A Trotter-Kato product formula for a class of non-autonomous evolution equations

In this article dedicated to Professor V. Lakshmikantham on the occasion of the celebration of his 84th birthday, we announce new results concerning the existence and various properties of an evolution system UA+B(t, s)(0 <= s <= t <= T) generated by the sum -(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing G(B) for the algebra of all linear bounded operators on B, we can express UA+B(t, s)(0 <= s <= t <= T) as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by -A(t) and -B(t), thereby getting a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D subset of boolean AND D-t epsilon[0,D-T](A(t)+B(t)) everywhere dense in B. We then mention several possible applications of our product formula to various classes of non-autonomous parabolic initial-boundary value problems, as well as to evolution problems of Schrodinger type related to the theory of time-dependent singular perturbations of self-adjoint operators in quantum mechanics. We defer all the proofs and all the details of the applications to a separate publication. (C) 2008 Elsevier Ltd. All rights reserved.

Evaluation of trace element levels in muscles, liver and gonad of fish species from Sao Francisco River of the Parana Brazilian state by using SR-TXRF technique

This study is focused on the analysis of an accumulation of inorganic elements in muscles, liver and gonad of seven fish species from Sao Francisco River located in the Parana state of Brazil. Concentrations of the elements were determined using the SR-TXRF technique. In the muscles of fish species, negative length dependent relationships were observed for chromium and zinc ion absorption. The obtained results showed that accumulated Cr ions values are above the limits defined in the Brazilian legislative norm on food. (C) 2010 Elsevier Ltd All rights reserved.

Major- and trace-element composition of olivine, perovskite, clinopyroxene, Cr-Fe-Ti oxides, phlogopite and host kamafugites and kimberlites, Alto Paranaiba, Brazil

An integrated whole-rock petrographic and geochemical study has been carried out on kamafugites and kimberlites of the Late Cretaceous Alto Paranaiba igneous province, in Brazil, and their main minerals, olivine, clinopyroxene, perovskite, phlogopite, spinels and ilmenite. Perovskite is by far the dominant repository for light lanthanides, Nb, Ta, Th and U, and occasionally other elements, reaching concentrations up to 3.4 x 10(4) chondrite values for light lanthanides and 105 chondrite for Th. A very strong fractionation between light and heavy lanthanides (chondrite-normalized La/Yb from similar to 175 to similar to 2000) is also observed. This is likely the first comprehensive dataset on natural perovskite. Clinopyroxene has variable trace-element contents. likely due to the different position of this phase in the crystallization sequence; Sc reaches values as high as 200 ppm whereas the lanthanides show very variable enrichment in light over heavy REE, and commonly show a negative Eu anomaly. The olivine, phlogopite (and tetra-ferriphlogopite), Cr-Ti oxide and ilmenite are substantially barren minerals for lanthanides and most other trace elements, with the exception of Ba, Cs and Rb in mica, and V, Nb and Ta in ilmenite. Estimated mineral/whole-rock partition coefficients for lanthanides in perovskite are similar to previous determinations, though much higher than those calculated in experiments with synthetic compositions, testifying once more to the complex behavior of these elements in a natural environment. The enormous potential for exploitation of lanthanides, Th, U and high-field-strength elements in the Brazilian kamafugites, kimberlites and related rocks is clearly shown.

Major- and trace-element composition of REE-rich turkestanite from peralkaline granites of the Morro Redondo Complex, Graciosa Province, south Brazil

Turkestanite, a rare Th- and REE-bearing cyclosilicate in the ekanite-steacyite group was found in evolved peralkaline granites from the Morro Redondo Complex, south Brazil. It occurs with quartz, alkali feldspar and an unnamed Y-bearing silicate. Electron microprobe analysis indicates relatively homogeneous compositions with maximum ThO(2), Na(2)O and K(2)O contents of 22.4%, 2.93% and 3.15 wt.%, respectively, and significant REE(2)O(3) abundances (5.21 to 11.04 wt.%). The REE patterns show enrichment of LREE over HREE, a strong negative Eu anomaly and positive Ce anomaly, the latter in the most transformed crystals. Laser ablation inductively coupled plasma mass spectrometry trace element patterns display considerable depletions in Nb, Zr, Hf, Ti and Li relative to whole-rock sample compositions. Observed compositional variations suggest the influence of coupled substitution mechanisms involving steacyite, a Na-dominant analogue of turkestanite, iraqite, a REE-bearing end-member in the ekanite-steacyite group, ekanite and some theoretical end-members. Turkestanite crystals were interpreted as having precipitated during post-magmatic stages in the presence of residual HFSE-rich fluids carrying Ca, the circulation of which was enhanced by deformational events.

Microgranitic Enclaves as Products of Self-mixing Events: a Study of Open-system Processes in the Maua Granite, Sao Paulo, Brazil, Based on in situ Isotopic and Trace Elements in Plagioclase

Felsic microgranular enclaves with structures indicating that they interacted in a plastic state with their chemically similar host granite are abundant in the Maua Pluton, SE Brazil. Larger plagioclase xenocrysts are in textural disequilibrium with the enclave groundmass and show complex zoning patterns with partially resorbed An-rich cores (locally with patchy textures) surrounded by more sodic rims. In situ laser ablation-(multi-collector) inductively coupled plasma mass spectrometry trace element and Sr isotopic analyses performed on the plagioclase xenocrysts indicate open-system crystallization; however, no evidence of derivation from more primitive basic melts is observed. The An-rich cores have more radiogenic initial Sr isotopic ratios that decrease towards the outermost part of the rims, which are in isotopic equilibrium with the matrix plagioclase. These profiles may have been produced by either (1) diffusional re-equilibration after rim crystallization from the enclave-forming magma, as indicated by relatively short calculated residence times, or (2) episodic contamination with a decrease of the contaminant ratio proportional to the extent to which the country rocks were isolated by the crystallization front. Profiles of trace elements with high diffusion coefficients would require unrealistically long residence times, and can be modeled in terms of fractional crystallization. A combination of trace element and Sr isotope data suggests that the felsic microgranular enclaves from the Maua Pluton are the products of interaction between end-member magmas that had similar compositions, thus recording `self-mixing` events.

U-Pb-Hf-trace element systematics and geochronology of zircon from a granulite-facies metamorphosed mafic-ultramafic layered complex in Central Brazil

Zircon recrystallization is a common process during high-grade metamorphism and promotes partial or complete resetting of the original isotopic and chemical characteristics of the mineral and thus complicates U-Pb geochronological interpretation. In Central Brazil, this complexity may be illustrated by three composite mafic-ultramafic intrusions metamorphosed under amphibolite-to-granulite conditions. Their ages of emplacement and metamorphic ages have been a matter of controversy for the last thirty years. The Serra da Malacacheta and Barro Alto complexes make up the southernmost of these layered bodies and four samples from distinct rock types were investigated in order to verify the consequences of metamorphic alteration of zircon for U-Pb dating. Cathodoluminescent imaging reveals internal features which are typical of concomitant dissolution-reprecipitation processes, such as convolute zoning and inward-moving recrystallization fronts, even in samples in which partially preserved igneous textures are observed. Due to this extensive alteration, LA-ICPMS U-Pb isotopic analysis yielded inconclusive data. However, in situ Hf isotopic and trace-element analyses help to clarify the real meaning of the geochronological data. Low Lu/Hf (<0.004) and homogeneous (176)Hf/(177)Hf(t) values imply that the zircon populations within individual samples have crystallized in a single episode, despite the observed variations in age values. Trace element signatures of zircon grains from garnet-bearing samples reveal that they were unreactive to the development of the peak metamorphism mineral assemblage and, thus, the main chemical feature in such grains is attributed to a coupled dissolution-reprecipitation process. However, in the Cafelandia amphibolite an additional alteration process is identified, probably related to the influx of late-stage fluids. Combined isotopic and geochemical investigation on zircon grains allowed the distinction of two magmatic events. The first corresponds to the crystallization of the Serra da Malacacheta Complex and characterizes a juvenile magmatism at similar to 1.3 Ga. The younger episode, recognized in the Barro Alto Complex, is dated at ca. 800 Ma and is represented by mafic and ultramafic rocks showing intense contamination with continental crust, implying that the emplacement took place, most likely, in a continental back-arc setting. Altered zircon domains as well as titanite grains date the metamorphic event at ca. 760-750 Ma. (C) 2011 Elsevier B.V. All rights reserved.

Trace element geochemistry and Sr-Nd characteristics of Mesoproterozoic mafic intrusive rocks from Rondonia, Brazil, SW Amazonian Craton: Petrogenetic and tectonic inferences

The Amazonian Craton comprises an Archean domain surrounded by four successively younger Proterozoic tectonic provinces. Within the Rio-Negro-Juruena province the Serra da Providencia Intrusive Suite (1.60 and 1.53 Ga) consists of A-type rapakivi granites, charnockites and mangerites genetically associated with diabase dikes, gabbros and amphibolites lites. The original mafic melts were derived from a depleted mantle source (epsilon(Nd(T)) + 2.5 to +2.8; epsilon(Sr(T)) - 12.1). Underplated mafic magma induced melting of a short-lived fielsic crust, thus originating coeval felsic-inafic magmatism in a continental intraplate setting. The Colorado Complex, assigned to the Rondonian-San Ignacio province, comprises 1.35-1.36 Ga intrusive bimodal magmatism represented by monzonite gneisses associated with amphibolite, gabbro and metadiabase dikes intercalated with metasediments with detrital zircon that yield U-Pb ages of 1.35 to 1.42 Ga. Mafic samples display juvenile signatures (epsilon(Nd(T)) 0.0 to +5.2; epsilon(Sr(T)) -5.0 to -30.7) and are less contaminated than the Serra da Previdencia and Nova Brasiladndia ones. The generation of the basaltic magma is related to the subduction of an oceanic slab below the peridotite wedge (intraoceanic arc setting). Fluids and/or small melts from the slab impregnated the mantle. The Nova Brasilandia Sequence (Sunsas-Aguapei province) comprises a metasedimentary sequence intruded by 1.10-1.02 Ga metadiabases, gabbros, meta-gabbros, and amphibolites associated with granitic plutons (bimodal magmatism). The original tholeiitic magmas, derived from a depleted source (epsilon(Nd(T)) = +3.1 to +5.0), in a proto-oceanic setting, underwent subsequent contamination by the host rocks, as indicated by the isotopic and trace element data.

THE CRYSTAL STRUCTURE OF A MICROLITE-GROUP MINERAL WITH A FORMULA NEAR NaCaTa(2)O(6)F FROM THE MORRO REDONDO MINE, CORONEL MURTA, MINAS GERAIS, BRAZIL

We present electron-microprobe and single-crystal X-ray-diffraction data for a microlite-group mineral with a formula near NaCaTa(2)O(6)F from the Morro Redondo mine, Coronel Murta, Minas Gerais, Brazil. On the basis of these data, the formula is A(Na(0.88)Ca(0.88)Pb(0.02)square(0.22))(Sigma 2.00) (B)(Ta(1.70)Nb(0.14)Si(0.12)As(0.04))(Sigma 2.00) (X)[(O(5.75)(OH)(0.25)](Sigma 6.00) (Y)(F(0.73)square(0.27))(Sigma 1.00). According to the new nomenclature for the pyrochlore-supergroup minerals, it is intermediate between fluornatromicrolite and "" fluorcalciomicrolite"". The crystal structure, F (d3) over barm, a = 10.4396(12) angstrom, has been refined to an R(1) value of 0.0258 (wR(2) = 0.0715) for 107 reflections (MoK alpha radiation). There is a scarcity of crystal-chemical data for pyrochlore-supergroup minerals in the literature. A compilation of these data is presented here.

Mineralogy and trace-element geochemistry of the high-grade iron ores of the Aguas Claras Mine and comparison with the Capao Xavier and Tamandua iron ore deposits, Quadrilahero Ferrifero, Brazil

Several major iron deposits occur in the Quadrilatero Ferrifero (QF), southeastern region of Brazil, where metamorphosed and heterogeneously deformed banded iron formation (BIF) of the Caue Formation, regionally called itabirite, was transformed into high- (Fe >64%) and lowgrade (30%trace elements. The high-grade ores at the Aguas Claras Mine have at least a dual origin, involving hypogene and supergene processes. The occurrence of the hard, massive high-grade ore within ""fresh"" dolomitic itabirite is evidence of its hypogene origin. Despite the contention about the origin of the dolomitic itabirite (if this rock is a carbonate-rich facies of the Caue Formation or a hematite-carbonate precursor of the soft high-grade ore), mineralogical and geochemical features of the soft high-grade ore indicate that it was formed by leaching of dolomite from the dolomitic itabirite by meteoric water. The comparison of the Aguas Claras, Capao Xavier and Tamandua orebodies shows that the original composition of the itabiritic protore plays a major role in the genesis of high- and low-grade soft ores in the QF. Under the same weathering and structural conditions, the dolomitic itabirite is the more favorable to form high-grade deposits than siliceous itabirite. Field relations at the Aguas Claras and Capao Xavier deposits suggest that it is not possible to form huge soft high-grade supergene deposits from siliceous itabirite, unless another control, such as impermeable barriers, had played an important role. The occurrence in the Tamandua Mine of a large, soft, high-grade orebody formed from siliceous itabirite and closely associated with hypogene hard ore suggests that large, soft, high-grade orebodies of the Quadrilatero Ferrifero, which occur within siliceous itabirite, have a hypogene contribution in their formation.

On a product formula for the Conley-Zelmder index of symplectic paths and its applications

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley-Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.

On a formula for the spectral flow and its applications

We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi-Riemannian geodesic, and we compute its value in terms of the Maslov index. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

A matrix formula for the skewness of maximum likelihood estimators

We give a general matrix formula for computing the second-order skewness of maximum likelihood estimators. The formula was firstly presented in a tensorial version by Bowman and Shenton (1998). Our matrix formulation has numerical advantages, since it requires only simple operations on matrices and vectors. We apply the second-order skewness formula to a normal model with a generalized parametrization and to an ARMA model. (c) 2010 Elsevier B.V. All rights reserved.

Covariance matrix formula for Birnbaum-Saunders regression models

The Birnbaum-Saunders regression model is commonly used in reliability studies. We derive a simple matrix formula for second-order covariances of maximum-likelihood estimators in this class of models. The formula is quite suitable for computer implementation, since it involves only simple operations on matrices and vectors. Some simulation results show that the second-order covariances can be quite pronounced in small to moderate sample sizes. We also present empirical applications.

A Character Formula for the Category (O)over-tilde of Modules for Affine sl(2)

One may construct, for any function on the integers, an irreducible module of level zero for affine sl(2) using the values of the function as structure constants. The modules constructed using exponential-polynomial functions realize the irreducible modules with finite-dimensional weight spaces in the category (O) over tilde of Chari. In this work, an expression for the formal character of such a module is derived using the highest weight theory of truncations of the loop algebra.

Versatile microanalytical system with porous polypropylene capillary membrane for calibration gas generation and trace gaseous pollutants sampling applied to the analysis of formaldehyde, formic acid, acetic acid and ammonia in outdoor air

The analytical determination of atmospheric pollutants still presents challenges due to the low-level concentrations (frequently in the mu g m(-3) range) and their variations with sampling site and time In this work a capillary membrane diffusion scrubber (CMDS) was scaled down to match with capillary electrophoresis (CE) a quick separation technique that requires nothing more than some nanoliters of sample and when combined with capacitively coupled contactless conductometric detection (C(4)D) is particularly favorable for ionic species that do not absorb in the UV-vis region like the target analytes formaldehyde formic acid acetic acid and ammonium The CMDS was coaxially assembled inside a PTFE tube and fed with acceptor phase (deionized water for species with a high Henry s constant such as formaldehyde and carboxylic acids or acidic solution for ammonia sampling with equilibrium displacement to the non-volatile ammonium ion) at a low flow rate (8 3 nLs(-1)) while the sample was aspirated through the annular gap of the concentric tubes at 25 mLs(-1) A second unit in all similar to the CMDS was operated as a capillary membrane diffusion emitter (CMDE) generating a gas flow with know concentrations of ammonia for the evaluation of the CMDS The fluids of the system were driven with inexpensive aquarium air pumps and the collected samples were stored in vials cooled by a Peltier element Complete protocols were developed for the analysis in air of NH(3) CH(3)COOH HCOOH and with a derivatization setup CH(2)O by associating the CMDS collection with the determination by CE-C(4)D The ammonia concentrations obtained by electrophoresis were checked against the reference spectrophotometric method based on Berthelot s reaction Sensitivity enhancements of this reference method were achieved by using a modified Berthelot reaction solenoid micro-pumps for liquid propulsion and a long optical path cell based on a liquid core waveguide (LCW) All techniques and methods of this work are in line with the green analytical chemistry trends (C) 2010 Elsevier B V All rights reserved