C(0) and bi-Lipschitz K-equivalence of mappings

In this paper we investigate the classification of mappings up to K-equivalence. We give several results of this type. We study semialgebraic deformations up to semialgebraic C(0) K-equivalence and bi-Lipschitz K-equivalence. We give an algebraic criterion for bi-Lipschitz K-triviality in terms of semi-integral closure (Theorem 3.5). We also give a new proof of a result of Nishimura: we show that two germs of smooth mappings f, g : R(n) -> R(n), finitely determined with respect to K-equivalence are C(0)-K-equivalent if and only if they have the same degree in absolute value.

Neoproterozoic geodynamic evolution of SW-Gondwana: a southern African perspective

Our current understanding of the tectonic history of the principal Pan-African orogenic belts in southwestern Africa, reaching from the West Congo Belt in the north to the Lufilian/Zambezi, Kaoko, Damara, Gariep and finally the Saldania Belt in the south, is briefly summarized. On that basis, possible links with tectono-stratigraphic units and major structures on the eastern side of the Rio de la Plata Craton are suggested, and a revised geodynamic model for the amalgamation of SW-Gondwana is proposed. The Rio de la Plata and Kalahari Cratons are considered to have become juxtaposed already by the end of the Mesoproterozoic. Early Neoproterozoic rifting led to the fragmentation of the northwestern (in today`s coordinates) Kalahari Craton and the splitting off of several small cratonic blocks. The largest of these ex-Kalahari cratonic fragments is probably the Angola Block. Smaller fragments include the Luis Alves and Curitiba microplates in eastern Brazil, several basement inliers within the Damara Belt, and an elongate fragment off the western margin, named Arachania. The main suture between the Kalahari and the Congo-So Francisco Cratons is suspected to be hidden beneath younger cover between the West Congo Belt and the Lufilian/Zambezi Belts and probably continues westwards via the Cabo Frio Terrane into the Goias magmatic arc along the Brasilia Belt. Many of the rift grabens that separated the various former Kalahari cratonic fragments did not evolve into oceanic basins, such as the Northern Nosib Rift in the Damara Belt and the Gariep rift basin. Following latest Cryogenian/early Ediacaran closure of the Brazilides Ocean between the Rio de la Plata Craton and the westernmost fragment of the Kalahari Craton, the latter, Arachania, became the locus of a more than 1,000-km-long continental magmatic arc, the Cuchilla Dionisio-Pelotas Arc. A correspondingly long back-arc basin (Marmora Basin) on the eastern flank of that arc is recognized, remnants of which are found in the Marmora Terrane-the largest accumulation of oceanic crustal material known from any of the Pan-African orogenic belts in the region. Corresponding foredeep deposits that emerged from the late Ediacaran closure of this back-arc basin are well preserved in the southern areas, i.e. the Punta del Este Terrane, the Marmora Terrane and the Tygerberg Terrane. Further to the north, present erosion levels correspond with much deeper crustal sections and comparable deposits are not preserved anymore. Closure of the Brazilides Ocean, and in consequence of the Marmora back-arc basin, resulted from a change in the Rio de la Plata plate motion when the Iapetus Ocean opened between the latter and Laurentia towards the end of the Ediacaran. Later break-up of Gondwana and opening of the modern South Atlantic would have followed largely along the axis of the Marmora back-arc basin and not along major continental sutures.

The Rondonian-San Ignacio Province in the SW Amazonian Craton: An overview

The Rondonian-San Ignacio Province (1.56-1.30 Ga) is a composite orogen created through successive accretion of arcs, ocean basin closure and final oblique microcontinent-continent collision. The effects of the collision are well preserved mostly in the Paragua Terrane (Bolivia and Mato Grosso regions) and in the Alto Guapore Belt and the Rio Negro-Juruena Province (Rondonia region), considering that the province was affected by later collision-related deformation and metamorphism during the Sunsas Orogeny (1.25-1.00 Ga). The Rondonian-San Ignacio Province comprises: (1) the Jauru Terrane (1.78-1.42 Ga) that hosts Paleoproterozoic basement (1.78-1.72 Ga), and the Cachoeirinha (1.56-1.52 Ga) and the Santa Helena (1.48-1.42 Ga) accretionary orogens, both developed in an Andean-type magmatic arc; (2) the Paragua Terrane (1.74-1.32 Ga) that hosts pre-San Ignacio units (>1640 Ma: Chiquitania Gneiss Complex, San Ignacio Schist Group and Lomas Manechis Granulitic Complex) and the Pensamiento Granitoid Complex (1.37-1.34 Ga) developed in an Andean-type magmatic arc; (3) the Rio Alegre Terrane (1.51-1.38 Ga) that includes units generated in a mid-ocean ridge and an intra-oceanic magmatic arc environments; and (4) the Alto Guapore Belt (<1.42-1.34 Ga) that hosts units developed in passive marginal basin and intra-oceanic arc settings. The collisional stage (1.34-1.32 Ga) is characterized by deformation, high-grade metamorphism, and partial melting during the metamorphic peak, which affected primarily the Chiquitania Gneiss Complex and Lomas Manechis Granulitic Complex in the Paragua Terrane, and the Colorado Complex and the Nova Mamore Metamorphic Suite in the Alto Guapore Belt. The Paragua Block is here considered as a crustal fragment probably displaced from its Rio Negro-Juruena crustal counterpart between 1.50 and 1.40 Ga. This period is characterized by extensive A-type and intra-plate granite magmatism represented by the Rio Crespo Intrusive Suite (ca. 1.50 Ga), Santo Antonio Intrusive Suite (1.40-1.36 Ga), and the Teotonio Intrusive Suite (1.38 Ga). Magmatism of these types also occur at the end of the Rondonian-San Ignacio Orogeny, and are represented by the Alto Candeias Intrusive Suite (1.34-1.36 Ga), and the Sao Lourenco-Caripunas Intrusive Suite (1.31-1.30 Ga). The cratonization of the province occurred between 1.30 and 1.25 Ga. (C) 2009 Elsevier Ltd. All rights reserved.

GEOCHEMISTRY AND U-PB ZIRCON AGE OF THE INTERNAL OSSA-MORENA ZONE OPHIOLITE SEQUENCES: A REMNANT OF RHEIC OCEAN IN SW IBERIA

The Early Paleozoic geodynamic evolution in SW Iberia is believed to have been dominated by the opening of the Rheic Ocean. The Rheic Ocean is generally accepted to have resulted from the drift of peri-Gondwanan terranes such as Avalonia from the northern margin of Gondwana during Late Cambrian-Early Ordovician times. The closure of the Rheic Ocean was the final result of a continent-continent collision between Gondwana and Laurussia that produced the Variscan orogen. The Ossa-Morena Zone is a peri-Gondwana terrane, which preserves spread fragments of ophiolites - the Internal Ossa-Morena Zones Ophiolite Sequences (IOMZOS). The final patchwork of the IOMZOS shows a complete oceanic lithospheric sequence with geochemical characteristics similar to the ocean-floor basalts, without any orogenic fingerprint and/or crustal contamination. The IOMZOS were obducted and imbricated with high pressure lithologies. Based on structural, petrological and whole-rock geochemical data, the authors argue that the IOMZOS represent fragments of the oceanic lithosphere from the Rheic Ocean. Zircon SHRIMP U-Pb geochronological data on metagabbros point to an age of ca. 480 Ma for IOMZOS, providing evidence of a well-developed ocean in SW Iberia during this period, reinforcing the interpretation of the Rheic Ocean as a wide ocean among the peri-Gondwanan terranes during Early Ordovician times.

Closing the Clymene ocean and bending a Brasiliano belt: Evidence for the Cambrian formation of Gondwana, southeast Amazon craton

We report new paleomagnetic and geochronological data from Ediacaran rift-drift carbonates in the Paraguai belt at the southern end of the suture zone between the Amazon craton and the Sao Francisco and Rio de Plata cratons, South America. Early thrusting resulted in remagnetization ca. 528 +/- 36 Ma or later; the mean age is established by (40)Ar/(39)Ar encapsulation dating of mixed authigenic and detrital illite from remagnetized carbonates from the unmetamorphosed fold-thrust belt. This remagnetization overlaps with a 525 Ma Gondwana reference pole. Metamorphic illite from the slate belt yields (40)Ar/(39)Ar ages of 496-484 Ma, the timing of peak regional metamorphism. Oroclinal bending of the Paraguai belt was caused by a 90 degrees clockwise rotation of the east-west limb after ca. 528 Ma, probably reflecting the irregular margin of the southeast Amazon craton. The age of the Paraguai belt overlaps with that of the Pampean orogeny farther south along the western margin of the Rio de Plata craton, suggesting a coeval closure for the Clymene ocean separating the Amazon craton from the Sao Francisco and Rio de Plata cratons.

Tectonic implications of a paleomagnetic study of the Sarmiento Ophiolitic Complex, southern Chile

A paleomagnetic study was carried out on the Late Jurassic Sarmiento Ophiolitic Complex (SOC) exposed in the Magallanes fold and thrust belt in the southern Patagonian Andes (southern Chile). This complex, mainly consisting of a thick succession of pillow-lavas, sheeted dikes and gabbros, is a seafloor remnant of the Late Jurassic to Early Cretaceous Rocas Verdes basin that developed along the south-western margin of South America. Stepwise thermal and alternating field demagnetization permitted the isolation of a post-folding characteristic remanence, apparently carried by fine grain (SD?) magnetite, both in the pillow-lavas and dikes. The mean ""in situ"" direction for the SOC is Dec: 286.9 degrees, Inc: -58.5 degrees, alpha-95: 6.9 degrees, N: 11 (sites). Rock magnetic properties, petrography and whole-rock K-Ar ages in the same rocks are interpreted as evidence of correlation between remanence acquisition and a greenschist facies metamorphic overprint that must have occurred during latest stages or after closure and tectonic inversion of the basin in the Late Cretaceous. The mean remanence direction is anomalous relative to the expected Late Cretaceous direction from stable South America. Particularly, a declination anomaly over 50 degrees is suggestively similar to paleomagnetically interpreted counter clockwise rotations found in thrust slices of the Jurassic El Quemado Fm. located over 100 km north of the study area in Argentina. Nevertheless, a significant ccw rotation of the whole SOC is difficult to reconcile with geologic evidence and paleogeographic models that suggest a narrow back-arc basin sub-parallel to the continental margin. A rigid-body 30 degrees westward tilting of the SOC block around a horizontal axis trending NNW, is considered a much simpler explanation, being consistent with geologic evidence. This may have occurred as a consequence of inverse reactivation of old normal faults, which limit both the SOC exposures and the Cordillera Sarmiento to the East. The age of tilting is unknown but it must postdate remanence acquisition in the Late Cretaceous. Two major orogenic events of the southern Patagonian Andes, in the Eocene (ca. 42 Ma) and Middle Miocene (ca. 12 Ma), respectively, could have caused the proposed tilting. (C) 2008 Elsevier B.V. All rights reserved.

Multiple Brake Orbits and Homoclinics in Riemannian Manifolds

Let (M, g) be a complete Riemannian manifold, Omega subset of Man open subset whose closure is homeomorphic to an annulus. We prove that if a,Omega is smooth and it satisfies a strong concavity assumption, then there are at least two distinct geodesics in starting orthogonally to one connected component of a,Omega and arriving orthogonally onto the other one. Using the results given in Giamb et al. (Adv Differ Equ 10:931-960, 2005), we then obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. Under a further symmetry assumption, the result is improved by showing the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinic orbits. In our proof we shall use recent deformation results proved in Giamb et al. (Nonlinear Anal Ser A: Theory Methods Appl 73:290-337, 2010).

Desingularization of singular Riemannian foliation

Let F be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F we construct a regular Riemannian foliation (F) over cap on a compact Riemannian manifold (M) over cap and a desingularization map (rho) over cap : (M) over cap -> M that projects leaves of (F) over cap into leaves of F. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F are compact, then, for each small epsilon > 0, we can find (M) over cap and (F) over cap so that the desingularization map induces an epsilon-isometry between M/F and (M) over cap/(F) over cap. This implies in particular that the space of leaves M/F is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {((M) over cap (n)/(F) over cap (n))}.

On the Multiplicity of Orthogonal Geodesics in Riemannian Manifold With Concave Boundary. Applications to Brake Orbits and Homoclinics

Let (M, g) be a complete Riemannian Manifold, Omega subset of M an open subset whose closure is diffeomorphic to an annulus. If partial derivative Omega is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in (Omega) over bar = Omega boolean OR partial derivative Omega starting orthogonally to one connected component of partial derivative Omega and arriving orthogonally onto the other one. The results given in [6] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a. class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.