mafic alkaline volcanic centres (basanites, tephrites. alkali basalts. hawaiites and rare benmoreites) of Middle Miocene-Pliocene age (similar to 12-2 Ma). Two types of trachytes and phonolites have been recognized on the basis of petrography, mineralogy and geochemistry. Type-1 trachytes and phonolites display a smooth spoon-shaped REE pattern without negative Europium anomalies. Type-2 trachytes and phonolites show a remarkable Eu negative anomaly, higher concentration in HFSE (Nb-Ta-Zr-Hf), REE and Ti than Type-1 rocks. The origin of Type-1 trachytes and phonolites is compatible with removal of clinopyroxene, plagioclase, alkali feldspar, amphibole. magnetite and titanite starting from benmoreitic magmas. found in the same outcrops. Type-2 trachytes and phonolites could be the result of extensive fractional crystallization starting from mafic alkaline magma, without removal of titanite. In primitive mantle-normalized diagrams, the mafic rocks (Mg#= 62-68, Cr up to 514 ppm, Ni up to 425 ppm) show peaks at Nb and Ta and troughs at K. These characteristics, coupled with low Sr-87/Sr-86(i) (0.7033-0.7038) and positive epsilon(Nd) (from +4.2 to + 5.3) features typical of the mafic anorogenic magmas of the northern African plate and of HIMU-OIB-like magma in general. The origin of the mafic rocks is compatible from a derivation from low degree partial melting (3-9%) shallow mantle sources in the spinel/gamet facies. placed just below the rigid plate in the uppermost low-velocity zone. The origin of the igneous activity is considered linked to passive lithospheric thinning related to the development of continental rifts like those of Sicily Channel (e.g.. Pantelleria and Linosa) and Sardinia (e.g., Campidano Graben) in the Central-Western Mediterranean Sea. (C) 2012 Elsevier B.V. All rights reserved.

Solvability near the characteristic set for a special class of complex vector fields

This work deals with the solvability near the characteristic set Sigma = {0} x S-1 of operators of the form L = partial derivative/partial derivative t+(x(n) a(x)+ ix(m) b(x))partial derivative/partial derivative x, b not equivalent to 0 and a(0) not equal 0, defined on Omega(epsilon) = (-epsilon, epsilon) x S-1, epsilon > 0, where a and b are real-valued smooth functions in (-epsilon, epsilon) and m >= 2n. It is shown that given f belonging to a subspace of finite codimension of C-infinity (Omega(epsilon)) there is a solution u is an element of L-infinity of the equation Lu = f in a neighborhood of Sigma; moreover, the L-infinity regularity is sharp.

Amazonian malaria: Asymptomatic human reservoirs, diagnostic challenges, environmentally driven changes in mosquito vector populations, and the mandate for sustainable control strategies

Across the Americas and the Caribbean, nearly 561,000 slide-confirmed malaria infections were reported officially in 2008. The nine Amazonian countries accounted for 89% of these infections; Brazil and Peru alone contributed 56% and 7% of them, respectively. Local populations of the relatively neglected parasite Plasmodium vivax, which currently accounts for 77% of the regional malaria burden, are extremely diverse genetically and geographically structured. At a time when malaria elimination is placed on the public health agenda of several endemic countries, it remains unclear why malaria proved so difficult to control in areas of relatively low levels of transmission such as the Amazon Basin. We hypothesize that asymptomatic parasite carriage and massive environmental changes that affect vector abundance and behavior are major contributors to malaria transmission in epidemiologically diverse areas across the Amazon Basin. Here we review available data supporting this hypothesis and discuss their implications for current and future malaria intervention policies in the region. Given that locally generated scientific evidence is urgently required to support malaria control interventions in Amazonia, we briefly describe the aims of our current field-oriented malaria research in rural villages and gold-mining enclaves in Peru and a recently opened agricultural settlement in Brazil. (C) 2011 Elsevier B.V. All rights reserved.

Interpolation estimate for a finite-element space with embedded discontinuities

We consider a recently proposed finite-element space that consists of piecewise affine functions with discontinuities across a smooth given interface Γ (a curve in two dimensions, a surface in three dimensions). Contrary to existing extended finite element methodologies, the space is a variant of the standard conforming Formula space that can be implemented element by element. Further, it neither introduces new unknowns nor deteriorates the sparsity structure. It is proved that, for u arbitrary in Formula, the interpolant Formula defined by this new space satisfies Graphic where h is the mesh size, Formula is the domain, Formula, Formula, Formula and standard notation has been adopted for the function spaces. This result proves the good approximation properties of the finite-element space as compared to any space consisting of functions that are continuous across Γ, which would yield an error in the Formula-norm of order Graphic. These properties make this space especially attractive for approximating the pressure in problems with surface tension or other immersed interfaces that lead to discontinuities in the pressure field. Furthermore, the result still holds for interfaces that end within the domain, as happens for example in cracked domains.