Pseudodifferential operators with C*-algebra-valued symbols: Abstract characterizations

Given a separable unital C*-algebra C with norm parallel to center dot parallel to, let E-n denote the Banach-space completion of the C-valued Schwartz space on R-n with norm parallel to f parallel to(2)=parallel to < f, f >parallel to(1/2), < f, g >=integral f(x)* g(x)dx. The assignment of the pseudodifferential operator A=a(x,D) with C-valued symbol a(x,xi) to each smooth function with bounded derivatives a is an element of B-C(R-2n) defines an injective mapping O, from B-C(R-2n) to the set H of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module E-n. In this paper, we construct a left-inverse S for O and prove that S is injective if C is commutative. This generalizes Cordes' description of H in the scalar case. Combined with previous results of the second author, our main theorem implies that, given a skew-symmetric n x n matrix J and if C is commutative, then any A is an element of H which commutes with every pseudodifferential operator with symbol F(x+J xi), F is an element of B-C(R-n), is a pseudodifferential operator with symbol G(x - J xi), for some G is an element of B-C(R-n). That was conjectured by Rieffel.

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.

Decomposing Blaschke Products And Polynomials

The goal of this paper is to contribute to the understanding of complex polynomials and Blaschke products, two very important function classes in mathematics. For a polynomial, $f,$ of degree $n,$ we study when it is possible to write $f$ as a composition $f=g\circ h$, where $g$ and $h$ are polynomials, each of degree less than $n.$ A polynomial is defined to be \emph{decomposable }if such an $h$ and $g$ exist, and a polynomial is said to be \emph{indecomposable} if no such $h$ and $g$ exist. We apply the results of Rickards in \cite{key-2}. We show that $$C_{n}=\{(z_{1},z_{2},...,z_{n})\in\mathbb{C}^{n}\,|\,(z-z_{1})(z-z_{2})...(z-z_{n})\,\mbox{is decomposable}\},$$ has measure $0$ when considered a subset of $\mathbb{R}^{2n}.$ Using this we prove the stronger result that $$D_{n}=\{(z_{1},z_{2},...,z_{n})\in\mathbb{C}^{n}\,|\,\mbox{There exists\,}a\in\mathbb{C}\,\,\mbox{with}\,\,(z-z_{1})(z-z_{2})...(z-z_{n})(z-a)\,\mbox{decomposable}\},$$ also has measure zero when considered a subset of $\mathbb{R}^{2n}.$ We show that for any polynomial $p$, there exists an $a\in\mathbb{C}$ such that $p(z)(z-a)$ is indecomposable, and we also examine the case of $D_{5}$ in detail. The main work of this paper studies finite Blaschke products, analytic functions on $\overline{\mathbb{D}}$ that map $\partial\mathbb{D}$ to $\partial\mathbb{D}.$ In analogy with polynomials, we discuss when a degree $n$ Blaschke product, $B,$ can be written as a composition $C\circ D$, where $C$ and $D$ are finite Blaschke products, each of degree less than $n.$ Decomposable and indecomposable are defined analogously. Our main results are divided into two sections. First, we equate a condition on the zeros of the Blaschke product with the existence of a decomposition where the right-hand factor, $D,$ has degree $2.$ We also equate decomposability of a Blaschke product, $B,$ with the existence of a Poncelet curve, whose foci are a subset of the zeros of $B,$ such that the Poncelet curve satisfies certain tangency conditions. This result is hard to apply in general, but has a very nice geometric interpretation when we desire a composition where the right-hand factor is degree 2 or 3. Our second section of finite Blaschke product results builds off of the work of Cowen in \cite{key-3}. For a finite Blaschke product $B,$ Cowen defines the so-called monodromy group, $G_{B},$ of the finite Blaschke product. He then equates the decomposability of a finite Blaschke product, $B,$ with the existence of a nontrivial partition, $\mathcal{P},$ of the branches of $B^{-1}(z),$ such that $G_{B}$ respects $\mathcal{P}$. We present an in-depth analysis of how to calculate $G_{B}$, extending Cowen's description. These methods allow us to equate the existence of a decomposition where the left-hand factor has degree 2, with a simple condition on the critical points of the Blaschke product. In addition we are able to put a condition of the structure of $G_{B}$ for any decomposable Blaschke product satisfying certain normalization conditions. The final section of this paper discusses how one can put the results of the paper into practice to determine, if a particular Blaschke product is decomposable. We compare three major algorithms. The first is a brute force technique where one searches through the zero set of $B$ for subsets which could be the zero set of $D$, exhaustively searching for a successful decomposition $B(z)=C(D(z)).$ The second algorithm involves simply examining the cardinality of the image, under $B,$ of the set of critical points of $B.$ For a degree $n$ Blaschke product, $B,$ if this cardinality is greater than $\frac{n}{2}$, the Blaschke product is indecomposable. The final algorithm attempts to apply the geometric interpretation of decomposability given by our theorem concerning the existence of a particular Poncelet curve. The final two algorithms can be implemented easily with the use of an HTML

Molecular cloning of exons II and III of the a-globin major gene from Odontophrynus americanus 2n and 4n (Amphibia, Anura)

The a-globin major genes from diploid and tetraploid Odontophrynus americanus were studied using PCR-based technology. The cloned and sequenced amplified fragments were shown to contain most of the exon II sequences as well as the whole exon III sequence of the a-globin gene. Unexpectedly, intron 2 was entirely absent in the amplified fragments of both 2n and 4n origin. High conservation was observed among the obtained sequences when compared to corresponding sequences from human and Xenopus laevis origin. The possibility that these sequences might be pseudogenes is raised

Evidence of chromosome fusion in Gymnotus sylvius Albert & Fernandes-Matoli, 1999 (Teleostei: Gymnotiformes) detected by telomeric probes and R-banding

Gymnotus cf. carapo and Gynznotus sylvius are two fish species inhabiting the Upper Parana River Basin, presenting respectively 2n =54 and 2n = 40 chromosomes. In the present cytogenetic analysis, R-banding and telomere-sequence hybridization were carried out in order to determine the possible relationship between the karyotipes of these two species. Incorporation bands (R-bands) obtained for the two species allowed the identification of chromosome similarities, showing to be an usefull alternative to the G-banding methods, which fail in producing satisfying results in most of analyzed fish species. This approach, associated with the hybridization of telomeric sequences, permited to identify chromosomal rearrangements that could be used as indicators of karyotypic evolution within the group. In the present case, telomeric sequences were detected in the centromeric region of two metacentric chromosome pairs of Gymnotus sylvius. The results obtained after hybridization with the telomere sequences, coupled with the chromosome homeologies detected by R-banding, showed that G. cf carapo and G. sylvius should present a common ancestor, and this may also be corroborated by the similarities found in three chromosome pairs, that seem to have been conserved during the evolution of the two species. Based on the data here presented we propose that G. sylvius may have undergone a recent process of chromosome fusion that resulted in the diminution of its chromosome number.

Carcass and meat quality traits of wild boar (Sus scrofa s. L.) with 2n=36 karyotype compared to those of phenotypically similar crossbreeds (2n=37 and 2n=38) raised under same farming conditions. 1. Carcass quantity and meat dressing

The aim of this study was to compare wild boar (chromosomal number 2n = 36) to phenotypically similar animals of 2n = 37 and 2n = 38 chromosomes (crossbreeds) with respect to live weight, carcass yield, meat yield, fat and weight of inner organs. All animals were born and raised on the same farm and slaughtered at 39 weeks. The final live weight of wild boar 2n = 36 was significantly lower (47.2 kg) as compared to crossbreeds (80.0 kg). Animals 2n = 36 had more carcass yields (65.5%) than 2n = 37 karyotype (64.9%) and 2n = 38 (64.4%). Wild boar had the highest yields for the cuts with bones and boneless cuts compared to crossbreeds. Therefore, variations in karyotype are accompanied by differences in some carcass quantitative traits, i.e., 2n = 36 grow and fatten slower than crossbreeds 2n = 37 and 2n = 38. (c) 2008 Elsevier Ltd. All rights reserved.

Sacrae Rotae Romanae decisiones Recentiores in compendium redactae ad modum indicis per materias, [et] tractatus : in quinque tomos a nonnullis Mediolanensis Athenaei Sociis distributae : tomus quartus continens literas N.O.P.Q.R.

Mode of access: Internet.

The integral Be⁹(n, 2n) cross section /

"MAy 14, 1964"--page 7.

R.P. Fr. Leandri de SS. Sacramento ... Quaestionum moralium theologicarum in decem Decalogi praecepta.

Mode of access: Internet.

Packing of R^2 by Crosses

A cross in Rn is a cluster of unit cubes comprising a central one and 2n arms. In their monograph Algebra and Tiling, Stein and Szabó suggested that tilings of ℝn by crosses should be studied. The question of the existence of such a tiling has been answered by various authors for many special cases. In this paper we completely solve the problem for ℝ2. In fact we do not only characterize crosses for which there exists a tiling of ℝ2 but for each cross we determine its maximum packing density.

Abundância de ramos reprodutivos e produção de sementes em populações de Ipomoea pes-caprae (L.) R. Br. na Ilha de Santa Catarina, Brasil

The variation in the proportion of reproductive branches, fruit, and seed production of Ipomoea pes-caprae (L.) R. Br. (Convolvulaceae) were evaluated at ten beaches on Santa Catarina Island, state of Santa Catarina, Brazil. Three patches per beach of Ipomoea pes-caprae were monitored, involving two reproductive cycles. Ipomoea pes-caprae presented initially an average length of patches of 14 m, with 9.6 branches/m² and 39% of reproductive branches. The proportion of reproductive branches varied between the cycles, but there was not noticed an alternation of reproductive effort between the subsequent cycles. There was a reduction in the percentage of reproductive branches at six localities. In four beaches where Ipomoea pes-caprae populations declined, occurred reduction in the reproductive vigor, and in the seed production, being these declines associated to strong sea erosion. In another hand, in one beach with population increase, there were little reproductive branches due to the occurrence of young stolons. Four patches never maturated fruits, being three of these located at small beaches. The fruit and seed productions in the patches showed values up to 40 fruits/m² and up to 140 seeds/m², respectively. Populations with great seed production were localized in areas adjacent to great coastal plains, which may represent potential seed sources for areas with small seed production in the island.

Padrões de distribuição geográfica das espécies de Schefflera J. R. Forst. & G. Forst. (Araliaceae) do Brasil extra-amazônico

Uma análise da distribuição geográfica de Schefflera no Brasil extra-amazônico foi realizada com base em mapas atualizados plotando as ocorrências conhecidas das 26 espécies do gênero encontradas nessa grande área: S. angustissima (Marchal) Frodin, S. aurata Fiaschi, S. botumirimensis Fiaschi & Pirani, S. burchellii (Seem.) Frodin & Fiaschi, S. calva (Cham.) Frodin & Fiaschi, S. capixaba Fiaschi, S. cephalantha (Harms) Frodin, S. cordata (Taub.) Frodin & Fiaschi, S. distractiflora (Harms) Frodin, S. fruticosa Fiaschi & Pirani, S. gardneri (Seem.) Frodin & Fiaschi, S. glaziovii (Taub.) Frodin & Fiaschi, S. grandigemma Fiaschi, S. kollmannii Fiaschi, S. longipetiolata (Pohl ex DC.) Frodin & Fiaschi, S. lucumoides (Decne. & Planch. ex Marchal) Frodin & Fiaschi, S. macrocarpa (Cham. & Schltdl.) Frodin, S. malmei (Harms) Frodin, S. morototoni (Aubl.) Maguire, Steyermark & Frodin, S. racemifera Fiaschi & Frodin, S. ruschiana Fiaschi & Pirani, S. selloi (Marchal) Frodin & Fiaschi, S. succinea Frodin & Fiaschi, S. villosissima Fiaschi & Pirani, S. vinosa (Cham. & Schltdl.) Frodin & Fiaschi e S. aff. varisiana Frodin. Dois centros de endemismo associados com áreas de altitude elevada foram reconhecidos: Cadeia do Espinhaço em Minas Gerais e florestas montanas do Estado do Espírito Santo. Os padrões de distribuição geográfica ilustrados são discutidos com base em dados obtidos para outros grupos de angiospermas e em estudos fitogeográficos das principais fitocórias do Brasil extra-amazônico. São apresentadas também hipóteses acerca de prováveis relações filogenéticas entre alguns táxons, visando à busca de possíveis correlações entre estas e a biogeografia do grupo.

Thaptomys Thomas 1915 (Rodentia, Sigmodontinae, Akodontini) with karyotypes 2n = 50, FN = 48, and 2n = 52, FN = 52: two monophyletic lineages recovered by molecular phylogeny

A novel karyotype with 2n = 50, FN = 48, was described for specimens of Thaptomys collected at Una, State of Bahia, Brazil, which are morphologically indistinguishable from Thaptomys nigrita, 2n = 52, FN = 52, found in other localities. It was hence proposed that the 2n = 50 karyotype could belong to a distinct species, cryptic of Thaptomys nigrita, once chromosomal rearrangements observed, along with the geographic distance, might represent a reproductive barrier between both forms. Phylogenetic analyses using maximum parsimony and maximum likelihood based on partial cytochrome b sequences with 1077 bp were performed, attempting to establish the relationships among the individuals with distinct karyotypes along the geographic distribution of the genus; the sample comprised 18 karyotyped specimens of Thaptomys, encompassing 15 haplotypes, from eight different localities of the Atlantic Rainforest. The intra-generic relationships corroborated the distinct diploid numbers, once both phylogenetic reconstructions recovered two monophyletic lineages, a northeastern clade grouping the 2n = 50 and a southeastern clade with three subclades, grouping the 2n = 52 karyotype. The sequence divergence observed between their individuals ranged from 1.9% to 3.5%.