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.

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.

Edge effects as the principal cause of area effects on birds in fragmented secondary forest

Bird communities in tropical forests are strongly affected by both patch area and habitat edges. The fact that both effects are intrinsically confounded in space raises questions about how these two widely reported ecological patterns interact, and whether they are independent or simply different spatial manifestations of the same phenomenon. Moreover, do small patches of secondary forest, in landscapes where the most sensitive species have gone locally extinct, exhibit similar patterns to those previously observed in fragmented and continuous primary forests? We addressed these questions by testing edge-related differences in vegetation structure and bird community composition at 31 sites in fragmented and continuous landscapes in the imperilled Atlantic forest of Brazil. Over a two-year period, birds were captured with mist nets to a standardized effort of 680 net-hours at each site (similar to 22 000 net-hours resulting in 3381 captures from 114 species). We found that the bird community in patches of secondary forest was degraded in species composition compared to primary continuous forest, but still exhibited a strong response to edge effects. In fragmented secondary forests, edge and area effects also interacted, such that the magnitude of edge to interior differences on bird community composition declined markedly with patch size. The change in bird species composition between forest interiors and edges was similar to the change in community composition between large and small patches (because species had congruent responses to edge and area), but after controlling for edge effects community composition was no longer affected by patch area. Our results show that although secondary forests hold an impoverished bird community, ecological patterns such as area and edge effects are similar to those reported for primary forests. Our data provide further evidence that edge effects are the main drivers of area effects in fragmented landscapes.

Ecological niche modeling and principal component analysis of Krameria Loefl. (Krameriaceae)

Krameria plants are found in arid regions of the Americas and present a floral system that attracts oil-collecting bees. Niche modeling and multivariate tools were applied to examine ecological and geographical aspects of the 18 species of this genus, using occurrence data obtained from herbaria and literature. Niche modeling showed the potential areas of occurrence for each species and the analysis of climatic variables suggested that North American species occur mostly in deserted or xeric ecoregions with monthly precipitation below 140 mm and large temperature ranges. South American species are mainly found in deserted ecoregions and subtropical savannas where monthly precipitation often exceeds 150 mm and temperature ranges are smaller. Principal Component Analysis (PCA) performed with values of temperature and precipitation showed that the distribution limits of Krameria species are primarily associated with maximum and minimum temperatures. Modeling of Krameria species proved to be a useful tool for analyzing the influence of the ecological niche variables in the geographical distribution of species, providing new information to guide future investigations. (C) 2011 Elsevier Ltd. All rights reserved.

An extension of the concept of gradient semigroups which is stable under perturbation

In this article we introduce the concept of a gradient-like nonlinear semigroup as an intermediate concept between a gradient nonlinear semigroup (those possessing a Lyapunov function, see [J.K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol. 25, Amer. Math. Soc., 1989]) and a nonlinear semigroup possessing a gradient-like attractor. We prove that a perturbation of a gradient-like nonlinear semigroup remains a gradient-like nonlinear semigroup. Moreover, for non-autonomous dynamical systems we introduce the concept of a gradient-like evolution process and prove that a non-autonomous perturbation of a gradient-like nonlinear semigroup is a gradient-like evolution process. For gradient-like nonlinear semigroups and evolution processes, we prove continuity, characterization and (pullback and forwards) exponential attraction of their attractors under perturbation extending the results of [A.N. Carvalho, J.A. Langa, J.C. Robinson, A. Suarez, Characterization of non-autonomous attractors of a perturbed gradient system, J. Differential Equations 236 (2007) 570-603] on characterization and of [A.V. Babin, M.I. Vishik, Attractors in Evolutionary Equations, Stud. Math. Appl.. vol. 25, North-Holland, Amsterdam, 1992] on exponential attraction. (C) 2009 Elsevier Inc. All rights reserved.

Stability of gradient semigroups under perturbations

In this paper we prove that gradient-like semigroups (in the sense of Carvalho and Langa (2009 J. Diff. Eqns 246 2646-68)) are gradient semigroups (possess a Lyapunov function). This is primarily done to provide conditions under which gradient semigroups, in a general metric space, are stable under perturbation exploiting the known fact (see Carvalho and Langa (2009 J. Diff. Eqns 246 2646-68)) that gradient-like semigroups are stable under perturbation. The results presented here were motivated by the work carried out in Conley (1978 Isolated Invariant Sets and the Morse Index (CBMS Regional Conference Series in Mathematics vol 38) (RI: American Mathematical Society Providence)) for groups in compact metric spaces (see also Rybakowski (1987 The Homotopy Index and Partial Differential Equations (Universitext) (Berlin: Springer)) for the Morse decomposition of an invariant set for a semigroup on a compact metric space).

Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

We consider the Dirichlet problem for the equation -Delta u = lambda u +/- (x, u) + h(x) in a bounded domain, where f has a sublinear growth and h is an element of L-2. We find suitable conditions on f and It in order to have at least two solutions for X near to an eigenvalue of -Delta. A typical example to which our results apply is when f (x, u) behaves at infinity like a(x)vertical bar u vertical bar(q-2)u, with M > a(x) > delta > 0, and I < q < 2. (C) 2007 Elsevier Inc. All rights reserved.

Decomposition of spectral density in individual eigenvalue contributions

The eigenvalue densities of two random matrix ensembles, the Wigner Gaussian matrices and the Wishart covariant matrices, are decomposed in the contributions of each individual eigenvalue distribution. It is shown that the fluctuations of all eigenvalues, for medium matrix sizes, are described with a good precision by nearly normal distributions.

On the generalized eigenvalue method for energies and matrix elements in lattice field theory

We discuss the generalized eigenvalue problem for computing energies and matrix elements in lattice gauge theory, including effective theories such as HQET. It is analyzed how the extracted effective energies and matrix elements converge when the time separations are made large. This suggests a particularly efficient application of the method for which we can prove that corrections vanish asymptotically as exp(-(E(N+1) - E(n))t). The gap E(N+1) - E(n) can be made large by increasing the number N of interpolating fields in the correlation matrix. We also show how excited state matrix elements can be extracted such that contaminations from all other states disappear exponentially in time. As a demonstration we present numerical results for the extraction of ground state and excited B-meson masses and decay constants in static approximation and to order 1/m(b) in HQET.

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.

A semi-Markov multistate model for estimation of the mean quality-adjusted survival for non-progressive processes

We discuss the estimation of the expected value of the quality-adjusted survival, based on multistate models. We generalize an earlier work, considering the sojourn times in health states are not identically distributed, for a given vector of covariates. Approaches based on semiparametric and parametric (exponential and Weibull distributions) methodologies are considered. A simulation study is conducted to evaluate the performance of the proposed estimator and the jackknife resampling method is used to estimate the variance of such estimator. An application to a real data set is also included.

Tori embedded in R(3) with dense principal lines

In this paper are given examples of tori T(2) embedded in R(3) with all their principal lines dense. These examples are obtained by stereographic projection of deformations of the Clifford torus in S(3). (C) 2008 Elsevier Masson SAS. 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.

An eigenvalue problem for the biharmonic operator on Z(2)-symmetric regions

In this work we show that the eigenvalues of the Dirichlet problem for the biharmonic operator are generically simple in the set Of Z(2)-symmetric regions of R-n, n >= 2, with a suitable topology. To accomplish this, we combine Baire`s lemma, a generalised version of the transversality theorem, due to Henry [Perturbation of the boundary in boundary value problems of PDEs, London Mathematical Society Lecture Note Series 318 (Cambridge University Press, 2005)], and the method of rapidly oscillating functions developed in [A. L. Pereira and M. C. Pereira, Mat. Contemp. 27 (2004) 225-241].

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