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.

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.

A general Trotter-Kato formula for a class of evolution operators

In this article we prove new results concerning the existence and various properties of an evolution system U(A+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 L(B) for the algebra of all linear bounded operators on B, we can express U(A+B)(t, s)0 <= s <= t <= T as the strong limit in C(8) of a product of the holomorphic contraction semigroups generated by -A (t) and - B(t), respectively, thereby proving 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(t is an element of)[0,T] D(A(t) + B(t)) everywhere dense in B. We obtain a special case of our formula when B(t) = 0, which, in effect, allows us to reconstruct U(A)(t, s)0 <=(s)<=(t)<=(T) very simply in terms of the semigroup generated by -A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of timedependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrodinger type in quantum mechanics.

Dependence of the product on the P-P ligand in reactions of [RuCl(3)(NO)(P-P)] complexes (P-P = aromatic diphosphines) with 2-mercaptopyridine

This study presents the syntheses and characterization of 2-mercaptopyridine (pyS(-)) complexes containing ruthenium(II) with the following general formula [Ru(pyS)(2)(P-P)], P-P = (c-dppen) = cis-1,2-bis(diphenylphosphino)ethylene) (1); (dppe)=1,2-bis(diphenylphosphino)ethane (2); (dppp)=1,3-bis(diphenylphosphino)propane (3) and (dppb) = 1,4-bis(diphenylphosphino)butane (4). The complexes were synthesized from the mer- or fac-[RuCl(3)(NO)(P-P)] precursors in the presence of triethylamine in methanol solution with dependence of the product on the P-P ligand. The reaction of pyS- with a ruthenium complex containing a bulky aromatic diphosphine dppb disclosed a major product with a dangling coordinated dppbO-P, the [Ru(pyS)(2)(NO)(eta(1)-dppbO-P)]PF(6) (5). In addition, this work also presents and discusses the spectroscopic and electrochemical behavior of 1-5. and report the X-ray structures for I and S. (C) 2009 Elsevier Ltd. All rights reserved.

Natural History of the Lutz`s Frog Cycloramphus lutzorum Heyer, 1983 (Anura: Cycloramphidae) in the Brazilian Atlantic Forest: Description of the Advertisement Call, Tadpole, and Karyotype

We describe the advertisement call, tadpole, karyotype, and additional information on the natural history of Cycloramphus lutzorum from southern Brazil. Sonograms were generated from digitally recorded calls. Tadpoles were collected in the field for description in the lab, and an adult was collected for karyotyping. Data on seasonal activity were gathered monthly from November 2005 to November 2007. All tadpoles (N = 21), juveniles (N = 18), and adults (N = 52) were found exclusively in streams. Reproduction, as identified by calling frogs, occurred from July through November. Frogs call all day long, but mostly at dusk, from rock crevices inside the stream edges near the splash zone. The call is short and loud, with 11 pulsed notes, of 491-641 ms, with a dominant frequency of 0.98-1.39 kHz. We describe the exotrophic and semiterrestrial tadpoles, always found in constantly humid vertical rock walls in the stream. Tadpoles of C. lutzorum are recognized by differences in labial tooth row formula, eye diameter, body shape, position of nares, and development of tail. Like congeneric species, the karyotype of C. lutzorum comprises 26 metacentric and submetacentric chromosomes. Cycloramphus lutzorum is restricted to and adapted for living in fast flowing streams, many of which are threatened by deforestation, pollution, and habitat loss. Therefore, we recommend the status of C. lutzorum be changed from its current ""Data Deficient"" to ""Near Threatened (NT)"" in the IUCN species red list.

Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves

We construct indecomposable and noncrossed product division algebras over function fields of connected smooth curves X over Z(p). This is done by defining an index preserving morphism s: Br(<(K(X))over cap>)` --> Br(K(X))` which splits res : Br(K (X)) --> Br(<(K(X))over cap>), where <(K(X))over cap> is the completion of K (X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over <(K(X))over cap>. (C) 2010 Elsevier Inc. All rights reserved.

Spallation product distributions and neutron multiplicities for accelerator-driven system using the CRISP code

Neutron multiplicities for several targets and spallation products of proton-induced reactions in thin targets of interest to an accelerator-driven system obtained with the CRISP code have been reported. This code is a Monte Carlo calculation that simulates the intranuclear cascade and evaporationl fission competition processes. Results are compared with experimental data, and agreement between each other can be considered quite satisfactory in a very broad energy range of incitant particles and different targets.

Quantum States Allowing Minimum Uncertainty Product of phi and L(z)

We provide necessary and sufficient conditions for states to have an arbitrarily small uncertainty product of the azimuthal angle phi and its canonical moment L(z). We illustrate our results with analytical examples.

Bendadaite, a new iron arsenate mineral of the arthurite group

Bendadaite, ideally Fe(2+)Fe(2)(3+)(AsO(4))(2)(OH)(2 center dot).4H(2)O, is a new member of the arthurite group It was found as a weathering product of arsenopyrite on a single hand specimen from the phosphate pegmatite Bendada. central Portugal (type locality) Co-type locality is the granite pegmatite of La via do Almerindo (Almerindo mine), Linopolis, Divmo das Laranjeiras county, Minas Gerais, Brazil Further localities are the Vein Negra mine, Copiapo province, Chile, mid-East, Bou Azzer district, Morocco, and Para Inferida yard, Fenugu Sibirt mine, Gonnosfanadiga, Medio Campidano Province, Sardinia. Italy Type bendadaite occurs as blackish green to dark brownish tufts (<0 1 mm long) and flattened radiating aggregates. in intimate association with an intermediate member of the scorodite-mansfieldite series It is monoclinic. space group P2(l/c). with a = 10 239(3) angstrom. b = 9 713(2) angstrom, c = 5 552(2) angstrom. beta = 94 11(2)degrees. = 550 7(2) angstrom(3). Z = 2 Electron-microprobe analysis yielded (wt %). CaO 0 04, MnO 0 03. CuO 006, ZnO 004. Fe(2)O(3) (total) 43 92, Al(2)O(3) 115. SnO(2) 0 10, As(2)O(5) 43 27. P(2)O(5) 1 86, SO(3) 0.03 The empirical formula is (Fe(0 52)(2+)Fe(0 32)(3+)rectangle(0 16))(Sigma 1 00)(Fe(1 89)(3+)Al(0 11))(Sigma 2 00)(As(1 87)P(0 13))(Sigma 2 00)O(8)(OH)(2 00) 4H(2)O based. CM 2(As,P) and assuming ideal 80, 2(OH), 4H2O and complete occupancy of the ferric on site by Fe(3+) and Al Optically, bendadaite is biaxial, positive, 2V(est) = 85+/-4 degrees, 2V(eale) = 88 degrees, with alpha 1 734(3). 13 1 759(3), 7 1 787(4) Pleochrosim is medium strong X pale reddish brown. Y yellowish brown, Z dark yellowish brown. absorption Z > V > X, optical dispersion weak, r > v. Optical axis plane Is parallel to (010), with X approximately parallel to a and Z nearly parallel to c Bendadaite has vitreous to sub-adamantine luster, is translucent and non-fluorescent It is brittle, shows irregular fracture and a good cleavage parallel to 1010} 3 15 0 10 g/cm(3), 3 193 g/cm3 (for the empirical formula) The five strongest powder diffraction lines [d in angstrom (I)(hkl] are 10 22 (10)(100), 7 036 (8)(110), 4 250 (5)(11 I), 2 865 (4)(311), 4 833 (3)(020,011) The d spacings are very similar to those of its Zn analogue, ojelaite The crystal structure of bendadaite was solved and refined using a crystal from the co-type locality with the composition (Fe(0 95)(2+)rectangle(0 05))(Sigma 1 00)(Fe(1 80)(3+)Al(0 20))Sigma(2 00)(As(1 48)P(0 52))(Sigma 2 00)O(8)) (OH)(2) 4H(2)O (R = 16%) and confirms an arthurite-type atomic arrangement

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.

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.

Thermal analysis of prednicarbate and characterization of thermal decomposition product

In the present work, the thermal behavior of prednicarbate was studied using DSC and TG/DTG. The solid product remaining at the first decomposition step of the drug was isolated by TG, in air and N(2) atmospheres and was characterized using LC-MS/MS, NMR, and IR spectroscopy. It was found that the product at the first thermal decomposition step of prednicarbate corresponds to the elimination of the carbonate group bonding to C(17), and a consequent formation of double bond between C(17) and C(16). Structure elucidation of this degradation product by spectral data has been discussed in detail.