Cantor-Bernstein Sextuples for Banach Spaces

Let X and Y be Banach spaces isomorphic to complemented subspaces of each other with supplements A and B. In 1996, W. T. Gowers solved the Schroeder-Bernstein (or Cantor-Bernstein) problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain a necessary and sufficient condition on the sextuples (p, q, r, s, u, v) in N with p + q >= 1, r + s >= 1 and u, v is an element of N*, to provide that X is isomorphic to Y, whenever these spaces satisfy the following decomposition scheme A(u) similar to X(P) circle plus Y(q) B(v) similar to X(r) circle plus Y(s). Namely, Phi = (p - u)(s - v) - (q + u)(r + v) is different from zero and Phi divides p + q and r + s. These sextuples are called Cantor-Bernstein sextuples for Banach spaces. The simplest case (1, 0, 0, 1, 1, 1) indicates the well-known Pelczynski`s decomposition method in Banach space. On the other hand, by interchanging some Banach spaces in the above decomposition scheme, refinements of the Schroeder-Bernstein problem become evident.

Reproductive ecology of dipsadine snakes, with emphasis on South American species

A relatively large amount of variation occurs in the reproductive ecology of tropical snakes, and this variation is generally regarded as being a consequence of seasonality in climate and prey availability. In some groups, even closely related species may differ in their reproductive ecology; however, in others it seems to be very conservative. Here we explore whether characters related to reproduction are phylogenetically constrained in a monophyletic group of snakes, the subfamily Dipsadinae, which ranges from Mexico to southern South America. We provide original data on reproduction for Leptodeira annulata, Imantodes cenchoa, and three species of Sibynomorphus from southern, southeastern and central Brazil, and data from literature for other species and populations of dipsadines. Follicular cycles were seasonal in Atractus reticulatus, Dipsa, albifrons, Hypsiglena torquata, Leptodeira maculata, L. punctata, Sibynomorphus spp. and Sibon sanniola from areas where climate is seasonal. In contrast, extended or continuous follicular cycles were recorded in Dipsas catesbyi, D. neivai, Imantodes cenchoa, Leptodeira annulata, and Ninia maculata from areas with seasonal and aseasonal climates. Testicular cycles also varied from seasonal (in H. torquiata) to continuous (in Dipsa,5 spp., Leptodeira annulata, L. maculata, N. maculata, and Sibynomorphus spp.). Most dipsadines are small (less than 500 rum SVL), and females attain sexual maturity with similar relative body size than males. Sexual dimorphism occurred in terms of SVL and tail length in most species, and clutch size tended to be small (less than five eggs). Combat behavior occurs in Imantodes cenchoa, which did not show sexual size dimorphism. Reproductive timing, for both females and males, varied among species but in general there were no differences between the tribes of Dipsadinae in most of the reproductive characteristics, such as mean body size, relative size at sexual maturity, sexual size and tail dimorphism, duration of vitellogenesis or egg-carrying in oviducts.

Comparative bioavailability of cefuroxime axetil suspension formulations administered with food in healthy subjects

Objective: To assess the comparative bioavailability of two formulations (250 mg/5 mL suspension) of cefuroxime axetil (CAS 64544-07-6), administered with food, in healthy volunteers of both sexes. Methods: The study was conducted using an open, randomized, two-period crossover design with a 1-week washout interval. Plasma samples were obtained for up to 12 h post dose. Plasma cefuroxime axetil concentrations were analyzed by liquid chromatography coupled with tandem mass spectrometry (LC-MS-MS) with negative ion electrospray ionization using multiple reactions monitoring (MRM). From the cefuroxime axetil plasma concentration vs. time curves, the following pharmacokinetic parameters were obtained: AUC(last) and C(max). Results: The limit of quantification was 0.1 mu g/mL for plasma cefuroxime axetil analysis. The geometric mean and 90% confidence interval CI of test/reference product percent ratios were: 106.1% (100.8%-111.8%) for C(max), 109.4% (104.8%-114.2%) for AUC(last). Conclusion: Since the 90% Cl for AUC(last) and C(max) ratios were within the 80-125 % interval proposed by the US FDA, it was concluded that cefuroxime axetil (test formulation, 250 mg/5 mL suspension) was bioequivalent to a reference formulation under fed conditions, for both the rate and extent of absorption.

On isomorphism classes of C(2(m) circle plus [0, alpha]) spaces

We provide a complete isomorphic classification of the Banach spaces of continuous functions on the compact spaces 2(m) circle plus [0, alpha], the topological sums of Cantor cubes 2(m), with m smaller than the first sequential cardinal, and intervals of ordinal numbers [0, alpha]. In particular, we prove that it is relatively consistent with ZFC that the only isomorphism classes of C(2(m) circle plus [0, alpha]) spaces with m >= N(0) and alpha >= omega(1) are the trivial ones. This result leads to some elementary questions on large cardinals.

Fibonacci bimodal maps

We introduce the Fibonacci bimodal maps on the interval and show that their two turning points are both in the same minimal invariant Cantor set. Two of these maps with the same orientation have the same kneading sequences and, among bimodal maps without central returns, they exhibit turning points with the strongest recurrence as possible.

Spaces of compact operators on C(2(m) circle plus [0, alpha]) spaces

We classify up to isomorphism the spaces of compact operators K(E, F), where E and F are Banach spaces of all continuous functions defined on the compact spaces 2(m) circle plus [0, alpha], the topological sum of Cantor cubes 2(m) and the intervals of ordinal numbers [0, alpha]. More precisely, we prove that if 2(m) and aleph(gamma) are not real-valued measurable cardinals and n >= aleph(0) is not sequential cardinal, then for every ordinals xi, eta, lambda and mu with xi >= omega(1), eta >= omega(1), lambda = mu < omega or lambda, mu is an element of [omega(gamma), omega(gamma+1)[, the following statements are equivalent: (a) K(C(2(m) circle plus [0, lambda]), C(2(n) circle plus [0, xi])) and K(C(2(m) circle plus [0, mu]), C(2(n) circle plus [0, eta]) are isomorphic. (b) Either C([0, xi]) is isomorphic to C([0, eta] or C([0, xi]) is isomorphic to C([0, alpha p]) and C([0, eta]) is isomorphic to C([0,alpha q]) for some regular cardinal alpha and finite ordinals p not equal q. Thus, it is relatively consistent with ZFC that this result furnishes a complete isomorphic classification of these spaces of compact operators. (C) 2010 Elsevier Inc. All rights reserved.

Henon-like maps with arbitrary stationary combinatorics

We extend the renormalization operator introduced in [A. de Carvalho, M. Martens and M. Lyubich. Renormalization in the Henon family, I: universality but non-rigidity. J. Stat. Phys. 121(5/6) (2005), 611-669] from period-doubling Henon-like maps to Henon-like maps with arbitrary stationary combinatorics. We show that the renonnalization picture also holds in this case if the maps are taken to be strongly dissipative. We study infinitely renormalizable maps F and show that they have an invariant Cantor set O on which F acts like a p-adic adding machine for some p > 1. We then show, as for the period-doubling case in the work of de Carvalho, Martens and Lyubich [Renormalization in the Henon family, I: universality but non-rigidity. J. Stat. Phys. 121(5/6) (2005), 611-669], that the sequence of renormalizations has a universal form, but that the invariant Cantor set O is non-rigid. We also show that O cannot possess a continuous invariant line field.